The Long-Run Effects of Monetary Policy

FEDERAL RESERVE BANK OF SAN FRANCISCO

WORKING PAPER SERIES

Òscar Jordà

Federal Reserve Bank of San Francisco

University of California, Davis

Sanjay R. Singh

Federal Reserve Bank of San Francisco

University of California, Davis

Alan M. Taylor

Columbia University

NBER and CEPR

May 2023

Working Paper 2020-01

https://www.frbsf.org/economic-research/publications/working-papers/2020/01/

Suggested citation:

Jordà, Òscar, Sanjay R. Singh, Alan M. Taylor. 2023. “The long-run effects of monetary

policy

,” Federal Reserve Bank of San Francisco Working Paper 2020-01.

https://doi.org/10.24148/wp2020-01

The views in this paper are solely the responsibility of the authors and should not be interpreted

as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors

of the Federal Reserve System.

The long-run eﬀects of monetary policy

⋆

Oscar Jord`a

†

Sanjay R. Singh

Alan M. Taylor

‡

May 2023

Abstract

We document that the real eﬀects of monetary shocks last for over a decade. Our approach

relies on (1) identiﬁcation of exogenous and non-systematic monetary shocks using the

trilemma of international ﬁnance; (2) merged data from two new international historical

cross-country databases; and (3) econometric methods robust to long-horizon inconsistent

estimates. Notably, the capital stock and total factor productivity (TFP) exhibit greater

hysteresis than labor. When we allow for asymmetry, we ﬁnd these eﬀects with tightening

shocks, but not with loosening shocks. When extending the horizon of the responses

reported in several recent studies that use alternative monetary shocks, we ﬁnd similarly

persistent real eﬀects, thus supporting our main ﬁndings.

JEL classiﬁcation codes: E01, E30, E32, E44, E47, E51, F33, F42, F44.

Keywords: monetary policy, money neutrality, hysteresis, trilemma, instrumental variables,

local projections.

⋆

We are thankful to Sushant Acharya, George-Marios Angeletos, Gadi Barlevy, Susanto Basu, Saki Bigio,

Edouard Challe, Lawrence Christiano, James Cloyne, Julian di Giovanni, St´ephane Dupraz, Martin Eichenbaum,

Charles Engel, Gauti Eggertsson, Antonio Fat´as, John Fernald, Luca Fornaro, Jordi Gal´ı, Georgios Georgiadis,

Yuriy Gorodnichenko, Pierre-Olivier Gourinchas, James Hamilton, Oleg Itskhoki, Jean-Paul L’Huillier, Silvia

Miranda-Agrippino, Dmitry Mukhin, Juan Pablo Nicolini, Pascal Paul, Albert Queralto, Valerie Ramey, Ina

Simonovska, J´on Steinsson, Andrea Tambalotti, Silvana Tenreyro, Mart´ın Uribe, Johannes Wieland, who

provided very helpful comments and suggestions, as did many seminar and conference participants at the

Barcelona Summer Forum, the NBER Summer Institute IFM and EFCE Program Meetings, the Econometric

Society World Congress, the SED Annual Meeting, the ASSA Meetings, Bank of England, the Federal Reserve

Banks of Richmond, San Francisco, St. Louis, and Board of Governors, CEPR MEF Seminar Series, the Midwest

Macro Conference, Northwestern University, Delhi School of Economics, ISI Delhi, IGIDR Mumbai, Salento

Macro, Claremont McKenna College, the Swiss National Bank, the Bank of Spain, the Norges Bank, the Bank

of Finland, Danmarks Nationalbank, Columbia SIPA, UC Berkeley, UC Davis, UC Irvine, UCLA, UC San

Diego, University of Wisconsin–Madison, Universit¨at Z¨urich, Vanderbilt University, Asian School of Business,

and Trinity College, Dublin. Antonin Bergeaud graciously shared detailed data from the long-term productivity

database created with Gilbert Cette and R´emy Lecat at the Banque de France. All errors are ours. The views

expressed herein are solely those of the authors and do not necessarily represent the views of the Federal Reserve

Bank of San Francisco or the Federal Reserve System.

†

Federal Reserve Bank of San Francisco and Department of Economics, University of California, Davis

([email protected]; o[email protected]).

Federal Reserve Bank of San Francisco and Department of Economics, University of California, Davis

(sanjay[email protected]; [email protected]).

‡

Columbia University; NBER; and CEPR (amt2314@columbia.edu).

Are there circumstances in which changes in aggregate demand can have an appreciable,

persistent eﬀect on aggregate supply?

— Yellen (2016)

1. Introduction

What is the eﬀect of monetary policy on the long-run productive capacity of the economy?

Since at least Hume (1752), macroeconomics has largely operated under the assumption that

money is neutral in the long-run, and a vast literature spanning centuries has gradually built

the case (see, for example, King and Watson, 1997, for a review). Contrary to this monetary

canon, we ﬁnd evidence rejecting long-run neutrality.

Our investigation of monetary neutrality rests on three pillars. First, it is essential to

identify exogenous movements in interest rates to obtain a reliable measure of monetary eﬀects

and avoid confounding. Second, because we focus on long-run outcomes, we rely on a panel

of countries observed over as long a period as we can to maximize data span and, hence, the

statistical power of our conclusions. Third, as we show below, the empirical method used can

make a big diﬀerence: common approaches are designed to maximize short-horizon ﬁt, but we

need methods that are consistent over longer spans of time. We discuss how we build on each

of these three pillars next.

On identiﬁcation, the ﬁrst pillar, in section 2 we exploit the trilemma of international ﬁnance

(see, for example, Obstfeld, Shambaugh, and Taylor, 2004, 2005; Shambaugh, 2004). The

key idea is that when a country pegs its currency to some base currency with free movement

of capital across borders, it loses control—at least to some degree—over its interest rate: a

correlation in home and base interest rates is thus induced, which is exact when the peg is

hard and arbitrage frictionless, but is generally less than one otherwise. Insofar as base rates

are determined by base country conditions alone, they provide a potential source of exogenous

variation in home rates. We theoretically ground this identiﬁcation strategy in a canonical New

Keynesian small open economy model (Schmitt-Groh´e and Uribe, 2016; Fornaro and Romei,

2019; Bianchi and Lorenzoni, 2022). Speciﬁcally, we derive analytical results to show formally,

for the ﬁrst time, how a trilemma-based identiﬁcation approach recovers the monetary policy

impulse response function of interest. Moreover, we exploit the model to provide bounds on

failures of the exclusion restriction due to spillover eﬀects.

Second, moving on to the data pillar, in section 3 we rely on two recent macro-history

databases spanning 125 years and 17 advanced economies. First, we use the data in Jord`a,

We leverage the insights of open-economy literature to show how theory maps rigorously into our trilemma

identiﬁcation scheme and guides an econometric approach that builds on earlier work in this vein (di Giovanni,

McCrary, and Von Wachter, 2009; Jord`a, Schularick, and Taylor, 2020).

Schularick, and Taylor (2017), available at

www.macrohistory.net/database

. This “JST

Database” contains key macroeconomic series, such as output, interest rates, as well as inﬂation,

credit, and many other potentially useful control variables for our analysis. Second, to allow

for a Solow decomposition of output into its components, we merge and incorporate data from

Bergeaud, Cette, and Lecat (2016), available at

http://www.longtermproductivity.com

Their data series include observations on investment in machines and buildings, number of

employees, and hours worked. With these variables, we construct measures of total factor

productivity (TFP), including with utilization adjustment, and then decompose impulse

responses for output into TFP, capital input, and labor input, to explore the channels of the

hysteresis that we have uncovered.

The third and ﬁnal pillar of our analysis in section 4 has to do with the econometric approach.

We use local projections (Jord`a, 2005) in order to get more accurate estimates of the impulse

response function (IRF) at longer horizons. As we show formally, as long as the truncation

lag in local projections is chosen to grow with the sample size (at a particular rate that we

make speciﬁc below), local projections (LPs) estimate the impulse response consistently at any

horizon. In fact, as recently shown by Xu (2023), local projections are semi-parametrically

eﬃcient in settings were the lag order may be inﬁnite and the truncation lag is allowed to grow

with the sample. Other procedures commonly used to estimate impulse responses do not have

this property (see, for example, Lewis and Reinsel, 1985; Kuersteiner, 2005), and this—among

several other reasons—may explain the failure of some of the prior literature to discern the

highly persistent eﬀects that we document here. See also, Plagborg-Møller and Wolf (2021);

Li, Plagborg-Møller, and Wolf (2022) for related results on comparison of LPs with vector

autoregressions (VARs).

Supported by these three pillars in section 5 we show that, surprisingly, monetary policy

aﬀects the productive capacity of the economy for a very long time. In response to an exogenous

monetary shock, output declines and does not return to its pre-shock trend even twelve years

thereafter. Next, we investigate the source of this hysteresis and ﬁnd that capital and TFP

experience similar trajectories to output. In contrast, total hours worked (both hours per

worker and number of workers) return more quickly to the original trend. Hence, our new

ﬁndings are distinct from the usual labor hysteresis mechanism previously emphasized in the

literature (see, for example, Blanchard and Summers, 1986; Gal´ı, 2015a; Blanchard, 2018; Gal´ı,

2020).

After a series of robustness checks, in section 6 we show that the responses display a key

asymmetry, or nonlinearity, with hysteresis forces much stronger after tightening shocks than

We are particularly thankful to Antonin Bergeaud for sharing some of the disaggregated series from their

database that we use to construct our own series of adjusted TFP.

loosening shocks, consistent with prior research on shorter-horizon response asymmetries (for

example, Tenreyro and Thwaites, 2016; Angrist, Jord`a, and Kuersteiner, 2018). Tight monetary

policy has long lasting eﬀects, but loose monetary policy does not stimulate growth. The

asymmetries that we ﬁnd echo results from models with downward nominal wage rigidity

(DNWR), though other mechanisms are possible.

How do our ﬁndings stack up against the state of knowledge? A voluminous literature

based on post-WW2 U.S. data has examined the causal eﬀects of monetary policy (see, for

example, Christiano, Eichenbaum, and Evans, 1999; Ramey, 2016; Nakamura and Steinsson,

2018, provide a detailed review), but the evidence on long-run neutrality is, at best, mixed

(King and Watson, 1997). An important exception is the work of Bernanke and Mihov (1998),

which fails to reject long-run neutrality, but ﬁnds that the point estimates of GDP response to

monetary innovations do not revert to zero even after ten years. Mankiw (2001) interprets this

non-reversal as potential evidence of long-run non-neutrality.

In section 7, we relate our work to recent studies that employ diﬀerent methods to identify

monetary shocks for the U.S. and the U.K. economies. Using their replication codes, we extend

the original estimates from published studies to eight-year horizons and, in fact, ﬁnd similar

evidence of long-run non-neutral eﬀects of monetary shocks. Miranda-Agrippino and Ricco

(2021) estimate a Bayesian VAR(12) for the U.S. economy with high-frequency market-based

monetary surprises around Federal Open Market Committee announcements (G¨urkaynak,

Sack, and Swanson, 2005) and Federal Reserve’s Greenbook forecasts. Brunnermeier, Palia,

Sastry, and Sims (2021) estimate a large-scale Bayesian SVAR model, for the U.S. economy,

with identiﬁcation based on heteroskedasticity. Cesa-Bianchi, Thwaites, and Vicondoa (2020)

estimate a proxy structural VAR, for the U.K. economy, with their constructed high-frequency

monetary surprises measured around monetary policy announcements of the Bank of England.

Finally, our paper has been followed by a more recent literature that examines long-run

eﬀects of transitory shocks. Antolin-Diaz and Surico (2022) document persistent eﬀects of

transitory government spending shocks. Cloyne, Mart´ınez, Mumtaz, and Surico (2022) ﬁnd

evidence for long-run eﬀects of transitory corporate tax shocks. Furlanetto, Lepetit, Robstad,

Rubio-Ram´ırez, and Ulvedal (2021) also ﬁnd that demand shocks have hysteresis eﬀects for

On DNWR see, for example, Akerlof, Dickens, and Perry (1996), Benigno and Ricci (2011), Schmitt-Groh´e

and Uribe (2016), Barnichon, Debortoli, and Matthes (2021), Bianchi, Ottonello, and Presno (Forthcoming),

Born, D’Ascanio, M¨uller, and Pfeifer (2022).

Mankiw notes (emphasis added): “Bernanke and Mihov estimate a structural vector autoregression and

present the impulse response functions for real GDP in response to a monetary policy shock. (See their Figure

III.) Their estimated impulse response function does not die out toward zero, as is required by long-run neutrality.

Instead, the point estimates imply a large impact of monetary policy on GDP even after ten years. Bernanke

and Mihov don’t emphasize this fact because the standard errors rise with the time horizon. Thus, if we look

out far enough, the estimated impact becomes statistically insigniﬁcant. But if one does not approach the data

with a prior view favoring long- run neutrality, one would not leave the data with that posterior. The data’s best

guess is that monetary shocks leave permanent scars on the economy.” See also Gal´ı (1998).

the U.S. economy using a structural VAR model identiﬁed with short-run sign and long-run

zero restrictions. A theoretical literature at the intersection of endogenous productivity growth

and business cycles following the seminal work by Stadler (1990) provides micro-foundations

for hysteresis eﬀects of transitory shocks. Cerra, Fat´as, and Saxena (2023) provide a recent

review of literature on hysteresis and business cycles. Complementary to our paper, theoretical

analyses by Benigno and Fornaro (2018) and Fornaro and Wolf (2022) link low nominal interest

rates to the rate of growth of productivity.

Going beyond our paper, hysteresis matters for

how we build models of monetary economies and what optimal monetary policy is in those

models: the welfare implications could be substantial (Benigno and Benigno, 2003; Benigno

and Woodford, 2012; Garga and Singh, 2021).

2. Identification via the trilemma

The trilemma of international ﬁnance gives a theoretically justiﬁed source of exogenous variation

in interest rates (Jord`a, Schularick, and Taylor, 2020). The logic is straightforward: under

a hard peg with perfect capital mobility, returns on similarly risky assets will be arbitraged

between the pegging (home) and pegged to (base) economies. In ideal frictionless settings,

strict interest parity would imply that rates are exactly correlated.

In reality this correlation is less than perfect, of course. But even under soft pegs (or dirty

ﬂoats), with frictions or imperfect arbitrage, a non-zero interest rate correlation between a home

economy and the base economy to which it pegs its exchange rate is enough for identiﬁcation

using instrumental variables. In this section we present an open economy model to make formal

the conditions for identiﬁcation, even in the presence of spillovers via non–interest rate channels

(or in the parlance of instrumental variables, violations of the exclusion restriction). This level

of detail allows us to construct econometric estimation procedures via propositions derived

from the model.

2.1. The identiﬁcation problem in a nutshell

In measuring the eﬀect of exogenous changes in domestic interest rates on output, consider

the simplest possible setup. For reasons that will become clear momentarily, we express all

variables in deviations from steady state (denoted with hats) so as to follow the same notation

See, among others, Fat´as (2000); Barlevy (2004); Anzoategui, Com´ın, Gertler, and Mart´ınez (2019); Bianchi,

Kung, and Morales (2019); Guerron-Quintana and Jinnai (2019); Queralto (2020); Schm¨oller and Spitzer (2021);

Vinci and Licandro (2021).

Our paper is also tangentially related to literature on productivity research. Baqaee and Farhi (2019)

construct a general framework where monetary shocks may aﬀect allocative eﬃciency. Meier and Reinelt

(Forthcoming) provide evidence of increased misallocation following contractionary monetary policy shocks.

of the economic model that will follow. Hence, let

denote output;

the home interest rate;

and

∗

as base-country interest rates to which the home economy pegs its exchange rate. The

idea is to estimate β in the following regression:

β + v

(1)

using

∗

as an instrument (di Giovanni, McCrary, and Von Wachter, 2009; Jord`a, Schularick,

and Taylor, 2020). Base country interest rates seem likely to be determined by base country

economic conditions alone. Hence variation might be assumed to be essentially exogenous with

respect to the home economy considered.

However, does the exclusion restriction hold? What if, aside from the interest rates, there

are spillover channels from the base to the home country? That is, is there a direct channel by

which

∗

aﬀects

? If that is the case, then the regression really should be:

β +

∗

θ + u

(2)

It is easy to show that the IV estimator in Equation 1 would have a bias given by:

β → β +

∗2

) θ

∗

)

(3)

which will be non-zero as long as

θ ̸

= 0. However, if

were known, then Equation 1 could be

estimated by instrumental variables by redeﬁning the left-hand side as:

(

−

∗

θ) =

β + u

(4)

This observation was made, for example, by Conley, Hansen, and Rossi (2012).

In what follows, we derive an economic model that allows us to carefully work out the

exogeneity conditions of

∗

and then determine the appropriate adjustments for potential

spillovers (the θ that results in the violation of the exclusion restriction).

2.2. Theory to guide identiﬁcation

We build on a standard open economy setup widely used today as in Schmitt-Groh´e and Uribe

(2016), Fornaro and Romei (2019), and Bianchi and Lorenzoni (2022).

Our aim is not a new

model, but how theory maps rigorously into our trilemma identiﬁcation scheme and guides our

econometric approach to identiﬁcation. As the model is standard, many details are relegated

Elements of this framework appear in Benigno, Fornaro, and Wolf (2020), Farhi and Werning (2017), and

Fornaro (2015), among others. A textbook treatment is Schmitt-Groh´e, Uribe, and Woodford (2022, Ch. 13).

to the Appendix. In Appendix H, we obtain similar results in a Mundell-Fleming-Dornbusch

model with additional ﬁnancial channels, as in Gourinchas (2018).

We assume that there is perfect foresight. The environment features incomplete international

markets with nominal rigidities. We focus on two countries: a large economy that we label the

base and a small open economy, the home country. We begin by describing a benchmark small

open economy. We want to recover the impulse response of output to a monetary shock in this

benchmark economy using trilemma identiﬁcation.

2.3. Benchmark economy

Households. There is a continuum of measure one of identical households. Each household

receives an endowment of tradables

T t

every period. The household preferences are given by

lifetime utility function

max

t+1

}

∞

t=0



log(C

) − φ

1+ν

1 + ν



where

denotes the discount factor,

is the (inverse) Frisch elasticity of labor supply,

is a

scaling parameter to normalize

= 1 in the steady state, and

[·]

is the expectation operator

conditional on information available until date 0. The composite good

is a Cobb-Douglas

aggregate



T t





1−ω



1−ω

of a tradable good

T t

and a non-tradable good

, where

ω ∈ (0, 1) is the tradable share.

Households can trade in one-period riskless real and nominal bonds. Real bonds are

denominated in units of the tradable consumption good and pay gross interest rate

, taken

as given (i.e., a world real interest rate). Nominal bonds issued by the domestic central bank

are denominated in units of domestic currency, and pay gross nominal interest rate

. The

households’ budget constraint in units of domestic currency is then

T t

+ P

T t

+ B

= W

+ P

T t

+ P

T t

t+1

+ T

+ Z

where

T t

and

are the prices of tradable and non-tradable goods in local currency;

the level of real debt in units of tradable good assumed in period

t −

1 and due in period

;

is the level of nominal debt in units of local currency assumed in period

t −

1 and due in period

;

is the nominal wage per unit of labor employed;

are nominal lump-sum transfers from

the government; and Z

nominal proﬁts from domestic ﬁrms owned by households.

The household chooses a sequence of

T t

, C

, l

, d

t+1

, B

t+1

}

to maximize lifetime utility

For ease of exposition, we consider a cashless economy.

subject to the budget constraint, taking initial bond holdings as given. Labor is immobile

across countries, so the wage level is local to each small open economy.

The ﬁrst-order conditions for the household’s optimization problem are

T t

T t+1

, (5)

T t

T t+1

T t

T t+1

, (6)

≡

T t

(1 − ω)C

T t

ωC

, (7)

φl

T t

. (8)

Tradable goods and bonds. We assume that law of one price holds for the tradable good.

Let

be the nominal exchange rate for home relative to the base, and let

∗

be the base price

of the tradable good denominated in base currency.

Then, we have that

T t

∗

. From

Equation 5 and Equation 6 we can then derive the interest rate parity condition,

= R

T t+1

T t

= R

t+1

∗

t+1

∗

. (9)

The world real interest rate is taken as given, so there can be dependence on initial conditions.

The home real interest rate on tradable bonds is equal to the world real interest in tradable

bonds:

= R

∗

. (10)

Production and nominal rigidity. The non-tradable consumption good is a Dixit-Stiglitz

aggregate over a continuum of products

(

) produced by monopolistically competitive

producers indexed by

, with

≡

(

)

(ϵ

− 1)

/ϵ

)

/(ϵ

− 1)

. Each ﬁrm

in the home

economy produces a homogenous good with technology given by

(

) =

(

), taking the

demand for its product as given by

(

) =

(

(i)

)

−ϵ

, where we use the price index

of the non-tradable good composite,

= (

(

)

1−ϵ

)

/(1 − ϵ

)

. We assume the presence

of relevant production subsidies to oﬀset monopoly distortions.

We consider an analytical tractable form of nominal rigidity commonly used in open-

economy literature following Obstfeld and Rogoﬀ (1995a). We assume that prices for all ﬁrms

are pre-ﬁxed in the ﬁrst period. Second period onwards, prices are fully ﬂexible.

It is common in the small open economy literature to treat price level in the base economy

∗

as synonymous

for price level of tradable goods in the base economy P

∗

T t

In Appendix E, we introduce a debt-elastic interest rate premium following Schmitt-Groh´e and Uribe

(2003) and Uribe and Schmitt-Groh´e (2017) along with staggered price setting to show robustness of our results.

Appendix E shows the robustness of our results using Calvo (1983) price setting in a stationary model.

Monetary and ﬁscal policy. The policy rate is the home nominal interest rate on one-period

domestic currency bonds. We want to derive the impulse response of domestic output to a

domestic monetary policy shock, i.e., the usual reference object of interest. We assume that

the home nominal interest rate follows an exogenous path subject to policy shocks ε

. (11)

Since we are simulating responses to one-time shocks, we interpret this policy rule assumption

as equivalent to that of temporary interest rate peg made in the zero lower bound (ZLB)

literature (Eggertsson and Woodford, 2003; Werning, 2011). Once the shock abates, a policy

rule that maintains local determinacy (Blanchard and Kahn, 1980) is expected to hold in those

environments with temporary interest rates at the zero lower bound. We will be invoking a

similar equilibrium selection device whereby the economy returns back to the same deterministic

steady state.

The portfolio allocation between the real and nominal bonds is not determinate in this type

of model. To ensure determinacy, and since all agents at home are identical, we now assume

that home domestic nominal bonds are in net zero supply, i.e.,

t+1

= 0. We also assume that

the home ﬁscal authority follows a balanced budget every period.

Market clearing. We impose that the non-tradable goods market has to clear at home,

implying that production of non-tradable goods must equal the consumption demand for

non-tradable goods. Therefore, we have that

= L

= Y

= C

. (12)

Finally, the external budget constraint of the economy must be satisﬁed every period, so

T t

+ d

= Y

T t

t+1

. (13)

Construction of small open economy GDP. Our key outcome variable of interest is the

real GDP in the small economy. To make the connection with our empirical counterparts, and

to keep our baseline discussion focused, for now we construct this real GDP variable using

constant aggregation weights implied by the Cobb-Douglas aggregator.

Similar solution methods to do counterfactual policy simulations have been developed for economies away

from the ZLB (Las´een and Svensson, 2011; Guerrieri and Iacoviello, 2015; Christiano, 2015). Embedding

an endogenous policy transmission through inﬂation targeting, while the shock is on, does not change our

theoretical results since we are identifying responses to non-systematic components of monetary policy.

We assume appropriate government subsidies ﬁnanced by lumpsum taxes to eliminate monopoly rents in

the intermediate goods sector.

Clearly, variation in aggregation weights can cause changes in real GDP in a multiple sector

economy, and this deﬁnition abstracts from such potential index number problems. That

said, we present analytical results in an environment with time-varying aggregation weights in

Appendix C.

The large economy. The small country takes the path of prices

∗

and real interest rates

∗

in the large (base) economy as given. Without loss of generality, we therefore assume rigid

prices in the base economy, with P

∗

= 1.

Equilibrium. We analyze the economy starting at steady state at date 0. We set the initial

external debt of the economy to zero,

= 0. There is a one-time unanticipated shock at date

1 to a domestic monetary policy rule. We assume that the world real interest rate is equal to

the inverse of the domestic discount factor

∗

...

−1

at all dates. Essentially, this is a

two-period economy with the ﬁrst period as the short-run, and subsequent time as the long-run.

We present the equilibrium conditions in Appendix A. For some analytical results, we will

log-linearize the economy around initial steady state and denote the corresponding variables

with hats.

Object of interest: response of GDP to a domestic monetary shock. We are interested

in the response of GDP to a domestic monetary shock. We denote this coeﬃcient as β.

2.4. Fixed exchange rate economy exposed to base rate shocks

We consider an identical small open economy to the one just described, but with a diﬀerent

policy conﬁguration. We assume that this economy’s nominal exchange rate is ﬁxed to the

currency of a large economy, termed as base. We then use the changes in the base economy

interest rate as instruments for domestic monetary shocks. We wish to study conditions under

which this instrument can recover the coeﬃcient β.

To begin, we consider the setup of a hard peg (we will shortly discuss the case of dirty

ﬂoat or soft peg): A hard peg ﬁxes the nominal exchange rate at a given level. Without loss of

generality, we assume the rule

= 1 . (14)

By Equation 10, there is perfect passthrough from base economy interest rate changes into

home nominal interest rates, hence

−

∗

−

∗

, where

and

∗

denote the steady

state levels of nominal interest rates in the home and base economies, respectively.

Our identiﬁcation strategy does not require that we isolate exogenous changes in base

country interest rates as long as they are determined by domestic conditions alone. For small

economies, this seems like a plausible assumption. However, in our empirical speciﬁcations, we

go one step further. Rather than using raw base country interest rates, we use the component

of these rates that cannot be predicted using base macroeconomic controls. That is, we buy

double insurance by using these residuals as the source of truly exogenous movements in home

economy interest rates. We refer the reader to the empirical methodology section below for

more details.

Based on this alternative regime, the question now is to determine the conditions under

which, using base interest rate shocks as instruments, one can recover exactly the same impulse

response as that generated by a standard domestic monetary policy shock in the benchmark

economy. Our focus is then the impulse response of small open economy output—under peg

—following a shock in

∗

, and how it compares to

, the impulse response of output in the

benchmark economy following a domestic policy shock.

In the peg conﬁguration, there is a one-time unanticipated shock to the foreign interest

rate

∗

. In order to denote subsequent dates 2, 3, 4, ..., as representing the long-run, we

assume that the world interest rate is equal to the inverse of the domestic discount factor

∗

...

−1

at all future dates. This assumption essentially reduces the ﬁxed-exchange

rate economy to a two-period model as in the textbook treatment of Schmitt-Groh´e, Uribe,

and Woodford (2022, Ch. 13).

2.5. Identiﬁcation via the trilemma

We now present the core theoretical results of our paper as a series of propositions. All of the

proofs in this section have been relegated to Appendix B.

We begin by noting that tradable goods consumption, as well as real debt choice, are

independent of the monetary policy regime.

A well-known result, this simpliﬁes the analysis:

Proposition 1. The responses to a base interest rate shock of tradable consumption and the

domestic real interest rate (on bonds denominated in tradable goods) do not depend on whether

the home economy pegs or ﬂoats.

The upshot of this result is that we can now separate the determination of all remaining

variables from {C

T t

, R

, d

t+1

This result is well noted in the literature at least since Obstfeld and Rogoﬀ (1995a, Appendix) in the

case with ﬁxed base economy interest rates. Uribe and Schmitt-Groh´e (2017, Section 9.5) present the general

result in settings where the inter-temporal elasticity of substitution is equal to the intra-temporal elasticity of

substitution between tradable and non-tradable goods.

The key diﬀerence between a peg and a ﬂoat comes from whether the nominal exchange rate is used

to counter the passthrough of foreign rates into domestic policy rates. There is an extant literature in open

economy macroeconomics that has emphasized this insight, most recently articulated by Farhi and Werning

(2012), Fornaro (2015), as well as Schmitt-Groh´e and Uribe (2016) upon which we build.

Hence consider the equilibrium of a small open economy under a (hard) peg and the

equilibrium of the benchmark economy with a domestic policy shock. Assume real GDP is

constructed with constant and identical aggregation weights in the two economies. Then the

following proposition holds:

Proposition 2 (impulse response equivalence: hard pegs). The response of real GDP to a base

interest rate shock in a peg is identical to the response of real GDP to a domestic policy shock

of the same magnitude in a benchmark economy.

2.6. Departures from the baseline model

We now extend the baseline model in three ways. In all cases, to sharpen the results, we keep

the benchmark economy ﬁxed to the baseline described above, and only vary the economy

conﬁguration in the peg economy. First, we allow for imperfect interest rate pass-through from

the base rate into the home economy. This can happen when the home economy is in a soft peg

or in dirty ﬂoat regime. Given this setting, we then show that one can still use base country

rates to recover

. Second, we allow for endogenous production of tradable goods in the peg

economy. We show that tradable good production increases in response to a base interest rate

shock through a labor reallocation mechanism. Consequently, the response of GDP to base

interest rate shocks is downward biased. Third, we consider other channels through which base

interest rate shocks can spill over into the home economy. In this case we show how one can

adjust the response to base country rates and how this correction still produces the equivalent

benchmark economy response to a monetary shock.

2.6.1 Soft pegs and dirty ﬂoats

We deﬁne imperfect pass-through (whether for a soft peg, or a dirty ﬂoat) of base rates to

home rates using a pass-through coeﬃcient 0

< λ ≤

1 such that:

−

(

∗

−

∗

). Then:

Proposition 3 (impulse response equivalence: imperfect pass-through). Consider the equilib-

rium of a small open economy with imperfect pass-through and the equilibrium of the benchmark

economy with a domestic policy shock (subsection 2.3). Assume real GDP is constructed with

constant and identical aggregation weights in the two economies. To a ﬁrst-order approximation,

the response of real GDP with imperfect pass-through to a base economy interest rate shock is a

fraction λ of the response of real GDP in a benchmark economy to a domestic policy shock of

same magnitude.

Presence of imperfect pass-through implies that domestic interest rate needs to be appropri-

ately scaled to allow interpretation. For this reason, we will later estimate the linear impulse

response function of output to a unit increase in domestic interest rate, instrumented with the

change in base economy interest rate. This normalization provides the appropriate scaling to

recover β.

2.6.2 Endogenous tradable good

We extend the baseline model in the peg economy (subsection 2.4) by allowing tradable output

to be produced with labor using a constant returns to scale technology. Prices are set ﬂexibly

in the tradable-good sector. Labor is fully mobile, within the economy, across the tradable and

the non-tradable sector. Economy-wide real wages (in units of tradable goods) are constant

T t

= 1 ∀t ≥ 0 .

Because the total labor supplied in the economy is divided between tradable and non-tradable

good sector, the labor market clearing condition is now modiﬁed as:

= L

T t

+ L

= Y

T t

+ L

Substituting this market clearing condition in the intra-temporal labor supply condition of

the household, we get:

φ (L

T t

+ L

)

T t

= 1

where

is fraction of total labor force allocated to the tradable goods sector in the steady

state. The rest of the equilibrium equations are same as in the baseline peg economy model

(subsection 2.4). We obtain the following result:

Proposition 4 (endogenous tradable goods). Consider the equilibrium of a hard peg economy

extended with endogenous production of tradable goods described above. And consider the

benchmark small open economy with domestic policy shock described in subsection 2.3. Assume

real GDP is constructed with constant and identical aggregation weights in the two economies.

The response of real GDP to a base economy interest rate shock is an upward biased estimate

of the response of real GDP in a benchmark economy to a domestic policy shock of the same

magnitude.

While the impulse response of non-tradable output is identical across the peg and the

benchmark economy, the total output response is biased upwards (i.e., towards zero, or smaller

absolute size) in the peg economy relative the benchmark economy. This upward bias emerges

due to labor reallocation to the tradable goods sector. We next formalize the empirically

relevant scenario when the exclusion restriction may fail.

2.6.3 Spillovers

If there are other channels through which base interest rates can aﬀect the model equilibrium,

these spillovers will aﬀect the previous results derived for pegs and imperfect pass-through

economies. The equivalency with the impulse response of output in the benchmark economy

will break down.

To see this, consider the following postulated relationship between tradable output and the

base real interest rate (log-linear approximation around initial steady state in hats),

T t

= α

∗

, (15)

where

α <

0. Such a relationship is often embedded into open economy models through

a modeling of export demand (e.g, see Gal´ı and Monacelli, 2016).

Intuitively, the home

economy’s ability to sell its export good to the base (or any economy pegged to the base)

is now demand constrained. This demand is not perfectly elastic, but depends on the state

of consumption demand in the base economy, which in turn depends on the base real rate.

Equation 15 is just a reduced-form expression of this dependence.

Then the following holds:

Proposition 5 (spillovers in a peg). Consider the log-linear equilibrium of a hard peg economy

with spillovers (i.e., extended with Equation 15), and the log-linear equilibrium of the benchmark

economy with a domestic policy shock described in subsection 2.3. Assume real GDP is

constructed with constant and identical aggregation weights in the two economies. Denote the

response of real GDP in a peg to a unit and i.i.d. base economy interest rate shock with

, and

the response of real GDP in the benchmark economy to a unit and i.i.d. domestic policy shock

with β, as in Equation 1. Then,

β = γ

− α



P Y

+ (1 − ζ)

P Y



. (16)

The spillover emerges through two channels: (i) through the direct eﬀect of foreign real

interest rates on tradable-output, and (ii) through a wealth eﬀect. Home consumers reduce

their consumption of tradable goods because their endowment of tradable output also falls.

Because of nominal rigidity, and ﬁxed exchange rates, non-tradable consumption co-moves with

tradable consumption. Hence this additional decline in tradable consumption propagates into

the demand for non-tradable goods.

Note that

can be positive in models with endogenous production of tradable goods (see subsubsection 2.6.2).

We think the case of

α <

0 is more realistic in a world where contractionary policy shocks in the U.S. economy

have contractionary spillovers into rest of the world.

A corollary of Proposition 5 applies to an imperfect pass-through economy.

Corollary 1. Consider the log-linear equilibrium of a small open imperfect pass-through economy

(extended with Equation 15 and the log-linear equilibrium of the benchmark economy with a

domestic policy shock described in subsection 2.3. Assume real GDP is constructed with constant

and identical aggregation weights in the two economies. Denote the response of real GDP in the

imperfect pass-through economy to a unit, i.i.d. base economy interest rate shock with

, and

the response of real GDP in the benchmark economy to a unit, i.i.d. domestic policy shock with

β. Then,

β =

− α



P Y

+ (1 − ζ)

P Y



To sum up, this last result shows that the same logic applies to the continuum of regimes

from hard peg (

= 1) to pure ﬂoat (

= 0), with appropriate scaling of responses by

. Thus,

for estimation purposes, we may draw on information from any economy within this continuum,

not just those with regimes at the extremes.

2.7. Model implications for econometric identiﬁcation

The ﬁnal model just introduced, with spillovers, explains how base country monetary policy

can aﬀect the output of tradable goods (via export demand shifts) as well as the output of

nontradable goods (via interest arbitrage and conventional domestic demand shifts). These

spillover eﬀects onto smaller open economies depend on the share of tradable output in their

GDP. Using the insights and notation from the model, in this section we explore its implications

for the identiﬁcation of our impulse responses.

Disciplining the spillover coeﬃcient. As in Equation 43 in the appendix, we assume

imperfect pass-through of base rates into home rates. In regression form, this can be expressed

= λ

∗

+ v

, (17)

where, as before,

, and

∗

are in deviations from steady state, and

λ ∈

1] is the pass-

through coeﬃcient, and is possibly diﬀerent for country-time pairs nominally classiﬁed as

pegs

versus

floats

. We omit the constant term without loss of generality and we assume that

is a

well-behaved, white noise error term. For now, it is convenient to leave more complex dynamic

speciﬁcations aside to convey the intuition simply.

Similarly, Equation 16 in regression form can be expressed as in Equation 2

β +

∗

θ + u

, (18)

where here too

, and

∗

are deviations from steady state. For now, we leave unspeciﬁed

whether

belongs to a peg or a ﬂoat. Note that under Equation 16, we have

θ =







P Y

|{z}

≡Φ

+(1 − ζ)

P Y

| {z }

≡1−Φ







α , (19)

that is, sum of (i) the share of tradable export output in GDP, which we denote Φ =

/P Y

scaled by the parameter

, which determined how

∗

aﬀects tradable output, add (ii) the share

of non-tradable output in GDP scaled by the parameters

and 1

− ζ

, which determined how

reduction in tradable output aﬀects consumption and savings. However, in commonly seen

calibrations in the literature,

is set at about 0

96. Hence, to a ﬁrst approximation, the second

channel is negligible with 1 − ζ ≈ 0.04, and we will neglect it below.

In reality, there are two main reasons we might expect

θ →

0. One is that output is

dominated by non-tradables. In the JST database for advanced economies, over 150 years of

history, tradable export shares are 30% at most, and usually in the 10%–20% range, so Φ

≤

is a reasonable upper bound. Next is

, the spillover eﬀect of base country rates

∗

on tradable

export demand at home. It is fair to assume that this eﬀect will be at most as strong as the

eﬀect of domestic rates on tradable output, so

α ≤ β

. These observations allow us to derive

bounds on the true value of β when there are, potentially, spillover eﬀects.

IV estimator with no spillovers. Our data come from two subpopulations, pegs and ﬂoats,

which principally diﬀer in the degree to which

λ →

1. In practice we hesitate to impose the

same parameters across both subpopulations thus allowing for diﬀerent

and

, so the reduced

form regressions are

= D

∗

+ D

∗

+ η

, (20)

= D

∗

+ D

∗

+ v

, (21)

where D

= 1 for pegs, 0 otherwise, and similarly D

= 1 for ﬂoats, 0 otherwise.

In other words, if there are no spillovers, the IV estimator of

will be the ratio of the

weighted average of the γ over the weighted average of the λ: we will be estimating a “model

average” β using information from both of the two subpopulations, pegs and ﬂoats.

IV estimator with spillovers. What happens if

θ ̸

= 0? In that case, we provide a bound

for the possible values that

= Φ

can take based on our model, as we discussed earlier. The

share of tradables in GDP is directly measurable and, as we argued above, falls typically in the

range Φ ∈ [0.1, 0.3] in the JST database. As we discussed earlier, we assume that eﬀect of

∗

on tradable output is no larger than the eﬀect of

; that is, we impose as the conservative

upper bound implied by α = β.

Based on these assumptions, we can write

= Φ

and employ the calibrated range of values

of Φ. Then it is easy to see that one can transform the original Equation 18 to get

= (

∗

Φ)β + u

, (22)

and one can estimate

with this expression using instrumental variables along the lines just

discussed using the subpopulations of pegs and ﬂoats, that is, with the ﬁrst stage given by

Equation 21.

To sum up, in the empirical work that follows, we will be focused on estimating the following

IV model,

= (

∗

Φ)β + u

, (23)

= D

∗

+ D

∗

+ v

, (24)

which we have shown will recover the true reference impulse response for the benchmark model

based on impulse responses for pegs and ﬂoats. Conley, Hansen, and Rossi (2012) derive a

generic spillover correction in IV estimation that is closely related to the results presented here.

We elaborate on this point in the empirical sections below.

3. Data and series construction

The empirical features motivating our analysis rest on two major international and historical

databases. Data on macro aggregates and ﬁnancial variables, including assumptions on exchange

rate regimes and capital controls, can be found in

www.macrohistory.net/data

. This database

covers 17 advanced economies reaching back to 1870 at annual frequency. Detailed descriptions

of the sources of the variables contained therein, their properties, and other ancillary information

are discussed in Jord`a, Schularick, and Taylor (2017) and Jord`a, Schularick, and Taylor (2020),

as well as references therein. Importantly, we will rely on a similar construction of the

trilemma instrument discussed in Jord`a, Schularick, and Taylor (2016), and more recently

Jord`a, Schularick, and Taylor (2020). This will be the source of exogenous variation in interest

rates. The instrument construction details will become clearer in the next section.

The second important source of data relies on the work by Bergeaud, Cette, and Lecat (2016)

and available at

http://www.longtermproductivity.com

. This historical database adds to

our main database observations on capital stock (machines and buildings), hours worked,

and number of employees, and the Solow residuals (raw TFP). In addition, we construct

time-varying capital and labor utilization corrected series using the procedure discussed in Imbs

(1999) with the raw data from Bergeaud, Cette, and Lecat (2016) to construct our own series

of utilization-adjusted TFP. We went back to the original sources so as to ﬁlter out cyclical

variation in input utilization rates in the context of a richer production function that allows for

factor hoarding. We explain the details of this correction in Appendix F.

Guided by our model and identiﬁcation strategy as discussed in the previous section, we

divide our sample into three subpopulations of country-year observations. The bases will refer

to those economies whose currencies serve as the currency anchor for the subpopulation of

pegging economies, labeled as the pegs. Other economies, the ﬂoats, allow their exchange to be

freely determined by the market.

Base and peg country codings can be found in Jord`a, Schularick, and Taylor (2020, Table

1 and Appendix A), and are based on updates to older, established deﬁnitions (Obstfeld,

Shambaugh, and Taylor, 2004, 2005; Shambaugh, 2004; Ilzetzki, Reinhart, and Rogoﬀ, 2019).

A country

is deﬁned to be a peg at time

, denoted with the dummy variable

i,t

= 1, if

it maintained a peg to its base at dates

t −

1 and

. This conservative deﬁnition serves to

eliminate opportunistic pegging, and it turns out that transitions from ﬂoating to pegging

and vice versa represent less than 5% of the sample, the average peg lasting over 20 years.

Interestingly, pegs are, on average, more open than ﬂoats.

Finally, let

i,t

= 1

− D

i,t

denote

a non-peg, i.e., ﬂoat. The choice of exchange rate regime is treated as exogenous, and indeed

we ﬁnd zero predictability of the regime based on macroeconomic observables in our advanced

economy sample. Regimes are also highly persistent in this sample which excludes emerging and

developing countries, in contrast to the ﬁndings of limited persistence for the full cross-section

of countries as in Obstfeld and Rogoﬀ (1995b).

Based on this discussion, we construct an adjusted instrument as follows using a cleaning

regression (Romer and Romer, 2004). Let ∆

i,t

denote changes in country

’s short-term

nominal interest rate, let ∆

b(i,t),t

denote the change in short term interest rate of country

’s

base country

(

i, t

), and let

∆

b(i,t),t

denote its predictable component explained by a vector of

Our construction of productivity assumes misallocation related-wedges are absent. We have not yet found

the data to take into account markups or sectoral heterogeneity in our productivity estimates. See Basu and

Fernald (2002) and Syverson (2011) for extensive discussions on what determines productivity.

In the full sample, the capital openness index averages 0.87 for pegs (with a standard deviation of 0.21)

and 0.70 for ﬂoats (with standard deviation 0.31). After WW2 there is essentially no diﬀerence between them.

The average is 0.76 for pegs and 0.74 for ﬂoats with a standard deviation of 0.24 and 0.30 respectively. See

Jord`a, Schularick, and Taylor (2020) for further details on the construction of the instrument.

Figure 1:

IV construction: residualized component ∆

b(i,t),t

of base country interest rates in the

historical sample

-4 -2 0 2 4 6

Residualized base country change in interest rate

1900 1920 1940 1960 1980 2000 2020

United Kingdom

Hybrid (U.K., U.S., France)

United States

Germany

Notes: ∆

b(i,t),t

= (∆

b(i,t),t

−∆

b(i,t),t

) where

(

i, t

) denotes the base for country

at time

and the ﬁnal term is the predicted

interest rate from a cleaning regression. See text.

base country macroeconomic variables. The list of controls used to construct

∆i

b(i,t),t

include

log real GDP; log real consumption per capita; log real investment per capita; log consumer

price index; short-term interest rate (usually a 3-month government bill); long-term interest rate

(usually a 5-year government bond); log real house prices; log real stock prices; and the credit

to GDP ratio. The variables enter in ﬁrst diﬀerences except interest rates. Contemporaneous

terms (except for the left-hand side variable) and two lags are included.

Hence, using the notation from the previous section, denote ∆

b(i,t),t

= (∆

b(i,t),t

−∆

b(i,t),t

In Figure 1, we display these constructed base-country interest rate residuals for the four types

of base as in Obstfeld, Shambaugh, and Taylor (2005) and Jord`a, Schularick, and Taylor

(2020): the United Kingdom during the classical Gold Standard era before WW1, a hybrid

base consisting of an average of U.K., France and U.S. short rates in the interwar years, the

United States after WW2, and Germany from the start of the European Monetary System

in the 1970s. These base countries are assumed to not take into account the state of the

economy in the smaller countries which are pegging to them. For the pegs we use one of these

bases, as appropriate; for the ﬂoats, we follow Ilzetzki, Reinhart, and Rogoﬀ (2019); Obstfeld,

Shambaugh, and Taylor (2005); and Shambaugh (2004) to determine the appropriate base.

Note that in the historical eras before the 1970s there do not exist data on private-sector or

central bank forecasts of future macroeconomic variables, so we cannot include these in the

control set. However, in section 7, we discuss evidence from an alternate identiﬁcation approach

by Miranda-Agrippino and Ricco (2021) that controls for both private-sector expectations and

central bank forecasts. We would argue that the monetary residuals appear reasonable; for

example, policy in various base countries is seen to be tight in the late 1920s before the Great

Crash; around 1980 in the era of tightening by Volcker and P¨ohl; just before 2000 in the U.S.

under the Greenspan Fed, or again in 2006 under the Bernanke Fed.

Finally, since countries in a given year may not be perfectly open to capital ﬂows, we

then scale the base shock, adjusting for capital mobility using the capital openness index of

Quinn, Schindler, and Toyoda (2011), denoted

i,t

∈

1]. The resulting trilemma instruments

adjusted for capital mobility, following Jord`a, Schularick, and Taylor (2020), are thus deﬁned as

i,t

≡ D

i,t

∆

b(i,t),t

; j = P, F , (25)

where P refers to pegs and F refers to ﬂoats.

4. Consistent long-horizon impulse responses

In thinking about the propagation of a shock, especially to distant horizons, it is generally

considered good practice to allow for generous lag structures—and in the limit, allowing

for possibly inﬁnite lags. Inﬁnite dimensional models have a long tradition in econometric

theory and form the basis for many standard results. For example, Berk (1974) considers the

problem of estimating the spectral density of an inﬁnite order process using ﬁnite autoregression.

In multivariate settings, Lewis and Reinsel (1985) establish the consistency and asymptotic

normality of ﬁnite order approximations to an inﬁnite order multivariate system. Kilian (1998)

shows that the ﬁnite sample biases of the underlying ﬁnite order autoregressions can induce

severe bias on impulse response bootstrap inference based on vector autoregressions (VARs).

In empirical practice, the well-known biases arising from impulse responses estimated with

ﬁnite VARs are further aggravated by having to choose relatively short lag lengths due to the

parametric loads required in their estimation as Kuersteiner (2005) shows. The solution that

we pursue in this paper to avoid these issues, however, is to calculate impulse responses using

local projections instead.

Suppose the data are generated by an invertible, reduced-form, inﬁnite moving average

process or

V MA

(

∞

)—the well-known impulse response representation. Invertibility here means

that the space of the vector

spans the space of the residual vector,

, and that the process can

alternatively be expressed as a reduced-form, inﬁnite vector autoregression or

V AR

(

∞

). This

assumption allows for very general impulse response trajectories with potentially interesting

dynamics at long-horizons.

We set aside any discussion on identiﬁcation since the main issues discussed here do not

depend on it. Let

∞

h=0

t−h

; h = 0, 1, . . . ; B

= I , (26)

be the

V MA

(

∞

) representation of the m-dimensional vector

(without loss of generality, we

omit the constant term). Under the well-known general invertibility assumptions explicitly

stated in Appendix D, the V AR(∞) is

∞

j=1

t−j

+ ϵ

; j = 1, 2, . . . . (27)

The moving average matrices, B

, and the autoregressive matrices, A

, follow the well-known

recursion due to Durbin (1959) given by

= A

h−1

+ A

h−2

+ . . . + A

k−h

+ A

k+1

k−h−1

+ . . . + A

h−1

+ A

| {z }

remainder term

. (28)

Lewis and Reinsel (1985) established that, under standard regularity assumptions, a

V AR

(

)

provides consistent estimates of

, . . . , A

with

p, T → ∞

as long as

grows at a rate

/T →

There are two practical implications of this result. First, if the truncation lag is too small,

k < p

the consistency assumption fails and hence, based on Equation 28, we will obtain inconsistent

impulse response estimates B

, even when h is relatively small.

The second and more subtle implication is the following. Suppose that indeed the truncation

lag is chosen so that

and hence the consistency condition is met. Then, as is clear from

Equation 28, estimates of the impulse response for horizons

= 1

, . . . , k

will be consistently

estimated, but not for horizons

h > k

. The reason is that for

h > k

, the expression for

involves the terms

, . . . , B

k−h−1

, A

k+1

, . . . , A

(i.e., the remainder term in Equation 28),

which have been truncated and hence their omission introduces inconsistency.

What about local projections? We extend the proof in Lewis and Reinsel (1985) in Appendix

D. We show that local projections are consistent for any horizon

, even when the lag structure

is truncated as long as

p, T → ∞

at rate

/T →

0. Lusompa (2019) derives a related result in

the context of generalized least-squares inference of local projections. Relatedly, Montiel Olea

and Plagborg-Møller (2021) use similar asymptotic arguments to show how lag-augmented

local projections provide asymptotically valid inference for both stationary and non-stationary

data over a wide range of response horizons. More recently Xu (2023) shows that in inﬁnite

dimensional settings, local projections achieve semiparametric eﬃciency.

Figure 2:

Estimating cumulative responses: autoregressive versus local projection biases at long

horizons.

(a) True responses

-.6 -.4 -.2 0 .2

0 5 10 15 20 25

Horizon

Impulse

Cumulative

(b) Lag truncation: 3, 6, 9, and 12 lags

-.6 -.4 -.2 0 .2

0 5 10 15 20 25

Horizon

True LP AR(3)

AR(6) AR(9) AR(12)

Notes: Sample size: 1,000. Monte Carlo replications: 1,000. The shaded error bands are 1 and 2 standard error bands based on the

local projection Monte Carlo average. LP refers to cumulative local projections using 2 lags. AR(k) refers to impulse responses

cumulated from an autoregressive model with k = 3, 6, 9, and 12 lags. See text.

Basically, local projections are direct estimates of the impulse response (moving average)

coeﬃcients. Truncating the lag structure, even when

h > k

, has asymptotically vanishing

eﬀects on the consistency of the estimator. Truncated VARs on the other hand, have to

be inverted to construct the impulse response. Hence the impulse response depends on the

entire dynamic speciﬁcation of the VAR. The cumulation of small sample inconsistencies over

increasing horizons can pile up and turn into non-negligible distortions to the impulse response,

specially at long horizons.

Of course, the solution would be to specify the VAR truncation lag,

, to be large as

the impulse response horizon (as long as

/T →

0). Setting aside the parametric burden

imposed in the estimation, this may not be enough to address the second of the practical issues

highlighted earlier, namely the truncation of the remainder term in Equation 28. To illustrate

these issues, Figure 2 shows a simple Monte Carlo exercise. We generate an MA process whose

coeﬃcients are determined by the impulse response function displayed in panel (a). The implied

cumulative response is also shown, as this is the object of interest in our application. This

impulse response is meant to loosely mimic the shape of the responses we ﬁnd later in the

paper. In cumulative terms, a shock has transitory, but long-lived eﬀects on the variable.

Further details on the setup of the Monte Carlo exercise along with the speciﬁcs of how the two panels of

Panel (b) of Figure 2 hence shows Monte Carlo averages from estimates of the cumulative

response from a simple AR model with 3, 6, 9, and 12 lags versus local projections using only 2

lags—a considerable handicap for the local projection. Again, to mimic the empirical analysis,

we assume a sample with 1,000 observations (results with 300 observations yield nearly identical

results). We repeat the experiment 1,000 times. The error bands displayed are the one and two

standard error bands of the local projection Monte Carlo averages.

As is evident from the ﬁgure, given the long-lived dynamics of our experiment, truncating

below 12 lags generates cumulative eﬀects that are relatively short-lived and far oﬀ the true

response. The reason is that fewer than 12 lags would generally capture the early stages of the

impulse response, where not much action has yet taken place, but it would miss entirely the

undoing of the dynamics of periods 1–12 that follows in periods 13–24.

In contrast, local projections provide quite a close estimate of the response even though the

truncation lag is quite severe. As we increase the AR lag length to 12 (the point at which the

original negative dynamics die-oﬀ as panel (a) illustrates), the AR model with 12 lags picks up

the shape of the response very nicely though it gets into trouble once the horizon goes beyond

12 lags, and especially at the tail end, as the theory predicted. In contrast, local projections

continue to approximate the response well, even at those long horizons.

Consider our application, which involves 9 variables. A 9-dimensional vector autoregression

with 12 lags (as in the Monte Carlo application) involves 108 regressors per equation. The

correct lag length, which is 24 in our D.G.P. involves a whopping 216 regressors. Compare

that to the 18 regressors for the local projection. Further, note that even truncating the AR

at 12 lags is really on the boundary of the order needed to capture the main features of the

theoretical impulse response given the D.G.P. Typical information criteria, specially commonly

used Bayesian (or Schwartz) information criteria, will tend to select lag lengths that are entirely

too small (see Kuersteiner, 2005). Even if long lag lengths are selected, the parametric loads

make the task of analyzing the data across subsamples (as we do) even more diﬃcult or often

times, impossible.

5. The data show that monetary shocks have long-lived effects

The empirical approach from this point forward relies on local projections, estimated with

instrumental variables (LPIV), based on Equation 23 and Equation 30. The instruments,

adjusted for capital mobility, are

i,t

and

i,t

, as deﬁned earlier, and we estimate the following

Figure 2 are generated are in Appendix D.

(cumulative) impulse responses for the baseline, no spillover case (Φ = 0 in Equation 19),

i,t+h

− y

i,t−1

= α

i,h

+ ∆

i,t

+ x

i,t

+ u

i,t+h

, (29)

∆R

i,t

= κ

+ z

i,t

+ z

i,t

+ x

i,t

ζ + v

, (30)

for

= 0

, . . . , H

;

= 1

, . . . , N

;

, . . . , T

, where

i,t+h

is the outcome variable, log real

GDP, for country

observed

periods from today,

i,h

are country ﬁxed eﬀects at horizon

∆

i,t

refers to the instrumented change in the short-term interest rate (usually government

bills), our stand-in for the policy rate;

is the cumulative impulse response function of variable

y for country i at horizon h relative to its value at horizon −1; and x

i,t

collects all additional

controls including lags of the outcome and interest rates, as well as lagged values of other macro

aggregates.

Moreover, we control for global business cycle eﬀects through a global world

GDP control variable to parsimoniously soak up common global ﬂuctuations. We calculate

heteroscedasticity and autocorrelation robust Driscoll and Kraay (1998) standard errors.

Table 1 reports the ﬁrst-stage regression of the pegging country’s short term interest rate

∆R

i,t

on the instruments

i,t

, z

i,t

and controls

i,t

, country ﬁxed eﬀects and robust standard

errors. The interest-rate passthrough is roughly 0.6 for pegs and 0.25 for ﬂoats. Thus, neither

represents a hard peg or a pure ﬂoat corner case, further bolstering the case for studying the

more general imperfect pass-through case discussed earlier. Both instruments are statistically

signiﬁcant. We ﬁnd that the peg instrument,

i,t

, has a

-statistic close to 9 in the full and

post-WW2 samples and is therefore not a weak instrument. The ﬂoat instrument,

i,t

, has a

-statistic close to 3 in the full and post-WW2 samples, a weaker instrument, as one would

expect. Nevertheless, we show that our results are robust to excluding the weaker instrument.

5.1. Main results

The main ﬁndings in our paper are shown by the response of real GDP to a shock to domestic

interest rates. We display these results graphically in Figure 3. This ﬁgure is organized into

two columns, charts (a) and (c) refer to full sample results, and columns (b) and (d) to the

post-WW2 sample. In addition, the top row—charts (a) and (b)—is based on using the peg and

ﬂoat instruments, whereas the second row—charts (c) and (d)—only use the peg instrument

The list of domestic macro-ﬁnancial controls used include log real GDP; log real consumption per capita;

log real investment per capita; log consumer price index; short-term interest rate (usually a 3-month government

security); long-term interest rate (usually a 5-year government security); log real house prices; log real stock prices;

and the credit to GDP ratio. The variables enter in ﬁrst diﬀerences except for interest rates. Contemporaneous

terms (except for the left-hand side variable) and two lags are included. We control for contemporaneous values

of other macro-ﬁnancial variables for two purposes a) base rate movements might be predictable by current

home macro-conditions, and b) we wanted to impose restrictions in the spirit of Cholesky ordering whereby real

GDP is ordered at the top. Results are robust to excluding contemporaneous home-country controls.

Figure 3: Baseline response to 100 bps shock: Real GDP.

(a) Full sample: 1900–2015.

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

(b) Post-WW2 sample: 1948–2015.

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

using only the peg IV.

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

(d) Post-WW2 sample: 1948–2015,

using only the peg IV.

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs. Full sample:

1900–2015 (World Wars excluded). LP-IV estimates displayed as a solid blue line with 68% and 95% standard error bands. Top

row uses both peg and ﬂoat instruments; bottom row uses only peg instrument. Estimation is robust with Driscoll and Kraay

(1998) standard errors. See text.

Table 1: Trilemma instruments: First stage evidence.

All years Post-WW2

0.59

∗∗∗

0.61

∗∗∗

t-statistic [9.47] [9.02]

0.27

∗∗∗

0.26

∗∗∗

t-statistic [3.30] [2.77]

Observations 1104 874

Notes:

∗∗∗

p <

01,

∗∗

p <

05,

∗

p <

1. Full sample: 1900–2015 excluding WW1: 1914–1919 and WW2: 1939–1947. Post

WW2 sample: 1948–2015. These regressions include country ﬁxed eﬀects as well as up to two lags of the ﬁrst diﬀerence in log real

GDP, log real consumption, investment to GDP ratio, credit to GDP, short and long-term government rates, log real house prices,

log real stock prices, and CPI inﬂation. In addition we include world GDP growth to capture global cycles. Estimation is robust

with Driscoll and Kraay (1998) standard errors. See text.

as a robustness check. Regardless of the sample used, a 1 percentage point shock in domestic

short-term interest rates has sizable and long-lasting counterfactual eﬀects on GDP. In the full

sample, GDP declines by 4.60 percent over 12 years. A similar eﬀect is found when we restrict

the sample to post-WW2. The drop 12 years after impact is 3.94 percent.

5.2. Inspecting the mechanism

The results in Figure 3 are a far cry from traditional notions of long-run neutrality found in the

literature. What is the source of this persistent decline? We employ a Solow decomposition of

GDP (

) into its components, using a Cobb-Douglas production function, to construct hours

worked (

, employees times number of hours per employee); capital stock (

, measured capital

in machines and buildings); and the Solow residual, labeled as total factor productivity (

T F P

Figure 4 displays the (cumulative) responses of each of these components to the same 100

bps shock in the domestic short-term interest rate using the trilemma instrument, both for the

full and the post-WW2 samples.

The chart displays each of the components with one and

two standard error conﬁdence bands.

Several features deserve mention. Figure 4a shows that there are similar declines in capital

and raw TFP. In terms of growth accounting, the capital response component accounts for

two-thirds and the TFP response component for about one-third of the decline in real GDP.

However, total hours worked exhibits a much ﬂatter response, with much lower labor hysteresis.

Because capital enters the production function with a smaller weight, it should be clear from

the ﬁgure that most of the decline in GDP is explained by the TFP variable, followed by capital,

with total hours worked mostly ﬂat.

For comparability, we use the same controls in estimating responses of various components of the production

function as those used for the GDP response.

Appendix I.2 provides the corresponding ﬁgure for post-WW2 sample with similar ﬁndings.

Figure 4:

Baseline response to 100 bps shock: Real GDP and Solow decomposition. Full sample,

1900–2015.

(a) Estimates using raw data

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Real GDP

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Labor

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Capital

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

TFP

(b) Estimates using Imbs correction for factor utilization

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Real GDP

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Labor

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Capital

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

TFP

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs (both peg and

ﬂoat instruments). Full sample: 1900–2015 (World Wars excluded). LP-IV estimates displayed as a thick lines and 68% and 95%

standard error bands. The upper panel uses raw data on capital stocks and total hours to construct TFP as a residual. The lower

panel adjusts the raw data on capital stock and total hours to obtain estimates of actual factor inputs by using the Imbs (1999)

correction. Estimation is robust with Driscoll and Kraay (1998) standard errors. See text.

Capital accumulation follows textbook dynamics in the short-run. The capital response is

initially muted but builds up over time. But unlike a textbook New Keynesian model (Gal´ı,

2015b), the capital stock does not recover even after 12 years. Similarly, TFP falls gradually

rather than suddenly, and also does not recover.

How accurate are estimates based on raw capital and labor data? One serious concern

with Solow decompositions, well known at least since the work of Basu and Kimball (1997),

is the issue of capacity utilization biases (See also Basu, Fernald, and Kimball 2006). When

measures the capital stock, as here, that is not capital input: input is only the capital

being used, possibly much lower in periods of slack when plant and equipment may be idling.

Likewise if

measures labor stock, even if it measures total hours, it may be biased upwards in

periods of slack if labor is hoarded, and not fully utilized. In such cases, na¨ıve use of the Solow

approach will result in mismeasured factor inputs that display cyclicality that is too weak,

leaving residual TFP with cyclicality that is too strong, a pervasive problem that exaggerates

the role of TFP shocks as a source of business cycles.

Therefore, following the literature, we revise the capital and labor raw data to account

for cyclicality in utilization, following the well-established method proposed by Imbs (1999).

The results are shown in Figure 4b, and reveal some subtle diﬀerences. Overall the responses

are similar in terms of shape and statistical signiﬁcance, so the qualitative story is the same.

But quantitatively, the TFP response is now muted in amplitude, as expected, and the factor

responses are accordingly larger, suggesting that the Imbs correction captures some utilization-

driven factor slack. In what follows, we use Imbs-corrected responses as our baseline. This is

clearly a more conservative choice that would tend to make the TFP response more muted.

Finally, we verify that the long-run responses we have found are not a simple mechanical

result of an unusually persistent response of interest rates to its own shock. A simple check

shows that this is not the case. In Appendix J, Figure A3 we show that the short-term nominal

interest rate returns to zero by the fourth year. For completeness, Figure A4 also in the

appendix, shows the response of the consumer price index. The response of the nominal interest

rate and the price level is typical of what has been reported often in the literature (see, e.g.,

Christiano, Eichenbaum, and Evans, 1999; Coibion, 2012; Nakamura and Steinsson, 2018).

5.3. Robustness and discussion

Our baseline speciﬁcation is quite saturated, including lags and current values of global GDP

growth. This rich speciﬁcation served multiple purposes. Global shocks that caused bases to

change interest rates are controlled for during instrument construction, and in the second-stage

LP. We now discuss further robustness checks to ensure that the persistent eﬀects that we ﬁnd

are not misattributed to monetary policy shocks.

Allowing for spillovers. A violation of the exclusion restriction could occur if base rates

aﬀect home outcomes through channels other than home rates. These spillover eﬀects could

happen if base rates proxy for factors common to all countries, but these factors would have to

persist despite having included global GDP as a control. Or they could occur for other reasons,

such as spillovers via trade. In addition to the control strategy used in our baseline speciﬁcation,

we address the issue more formally by estimating a spillover-corrected IV speciﬁcation developed

in Section 2.7. Equations 23 and 30 generalize our baseline IV estimator to accommodate

spillovers that vary with size of export share in the peg economies. With a range of values for

∈

3], we show in Figure 5 the spillover-corrected estimates of the response of output to

a 100 bps monetary policy shock. A light-green shaded area with dashed border shows the

correction implied by this range of Φ. While the responses at year 12 are somewhat smaller

than the baseline estimates (solid blue line), monetary policy shocks still exert a sizable and

persistent eﬀect on output.

Controls for external factors. A cruder approach to validate the exclusion restriction is

by directly controlling for a primary channel through which the spillover eﬀects may originate.

A monetary tightening in the base country may reduce the demand for goods from the pegging

economy. This eﬀect would amplify the eﬀect of the trilemma shock on home output. With

soft peg regimes, there may be further eﬀects through changes in nominal exchange rates. To

account for these eﬀects, we control for global GDP growth rate, base country GDP growth

rate, exchange rate of the pegging economy with respect to the U.S., and the current account of

the peg. Since we do not have exchange rate data with respect to other countries, we indirectly

control for those spillovers using the current account of the peg country. Figure 6a plots the

responses to the trilemma identiﬁed shock. Directly controlling for open-economy variables,

motivated by export demand channels, does not aﬀect our main result: monetary shocks still

have a large and very persistent eﬀect on real GDP.

Check for structural breaks. Fernald (2014) and Gordon (2016) have convincingly argued

that there are structural breaks in U.S. TFP growth. One may suspect that there are structural

breaks in other economies’ TFP growth rates as well. If such structural breaks coincide in time

with monetary shocks of the same sign, they could bias our results. To address this concern, we

ﬁrst estimate up to ﬁve structural breaks in TFP growth for each country in our sample using

the UD-max statistic of Bai and Perron (1998). We report these estimated structural break

dates in Appendix I.3. Then in our baseline speciﬁcation, we allow output growth to lie in

either of the ﬁve regimes at horizon zero. Figure 6b plots the estimated impulse response when

including structural breaks in TFP growth. As evident, our results are robust to accounting for

structural breaks.

Figure 5:

Response to 100 bps trilemma shock with spillover corrections: Real GDP. Full sample,

1900–2015.

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

IV IV spillover corrected

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs. The LP-IV point

estimates displayed as a solid blue line with 68% and 95% standard error bands bands, and the range of LP-IV spillover corrected

point estimates displayed as a light green shaded area with dashed border, using Φ

∈

3]. Estimation is robust with Driscoll

and Kraay (1998) standard errors.See text.

Figure 6:

Response to 100 bps trilemma shock with additional controls: Real GDP. Full sample,

1900–2015.

(a) Open economy model based controls

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

(b) Structural breaks in TFP growth

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs. Full sample:

1900–2015 (World Wars excluded). LP-IV estimates displayed as a solid blue line with 68% and 95% standard error bands.

Estimation is robust with Driscoll and Kraay (1998) standard errors. See text.

Check for pre-trends. In Figure 7, we check for pre-trends by showing the baseline estimates

extended with negative time horizons up to

−

8 years before the shock. Pre-trends might

cast doubt on the parallel trends assumption, and would suggest that the trilemma identiﬁed

surprise was anticipated prior to the treatment or the treatment was in response to some other

confounder. As described earlier, in our estimations, we have included contemporaneous values

and two lags of domestic macro-ﬁnance variables (including the outcome variable) in the control

set along with controls for global and base country GDP respectively. As the ﬁgure shows,

there are no noticeable pre-trends detected. In the absence of data on forecasts, absence of

pre-trends is reassuring.

Check for outlier countries or events. Another concern may be that our baseline result

is driven by outliers, such as a speciﬁc country or am unusual event in the historical data. We

already reported results only on the post-World War 2 sample. In Figure 8, we conduct two

further robustness exercises when estimating response of GDP to a 100 bps monetary shock

using the trilemma IV. Figure 8a shows “hairplot” responses where we drop one country at

a time from the full sample and re-estimate the impulse response of GDP. These alternate

responses are displayed as dashed lines. The baseline estimate is shown as solid blue line

for reference. The main takeaway from this exercise is that no single country trajectory is

driving our baseline results. Moreover, all the alternate responses exhibit the long-run eﬀects

in response to the trilemma monetary shock. Similarly, Figure 8b shows a “hairplot” where we

drop a 5-year window at a time and re-estimate the impulse response of GDP. The hair plots

show remarkable stability in long-run responses.

6. No free monetary lunch: asymmetric sign-dependent

responses

Our results provide evidence that monetary policy shocks have long run eﬀects, with signs of

hysteresis out to the 10+ year horizon. However, we assumed a symmetric estimation technique

in our empirical results. Is this a valid assumption? Can the central bank boost the economy’s

potential with acommodative monetary policy in the same manner that contractionary policy

appears to reduce its long-term productive capacity? Some standard theoretical extensions

can deliver such asymmetry properties (e.g., downward nominal wage rigidity, or DNWR as in

Schmitt-Groh´e and Uribe (2016)), and many empirical papers have found supportive evidence

of short-horizon “pushing on a string” features of expansionary monetary policy shocks (e.g.,

Angrist, Jord`a, and Kuersteiner, 2018; Tenreyro and Thwaites, 2016).

Beyond monetary policy, Barnichon, Debortoli, and Matthes (2021) ﬁnd asymmetry in the ﬁscal multiplier.

Figure 7:

Response to 100 bps trilemma shock: Real GDP. Full sample, 1900–2015. With pre-trends.

-8 -6 -4 -2 0 2

Percent

-8 -4 0 4 8 12

Year

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs. LP-IV estimates

displayed as a solid blue line with 68% and 95% standard error bands. Estimation is robust with Driscoll and Kraay (1998) standard

errors. See text.

Figure 8: Response to 100 bps trilemma shock: Real GDP. Full sample, 1900–2015. Hairplots.

(a) Dropping each country one at a time

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

(b) Dropping successive 5-year windows one at a time

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs. LP-IV estimates

displayed as a solid blue line for baseline, other estimates shown by dashed lines. See text.

Figure 9:

Baseline asymmetric responses to 100 bps loosening and tightening shocks: Real GDP and

Solow decomposition. Full sample, 1900–2015.

(a) Estimates using Imbs correction for factor utilization, loosening shock

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Real GDP

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Labor

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Capital

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

TFP

(b) Estimates using Imbs correction for factor utilization, tightening shock

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Real GDP

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Labor

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Capital

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

TFP

Notes: Response to a 100 bps loosening and tightening shock in domestic short-term interest rate instrumented with the trilemma

IVs. Full sample: 1900–2015 (World Wars excluded). LP-IV estimates displayed as a thick lines with 68% and 95% standard error

bands. Both panels adjusts the raw data on capital stock and total hours to obtain estimates of actual factor inputs by using

the Imbs (1999) correction. In the upper panel the instrument

is replaced with zero when

z >

0 or ∆

R >

0, to include only

loosening shocks. In the lower panel the instrument

is replaced with zero when

z <

0 or ∆

R <

0,to include only tightening shocks.

Estimation is robust with Driscoll and Kraay (1998) standard errors. See text.

It is therefore natural to explore whether such symmetries appear at the longer 10+ year

horizon that we study here. Figure 9 presents some additional ﬁndings in this regard for our

baseline, full-sample, Solow decomposition responses. In the upper panel, the trilemma shocks

are restricted to include only strict loosening shocks: the instrument

is replaced with zero

when

z >

0 or ∆

R >

0, so both the shock in the foreign base and the rate change in the home

peg move in a negative direction. In the lower panel, the trilemma shocks are restricted to

include only strict tightening shocks: the instrument

is replaced with zero when

z <

0 or

∆

R <

0, so both the shock in the foreign base and the rate change in the home peg move in a

positive direction.

For loosening shocks in Figure 9a, we see that, relative to baseline, any strong evidence for

hysteresis virtually disappears. None of the responses is negative and statistically signiﬁcant

at long horizons, although some of the point estimates still go in the same direction. For

tightening shocks in Figure 9b, we see that relative to baseline, the evidence for hysteresis is

even stronger. All of the responses are negative and statistically signiﬁcant at long horizons,

larger than in the baseline, and clearly this is what was driving our main result.

The lesson of this exercise highlights a plausible, but important caveat to our main results.

Central banks can’t manipulate the supply-side of the economy to increase its long-run capacity

by exploiting loose policy. However, when policy is kept too tight, monetary policy can decrease

the long-run productive capacity of the economy. This evidence of asymmetric hysteresis

therefore carries important lessons for policy. The balance of risks for monetary policy mistakes

are not evenly weighted when it comes to the long-run productive path of the economy:

unusually tight policy risks signiﬁcant downside damage, but unusually loose policy can’t do

much to deliver any upside beneﬁt.

7. Real Effects of Monetary Shocks in the Literature

We now brieﬂy compare our ﬁnding of persistent eﬀects of monetary shocks to the literature.

We ﬁnd that, when we extend the estimation horizon, some of the recent prominent studies in

this literature yield reassuringly similar ﬁndings of longer-run non-neutrality, even though the

main focus of these studies was quite diﬀerent and looked at a shorter horizon. These results

for U.S. and U.K. economies in recent periods is consistent with relatively long-lived eﬀects of

monetary policy across diﬀerent identiﬁcation schemes.

In Figures 10–12, we replicate and extend estimates from the published studies of Miranda-

Agrippino and Ricco (2021) and Brunnermeier, Palia, Sastry, and Sims (2021) for the U.S.

economy, and Cesa-Bianchi, Thwaites, and Vicondoa (2020) for the U.K. economy. Using their

published replication codes, we increase estimation horizon to 8-years or 96-months.

Figure 10: Eﬀects of monetary shocks in the U.S., Miranda-Agrippino and Ricco (2021)

(a) Industrial Production

-2.5 -2 -1.5 -1 -.5 0

Percent

0 24 48 72 96

Months

(b) One-year T-bond

-.5 0 .5 1

Percent

0 24 48 72 96

Months

Notes: The ﬁgure plots the estimated IRFs for U.S. Industrial Production and One-year Treasury Bond to monetary policy shocks.

Panels (a) and (b) display our estimates from a Bayesian VAR(12) on monthly data (1979:1–2014:12) using Miranda-Agrippino and

Ricco (2021, Figure 3)’s informationally robust monetary policy instruments. IRFs are traced following a one-time exogenous shock

in the interest rate of about 100 basis points. Shaded areas denote 90% posterior coverage bands in panels (a) and (b).

Miranda-Agrippino and Ricco (2021) estimate a six-variable monthly Bayesian VAR(12)

for the U.S. economy with high-frequency market-based monetary surprises around Federal

Open Market Committee announcements following the work of G¨urkaynak, Sack, and Swanson

(2005).

They propose an identiﬁcation strategy that makes use of Greenbook forecasts to

control for the central bank’s information set. They label their instrument as informationally

robust monetary policy instrument (MPI). Figure 10 reports impulse responses estimated from

using the methodology reported in their Figure 3. The shock is normalized to induce a 100

basis points increase in the one-year rate. Shaded areas denote 90% posterior coverage bands

in both panels.

Brunnermeier, Palia, Sastry, and Sims (2021) estimate a ten-variable monthly Bayesian

SVAR(10) model identiﬁed by heteroskedasticity (Rigobon, 2003) to study the relationship

between credit, output, and monetary policy.

Using time-varying variance across historical

episodes in the U.S. economy from the 1970s, they identify monetary policy shocks. Figure 11

reports impulse responses using their replication code for column 1 of their Figures 1 through

4. The shock is scaled to draws from a unit-scale

distribution with 5.7 degrees of freedom.

Shaded areas denote 90% posterior coverage bands in both panels.

Their Bayesian VAR is based on work by Coibion (2012) and Gertler and Karadi (2015). The vector

includes industrial production, the unemployment rate, the consumer price index, a commodity price index, the

excess bond premium of Gilchrist and Zakrajˇsek (2012), and the one-year nominal rate as the policy rate.

The vector includes industrial production, personal consumption expenditure price index, sum of commercial

bank real estate and consumer loans, commercial bank commercial & industrial loans, M1 money supply, Federal

funds rate, a commodity price index, term spread of 10 year over 3 month Treasuries, Gilchrist and Zakrajˇsek

(2012)’s bond spread, and “TED spread” of 3-month Eurodollars over 3 month Treasuries.

Figure 11: Eﬀects of monetary shocks in the U.S., Brunnermeier et al. (2021)

(a) Industrial Production

-4 -3 -2 -1 0 1

Percent

0 24 48 72 96

Months

(b) Fed Funds Rate

-.5 0 .5 1 1.5

Percent

0 24 48 72 96

Months

Notes: The ﬁgure plots the estimated IRFs for U.S. Industrial Production and Eﬀective Federal Funds Rate to monetary

policy shocks. Panels (a) and (b) display estimates from a large-scale Bayesian VAR with heteroskedasticity on monthly data

(1973:1–2015:6) from Brunnermeier et al. (2021). IRFs are traced following a one-time exogenous shock in the interest rate of

about 40 basis points. Shaded areas denote 90% posterior coverage bands in panels (a) and (b).

Figure 12: Eﬀects of monetary shocks in the U.K., Cesa-Bianchi et al. (2020)

(a) Monthly estimate of GDP

-3 -2 -1 0 1

Percent

0 24 48 72 96

Months

(b) One-year Gilt Rate

-.5 0 .5 1 1.5

Percent

0 24 48 72 96

Months

Notes: Extended estimates from Figures 2 and 3 from high-frequency identiﬁcation of U.K. monetary shocks Cesa-Bianchi, Thwaites,

and Vicondoa (2020). Shaded areas denote 68% conﬁdence intervals computed using moving block bootstrap with 5,000 replications.

Sample: 1992:1–2015:1.

Cesa-Bianchi, Thwaites, and Vicondoa (2020) estimate a monthly SVAR with external

instruments following Mertens and Ravn (2013) and Gertler and Karadi (2015). They construct

a new series of monetary policy surprises for the U.K. using intra-day data on the price of three-

month Sterling futures contracts around announcements by the Monetary Policy Committee of

the Bank of England.

Figure 12 reports impulse responses estimated using the methodology

in their Figure 3. The shock is normalized to induce a 100 basis points increase in the one-year

rate. Shaded areas denote 68% conﬁdence intervals computed using moving block bootstrap

with 5,000 replications (Jentsch and Lunsford, 2019).

Based on just three recent papers, these ﬁgures show strikingly persistent response of output

to transitory monetary interventions. But evidence consistent with our ﬁndings can also be

found in a range of other studies in the literature. As noted in the introduction, Bernanke

and Mihov (1998) also found similar persistent real eﬀects. We report a screenshot of their

results in Figure A2 in the appendix. Moran and Queralto (2018) estimate a causal eﬀect

of monetary policy shocks on TFP growth using a three-equation VAR model and Cholesky

identiﬁcation for the U.S. economy. Plagborg-Møller (2019) ﬁnds long-run eﬀects of monetary

shocks using Bayesian inference on Structural Vector Moving Average representation of the U.S.

data. Willems (2020) documents persistent eﬀects of large monetary tightenings in a panel

data spanning 162 countries over 1970–2017. Palma (2022) shows exogenous increases in money

supply in the early modern period had a persistent eﬀect of real activity. Cloyne and H¨urtgen

(2014, Fig 3), in their Bank of England working paper, document very persistent real eﬀects

of monetary shocks in the U.K. economy using an auto-regressive distributed lag framework,

also used by Romer and Romer (2004). Cloyne and H¨urtgen (2014) ﬁnd that the dynamics

of the underlying shock process play an important role in recovering long-run eﬀects in their

estimation method. In a similar vein, McKay and Wieland (2021) document a boom-bust cycle

to the narrative monetary shocks series of Romer and Romer (2004), but in that setting the

interest-rate tightening cycle is followed by an easing cycle in their estimation. A rationalization

of heterogeneity in estimated responses was oﬀered by Coibion (2012) who demonstrated that

diﬀerent estimations of causal eﬀects of monetary shocks may produce diﬀerent results as the

underlying shock process vary in terms of size and persistence.

The vector includes the consumer price index, the unemployment rate, the nominal eﬀective exchange rate,

the mortgage and corporate bond spreads, the nominal yield on the 1-year gilt as the policy rate, the spread

between the Moody’s BAA corporate bond and the U.S. 10-year government bond, a monthly estimate of GDP,

a measure of credit quantities, equity prices, and the trade balance. The policy surprises are available from

1997:6 to 2015:1. They use two lags in their VAR, we report results from a four lag speciﬁcation to allow more

persistence following the econometric lessons from section 4. Estimates from VAR(2) are also persistent, but

produce a relatively sharper recovery to zero.

8. Conclusion

This paper challenges the view that money is neutral in the long-run. We ﬁnd that monetary

policy has real eﬀects that last for a decade or more. In an important caveat, we ﬁnd responses

are asymmetric: strong for a tightening shock, weak for a loosening shock. We spent considerable

time and energy with the three pillars of our empirical strategy—identiﬁcation, data, and

methods—to assure the reader that our results are solid.

The source of the main hysteresis result—that monetary policy shocks have long-lasting

eﬀects on output—was striking to us even though a careful read of the literature suggests

that the evidence had been mounting for years. We ﬁnd that capital and TFP growth are

the main drivers of this result, but not hours worked, in contrast to standard models of labor

hysteresis. Eventually the labor market returns to its pre-shock datum, but the levels of capital

and TFP remain scarred. Our ﬁndings do not negate the inﬂuence of labor frictions in shaping

the business cycle at shorter horizons. Instead, after a few years, we do not ﬁnd a strong role

for such labor scarring in explaining why monetary policy has such long-lived eﬀects.

There is much that is left unexplored in this paper as it is already quite long. Determining the

micro-foundations that explain TFP growth hysteresis would require a diﬀerent paper devoted

to the topic with a completely diﬀerent data set. Exploring the optimality of the monetary

policy rule in more general settings, and the welfare consequences of the hysteresis results

documented here are of ﬁrst order importance for policymakers. Perhaps more importantly,

our paper challenges long-held views that require a reexamination of standard business cycle

models and evidence.

References

Akerlof, George A., William T. Dickens, and George L. Perry. 1996. The Macroeconomics of Low

Inﬂation. Brookings Papers on Economic Activity 1996(1): 1–76.

Angrist, Joshua D.,

Oscar Jord`a, and Guido M. Kuersteiner. 2018. Semiparametric Estimates of

Monetary Policy Eﬀects: String Theory Revisited. Journal of Business and Economic Statistics

36(3): 371–387.

Antolin-Diaz, Juan, and Paolo Surico. 2022. The Long-Run Eﬀects of Government Spending. CEPR

Discussion Paper DP17433.

Anzoategui, Diego, Diego Com´ın, Mark Gertler, and Joseba Mart´ınez. 2019. Endogenous Technology

Adoption and R&D as Sources of Business Cycle Persistence. American Economic Journal:

Macroeconomics 11(3): 67–110.

Bai, Jushan, and Pierre Perron. 1998. Estimating and Testing Linear Models with Multiple Structural

Changes. Econometrica 66(1): 47–78.

Baqaee, David Rezza, and Emmanuel Farhi. 2019. Productivity and Misallocation in General

Equilibrium. Quarterly Journal of Economics 135(1): 105–163.

Barlevy, Gadi. 2004. The Cost of Business Cycles under Endogenous Growth. American Economic

Review 94(4): 964–990.

Barnichon, Regis, Davide Debortoli, and Christian Matthes. 2021. Understanding the Size of the

Government Spending Multiplier: It’s in the Sign. Review of Economic Studies 89(1): 87–117.

Basu, Susanto, and John G Fernald. 2002. Aggregate Productivity and Aggregate Eechnology.

European Economic Review 46(6): 963–991.

Basu, Susanto, John G. Fernald, and Miles S. Kimball. 2006. Are Technology Improvements Contrac-

tionary? American Economic Review 96(5): 1418–1448.

Basu, Susanto, and Miles S. Kimball. 1997. Cyclical Productivity with Unobserved Input Variation.

NBER Working Paper 5915.

Benigno, Gianluca, and Pierpaolo Benigno. 2003. Price Stability in Open Economies. Review of

Economic Studies 70(4): 743–764.

Benigno, Gianluca, and Luca Fornaro. 2018. Stagnation Traps. Review of Economic Studies 85(3):

1425–1470.

Benigno, Gianluca, Luca Fornaro, and Martin Wolf. 2020. The Global Financial Resource Curse.

Federal Reserve Bank of New York Staﬀ Report 915.

Benigno, Pierpaolo, and Luca Antonio Ricci. 2011. The Inﬂation-Output Trade-oﬀ with Downward

Wage Rigidities. American Economic Review 101(4): 1436–66.

Benigno, Pierpaolo, and Michael Woodford. 2012. Linear-Quadratic Approximation of Optimal Policy

Problems. Journal of Economic Theory 147(1): 1–42.

Bergeaud, Antonin, Gilbert Cette, and R´emy Lecat. 2016. Productivity Trends in Advanced Countries

between 1890 and 2012. Review of Income and Wealth 62(3): 420–444.

Berk, Kenneth N. 1974. Consistent autoregressive spectral estimates. Annals of Statistics 2(3):

489–502.

Bernanke, Ben S., and Ilian Mihov. 1998. The Liquidity Eﬀect and Long-run Neutrality. Carnegie-

Rochester Conference Series on Public Policy 49(1): 149–194.

Bianchi, Francesco, Howard Kung, and Gonzalo Morales. 2019. Growth, Slowdowns, and Recoveries.

Journal of Monetary Economics 101(C): 47–63.

Bianchi, Javier, and Guido Lorenzoni. 2022. The prudential use of capital controls and foreign currency

reserves. In Handbook of International Economics: International Macroeconomics, Volume 6, edited

by Gopinath, Gita, Elhanan Helpman, and Kenneth Rogoﬀ, 237–289. Elsevier.

Bianchi, Javier, Pablo Ottonello, and Ignacio Presno. Forthcoming. Fiscal Stimulus under Sovereign

Risk. Journal of Political Economy .

Blanchard, Olivier. 2018. Should We Reject the Natural Rate Hypothesis? Journal of Economic

Perspectives 32(1): 97–120.

Blanchard, Olivier J., and Lawrence H. Summers. 1986. Hysteresis and the European Unemployment

Problem. In NBER Macroeconomics Annual 1986, volume 1, 15–90. Cambridge, Mass.: MIT Press.

Blanchard, Olivier Jean, and Charles M. Kahn. 1980. The Solution of Linear Diﬀerence Models under

Rational Expectations. Econometrica 48(5): 1305–1311.

Born, Benjamin, Francesco D’Ascanio, Gernot J M¨uller, and Johannes Pfeifer. 2022. Mr. Keynes

meets the Classics: Government spending and the Real Exchange Rate. CEPR Discussion Paper

DP14073.

Brunnermeier, Markus, Darius Palia, Karthik A. Sastry, and Christopher A. Sims. 2021. Feedbacks:

Financial Markets and Economic Activity. American Economic Review 111(6): 1845–79.

Calvo, Guillermo A. 1983. Staggered Prices in a Utility-maximizing Framework. Journal of Monetary

Economics 12(3): 383–398.

Cerra, Valerie, Antonio Fat´as, and Sweta C. Saxena. 2023. Hysteresis and Business Cycles. Journal

of Economic Literature 61(1): 181–225.

Cesa-Bianchi, Ambrogio, Gregory Thwaites, and Alejandro Vicondoa. 2020. Monetary Policy Trans-

mission in the United Kingdom: A High Frequency Identiﬁcation Approach. European Economic

Review 123: 103375.

Christiano, Lawrence. 2015. Applications of DSGE Model: What Happens if we Switch from Fixed to

Flexible Interest Rate Regime? Northwestern University. Unpublished.

Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans. 1999. Monetary Policy Shocks:

What Have We Learned and To What End? In Handbook of Macroeconomics, edited by John

B. Taylor and Michael Woodford, volume 1 of Handbook of Macroeconomics, chapter 2, 65–148.

Amsterdam: Elsevier.

Cloyne, James, and Patrick H¨urtgen. 2014. The Macroeconomic Eﬀects of Monetary Policy: A New

Measure for the United Kingdom. Bank of England Working Paper 493.

Cloyne, James, Joseba Mart´ınez, Haroon Mumtaz, and Paolo Surico. 2022. Short-Term Tax Cuts,

Long-Term Stimulus. NBER Working Paper 30246.

Coibion, Olivier. 2012. Are the eﬀects of monetary policy shocks big or small? American Economic

Journal: Macroeconomics 4(2): 1–32.

Conley, Timothy G., Christian B. Hansen, and Peter E. Rossi. 2012. Plausibly Exogenous. Review of

Economics and Statistics 94(1): 260–272.

di Giovanni, Julian, Justin McCrary, and Till Von Wachter. 2009. Following Germany’s lead: Using

international monetary linkages to estimate the eﬀect of monetary policy on the economy. Review

of Economics and Statistics 91(2): 315–331.

Driscoll, John C., and Aart C. Kraay. 1998. Consistent covariance matrix estimation with spatially

dependent panel data. Review of Economics and Statisticsfor 80(4): 549–560.

Durbin, J. 1959. Eﬃcient Estimation of Parameters in Moving-Average Models. Biometrika 46(3/4):

306–316.

Eggertsson, Gauti B., and Michael Woodford. 2003. The Zero Bound on Interest Rates and Optimal

Monetary Policy. Brookings Papers on Economic Activity 2003(1): 139–211.

Farhi, Emmanuel, and Iv´an Werning. 2012. Dealing with the Trilemma: Optimal Capital Controls

with Fixed Exchange Rates. NBER Working Paper 18199.

Farhi, Emmanuel, and Iv´an Werning. 2017. Fiscal Unions. American Economic Review 107(12):

3788–3834.

Fat´as, Antonio. 2000. Do business cycles cast long shadows? Short-run persistence and economic

growth. Journal of Economic Growth 5(2): 147–162.

Fernald, John G. 2014. A quarterly, utilization-adjusted series on total factor productivity. Federal

Reserve Bank of San Francisco Working Paper 2012-19.

Fornaro, Luca. 2015. Financial crises and exchange rate policy. Journal of International Economics

95(2): 202–215.

Fornaro, Luca, and Federica Romei. 2019. The Paradox of Global Thrift. American Economic Review

109(11): 3745–79.

Fornaro, Luca, and Martin Wolf. 2022. The Scars of Supply Shocks: Implications for Monetary Policy.

CEPR Discussion Paper 15423.

Furlanetto, Francesco, Antoine Lepetit, Ørjan Robstad, Juan Rubio-Ram´ırez, and P˚al Ulvedal. 2021.

Estimating hysteresis eﬀects. Norges Bank Working Paper 2020/13.

Gal´ı, Jordi. 1998. The liquidity eﬀect and long-run neutrality: A comment. Carnegie-Rochester

Conference Series on Public Policy 49: 195–206.

Gal´ı, Jordi. 2015a. Hysteresis and the European Unemployment Problem Revisited. NBER Working

Paper 21430.

Gal´ı, Jordi. 2015b. Monetary policy, inﬂation, and the business cycle: An introduction to the new

Keynesian framework and its applications. Princeton, N.J.: Princeton University Press.

Gal´ı, Jordi. 2020. Insider-Outsider Labor Markets, Hysteresis and Monetary Policy. NBER Working

Paper 27385.

Gal´ı, Jord´ı, and Tommaso Monacelli. 2016. Understanding the Gains from Wage Flexibility: The

Exchange Rate Connection. American Economic Review 106(12): 3829–68.

Garga, Vaishali, and Sanjay R. Singh. 2021. Output Hysteresis and Optimal Monetary Policy. Journal

of Monetary Economics 117: 871–886.

Gertler, Mark, and Peter Karadi. 2015. Monetary Policy Surprises, Credit Costs, and Economic

Activity. American Economic Journal: Macroeconomics 7(1): 44–76.

Gilchrist, Simon, and Egon Zakrajˇsek. 2012. Credit spreads and business cycle ﬂuctuations. American

Economic Review 102(4): 1692–1720.

Gordon, Robert J. 2016. The Rise and Fall of American Growth: The U.S. Standard of Living since

the Civil War. Princeton, N.J.: Princeton University Press.

Gourinchas, Pierre-Olivier. 2018. Monetary Policy Transmission in Emerging Markets: An Application

to Chile. In Monetary Policy and Global Spillovers: Mechanisms, Eﬀects and Policy Measures,

edited by Enrique G. Mendoza, Ernesto Past´en, and Diego Saravia, volume 25 of Central Banking,

Analysis, and Economic Policies Book Series, chapter 8, 279–324. Santiago: Central Bank of Chile.

Guerrieri, Luca, and Matteo Iacoviello. 2015. OccBin: A toolkit for solving dynamic models with

occasionally binding constraints easily. Journal of Monetary Economics 70: 22–38.

Guerron-Quintana, Pablo A., and Ryo Jinnai. 2019. Financial frictions, trends, and the great recession.

Quantitative Economics 10(2): 735–773.

G¨urkaynak, Refet S, Brian Sack, and Eric Swanson. 2005. Do Actions Speak Louder Than Words?

The Response of Asset Prices to Monetary Policy Actions and Statements. International Journal

of Central Banking 1(1): 55–94.

Hume, David. 1752. ‘Of Money’, ‘Of Interest’ and ‘Of the Balance of Trade’. In Essays Moral, Political,

and Literary, edited by Eugene F. Miller. Indianapolis, Ind.: Liberty Fund.

Ilzetzki, Ethan, Carmen M. Reinhart, and Kenneth S. Rogoﬀ. 2019. Exchange Arrangements Entering

the Twenty-First Century: Which Anchor will Hold? Quarterly Journal of Economics 134(2):

599–646.

Imbs, Jean M. 1999. Technology, growth and the business cycle. Journal of Monetary Economics

44(1): 65–80.

Jentsch, Carsten, and Kurt G Lunsford. 2019. The dynamic eﬀects of personal and corporate income

tax changes in the United States: Comment. American Economic Review 109(7): 2655–78.

Jord`a,

Oscar. 2005. Estimation and Inference of Impulse Responses by Local Projections. American

Economic Review 95(1): 161–182.

Jord`a,

Oscar, Moritz Schularick, and Alan M. Taylor. 2016. The great mortgaging: housing ﬁnance,

crises and business cycles. Economic Policy 31(85): 107–152.

Jord`a,

Oscar, Moritz Schularick, and Alan M. Taylor. 2017. Macroﬁnancial History and the New

Business Cycle Facts. NBER Macroeconomics Annual 31(1): 213–263.

Jord`a,

Oscar, Moritz Schularick, and Alan M. Taylor. 2020. The eﬀects of quasi-random monetary

experiments. Journal of Monetary Economics 112: 22–40.

Kilian, Lutz. 1998. Small-sample Conﬁdence Intervals for Impulse Response Functions. Review of

Economics and Statistics 80(2): 218–230.

King, Robert G., and Mark W. Watson. 1997. Testing Long-Run Neutrality. Federal Reserve Bank of

Richmond Economic Quarterly 83(3): 69–101.

Kuersteiner, Guido M. 2005. Automatic Inference For Inﬁnite Order Vector Autoregressions. Econo-

metric Theory 21(1): 85–115.

Las´een, Stefan, and Lars E. O. Svensson. 2011. Anticipated Alternative Policy Rate Paths in Policy

Simulations. International Journal of Central Banking 7(3): 1–35.

Lewis, Richard, and Gregory C. Reinsel. 1985. Prediction of multivariate time series by autoregressive

model ﬁtting. Journal of Multivariate Analysis 16(3): 393–411.

Li, Dake, Mikkel Plagborg-Møller, and Christian K Wolf. 2022. Local Projections vs. VARs: Lessons

From Thousands of DGPs. NBER Working Paper 30207.

Lusompa, Amaze B. 2019. Local projections, autocorrelation, and eﬃciency. UC Irvine. Unpublished.

Mankiw, N. Gregory. 2001. The inexorable and mysterious tradeoﬀ between inﬂation and unemployment.

Economic Journal 111(471): 45–61.

McKay, Alisdair, and Johannes F. Wieland. 2021. Lumpy Durable Consumption Demand and the

Limited Ammunition of Monetary Policy. Econometrica 89(6): 2717–2749.

Meier, Matthias, and Timo Reinelt. Forthcoming. Monetary Policy, Markup Dispersion, and Aggregate

TFP. Review of Economics and Statistics .

Mertens, Karel, and Morten O. Ravn. 2013. The Dynamic Eﬀects of Personal and Corporate Income

Tax Changes in the United States. American Economic Review 103(4): 1212–1247.

Miranda-Agrippino, Silvia, and Giovanni Ricco. 2021. The transmission of monetary policy shocks.

American Economic Journal: Macroeconomics 13(3): 74–107.

Montiel Olea, Jos´e Luis, and Mikkel Plagborg-Møller. 2021. Local Projection Inference Is Simpler and

More Robust Than You Think. Econometrica 89(4): 1789–1823.

Moran, Patrick, and Albert Queralto. 2018. Innovation, productivity, and monetary policy. Journal

of Monetary Economics 93: 24–41.

Nakamura, Emi, and J´on Steinsson. 2018. Identiﬁcation in Macroeconomics. Journal of Economic

Perspectives 32(3): 59–86.

Obstfeld, Maurice, and Kenneth Rogoﬀ. 1995a. Exchange rate dynamics redux. Journal of Political

Economy 103(3): 624–660.

Obstfeld, Maurice, and Kenneth Rogoﬀ. 1995b. The Mirage of Fixed Exchange Rates. Journal of

Economic Perspectives 9(4): 73–96.

Obstfeld, Maurice, Jay C. Shambaugh, and Alan M. Taylor. 2004. Monetary sovereignty, exchange

rates, and capital controls: The trilemma in the interwar period. IMF Staﬀ Papers 51(S1): 75–108.

Obstfeld, Maurice, Jay C. Shambaugh, and Alan M. Taylor. 2005. The Trilemma in History: Tradeoﬀs

Among Exchange Rates, Monetary Policies, and Capital Mobility. Review of Economics and

Statistics 87(3): 423–438.

Palma, Nuno. 2022. The real eﬀects of monetary expansions: evidence from a large-scale historical

experiment. Review of Economic Studies 89(3): 1593–1627.

Plagborg-Møller, Mikkel. 2019. Bayesian inference on structural impulse response functions. Quanti-

tative Economics 10(1): 145–184.

Plagborg-Møller, Mikkel, and Christian K Wolf. 2021. Local projections and VARs estimate the same

impulse responses. Econometrica 89(2): 955–980.

Queralto, Albert. 2020. A model of slow recoveries from ﬁnancial crises. Journal of Monetary

Economics 114: 1–25.

Quinn, Dennis, Martin Schindler, and A. Maria Toyoda. 2011. Assessing measures of ﬁnancial openness

and integration. IMF Economic Review 59(3): 488–522.

Ramey, Valerie A. 2016. Macroeconomic Shocks and Their Propagation. In Handbook of Macroeco-

nomics, edited by John B. Taylor and Harald Uhlig, volume 2 of Handbook of Macroeconomics,

chapter 2, 71–162. Amsterdam: Elsevier.

Rigobon, Roberto. 2003. Identiﬁcation through heteroskedasticity. Review of Economics and Statistics

85(4): 777–792.

Romer, Christina D., and David H. Romer. 2004. A New Measure of Monetary Shocks: Derivation

and Implications. American Economic Review 94(4): 1055–1084.

Schmitt-Groh´e, Stephanie, and Mart´ın Uribe. 2003. Closing small open economy models. Journal of

International Economics 61(1): 163–185.

Schmitt-Groh´e, Stephanie, and Mart´ın Uribe. 2016. Downward Nominal Wage Rigidity, Currency.

Pegs, and Involuntary Unemployment. Journal of Political Economy 124(5): 1466–1514.

Schmitt-Groh´e, Stephanie, Mart´ın Uribe, and Michael Woodford. 2022. International Macroeconomics:

A Modern Approach. Princeton, N.J.: Princeton University Press.

Schm¨oller, Michaela Elfsbacka, and Martin Spitzer. 2021. Deep recessions, slowing productivity and

missing (dis-)inﬂation in the euro area. European Economic Review 134: 103708.

Shambaugh, Jay C. 2004. The Eﬀect of Fixed Exchange Rates on Monetary Policy. Quarterly Journal

of Economics 119(1): 301–352.

Stadler, George W. 1990. Business Cycle Models with Endogenous Technology. American Economic

Review 80(4): 763–78.

Syverson, Chad. 2011. What determines productivity? Journal of Economic Literature 49(2): 326–365.

Tenreyro, Silvana, and Gregory Thwaites. 2016. Pushing on a String: US Monetary Policy is Less

Powerful in Recessions. American Economic Journal: Macroeconomics 8(4): 43–74.

Uribe, Mart´ın, and Stephanie Schmitt-Groh´e. 2017. Open Economy Macroeconomics. Princeton, N.J.:

Princeton University Press.

Vinci, Francesca, and Omar Licandro. 2021. Switching-track after the Great Recession. CESifo

Working Paper 9107.

Werning, Iv´an. 2011. Managing a Liquidity Trap: Monetary and Fiscal Policy. NBER Working Paper

17344.

Willems, Tim. 2020. What do monetary contractions do? Evidence from large tightenings. Review of

Economic Dynamics 38: 41–58.

Xu, Ke-Li. 2023. Local Projection Based Inference under General Conditions. CAEPR Working

Papers 2023-001, Department of Economics, Indiana University Bloomington.

Yellen, Janet L. 2016. Macroeconomic Research After the Crisis. Speech at the conference “The

Elusive ‘Great’ Recovery: Causes and Implications for Future Business Cycle Dynamics,” Federal

Reserve Bank of Boston, October.

ONLINE APPENDIX

A. Equilibrium conditions in the baseline model

A perfect foresight equilibrium in the baseline model (subsection 2.3) is given by a sequence of

11 processes

, C

T t

, C

, d

t+1

, R

, l

, W

, P

T t

, E

, P

, }

that satisfy the following equilibrium

conditions for a given sequence of exogenous processes {Y

T t

, ε

, P

∗

} and initial values {d



T t





1 − ω



1−ω

, (31)

T t

+ d

= Y

T t

t+1

, (32)

−1

T t

= ζE



−1

T t+1



, (33)

= R

∗

, (34)

T t

(1 − ω)C

T t

ωC

, (35)

−1

T t

= ζE

(

−1

T t+1

T t

)

, (36)

φl

T t

(37)

= C

(38)

T t

= E

∗

(39)

; and P

= W

∀t > 1 (40)

and R

∗

= ζ

−1

. (benchmark)

We label this the benchmark economy, and maintain it as is in exercises shown in Section 2.4

In the baseline peg economy conﬁguration (subsection 2.4), equilibrium is given by the same

equations as above, except we replace the ﬁnal equation with the following policy regime equation

(and shock process):

= 1 and R

∗

= ζ

−1

(peg)

B. Solution for the baseline model and Proofs for Section 2

Notation To derive some of the results, we consider a ﬁrst-order approximation of the equilibrium

conditions around the initial steady state (date 0 economy). For a variable

, we deﬁne:

ˆx

−¯x

¯x

In the case of d

t+1

, we deﬁne

t+1

= d

t+1

−

B.1. Baseline model

We assume the economy starts in a ﬂexible price steady state equilibrium indexed by initial debt

. We set this initial external debt of the economy to zero,

= 0. Real interest rates are equal

to inverse of the domestic discount factor:

t−1

∗

t−1

−1

, and tradable good consumption is

given by: C

T 0

= Y

. For tradable consumption to be positive, we assume

d < Y

(1 − ζ)

−1

World price of tradable good,

∗

, is normalized to one, and initial nominal exchange rate is

also normalized to one,

= 1. We normalize

such that

= 1 in the steady

state. That is,

= 1

− ω

. Real wages in units of tradables equal relative price of non-tradables:

1−ω



− (1 − ζ)



There is a one-time unanticipated shock at date 1. In the peg conﬁguration, the shock is to foreign

interest rate

∗

. In the benchmark conﬁguration, the shock is to

in the policy rule. The tradable

output endowment is constant at all dates: Y

We assume that non-tradable goods prices are pre-ﬁxed to

1−ω



− (1 − ζ)



for all non-tradable

ﬁrms at date 1. At future dates, non-tradable good prices are set ﬂexibly to equal nominal wage

(marginal product of labor in domestic currency).

B.2. Proof for Proposition 1

Because of equal inter- and intra-temporal elasticities, loans market problem can be solved without

reference to labor market and non-tradable goods market.

Proof.

The proof follows directly from Equation 5, Equation 10, and Equation 13, which deﬁne the

competitive equilibrium for {C

T t

, R

, d

t+1

} a given sequence of {R

∗

Date 2 onwards, the economy is in a steady state. The country’s external budget constraint at

dates 1 and 2:

T 1

= Y

− d

, (41)

T 2

= Y

− (1 − ζ)d

(42)

Combining the last two equations:

(1 − ζ) (C

T 1

− Y

+ d

) +

T 2

− Y

= 0

The demand for loans is given by :

ζR

− (Y

− d

)

Home real interest rate is equal to the world real interest rate

= R

∗

Equilibrium is then

= (1 − ζR

∗

+ ζR

∗

Tradable good consumption is given by:

T 1

= Y



1 − ζ +

∗



− (1 − ζ) d

; C

T 2

= Y

ζ (R

∗

(1 − ζ) + 1) − (1 − ζ)ζR

∗

B.3. Proof for Proposition 2

Proof. Peg economy:

Since

∗

is normalized to one, and nominal exchange rate is ﬁxed (and normalized to one),

T t

= E

= P

∗

= 1.

We consider a one-time unanticipated increase in

0. Hence,

T 1



1 − ζ +

∗



−

(1 − ζ) d

< C

T 0

. Date 2 onwards, W

= P

= p

(1−ω)C

T 2

Non-tradable good production at date t falls below the steady state:

T 1

T 0

1. To a

ﬁrst order approximation,, and substituting

= 0, non-tradable output in the benchmark economy is

given by

= −ζε

where hats denote log-linear approximation around the initial steady state.

Benchmark economy:

There is a one-time unanticipated shock at date 1 to home nominal interest rate

−1

, and

it returns to

−1

next period onwards. Consumption of tradable goods, debt and real interest

rate on tradable bond are not aﬀected by this shock. Date 2 onwards, p

(1−ω)



− (1 − ζ)



We can construct path for

, P

T t

}

that solves the system of equations such that

}

is ex-

actly same as in the peg economy. Given the nominal rigidity,

T 1



1−ζ+



−(1−ζ)

From the consumption Euler equation for tradable good,

T 2

ζP

T 1

. Finally, we obtain

(1−ω)



− (1 − ζ)



T 2

The policy rule under a peg prevents any adjustment in nominal exchange rates, i.e.,

Hence the path of nominal interest rates in a peg economy,

, is identical to the path in the

benchmark economy for a given shock.

To a ﬁrst order approximation, and substituting

= 0, non-tradable output in the benchmark

economy is given by

= −ζε

where hats denote log-linear approximation around the initial steady state.

Since the tradable output is an exogenous endowment, and we have assumed constant aggregation

weights, the impact-response of real GDP is identical across the two economies.

B.4. Proof of Proposition 3

Proof.

From interest rate Euler equation, and ﬁxed foreign tradable prices, we obtain the interest

parity relationship between home rate and foreign rate

= R

∗

t+1

In an imperfect pass-through economy, after log-linearizing around initial steady state, we can write

= λ

∗

; λ ∈ (0, 1] . (43)

the expected exchange rate appreciation is now (1

− λ

)

∗

. With same steps as in the proof provided

in subsection B.3, we obtain Proposition 3.

B.5. Proof of Proposition 4

Proof.

The solution for benchmark economy is as discussed in subsection B.3. In the peg economy,

there is now endogenous production of tradables. Because of common labor markets and constant

returns to scale production in tradable good sector, real wages (in units of tradable goods) are equal

to one. Because the total labor supplied in the economy is divided between tradable and non-tradable

good sector, the labor market clearing condition is now modiﬁed as:

= L

T t

+ L

Substituting this market clearing condition in the intra-temporal labor supply condition of the

household, we get:

φ (L

T t

+ L

)

T t

= 1

We solve for the steady state value such that

0 and

= 1. From the

intra-temporal condition on choice between tradables and nontradables, we then get the steady state

tradable output:

1 − ω

+ (1 − ζ)

Tradable consumption is

1−ω

. Finally, the labor disutility parameter is set such that labor market

clears:

φ =

1 − ω



+ 1



Total labor employed in the initial steady state (indexed with

d) is thus

L = 1 +

1 − ω

+ (1 − ζ)

We assume that

d = 0.

Because of the assumed nominal rigidity, tradables and non-tradables consumption co-move one

to one. As a result, the intra-temporal labor supply condition becomes



T t

1−ω

T t



T t

= 1

Since

ν >

0, tradable output co-moves negatively with tradable (and non-tradable) consumption, and

tradable output is more volatile than non-tradable output.

From the following system of equations, we can ﬁnd the solution for {C

T t

, Y

T t

, R

, d

t+1

, C

}

T t

+ d

= Y

T t

t+1

, (44)

−1

T t

= ζE



−1

T t+1



, (45)

= R

∗

(46)

1 =



T t

1−ω

T t



T t

(47)

1 − ω

T t

(48)

As before, the solution takes the following form: date 2 onwards the economy is in a new long-run.

We log-linearize the equilibrium around the initial steady state to analytically derive the sign of the

bias from trilemma identiﬁcation upto a ﬁrst-order approximation for small shocks and initial debt

equal to zero.

T t

t+1

, (49)

T t

T t+1

−

, (50)

∗

(51)

T t

= −

(1 + ν(1 − ω))

νω

T t

(52)

T t

(53)

where

≡

1−ω

The date-1 solution for non-tradable output and tradable-outptut is given by:

T 1

= −ζ

< 0;

T 1

(1 + ν(1 − ω))

νω

> 0

Tradable output goes up in response to a one-time increase in

∗

in the peg economy. There is an

increase in labor supply in the tradable goods sector following a contraction in demand for labor in

the non-tradable sector.

While the impact response of non-tradable output is identical across the peg and the benchmark

economy, total output’s response at date 1 is biased upwards (i.e., towards zero, or smaller absolute

size) in the peg economy relative the benchmark economy.

B.6. Proof of Proposition 5

Proof.

The log-linear equilibrium (around initial steady state with zero debt) conditions for tradable-

goods consumption, output, and real interest rate are given by:

T t

t+1

, (54)

T t

T t+1

−

, (55)

∗

(56)

T t

= −α

∗

. (57)

where

≡

1−ω

. We can solve these equations to derive the response of tradable consumption at

date 1:

T 1

= −ζ

∗

− α(1 − ζ)

∗

Rest of the economy is same as studied in the baseline peg economy. Given the nominal rigidity

and ﬁxed exchange rate regime, tradable and non-tradable consumption perfectly co-move:

T 1

The non-tradable output in the peg economy falls more than in the benchmark economy. The diﬀerence

is equal to α(1 − ζ)

∗

In the presence of the spillover, tradable output contracts with an increase in base interest rates,

while it is unaﬀected in the benchmark economy.

Using the construction of real GDP described in Section 2.3, we can compute the approximate

diﬀerence in the impulse responses of real GDP as

peg

−

benchmark

P Y

|{z}

tradable output

share in GDP

× (

peg

T t

−

benchmark

T t

) +

P Y

| {z }

non-tradable output

share in GDP

× (

peg

−

benchmark

)

(58)

P Y

∗

P Y

α(1 − ζ)

∗

. (59)

Now we assume that the base shock equals the benchmark policy shock,

∗

= ε

, so we have that

peg

∗

|{z}

≡γ

−

benchmark

| {z }

≡β



P Y

+ (1 − ζ)

P Y



α .

Hence,

β = γ

−



P Y

+ (1 − ζ)

P Y



α . (60)

C. Extension: time-varying aggregation weights

We consider the more general extension of the baseline model (subsection 2.3) allowing for time-

variation in aggregation weights in the construction of total output. The consumption aggregator is :

= Ψ

T t

1−ω

, where Ψ

≡ ω

−ω

− ω

)

1−ω

is a scaling factor. This implies that domestic CPI is

given by

T t

1−ω

. Total nominal output is

T t

. Let total output be denoted with

, and is given by:

T t

+ P

= p

ω−1

T t

+ p

where p

≡

T t

. From the optimality conditions, we have that

(1 − ω)C

T t

ωC

In terms of log-deviations from steady state, total output is given by



(ω − 1)p

ω−1

+ ωp





T t

−



P Y

T t

P Y

When ω → 0 (tradable goods share is inﬁnitesimally small),

= p

−1



−

T t



P Y

T t

P Y

In the baseline model, with exogenous endowment of tradable goods,

= p

−1



−

T t



P Y

Recall that

is identical across the peg and the benchmark economy as proved in Proposition

2. From results in Appendix B sequence of

T t

0 under a peg and equal to 0 under benchmark

economy. Hence the response of total output under a peg is downward biased relative to that under

the benchmark economy.

D. Proofs of consistency for impulse responses

This section provides the basic ideas behind the proofs of consistency for truncated VARs and LPs

when the true DGP is an invertible MA(

∞

). The reader is referred to the references cited for additional

details.

D.1. Data generating process and main assumptions

Assume the data generating process for the m–dimensional vector process y

is:

∞

j=0

t−j

; B

= I;

∞

j=0

||B

|| < ∞ , (61)

where

||B

(

′

) and

(

) =

∞

j=0

such that

det{B

(

)

} ̸

= 0 for

|z| ≤

Under these

assumptions, this invertible M A(∞) can also be expressed as:

∞

j=1

t−j

+ ϵ

;

∞

j=1

||A

|| < ∞; det{A(z)} ̸= 0 for |z| ≤ 1 .

Further, we make assumptions 1–4 following Lewis and Reinsel (1985), and Lusompa (2019) (Kuer-

steiner (2005) makes somewhat stronger assumptions because he later derives testing procedures to

determine the optimal lag length). These assumptions are:

Assumption 1 {y

} is generated by Equation 61.

Assumption 2 E|ϵ

| ≤ γ

< ∞ for 1 ≤ i, j, k, l ≤ m.

Assumption 3 The truncation lag

is chosen as a function of the sample size

such that

/T → 0 as p, T → ∞.

Assumption 4 p is chosen as a function of T such that

1/2

∞

j=p+1

||A

|| → 0 as p, T → ∞.

Then, as discussed in the text, Lewis and Reinsel (1985) show:

− A

−→ 0 as p, T → ∞.

This well-known result says that even when the data are generated by an inﬁnite-order process, the

coeﬃcients of the ﬁrst

terms are consistently estimated. We show next that despite this result,

inconsistencies in the estimation of impulse responses can crop up.

D.2. Potential sources of bias in truncated VARs

In ﬁnite samples, inconsistent estimates of the impulse response function can arise from at least two

sources that we now quantify: (1) the truncation lag is too short given Assumptions 1–4; and (2) the

truncation lag is appropriate, but the impulse response is calculated for periods that extend beyond

the truncation lag. To investigate the ﬁrst source of inconsistency, rewrite the V AR(∞) as

j=1

t−j

+ u

j=k+1

t−j

∞

j=p+1

t−j

+ ϵ

where we assume

k < p

and

is the truncation lag that meets Assumptions 1–4 of the proof of

consistency. Hence rewrite the previous expression as

= A(k)X

k,t−1

+ u

; A(k) = (A

, . . . , A

); X

k,t−1

= (y

t−1

, . . . , y

t−k

)

′

The least-squares estimate of A(k) is therefore

A(k) =

T − k

′

k,t−1

T − k

k,t−1

′

k,t−1

−1

Hence

A(k) = A(k) +

T − k

′

k,t−1

T − k

k,t−1

′

k,t−1

−1

Given the three components of

, it is easy to see that the source of inconsistency in estimates of the

ﬁrst k autoregressive terms will come from the component





T − k

j=k+1

t−j

′

k,t−1





T − k

k,t−1

′

k,t−1

−1

since the proof of consistency in Lewis and Reinsel (1985) shows that the other two terms vanish

asymptotically. The source of inconsistency can be quantiﬁed by noticing that

T − k

k,t−1

′

k,t−1

−1

→







Γ(0) Γ(1) · · · Γ(k)

Γ(1) Γ(0) · · · Γ(k − 1)

Γ(k) Γ(k − 1) · · · Γ(0)







−1

→ Γ

−1

as shown in Lewis and Reinsel (1985), where

(

′

t−j

) = Γ(

) and Γ(

−j

) = Γ(

)

′

. Hence, asymptoti-

cally, the source of inconsistency is

k+1

(Γ(j − 1), . . . , Γ(j − k)) Γ

−1

However, even when the lag-length

is chosen to be suﬃciently large, another source of bias can crop

up into the estimation of the impulse response. In particular, following Durbin (1959), we know that

= A

h−1

+ A

h−2

+ · · · + A

h−1

+ A

If the VAR is truncated at lag k, for k ≤ p, it is easy to see that the previous expression becomes

= A

h−1

+ A

h−2

+ · · · + A

k−h

+ A

k+1

k−h−1

+ · · · + A

h−1

+ A

{z }

remainder

and, hence,

− B

|| = ||A

k+1

(

k−h−1

− B

k−h−1

) + . . . + A

h−1

(

− B

) + A

|| ̸→ 0 ,

since ||

k+j

− B

k+j

|| for j ≥ 1 is not guaranteed to vanish asymptotically.

Next notice that the impulse response for horizons h > k will be estimated using the recursion

k+j

k+j−1

+ . . . +

; j = 1, . . . , H .

Even if k = p, and hence ||

|| → A

for j = 1, . . . , k, the fact remains that the remainder term

k+1

k−h−1

+ · · · + A

h−1

+ A

cumulates increasing sums of coeﬃcients that are not estimated in the model. As the Monte Carlo

exercise showed earlier, the inconsistency at longer horizons tends to accumulate.

D.3. The consistency of the local projections estimator

In this section we use the same assumptions as in the previous section to establish the consistency of

the local projections estimator at any horizon.

Using the VAR(∞) representation of the DGP and recursive substitution, it is easy to see that

t+h

= B

h+1

t−1

+ {C

h+2

t−2

+ C

h+3

t−3

+ · · · } + ϵ

t+h

+ B

t+h−1

+ ϵ

where

h+2

= B

+ · · · + B

+ A

h+1

h+3

= B

+ · · · + B

h+1

+ A

h+2

h+k

= B

k−1

+ · · · + B

h+k−2

+ A

h+k−1

Now, consider truncating the lag of the local projection at

, where

meets Assumptions 1–4 of

the Lewis and Reinsel (1985) consistency theorem discussed in the previous section.

Then the truncated local projection can be written as

t+h

= B

h+1

t−1

+ C

h+2

t−2

+ C

h+3

t−3

· · · + C

h+k

t−k

+ u

t+h

= ϵ

t+h

+ {B

t+h−1

+ B

t+h−2

+ · · · + B

} + {C

h+k+1

t−k−1

+ C

h+k+2

t−k−2

+ · · · } .

Let

= (

, C

h+2

, . . . , C

h+k

) and

t−1

= (

t−1

, . . . , y

t−k

)

′

as deﬁned earlier but where the

subscript k is omitted here for simplicity. Then the local projection can be compactly written as

t+h

= DX

t−1

+ u

t+h

The least-squares estimate of D is simply

D =

T − h − k

T −h

t+h

′

t−1

T − h − k

T −h

t−1

′

t−1

−1

from where consistency can be determined from the following expression

D = D +

T − h − k

T −h

t+h

′

t−1

T − h − k

T −h

t−1

′

t−1

−1

Lewis and Reinsel (1985) show that

−1

is uniformly bounded where we use the fact that

||AB||

≤ ||A||

||B||

; as well as

||AB||

≤ ||A||

||B||

where

||C||

sup

l̸=0

′

Cl/l

′

, the largest

eigenvalue of C

′

C (see Wiener and Masani, 1958).

Now we turn our focus to the terms

T − h − k

T −h

t+h

′

t−1

T − h − k

T −h

(ϵ

t+h

+ B

t+h−1

+ · · · + B

′

t−1

T − h − k

T −h

h+k

t−k−1

+ C

h+k+1

t−k−2

+ · · · )X

′

t−1

It is easy to see that

T − h − k

T −h

t+h

′

t−1

→ 0 ,

T − h − k

T −h

t+h−j

′

t−1

→ 0 ,

since ||B

|| < ∞ for j = 1, · · · , h. Hence, the only tricky part is to examine the terms

h+k+j

T − h − k

T −h

t−k−(j+1)

′

t−1

for j = 0, 1, . . . .

Note that

h+k+j

= B

k+k

+ · · · + B

h+k+j−1

+ A

h+k+j

j = 0, 1, . . . ,

A10

hence

∞

j=0

||C

h+k+j

|| =

∞

j=0

||B

k+k

+ · · · + B

h+k+j−1

+ A

h+k+j

≤

∞

j=0

||B

k+j

|| + · · · +

∞

j=1

||B

h+k+j−1

|| +

∞

j=0

||A

h+j+j

= ||B

∞

j=0

||A

k+j

|| + · · · + ||B

∞

j=1

||A

h+k+j−1

|| +

∞

j=0

||A

h+k+j

||.

From the assumptions we know that the ||B

are uniformly bounded, and also that

1/2

∞

j=0

||A

k+j

|| → 0 =⇒ k

1/2

∞

j=0

||C

h+k+j

|| → 0 ,

and this condition can now be used to show that

∞

j=0

h+k+j

T − h − k

T −h

t−k−(j+1)

′

t−1

→ 0, as k, T → ∞ .

Summarizing, these derivations show that the same conditions that ensure consistency of the coeﬃcients

estimates in a truncated VAR also ensure consistency of the local projections with truncated lag

length. However, because the coeﬃcient for

t−1

in the local projection is a direct estimate of the

impulse response coeﬃcient, then we directly get a proof of consistency for the coeﬃcients of the

impulse response at any horizon regardless of truncation.

D.4. Monte Carlo results for impulse response estimators

This section provides details of the Monte Carlo experiments reported in the main text in addition to

presenting complementary Monte Carlo experiments based on the same simulated data, but presenting

the impulse response (rather than the cumulated response itself).

The data are generated as a

(25) model whose coeﬃcients are generated by the following

Gaussian Basis Function:

a exp

(

−

((

j − b

)

) for

= 1

, . . . ,

25 and for

−

= 12; and

= 6. This results in the impulse and cumulative responses shown in panel (a) of Figure 2. The error

terms are assumed to be standard Gaussian. The left hand side variable is expressed in the diﬀerences

to replicate exactly the estimation of the cumulative response in the empirical section. We simulate

samples of size 1,500, but the ﬁrst 500 observations are then discarded to avoid initialization issues.

Using these data, we then estimate

(

) models for

= 3

12 and local projections using 2 lags.

As a complement to Figure 2, Figure A1 presents the experiments based on the impulse response

itself to illustrate the consistency of the

(

) estimators up to horizon

h ≤ k

but not beyond. The

solid blue line is the true response based on our parameter choices for the D.G.P. The dashed blue

line with Monte Carlo one and two standard error bands are the local projections using two lags only.

The dotted maroon lines are the impulse responses from AR models with 3, 6, 9, and 12 lags as in the

Monte Carlo in the main text. As the ﬁgure clearly shows, impulse response coeﬃcients are estimated

well using the

(

) models up to horizon

, as the asymptotic theory just presented showed. In

contrast, the local projection estimator does well across all horizons. The cumulative versions of these

responses are the experiments reported in Figure 2 in the main text.

A11

Figure A1:

Estimating non-cumulative responses: autoregressive versus local projection biases at

long horizons.

-.15 -.1 -.05 0 .05 .1

0 5 10 15 20 25

Horizon

True LP AR(3)

AR(6) AR(9) AR(12)

Notes: sample size = 1,000. Monte Carlo replications: 1,000. Error bands in light blue are 1 and 2 standard error bands based on

the local projection Monte Carlo average.

(

) for k = 3, 6, 9, 12 refers to impulse responses from an autoregressive model with

k lags. See text.

E. Trilemma Identification with Calvo Staggered Price Setting

& Debt-Elastic Interest Premium

This appendix shows how identiﬁcation works with Calvo (1983) staggered price setting assumption

instead of the Obstfeld and Rogoﬀ (1995) one-period sticky price assumption. There are two

modiﬁcations relative to the model described in subsection 2.3: (i) non-tradable good ﬁrms set prices

in a staggered fashion ala Calvo (1983), and (ii) we assume a debt-elastic interest rate premium

in order to stationarize the economy. The main takeaway from this section is that the trilemma

identiﬁcation of output response attenuates the object of interest: response of output to a domestic

monetary shock in the benchmark economy.

E.1. Baseline model with Calvo price rigidities

Equilibrium conditions in the Calvo model

A perfect foresight equilibrium in the baseline model, presented in the paper, is given by a sequence of

15 processes

T t

, C

, d

t+1

, p

T t

, R

, w

, L

∆

pNt

, E

, ˜p

, K

Npt

, Z

Npt

}

that satisfy the

following equilibrium conditions for a given sequence of exogenous processes

T t

, R

∗

}

and initial

values {d

, E

−1

, p

−1

, ∆

pN−1

T t

+ d

= Y

T t

t+1

, (62)

A12

(1 − ω)C

T t

ωC

, (63)

−1

T t

= ξE

(

−1

T t+1

T t

)

, (64)

−1

T t

= ξE



−1

T t+1



, (65)

= R

∗

+ ψ(e

t+1

−

− 1) , (66)

˜p

Npt

, (67)

Npt

= w

+ θ

T t

T t+1

Nt+1

Npt+1

, (68)

Npt

= p

+ θ

T t

T t+1

−1

Nt+1

Npt+1

, (69)

1 = θ

−1

+ (1 − θ

)˜p

1−ϵ

, (70)

φL

T t

= w

, (71)

∆

= C

, (72)

∆

= (1 − θ

)˜p

−ϵ

+ θ

∆

Nt−1

, (73)

t−1

T t

, (74)

T t

t−1

∗

. (75)

and one of the following two equations for the respective policy regime:

= 1 (peg)

(benchmark)

Steady State Equilibrium

We solve for a deterministic steady state of the economy indexed with level of debt

. We assume

that such a steady state exists. This requires that

ψ >

0 is suﬃciently large that the economy may

return to its steady state equilibrium, described by:

R =

∗

= ζ

−1

− (1 − ζ)

L = 1

= Π

= p = w = ∆

= ˜p

= E = 1

φ = 1 − ω

Note that we have assumed that there are steady state government subsidies that oﬀset ﬁrm

markups.

A13

First order approximation equilibrium

We consider a ﬁrst-order approximation of the open economy model with Calvo price setting around

the steady state just described. For a variable x, we deﬁne:

ˆx

− ¯x

¯x

In the case of d

t+1

, we deﬁne

t+1

= d

t+1

−

T t

= ζ

t+1

− ζ

, (76)

ˆp

T t

−

, (77)

T t

= E

T t+1

−



− E

T t+1

i

, (78)

T t

= E

T t+1

−

, (79)

∗

+ ζψ

t+1

, (80)

= ζ

Nt+1

+ κ

, (81)

ˆp

− ˆp

t−1

−

T t

, (82)

T t

−

t−1

. (83)

and one of the following three equations for the respective policy regime:

= 0 (peg)

= 0 (ﬂoat)

∗

= 0;

= ε

(benchmark)

Relative to the model presented in subsection 2.3, the main change is that nominal rigidities are now

captured in the new Keynesian Phillips curve (Equation 81), where

κ >

0 is the slope of the Phillips

curve.

Proposition 1 holds in this economy. We can determine the equilibrium for

{

T t

t+1

}

for

given sequence of {

∗

, ε

} and initial value of

= 0.

Using method of undetermined coeﬃcients, we ﬁnd the following solution for {

T t

t+1

T t







0 if t < 1;

if t = 1;

if t > 1.

(84)

t+1











0 if t < 1;

(1−ζψ

d)ζ

if t = 1;

(1−ζψ

d)ζ

if t > 1.

(85)

peg







0 if t < 1;

+ ζψd

if t = 1;

ζψd

t+1

if t > 1.

(86)

A14

where D

, and D

are deﬁned as follows:

= −



1 − ζ(ψ

+ 1 − ζψ





1 − ζ(ψ

+ 1 − ζψ



+ 4ψζ

= −

ζ(1 − D

ζ(1 − ζψ

d) − (D

− ψζ)

Assumption 1 (Regularity assumptions). We make regularity assumptions on parameters such that

following restrictions are satisﬁed:

1. We assume that 1 − ζψ

d > 0.

In a stationary equilibrium, debt converges back to steady state. This implies that parameters

be such that



(1−ζψ

d)ζ



< 1 is satisﬁed.

For monotonicity condition to be satisﬁed, i.e. interest rate at home to positively co-move with

foreign interest rate, we require that 1 + ψ

(1−ζψ

> 0

The passthrough from change in foreign rate to change in home interest rate is not exactly one,

but is instead given by: 1 +

(1−ζψ

. When

= 0, it can be shown that this passthrough is less

than one for values ψ > 0. For suﬃciently large

d, the passthrough can be greater than one.

Assumption 2 (Closed-economy Limit). We assume that debt position of the economy in steady

state is such that

d =

This assumption ensures that, in response to foreign interest rate shock, the passthrough of foreign

rate to home rate is exactly equal to one. Under Assumption 2, D

= −1. The solution for tradable

consumption block in this case is then

T 1

= −

∗

T t

= 0 ∀t ≥ 2

For values of

1, date-1 response of tradable consumption to foreign interest rate is less than

one. That is,

> −

1. Finally, for values

1, date-1 response of tradable consumption to foreign

interest rate is larger than one. That is, D

< −1.

T 1

> −ε

;

< 0;

∗

< 1;

T 1

= −ε

;

= 0;

∗

= 1;

T 1

< −ε

;

> 0;

∗

> 1.

(87)

We consider

≤

1 to be the empirically relevant case for the sample of our economies. This

restriction requires that foreign debt to tradable output ratio ratio of an economy be less than 100%,

which is reasonable in the case of advanced economies. For example, Fornaro and Romei (2019)

calibrate this debt to GDP ratio at 9.4% in their quantitative exercise for a sample of advanced

economies.

A15

peg economy

We assume that

for exposition purposes. Later, we will describe how results are aﬀected in

the more general empirically relevant case of

d ≤

We can then construct an equilibrium in the peg economy such that following equations determine

a path for {

= E

Nt+1

−



− E

Nt+1

i

, (88)

= ζ

Nt+1

+ κ

, (89)

(90)

along with a restriction that

T 1

−

The solution involves ﬁnding

such that the system satisﬁes the sequence implied by the above

equations, and results in the economy converging back to the steady state.

The solution of this system at date 1 is as follows:

= −



1 +

ζ[κ+ζ(1−ϕ)]

1+κ+ζ(1−ϕ)



κ + 1 +

ζ[κ+ζ(1−ϕ)]

1+κ+ζ(1−ϕ)

;

= −

κ + 1 +

ζ[κ+ζ(1−ϕ)]

1+κ+ζ(1−ϕ)

where ϕ ≡

1+κ+ζ−

√

4ζ

2ζ

≥ 0.

At Date 2,

1+κ+ζ(1−ϕ)

> 0. Date 3 onwards, the solution takes the form:

= ϕ

Nt−1

When the Phillips curve does not involve a forward looking term (

the initial value

of non-tradable output that satisﬁes the sequence is

= −

T 1

1 + κ

> −

∗

At Date 2 ,

−

∗

1+κ

0. Date 3 onwards, solution takes the following form:

= −

Nt−1

1+κ

> 0 for all t ≥ 3.

When prices are perfectly rigid (κ → 0),

= −

benchmark economy

We construct the standard Taylor rule equilibrium in the benchmark economy using the solution

method that as soon as the shock is over, the economy returns back to the steady state. With a

one-period shock to monetary policy rule, this solution implies that date 2 onwards non-tradable

consumption and inﬂation are back to steady state. At date 1:

= −ε

;

= κ

Such a Phillips curve can be derived from explicit microfoundations by choosing an appropriate indexation

for ﬁrms that do not reset prices in a given period. See for example, Bhattarai, Eggertsson, and Gafarov

(Forthcoming), Bilbiie (2019), Bilbiie (2020).

A16

Identiﬁcation when prices are rigid (κ = 0)

Assume Assumption 2 holds. When prices are perfectly rigid (

κ →

0), the trilemma identiﬁcation

exactly recovers the impulse response of GDP in the benchmark economy.

Upward bias in the peg economy when prices are sticky but not perfectly rigid

Assume Assumption 2 holds. The impulse response of GDP to a foreign interest rate shock in the peg

economy suﬀers from an upward bias relative to the impulse response of GDP to an equivalent sized

domestic monetary shock in the benchmark economy. In the peg economy, nontradable consumption

response is equal to

−



ζ[κ+ζ(1−ϕ)]

1+κ+ζ(1−ϕ)



κ+1+

ζ[κ+ζ(1−ϕ)]

1+κ+ζ(1−ϕ)

which is less (in absolute terms) than the response of

nontradable consumption in the benchmark economy to an equivalent sized domestic shock.

Upward bias in the peg economy when

d ≤ ζ

−1

Assume

d ≤

. The impulse response of GDP to a foreign interest rate shock in the peg economy

suﬀers from an upward bias relative to the impulse response of GDP to an equivalent sized domestic

monetary shock in the benchmark economy. In the peg economy, nontradable consumption response

is equal to

T 1

−

ζ[κ+ζ(1−ϕ)]

1+κ+ζ(1−ϕ)

κ+1+

ζ[κ+ζ(1−ϕ)]

1+κ+ζ(1−ϕ)

which is less (in absolute terms) than the response of nontradable

consumption in the benchmark economy to an equivalent sized domestic shock.

E.2. Production of tradables

As in the manuscript, we extend the baseline model (E.1) by allowing tradable output to be produced

with labor using a constant returns to scale technology:

T t

, where

T t

is labor used in

production of tradable goods. Prices are set ﬂexibly in the tradable-good sector. Labor is fully mobile,

within the economy, across the tradable and the non-tradable sector. Economy-wide real wages (in

units of tradable goods) are constant.

= 1 ∀t ≥ 0

Because the total labor supplied in the economy is divided between tradable and non-tradable

good sector, the labor market clearing condition is now modiﬁed as:

= L

T t

+ L

= Y

T t

+ L

Substituting this market clearing condition in the intra-temporal labor supply condition of the

household, we get:

φ (L

T t

+ L

)

T t

= w

= 1

Equilibrium conditions with production of tradables

A perfect foresight equilibrium in the baseline model, presented in the paper, is given by a sequence of

16 processes

T t

, C

, d

t+1

, p

T t

, R

, Y

T t

, L

T t

, L

∆

pNt

, E

, ˜p

, K

Npt

, Z

Npt

}

that satisfy

the following equilibrium conditions for a given sequence of exogenous processes

∗

}

and initial

values {d

, E

−1

, p

−1

, ∆

pN−1

T t

+ d

= Y

T t

t+1

, (91)

A17

(1 − ω)C

T t

ωC

, (92)

−1

T t

= ξE

(

−1

T t+1

T t

)

, (93)

−1

T t

= ξE



−1

T t+1



, (94)

= R

∗

+ ψ(e

t+1

−

− 1) , (95)

˜p

Npt

, (96)

Npt

= C

+ θ

T t

T t+1

Nt+1

Npt+1

, (97)

Npt

= p

+ θ

T t

T t+1

−1

Nt+1

Npt+1

, (98)

1 = θ

−1

+ (1 − θ

)˜p

1−ϵ

, (99)

φ (L

T t

+ L

)

T t

= 1 , (100)

T t

= L

T t

(101)

∆

= C

, (102)

∆

= (1 − θ

)˜p

−ϵ

+ θ

∆

Nt−1

, (103)

t−1

T t

, (104)

T t

t−1

∗

. (105)

and the peg economy policy regime

= 1 (106)

Note we have substituted the labor demand in tradable good sector,

= 1, into the labor supply

equation.

Steady State Equilibrium

We solve for a deterministic steady state of the economy indexed with level of debt

. We assume

that such a steady state exists. This requires that

ψ >

0 is suﬃciently large that the economy may

return to its steady state equilibrium. We solve for the steady state value such that

0 and

= 1. From the intra-temporal condition on choice between tradables and nontradables, we

then get the steady state tradable output:

1 − ω

+ (1 − ζ)

Finally, the labor disutility parameter is set such that labor market clears:

φ =

1 − ω



+ 1



A18

Total labor employed in the initial steady state (indexed with

d) is thus

L = 1 +

1 − ω

+ (1 − ζ)

R =

∗

= ζ

−1

1 − ω

+ (1 − ζ)

1 − ω

L = 1

= Π

= p = w = ∆

= ˜p

= E = 1

= Z

1 − θ

φ = 1 − ω

Note that we have assumed that there are steady state government subsidies that oﬀset ﬁrm

markups in the non-tradable sector.

First order approximation equilibrium

We consider a ﬁrst-order approximation of the equilibrium conditions around the steady state just

described. For a variable x, we deﬁne:

ˆx

− ¯x

¯x

In the case of d

t+1

, we deﬁne

t+1

= d

t+1

−

d. Other steady state equilibrium values are given by

T t

t+1

−

, (107)

ˆp

T t

−

, (108)

T t

= E

T t+1

−



− E

T t+1

i

, (109)

T t

= E

T t+1

−

, (110)

∗

+ ζψ

t+1

, (111)

0 = ν

T t

+ ν



1 −



T t

(112)

= ζ

Nt+1

+ κ



−

T t



, (113)

ˆp

− ˆp

t−1

−

T t

, (114)

T t

−

t−1

(115)

= 0 (116)

Solution

We derive an equilibrium of the peg economy. We guess that non-tradable goods inﬂation is zero.

That is,

= 0. We will verify that this condition is satisﬁed in the solution.

A19

From the Phillips curve equation,

= 0

∀t ≥

0 imples that

T t

. Thus, tradable and

nontradable consumption positively co-move one-to-one. The intra-temporal labor supply condition

simpliﬁes to

T t

= −



(1 + ν

−1

)

− 1



T t

Since

ν >

0, tradable output co-moves negatively with tradable (and non-tradable) consumption, and

tradable output is more volatile than non-tradable output.

Using the method of undetermined coeﬃcients, we can construct the solution for

{

T t

t+1

}

from the following system:

T t

= ζ

t+1

− ζ

T t

= E

T t+1

−

∗

+ ζψ

t+1

where B ≡

T t

+ ν

−1

L + 1.

The solution is as follows:

T t







0 if t < 1;

if t = 1;

if t > 1.

(117)

t+1











0 if t < 1;

d+BH

(1−ζψ

d)ζ

if t = 1;

1+BH

(1−ζψ

d)ζ

if t > 1.

(118)

peg







0 if t < 1;

+ ζψd

if t = 1;

ζψd

t+1

if t > 1.

(119)

where H

, and H

are deﬁned as follows:

= −



1 − ζ(ψB + 1 − ζψ





1 − ζ(ψB + 1 − ζψ



+ 4ψζ

= −

ζ(1 − H

ζ(1 − ζψ

d) − (H

− ψζ)B

Assumption 3 (Regularity assumptions). We make regularity assumptions on parameters such that

following restrictions are satisﬁed:

In a stationary equilibrium, debt converges back to steady state. This implies that parameters

be such that



1+BH

(1−ζψ

d)ζ



< 1 is satisﬁed.

For monotonicity condition to be satisﬁed, i.e. interest rate at home to positively co-move with

foreign interest rate, we require that 1 + ψ

d+BH

(1−ζψ

> 0

3. We assume that 1 − ζψ

d > 0.

The passthrough from change in foreign rate to change in home interest rate is not exactly one,

but is instead given by: 1 +

d+BD

(1−ζψ

. When

= 0, it can be shown that this passthrough is less than

one for values ψ > 0. For suﬃciently large

d, the passthrough can be greater than one.

A20

Assumption 4 (Closed-economy Limit). We assume that debt position of the economy in steady

state is such that

d =

(1 + ν

−1

)



1 +

1−ω



ζ + ν

−1

(1 − ζ)

This assumption is equivalent to

. It ensures that, in response to foreign interest rate shock,

the passthrough of foreign rate to home rate is exactly equal to one. Under Assumption 4,

−

The solution for tradable consumption block in this case is then

T 1

= −

∗

T t

= 0 ∀t ≥ 2

For values of

d <

(1+ν

−1

)

(

1−ω

)

ζ+ν

−1

(1−ζ)

, date-1 response of tradable consumption to foreign interest rate

is less than one. That is,

> −

1. Finally, for values

d >

(1+ν

−1

)

(

1−ω

)

ζ+ν

−1

(1−ζ)

, date-1 response of tradable

consumption to foreign interest rate is larger than one. That is, H

< −1.

T 1

> −ε

;

< 0;

∗

d <

(1+ν

−1

)

(

1−ω

)

ζ+ν

−1

(1−ζ)

;

T 1

= −ε

;

= 0;

∗

d =

(1+ν

−1

)

(

1−ω

)

ζ+ν

−1

(1−ζ)

;

T 1

< −ε

;

> 0;

∗

d >

(1+ν

−1

)

(

1−ω

)

ζ+ν

−1

(1−ζ)

(120)

We consider

d ≤

(1+ν

−1

)

(

1−ω

)

ζ+ν

−1

(1−ζ)

to be the empirically relevant case for our sample of advanced

economies. This condition is equivalently expressed as

d <

T t

−1

+ 1, which implies that Debt

to GDP ratio of a country be below:

T t

−1

. For standard macro calibration values of Frisch

elasticity equal to or greater than one, this implies an upper bound of atleast 100%.

The tradable output in this economy responds in the opposite direction to tradable consumption,

and its response is larger in magnitude relative to the tradable consumption response. Solution for

remaining variables in the peg economy is as follows:



0 if t < 1;

T t

if t ≥ 1.

(121)

T t



0 if t < 1;

−B

T t

if t ≥ 1.

(122)

ˆw

= ˆp

T t

= 0. (123)

E.3. Upward bias in peg economy with tradable production

We compare response of total output in two economies. One is the peg economy with tradable

production subject to a one time foreign interest rate shock. Second is the benchmark economy

considered in Section E.1 subject to a one-time domestic interest rate shock. The response of non-

tradable output and total output in the benchmark economy is exactly equal to the shock. The

response of nontradable output in the peg economy is exactly equal to the shock when the steady

state debt equals

closed

≡

(1+ν

−1

)

(

1−ω

)

ζ+ν

−1

(1−ζ)

. The tradable output moves in the opposite direction to

the nontradable output. Thus, total output response in the peg economy is less than the magnitude

of the shock.

A21

For values of steady state debt less than

closed

,the nontradable output in peg economy responds

by less than the magnitude of the shock. Even in this case, total output response in the peg economy

is less than the magnitude of the shock.

F. Imbs correction

We follow Imbs (1999) and adjust TFP for utilization of capital and labor inputs. See Paul (2020)

for a related construction of utilization-adjusted TFP in the historical data. We assume perfectly

competitive factor markets and a technology which is constant returns to scale in eﬀective capital and

labor. In aggregate, and for the representative ﬁrm, the production function is

= A

)

1−α

where

is output,

is capital stock,

is total hours worked, and

and

denote the respective

factor utilizations.

is the utilization adjusted TFP. We assume perfect competition in the input

and the output markets. Higher capital utilization increases the depreciation of capital

δu

where

ϕ >

1. As a result, ﬁrms choose capital utilization rate optimally. Labor hoarding is calculated

assuming instantaneous adjustment of eﬀort

against a payment of a higher wage

(

), while keeping

ﬁxed employment (determined one period in advance). The ﬁrm’s optimization problem is given by:

max

)

1−α

− w(e

− (r

+ δu

))K

Households choose consumption, labor supply and eﬀort to maximize their lifetime utility subject to

their budget constraint (with complete asset markets)

max

∞

t=0

ln C

−

)

1+ν

1 + ν

−

)

1+ν

1 + ν

Normalizing the long-run capital-utilization and labor-utilization rates to one, the utilization rates

can be derived from



Y/K



r+δ

; e





1+ν

;

where Y , C, L and K are the steady-state values of output, consumption, labor, and capital.

The Solow residual then can be decomposed into utilization-adjusted TFP and utilization correc-

tions, with

T FP

≡

1−α

= A

× u

1−α

To construct country-speciﬁc steady state values of

Y/K

, we extract a HP-ﬁlter trend from the data

series.

In the utilization adjusted series used in the main text, we set

= 0

33, and

= 1. Results

are robust to constructing country speciﬁc values of these parameters.

Our empirical results are robust to computing moving averages over a 10 year window, using time-varying

values of

constructed from labor-income data, and reasonable parameters of the aggregate capital depreciation

rate. Bergeaud, Cette, and Lecat (2016) constructed capital stock for machines and buildings separately using

the perpetual inventory method with data on investment in machines and buildings and diﬀerent depreciation

rates. Our results are robust to choosing diﬀerent depreciation parameters.

A22

G. Persistent effects of monetary shocks in the literature

Figure A2: Bernanke and Mihov (1998), Figures 3 and 4, U.S. Economy

Notes: Screenshot of Figures 3 and 4 from Bernanke and Mihov (1998). Sample: 1966:1–1996:12. Dashed lines indicate two

standard error bands.

H. Mundell-Fleming-Dornbusch model

Although the arbitrage mechanism behind the trilemma is easily grasped, in this section we investigate

the economic underpinnings of our identiﬁcation strategy more formally with a variant of the well

known Mundell-Fleming-Dornbusch model. In particular, we incorporate the extensions to the

model discussed in Blanchard (2016) and Gourinchas (2018), which embed various ﬁnancial spillover

mechanisms.

Speciﬁcally, consider a framework made of two countries: a small domestic economy and a large

foreign economy, which we can call the United States, for now. Foreign (U.S.) variables are denoted

with an asterisk. Assume prices are ﬁxed.

Given interest rates, the following equations describe the setup:

Y = A + NX ,

A = ℵ − ci − fE ,

NX = a(Y

∗

− Y ) + bE ,

∗

= A

∗

= ℵ

∗

− c

∗

E = d(i

∗

− i) + gi

∗

+ χ ,

where

a, b, c, c

∗

, d, f, g, χ ≥

0. Domestic output

is equal to the sum of domestic absorption

and

net exports

. Domestic absorption depends on an aggregate demand shifter

, and negatively on

the domestic (policy) nominal interest rate

denotes ﬁnancial spillovers through the exchange

rate (e.g., balance sheet exposure of domestic producers in a dollarized world).

f ≥

0, then a

depreciation of the exchange rate E hurts absorption.

Jiang, Krishnamurthy, and Lustig (2020) provide a micro-foundation to generate these spillovers associated

with the global ﬁnancial cycle (Rey, 2015).

A23

Net exports depends positively on U.S. output

∗

, negatively on domestic output

, and positively

on the exchange rate. U.S. output is determined in similar fashion except that the U.S. is considered a

large country, so it is treated as a closed economy. Finally, the exchange rate depends on the diﬀerence

between domestic and U.S. interest rates and on a risk-premium shock. The term

is intended to

capture risk-premium eﬀects associated with U.S. monetary policy.

In order to make the connection between the instrument as we deﬁned it earlier and this stylized

model, we now think of ∆

∗

as the instrument

j,t

≡ k

j,t

(∆

b(j,t),t

−

∆i

b(j,t),t

) described earlier. The

proposition below explores the benchmark setting of the trilemma to derive the basic intuition.

The textbook speciﬁcation with hard pegs

Under the assumption that

= 0, many interesting channels are switched oﬀ and the model

just introduced reduces to the textbook Mundell-Fleming-Dornbusch version. Consider what happens

when the U.S. changes its interest rate, ∆

∗

. Since

= 0, to maintain the peg it must be that

∆

= ∆

∗

. The one-to-one change in the home interest rate has a direct eﬀect on domestic absorption

given by −c∆i.

However, notice that changes in the U.S. rate aﬀect U.S. absorption and in turn net exports.

Piecing things together:

∆Y = ∆A + ∆NX ,

∆Y = −

1 + a

∆i −

∗

1 + a

∆i

∗

As is clear from the expression, ∆

∗

aﬀects domestic output directly (and not just through ∆

), resulting

in a violation of the exclusion restriction central to instrumental variable estimation. However, note

that this violation is easily resolved by including net exports as a control, or even just base country

output, something we do later in the estimation. Moreover, in this simple static model, all eﬀects are

contemporaneous. However, in practice the feedback loop of higher U.S. interest rates to lower net

exports to lower output will take place gradually, in large part alleviating the exclusion restriction

violation.

Financial spillovers with soft pegs

Consider now a more general setting with ﬁnancial spillovers, that is,

g >

0 and

f >

0 and a soft peg.

That is, the central bank may adjust using interest rates and allow some movement of the exchange

rate.

This will aﬀect the pass through of U.S. interest rates to domestic rates since now:

∆i =

∆ϵ +

d + g

∆i

∗

where ∆E ∈ ±∆ϵ refers to some band within which the exchange rate is allowed to ﬂuctuate.

The eﬀect on output from changes in U.S. interest rates is very similar, but with an added term:

∆Y =

1 + a

∆i −

∗

1 + a

∆i

∗

+ (b − f)∆ϵ .

Itskhoki and Mukhin (2021) argue that such risk-premia violations of UIP are smaller under exchange rate

pegs, i.e., g is smaller.

A24

Under a hard peg policy, with ∆

= 0, an increase in U.S. interest rates boosts home interest rates

but it no longer does so one-to-one, as explained earlier. Partial ﬂexibility in exchange rates under a

soft peg, with

∆

ϵ| >

0, gives some further monetary autonomy to the home economy, and reduces

the pass-through to home interest rates, all else equal. This additional ﬂexibility in exchange rates,

however, results in other ﬁnancial and trade spillovers due to dependence of domestic absorption and

net exports on the exchange rate as shown by the term (b − f)∆ϵ.

Summarizing our discussion, it is important to recognize that exogenous variation in interest rates

(induced either through the trilemma mechanism as just discussed, or through alternative channels)

has eﬀects through domestic absorption and through net exports. This secondary channel, if not

properly controlled for, generates violations of the exclusion restriction.

A25

I. Additional figures

I.1. LPIV responses for short term nominal interest and CPI

Figure A3: Baseline short term nominal interest rate own response to 100 bps shock.

(a) Full sample, 1900–2015

-1 -.5 0 .5 1 1.5

Percent

0 4 8 12

Year

(b) Post ww2 sample, 1948–2015

-1 -.5 0 .5 1 1.5

Percent

0 4 8 12

Year

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs. Full sample:

1900–2015 (World Wars excluded). Post-WW2 sample: 1948–2015. LP-IV estimates displayed as a thick line and 68% and 95%

standard error bands. Estimation is robust with Driscoll and Kraay (1998) standard errors. See text.

A26

Figure A4: Baseline consumer price index response to 100 bps shock.

(a) Full sample, 1900–2015

-15 -10 -5 0 5 10

Percent

0 4 8 12

Year

(b) Post ww2 sample, 1948–2015

-15 -10 -5 0

Percent

0 4 8 12

Year

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs. Full sample:

1900–2015 (World Wars excluded). Post-WW2 sample: 1948–2015. LP-IV estimates displayed as a thick line and 68% and 95%

standard error bands. Estimation is robust with Driscoll and Kraay (1998) standard errors. See text.

A27

I.2. LPIV responses for components in Post WW2 sample

Figure A5 plots responses of real GDP and Solow decomposition for the post-WW2 sample.

Figure A5:

Baseline response to 100 bps shock: Real GDP and Solow decomposition. Post ww2

sample, 1948–2015.

(a) Estimates using raw data

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Real GDP

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Labor

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Capital

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

TFP

(b) Estimates using Imbs correction for factor utilization

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Real GDP

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Labor

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

Capital

-8 -6 -4 -2 0 2

Percent

0 4 8 12

Year

TFP

Notes: Response to a 100 bps shock in domestic short-term interest rate instrumented with the trilemma IVs. Post-WW2 sample:

1948–2015. LP-IV estimates displayed as a thick lines and 68% and 95% standard error bands. The upper panel uses raw data on

capital stocks and total hours to construct TFP as a residual. The lower panel adjusts the raw data on capital stock and total

hours to obtain estimates of actual factor inputs by using the Imbs (1999) correction. See text.

A28

I.3. Structural break dates in TFP growth

-10 -5 0 5 10

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for AUS

-40 -20 0 20 40

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for BEL

-20 -10 0 10 20

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for CAN

-10 0 10 20 30

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for CHE

-100 -50 0 50

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for DEU

-20 -10 0 10 20

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for DNK

-30 -20 -10 0 10

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for ESP

-20 -10 0 10 20

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for FIN

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-40 -20 0 20 40

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for FRA

-10 -5 0 5

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for GBR

-20 -10 0 10 20 30

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for ITA

-80 -60 -40 -20 0 20

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for JPN

-40 -20 0 20 40 60

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for NLD

-10 -5 0 5 10 15

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for NOR

-10 -5 0 5 10 15

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for PRT

-10 -5 0 5 10 15

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for SWE

A30

-10 -5 0 5 10 15

1900 1920 1940 1960 1980 2000 2020

Year

TFP growth for USA

A31

Appendix references

Bergeaud, Antonin, Gilbert Cette, and R´emy Lecat. 2016. Productivity Trends in Advanced Countries

between 1890 and 2012. Review of Income and Wealth 62(3): 420–444.

Bernanke, Ben S., and Ilian Mihov. 1998. The Liquidity Eﬀect and Long-run Neutrality. Carnegie-

Rochester Conference Series on Public Policy 49(1): 149–194.

Bhattarai, Saroj, Gauti B. Eggertsson, and Bulat Gafarov. Forthcoming. Time Consistency and

Duration of Government Debt: A Model of Quantitative Easing. Review of Economic Studies .

Bilbiie, Florin O. 2019. Optimal Forward Guidance. American Economic Journal: Macroeconomics

11(4): 310–45.

Bilbiie, Florin O. 2020. The New Keynesian cross. Journal of Monetary Economics 114: 90–108.

Blanchard, Olivier. 2016. Currency Wars, Coordination, and Capital Controls. NBER Working Paper

22388.

Calvo, Guillermo A. 1983. Staggered Prices in a Utility-maximizing Framework. Journal of Monetary

Economics 12(3): 383–398.

Driscoll, John C., and Aart C. Kraay. 1998. Consistent covariance matrix estimation with spatially

dependent panel data. Review of Economics and Statisticsfor 80(4): 549–560.

Durbin, J. 1959. Eﬃcient Estimation of Parameters in Moving-Average Models. Biometrika 46(3/4):

306–316.

Fornaro, Luca, and Federica Romei. 2019. The Paradox of Global Thrift. American Economic Review

109(11): 3745–79.

Gourinchas, Pierre-Olivier. 2018. Monetary Policy Transmission in Emerging Markets: An Application

to Chile. In Monetary Policy and Global Spillovers: Mechanisms, Eﬀects and Policy Measures,

edited by Enrique G. Mendoza, Ernesto Past´en, and Diego Saravia, volume 25 of Central Banking,

Analysis, and Economic Policies Book Series, chapter 8, 279–324. Santiago: Central Bank of Chile.

Imbs, Jean M. 1999. Technology, growth and the business cycle. Journal of Monetary Economics

44(1): 65–80.

Itskhoki, Oleg, and Dmitry Mukhin. 2021. Mussa Puzzle Redux. Working Paper 28950, National

Bureau of Economic Research.

Jiang, Zhengyang, Arvind Krishnamurthy, and Hanno Lustig. 2020. Dollar Safety and the Global

Financial Cycle. NBER Working Paper 27682.

Kuersteiner, Guido M. 2005. Automatic Inference For Inﬁnite Order Vector Autoregressions. Econo-

metric Theory 21(1): 85–115.

Lewis, Richard, and Gregory C. Reinsel. 1985. Prediction of multivariate time series by autoregressive

model ﬁtting. Journal of Multivariate Analysis 16(3): 393–411.

Lusompa, Amaze B. 2019. Local projections, autocorrelation, and eﬃciency. UC Irvine. Unpublished.

A32

Obstfeld, Maurice, and Kenneth Rogoﬀ. 1995. Exchange rate dynamics redux. Journal of Political

Economy 103(3): 624–660.

Paul, Pascal. 2020. Historical patterns of inequality and productivity around ﬁnancial crises. Federal

Reserve Bank of San Francisco Working Paper 2017-23.

Rey, H´el`ene. 2015. Dilemma not Trilemma: The Global Financial Cycle and Monetary Policy

Independence. NBER Working Paper 21162.

Wiener, Norbert, and Pesi Masani. 1958. The prediction theory of multivariate stochastic processes,

II: The linear predictor. Acta Mathematica 99(1): 93–137.

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