Functional Approximation of Impulse Responses: Insights into the Effects of Monetary Shocks

Transcription

1 Functional Approximation of Impulse Responses: Insights into the Effects of Monetary Shocks Regis Barnichon a, Christian Matthes b 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: First draft: March 214. 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 214, the 214 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 215 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, 217

2 1 1. Introduction The impulse response function (IRF) is an important tool used to summarize the dynamic effects of shocks on macroeconomic time series. Since Sims (198), researchers interested in estimating IRFs without imposing a specific economic model have traditionally relied on Vector AutoRegressions (VAR), even though the VAR per se is often of no particular interest. This paper proposes a new method of Impulse Responses (FAIR) to estimate the dynamic effects of structural shocks. The FAIR methodology has two distinct features. First, and instead of assuming the existence of a VAR representation, FAIR works directly with the moving-average representation of the data, that is FAIR directly estimates the object of interest the impulse response function. This confers a number of advantages, notably the ability to impose prior information and structural identifying restrictions in a flexible and transparent fashion, as well as the possibility to allow for a large class of non-linear effects. Second, FAIR approximates the impulse response functions with a few basis functions, which serves as a dimension reduction tool that makes the estimation of the moving-average representation feasible. While different families of basis functions are possible, Gaussian basis functions are of particular interest for macro applications. Intuitively, IRFs are often found (or theoretically predicted) to be monotonic or hump-shaped, and in such cases, just one Gaussian basis function can already provide a parsimonious and intuitive way to summarize an impulse response function with only three parameters, each capturing a separate and easily interpretable feature of the IRF: (a) the magnitude of the peak effect of a shock, (b) the time to that peak effect, and (c) the persistence of that peak effect. This interpretability of the FAIR coefficients further reinforces the benefits of directly working with the moving-average representation: imposing prior information and structural identifying restrictions is easy and transparent, and the model can easily be generalized to introduce non-linearities. More generally, while impulse response functions are large dimensional objects whose important 2

3 27 features can be hard to summarize and assess statistically, 1 the ability to summarize an impulse response function with a few key parameters amenable to statistical inference can make FAIR particularly helpful to evaluate and guide the development of successful models. 2 After establishing the good finite sample properties of FAIR with Monte-Carlo simulations, we illustrate the benefits of FAIR by studying the effects of monetary shocks. FAIR can easily incorporate the different identification schemes found in the structural VAR literature, and we re-visit the effects of monetary shocks using three popular identification schemes: a recursive identification scheme, (ii) a set identification scheme based on sign restrictions, and (iii) a narrative identification scheme where a series of shocks has been previously identified (possibly with measurement error) from narrative accounts. First, we focus on linear models and use FAIR to re-assess some of the key features of the effects of monetary shocks that have been instrumental in the development of monetary models over the past 2 years. We highlight three stylized facts that emerge across the three different identification schemes: (i) a 1 basis point contractionary monetary shock has a peak effect of unemployment of about.3 ppt after about ten quarters, (ii) the peak response of inflation lags that of unemployment by two-to-four quarters, (iii) the responses of inflation and unemployment display similar persistence, with a half-life of the peak effect the time for the response to return to half of its peak value at about five quarters for both inflation and unemployment. Second, we use FAIR to explore whether monetary shocks have asymmetric effects on unemployment. Specifically, we study whether a contractionary monetary shock has a stronger 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, 217): they are simple yet general moments relevant to any model and they can easily be used by researchers to discipline and evaluate models. 3

4 on a string. 3 While this question is central to the conduct of monetary policy, the evidence for asymmetric effects is relatively thin and inconclusive, 4 and part of this state of affairs reflects the fact that estimating the asymmetric effects of a monetary shock is very hard within a VAR framework. In contrast, FAIR can easily accommodate such non-linearities. Consistent with the string metaphor, we find evidence of strong asymmetries in the effects of monetary shocks. Regardless of whether we identify monetary shocks from a recursive ordering, from sign restrictions or from a narrative approach, we find that a contractionary shock has a strong adverse effect on unemployment, while an expansionary shock has little effect on unemployment. The response of inflation is also asymmetric: inflation responds less strongly following a contractionary shock than following an expansionary shock. However, the asymmetric reaction of inflation is the opposite to that of unemployment, so that unemployment reacts less when prices react more and vice-versa. Interestingly, this is the pattern that one would expect if (i) nominal rigidities were behind the real effects of monetary policy, and (ii) downward nominal rigidities were behind the asymmetric effects of monetary shocks on unemployment. A number of methods have been proposed to estimate impulse response functions, and we see FAIR as a useful addition to researchers toolkit. While VARs have been the main approach since Sims (198), an increasing number of papers are relying on Local Projections (LP, Jorda 25) themselves closely related to Autoregressive Distributed Lags (ADL, see Hendry, 1984) to directly estimate impulse response functions. As we describe in details in the main text, FAIR builds on this earlier ADL literature, notably on Hansen and Sargent (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, 217), a possibility supported by recent evidence from Bunn et al. (217) and Christelis et al. (217). 4 For instance, while Cover (1992) and Tenreyro and Thwaites (216) find evidence of asymmetric effects, Ravn and Sola (24) and Weise (1999) instead find nearly symmetric effects. 4

5 ability to summarize the dynamic effects of shocks with a few key interpretable moments that can directly inform model building, (iii) ease of prior elicitation and structural identifi- cation, and (iv) preserving efficiency when allowing for non-linear effects of shocks, notably 74 asymmetric effects. Finally, our approach relates to the recent work of Plagborg-Moller (217), who proposes a Bayesian method to directly estimate the structural moving-average representation of the data by using prior information about the shape and the smoothness of the impulse responses. Section 2 presents our functional approximation of impulse responses and discusses the benefits of using Gaussian basis functions, Section 3 re-visits the linear effects of monetary shocks with FAIR; Section 4 generalizes FAIR to non-linear models; Section 5 studies the asymmetric effects of monetary shocks; Section 6 concludes Functional Approximation of Impulse Responses (FAIR) We first introduce FAIR in univariate setting and then generalize it to a multivariate setting Univariate setting For a stationary univariate data generating process, the IRF corresponds to the coeffi- cients of the moving-average model y t = H ψ(h)ε t h (1) h= 88 where ε t is an i.i.d. innovation with Eε t = and Eε 2 t = 1, and H is the number of lags, which can be finite or infinite. The lag coefficients ψ(h) is the impulse response of y t at horizon h to innovation ε t. 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

6 93 AR( ) A(L)y t = ε t (2) where A(L) = Ψ(L) 1 with Ψ(L) = H ψ(h)l h and L the lag operator. 5 h= Since estimating an AR( ) is not possible in finite sample, in practice researchers esti- mate a finite order AR(p) meant to approximate the AR( ), and the IRF is recovered by inverting that AR(p). In effect, the AR(p) approximation serves as a dimension reduction tool that makes the estimation of (1) feasible. For instance, an AR(1) approximates the coefficients of the MA( ) as an exponential function of a single parameter. In this paper, we propose an alternative dimension-reducing tool Functional Approximation of Impulse Responses or FAIR, which consists in representing the impulse response function as an expansion in basis functions. Instead of estimating an intermediate model the AR(p) and then recovering the impulse response, FAIR directly estimates the object of scientific interest the impulse response function. Specifically, a functional approximation of ψ consists in decomposing ψ into a sum of basis functions, i.e., in modeling ψ(h) as a 16 basis function expansion N ψ(h) = a n g n (h), h > (3) n= with g n : 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 19 functions and posit ψ(h) = N n=1 h bn ( a n e cn )2, h > (4) with a n, b n, and c n parameters to be estimated. Since model (4) uses N Gaussian basis functions, we refer to this model as a FAIR G 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 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 > or for h. In this paper, we treat ψ() 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

7 Multivariate setting While the discussion has so far concentrated on a univariate process, we can easily generalize the FAIR approach to a multivariate setting. Consider the structural vector movingaverage model of y t, a vector of stationary variables, y t = Ψ(L)ε t, where Ψ(L) = H Ψ h L h (5) h= where ε t is the vector of i.i.d. structural innovations with Eε t = and Eε t ε t = I, and H is the number of lags, which can be finite or infinite. The matrix of lag coefficients Ψ h contains the impulse responses of y t at horizon h to the structural shocks ε t. Setting aside the issue of identification for now, the common strategy to recover (5) is identical to the univariate case: rewrite (5) as a VAR( ) and then estimate its VAR(p) approximation. In this paper, we propose to use FAIR and directly estimate the impulse response functions by approximating the elements ψ(h) of Ψ h as in (4) FAIR benefits We now argue that FAIR G FAIR models with Gaussian basis functions can be particularly attractive in macro applications for two reasons: (i) parsimony, and (ii) interpretability of the FAIR G parameters as summary features of the IRFs. We postpone a third important benefit of FAIR G the possibility to allow for non-linearities in a flexible yet parsimonious fashion to a later section dedicated to non-linearities. We will pay particular attention to the FAIR G1 a FAIR with one Gaussian basis function ψ(h) = ae (h b)2 c 2, h >, (6) which captures an IRF with only 3 parameters: a, b and c, as illustrated in the top panel of Figure 1. sum of Gaussian basis functions. 7

8 Parsimony of FAIR A first advantage of using Gaussian basis functions is that a small number of Gaussians can already approximate a large class of IRFs, in fact most impulse responses encountered in macro applications. Intuitively, IRFs are often found (or theoretically predicted) to be monotonic or hump-shaped (e.g., Christiano, Eichenbaum, and Evans, 1999). 8 In such cases, one or two Gaussian basis functions can already provide a very good approximate description of the impulse response. With the one-gaussian and the two-gaussian model summarizing an IRF with only 3 and 6 parameters respectively, Gaussian basis functions offer an efficient dimension reduction tool; reducing the number of parameters substantially while still allowing for a large class of impulse responses. To illustrate this point, Figure 2 plots the IRFs of unemployment, inflation and the fed funds rate to a shock to the fed funds rate estimated from a standard VAR specification with a recursive ordering, along with the IRFs approximated with Gaussian basis functions. 9 For unemployment and the fed funds rate we use a FAIR G with only one Gaussian basis function, 146 a FAIR G1. We can see that this simple model already does a good job at capturing the responses of unemployment and the fed funds rate implied by the VAR, while reducing the dimension of each IRF in Figure 2 from 25 to only 3 parameters. To capture the oscillating pattern of inflation following a monetary shock, two Gaussian basis functions are necessary, and we can see in Figure 2 that the IRF approximated with two Gaussian basis functions a FAIR G2 does a good job there as well. With a small number of Gaussian basis functions, the FAIR G model is parsimonious, 1 8 In New-Keynesian models, the IRFs are generally monotonic or hump-shaped (see e.g., Walsh, 21). 9 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. 1 For instance, for a trivariate model with three structural shocks, a FAIR G1 would need 27 parameters (9 impulse responses times 3 parameters per impulse response, ignoring intercepts) to capture the whole set of impulse responses {Ψ h } H h=1. In the example of Figure 2 where the impulse responses of inflation are approximated by a richer FAIR G2, there would be 36 parameters ( = 36). In comparison, a quarterly VAR with 3 variables and 4 lags has = 36 free parameters, and a monthly VAR with 12 lags has = 18 free parameters. 8

9 and this will allow us to directly estimate a vector moving-average representation of the data. Finally, we note that, unlike VARs, the number of FAIR parameters to estimate need not increase with the frequency of the data, and a FAIR G1 summarizes an IRF with only 3 parameters regardless of whether H = 25 or H = 6. Stated differently, while a monthly VAR would typically require much more lags than the corresponding quarterly VAR to capture the dynamic correlations present in the data, the number of FAIR parameters would remain the same with monthly or quarterly data, as long as the shape of the impulse response is not altered by time aggregation Interpretability and portability of FAIR coefficients A second advantage of using Gaussian basis functions is that the estimated coefficients can have a direct economic interpretation in terms of features of the IRFs. This stands in contrast to VARs where the IRFs are non-linear transformations of the VAR coefficients. Coefficient interpretability can make FAIR priors easier to elicit and make FAIR results particularly helpful to evaluate and guide the development of successful models. 167 The ease of interpretation is most salient in a FAIR G1 model (6) where the a, b and c coefficients have a direct economic interpretation, and in fact capture three separate characteristics of a hump-shaped impulse response: the peak effect of a shock, the time to peak effect, and the persistence of that peak effect. To see that graphically, the top panel of Figure 1 shows a hump-shape impulse response parametrized with a one Gaussian basis function: parameter a is the height of the impulse-response, which corresponds to the maximum effect of a unit shock, parameter b is the timing of this maximum effect, and parameter c captures the persistence of the effect of the shock, as the amount of time τ required for the effect of a shock to be 5% of its maximum value is given by τ = c ln 2. These a-b-c coefficients are generally considered the most relevant characteristics of an impulse response function and are generally the most discussed in the literature. Take the example of the literature on the effects of monetary shocks. In two recent influential papers Coibion (212) and Ramey (216), the authors compare the effects of monetary shocks across different specifications 9

10 by comparing the peak response of the variable of interest. In the FAIR G1 model, this is simply the a parameter. In Christiano et al. (25, p8), the authors visually inspect the impulse response functions and summarize the dynamics effects of a monetary shock with two stylized facts: (i) the response of real variables peaks after six quarters and return to 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 FAIR G1 model However, unlike earlier approaches, FAIR G1 would directly estimate these summary statistics and provide confidence intervals around these statements. With more than one basis function, the ease of interpretation of the estimated parameters is no longer guaranteed. However, in some cases a FAIR G model with multiple Gaussian-basis functions can retain its interpretability. For instance, consider an oscillating pattern as in the bottom panel of Figure 1, which is typical of the response of inflation to a contractionary monetary shock. In that case, the FAIR G parameters can retain their interpretability if the first Gaussian basis function captures the first-round effect of the shock a positive hump-shaped response of inflation usually refereed to as the price puzzle (Christiano et al. 1999), while the second Gaussian function captures the larger second-round effect a 196 negative hump-shaped effect. Going back to our VAR example from Figure 2, the two Gaussian basis functions used to match the impulse response of inflation show little overlap. In particular, one can summarize the 2 nd -round effect of the shock the main object of scientific interest in the case of the inflation response to monetary shocks with the a-b-c parameters of the second basis function: in that case, the a 2, b 2 and c 2 capture the peak effect, the time to peak effect, and the persistence of the 2 nd -round effect of the shock Formally, one can write a no-overlap condition as h 2 g 1 (h)dh ɛ with h 2 such that h 2 g 2 (h)dh = α 1 (7) with g n denoting the nth basis function. That condition is a restriction on the b n and c n coefficients that ensures that the two basis functions g 1 and g 2 have limited overlap. Intuitively, for say ɛ =.5 and α =.9, the condition states that the first Gaussian function (g 1 ) the first-round effect of the shock has a negligible (5 percent or less) overlap with 9 percent of the second Gaussian function (g 2 ) the second-round effect of the 1

11 Going beyond interpretation and ease of prior elicitation, the ability to capture the key properties of the dynamic effects of a shock makes FAIR G particularly helpful to evaluate and guide the development of successful models. IRFs are large dimensional objects whose important features can be hard to summarize and assess statistically. Moreover, when multiple identification strategies, different specifications and different sample periods are considered, comparing the features of the different sets of impulse responses becomes quickly very 28 difficult. 12 In contrast, FAIR G, and most notably FAIR G1, considerably reduces the dimen sionality of the IRFs by capturing only the first-order properties of the impulse responses. By providing simple and easily interpretable summary statistics amenable to statistical inference, FAIR estimates can facilitate the emergence of a set of robust findings and help the development of successful models. More generally, when summarizing the key characteristics of the dynamic effects of a structural shock, the a-b-c parameters can be seen as portable identified moments (as advocated by Nakamura and Steinsson, 217) that can easily be used by researchers to discipline and evaluate models: They are portable in the sense that they are simple yet general moments relevant to any model, and they are identified when the FAIR model is identified with structural identifying assumptions (which, as we will see, are easy to impose in FAIR) FAIR estimation FAIR models can be estimated using maximum likelihood or Bayesian methods. While 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 FAIR G2 response of inflation, that condition would be g 1 (b 2 ) : 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 Amir- Ahmadi et al. (216). 11

12 and relatively easy thanks to modern computational capabilities. The problem boils down to the estimation of a truncated moving-average model, and we recursively construct the likelihood by using the prediction error decomposition and assuming that the structural innovations {ε t } are Gaussian with mean zero and variance 226 one (see the appendix for more details). To initialize the recursion, we set the first H innovations {ε j } j= H to zero.13 Moreover, since the structural innovations {ε t } are not fully identified by the first two moments of the data, it is also at this stage that we impose the identifying restriction that we describe in the next section. A potentially important problem when estimating moving-average models is the issue of under-identification: In linear moving average models, different representations (i.e., different sets of coefficients and innovation variances) can exhibit the same first two moments, so that 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 could remedy this limitation and estimate non-invertible moving-average representations by using the procedure recently proposed by Plagborg-Moller (217), 15 provided that one has enough prior information to favor one moving-average representation over the others. To choose N, the order of the FAIR model, the researcher can either choose to restrict himself to a class of functions if prior knowledge on the shape of the impulse response is available (for instance, using only one Gaussian basis function) or use likelihood ratio 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. 14 If our estimation algorithm chose parameter estimates that implied non-invertibility, our {ε t } estimates would ultimately grow very large, and this situation would lead to very low likelihoods, since we assume that the {ε t } 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...ε }, 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. 12

13 244 with increasing number of basis functions. 16 Finally, note that FAIR models can only be estimated for stationary series (so that the moving-average can be truncated). If the data are non-stationary, we can (i) allow for a deterministic trend in equation (5) and/or (ii) difference the data, and then proceed exactly as described above Imposing structural identifying assumptions As with structural VARs, in order to give a structural interpretation to some of the 25 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 fact, since FAIR models work directly with the structural moving-average representation, imposing prior information on the IRFs is arguably easier and more transparent in FAIR than in VARs, where imposing certain prior information can be challenging. In particular, because the impulse-responses are non-linear transformations of the VAR coefficients, some prior information on the IRFs may be hard to impose in a VAR framework. Moreover, some identification schemes may impose unacknowledged and/or unintended prior information on the impulse responses. 18 Given our later focus on the effects of monetary policy, we will illustrate how FAIR can easily incorporate identifying restrictions using three popular approaches in the monetary 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 for co-integration by testing whether the matrix sum of moving-average coefficients ( H h Ψ l ) is of reduced h=1 l= rank (Engle and Yoo, 1987). 18 See for instance, Baumeister and Hamilton (215) in the case of sign restrictions. A simple yet illuminating example can be found in Plagborg-Moller (217) 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

14 a series of shocks has been previously identified (possibly with measurement error) from narrative accounts, and (iii) a set identification scheme based on sign restrictions. 19 We also mention how FAIR could open the door for more general identification schemes based on shape-restrictions. Short-run restrictions and recursive ordering: Short-run restrictions in a fully identified model consists in imposing L(L 1) 2 restrictions on the contemporaneous matrix Ψ (of dimension L L), and a common approach is to impose that Ψ is lower triangular, so that the different shocks are identified from a timing restriction. To identify only one shock, the weaker restriction is that only a subset of the variables (ordered first in the vector y t ) do not react on impact to that shock, which is ensured by setting the upper-right block of Ψ to zero. Otherwise, Ψ is left unrestricted apart from invertibility. Narrative identification: In a narrative identification scheme, a series of shocks has been previously identified from narrative accounts. For that case, we can proceed as with 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 y t, and we assume that Ψ has its first row filled with except for the diagonal coefficient, which implies that the narratively identified shock does not react contemporaneously to other shocks. In other words, we are assuming that the narrative shocks are contemporaneously correlated with the true monetary shocks and uncorrelated with other structural shocks. 2 Sign restrictions: Set identification through sign restrictions consists in imposing sign- 282 restrictions on the sign of the Ψ h matrices, i.e., on the impulse response coefficients at different horizons. Again, because a FAIR model works directly with the moving average representation and the Ψ h matrices, imposing sign-restrictions in FAIR is straightforward. 19 Other identification schemes, for instance long-run restrictions, are also straightforward to impose. 2 Note that this assumption is weaker than the common assumption (e.g., Romer and Romer, 24) 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 (28). 14

15 One can impose sign-restrictions on only the impact coefficients (captured by Ψ, which could be left as a free parameter in this case) and/or sign restrictions on the impulse response. 287 This is straightforward in a FAIR G1 model where the sign restriction applies for the entire horizon of the impulse response. With oscillating pattern and a higher-order FAIR G model, we can impose sign restrictions over a specific horizon by using priors on the location and the sign of the loading of the basis functions. 21 Identification restrictions through priors: When the FAIR parameters can be interpreted as features of the impulse responses, one can go beyond sign-restrictions and envision set identification through shape restrictions. Using the insights from Baumeister and Hamilton (215), one could implement shape restrictions through informative priors on the coefficients of Ψ and on the a-b-c coefficients. For instance, one could posit priors on the location of the peak effect or posit priors on the persistence of the effect of the shock, among other possibilities Assessing FAIR performances from Monte-Carlo simulations As described in details in the appendix, we conducted a number of Monte-Carlo simu- lations to study the performance of FAIR in finite sample. For a trivariate system with a 31 linear data generating process, we found that a well-specified FAIR G1 model can generate substantially more accurate impulse response estimates (in a mean-squared error sense) than a VAR model, even when the VAR includes many lags. Importantly, these results were ob- 34 tained without the use of any informative prior. 23 Moreover, in a second set of simulations 35 where the data generating process is a VAR, we found that a mis-specified FAIR performs 21 To implement parameter restrictions on Ψ and/or {a n, b n, c n }, we assign a minus infinity value to the log-likelihood whenever the restrictions are not met. 22 See Plagborg-Moller (217) 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 FAIR G model can provide a useful alternative to estimate IRFs. 15

16 36 just as well (even slightly better) than a well-specified VAR model. 24 That being said, our goal is not to claim that FAIR models are superior to VARs. Instead, the simulations are meant to convey that FAIR models can provide a useful alternative to VARs Relation to alternative IRF estimators While VARs have been the main approach to estimate IRFs since Sims (198), an increasing number of papers are now relying on Local Projections (LP, Jorda 25) themselves closely related to Autoregressive Distributed Lags (ADL)) to directly estimate impulse response functions. FAIR aims to straddle between the parametric parsimony of VARs and the flexibility of LP. Indeed, while LP (or ADL in its naive form) is model-free not imposing any underlying dynamic system, this can come at an efficiency cost (Ramey, 212), which can make inference difficult. In contrast, by positing that the response function can be approximated by one (or a few) Gaussian functions, FAIR imposes strong dynamic restrictions between the parameters of the impulse response function, which can improve efficiency. Moreover, FAIR alleviates another source of inefficiency in ADL/LP, namely the presence of serial correlation in the ADL/LP regression residuals. By jointly modeling the behavior of key macroeconomic variables (in the spirit of a VAR), FAIR is effectively modeling the serial correlation present in ADL/LP residuals, which can further improve efficiency. 25 Our functional approximation of the IRFs has intellectual precedents in the ADL literature, 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 FAIR G 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

17 FAIR G approximation notably FAIR G1 aims at summarizing the distributed lag function with a parsimonious and interpretable representation. As we saw, the interpretability is important for the portability of the FAIR estimates (and their usefulness for model building) as well as for prior elicitation. That being said, the Gaussian basis approximation is not the only possibility for a FAIR model, and alternative functional approximations say an Almon polynomial or a Jorgenson rational function could be preferred in different situations or could be used as robustness checks. Another benefit of FAIR over VAR and LP/ADL is the ease of prior elicitation and structural identification. While we saw that identification can be thorny in VARs, the problem 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 schemes found in the structural VAR literature (e.g., sign restrictions). Finally, our direct estimation of vector moving-average models is related to the recent work of Plagborg-Moller (PM, 217), who proposes a Bayesian method to directly estimate 343 the structural moving-average representation of the data. Different from our functional approximation approach, PM reduces estimation variance by using prior information about the shape and the smoothness of the impulse responses A FAIR summary of the linear effects of monetary shocks In this section, we show how the FAIR estimates can summarize the dynamics effects of monetary shocks and provide key identified moments to discipline models. We consider a model of the US economy in the spirit of Primiceri (25), where y t includes the unemployment rate, the PCE inflation rate and the federal funds rate 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 Ψ h are close enough to zero for h > H. We leave the non-zero coefficients of the contemporaneous impact matrix Ψ 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 σ a = 1 ppt, σ b = 1 quarters and σ c = 2 quarters with the constraint c > (in half-life units, this σ c 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

18 We use one Gaussian basis function to parametrize the impulse responses of unemployment and the fed funds rate. For the response of inflation, we use two Gaussian functions. We do so for two reasons: first, to allow for the possibility of a price puzzle in which inflation displays an oscillating pattern, and second to showcase how FAIR can capture oscillating impulse responses while preserving the interpretability of the coefficients. To put our results in the context of the literature (and also showcase how FAIR can easily accommodate popular identifying schemes), we identify monetary shocks using three different schemes: (i) a timing restriction, (ii) a narrative approach, and (iii) sign restrictions. To ensure the same sample period across identification schemes, the data cover 1969Q1 to 27Q4. 27 Specifically, we first assume that monetary policy affects macro variables with a one period lag (e.g., Christiano et al., 1999), so that the matrix Ψ has its last column filled with except for the diagonal coefficient. Second, we use the narrative approach of Romer and Romer (24), extended until 27 by Tenreyro and Thwaites (216). For this identification scheme, we expand our trivariate model by having 4 variables included in the following order: the Romer and Romer shocks, unemployment, inflation and the fed funds rate, and 367 we posit that the contemporaneous matrix Ψ has its first row filled with except for the diagonal coefficient, which implies that the narratively identified shock does not react contemporaneously to other shocks. This specification allows us to treat the Romer and Romer shocks as an instrument for the true monetary shocks. Third, we evaluate the presence of asymmetry using monetary shocks identified through sign restrictions (see e.g., Uhlig, 25). We posit that positive monetary shocks are the only shocks that (i) raise the fed funds rate and (ii) lower inflation roughly two years after the shock. Specifically, with a two Gaussian basis function specification for the response of inflation, we impose that the loading on the second basis function is negative (a π,2 < ), 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

19 376 (meant to capture a possible price puzzle) can load positively or negatively but is restricted 377 to peak within a year (b π,1 4) with a half-life of at most a year (c π,1 ln 2 4). In 378 words, our identification restriction is that the price puzzle cannot last for too long, so that the response of inflation must be negative after roughly two years. In the appendix, we plot the prior IRF of inflation implied by these priors Results We first display our results in the usual way, and Figure 3 plots the impulse response functions of unemployment, inflation and the fed funds rate to a monetary shock that raises the fed funds rate by 1 basis points at its peak. Each column corresponds to a different identification scheme. Next, Figure 4 presents the same results but through the lens of the FAIR parameter estimates: the figure shows the a-b-c summary of the dynamic effects of a monetary shock on unemployment and inflation. 28 A few noteworthy results stand out across the three identification schemes. First, once we take estimation uncertainty into account, the quantitative effects of a monetary shock are consistent across identification schemes. While earlier research reported that narrative-identified monetary shocks have larger effects on unemployment and inflation than recursively-identified monetary shocks (Coibion 212, Nakamura and Steinsson 217), 29 once we hold the methodology constant and control for the peak response of the fed funds rate to the shock (so that the monetary impetus is similar across identification schemes), the peak effect of a monetary shock on unemployment is similar across identification schemes: as shown in Figure 4 (top-left panel), we find that a rec u =.24 [.19,.3], a nar u =.31 [.2,.41] and a sgn u =.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 ɛ =.5). This can be seen graphically in the two median basis functions plotted in Figure Coibion (212) 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

20 the 9 percent credible interval. Since the intervals for the a u estimates display substantial overlap, we conclude that the a u estimates are consistent across methods. Looking at the peak response of inflation (a π, top-right panel of Figure 4) gives similar conclusions: while the response of inflation appears larger following narrative shocks, the a π coefficients are not inconsistent across schemes once we take estimation uncertainty into account. 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, the 9 % confidence bands show considerable overlap across identification schemes (middle and bottom row, Figure 4). This similarity is most striking for the persistence of the IRFs: for both inflation and unemployment, the IRF returns to half of its peak value in about five quarters. For b the time to peak value, the response of unemployment peaks after nine- to-eleven quarters with b rec u = 9.6 [8.1,12.], b nar u = 11.5 [9.8,13.4] and b sgn u = 9.1 [7.,1.8] quarters, while the response of inflation peaks two-to-four quarters later with π,u b rec = 3.8 [1.,7.1], π,u b nar = 1.8 [.1,3.] and π,u b sgn = 3.1 [.2,5.5] where π,u b = b π b u. Given the importance of the difference between b u and b π for the source of monetary non-neutrality (e.g., Mankiw and Reis, 22), we can properly test that b u < b π, i.e., that the difference between the two peak times is statistically significant. To do so, Figure 5 plots the joint posterior distribution of b u (x-axis) and b π (y-axis). The dashed red line denotes identical peak times, so that the figure can be seen as a test of no difference in peak times: a posterior density lying above or below the red line indicates statistical evidence for different peak times. Across the three identification schemes, more than 96, 88 and 92 percent of the posterior probability lies above the dashed-red line, indicating that the peak of the unemployment response occurs significantly before that of inflation. Overall, we conclude from this exercise that (i) a 1 basis point contractionary monetary shock has a peak effect of unemployment of about.3 ppt after about 1 quarters, (ii) the peak response of inflation lags that of unemployment by two-to-four quarters, (iii) the responses of inflation and unemployment display similar persistence, with the half-life of 2

21 425 the peak response at about 5 quarters for both inflation and unemployment Non-linearities with FAIR: assessing the asymmetric effects of shocks Since FAIR models work directly with the moving-average representation of the data, FAIR can easily allow for non-linear effects of shocks. Moreover, the parsimonious nature of FAIR makes it a good starting point to explore the presence of non-linearities while preserving degrees of freedom. Different non-linear effects of shocks are possible, and in this section, we will focus on extending FAIR to estimate possible asymmetric effects of shocks, whereby a positive shock can trigger a different impulse response than a negative shock. 3 The question of asymmetry is particularly interesting for two reasons. First, the possible asymmetric effect of monetary policy the pushing on a string dictum is an important, yet under-studied, question with substantial policy implications. Second, the question of asymmetry is very hard to address within a VAR framework, but it can be addressed relatively easily with a FAIR model Introducing asymmetry With asymmetric effects of shocks, the matrix of impulse responses Ψ h depends on the sign of the structural shocks, i.e., we let Ψ h take two possible values: Ψ + 44 h model with asymmetric effects of shocks would be or Ψ h, so that a 441 y t = H h= [ Ψ + h (ε t h 1 εt h >) + Ψ h (ε t h 1 εt h <) ] (8) with Ψ h and Ψ h the lag matrices of coefficients for, respectively, positive and negative shocks and denoting element-wise multiplication Other non-linearities are naturally possible. For instance, in related work on fiscal policy, we show how to extend FAIR to allow for state dependence, whereby the effect of a shock depends on some state variable at that time of the shock (Barnichon and Matthes, 217). 21