Functional Approximation of Impulse Responses: Insights into the ...

Functional Approximation of Impulse Responses: Insights into the Effects of Monetary ShocksI Regis Barnichona , Christian Matthesb a

FRB San Francisco b FRB Richmond

Abstract This paper proposes a new method –Functional Approximation of Impulse Responses (FAIR)– to estimate the dynamic effects of structural shocks. FAIR approximates impulse responses with a few basis functions and then directly estimates the moving average representation of the data. FAIR can offer a number of benefits over earlier methods, including VAR and Local Projections: (i) parsimony and efficiency, (ii) ability to summarize the dynamic effects of shocks with a few key moments that can directly inform model building, (iii) ease of prior elicitation and structural identification, and (iv) flexibility in allowing for non-linear effects of shocks while preserving efficiency. We illustrate these benefits by re-visiting the effects of monetary shocks, notably their asymmetric effects. JEL classifications: C14, C32, C51, E32, E52 Keywords:

I

First draft: March 2014. We would like to thank Luca Benati, Francesco Bianchi, Christian Brownlees, Fabio Canova, Tim Cogley, Davide Debortoli, Wouter den Haan, Jordi Gali, Yuriy Gorodnichenko, Eleonora Granziera, Oscar Jorda, Thomas Lubik, Jim Nason, Kris Nimark, Mikkel Plagborg-Moller, Giorgio Primiceri, Ricardo Reis, Jean-Paul Renne, Barbara Rossi, Konstantinos Theodoridis, Mark Watson, Yanos Zylberberg and seminar participants at the Barcelona GSE Summer Forum 2014, the 2014 NBER/Chicago Fed DSGE Workshop, William and Mary college, EUI Workshop on Time-Varying Coefficient Models, Oxford, Bank of England, NYU Alumni Conference, SED Annual Meeting (Warsaw), Universitaet Bern, Econometric Society World Congress (Montreal), the Federal Reserve Board, the 2015 SciencesPo conference on Empirical Monetary Economics and the San Francisco Fed for helpful comments. The views expressed here do not necessarily reflect those of the Federal Reserve Bank of Richmond or San Francisco or of the Federal Reserve System. Any errors are our own. Preprint submitted to Journal of Monetary Economics

November 29, 2017

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1. Introduction

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The impulse response function (IRF) is an important tool used to summarize the dynamic

3

effects of shocks on macroeconomic time series. Since Sims (1980), researchers interested

4

in estimating IRFs without imposing a specific economic model have traditionally relied on

5

Vector AutoRegressions (VAR), even though the VAR per se is often of no particular interest.

6

This paper proposes a new method – of Impulse Responses (FAIR)– to estimate the

7

dynamic effects of structural shocks.

8

The FAIR methodology has two distinct features. First, and instead of assuming the exis-

9

tence of a VAR representation, FAIR works directly with the moving-average representation

10

of the data, that is FAIR directly estimates the object of interest –the impulse response

11

function–. This confers a number of advantages, notably the ability to impose prior infor-

12

mation and structural identifying restrictions in a flexible and transparent fashion, as well

13

as the possibility to allow for a large class of non-linear effects.

14

Second, FAIR approximates the impulse response functions with a few basis functions,

15

which serves as a dimension reduction tool that makes the estimation of the moving-average

16

representation feasible. While different families of basis functions are possible, Gaussian basis

17

functions are of particular interest for macro applications. Intuitively, IRFs are often found

18

(or theoretically predicted) to be monotonic or hump-shaped, and in such cases, just one

19

Gaussian basis function can already provide a parsimonious and intuitive way to summarize

20

an impulse response function with only three parameters, each capturing a separate and easily

21

interpretable feature of the IRF: (a) the magnitude of the peak effect of a shock, (b) the time

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to that peak effect, and (c) the persistence of that peak effect. This interpretability of the

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FAIR coefficients further reinforces the benefits of directly working with the moving-average

24

representation: imposing prior information and structural identifying restrictions is easy

25

and transparent, and the model can easily be generalized to introduce non-linearities. More

26

generally, while impulse response functions are large dimensional objects whose important

2

27

features can be hard to summarize and assess statistically,1 the ability to summarize an

28

impulse response function with a few key parameters amenable to statistical inference can

29

make FAIR particularly helpful to evaluate and guide the development of successful models.2

30

After establishing the good finite sample properties of FAIR with Monte-Carlo simula-

31

tions, we illustrate the benefits of FAIR by studying the effects of monetary shocks. FAIR can

32

easily incorporate the different identification schemes found in the structural VAR literature,

33

and we re-visit the effects of monetary shocks using three popular identification schemes: a

34

recursive identification scheme, (ii) a set identification scheme based on sign restrictions, and

35

(iii) a narrative identification scheme where a series of shocks has been previously identified

36

(possibly with measurement error) from narrative accounts.

37

First, we focus on linear models and use FAIR to re-assess some of the key features of

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the effects of monetary shocks that have been instrumental in the development of monetary

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models over the past 20 years. We highlight three stylized facts that emerge across the three

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different identification schemes: (i) a 100 basis point contractionary monetary shock has a

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peak effect of unemployment of about 0.3 ppt after about ten quarters, (ii) the peak response

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of inflation lags that of unemployment by two-to-four quarters, (iii) the responses of inflation

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and unemployment display similar persistence, with a “half-life” of the peak effect –the time

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for the response to return to half of its peak value– at about five quarters for both inflation

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and unemployment.

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Second, we use FAIR to explore whether monetary shocks have asymmetric effects on un-

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employment. Specifically, we study whether a contractionary monetary shock has a stronger

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effect –being akin to pulling on a string– than an expansionary shock –being akin to pushing 1

In fact, researchers traditionally resort to large panels of IRF plots to represent the responses of many variables to different shocks. This overload of information can blur the key implications for theoretical modeling and make comparison of the IRF estimates across different schemes, different model specifications and/or different sample periods difficult. 2 More generally, by providing summary statistics for the dynamic effects of a structural shock, the FAIR parameters can be seen as portable moments (Nakamura and Steinsson, 2017): they are simple yet general moments relevant to any model and they can easily be used by researchers to discipline and evaluate models.

3

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on a string–.3 While this question is central to the conduct of monetary policy, the evidence

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for asymmetric effects is relatively thin and inconclusive,4 and part of this state of affairs

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reflects the fact that estimating the asymmetric effects of a monetary shock is very hard

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within a VAR framework. In contrast, FAIR can easily accommodate such non-linearities.

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Consistent with the string metaphor, we find evidence of strong asymmetries in the effects

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of monetary shocks. Regardless of whether we identify monetary shocks from a recursive

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ordering, from sign restrictions or from a narrative approach, we find that a contractionary

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shock has a strong adverse effect on unemployment, while an expansionary shock has little

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effect on unemployment. The response of inflation is also asymmetric: inflation responds less

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strongly following a contractionary shock than following an expansionary shock. However,

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the asymmetric reaction of inflation is the opposite to that of unemployment, so that unem-

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ployment reacts less when prices react more and vice-versa. Interestingly, this is the pattern

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that one would expect if (i) nominal rigidities were behind the real effects of monetary policy,

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and (ii) downward nominal rigidities were behind the asymmetric effects of monetary shocks

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on unemployment.

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A number of methods have been proposed to estimate impulse response functions, and

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we see FAIR as a useful addition to researchers’ toolkit. While VARs have been the main

66

approach since Sims (1980), an increasing number of papers are relying on Local Projections

67

(LP, Jorda 2005) –themselves closely related to Autoregressive Distributed Lags (ADL, see

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Hendry, 1984)– to directly estimate impulse response functions. As we describe in details in

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the main text, FAIR builds on this earlier ADL literature, notably on Hansen and Sargent

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(1981), to offer a number of benefits over VAR and LP: (i) parsimony and efficiency, (ii) 3

Theoretically, there are at least two reasons why monetary policy could have asymmetric effects. First, with downward wage/price rigidity, economic activity can display asymmetric responses to monetary changes (e.g., Morgan, 1993). Second, with borrowing constraints, increasing borrowing costs may curb investment and/or consumption, but lowering borrowing costs need not stimulate investment or consumption. This can happen if borrowing-constrained agents respond more to income drops than to income gains (Auclert, 2017), a possibility supported by recent evidence from Bunn et al. (2017) and Christelis et al. (2017). 4 For instance, while Cover (1992) and Tenreyro and Thwaites (2016) find evidence of asymmetric effects, Ravn and Sola (2004) and Weise (1999) instead find nearly symmetric effects.

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ability to summarize the dynamic effects of shocks with a few key interpretable moments

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that can directly inform model building, (iii) ease of prior elicitation and structural identifi-

73

cation, and (iv) preserving efficiency when allowing for non-linear effects of shocks, notably

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asymmetric effects. Finally, our approach relates to the recent work of Plagborg-Moller

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(2017), who proposes a Bayesian method to directly estimate the structural moving-average

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representation of the data by using prior information about the shape and the smoothness

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of the impulse responses.

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Section 2 presents our functional approximation of impulse responses and discusses the

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benefits of using Gaussian basis functions, Section 3 re-visits the linear effects of monetary

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shocks with FAIR; Section 4 generalizes FAIR to non-linear models; Section 5 studies the

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asymmetric effects of monetary shocks; Section 6 concludes.

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2. Functional Approximation of Impulse Responses (FAIR)

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We first introduce FAIR in univariate setting and then generalize it to a multivariate

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setting.

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2.1. Univariate setting

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For a stationary univariate data generating process, the IRF corresponds to the coefficients of the moving-average model

yt =

H X

ψ(h)εt−h

(1)

h=0

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where εt is an i.i.d. innovation with Eεt = 0 and Eε2t = 1, and H is the number of lags,

89

which can be finite or infinite. The lag coefficients ψ(h) is the impulse response of yt at

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horizon h to innovation εt .

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Since ψ(h) is a large (possibly infinite) dimensional object, estimation of (1) can be

92

difficult. Instead, one can invoke the Wold decomposition theorem to re-write (1) as an

5

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AR(∞) A(L)yt = εt

94

H P

where A(L) = Ψ(L)−1 with Ψ(L) =

(2)

ψ(h)Lh and L the lag operator.5

h=0 95

Since estimating an AR(∞) is not possible in finite sample, in practice researchers esti-

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mate a finite order AR(p) meant to approximate the AR(∞), and the IRF is recovered by

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inverting that AR(p). In effect, the AR(p) approximation serves as a dimension reduction

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tool that makes the estimation of (1) feasible. For instance, an AR(1) approximates the

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coefficients of the MA(∞) as an exponential function of a single parameter.

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In this paper, we propose an alternative dimension-reducing tool –Functional Approxi-

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mation of Impulse Responses or FAIR–, which consists in representing the impulse response

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function as an expansion in basis functions. Instead of estimating an intermediate model

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–the AR(p)– and then recovering the impulse response, FAIR directly estimates the object of

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scientific interest –the impulse response function–. Specifically, a functional approximation

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of ψ consists in decomposing ψ into a sum of basis functions, i.e., in modeling ψ(h) as a

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basis function expansion ψ(h) =

N X

∀h > 0

an gn (h),

(3)

n=1 107

with gn : R → R the nth basis function, n = 1, .., N .6 Different families of basis functions are possible, and in this paper we use Gaussian basis

108

109

functions and posit ψ(h) =

N X

an e−(

h−bn 2 ) cn

,

∀h > 0

(4)

n=1 110

with an , bn , and cn parameters to be estimated. Since model (4) uses N Gaussian basis

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functions, we refer to this model as a FAIRG of order N .7 5

A caveat of this approach is that it will only recover the fundamental, i.e., invertible, representation of (1). We come back to this point in section 2.4. 6 The functional approximation of ψ may or may not include the contemporaneous impact coefficient, that is one may choose to use the approximation (4) for h > 0 or for h ≥ 0. In this paper, we treat ψ(0) as a free parameter for additional flexibility. 7 We show in the appendix that any mean-reverting impulse response function can be approximated by a

6

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2.2. Multivariate setting

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While the discussion has so far concentrated on a univariate process, we can easily gen-

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eralize the FAIR approach to a multivariate setting. Consider the structural vector moving-

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average model of yt , a vector of stationary variables,

yt = Ψ(L)εt ,

where Ψ(L) =

H X

Ψh L h

(5)

h=0

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where εt is the vector of i.i.d. structural innovations with Eεt = 0 and Eεt ε0t = I, and H is

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the number of lags, which can be finite or infinite. The matrix of lag coefficients Ψh contains

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the impulse responses of yt at horizon h to the structural shocks εt .

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Setting aside the issue of identification for now, the common strategy to recover (5) is

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identical to the univariate case: rewrite (5) as a VAR(∞) and then estimate its VAR(p)

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approximation. In this paper, we propose to use FAIR and directly estimate the impulse

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response functions by approximating the elements ψ(h) of Ψh as in (4).

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2.3. FAIR benefits

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We now argue that FAIRG –FAIR models with Gaussian basis functions– can be particu-

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larly attractive in macro applications for two reasons: (i) parsimony, and (ii) interpretability

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of the FAIRG parameters as summary features of the IRFs. We postpone a third important

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benefit of FAIRG –the possibility to allow for non-linearities in a flexible yet parsimonious

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fashion– to a later section dedicated to non-linearities.

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We will pay particular attention to the FAIRG1 –a FAIR with one Gaussian basis function– ψ(h) = ae−

(h−b)2 c2

, ∀h > 0,

(6)

130

which captures an IRF with only 3 parameters: a, b and c, as illustrated in the top panel of

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Figure 1. sum of Gaussian basis functions.

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Parsimony of FAIR

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A first advantage of using Gaussian basis functions is that a small number of Gaussians

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can already approximate a large class of IRFs, in fact most impulse responses encountered

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in macro applications. Intuitively, IRFs are often found (or theoretically predicted) to be

136

monotonic or hump-shaped (e.g., Christiano, Eichenbaum, and Evans, 1999).8 In such cases,

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one or two Gaussian basis functions can already provide a very good approximate description

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of the impulse response. With the one-Gaussian and the two-Gaussian model summarizing

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an IRF with only 3 and 6 parameters respectively, Gaussian basis functions offer an “effi-

140

cient” dimension reduction tool; reducing the number of parameters substantially while still

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allowing for a large class of impulse responses.

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To illustrate this point, Figure 2 plots the IRFs of unemployment, inflation and the fed

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funds rate to a shock to the fed funds rate estimated from a standard VAR specification with

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a recursive ordering, along with the IRFs approximated with Gaussian basis functions.9 For

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unemployment and the fed funds rate we use a FAIRG with only one Gaussian basis function,

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a FAIRG1 . We can see that this simple model already does a good job at capturing the

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responses of unemployment and the fed funds rate implied by the VAR, while reducing the

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dimension of each IRF in Figure 2 from 25 to only 3 parameters. To capture the oscillating

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pattern of inflation following a monetary shock, two Gaussian basis functions are necessary,

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and we can see in Figure 2 that the IRF approximated with two Gaussian basis functions –a

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FAIRG2 – does a good job there as well. With a small number of Gaussian basis functions, the FAIRG model is parsimonious,10

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8

In New-Keynesian models, the IRFs are generally monotonic or hump-shaped (see e.g., Walsh, 2010). We describe the exact VAR specification in section 5. In Figure 2, the parameters of the Gaussian basis functions were set to minimize the discrepancy (sum of squared residuals) with the VAR-based impulse responses. Importantly, this is not how we estimate FAIR models. 10 For instance, for a trivariate model with three structural shocks, a FAIRG1 would need 27 parameters (9 impulse responses times 3 parameters per impulse response, ignoring intercepts) to capture the whole H set of impulse responses {Ψh }h=1 . In the example of Figure 2 where the impulse responses of inflation are approximated by a richer FAIRG2 , there would be 36 parameters (32 ∗ 2 + 6 ∗ 3 = 36). In comparison, a quarterly VAR with 3 variables and 4 lags has 4 ∗ 32 = 36 free parameters, and a monthly VAR with 12 lags has 12 ∗ 32 + 6 = 108 free parameters. 9

8

153

and this will allow us to directly estimate a vector moving-average representation of the data.

154

Finally, we note that, unlike VARs, the number of FAIR parameters to estimate need

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not increase with the frequency of the data, and a FAIRG1 summarizes an IRF with only 3

156

parameters regardless of whether H = 25 or H = 60. Stated differently, while a monthly VAR

157

would typically require much more lags than the corresponding quarterly VAR to capture

158

the dynamic correlations present in the data, the number of FAIR parameters would remain

159

the same with monthly or quarterly data, as long as the shape of the impulse response is

160

not altered by time aggregation.

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Interpretability and portability of FAIR coefficients

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A second advantage of using Gaussian basis functions is that the estimated coefficients

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can have a direct economic interpretation in terms of features of the IRFs. This stands in

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contrast to VARs where the IRFs are non-linear transformations of the VAR coefficients.

165

Coefficient interpretability can make FAIR priors easier to elicit and make FAIR results

166

particularly helpful to evaluate and guide the development of successful models.

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The ease of interpretation is most salient in a FAIRG1 model (6) where the a, b and c

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coefficients have a direct economic interpretation, and in fact capture three separate char-

169

acteristics of a hump-shaped impulse response: the peak effect of a shock, the time to peak

170

effect, and the persistence of that peak effect. To see that graphically, the top panel of Figure

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1 shows a hump-shape impulse response parametrized with a one Gaussian basis function:

172

parameter a is the height of the impulse-response, which corresponds to the maximum effect

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of a unit shock, parameter b is the timing of this maximum effect, and parameter c captures

175

the persistence of the effect of the shock, as the amount of time τ required for the effect of √ a shock to be 50% of its maximum value is given by τ = c ln 2. These a-b-c coefficients

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are generally considered the most relevant characteristics of an impulse response function

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and are generally the most discussed in the literature. Take the example of the literature on

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the effects of monetary shocks. In two recent influential papers –Coibion (2012) and Ramey

179

(2016)–, the authors compare the effects of monetary shocks across different specifications

174

9

180

by comparing the peak response of the variable of interest. In the FAIRG1 model, this is

181

simply the a parameter. In Christiano et al. (2005, p8), the authors visually inspect the

182

impulse response functions and summarize the dynamics effects of a monetary shock with

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two stylized facts: (i) the response of real variables peaks after six quarters and return to

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pre-shock levels after about three years, and (ii) the response of inflation peaks after two

185

years. These two statements refer precisely to the b and c coefficients of a FAIRG1 model.

186

However, unlike earlier approaches, FAIRG1 would directly estimate these summary statistics

187

and provide confidence intervals around these statements.

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With more than one basis function, the ease of interpretation of the estimated parameters

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is no longer guaranteed. However, in some cases a FAIRG model with multiple Gaussian-basis

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functions can retain its interpretability. For instance, consider an oscillating pattern as in

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the bottom panel of Figure 1, which is typical of the response of inflation to a contractionary

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monetary shock. In that case, the FAIRG parameters can retain their interpretability if

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the first Gaussian basis function captures the first-round effect of the shock –a positive

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hump-shaped response of inflation usually refereed to as the price puzzle (Christiano et

195

al. 1999)–, while the second Gaussian function captures the larger second-round effect –a

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negative hump-shaped effect–. Going back to our VAR example from Figure 2, the two

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Gaussian basis functions used to match the impulse response of inflation show little overlap.

198

In particular, one can summarize the 2nd -round effect of the shock –the main object of

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scientific interest in the case of the inflation response to monetary shocks– with the a-b-c

200

parameters of the second basis function: in that case, the a2 , b2 and c2 capture the peak

201

effect, the time to peak effect, and the persistence of the 2nd -round effect of the shock.11 11

Formally, one can write a “no-overlap” condition as Z ∞ Z g1 (h)dh ≤  ' 0 with h2 such that h2

∞

g2 (h)dh = α ' 1

(7)

h2

with gn denoting the nth basis function. That condition is a restriction on the bn and cn coefficients that ensures that the two basis functions g1 and g2 have limited overlap. Intuitively, for say  = .05 and α = .9, the condition states that the first Gaussian function (g1 ) –the first-round effect of the shock– has a negligible (5 percent or less) overlap with 90 percent of the second Gaussian function (g2 ) –the second-round effect of the

10

202

Going beyond interpretation and ease of prior elicitation, the ability to capture the key

203

properties of the dynamic effects of a shock makes FAIRG particularly helpful to evaluate

204

and guide the development of successful models. IRFs are large dimensional objects whose

205

important features can be hard to summarize and assess statistically. Moreover, when mul-

206

tiple identification strategies, different specifications and different sample periods are consid-

207

ered, comparing the features of the different sets of impulse responses becomes quickly very

208

difficult.12 In contrast, FAIRG , and most notably FAIRG1 , considerably reduces the dimen-

209

sionality of the IRFs by capturing only the first-order properties of the impulse responses.

210

By providing simple and easily interpretable summary statistics amenable to statistical in-

211

ference, FAIR estimates can facilitate the emergence of a set of robust findings and help the

212

development of successful models. More generally, when summarizing the key characteristics

213

of the dynamic effects of a structural shock, the a-b-c parameters can be seen as portable

214

identified moments (as advocated by Nakamura and Steinsson, 2017) that can easily be used

215

by researchers to discipline and evaluate models: They are portable in the sense that they

216

are simple yet general moments relevant to any model, and they are identified when the

217

FAIR model is identified with structural identifying assumptions (which, as we will see, are

218

easy to impose in FAIR).

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2.4. FAIR estimation

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FAIR models can be estimated using maximum likelihood or Bayesian methods. While

221

the computational cost is not as trivial as OLS in the case of VARs, the estimation is simple shock–. One can either impose the “no-overlap” condition in the estimation stage, or leave the parameters free but only interpret the coefficients if the “no-overlap” condition holds. For researchers mostly interested in the peak effects of a shock and thus only interested in retaining interpretability of the a coefficients, a weaker condition is that only one basis function contributes to the peak value of the impulse response. In the case of the FAIRG2 response of inflation, that condition would be g1 (b2 ) ' 0: the value of the first basis function is close to zero at the peak of the second basis function. 12 In fact, researchers traditionally resort to large panels of IRF plots to represent the responses of many variables to different shocks. This overload of information can blur the key implications for theoretical modeling and make comparison of IRF estimates across different schemes, different model specifications and/or different sample periods difficult. To give an example from our own work, see for instance AmirAhmadi et al. (2016).

11

222

and relatively easy thanks to modern computational capabilities.

223

The problem boils down to the estimation of a truncated moving-average model, and

224

we recursively construct the likelihood by using the prediction error decomposition and

225

assuming that the structural innovations {εt } are Gaussian with mean zero and variance

226

one (see the appendix for more details). To initialize the recursion, we set the first H

227

innovations {εj }0j=−H to zero.13 Moreover, since the structural innovations {εt } are not fully

228

identified by the first two moments of the data, it is also at this stage that we impose the

229

identifying restriction that we describe in the next section.

230

A potentially important problem when estimating moving-average models is the issue of

231

under-identification: In linear moving average models, different representations (i.e., different

232

sets of coefficients and innovation variances) can exhibit the same first two moments, so that

233

with Gaussian-distributed innovations, the likelihood can display multiple peaks, and the

234

moving average model is inherently underidentified (e.g., Lippi and Reichlin, 1994). By

235

constructing the likelihood recursively using past observations, our algorithm will effectively

236

estimate the fundamental moving-average representation of the data.14 In principle, one

237

could remedy this limitation and estimate non-invertible moving-average representations by

238

using the procedure recently proposed by Plagborg-Moller (2017),15 provided that one has

239

enough prior information to favor one moving-average representation over the others.

240

To choose N , the order of the FAIR model, the researcher can either choose to restrict

241

himself to a class of functions if prior knowledge on the shape of the impulse response

242

is available (for instance, using only one Gaussian basis function) or use likelihood ratio

243

tests/posterior odds ratios (assigning equal probability to any two models) to compare models 13

Alternatively, we could use the first H values of the shocks recovered from a structural VAR. If our estimation algorithm chose parameter estimates that implied non-invertibility, our {εt } estimates would ultimately grow very large, and this situation would lead to very low likelihoods, since we assume that the {εt } are standard normal. As a result, our estimation procedure will effectively estimate the invertible representation. 15 By using the Kalman filter with priors on the H initial values of the shocks {ε−H ...ε0 }, Plagborg-Moller’s procedure can handle the estimation of both invertible and non-invertible representations and thus does not restrict the researcher to the invertible moving-average representation. However, unlike our proposed approach, that procedure would be difficult to implement in non-linear models. 14

12

244

with increasing number of basis functions.16 Finally, note that FAIR models can only be

245

estimated for stationary series (so that the moving-average can be truncated). If the data

246

are non-stationary, we can (i) allow for a deterministic trend in equation (5) and/or (ii)

247

difference the data, and then proceed exactly as described above.17

248

2.5. Imposing structural identifying assumptions

249

As with structural VARs, in order to give a structural interpretation to some of the

250

shocks, researchers must use identifying assumptions. We will see that FAIR can easily

251

incorporate the main identification schemes found in the structural VAR literature. In

252

fact, since FAIR models work directly with the structural moving-average representation,

253

imposing prior information on the IRFs is arguably easier and more transparent in FAIR

254

than in VARs, where imposing certain prior information can be challenging. In particular,

255

because the impulse-responses are non-linear transformations of the VAR coefficients, some

256

prior information on the IRFs may be hard to impose in a VAR framework. Moreover, some

257

identification schemes may impose unacknowledged and/or unintended prior information on

258

the impulse responses.18

259

Given our later focus on the effects of monetary policy, we will illustrate how FAIR can

260

easily incorporate identifying restrictions using three popular approaches in the monetary

261

literature: a recursive identification scheme, (ii) a narrative identification scheme where 16

This second approach can be seen as analogous to the choice of the parameter lag in VAR models. The usual approach is to use information criteria such as AIC and BIC, the latter being similar to using posterior odds ratio. 17 Importantly, the presence of co-integration does not imply that a FAIR model in first-difference is misspecified. The reason is that a FAIR model directly works with the moving-average representation and does not require inversion of the moving-average, unlike VAR models. After estimation, one can even test H P h P for co-integration by testing whether the matrix sum of moving-average coefficients ( Ψl ) is of reduced h=1 l=0

rank (Engle and Yoo, 1987). 18 See for instance, Baumeister and Hamilton (2015) in the case of sign restrictions. A simple yet illuminating example can be found in Plagborg-Moller (2017) in the context of sign restrictions for an AR(1) model. Imposing that the contemporaneous response to a shock is positive mechanically imposes that all the IRF coefficients are positive. While a richer AR model is more flexible, the mapping from parameters to IRF becomes more complicated, and this raises the possibility of inadvertently imposing unintended restrictions on the IRFs.

13

262

a series of shocks has been previously identified (possibly with measurement error) from

263

narrative accounts, and (iii) a set identification scheme based on sign restrictions.19 We also

264

mention how FAIR could open the door for more general identification schemes based on

265

shape-restrictions. Short-run restrictions and recursive ordering: Short-run restrictions in a fully

266

L(L−1) 2

restrictions on the contemporaneous matrix Ψ0

267

identified model consists in imposing

268

(of dimension L × L), and a common approach is to impose that Ψ0 is lower triangular, so

269

that the different shocks are identified from a timing restriction. To identify only one shock,

270

the weaker restriction is that only a subset of the variables (ordered first in the vector yt )

271

do not react on impact to that shock, which is ensured by setting the upper-right block of

272

Ψ0 to zero. Otherwise, Ψ0 is left unrestricted apart from invertibility.

273

Narrative identification: In a narrative identification scheme, a series of shocks has

274

been previously identified from narrative accounts. For that case, we can proceed as with

275

the recursive identification, because the use of narratively identified shocks can be cast as

276

a partial identification scheme. We order the narratively identified shocks series first in

277

yt , and we assume that Ψ0 has its first row filled with 0 except for the diagonal coefficient,

278

which implies that the narratively identified shock does not react contemporaneously to other

279

shocks. In other words, we are assuming that the narrative shocks are contemporaneously

280

correlated with the true monetary shocks and uncorrelated with other structural shocks.20

281

Sign restrictions: Set identification through sign restrictions consists in imposing sign-

282

restrictions on the sign of the Ψh matrices, i.e., on the impulse response coefficients at

283

different horizons. Again, because a FAIR model works directly with the moving average

284

representation and the Ψh matrices, imposing sign-restrictions in FAIR is straightforward. 19

Other identification schemes, for instance long-run restrictions, are also straightforward to impose. Note that this assumption is weaker than the common assumption (e.g., Romer and Romer, 2004) that the narratively identified shocks are perfectly correlated with the true monetary shocks. Our procedure allows the narrative shocks to contain measurement error, as long as the measurement error is independent of structural shocks. In fact, this approach amounts to an external instrumental approach, in the sense of Stock (2008). 20

14

285

One can impose sign-restrictions on only the impact coefficients (captured by Ψ0 , which could

286

be left as a free parameter in this case) and/or sign restrictions on the impulse response.

287

This is straightforward in a FAIRG1 model where the sign restriction applies for the entire

288

horizon of the impulse response. With oscillating pattern and a higher-order FAIRG model,

289

we can impose sign restrictions over a specific horizon by using priors on the location and

290

the sign of the loading of the basis functions.21

291

Identification restrictions through priors: When the FAIR parameters can be in-

292

terpreted as “features” of the impulse responses, one can go beyond sign-restrictions and

293

envision set identification through shape restrictions. Using the insights from Baumeister

294

and Hamilton (2015), one could implement shape restrictions through informative priors on

295

the coefficients of Ψ0 and on the a-b-c coefficients. For instance, one could posit priors on

296

the location of the peak effect or posit priors on the persistence of the effect of the shock,

297

among other possibilities.22

298

2.6. Assessing FAIR performances from Monte-Carlo simulations

299

As described in details in the appendix, we conducted a number of Monte-Carlo simu-

300

lations to study the performance of FAIR in finite sample. For a trivariate system with a

301

linear data generating process, we found that a well-specified FAIRG1 model can generate

302

substantially more accurate impulse response estimates (in a mean-squared error sense) than

303

a VAR model, even when the VAR includes many lags. Importantly, these results were ob-

304

tained without the use of any informative prior.23 Moreover, in a second set of simulations

305

where the data generating process is a VAR, we found that a mis-specified FAIR performs 21 To implement parameter restrictions on Ψ0 and/or {an , bn , cn }, we assign a minus infinity value to the log-likelihood whenever the restrictions are not met. 22 See Plagborg-Moller (2017) for a related idea. 23 Intuitively, when estimating IRFs from a finite-order VAR, researchers faces a difficult bias-variance trade-off between the need to estimate a high-enough order VAR and the need to keep the number of free parameters small. In situations where the order of the VAR must be large to guarantee a good approximation of the DGP (as in our Monte Carlo simulation), a FAIRG model can provide a useful alternative to estimate IRFs.

15

306

just as well (even slightly better) than a well-specified VAR model.24 That being said, our

307

goal is not to claim that FAIR models are superior to VARs. Instead, the simulations are

308

meant to convey that FAIR models can provide a useful alternative to VARs.

309

2.7. Relation to alternative IRF estimators

310

While VARs have been the main approach to estimate IRFs since Sims (1980), an increas-

311

ing number of papers are now relying on Local Projections (LP, Jorda 2005) –themselves

312

closely related to Autoregressive Distributed Lags (ADL))– to directly estimate impulse

313

response functions.

314

FAIR aims to straddle between the parametric parsimony of VARs and the flexibility of

315

LP. Indeed, while LP (or ADL in its naive form) is model-free –not imposing any underlying

316

dynamic system–, this can come at an efficiency cost (Ramey, 2012), which can make infer-

317

ence difficult. In contrast, by positing that the response function can be approximated by

318

one (or a few) Gaussian functions, FAIR imposes strong dynamic restrictions between the

319

parameters of the impulse response function, which can improve efficiency. Moreover, FAIR

320

alleviates another source of inefficiency in ADL/LP, namely the presence of serial correlation

321

in the ADL/LP regression residuals. By jointly modeling the behavior of key macroeconomic

322

variables (in the spirit of a VAR), FAIR is effectively modeling the serial correlation present

323

in ADL/LP residuals, which can further improve efficiency.25

324

Our functional approximation of the IRFs has intellectual precedents in the ADL lit-

325

erature, notably Almon (1965) or Jorgenson (1966) approximation of the distributed lag

326

function with polynomial or rational functions. In fact, Hansen and Sargent (1981) and

327

Ito and Quah (1989) estimate structural moving average models using a Jorgenson rational

328

function approximation of the IRFs. Unlike these earlier functional approximations, the 24

Intuitively, because VAR-based IRFs are linear combinations of damped sine-cosine functions, the VARbased IRFs can display counter-factual oscillations. With its tighter parametrization, a FAIRG with only a few basis functions avoids this problem. 25 Another advantage of FAIR over ADL/LP is that it can be used for model selection and model evaluation through marginal data density comparisons.

16

329

FAIRG approximation –notably FAIRG1 – aims at summarizing the distributed lag function

330

with a parsimonious and interpretable representation. As we saw, the interpretability is im-

331

portant for the portability of the FAIR estimates (and their usefulness for model building)

332

as well as for prior elicitation. That being said, the Gaussian basis approximation is not the

333

only possibility for a FAIR model, and alternative functional approximations –say an Almon

334

polynomial or a Jorgenson rational function– could be preferred in different situations or

335

could be used as robustness checks.

336

Another benefit of FAIR over VAR and LP/ADL is the ease of prior elicitation and struc-

337

tural identification. While we saw that identification can be thorny in VARs, the problem

338

is even more salient for LP/ADL, which typically require a series of previously identified

339

shocks (or instruments). In contrast, FAIR can easily accommodate all the identifying

340

schemes found in the structural VAR literature (e.g., sign restrictions).

341

Finally, our direct estimation of vector moving-average models is related to the recent

342

work of Plagborg-Moller (PM, 2017), who proposes a Bayesian method to directly estimate

343

the structural moving-average representation of the data. Different from our functional

344

approximation approach, PM reduces estimation variance by using prior information about

345

the shape and the smoothness of the impulse responses.

346

3. A FAIR summary of the linear effects of monetary shocks

347

In this section, we show how the FAIR estimates can summarize the dynamics effects of

348

monetary shocks and provide key identified moments to discipline models. We consider a

349

model of the US economy in the spirit of Primiceri (2005), where yt includes the unemploy-

350

ment rate, the PCE inflation rate and the federal funds rate.26 26

When constructing the likelihood, we truncate the moving-average at H = 45, chosen to be large enough such that the coefficients of the lag matrix Ψh are close enough to zero for h > H. We leave the non-zero coefficients of the contemporaneous impact matrix Ψ0 as free parameters. As priors, we use very loose Normal priors on the a-b-c coefficients that are centered on the values obtained by matching the impulse responses obtained from the VAR and with standard-deviations σa = 10 ppt, σb = 10 quarters and σc = 20 quarters with the constraint c > 0 (in “half-life” units, this σc corresponds to a half-life of about 4 years, a very persistent IRF). To illustrate that these are very loose priors, in the appendix we show some corresponding

17

351

We use one Gaussian basis function to parametrize the impulse responses of unemploy-

352

ment and the fed funds rate. For the response of inflation, we use two Gaussian functions.

353

We do so for two reasons: first, to allow for the possibility of a price puzzle in which inflation

354

displays an oscillating pattern, and second to showcase how FAIR can capture oscillating

355

impulse responses while preserving the interpretability of the coefficients.

356

To put our results in the context of the literature (and also showcase how FAIR can

357

easily accommodate popular identifying schemes), we identify monetary shocks using three

358

different schemes: (i) a timing restriction, (ii) a narrative approach, and (iii) sign restrictions.

359

To ensure the same sample period across identification schemes, the data cover 1969Q1 to

360

2007Q4.27

361

Specifically, we first assume that monetary policy affects macro variables with a one

362

period lag (e.g., Christiano et al., 1999), so that the matrix Ψ0 has its last column filled with

363

0 except for the diagonal coefficient. Second, we use the narrative approach of Romer and

364

Romer (2004), extended until 2007 by Tenreyro and Thwaites (2016). For this identification

365

scheme, we expand our trivariate model by having 4 variables included in the following

366

order: the Romer and Romer shocks, unemployment, inflation and the fed funds rate, and

367

we posit that the contemporaneous matrix Ψ0 has its first row filled with 0 except for

368

the diagonal coefficient, which implies that the narratively identified shock does not react

369

contemporaneously to other shocks. This specification allows us to treat the Romer and

370

Romer shocks as an instrument for the true monetary shocks. Third, we evaluate the presence

371

of asymmetry using monetary shocks identified through sign restrictions (see e.g., Uhlig,

372

2005). We posit that positive monetary shocks are the only shocks that (i) raise the fed

373

funds rate and (ii) lower inflation roughly two years after the shock. Specifically, with a

374

two Gaussian basis function specification for the response of inflation, we impose that the

375

loading on the second basis function is negative (aπ,2 < 0), while the first basis function prior IRFs. 27 We exclude the latest recession where the fed funds rate was constrained at zero and no longer captured variations in the stance of monetary policy.

18

377

(meant to capture a possible price puzzle) can load positively or negatively but is restricted √ to peak within a year (bπ,1 ≤ 4) with a “half-life” of at most a year (cπ,1 ln 2 ≤ 4). In

378

words, our identification restriction is that the price puzzle cannot last for too long, so that

379

the response of inflation must be negative after roughly two years. In the appendix, we plot

380

the prior IRF of inflation implied by these priors.

381

3.1. Results

376

382

We first display our results in the usual way, and Figure 3 plots the impulse response

383

functions of unemployment, inflation and the fed funds rate to a monetary shock that raises

384

the fed funds rate by 100 basis points at its peak. Each column corresponds to a different

385

identification scheme. Next, Figure 4 presents the same results but through the lens of the

386

FAIR parameter estimates: the figure shows the a-b-c summary of the dynamic effects of a

387

monetary shock on unemployment and inflation.28

388

A few noteworthy results stand out across the three identification schemes.

389

First, once we take estimation uncertainty into account, the quantitative effects of a

390

monetary shock are consistent across identification schemes. While earlier research reported

391

that narrative-identified monetary shocks have larger effects on unemployment and inflation

392

than recursively-identified monetary shocks (Coibion 2012, Nakamura and Steinsson 2017),29

393

once we hold the methodology constant and control for the peak response of the fed funds

394

rate to the shock (so that the monetary impetus is similar across identification schemes), the

395

peak effect of a monetary shock on unemployment is similar across identification schemes: as

396

nar shown in Figure 4 (top-left panel), we find that arec = .31[.20,.41] and asgn = u = .24[.19,.30] , au u

397

.29[.21,.38] where the main entry denotes the median value and the subscript entry denotes 28

For the impulse response of inflation, the a-b-c parameters of the second basis functions retain a useful interpretation, because the “no-overlap” condition (footnote 11) is satisfied by more than 99% of MCMC draws (taking with α = .9 and  = .05). This can be seen graphically in the two median basis functions plotted in Figure 3. 29 Coibion (2012) highlighted that part of this difference was due to different empirical methods. By being able to accommodate different identification schemes using the same methodology, FAIR not only allows for a cleaner comparison across results but also allows us to include in the picture the effects of sign-identified shocks.

19

398

the 90 percent credible interval. Since the intervals for the au estimates display substantial

399

overlap, we conclude that the au estimates are consistent across methods. Looking at the

400

peak response of inflation (aπ , top-right panel of Figure 4) gives similar conclusions: while

401

the response of inflation appears larger following narrative shocks, the aπ coefficients are not

402

inconsistent across schemes once we take estimation uncertainty into account.

404

Second, the dynamic behavior of unemployment and inflation is also similar across iden√ tification schemes: whether we consider the time to peak effect b or its “half-life” c ln 2,

405

the 90 % confidence bands show considerable overlap across identification schemes (middle

406

and bottom row, Figure 4). This similarity is most striking for the persistence of the IRFs:

407

for both inflation and unemployment, the IRF returns to half of its peak value in about

408

five quarters. For b the time to peak value, the response of unemployment peaks after nine-

409

= 9.1[7.0,10.8] quarters, = 11.5[9.8,13.4] and bsgn = 9.6[8.1,12.0] , bnar to-eleven quarters with brec u u u

410

while the response of inflation peaks two-to-four quarters later with ∆π,u brec = 3.8[1.0,7.1] ,

411

∆π,u bnar = 1.8[−.1,3.0] and ∆π,u bsgn = 3.1[0.2,5.5] where ∆π,u b = bπ − bu . Given the importance

412

of the difference between bu and bπ for the source of monetary non-neutrality (e.g., Mankiw

413

and Reis, 2002), we can properly test that bu < bπ , i.e., that the difference between the two

414

peak times is statistically significant. To do so, Figure 5 plots the joint posterior distribution

415

of bu (x-axis) and bπ (y-axis). The dashed red line denotes identical peak times, so that the

416

figure can be seen as a test of no difference in peak times: a posterior density lying above

417

or below the red line indicates statistical evidence for different peak times. Across the three

418

identification schemes, more than 96, 88 and 92 percent of the posterior probability lies

419

above the dashed-red line, indicating that the peak of the unemployment response occurs

420

significantly before that of inflation.

403

421

Overall, we conclude from this exercise that (i) a 100 basis point contractionary monetary

422

shock has a peak effect of unemployment of about 0.3 ppt after about 10 quarters, (ii)

423

the peak response of inflation lags that of unemployment by two-to-four quarters, (iii) the

424

responses of inflation and unemployment display similar persistence, with the “half-life” of

20

425

the peak response at about 5 quarters for both inflation and unemployment.

426

4. Non-linearities with FAIR: assessing the asymmetric effects of shocks

427

Since FAIR models work directly with the moving-average representation of the data,

428

FAIR can easily allow for non-linear effects of shocks. Moreover, the parsimonious nature of

429

FAIR makes it a good starting point to explore the presence of non-linearities while preserving

430

degrees of freedom. Different non-linear effects of shocks are possible, and in this section, we

431

will focus on extending FAIR to estimate possible asymmetric effects of shocks, whereby a

432

positive shock can trigger a different impulse response than a negative shock.30 The question

433

of asymmetry is particularly interesting for two reasons. First, the possible asymmetric effect

434

of monetary policy –the pushing on a string dictum– is an important, yet under-studied,

435

question with substantial policy implications. Second, the question of asymmetry is very

436

hard to address within a VAR framework, but it can be addressed relatively easily with a

437

FAIR model.

438

4.1. Introducing asymmetry

439

With asymmetric effects of shocks, the matrix of impulse responses Ψh depends on the

440

− sign of the structural shocks, i.e., we let Ψh take two possible values: Ψ+ h or Ψh , so that a

441

model with asymmetric effects of shocks would be H X   + (ε 1 ) yt = Ψh (εt−h 1εt−h >0 ) + Ψ− t−h ε au ). The evidence for asymmetry in the response of

521

− inflation is also good, although slightly less strong: the posterior probability that a+ π < aπ

522

is 0.93, 0.87 and >0.99 for the three identification schemes.

523

Moreover, Figure 9 shows that the asymmetry in inflation is the mirror image of the

524

asymmetry in unemployment: most of the posterior mass is above the 45 degree line for

525

the peak response of unemployment, but most of the posterior mass is below the 45 degree

526

line for the peak response of inflation. In other words, unemployment reacts less when

527

prices react more, and vice-versa. Interestingly, this is the pattern that one would expect

528

if (i) nominal rigidities were behind the real effects of monetary policy, and (ii) downward

529

nominal rigidities were behind the asymmetric effects of monetary shocks on unemployment.

530

In terms of magnitude, for the recursive scheme, a 100 basis point contractionary mon-

531

etary shock increases unemployment by arec,+ = .22[.16,.28] ppt whereas the corresponding u

532

expansionary monetary shock has little effect on unemployment (arec,− = .04[.0, 08] ppt). In u

533

comparison, for the narrative and sign-based identification schemes, contractionary shocks

534

increase unemployment by anar,+ = .35[.22,.43] or asgn,+ = .36[.21,.47] ppt at the peak, while u u

535

expansionary shocks have weaker (and only marginally significant) peak effects on unem-

536

ployment with anar,− = .09[.0, .20] and asgn,− = .14[.004, .33] . Thus, although the recursively u u

25

537

identified shocks have the smallest effects overall as in the linear model, the results across

538

identification schemes are consistent once we take estimation uncertainty into account.

539

Turning to the peak effect of inflation, the direction of the asymmetry is consistent

540

across identification schemes, but the magnitudes differ somewhat. In particular, the sign-

541

based identification points to a starker asymmetry in inflation that the other identification

542

schemes. Notably, the response of inflation to a contractionary shock is estimated to be

543

much more muted with sign restrictions (asgn,+ = −.03[−.04,−.02] ) than with the other two π

544

schemes (anar,+ = −.15[−.34,.0] and arec,+ = −.08[−.14,.0] ). π π

545

6. Conclusion

546

547

This paper proposes a new method to estimate the dynamic effects of structural shocks by using a functional approximation of the impulse response functions.

548

FAIR offers a number of benefits over other methods, including VAR and Local Projec-

549

tions: (i) parsimony and efficiency, (ii) ability to summarize the dynamic effects of shocks

550

with a few key moments that can directly inform model building, (iii) ease of prior elicitation

551

and structural identification, and (iv) flexibility in allowing for non-linearities while preserv-

552

ing efficiency. We illustrate these benefits by re-visiting the effects of monetary shocks,

553

notably their asymmetric effects.

554

Although this paper studies the effects of monetary shocks, Functional Approximation

555

of Impulse Responses may be useful in many other contexts, notably when the sample size

556

is small and/or the data are particularly noisy. FAIR could also be used to explore the

557

non-linear effects of other important shocks; notably where the existence of non-linearities

558

remains an important and resolved question, such as fiscal policy shocks (e.g., Auerbach and

559

Gorodnichenko, 2013) or credit supply shocks (Gilchrist and Zakrajsek, 2012).

26

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[3] Auclert, A, ”Monetary Policy and the Redistribution Channel,” Manuscript, May 2017

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February 1999

30

FAIR, 1 Gaussian ψ(h) = ae−(

h−b 2 c

)

b a 2

a

√ c ln 2

0

h

FAIR, 2 Gaussians ψ(h) = a1 e

 2 h−b − c 1 1

 2 h−b − c 2

+ a2 e

0

2

h

Figure 1: Functional Approximation of Impulse Responses (FAIR) with one Gaussian basis function (top panel) or two Gaussian basis functions (bottom panel).

31

Unemployment

0.1

0

5

10

15

20

Inflation

0.1

25 GB1 GB2

0 −0.1

Interest rate

5

10

15

20

1

25 VAR FAIR

0.5 0

5

10

15

20

25

Quarters

Figure 2: Approximating IRFs with Gaussian basis functions: IRFs of the unemployment rate (in ppt), annualized PCE inflation (in ppt) and the federal funds rate (in ppt) to 100 basis points monetary shock, as estimated from a VAR or approximated using one Gaussian basis function (top and bottom panels) or two Gaussian basis functions (middle panel). The two basis functions in the middle panel (dashed-green and dashed-red lines) are appropriately weighted so that their sum gives the functional approximation of the impulse response function (thick blue line). Estimation using data covering 1959-2007.

32

Inflation

Unemployment

Recursive

Narrative

Sign

0.4

0.4

0.4

0.2

0.2

0.2

0

0

10

20

0

0

10

20

0

0.4

0.4

0.4

0

0

0

−0.4

−0.4

−0.4

GB1

0

10

20

0

10

20

0

10 Quarters

20

GB

2

Interest rate

0

10

20

0

10

20

1

1

1

0.5

0.5

0.5

0

0

10 Quarters

20

0

0

10 Quarters

20

0

Figure 3: FAIR Impulse Response Functions: IRFs of the unemployment rate (in ppt), inflation (in ppt) and the federal funds rate (in ppt) to a 100bp monetary shock identified from a recursive ordering (left column), a narrative approach (middle column), and sign-restrictions (right column). Shaded bands denote the 5th and 95th posterior percentiles. Sample 1969-2007.

33

Unemployment

Inflation

0.4 −0.4 0.2 −0.8 0

Peak time (qtrs) (b)

Peak effect (ppt) (a)

0

Rec.

Narr.

Sign

20

20

15

15

10

10

Half−life (qtrs) √ (c ln2)

5

Rec.

Narr.

5

Sign

8

8

6

6

4

4

2

Rec.

Narr.

2

Sign

Rec.

Narr.

Sign

Rec.

Narr.

Sign

Rec.

Narr.

Sign

Figure 4: A FAIR summary of the effects of monetary shocks: 90th posterior range of the a-b-c parameters for the IRFs of unemployment and inflation to a 100 bp monetary shocks identified from a recursive ordering (“Rec.”), a narrative approach (“Narr.”), and sign-restrictions (“Sign”). The red square marks the median value. Sample 1969-2007. Recursive

Narrative

Sign

15

15

15

b

b

bπ

20

π

20

π

20

10

5

10

5

10

15 bu

20

5

10

5

10

15 bu

20

5

5

10

15

20

bu

Figure 5: FAIR estimates of the time to peak effect: 90th joint posterior density of bu (x-axis) and bπ (y-axis), the time to peak effects of respectively unemployment and inflation following a monetary shock identified from a recursive ordering, a narrative approach, and sign-restrictions. The 45◦ dashed-red line denotes identical times to peak response. Sample 1969-2007.

34

Unemployment

Price level

Interest rate

Contractionary shock

1.5 0

0.2

1

0.1

−0.5

0.5

0 0

10

20

−1

0

(−) Unemployment

10

20

0

(−) Price level

10

20

(−) Interest rate

0.3 Expansionary shock

0

1.5 0

0.2

1

0.1

−0.5

0.5

0 0

10

20

−1

0

10

20

0

0

10

20

Figure 6: Asymmetric IRFs, recursive identification: FAIR estimates of the IRFs of unemployment (in ppt), the (log) price level (in percent) and the federal funds rate (in ppt) to a 100bp monetary shock identified from a recursive ordering. Shaded bands denote the 5th and 95th posterior percentiles. For ease of comparison, responses to the expansionary shock are multiplied by -1. Sample 1959-2007.

35

Unemployment

Price level

Interest rate

Contractionary shock

1.5 0.5

0.4 0.3

0

0.2

1

−0.5

0.1

0.5 −1

0 −0.1

0

10

20

−1.5

0

(−) Unemployment

10

20

0

0

(−) Price level

10

20

(−) Interest rate

Expansionary shock

1.5 0.5

0.4 0.3

0

0.2

1

−0.5

0.1

0.5 −1

0 −0.1

0

10

20

−1.5

0

10

20

0

0

10

20

Figure 7: Asymmetric IRFs, narrative identification: FAIR estimates of the IRFs of unemployment (in ppt), the (log) price level (in percent) and the federal funds rate (in ppt) to 100bp monetary shock identified by Romer and Romer (2004). Shaded bands denote the 5th and 95th posterior percentiles. For ease of comparison, responses to the expansionary shock are multiplied by -1. Sample 1969-2007

36

Contractionary shock

Unemployment

Price level

Interest rate 2

0.6 0

1.5 0.4 0.2 0 0

10

20

−0.5

1

−1

0.5

−1.5

0 0

(−) Unemployment

10

20

(−) Price level

10

20

(−) Interest rate 2

0.6 Expansionary shock

0

0 1.5 0.4 −0.5 0.2

1

−1

0 0

10

20

−1.5

0.5 0 0

10

20

0

10

20

Figure 8: Asymmetric IRFs, sign-restrictions identification: FAIR estimates of the IRFs of unemployment (in ppt), the (log) price level (in percent) and the federal funds rate (in ppt) to a 100bp monetary shock identified with sign restrictions. Estimation from a FAIR with asymmetry (plain line). Shaded bands denote the 5th and 95th posterior percentiles. For ease of comparison, responses to the expansionary shock are multiplied by -1. Sample 1959-2007.

37

Unemployment

Inflation

0.6 0.4

−a+ π

a+ u

Recursive

0.4

0.2

0.2

0

0 0

0.2

0.4

0.6

0

a− u

0.2

0.4

−a− π

0.6 0.4

−a+ π

a+ u

Narrative

0.4

0.2

0.2

0

0 0

0.2

0.4

0.6

0

a− u

0.2

0.4

−a− π

0.6 0.4

−a+ π

a+ u

Sign

0.4

0.2

0.2

0

0 0

0.2

0.4

0.6

a− u

0

0.2

0.4

−a− π

Figure 9: A FAIR summary of the asymmetric effects of monetary shocks: posterior distribution of the peak responses of unemployment (au , left panel) and negative inflation (−aπ , right panel) to a 100 bp monetary shock. a+ denotes the peak response to a contractionary shock (a +100bp shock to the fed funds rate) and a− denotes the peak response to an expansionary shock (a −100bp shock). The dashed red line denotes symmetric peak responses. Results from a recursive identification scheme over 1959-2007 (“Recursive”, top row), a narrative identification scheme over 1966-2007 (“Narrative”, middle row), and a set identification scheme with sign restrictions over 1959-2007 (“Sign”, bottom row).

38