Dynamic Effects of Persistent Shocks
aa r X i v : . [ ec on . E M ] J un Dynamic Effects of Persistent Shocks
Mario Alloza ∗ Jes´us Gonzalo ∗ Carlos Sanz ∗∗ Bank of Spain Universidad Carlos III de Madrid Bank of Spain
June 26, 2020
Abstract
We provide evidence that many narrative shocks used by prominent literature arepersistent. We show that the two leading methods to estimate impulse responses toan independently identified shock (local projections and distributed lag models) treatpersistence differently, hence identifying different objects. We propose corrections tore-establish the equivalence between local projections and distributed lag models, pro-viding applied researchers with methods and guidance to estimate their desired objectof interest. We apply these methods to well-known empirical work and find that howpersistence is treated has a sizable impact on the estimates of dynamic effects.
Keywords : impulse response function, local projection, shock, fiscal policy, monetary policy.
JEL classification: C32, E32, E52, E62. ∗ We thank Fabio Canova, Jes´us Fern´andez-Villaverde, Alessandro Galesi, Gergely G´anics, Juan F. Ji-meno, Mikkel Plagborg-Møller, Juan Rubio-Ram´ırez, Enrique Sentana, and seminar participants at the IWorkshop of the Spanish Macroeconomics Network (Universidad P´ublica de Navarra), Bank of Spain, CFE2018 (University of Pisa), II Workshop in Structural VAR models (Queen Mary University of London), VIIWorkshop on Empirical Macroeconomics (Ghent University), 2019 American Meeting of the EconometricSociety (University of Washington), and 2019 edition of the Padova Macro Talks for insightful comments.Alloza: [email protected]. Gonzalo: [email protected]. Sanz: [email protected].
Introduction
Estimating the impact of economic shocks is a crucial aspect of macroeconomics. To identifyeconomically meaningful shocks, the literature has traditionally relied on systems of equationscoupled with restrictions implied by economic theory. Recently, researchers are increasinglyusing narrative identification, e.g., looking at written official documentation or newspapersand exploiting arguably exogenous variation in these series. While its focus on identifyingexogenous variation is appealing, the lack of restrictions in narrative methods yields objectswith less standard time series properties.In this paper, we analyze how the presence of persistence in narrative shocks affects theidentification and estimation of their dynamic effects, providing empirical researchers withmethods and guidance to deal with this issue. We begin by showing that many narrative shocks used by prominent literature are seriallycorrelated. In particular, we systematically test for serial correlation in eight shocks usedin leading economics journals. We find evidence of serial correlation in seven of them. Thepresence of persistence in the shock does not necessarily preclude these variables from beingcategorized as “shocks” following standard definitions of aggregate shocks. More concretely,according to Ramey (2016), a shock should represent unanticipated movements. What thiscondition implies is that shocks are unforecastable, i.e., they are forecast errors. In partic-ular, when the forecasting loss function is not quadratic, for instance, the check function,the forecasting errors may not be a martingale difference sequence (m.d.s) and thereforecould be serially correlated. However, serial correlation poses additional challenges for theidentification of the macroeconomic experiment of interest.When estimating the dynamic response of some variable to a serially correlated shock,some part of this persistence may be passed on to the impulse response function (IRF). See Romer and Romer (2004), Romer and Romer (2010), or Ramey and Zubairy (2018) for prominentexamples of narrative identification. Throughout the paper we use the term persistence as a phenomenon captured or reflected by serialcorrelation , a testable condition. We use both terms interchangeably. as if theshock were uncorrelated, i.e., to a counter-factual serially uncorrelated shock ( R ( h ) ∗ ), or theresponse to the shock as it is, i.e., including the effect of persistence in the IRF ( R ( h )).Deciding for one or the other depends on what specific question the researcher is tryingto address. On the one hand, R ( h ) ∗ allows to compare effects with those obtained from atheoretical or empirical model, and facilitates comparisons across different types of shocks(e.g., monetary versus fiscal shocks) or across countries. On the other hand, R ( h ) is moreappropriate if the researcher is interested in evaluating the most likely dynamic response ofa variable to a shock based on historical data. Regardless of which object is preferred bythe researcher, the difference between R ( h ) and R ( h ) ∗ is informative about how much of thedynamic transmission of a shock is due to the presence of persistence.We consider the two most popular methods to estimate impulse responses when a shockhas already been identified (e.g., using narrative methods). These are local projections (LPs)(Jord`a (2005)) and distributed lag models (DLMs). We show that, if there is no serialcorrelation, the two methods identify the same object. However, we demonstrate that thisequivalence breaks down in the presence of serial correlation. In this case, LPs identify R ( h )while DLMs regressions identify R ( h ) ∗ . The intuition is that LPs compute the response athorizon h by regressing the outcome variable in t + h against the shock in time t . Sincethe standard setting does not account for how the shock evolves between t and t + h , theresponses include two components: an economic effect (the economic impact of the shockon the endogenous variables) and an effect that exclusively depends on the degree of serialcorrelation of the shock. By contrast, DLMs implicitly account for the evolution of the shock,hence identifying the effect as if the shock were not persistent.While this result might seem discouraging, we then show that it is possible to adjust By DLMs we refer to single-equation regressions of an outcome variable against the contemporaneousvalue and lags of the shock with or without an autoregressive component. These methods are also knownas truncated moving average regressions. These specifications are frequent in the applied literature—see,e.g., Romer and Romer (2004), Cerra and Saxena (2008), Romer and Romer (2010), Alesina et al. (2015),Arezki et al. (2017), and Coibion et al. (2018). R ( h ) ∗ . As mentioned, if she runs standardLPs with a persistent shock, she will identify R ( h ) instead. Perhaps surprisingly, the mostobvious solution of including lags of the shock will not address this issue. However, we showthat, by including leads of the shock, she will recover R ( h ) ∗ . Likewise, we show how standardDLMs can be adapted so that they identify R ( h ).To illustrate how our methods work, we consider an actual empirical application, whichalso serves to assess the quantitative relevance of persistence in a real case by comparingestimates of R ( h ) and R ( h ) ∗ . In particular, we consider Ramey and Zubairy (2018)’s LPsestimation of the dynamic effects to a shock constructed from news about future changesin defense spending. We find that, after two years, the responses that exclude the effect ofpersistence in the shock are about 40% lower than the original Ramey and Zubairy (2018)’sestimates. The effect of serial correlation also seems to have an effect on the short-runresponse of fiscal multipliers during recessions. In the appendix, we consider additionalapplications, based on Guajardo et al. (2014), Romer and Romer (2004), Gertler and Karadi(2015), and Romer and Romer (2010). Overall, we find that how persistence is treated canhave a sizable impact on the estimated effects.The results of this paper generalize in at least three important aspects. First, the resultsof the (lack of) equivalence between LPs and DLMs when the shock is persistent carry overto multivariate settings popularly used in the empirical literature. Building on a result byPlagborg-Møller and Wolf (2019), we show that the dynamic response from a VAR with theshock embedded as an endogenous variable is equivalent to that of a VAR with the shockincluded as an exogenous variable only when that shock has no serial correlation. We believethis result has relevant practical implications for applied macroeconomic researchers. Second, This result arises because a VAR with a shock as an exogenous variable (often known as VAR-X) can beseen as multivariate generalization of a DLM (see Mertens and Ravn (2012) or Favero and Giavazzi (2012)for examples of VAR-X specifications). Furthermore, Plagborg-Møller and Wolf (2019) show that, undersome assumptions, LPs are equivalent to a VAR when the shock is included as an endogenous variable (as inBloom (2009) or Ramey (2011)).
4e also show that our results generalize to specific contexts where a researcher employs aninstrument in a LP setting (also known as LP-IV). Lastly, a researcher interested in usingLPs to uncover the dynamic relations of two variables may be interested in including leads ofa third variable to construct counterfactual responses as if the behavior of that third variablehad remained constant over the response horizon. This can be seen as the LP counterpartof constructing counterfactual responses in a VAR that allow to separate a direct effect ofa regressor on a dependent variable from other indirect effects. This procedure has beenfrequently used in the empirical VAR literature. Our paper makes four contributions to the literature. First, we formally and systemati-cally test for the presence of serial correlation in shocks used by previous work. Although theissue of persistence in shocks has been noted before, we believe we are the first to formallyand systematically test for serial correlation in prominent narratively-identified shocks.Our second contribution is to show that, while both LPs and DLMs identify the sameobject if the shock is serially uncorrelated, this equivalence breaks down in the presence ofpersistence. Plagborg-Møller and Wolf (2019) prove that LPs and VAR methods identifythe same impulse responses when both methods have an unrestricted lag structure. Thisresult formalizes some of the examples provided in Ramey (2016), which implies that differentidentification schemes in a VAR setting can be implemented in a LP context. Our result buildson a different premise: we consider the cases where the shock has already been identified usingnarrative measures and the researcher wants to use LPs or DLMs to estimate dynamic effects.Our third contribution is to provide methods to re-establish the LP-DLM equivalencewhen there is persistence, providing applied researchers with a menu of options to identifytheir desired object of interest. In this regard, our method of adding leads to LPs is related tothe tradition in factor analysis by Geweke and Singleton (1981) and on the DOLS estimation See, for example, Bernanke et al. (1997), Sims and Zha (2006), or Bachmann and Sims (2012). In recentresearch, Cloyne et al. (2020) propose an alternative method based on a Blinder-Oaxaca-type decomposition. Ramey (2016) finds that the time aggregation required to convert the shock in Gertler and Karadi (2015)to monthly frequency, inserts serial correlation. Miranda-Agrippino and Ricco (2018) corroborate this finding,by regressing the shock on four lags and testing their joint significance. They also find that other measuresof monetary shocks such as Romer and Romer (2004) exhibit serial correlation.
5f cointegration vectors (Stock and Watson (1993)). Dufour and Renault (1998) introduceleads in some of their IRFs to study causality at different horizons. Faust and Wright (2011)find that including ex-post forecast errors results in an accuracy improvement when forecast-ing excess bond and equity returns. More recently, Teulings and Zubanov (2014) find thatestimating dynamic effects of a dummy variable (e.g., banking crisis) in a panel data contextwith fixed effects and LPs suffers from a negative small-sample bias, since the estimation ofthe fixed effect picks up the value of future realization of the dummy variable. The authorsshow that this bias is attenuated either by increasing the sample size or by including futurerealizations of the dummy variable over the response horizon. Finally, we speak to some recent and well-known empirical work on the effects of mone-tary and fiscal policy (Ramey and Zubairy (2018) Guajardo et al. (2014), Romer and Romer(2004), and Gertler and Karadi (2015)). Our contribution is to apply our methods to theseworks and re-assess their empirical evidence. We do not claim that any of these papers is“wrong”. Rather, what our results indicate is that the correct interpretation of their resultsdepends on the desired object of interest and the employed estimating method.The rest of the paper proceeds as follows. Section 2 provides evidence of serial correlationin shocks used by previous work. Section 3 describes that LPs and DLMs treat persistencedifferently, and proposes a solution to re-establish the equivalence between them. It alsoprovides simulations to help understand the results. Section 4 discusses the previous findingsand the options available to applied researchers working with a persistent shock. Section 5lays out an application. Section 6 concludes. The online appendix contains proofs of thetheoretical results and further material, including the generalization of the results to VARand IV settings, additional robustness exercises, and other empirical applications. By contrast, the difference between LPs and DLMs that we identify is not due to a bias in the estimates,but instead to differences in identification due to the persistence of the shock. Since our problem still persistsasymptotically, increasing the sample does not reduce the LP-DLM difference. Additionally, this differenceis not necessarily negative, but will depend on the nature of the data generating process that drives thepersistence. Evidence and implications of serial correlation in shocks
When shocks are identified from within an empirical model, the researcher imposes a set ofrestrictions to recover shocks that can be economically meaningful. Typically, this impliesthat the resulting shocks are well-behaved and display some statistical features that mightbe seen as desirable—in particular, no persistence. Alternatively, shocks may be identifiedwithout the use of a model, for example, by using narrative methods. This alternativeidentification relies on the existence of historical sources, such as official documentation,periodicals, etc., from which a shock variable is constructed. In this section, we provideevidence that it is common that shocks identified this way are persistent. We then take stockon this finding in light of Ramey (2016)’s canonical definition of a shock.We study eight aggregate shocks used by prominent literature on monetary and fiscalpolicy. Some of these shocks are identified using narrative methods, while some employalternative strategies such as timing restrictions using high-frequency methods. To test for the presence of persistence we use a portmanteau -type test following Box and Pierce(1970). The null hypothesis is that the data are not serially correlated. We test for the pres-ence of autocorrelation in 40 periods, although results are robust to different horizons (seeTable D.1). In particular, Romer and Romer (2010) and Cloyne (2013) construct measures of exogenous tax changesfor the US and the UK, respectively. The authors classify legislated tax measures according to the motivation,as reflected in official documentation, and consider those tax changes that are the result of causes non-relatedto the state of the economy. In a similar vein, Ramey and Zubairy (2018) construct a measure of governmentspending shocks by looking at the announcements of future changes in defense spending. Guajardo et al.(2014) construct a series of fiscal consolidations in OECD countries motivated by a desire to reduce thedeficit (as opposed to motivated by current or prospective economic conditions). Romer and Romer (2004)and Cloyne and H¨urtgen (2016) identify exogenous changes in monetary policy by looking at the minutes anddiscussion of the monetary policy committees of the Federal Reserve and Bank of England, respectively (theyalso orthogonalize the resulting series using forecastable information available at that time). Alternatively,Gertler and Karadi (2015) identify a proxy of monetary policy shocks using high frequency surprises aroundpolicy announcements. Lastly, Arezki et al. (2017) construct a measure of news shocks based on the dateand size of worldwide giant oil discoveries. While some of these papers employ auxiliary regressions to isolateforecastable information, all have in common that the shocks have not been exclusively identified from a timeseries model. We implement the small sample correction following Ljung and Box (1978). For the cases of Arezki et al.(2017) and Guajardo et al. (2014), which refer to panel data, we test serial correlation using a generalizedversion of the autocorrelation test proposed by Arellano and Bond (1991) that specifies the null hypothesisof no autocorrelation at a given lag order. As further evidence of the presence of serial correlation in the aboveseries, Figure D1 plots the associated correlograms. Romer and Romer (2010) constitutesthe only considered shock for which we fail to detect the presence of persistence. According to the canonical definition (Ramey (2016)), empirical shocks should (i) beexogenous to current and lagged endogenous variables, (ii) be uncorrelated to other exogenousshocks, and (iii) represent unanticipated movements (or news about future shocks). Whileone might think that the presence of persistence violates the third condition, this is notnecessarily the case. When the forecasting loss function is the quadratic one, it is well knownthat the forecasting errors must be a m.d.s with respect to some information set and thereforeuncorrelated. This is the case when the shocks come directly from a conditional expectationmodel, like a VAR model. When the forecasting loss function is not quadratic, for instance,the check function (popular in quantile regressions), the forecasting errors are not a m.d.sand therefore they could be serially correlated. They still are forecasting errors (satisfy (iii))but are serially correlated.This indicates that serially-correlated shocks can still be labeled “shocks” according tothe previous definition. However, even if a researcher always operates under the quadratic The hypothesis of serial uncorrelation is rejected for significance levels below 5% when considering fewerlags in the test or when considering a longer series (with updated data) from Coibion (2012). The presenceof some degree of autocorrelation is shown in Panel E of Figure D1. Persistence may have different origins. In some instances, it arises because of the method used to converta nominal series into real terms. For example, Cloyne (2013) and Arezki et al. (2017) divide their seriesby lagged GDP, while Ramey and Zubairy (2018) use the GDP deflator and a measure of trend GDP. Inother instances, the serial correlation arises because of the mapping between different time frequencies. Thisis usually the case with the identification of monetary policy shocks, such as Romer and Romer (2004),Gertler and Karadi (2015), or Cloyne and H¨urtgen (2016), where daily monetary changes are converted intomonthly series. Finally, there are other shocks that are more likely to appear together, because of their multi-period nature (for example, episodes of fiscal consolidations, as identified by Guajardo et al. (2014), tend to bespread over the course a few years) or because they cluster around events like wars (as in Ramey and Zubairy(2018)). See Granger and Machina (2006) and Lee (2008) for a description and analysis of loss functions. paper type of shock Box-Pierce (40) test p-valueArezki et al. (2017) news about oil discoveries 177.903 0.000Cloyne (2013) tax (UK) 98.751 0.000Cloyne and H¨urtgen (2016) monetary policy (UK) 84.422 0.000Gertler and Karadi (2015) monetary policy (US) 124.568 0.000Guajardo et al. (2014) fiscal consolidations 185.810 0.000Ramey and Zubairy (2018) government spending 182.950 0.000Romer and Romer (2004) monetary policy (US) 53.758 0.072Romer and Romer (2010) tax (US) 19.023 0.998
The third column implements the Box and Pierce (1970) test of serial correlation using the small samplecorrection following Ljung and Box (1978). The null hypothesis of this test assumes that the data are notserially correlated within 40 periods. For Arezki et al. (2017) and Guajardo et al. (2014), which refer topanel data, we use a generalized version of the autocorrelation test proposed by Arellano and Bond (1991).The serial correlation test yields p-values smaller than 0.05 when testing the shocks of Romer and Romer(2004) with fewer lags or when using the updated data from Coibion (2012) (p-value drops to 0.0041).Ramey and Zubairy (2018) use extended data from Ramey (2011).
We consider the following VAR as the data generating process: y t = ∞ X ℓ =1 A ℓ y t − ℓ + ∞ X q =0 δ q x t − q + u t x t = ∞ X r =1 γ r x t − r + ε t , (1)where y t is a vector of endogenous time series, x t is a strictly exogenous variable suchthat E ( u t | y t − s , x t − p ) ∀ s > p R
0, and u t and ε t are a vector and a scalar i.i.d. variables,with mean and variance given by u t ∼ ( , Σ u ) and ε t ∼ (0 , σ ε ), respectively. Following theevidence discussed in the previous section, x t is considered to be a shock identified usingnarrative methods and is allowed to be persistent.This general framework encompasses several empirical specification often found in theliterature. For example, when ignoring the second equation, system (1) becomes a VAR withan exogenous variable (or VAR-X). Additionally, when y t is a scalar and A ℓ = ∀ ℓ , system(1) becomes a DLM. Alternatively, when x t is instead included in the vector of endogenousvariables y t , system (1) becomes a standard VAR. We explore the implications of this lastrepresentation in Appendix B.1.Without loss of generality, we consider a simpler version of system (1) with A ℓ = ∀ ℓ , See, for example, Mertens and Ravn (2012) or Favero and Giavazzi (2012), which assume ℓ and q arefinite numbers. As in Romer and Romer (2004) or Romer and Romer (2010). As in Bloom (2009) or Ramey (2011). q = 0 ∀ q > γ r = 0 ∀ r > y t = δx t + u t x t = γx t − + ε t , (2)where y t is now the economic outcome variable for interest (for example, GDP), x t is aneconomic shock (e.g., a fiscal or monetary policy shock) which is strictly exogenous E ( u t | x t − p ) ∀ p R
0, and u t and ε t are i.i.d variables with mean and variance given by u t ∼ (0 , σ u ) and ε t ∼ (0 , σ ε ), respectively. δ measures the contemporaneous impact of variable x t on y t andis the main parameter of interest.The data generating process described by system (2) is intentionally simple to illustratehow the dynamic relationship between the dependent variable y t and the shock x t dependson the persistence of the latter. Importantly, the obtained results also arise in more complexsettings when we incorporate more general characteristics as in system (1). We are interested in recovering the response of our variable of interest y t when a shock x t hits the system in period t . We consider two different IRFs. The first one, denoted by R ( h )for period h , is: R ( h ) = E [ y t + h | x t = 1 , Ω t − ] − E [ y t + h | x t = 0 , Ω t − ] , (3)where Ω t − represents all the history of previous realizations of ε t and x t up to period t −
1. Importantly, note that the above definition does not condition for future realizationsof x t . Hence, if γ = 0, an initial unit impulse in x t does not imply that x t + j = 0. In otherwords, equation (3) describes dynamic responses that include the possible persistence of the For example, in Subsection 3.3, we consider models that also include persistence in the dependent variableand lagged effects of the shock. Appendix B.2 proposes a DGP that calls for the use of instruments in LPregressions. Appendix B.3 provides an alternative specification where the degree of serial correlation in theshock x t is taken from the actual data, instead of following an autoregressive process. This impulse response is equivalent to R ( h ) = E [ y t + h | ε t = 1 , ε t +1 = 0 , ..., ε t + h = 0 , Ω t − ] − E [ y t + h | ε t = 0 , ε t +1 = 0 , ..., ε t + h = 0 , Ω t − ]. See, for example, Koop et al. (1996). x t . For example: R (0) = ∂y t ∂x t = δ R (1) = ∂y t +1 ∂x t = δγ R (2) = ∂y t +2 ∂x t = δγ . . . However, the researcher might also be interested in the response to the shock as if theshock had no persistence. We call this second IRF R ( h ) ∗ and define it as: R ( h ) ∗ = E [ y t + h | x t = 1 , x t +1 , ..., x t + h , Ω t − ] − E [ y t + h | x t = 0 , x t +1 , ..., x t + h , Ω t − ] . (4)Contrary to R ( h ), R ( h ) ∗ explicitly controls for future realizations of x t so that it describesdynamic responses that do not incorporate the effect of persistence (regardless of the valueof γ ), i.e., the responses are observationally equivalent to those that would arise from a datagenerating process with γ = 0: R (0) ∗ = ∂y t ∂x t = δ R (1) ∗ = ∂y t +1 ∂x t (cid:12)(cid:12)(cid:12) x t +1 = 0 R (2) ∗ = ∂y t +2 ∂x t (cid:12)(cid:12)(cid:12) x t +1 ,x t +2 = 0 . . . Note that, if γ = 0 (the shock is not persistent), then R ( h ) = R ( h ) ∗ ∀ h . By contrast, if γ = 0, then R ( h ) = R ( h ) ∗ ∀ h > The definition of R ( h ) ∗ is not new. When x t is the shock variable of interest, this impulse re-sponse is referred to as the “traditional impulse response function” by Koop et al. (1996): R ( h ) ∗ = E [ y t + h | x t = 1 , x t +1 = 0 , ..., x t + h = 0 , Ω t − ] − E [ y t + h | x t = 0 , x t +1 = 0 , ..., x t + h = 0 , Ω t − ]. It provides an an-swer to the question “what is the effect of a shock of size 1 hitting the system at time t on the state of thesystem at time t + h given that no other shocks hit the system?”. .1 Differences between DLMs and LPs under persistence We now consider the two most frequently used methods to estimate impulse responses when ashock is independently identified, DLMs and LPs, and compare the objects that they identifywhen the shock is persistent. We first consider the case of DLMs. The use of these models iswidespread in applied macroeconomics. In the case of system (2), note that we can recoverthe response function R ( h ) DLM using the following regression: y t = θ x t + θ x t − + θ x t − + θ x t − + θ x t − + . . . + e t , (5)and it follows that R ( h ) DLM = ∂y t + h ∂x t = θ h ∀ h .The second main method to compute impulse responses is LPs, proposed by Jord`a (2005).LPs are more robust to certain sources of misspecification and for this reason, their use hasincreased in recent times (see Ramey (2016) for examples). LPs compute impulse responsesby estimating an equation for each response horizon h = 0 , , . . . , H : y t + h = δ h x t + ξ t + h , (6)where the sequence of coefficients { δ h } Hh =0 determines the response of the variable of interest R ( h ) LP = δ h for each horizon h . We now consider under which conditions both methods identify the same objects.
Proposition 1.
Given the data generating process described by system (2) , if the shock x t isserially uncorrelated, then the response functions identified by DLMs and LPs are equal forall response horizons, that is: See, for example, Romer and Romer (2004), Cerra and Saxena (2008), Romer and Romer (2010),Alesina et al. (2015), Arezki et al. (2017), Coibion et al. (2018) for interesting applications based on DLMmethods, or Baek and Lee (2020) for a discussion of their properties. As mentioned in the introduction, thesemethods are also a special case of more general specifications such as VARs with exogenous variables (orVAR-X). We develop this point further in Appendix B.1, when generalizing some of the results of the paper. This regression should include as many lags as the response horizon h = 0 , , . . . , H . Unrelated to our case at hand, note that the structure of the LPs induce serial correlation in the residuals ξ t + h . This is usually corrected by computing autocorrelation-robust standard errors (Jord`a (2005)). SeeOlea and Plagborg-Møller (2020) for a recent contribution on inference in LPs. f γ = 0 , then R ( h ) DLM = R ( h ) LP = R ( h ) ∗ = R ( h ) ∀ h .If the shock is serially correlated, then the response functions identified by DLMs and LPsare different for all h > :If γ = 0 and h = 0 , then R ( h ) DLM = R ( h ) LP = R ( h ) ∗ = R ( h ) .If γ = 0 and h ≥ , then R ( h ) DLM = R ( h ) ∗ = R ( h ) LP = R ( h ) .Proof. See Appendix A.1.Following the above proposition, when γ = 0, LPs recover a dynamic response thatincludes three dynamic effects: (i) the effect that x t has directly on y t + h (due to a laggedimpact of the shock), (ii) the effect that x t has through the persistence of y t , and (iii) theeffect that x t has on y t + h through x t + h (since cov ( x t , x t + h ) = 0 when γ = 0). The first twoeffects are independent of γ and are shut down in our simple specification of system (2) (wewill incorporate them in our simulation exercises in the next subsection). The last effect (the persistence effect of x t ) drives the difference between R ( h ) DLM and R ( h ) LP . In particular, R ( h ) LP = R ( h ) = δγ h , while R ( h ) DLM = R ( h ) ∗ = 0 for all h ≥ h = 1: y t +1 = δ x t + ξ t +1 , (7)where δ = R (1) LP . The direct effect of x t on y t +1 is 0. If x t had no persistence, then δ would be 0. However, when γ = 0, we can use system (2) to express y t +1 as a function of x t : y t +1 = δx t +1 + u t +1 = δ ( γx t + ε t +1 ) + u t +1 = δγx t + u ∗ t +1 , where u ∗ t +1 = δε t +1 + u t +1 . This shows that the coefficient δ in equation (7) will also recoverthe persistence effect of x t : δ = δγ . The intuition is that between period t and period t + 1,14 t affects x t +1 when γ = 0. Since x t +1 is not a regressor in equation (7), then this effect isabsorbed by δ . When impulse responses are identified using DLMs, the treatment of the persistence of x t is different. Consider a version of equation (5) expressed in terms of t + 1: y t +1 = θ x t +1 + θ x t + θ x t − + θ x t − + θ x t − + . . . + e t +1 . (8)As noted earlier, the sequence of coefficients θ h determines the response function. Considerthe response when h = 1, i.e., R (1) DLM = θ . Note that, while we know from system (2)that ∂y t +1 ∂x t = δγ , the coefficient recovered by θ is indeed ∂y t +1 ∂x t (cid:12)(cid:12)(cid:12) x t +1 = 0. That is, since theDLM controls for x t +1 , the persistence effect of x t is accounted for.In other words, DLMs identify: R ( h ) DLM = E [ y t + h | x t = 1 , Ω t − , x t + h − , ..., x t +1 ] − E [ y t + h | x t = 0 , Ω t − , x t + h − , ..., x t +1 ] , while LPs identify: R ( h ) LP = E [ y t + h | x t = 1 , Ω t − ] − E [ y t + h | x t = 0 , Ω t − ] . Note that the difference between R LP and R DLM is positive (negative) when γ > γ < γ may be positive or negative. In this subsection we lay out two methods that can render the responses from DLMs andLPs identical, even under the presence of persistence. This omitted variables problem is also briefly mentioned in Alesina et al. (2015) in the particular contextof fiscal consolidation plans. For example, γ seems to be positive in Ramey and Zubairy (2018), and negative in Romer and Romer(2004). .2.1 Adapting LPs to exclude the effect of serial correlation A researcher may be interested in recovering responses as if the shock were serially uncorre-lated ( R ( h ) ∗ ). (We discuss in Section 4 when the object of interest may be R ( h ) ∗ , or R ( h )instead.) However, we have shown that R LP ( h ) = R ( h ) ∗ if γ = 0 and h ≥ x t are: (i) toinclude lags in the regression (6), or (ii) to replace x t with the error term that purges outthe persistence: ε t = x t − γx t − . (9)However, neither of these methods yields R ∗ ( h ). The reason is that replacing x t with ε t doesnot include any further information between t and t + h , so the responses of the dependentvariable will still be affected by x t + h . This point is further developed in Appendix B.4.A third potential method to exclude the effect of persistence would be recasting system (2)as a VAR that includes the shock as an endogenous variable. However, since in this caseLPs and a VAR would identify the same impulse responses (see Plagborg-Møller and Wolf(2019)) the VAR responses would also include an effect due to the persistence of the shock—we explore this in more detail in Appendix B.1.Instead, we propose a method based on the inclusion of leads of the persistent shockvariable. In particular, given that the DGP of system (2) poses an AR(1) for x t , one shouldregress: y t + h = δ h, x t + δ h, x t +1 + ξ t + h , (10)where δ h, is the h -horizon response identified by LPs that include leads of the shock x t , whichwe denote as R F ( h ). In more general processes, in which the autocorrelation of the shockmay be of an order larger than one, the optimal choice of leads can be derived adapting theprocedure from Choi and Kurozumi (2012). The most conservative procedure would be toinclude h leads of the shock in each period h . This is the choice implemented in Section 5, See also Lee (2020) for lag order selection in LPs.
Proposition 2.
Given the data generating process described by system (2) , the responsefunction identified by modified LPs to a shock x t as described in equation (10) is equal to theresponse as if the shock had no persistence (and to the response obtained from DLMs as inequation (5) ), that is: R ( h ) F = R ( h ) ∗ = R ( h ) DLM ∀ γ and h .Proof. See Appendix A.2.Intuitively, leads of x t in equation (10) act as controls for the persistence of the shockthroughout the response horizon, so that the parameter δ h, reflects the dynamic response toa counterfactual serially-uncorrelated shock, that is, controlling for the effect due to ∂x t +1 ∂x t = 0built in system (2) when γ = 0. As noted earlier, R ( h ) DLM = R ( h ) ∗ regardless of the value of γ . However, in some instances,the researcher may be interested in the response that includes the effect of persistence ( R ( h )).In this subsection, we show how to adapt DLMs to recover these responses. Intuitively, theidea is to compute the impulse responses in system (2) with respect to ε t instead of x t .Consider a recursive substitution of x t in system (2): y t = δγ t x + δ t X i =0 γ i ε t − i + u t . (11)The responses of y t to ε t , which we denote by R ( h ) DLM − per , can be obtained from thecoefficients ˜ θ h in: y t = ˜ θ ε t + ˜ θ ε t − + ˜ θ ε t − + ˜ θ ε t − + ˜ θ ε t − + . . . + e t . (12)17 roposition 3. Given the data generating process described by system (2) , the responsefunction identified by DLMs of y t to the innovation ε t as described in equation (12) is equiva-lent to the response that includes the effects of persistence (and to the response obtained fromLPs as in equation (6) ): R ( h ) DLM − per = R ( h ) = R ( h ) LP ∀ γ and h .Proof. See Appendix A.3.Proposition 3 establishes a direct equivalence between the coefficients obtained from equa-tion (12) and those obtained from LPs in equation (6): ˜ θ h = δ h ∀ h . The former arealso related to the coefficients estimated from the DLM in terms of x t , as in equation (5): θ = ˜ θ = δ , θ = ˜ θ − γ ˜ θ , . . . , θ h = ˜ θ h − γ ˜ θ h − . Intuitively, the response of y t +1 to x t has anoverall effect of δ = ˜ θ , which includes (i) the direct effect of x t on y t +1 (0, in our simple case)and (ii) the effect on y t +1 that is due to the persistence in x t (given by γδ ). The standard DLM estimation from equation (5), since it accounts for the evolution of x t over the responsehorizon, is implicitly subtracting the part of the response that is given by the persistence of x t from the overall effect. In this subsection, we perform stochastic simulations of the asymptotic behavior of the im-pulse response functions using both LPs and DLMs. Our goal is twofold. First, to evaluatequantitatively the conclusions reached in the previous subsection using a plausible calibrationof the parameters that determine the model. Second, to consider a slightly more complex(and realistic) version of the data generating process that includes richer features frequentlypresent in real empirical applications. In particular, we consider the following process: y t = ρy t − + B x t + B x t − + u t x t = γx t − + ε t , (13)18here E ( ε t − s u t − r ) = 0 ∀ s, r R
0, and u t and ε t follow N (0 ,
1) distributions. We set B = 1 . B = 1 and ρ = 0 . ρ , and allows the shock x t to have lagged effects on y t through B . We simulate system (13) for 100 million periods and recover the dynamic responses of y t to the shock x t using LPs: y t + h = ρy t − + β h, x t + β h, x t − + β h,f x t +1 + ξ t + h . (14)We consider three cases: (i) no persistence ( γ = 0), without including leads in the es-timation (i.e., setting β h,f = 0); (ii) some persistence ( γ = 0 .
2) and still β h,f = 0; (iii)some persistence ( γ = 0 .
2) and including a lead of the explanatory variable (i.e., allowing β h,f = 0). Note that equation (14) must include a lag of shock x t to capture the effect of B insystem (13). However, this does not control for the potential persistence of shock x t , as willbe apparent in the simulations.Figure 1 shows the results of our simulations. In case (i) (dark-blue solid line), theresponse has a contemporaneous effect of ˆ β , = 1 . ρ and B have positive values. Using the language of the previoussection, the impulse response function estimated by LPs with no persistence is asymptoticallyequivalent to the one obtained directly from equation (14), that is, ˆ R ( h ) LP → R ( h ) ∗ .In case (ii) (red solid line), the introduction of persistence in the shock x t results in alarger effect on y t on all horizons after impact. This has potentially important implications:if a macroeconomist is interested in the effects of a serially-uncorrelated shock (as in most We introduce this extra lag of the shock to make explicit the distinction between the effect due to thepersistence of the shock and the effect of lagged values of the shock on current outcomes. The choice of γ = 0 . ρ would yield higher biases due to the persistence of the process. = 0 0 0 & leads This figure shows the response of a simulated outcome variable to a shock with different degrees of persistence,using LPs. The dark blue line shows the results of estimating equation (14) assuming γ = 0 in equation (13).The red line shows the same estimation when γ = 0 .
2. The dashed grey line shows the response after includingleads of the shock as in equation (14) and still assuming γ = 0 . general equilibrium models), but naively estimates equation (14), implicitly setting β h,f = 0,then the dynamic response is upwardly biased due to the persistence of the shock, i.e.,ˆ R ( h ) LP > R ( h ) ∗ for h >
0. Given the assumptions on the autocorrelation of the process x t ,the bias is particularly large in the short and medium run. Higher values of the persistenceparameters γ and ρ would increase the difference between both responses (blue and red linesin Figure 1).In case (iii) (dashed grey line in Figure 1), we see that the inclusion of leads of x t rendersthe response of the outcome variable to a persistent shock identical to the one obtained whenconsidering a shock without persistence, i.e., ˆ R ( h ) F → R ( h ) ∗ . In Appendix B.3 we providean alternative simulation where the shock x t in (13) is not assumed to follow an AR(1)process but it is instead taken from actual data.Next, we use these simulations to show that the computation of impulse responses us-ing DLMs always yields the same estimates regardless of the persistence in x t , that is,20 R DLM ( h ) → R ∗ ( h ) for any value of γ .First, note that, since ρ <
1, system (13) can be inverted and re-written as: y t = (1 − ρL ) − ( B + B L ) x t + (1 − ρL ) − u t , (15)where L represents the lag operator.Given the independence of u t and x t , the representation from equation (15) suggests thatthe dynamic responses of y t from x t can be obtained from the coefficients ϑ h in the followingregression: y t = ϑ x t + ϑ x t − + ϑ x t − + ϑ x t − + . . . + ϑ H x t − H + ξ t , (16)where H is the response horizon. We estimate equation (16) fo three different cases: (i) assuming that γ = 0 in the datagenerating process described in system (13), (ii) assuming that γ = 0 . x t with ˆ ε t in equation (16) (i.e., following equation (12)).The results are shown in Figure 2. Cases (i) and (ii) are displayed in blue and dashed greylines, respectively. As argued earlier, since equation (16) controls for all potential dynamiceffects of x t , including its persistence, the coefficients ϑ h reflect the responses to a shock asif the variable x t showed no persistence, regardless of the value of γ . Hence, we have thatˆ R ( h ) DLM → R ( h ) ∗ for any γ . Note that these impulse response functions are the same asthose obtained with LPs ( ˆ R ( h ) LP ) when γ = 0, or when we include leads in the LPs ( ˆ R ( h ) F ).Case (iii) is shown in the red line in Figure 2. As argued in the previous subsection,when computing the impulse response with respect to ε t , we are allowing the DLMs to pickup the effect that is due to the persistence in x t . In other words, since we do not implicitlycontrol for the leads of x t but for those of ε t in the DLM, we are not taking into accountthe persistence of x t . In this case, the responses are equal to those obtained from LPs when Baek and Lee (2020) show that for autoregressive distributed lag models, setting the lag order to H is anecessary condition to achieve consistency.
This figure shows the response of a simulated outcome variable to a shock with different degrees of persistence,using DLMs . The dark blue line shows the results of estimating equation (16) assuming γ = 0 in system (13).The dashed grey line shows the same estimation when γ = 0 .
2. The red line shows the response whensubstituting x t in equation (16) by ˆ ε t , an OLS estimate of ε t (see equation (9)), where serial correlation hasbeen removed. R ( h ) ∗ ) include leads no action neededResponse with persistence ( R ( h )) no action needed replace x t with ε t γ = 0: ˆ R ( h ) DLM − per = ˆ R ( h ) LP → R ( h ). In the presence of a persistent shock, a researcher needs to determine what object to identify.Table 2 summarizes the adjustments required in LPs and DLMs depending on the choice ofthe object of interest.The researcher faces two options: to identify the response as if the shock were uncor-related ( R ( h ) ∗ ) or the response that includes the effect of persistence ( R ( h )). There arearguments in favor of both. Ultimately, deciding for one or the other may depend on whatspecific question the researcher is trying to address.Since R ( h ) ∗ can be understood as the IRF resulting from a standardized shock (so thatit becomes serially uncorrelated), it should be the desired object when the researcher wantsto establish comparisons across dynamic responses. There are at least three instances when R ( h ) ∗ can facilitate comparisons. First, a shock identified from within a model (say, a struc-tural VAR) or the innovation to a stochastic process in a DSGE model are, by construction, am.d.s. (they are non-persistent). Given the absence of serial correlation, the thought experi-ment carried out in such cases is equivalent to constructing and IRF such as the shock takesthe value of 1 on impact and 0 afterwards. Contrary to VAR-identified shocks or innovationsin a DSGE model, narratively-identified shocks may display serial correlation. If this is thecase, R ( h ) (resulting, for example, from standard LP) will identify a different object, sincethe effect of serial correlation is included in the IRFs. In this instance, R ( h ) ∗ will provide23he same macroeconomic experiment as, for example, a DSGE model. Second, R ( h ) ∗ can also be an object of interest when the researcher wants to comparethe effects of different shocks, e.g., whether fiscal or monetary policy is more effective instimulating output. For example, it may be the case that fiscal shocks tend to show morepersistence or that a given identification procedure tends to generate shocks with differentdegree of serial correlation. If the effect of persistence amounts to a non-negligible amountof the dynamic response, this could wrongly lead to the conclusion that one shock is moreeffective than the other when the true underlying cause is that the DGP of both shocks isdifferent. Since R ( h ) ∗ effectively standardizes the dynamic responses to shocks with differentdata generating processes, this would facilitate such comparison.Third, in a similar vein, R ( h ) ∗ can be useful when the researcher wants to compare theeffects of the same shock using data from different countries. This is because R ( h ) ∗ providesa standardization of the data generating processes of the shocks, which may be heterogeneousacross countries. On the other hand, R ( h ) should be the object of interest when the researcher is interestedin estimating the most likely dynamic response of a variable to a shock according to thehistorical data. This argument is similar to the one posed by Fisher and Peters (2010) andRamey and Zubairy (2018) to support the use of the cumulative multiplier (the ratio of theintegral of the output response to that of the government spending response) to evaluatethe effectiveness of fiscal policies. If we consider the effects of a monetary policy shock thatcuts the policy rate by one percentage point, it is important to note that, if that shockdisplays persistence, then the total monetary policy action (the evolution of the nominal As mentioned earlier, when x t is the shock of interest, R ( h ) ∗ is defined as the “traditional impulseresponse” in Koop et al. (1996). Consider the following example: we want to compare the effects of fiscal policy in the US (using a newsvariable) and in another country (where we have availability of an alternative news variables). Considerthe case that the news variables have different amounts of serial correlation and we obtain estimates of thegovernment spending multipliers in both countries. Could we conclude that government spending is moreeffective in one country versus the other? Potentially, both policies could be equally effective but their sourcesof identification (news variables) may have different DGPs (one with more serial correlation than other), whatleads to different multipliers. R ( h ) and R ( h ) ∗ is informative by itself, as it speaks about how much ofthe dynamic response is due to the implied DGP of the shock variable. Put differently, itinforms the researcher of a propagation mechanism: R ( h ) includes the propagation throughthe persistence of x t while R ( h ) ∗ does not.Further to this, the methods that underlie the construction of R ( h ) ∗ when using localprojections can be exported to more general uses. Hence the inclusion of leads of differentvariables can help in decomposing an IRF in different channels of propagation (where serialcorrelation is just one of them). This avenue could be particularly informative in highlightingwhat economics models can bring the dynamic responses closer to the data. In this section we use the empirical work of Ramey and Zubairy (2018) to show the quanti-tative relevance of serial correlation in an actual example. We do so by computing two typesof IRFs, R ( h ) and R ( h ) ∗ , as described above. Ramey and Zubairy (2018), building on previous work by Ramey (2011) and Owyang et al.(2013), produce a series of announces about future defense spending between 1890q1-2014q1,scaled by previous quarter trend real GDP. This series, plotted in panel D of Figure D2,has a positive autocorrelation of 18 .
4% (47.0% in the subsample after WWII). In the appendix, we consider additional applications, based on Guajardo et al. (2014), Romer and Romer(2004), Gertler and Karadi (2015), and Romer and Romer (2010). Ramey and Zubairy (2018) estimate trend GDP as sixth degree polynomial for the logarithm of GDPand multiplier by the GDP deflator. In fact, it is the use of the GDP deflator and trend GDP as a wayto scale the shocks what seems to induce the persistence. The persistence is also present when the shock isscaled by previous-quarter GDP, as in Owyang et al. (2013). This positive autocorrelation is significant at a confidence level of 90% when considering standard errorsthat are robust to the presence of heteroskedasticity and persistence (with more than one lag) for the wholesample. For the subsample starting after WWII, the autocorrelation is significant at any level. y t ) and government spending ( g t ): y t,h = β yh shock t + P X j =1 ρ zj,h z t − j + h X f =1 γ f,h shock t + f + ξ t g t,h = β gh shock t + P X j =1 ρ zj,h z t − j + h X f =1 γ f,h shock t + f + ε t , (17)where z t includes P lags of y t , g t and shock t . Note that, following the discussion in previoussections, we include h leads of the variable shock t . In particular, for each horizon h we include h leads.To replicate Ramey and Zubairy (2018)’s estimates, we set γ f,h = 0, ∀ f, h . The black,solid line in Figure 3 represents the estimated responses of output (left panel) and governmentspending (right panel) to the shock. As noted in Section 3, these dynamic responses are theequivalent to the R ( h ) as defined in equation (3) (with the only difference being that Ω t − includes now the past history of z t ). The results closely resemble those in Ramey and Zubairy(2018) (Figure 5 of their paper). Next, we allow γ f,h = 0. As discussed in Section 3, this amounts to estimating R ( h ) ∗ asdefined in equation (4). In the red lines in Figure 3, we observe that the dynamic responseschange considerably when the leads are included. For example, after two years, output andgovernment spending are 40% lower than in Ramey and Zubairy (2018)’s estimates. Thelarge observed difference between R ( h ) and R ( h ) ∗ suggests that the persistence of the newsvariable plays a non-negligible role in explaining the dynamic transmission of the fiscal shockto output and government spending.Whether to include leads or not also has implications for inference. The 95% confidence Figure D3 also replicates the original 95% confidence intervals computed using the Newey-West correction. We drop the last h observations of the sample, so that the specifications with and without leads canbe fully comparable. This does not have any discernible effect when replicating the original results fromRamey and Zubairy (2018). quarter -0.0500.050.10.150.20.250.30.35 GDP quarter -0.0500.050.10.150.20.250.30.350.40.45
GOV
Black lines show the results of estimating the system (17) without including any lead (as inRamey and Zubairy (2018)). Red solid lines represent the results of estimations when including h leadsof the Ramey and Zubairy (2018) news variable (with 95% confidence intervals). intervals when leads are included (shown in dashed lines in Figure 3) are substantially nar-rower than when they are not (grey areas in Figure D3). The latter are around 50% broaderafter two years, and more than twice as big after three years.The dynamic responses of output and government spending are informative about theexpected path of these variables after a shock. To obtain a measure of the efficiency offiscal policy (i.e., the increase of output per each dollar increase in government spending),Ramey and Zubairy (2018) use the cumulative multiplier, computed as: M t,h = P hi =1 β yh P hi =1 β gh . (18)We find that this statistic is not substantially affected by persistence of the shock (Fig-ure D4). Given that both output and government spending react similarly when including Ramey and Zubairy (2018) show that the cumulative multiplier can be obtained in one step yieldingidentical results to those obtained combining equations (17) and (18). Non-linear effects.
We now investigate whether the effect of persistence in the shockcan affect the responses in a non-linear setting, i.e., if government spending multipliers aredifferent in expansions and recessions. For this, we follow Ramey and Zubairy (2018) andestimate a series of non-linear LPs: x t + h = S t − " α A,h + P X j =1 ρ A,j,h z t − j + β A,h shock t + h X f =1 δ A,f,h shock t + f +(1 − S t − ) " α B,h + P X j =1 ρ A,j,h z t − j + β B,h shock t + h X f =1 δ B,f,h shock t + f + ξ t + h , (19)where x t is either output or government spending and S t is a binary variable indicating thestate of the economy. When S t = 1, the economy is booming and, when S t = 0, the economyis in recession, which is defined as when the unemployment rate is above the threshold of 6.5.In this setting, all the variables (and the constant), are allowed to have differential effectsduring expansions and recessions.We first replicate the non-linear responses of output and government spending duringbooms and recessions obtained by Ramey and Zubairy (2018). Hence, we estimate equa-tion (19) setting δ A,f,h = δ B,f,h = 0 ∀ f, h , which identifies R ( h ). Our results, shown inFigure 4 in black lines, resemble very closely those from the authors. Next, we repeat theexperiment accounting for potential persistence, that is, including leads of the shock (iden- Even though the multiplier does not change much when accounting for persistence, the fact that theexpected responses of output and government spending do change substantially is very relevant from a policy-maker point of view. For example, a higher response of government spending can affect other importantvariables such as public debt or future changes in tax liabilities. See Ramey (2019) for a recent summary of this debate. For example, an influential study byAuerbach and Gorodnichenko (2012) finds that government spending multipliers are higher during reces-sions using a non-linear VAR. Alloza (2018) highlights the role of the information used to define a period ofrecession, and finds that output responds negatively to government spending shocks in a post-WWII sampleunder different identification and estimation approaches. quarter -0.100.10.20.3
GDP - EXPANSION quarter -0.100.10.20.30.4
GOV - EXPANSION quarter -0.4-0.200.20.40.60.8
GDP - RECESSION quarter -0.500.51
GOV - RECESSION
Black lines show the results from system of equations (19) without including any lead (as inRamey and Zubairy (2018)). Grey areas represent 68 and 95% Newey-West confidence intervals for theseestimates. Red solid lines represent the results of estimations when including h leads of the Ramey’s newsvariable. Red dashed lines represent the 95% Newey-West confidence intervals for these estimates. tifying R ( h ) ∗ ). The results are shown in red lines in Figure 4. While relatively similar inthe case of expansions, the responses are quantitatively different during recessions. The esti-mates that include leads lie outside of the 95% confidence bands during much of the responsehorizon. The results suggest that ignoring the effect of persistence could yield responsesduring recessions that, after 2–3 years, are twice as large as the responses that account forthe effect of persistence. Or, in other words, persistence in the shock is responsible for up to50% of the dynamic transmission of the shock during recessions.In Figure 5, we show how these responses map into estimates of non-linear fiscal multi-pliers. In the case of expansions, the results do not change much depending on whether thepersistence is accounted for (red solid line, R ( h ) ∗ ) or not (black solid line, R ( h )). In eithercase, they resemble those in Ramey and Zubairy (2018) (see Figure 6 of their paper). Inrecessions, however, the results change substantially depending on whether the persistence is29igure 5: Government spending multiplier during expansions and recessions, with and with-out leads quarter -2-1.5-1-0.500.51 c u m u l a t i v e m u l t i p li e r EXP, no leadsEXP, with leadsREC, no leadsREC, with leads
The black solid and dashed lines show the cumulative multiplier during periods of expansion and recession,respectively, without including any lead (as in Ramey and Zubairy (2018)). The red solid and dashed linesshow the cumulative multiplier during periods of expansion and recession, respectively, when including leadsof the shock. controlled for or not. If it is not (black solid line), the multiplier has a negative value uponimpact and substantially falls in the following quarter to a value of -2. It becomes positivebefore the end of the first year, and fully converges to the value of the multiplier duringexpansions after six quarters. If the persistence is excluded from the dynamic responses (reddashed line), the cumulative multiplier is -1 (instead of -2) and becomes positive after thefirst year. Furthermore, the multiplier during recessions remains lower than the multiplierduring expansions for a much longer period. When the persistence is not accounted for, thisconvergence is achieved after 6 quarters, as mentioned above. However, when including leadsof the shock, this convergence is not fully reached during our considered response horizon.These results suggest that during the short and medium-run the government spending mul-tiplier could be lower during recessions than during expansions, and part of this differencemay be attributable to the presence of persistence in the shock.One of the main advantages of LPs is that they allow to accommodate non-linear settings,30s those in equation (19). This is particularly useful since, contrary to threshold VARs, LPsdo not impose any restriction on the evolution of state S t (while non-linear VARs that interactthe shock with a state dummy do assume that S t remains fixed during the response horizon).The framework explained in the previous section allows to consider additional macroeconomicexperiments that can help understand how restrictive this condition is. In particular, byincluding leads of the state S t in equation (19) we are identifying the counterfactual responseto a fiscal shock when the underlying state of the economy is not allowed to change (as inthreshold VARs). We perform this experiment and report the multipliers during booms inrecessions in green lines in Figure D5. We observe that, when the state is not allowed tochange, the multiplier during recessions is slightly higher in the short run, but essentiallyunchanged at medium and longer horizons. This exercise allows us to illustrate how the useof leads of variables in conjunction with LPs can help understand interesting counterfactualexercises and shed light on the dynamic transmission of shocks. We have shown that persistence results in the estimation of different responses when usingLPs versus traditional methods based on DLMs . For a researcher interested in the response as if the shock were not persistent, DLMs yield the desired object, but LPs need to beadapted. The opposite is true if the object of interest is the response to the shock “as it is”.Regardless of which is the thought experiment that the researcher seeks to carry out, thedifference between both types of responses is informative about how much of the dynamictransmission of a shock is due to the presence of persistence.The use of leads can be generalized to other interesting contexts, as it allows to shut downchannels of transmission. For example, one may be interested in the effects of monetarypolicy shocks on output due to a particular instrument while holding other variables (e.g.,changes to fiscal policy) constant. In the context of LPs, leads of a selected variable (e.g., tax31hanges) will deliver responses holding that variable constant. This methodology allows toseparate the direct (due to the impact through the regressor of interest) and indirect effects(due to other variables in the regression). This has often been used in the context of VARs,by imposing restrictions on the coefficients of selected impulse responses. The inclusion ofleads achieves a similar goal in LPs, hence allowing to construct interesting macroeconomicexperiments. We leave these questions for future research.
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Journal of Applied Econometrics , 29(3):497–514.37 nline AppendicesA Proofs
A.1 Proof of Proposition 1
Consider equation (6) (rewritten here for convenience): y t + h = δ h x t + ξ t + h , (A.1)where δ h = R ( h ) LP represents the impact of variable x t on y t + h (the response function).Since δ h is the linear projection coefficient of equation (A.1): δ h = cov ( y t + h , x t ) var ( x t ) . (A.2)The dynamic effect of x t on y t + h can also be obtained from DLMs as in equation (5): y t = θ x t + θ x t − + θ x t − + θ x t − + θ x t − . . . + u t . Since this expression holds ∀ t , it can be written as: y t + h = θ x t + h + θ x t + h − + θ x t + h − + θ x t + h − + . . . + θ h x t + u t , where the coefficient θ h = R ( h ) DLM represents the impulse response in period h , obtainedfrom: θ h = cov ( y t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h . (A.3)When γ = 0 in the process described by system (2), we have that x t = ε t ∼ white noise ( µ ε , σ ε ),so: θ h = cov ( y t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h = cov ( y t + h , x t ) var ( x t ) .
1n this case, δ h = θ h and LPs and DLMs yield the same responses: R ( h ) LP = R ( h ) DLM ∀ h . Note that R ( h ) DLM = R ( h ) ∗ ∀ h, γ since (under linearity): R ( h ) ∗ = ∂y t + h ∂x t (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h = cov ( y t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h . When γ = 0, equation (A.2) becomes (using the equations in system (2)): δ h = cov ( y t + h , x t ) var ( x t ) = cov ( δx t + h + u t + h , x t ) var ( x t ) = δ cov ( x t + h , x t ) var ( x t ) = δγ h , (A.4)using the expression: x t + h = γ h x t + h − X j =0 γ j ε t + h − j . (A.5)However, the dynamic response obtained from DLMs is: θ h = cov ( y t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h = δ cov ( x t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h . When h = 0, the above expression becomes θ = δ cov ( x t ,x t ) var ( x t ) = δ . For h >
0, we havethat θ h = δ cov ( x t + h ,x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h = 0. This shows that when γ = 0, we have that R ( h ) LP = R ( h ) DLM if and only if h = 0. (cid:4) A.2 Proof of Proposition 2
Consider equation (10) (rewritten here for convenience): y t + h = δ h, x t + δ h, x t +1 + ξ t + h , (A.6)where δ h, = R ( h ) F represents the impact of the shock x t on y t + h when including leadsof the former. Since δ h, is the linear projection coefficient of equation (A.6), then: δ h, = cov ( y t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 = δ cov ( x t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 . (A.7)2ote that the data generating process in system (2) considers that x t is an AR(1) so itcan be represented in terms of x t +1 (see equation (A.5)). Then, we have that DLMs andLPs with leads recover the same object: θ h = cov ( y t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h = δ cov ( x t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 = δ h, . To see this, note that in period h = 0 we have that: θ = cov ( y t , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h = cov ( y t , x t ) var ( x t ) = δ = δ , . In periods h >
0, we can rewrite equation (A.7) as: δ h, = δ cov ( x t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 = δγ h − cov ( x t +1 , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 = 0 . (A.8)Similarly, equation (A.3) becomes: θ h = cov ( y t + h , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 ,...,x t + h = δγ h − cov ( x t +1 , x t ) var ( x t ) (cid:12)(cid:12)(cid:12) x t +1 = 0 . (A.9)So we have R ( h ) F = R ( h ) DLM ∀ h , γ . And we know that R ( h ) DLM = R ( h ) ∗ , from thesection above. (cid:4) A.3 Proof of Proposition 3
Consider a version of equation (12) rewritten here for convenience: y t = ˜ θ ε t + ˜ θ ε t − + ˜ θ ε t − + ˜ θ ε t − + ˜ θ ε t − . . . + u t , (A.10)where ˜ θ = R ( h ) DLM − per represents the impact of variable ε t on y t + h . Note that ε t isnot observable but can be obtained if we know the data generating process described in Note also that the results easily generalize to cases when x t is an autoregressive process of higher order. y t on ε t and its lags,with ε t ∼ iid ( µ ε , σ ε ), we have:˜ θ h = cov ( y t + h , ε t ) var ( ε t ) (cid:12)(cid:12)(cid:12) ε t +1 ,...,ε t + h = cov ( y t + h , ε t ) var ( ε t ) = δ cov ( x t + h , ε t ) var ( ε t ) . (A.11)This expression is equivalent to equation (A.2) which implies that ˜ θ h = δ h and R ( h ) DLM − per = R ( h ) LP . To see this, substitute for x t + h in equation (A.11) using expression (A.5) and sys-tem (2): ˜ θ h = δγ h cov ( x t + P h − j =0 γ j ε t + h − j , ε t ) var ( ε t ) = δγ h cov ( x t , ε t ) var ( ε t ) = δγ h . (A.12)Note that the above expression yields the same result as equation (A.4), which showsthat ˜ θ h = δ h ∀ h . (cid:4) Additional results
B.1 Including the shocks as endogenous variables in a VAR
A researcher may consider including a shock with persistence as an endogenous variable in aVAR. Does this approach eliminate the effect of the persistence of the shock on the impulseresponses? A VAR, since it explicitly models the persistence of the shock, includes thiseffect in the estimated impulse responses and, hence, yields the same dynamic effects as LPs(contrary to what is obtained when including the shock as a distributed lag structure withina VAR). To see this in an intuitive way, consider the data generating process given by system (13)and rewritten here for convenience (with a slightly different notation): y t = ρy t − + δ x t + δ x t − + ε yt x t = γx t − + ε xt . (B.1)This process can be recast as a structural VAR of the form A Y t = B ∗ Y t − + ε t , with: − δ x t y t = γ δ ρ x t − y t − + ε xt ε yt . (B.2)An econometrician would estimate the following reduced-form VAR: Y t = BY t − + u t , (B.3)where B = A − B ∗ ; and u t = A − ε t are reduced-form residuals. Since the data generatingprocess given by equation (B.1) already incorporates restrictions on the contemporaneous Bloom (2009), Romer and Romer (2010), and Ramey (2011) are examples of studies that include shocksas endogenous variables in a VAR. These specifications are also known as hybrid VARs (see Coibion (2012)).Plagborg-Møller and Wolf (2019) formally show that VARs and LPs identify the same impulse responses.Here we illustrate that when one of the endogenous variables in the VAR is a persistent shock, this effect willbe carried over to the dynamic responses. ε xt Panel A) Response of y t to ε xt Panel B) Response of x t to ε xt LP = 0LP = 0.2VAR = 0VAR = 0.2
VAR = 0VAR = 0.2
The figure shows the VAR responses of y t (Panel A) and x t (Panel B) to ε xt estimated from (B.3), underdifferent assumptions of the persistence parameter γ : dashed orange lines for responses when there is nopersistence ( γ = 0) and dashed green lines for responses when there is persistence ( γ = 0 . γ = 0 (solid blueline) and γ = 0 . behavior of the variables, a researcher may identify the structural impulse responses bycomputing the Choleski decomposition (when x t is ordered first) of the variance-covariancematrix of reduced-form residuals u t .However, note that, when γ = 0, the response of y t to ε xt will include an effect due to thepersistence of the shock x t . Intuitively, consider the case of ρ = δ = 0. In this scenario,a researcher may be interesting in recovering a one-off shock to x t . However, the responseof y t will be given by R ( h ) V AR = δ P ∞ k =0 γ k ε xt − k , that is, the one-off shock will still haveeffects along the response horizon because of the peristence of x t (when γ = 0).To see this point in a more general way, consider the same calibration as used in the maintext (i.e., ρ = 0 . δ = 1 . δ = 1, and either γ = 0 or γ = 0 . y t and x t (the measured shock) to ε xt , shown in Figure B1. We alsoestimate the same impulse-response functions for y t using LPs, as in Section 3.3.The results illustrate that, regardless of the value γ , a VAR that considers x t as endoge-nous variable and LPs estimate the same impulse responses. When there is some persistence6n the shock x t , both the VAR (that considers x t as an endogenous variable) and LPs includea dynamic effect due to the persistence of x t .Importantly, when x t displays persistence, the response function estimated by a VARwill vary depending on whether x t is included as an endogenous variable (as shown be-fore) or an exogenous one (e.g., with a distributed lag or moving average structure, as inMertens and Ravn (2012) or Favero and Giavazzi (2012)). To show this point, Figure B2displays the response of y t when (i) x t is included as an endogenous variable in the VAR, or(ii) when estimating a regression of y t on x t and a lag of both variables. As it can be seen,considering x t as an exogenous regressor always delivers the same dynamic responses as if x t were serially uncorrelated, regardless of the actual value of γ .The discussion above highlights that the result regarding the equivalence of LPs andDLMs under no serial correlation of the shock can be generalized to multivariate settingsof VARs with the shock included as an endogenous variable (in the case of LPs) or as anexogenous variable (in the case of DLMs). 7igure B2: Responses when the shock is included as an endogenous or exogenous variable ina VAR VAR = 0VAR = 0.2VAR-X = 0VAR-X = 0.2
The figure shows the VAR responses of y t to ε xt estimated from (B.3) under different assumptions of thepersistence parameter γ and two different specifications: solid lines display the responses when the shock isincluded as an endogenous variable in the VAR and dashed line shows the responses when the same shock isincluded as an exogenous variable in the VAR. .2 Local projections with instrumental variables Recently, there has been an increased attention to the use of external sources of variation asinstruments in LPs. In this section, we investigate how persistence may affect the estimationof dynamic effects when using instrumental variables in local projections (LP-IV).Stock and Watson (2018) provide the conditions under which a researcher can exploitexternal variation to estimate impulse response functions. A valid instrument z t shouldbe both relevant and contemporaneously exogenous, that is, z t should not be correlated toany shock in the system except with the one that the researcher is interested in. Lastly,Stock and Watson (2018) impose a restriction called lead/lag exogeneity, which implies thatthe instrument should not be correlated with any lead or lag of any of the shocks in thesystem.Consider the following data generating process: y t = βg t + u t u t = m t + a t g t = λx t + (1 − λ ) m t (B.4) z t = x t + ν t x t = γx t − + ε t , where a t , ν t , m t and ε t follow independent N (0 ,
1) distributions. A researcher may beinterested in estimating the dynamic effects of variable g t on y t (e.g., the effects of governmentspending on output). However, g t is endogenous due to the presence of an omitted variable m t .The researcher may have the availability of an instrument z t , which is contemporaneouslyexogenous by construction and relevant when λ = 0. This instrument, since it depends See Ramey and Zubairy (2018) for an example. Related to this, Ramey (2016) discusses the distinctionbetween shock, innovation, and instrument. Barnichon and Mesters (2020) show how independently identifiedshocks can be used as instruments to estimate the coefficients of structural forward looking macroeconomicequations. x t , displays persistence when γ = 0. When there is persistence inthe instrument (and the shock), the lead/lag exogeneity condition mentioned above is notsatisfied. To illustrate this point, we simulate system (B.4) setting β = 2 (and differentvalues of λ and γ ) for 100 million periods, and estimate the dynamic effects of g t on y t usingLP.We first consider the case of γ = 0 and λ = 1, that is, there is no problem of endogeneityor persistence. The estimated effect of g t on y t recovered by LPs is represented by a solidblue line in Panel A of Figure B3. As expected, the contemporaneous impact of governmentspending on output is equal to 2. When considering λ = 0 . γ = 0), LPs that employ OLS will deliver biased estimates of the contemporaneous effect of g t (solid red line). The difference between the red and the blue lines in the first period is ameasure of the endogeneity bias. The problem of endogeneity can be addressed by using z t as an instrument for g t to recover the exogenous variation in government spending (given by x t ). This result (still considering γ = 0) is represented by the dashed grey line in Panel Aof Figure B3, which shows how the use of LP-IV can overcome the presence of endogeneity,delivering a response function identical to the benchmark case without omitted variablesbias.Next, we repeat the previous exercise but now we allow for persistence in the instrument(due to persistence of the shock); in particular, we set γ = 0 .
2. The results are shown in PanelB of Figure B3 (we still represent, in solid blue line, the benchmark case of γ = λ = 1 forreference). When there is endogeneity and persistence, LPs estimates of the dynamic effectsof g t are affected by both an endogeneity bias on impact, and by the effect of persistence inthe instrument during the rest of the response horizon (as shown in the previous section).This result is displayed by the solid red line in Panel B of Figure B3, which is differentfrom zero after impact. Now consider estimating the dynamic effects using LP-IV with aninstrument z t (that displays persistence). The results (dashed green line) show that the useof the instrument addresses the problem of endogeneity (on impact, the effect from the LP-IV10igure B3: LPs with instrumental variablesPanel A) γ = 0 Panel B) γ = 0 . No endogeneityOLSIV
No endogeneity, =0OLS, =0.2IV, =0.2IV with leads, =0.2
This figure shows the response of a simulated outcome variable to a shock using local projections withinstruments, with an underlying DGP given by system (B.4) and calibrated for different degrees ofpersistence in the shock ( γ = 0 in panel A and γ = 0 . estimates is able to recover the true effect of β = 2). However, the dynamic effect from therest of the response horizon still reflects the presence of persistence.As discussed above, persistence in the instrument violates the lead/lag exogeneity condi-tion. Stock and Watson (2018) state that, in general, this condition could be satisfied by theinclusion of further controls in the LP-IV regression. If the source of persistence is strictlyrestricted to the instrument, Stock and Watson (2018) show that the lead/lag exogeneitycondition could be reestablished by including lags of the instrument. However, in cases likesystem (B.4), where the instrument inherits its persistence from the shock, lags of the in-strument will not satisfy the lead-lag exogeneity condition. We build on the intuition fromStock and Watson (2018) and adapt it to the problem of persistence by including leads ofthe instrument in the set of exogenous variables in the LP-IV estimates. The results, shownin dashed grey lines in Panel B of Figure B3, corroborate this intuition: despite the presenceof both endogeneity and persistence, enhancing the LP-IV estimates with leads of the shock11llows to recover the dynamic effects as if the instrument were not serially correlated, i.e., R ( h ) ∗ .In sum, the presence of persistence can potentially violate the lead-lag exogeneity as-sumption, invalidating inference under LP-IV. The solution to reestablish this condition willdepend on the source of persistence in the model. When the instrument inherits its persistencefrom the shock, our proposed solution builds on the general intuition from Stock and Watson(2018), showing that the inclusion of leads of the instrument can deliver valid inference underLP-IV. B.3 Alternative simulation: using the persistence from an actualshock
In this subsection we compute the impulse response of a simulated variable y t to a shock x t with the following DGP: y t = ρy t − + B x t + B x t − + u t , (B.5)where x t is the actual government spending shock from Ramey and Zubairy (2018) as shownin Panel D of Figure D2. u t is a random variable following u t ∼ N (0 , ρ = 0 . B = 1 .
5, and B = 1.Equation (B.5) is simulated for 497 periods (the length of Ramey and Zubairy (2018)’sshock), and we then compute the relevant IRFs. We repeat this process 10,000 times, andcompute the average impulse responses across all repetitions. The results are shown inFigure B4.When computing the dynamic response with standard LPs (i.e., without including anylead), the estimates diverge from the expected response when the shock has no persistence(distance between red and dark blue lines in Figure B4). Adding one lead improves the esti-mates, bringing the impulse-response into line with the theoretical response in the first period(green line). The accuracy of the impulse-response converges to the theoretical response when12igure B4: Simulated responses using LPs with persistence from an actual shock True IRFLP - 0 leadsLP - 1 leadLP - 20 leads
This figure shows the response of a simulated outcome variable to the government spending shock fromRamey and Zubairy (2018). The dark blue line is the theoretical impulse-response to a shock that showsno persistence. The red line shows the LPs estimation of the impulse-response to the Ramey and Zubairy(2018) without including any lead. Green line repeats the same estimation adding one lead. Dashed greyline shows the response when including 20 leads. more leads are included. When we include as many leads as periods in the response horizon(20), the dynamic response estimated from LPs using the actual shock (with persistence) isequivalent to the response to a non-serially correlated shock (dashed grey line).
B.4 Responses in LPs using variables adjusted for serial correla-tion
An apparent potential alternative to the use of leads proposed in the main text might be toadjust the shock x t so that it does not display persistence (e.g., by regressing x t on its ownlags and using the resulting residual). Once the persistence is removed, one may expect thedynamic responses not to include the effect due to the persistence of the shock. However,this is not the case in a LPs setting, as we show next.Consider the case where we obtain a variable adjusted for serial correlation: ε t = x t − x t − , as shown in equation (9). Then, ε t can be used as substitute of the original shock x t .Assuming B = 0 in system (13) (for simplicity) consider the following series of LPs: y t + h = ρ h y t − + λ h ε t + ξ t + h . (B.6)To obtain the dynamic responses of y t to the shock ε t (adjusted for persistence), werewrite the first equation in system (13) as a function of ε t and compute the relevant partialderivatives. For the cases of h = 0 and h = 1 these are: λ = ∂y t +1 ∂ε t = B λ = ∂y t +1 ∂ε t = ρ ∂y t ∂ε t + B ∂x t +1 ∂ε t = ρB + B γ = B ( γ + ρ ) . (B.7)That is, even after correcting for the persistence in shock x t , conventional LPs yieldresponses R ( h ), i.e., still containing the effect of persistence of the shock.While this result may seem counter-intuitive, it arises from the fact that LPs do not havean explicit dynamic structure as a DLM . Hence, removing the persistence from x t does noteliminate its effect on y t +1 , y t +2 , etc.To empirically show this point, we simulate series of y t and x t following system (13) andthe calibration used in Section 3.3 (we now allow B = 0). We then obtain the residuals ˆ ε t as an estimate of ε t described above and estimate the following equation: y t + h = ρy t − + λ h, ˆ ε t + λ h, ˆ ε t − + ξ t + h . (B.8)Results are shown in Figure B5. The simulations corroborate the above results and wefind that the use of a variable adjusted for serial correlation as ˆ ε t in equation (B.6) fails toretrieve an impulse response as the one obtained when γ = 0 in equation (13).14igure B5: Simulated responses using ˆ ε t This figure shows the response of a simulated outcome variable to a shock with different degrees of persistence.The dark blue line shows the results of estimating equation (B.8) assuming γ = 0 in equation (13). The redline shows the same estimation when γ = 0 .
2. The dashed grey line shows the response when including apredicted regressor where persistence has been removed as explanatory variable (as in equation (B.6)). Additional empirical applications
C.1 Guajardo et al. (2014)
In this subsection we explore the relevance of our results in the context of episodes of fiscalconsolidation, as produced in Guajardo et al. (2014). The authors employ a panel of OECDeconomies to analyze the response of economic activity to discretionary changes in fiscalpolicy motivated by a desire to reduce the budget deficit and not correlated with the short-term economic outlook. As mentioned in Table 1, this measure of fiscal changes exhibitssome degree of persistence. To explore the effects of persistence in this context, we compute the responses estimatinga series of LPs: y i,t + h = µ h,i + λ h,t + β h, shock i,t + h X f =1 β h,f shock i,t + f + β h,s X i,t + ξ i,t + h , (C.1)where y i,t is a measure of economic activity (either private consumption or real GDP), µ h,i and λ h,t represent country and time fixed effects, respectively, and X it is a vector of variablesthat includes a lag of the shock, output, and private consumption, and a deterministic trend.In our setting, responses to the fiscal shocks are given by the estimates of coefficients β h, fordifferent horizons h .We first estimate equation (C.1) by setting β h,f = 0 ∀ h, f . The results, shown in blacksolid lines in Figure C1 qualitatively replicate the benchmark results of Guajardo et al.(2014), with a fiscal consolidation shock significantly reducing output during the first 6 A detailed description of these shocks can be found in Devries et al. (2011). Regressions of the fiscal consolidations measure (expressed as % of GDP) on its own lags and includingtime and country fixed effects reveal persistence in the previous two or three years (depending on the numberof lags included). Intuitively, some degree of persistence is expected in these series since they often involvedmulti-year plans, as noted in Alesina et al. (2015) and Alesina et al. (2017). Note that Guajardo et al. (2014) do not construct responses using LPs and hence their computed re-sponses do not show the effect of persistence, as noted in the previous section. There are, however, a numberof studies that employ their fiscal consolidations dataset with LPs (see, for example, Barnichon and Matthes(2017) or Goujard (2017)). year -2-1.5-1-0.500.5 pe r c en t GDP
Black lines show the results from equation (C.1) with output as dependent variable and setting β h,f = 0 ∀ h, f ,i.e., without including any leads of the shock. Grey areas represent 90% Newey-West confidence intervalsfor these estimates (as in Guajardo et al. (2014)). Red solid lines represent the results of estimations whenallowing β h,f = 0 and including h leads of the consolidations variable. years. Next, we estimate equation (C.1) but allow β h,f = 0 (red lines in Figure C1). Threepoints are worth noting regarding these results. First, when accounting for the effects ofpersistence, the point estimates are smaller in absolute value. On average, the new responsesare 35% lower during the first six years after the shock. Two years after a fiscal consolidation,output is almost 60% smaller when accounting for persistence (-0.2 vs -0.5).Second, when including leads of the shock, the estimates are more precise, which translatesinto smaller confidence intervals (set at 90% as in the original paper of Guajardo et al.(2014)). During the first six years, these intervals are about 20% smaller on average in thespecifications that include leads of the shock. Guajardo et al. (2014) focus on the dynamic effects of output and private consumption during 6 yearsafter the shock. We also compute results for private consumption, shown in Figure D6 in Appendix D. Asin the original paper, we also find a significant reduction in this variable during the first 6 years after aconsolidation shock. R ( h ) vs. R ( h ) ∗ . C.2 Romer and Romer (2010)
What happens when including leads of non-persistent shocks? In this section we conducta placebo test based on Romer and Romer (2010), who investigate the output effects oflegislated tax changes. Romer and Romer (2010) identify exogenous changes in tax revenuesby classifying fiscal reforms according to their motivation (i.e., whether or not they are theresponse to changing macroeconomic conditions). As discussed in Section 2, it is the onlyshock considered here for which we unambiguously fail to reject the null hypothesis of nopersistence. Hence, the inclusion of leads of the shock should not have a discernible impacton the estimation of dynamic responses. Beyond corroborating the previous statement, thissubsection shows that the unnecessary inclusion of leads does not negatively affect inferencein this application.We estimate the response of output to exogenous tax changes following Romer and Romer(2010). We adapt the original estimation from the authors to the LPs setting: y t + h − y t − y t − = β h, shock t + h X f =1 β h,f shock t + f + ξ t + h . (C.2)In our first exercise, we set β h,f = 0 ∀ h, f in equation (C.2) to replicate the results fromRomer and Romer (2010). The results are shown Figure C2 (black lines). The response of Adding controls such as lags of output or the own shock do not affect the obtained results shown next. quarter -5-4-3-2-10123 pe r c en t GDP
Black solid line shows the responses to a tax shock estimated from equation(C.2) with β h,f = 0, i.e., withoutincluding any lead. Grey areas represent 68 and 95% Newey-West confidence intervals for these estimates.Red solid line shows the responses to a tax shock estimated from equation (C.2) with β h,f = 0 and including h leads of the shock. Red dashed lines represent 95% Newey-West confidence intervals for these estimates. output is similar to that in Romer and Romer (2010): it falls persistently after a tax hike of1% of GDP, with a peak effect reached in the 10th quarter. Next, we allow for β h,f = 0. The results, shown in Figure C2 (red lines), suggest thatthe inclusion of leads does not significantly affect the results. The point estimations withand without leads of the shock overlap each other for most of the response horizon and onlydiverge slightly during the quarters 8 to 11th.While, given the results of Table 1 we should not expect a change in the point esti-mates (which we have corroborated) the same cannot be say about issues regarding infer-ence. However, Figure C2 shows that confidence bands are not distinguishable between bothspecifications during the first seven quarters and differ only slightly afterwards. The difference with the original estimations from Romer and Romer (2010) are only quantitative: thepeak tax multiplier is about 3 in the 10th quarter. Our estimations suggest a peak multiplier of 2.25 alsoreached in the same quarter.
19n sum, this placebo exercise is reassuring in that the inclusion of leads only matters whenthe explanatory variable displays some persistence. These results suggest that including leadsin LPs is a conservative way to address the effects of persistence when there is a suspicionthat the shock is persistent and the researcher wants to identify R ( h ) ∗ . C.3 Romer and Romer (2004)
We now consider the measure of monetary policy shocks produced by Romer and Romer(2004). The authors identify exogenous monetary policy changes following a three-step pro-cedure. First, they follow narrative methods to identify the Federal Reserve’s intentions forthe federal funds rate around FOMC meetings. Second, they regress the resulting measure onthe Federal Reserve’s internal forecasts (Greenbook) to account for all relevant informationused by the Fed. Lastly, the series is aggregated from FOMC frequency to monthly frequency.As shown in Table 1, the resulting measure displays some degree of persistence. In-terestingly, the correlogram of the series seems to show a pattern consistent with negative persistence (see Panel E in Figure D1). This implies that standard LPs that do not accountfor persistence in the shock will identify R ( h ).Romer and Romer (2004) estimate the response of output to the monetary policy shockusing a lag-distributed regression of the log of industrial output and the measure of monetarypolicy shocks. Here, we adapt the estimation to a LPs setting by following the exact dataand specification from Ramey (2016) (adapted in turn from Coibion (2012)) for the originalsample of 1969m3-1996m12): y t + h = β h, shock t + θ h ( L ) x t + β h,f h X f =1 shock t + f + ξ t + h , (C.3)where y t is either the federal funds rates, the log of industrial production, the log of consumerprice index, the unemployment rate, or the log of a commodity price index, and shock t is See Alloza and Sanz (2020) for another example that adds leads to LPs using a non-persistent shock.Similarly to the evidence provided in this section, they also show that adding leads does not affect inference. The degree of persistence is higher when using the updated series produced by Coibion (2012). x t , with two lags and the contemporaneous values of all dependentvariables, and two lags of the shock. By including the contemporaneous values of the thedependent variables, we are implementing the recursiveness assumption often used in VARsto identify monetary policy shocks. The results of estimating equation (C.3) when we set β h,f = 0 ∀ h, f are shown in solidblack lines (with 90% confidence bands) in Figure C3. Since we employ the same dataand specification, they replicate the results from Ramey (2016) (Figure 2B). Ramey (2016)argues that the responses using LPs show more plausible dynamics than those obtained froma standard VAR (a persistent fall in industrial output and a rise in unemployment thatslowly converge to 0). The drop in output after a monetary shock is broadly consistent withthe original results from Romer and Romer (2004) but there are, however, two importantdifferences. First, the trough in the response of output is reached after two years. In theestimates of Figure C3 and Ramey (2016), the trough is reached after a first year and lasts forabout twelve months with a slight rebound in between. Second, although both results refer tothe same impulse (a realization of the policy measure of one percentage point), the magnitudeof the output fall in the original estimates of Romer and Romer (2004) is substantially biggerthan when using LPs (-4.3 vs -1.7).Next, we investigate whether accounting for the persistence in the shock has an effect onthe dynamic responses. We re-estimate equation (C.3), but allowing β h,f = 0. The resultsare shown in red lines in Figure C3. We observe that the dynamics of output are closer tothe original estimates of Romer and Romer (2004): a continuous drop in output that reachesthe trough after the second year. Furthermore, the magnitude of the fall is now substantiallyhigher (-3.1) and closer to the results from Romer and Romer (2004). Another noticeabledifference is that the effects on unemployment and the initial positive reaction on prices (the This assumption implies that the monetary shock does not affect macroeconomic variables (such asoutput, prices, employment...) contemporaneously, and monetary variables (e.g., money stock, reserves...)do not affect the federal funds rates within a month. See Christiano et al. (1999) for further details. Lateron, we show estimates that relax this assumption (Figure C4). month -3-2-10123 pe r c en t FEDERAL FUNDS month -4-3-2-1012 pe r c en t INDUSTRIAL PRODUCTION month -0.500.51 pe r c en t UNEMPLOYMENT month -4-3-2-101 pe r c en t CPI
Black solid lines refer to a benchmark specification that preserves the recursive assumption and does notinclude leads of the shock. Grey areas show 90% Newey-West confidence intervals. Red solid lines include h leads of the monetary shocks. price puzzle ) are now larger. All in all, the results from Figure C3 suggest thataccounting for the persistence of the monetary policy shock can lead to larger estimates ofthe dynamic responses.Ramey (2016) also investigates the role of the recursiveness assumption in the dynamicresponses (the inclusion of contemporaneous values for some variables in the LPs estimationto replicate the identification in a VAR). She finds that relaxing this assumption results inweird dynamics of unemployment in the short run. We replicate these results by droppingthe contemporaneous values in x t in equation (C.3) and setting β h,f = 0. We indeed find thatunemployment rate significantly drops in the first months after a monetary policy contraction(black solid lines in Figure C4). We investigate whether these strange dynamics may be theresult of the persistence in the monetary policy shock. We estimate again equation (C.3)relaxing both the recursiveness assumption and allowing β h,f = 0. The results (red solidlines in Figure C4) are very similar to those from Figure C3. Interestingly, unemploymentresponds positively to the monetary policy contraction.23igure C4: Responses to monetary policy shock from Romer and Romer (2004) with norecursive assumption, with and without leads month -3-2-10123 pe r c en t FEDERAL FUNDS month -4-3-2-1012 pe r c en t INDUSTRIAL PRODUCTION month -0.500.511.5 pe r c en t UNEMPLOYMENT month -4-3-2-1012 pe r c en t CPI
Black solid lines refer to a benchmark specification that relaxes the recursive assumption and does not includeleads of the shock. Grey areas show 90% Newey-West confidence intervals. Red solid lines include h leads ofthe monetary shocks. C.4 Gertler and Karadi (2015)
In this section we explore another application of the effects of monetary policy shocks basedon Gertler and Karadi (2015). The authors identify exogenous changes in monetary policyby looking at variations in the 3-month-ahead futures of the federal funds within a 30-minutewindow of a FOMC announcement. By relying on this identification scheme, rather than onstandard timing assumptions (e.g., Christiano et al. (1999)), the authors are able to explorethe effects on measures of financial market frictions or other variables that are often assumedto be contemporaneously invariant to a monetary policy shock.As shown in Table 1, this measure of monetary policy shock displays some persistence.This was first noted by Ramey (2016), who highlights that the procedure followed by Gertler and Karadi(2015) to convert FOMC shocks (expressed at FOMC frequency) to monthly frequency in- This scheme is often denoted as High-Frequency Identification (HFI). See Ramey (2016) for a comparisonwith other identification procedures. Gertler and Karadi (2015) embedded the identified monetary policy shocks in a VAR,using the measure of monetary policy surprises as an instrument of the residuals in the VAR.Here we explore what consequences the persistence of the shock might have if the researcherwere to use standard LPs (and estimate R ( h )).To do so, we implement the following specification, suggested by Ramey (2016): y t + h = β h, shock t + θ h ( L ) x t + β h,f min { h, } X f =1 shock t + f + ξ t + h , (C.4)where y t is either the 1-year government bond rate, the log of industrial production, the excessbond premium spread from Gilchrist and Zakrajek (2012), or the log of consumer price index,and shock t is the measure of monetary policy shocks from Gertler and Karadi (2015). Theregressions also include a set of controls x t , with two lags and the contemporaneous values ofall dependent variables, and two lags of the shock. Following Ramey (2016), we estimateequation (C.4) for a sample of 1991m1-2012m6. Given this reduced sample, we limit thenumber of leads introduced in the estimation to a maximum of 12 (i.e., we use h leads for h <
12 and 12 leads for longer horizons). The results of these estimations are shown in Figure C5 for two cases: setting β h,f = 0 ∀ h, f (black solid lines with 68 and 90% confidence intervals) and allowing β h,f = 0 (red In particular, Gertler and Karadi (2015) cumulate the surprises on any FOMC days during the last 31days, effectively introducing a first-order moving-average structure. This is a variation of the procedurefollowed by Romer and Romer (2004) and that also results in a measure of monetary policy shocks thatdisplays persistence. Note that while the inclusion of lags of the shocks are meant to account for persistence in the shock, ouranalysis from Section 3 shows that they are not effective for this role. In our results, the inclusion of lags ofthe shock did not have any noticeable effect. We keep them here in order to replicate the results from Ramey(2016). Gertler and Karadi (2015) proceed in two steps: they first estimate the dynamic coefficients and residualsfrom a VAR during the period 1979-2012. Then they estimate the contemporaneous effects of monetary policyusing both the residuals from the previous step and the monetary policy instrument in a proxy VAR during1991-2012. We also preserve the sample at the end of the period until 2012m06 (which will be otherwise reducedwhen including leads) by considering values of the leads of the shock equal to 0 for the last 12 periods.Although this is not important for our results, it allows us to compare our estimates to those from Ramey(2016). Alternatively, Ramey (2016) concludes that these differences may be due to the fact that the reduced-formparameters (used to construct the impulse responses) are estimated for a longer sample (1970-2012 insteadof 1991-2012) or to potential misspecification of the original VAR estimates due to the rising importance offorward guidance, which may lead to a problem of non-fundamentalness in the VAR. month -202468 pe r c en t ONE-YEAR RATE month -0.0500.050.10.150.20.25 pe r c en t INDUSTRIAL PRODUCTION month -4-2024 pe r c en t EXCESS BOND PREMIUM month -0.06-0.04-0.0200.020.04 pe r c en t CPI
Black solid lines refer to a benchmark specification that does not include leads of the shock. Grey areas show68 and 90% Newey-West confidence intervals. Red solid lines include h leads of the monetary shocks up tohorizon h = 12, after then, the number of leads is kept to 12. Additional Tables and Figures
Table D.1: Robustness: different lag structures for tests
The columns report the values of a Box and Pierce (1970) test (with Ljung and Box (1978) correction)including different lags. P-values are shown in brackets. in Arezki et al. (2017) and Guajardo et al. (2014)is tested using a generalized version of the autocorrelation test proposed by Arellano and Bond (1991) thatspecifies the null hypothesis of no autocorrelation at a given lag order. − . − . − . . . . A u t o c o rr e l a t i on s o f s ho ck − . − . . . . A u t o c o rr e l a t i on s o f s ho ck Panel C: Gertler and Karadi (2015) Panel D: Ramey and Zubairy (2018) − . − . . . . . A u t o c o rr e l a t i on s o f s ho ck − . . . . . A u t o c o rr e l a t i on s o f s ho ck Panel E: Romer and Romer (2004) Panel F: Romer and Romer (2010) − . − . . . . A u t o c o rr e l a t i on s o f s ho ck − . − . . . . A u t o c o rr e l a t i on s o f s ho ck
95% confidence intervals computed using Bartlett’s formula for MA(q) processes. date -1.5-1-0.500.511.52 pe r c en t date -2-1.5-1-0.500.511.522.53 pe r c en t Panel C: Gertler and Karadi (2015) Panel D: Ramey and Zubairy (2018) date -0.3-0.25-0.2-0.15-0.1-0.0500.05 r a t e date -0.2-0.100.10.20.30.40.50.60.7 pe r c en t Panel E: Romer and Romer (2004) Panel F: Romer and Romer (2010) date -3.5-3-2.5-2-1.5-1-0.500.511.52 pe r c en t age po i n t s date -2-1.5-1-0.500.51 pe r c en t quarter -0.100.10.20.30.40.5 GDP quarter -0.100.10.20.30.40.50.60.7
GOV
Black lines show the results of estimating the system (17) without including any lead (as inRamey and Zubairy (2018)). Grey areas represent 68 and 95% Newey-West confidence intervals for these es-timates. Red solid lines represent the results of estimations when including h leads of the Ramey and Zubairy(2018) news variable (with 95% Newey-West confidence intervals). quarter c u m u l a t i v e m u l t i p li e r Black lines show the cumulative multiplier without including any lead. Red solid lines represent the estimatesof the cumulative multiplier when including a number of leads of the Ramey and Zubairy (2018)) newsvariable that increase with the response horizon. quarter -2-1.5-1-0.500.51 c u m u l a t i v e m u l t i p li e r EXP, no leadsEXP, with leadsEXP, with leads (fixed state)REC, no leadsREC, with leadsREC, with leads (fixed state)
The black solid and dashed lines show the cumulative multiplier during periods of expansion and recession,respectively, without including any lead (as in Ramey and Zubairy (2018)). The red solid and dashed linesshow the cumulative multiplier during periods of expansion and recession, respectively, when including leadsof the shocks and the state. Green solid and dashed lines refer to estimates of the expansion and recessionmultipliers, respectively, when including leads of the shock and the regime. year -1.6-1.4-1.2-1-0.8-0.6-0.4-0.200.2 pe r c en t CONSUMPTION
Black lines show the results from equation (C.1) with private consumption as dependent variable and setting β h,f = 0, i.e., without including any lead of the shock. Grey areas represent 90% Newey-West confidenceintervals for these estimates (save interval as reported in Guajardo et al. (2014)). Red solid lines representthe results of estimations when allowing β h,f = 0 and including h leads of the consolidations variable.leads of the consolidations variable.