Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
74,624 characters · 11 sections · 142 citation commands
Dynamic Effects of Persistent Shocks
Estimating the impact of economic shocks is a crucial aspect of macroeconomics. To identify economically meaningful shocks, the literature has traditionally relied on systems of equations coupled with restrictions implied by economic theory. Recently, researchers are increasingly using narrative identification, e.g., looking at written official documentation or newspapers and exploiting arguably exogenous variation in these series.\footnote{See romer2004measure, romer2010macroeconomic, or ramey2018government for prominent examples of narrative identification.} While its focus on identifying exogenous variation is appealing, the lack of restrictions in narrative methods yields objects with less standard time series properties.
In this paper, we analyze how the presence of persistence in narrative shocks affects the identification and estimation of their dynamic effects, providing empirical researchers with methods and guidance to deal with this issue.\footnote{Throughout the paper we use the term persistence as a phenomenon captured or reflected by serial correlation, a testable condition. We use both terms interchangeably. }
We begin by showing that many narrative shocks used by prominent literature are serially correlated. In particular, we systematically test for serial correlation in eight shocks used in leading economics journals. We find evidence of serial correlation in seven of them. The presence of persistence in the shock does not necessarily preclude these variables from being categorized as “shocks” following standard definitions of aggregate shocks. More concretely, according to ramey2016macroeconomic, a shock should represent unanticipated movements. What this condition implies is that shocks are unforecastable, i.e., they are forecast errors. In particular, 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 therefore could be serially correlated. However, serial correlation poses additional challenges for the identification 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). Hence, a researcher may want to identify two objects of interest: the response as if the shock were uncorrelated, i.e., to a counter-factual serially uncorrelated shock ($\mathcal{R}(h)^{*}$), or the response to the shock as it is, i.e., including the effect of persistence in the IRF ($\mathcal{R}(h)$). Deciding for one or the other depends on what specific question the researcher is trying to address. On the one hand, $\mathcal{R}(h)^{*}$ allows to compare effects with those obtained from a theoretical or empirical model, and facilitates comparisons across different types of shocks (e.g., monetary versus fiscal shocks) or across countries. On the other hand, $\mathcal{R}(h)$ is more appropriate if the researcher is interested in evaluating the most likely dynamic response of a variable to a shock based on historical data. Regardless of which object is preferred by the researcher, the difference between $\mathcal{R}(h)$ and $\mathcal{R}(h)^{*}$ is informative about how much of the dynamic transmission of a shock is due to the presence of persistence.
We consider the two most popular methods to estimate impulse responses when a shock has already been identified (e.g., using narrative methods). These are local projections (LPs) (jorda2005estimation) and distributed lag models (DLMs).\footnote{By DLMs we refer to single-equation regressions of an outcome variable against the contemporaneous value and lags of the shock with or without an autoregressive component. These methods are also known as truncated moving average regressions. These specifications are frequent in the applied literature---see, e.g., romer2004measure, cerra2008myth, romer2010macroeconomic, alesina2015output, arezki2017news, and coibion2018cyclical.} We show that, if there is no serial correlation, the two methods identify the same object. However, we demonstrate that this equivalence breaks down in the presence of serial correlation. In this case, LPs identify $\mathcal{R}(h)$ while DLMs regressions identify $\mathcal{R}(h)^{*}$. The intuition is that LPs compute the response at horizon $h$ by regressing the outcome variable in $t+h$ against the shock in time $t$. Since the standard setting does not account for how the shock evolves between $t$ and $t+h$, the responses include two components: an economic effect (the economic impact of the shock on the endogenous variables) and an effect that exclusively depends on the degree of serial correlation 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 both estimating methods to obtain the desired object of interest. Consider a researcher who wants to use LPs and is interested in identifying $\mathcal{R}(h)^{*}$. As mentioned, if she runs standard LPs with a persistent shock, she will identify $\mathcal{R}(h)$ instead. Perhaps surprisingly, the most obvious solution of including lags of the shock will not address this issue. However, we show that, by including leads of the shock, she will recover $\mathcal{R}(h)^{*}$. Likewise, we show how standard DLMs can be adapted so that they identify $\mathcal{R}(h)$.
To illustrate how our methods work, we consider an actual empirical application, which also serves to assess the quantitative relevance of persistence in a real case by comparing estimates of $\mathcal{R}(h)$ and $\mathcal{R}(h)^{*}$. In particular, we consider ramey2018government's LPs estimation of the dynamic effects to a shock constructed from news about future changes in defense spending. We find that, after two years, the responses that exclude the effect of persistence in the shock are about 40% lower than the original Ramey and Zubairy (2018)'s estimates. The effect of serial correlation also seems to have an effect on the short-run response of fiscal multipliers during recessions. In the appendix, we consider additional applications, based on guajardo2014expansionary, romer2004measure, gertler2015monetary, and romer2010macroeconomic. Overall, we find that how persistence is treated can have a sizable impact on the estimated effects.
The results of this paper generalize in at least three important aspects. First, the results of the (lack of) equivalence between LPs and DLMs when the shock is persistent carry over to multivariate settings popularly used in the empirical literature. Building on a result by plagborg2018local, we show that the dynamic response from a VAR with the shock embedded as an endogenous variable is equivalent to that of a VAR with the shock included as an exogenous variable only when that shock has no serial correlation.\footnote{This result arises because a VAR with a shock as an exogenous variable (often known as VAR-X) can be seen as multivariate generalization of a DLM (see mertens2012empirical or favero2012tax for examples of VAR-X specifications). Furthermore, plagborg2018local show that, under some assumptions, LPs are equivalent to a VAR when the shock is included as an endogenous variable (as in bloom2009uncertainty or ramey2011identifying).} We believe this result has relevant practical implications for applied macroeconomic researchers. Second, we also show that our results generalize to specific contexts where a researcher employs an instrument in a LP setting (also known as LP-IV). Lastly, a researcher interested in using LPs to uncover the dynamic relations of two variables may be interested in including leads of a third variable to construct counterfactual responses as if the behavior of that third variable had remained constant over the response horizon. This can be seen as the LP counterpart of constructing counterfactual responses in a VAR that allow to separate a direct effect of a regressor on a dependent variable from other indirect effects. This procedure has been frequently used in the empirical VAR literature.\footnote{See, for example, bernanke1997systematic, sims2006does, or bachman2012confidence. In recent research, cloyne2020decomposing propose an alternative method based on a Blinder-Oaxaca-type decomposition.}
Our paper makes four contributions to the literature. First, we formally and systematically test for the presence of serial correlation in shocks used by previous work. Although the issue of persistence in shocks has been noted before,\footnote{ramey2016macroeconomic finds that the time aggregation required to convert the shock in gertler2015monetary to monthly frequency, inserts serial correlation. miranda2018transmission corroborate this finding, by regressing the shock on four lags and testing their joint significance. They also find that other measures of monetary shocks such as romer2004measure exhibit serial correlation.} we believe we are the first to formally and systematically test for serial correlation in prominent narratively-identified shocks.
Our second contribution is to show that, while both LPs and DLMs identify the same object if the shock is serially uncorrelated, this equivalence breaks down in the presence of persistence. plagborg2018local prove that LPs and VAR methods identify the same impulse responses when both methods have an unrestricted lag structure. This result formalizes some of the examples provided in ramey2016macroeconomic, which implies that different identification schemes in a VAR setting can be implemented in a LP context. Our result builds on a different premise: we consider the cases where the shock has already been identified using narrative 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 equivalence when there is persistence, providing applied researchers with a menu of options to identify their desired object of interest. In this regard, our method of adding leads to LPs is related to the tradition in factor analysis by geweke1981maximum and on the DOLS estimation of cointegration vectors (stock1993simple). dufour1998causality introduce leads in some of their IRFs to study causality at different horizons. faust2011efficient find that including ex-post forecast errors results in an accuracy improvement when forecasting excess bond and equity returns. More recently, teulings2014economic find that estimating dynamic effects of a dummy variable (e.g., banking crisis) in a panel data context with fixed effects and LPs suffers from a negative small-sample bias, since the estimation of the fixed effect picks up the value of future realization of the dummy variable. The authors show that this bias is attenuated either by increasing the sample size or by including future realizations of the dummy variable over the response horizon.\footnote{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 persists asymptotically, increasing the sample does not reduce the LP-DLM difference. Additionally, this difference is not necessarily negative, but will depend on the nature of the data generating process that drives the persistence.}
Finally, we speak to some recent and well-known empirical work on the effects of monetary and fiscal policy (ramey2018government guajardo2014expansionary, romer2004measure, and gertler2015monetary). Our contribution is to apply our methods to these works 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 results depends on the desired object of interest and the employed estimating method.
The rest of the paper proceeds as follows. Section (ref) provides evidence of serial correlation in shocks used by previous work. Section (ref) describes that LPs and DLMs treat persistence differently, and proposes a solution to re-establish the equivalence between them. It also provides simulations to help understand the results. Section (ref) discusses the previous findings and the options available to applied researchers working with a persistent shock. Section (ref) lays out an application. Section (ref) concludes. The online appendix contains proofs of the theoretical results and further material, including the generalization of the results to VAR and IV settings, additional robustness exercises, and other empirical applications.
When shocks are identified from within an empirical model, the researcher imposes a set of restrictions to recover shocks that can be economically meaningful. Typically, this implies that the resulting shocks are well-behaved and display some statistical features that might be seen as desirable---in particular, no persistence. Alternatively, shocks may be identified without the use of a model, for example, by using narrative methods. This alternative identification relies on the existence of historical sources, such as official documentation, periodicals, etc., from which a shock variable is constructed. In this section, we provide evidence that it is common that shocks identified this way are persistent. We then take stock on this finding in light of ramey2016macroeconomic's canonical definition of a shock.
We study eight aggregate shocks used by prominent literature on monetary and fiscal policy. Some of these shocks are identified using narrative methods, while some employ alternative strategies such as timing restrictions using high-frequency methods.\footnote{ In particular, romer2010macroeconomic and cloyne2013tax construct measures of exogenous tax changes for 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-related to the state of the economy. In a similar vein, ramey2018government construct a measure of government spending shocks by looking at the announcements of future changes in defense spending. guajardo2014expansionary construct a series of fiscal consolidations in OECD countries motivated by a desire to reduce the deficit (as opposed to motivated by current or prospective economic conditions). romer2004measure and cloyne2016monetary identify exogenous changes in monetary policy by looking at the minutes and discussion of the monetary policy committees of the Federal Reserve and Bank of England, respectively (they also orthogonalize the resulting series using forecastable information available at that time). Alternatively, gertler2015monetary identify a proxy of monetary policy shocks using high frequency surprises around policy announcements. Lastly, arezki2017news construct a measure of news shocks based on the date and size of worldwide giant oil discoveries. While some of these papers employ auxiliary regressions to isolate forecastable information, all have in common that the shocks have not been exclusively identified from a time series model.}
To test for the presence of persistence we use a portmanteau-type test following box1970distribution.\footnote{We implement the small sample correction following ljung1978measure. For the cases of arezki2017news and guajardo2014expansionary, which refer to panel data, we test serial correlation using a generalized version of the autocorrelation test proposed by arellano1991tests that specifies the null hypothesis of no autocorrelation at a given lag order.} The null hypothesis is that the data are not serially correlated. We test for the presence of autocorrelation in 40 periods, although results are robust to different horizons (see Table (ref)).
The results from these tests are displayed in Table (ref). Out of the eight considered shocks, six show very large test statistics that result in rejections of the hypothesis of serial uncorrelation for any level of significance. One of them (romer2004measure) displays some degree of serial correlation which leads to failure to reject the null hypothesis only for significance levels above 5%.\footnote{The hypothesis of serial uncorrelation is rejected for significance levels below 5% when considering fewer lags in the test or when considering a longer series (with updated data) from coibion2012monetary. The presence of some degree of autocorrelation is shown in Panel E of Figure (ref).} As further evidence of the presence of serial correlation in the above series, Figure (ref) plots the associated correlograms. romer2010macroeconomic constitutes the only considered shock for which we fail to detect the presence of persistence.\footnote{Persistence may have different origins. In some instances, it arises because of the method used to convert a nominal series into real terms. For example, cloyne2013tax and arezki2017news divide their series by lagged GDP, while ramey2018government use the GDP deflator and a measure of trend GDP. In other instances, the serial correlation arises because of the mapping between different time frequencies. This is usually the case with the identification of monetary policy shocks, such as romer2004measure, gertler2015monetary, or cloyne2016monetary, where daily monetary changes are converted into monthly 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 guajardo2014expansionary, tend to be spread over the course a few years) or because they cluster around events like wars (as in ramey2018government). }
According to the canonical definition (ramey2016macroeconomic), empirical shocks should (i) be exogenous to current and lagged endogenous variables, (ii) be uncorrelated to other exogenous shocks, and (iii) represent unanticipated movements (or news about future shocks). While one might think that the presence of persistence violates the third condition, this is not necessarily the case. When the forecasting loss function is the quadratic one, it is well known that the forecasting errors must be a m.d.s with respect to some information set and therefore uncorrelated.\footnote{ See granger2006forecasting and lee2008loss for a description and analysis of loss functions.} This is the case when the shocks come directly from a conditional expectation model, 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.s and 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 to the previous definition. However, even if a researcher always operates under the quadratic loss function and considers that serially-correlated shocks should not be called “shocks”, in the rest of the paper we show that such shocks can still provide valuable information for empirical analysis.
We consider the following VAR as the data generating process:
where $\bm{y_t}$ is a vector of endogenous time series, $x_t$ is a strictly exogenous variable such that $\operatorname{\mathbb{E}} \left( \bm{u_t} | \bm {y_{t-s}}, x_{t-p} \right) $ $\forall s>0$, $p \gtreqless 0$, and $\bm{u_t}$ and $\varepsilon_t$ are a vector and a scalar i.i.d. variables, with mean and variance given by $\bm{u_t} \sim (\bm{0},\,\bm{\Sigma_u}^{2})$ and $\varepsilon_t \sim (0,\sigma_\varepsilon^{2})$, respectively. Following the evidence discussed in the previous section, $x_t$ is considered to be a shock identified using narrative methods and is allowed to be persistent.
This general framework encompasses several empirical specification often found in the literature. For example, when ignoring the second equation, system (ref) becomes a VAR with an exogenous variable (or VAR-X).\footnote{See, for example, mertens2012empirical or favero2012tax, which assume $\ell$ and $q$ are finite numbers.} Additionally, when $\bm{y_t}$ is a scalar and $\bm{A_\ell} = \bm{0}$ $\forall \ell$, system (ref) becomes a DLM.\footnote{As in romer2004measure or romer2010macroeconomic.} Alternatively, when $x_t$ is instead included in the vector of endogenous variables $\bm{y_t}$, system (ref) becomes a standard VAR.\footnote{As in bloom2009uncertainty or ramey2011identifying.} We explore the implications of this last representation in Appendix (ref).
Without loss of generality, we consider a simpler version of system (ref) with $\bm A_\ell=\bm{0} $ $\forall \ell$, $\bm \delta_q=0 $ $\forall q>0$ and $\gamma_r=0 $ $\forall r>1$:
where $y_t$ is now the economic outcome variable for interest (for example, GDP), $x_t$ is an economic shock (e.g., a fiscal or monetary policy shock) which is strictly exogenous $\operatorname{\mathbb{E}} \left( {u_t} | x_{t-p} \right) $ $\forall p \gtreqless 0$, and $u_t$ and $\varepsilon_t$ are i.i.d variables with mean and variance given by $u_t \sim (0,\,\sigma_u^{2})$ and $\varepsilon_t \sim (0,\sigma_\varepsilon^{2})$, respectively. $\delta$ measures the contemporaneous impact of variable $x_t$ on $y_t$ and is the main parameter of interest.
The data generating process described by system (ref) is intentionally simple to illustrate how the dynamic relationship between the dependent variable $y_t$ and the shock $x_t$ depends on the persistence of the latter. Importantly, the obtained results also arise in more complex settings when we incorporate more general characteristics as in system (ref).\footnote{For example, in Subsection 3.3, we consider models that also include persistence in the dependent variable and lagged effects of the shock. Appendix (ref) proposes a DGP that calls for the use of instruments in LP regressions. Appendix (ref) provides an alternative specification where the degree of serial correlation in the shock $x_t$ is taken from the actual data, instead of following an autoregressive process.}
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 $\mathcal{R}(h)$ for period $h$, is:
where $\Omega_{t-1}$ represents all the history of previous realizations of $\varepsilon_t$ and $x_t$ up to period $t-1$. Importantly, note that the above definition does not condition for future realizations of $x_t$. Hence, if $\gamma \neq 0$, an initial unit impulse in $x_t$ does not imply that $x_{t+j} = 0$.\footnote{This impulse response is equivalent to $\mathcal{R}(h) = \operatorname{\mathbb{E}} \left[ y_{t+h} | \varepsilon_t=1, \varepsilon_{t+1}=0,...,\varepsilon_{t+h}=0, \Omega_{t-1} \right] - \operatorname{\mathbb{E}} \left[ y_{t+h} | \varepsilon_t=0, \varepsilon_{t+1}=0,...,\varepsilon_{t+h}=0, \Omega_{t-1} \right]$. See, for example, koop1996impulse.} In other words, equation (ref) describes dynamic responses that include the possible persistence of the shock $x_t$. For example:
However, the researcher might also be interested in the response to the shock as if the shock had no persistence. We call this second IRF $\mathcal{R}(h)^{*}$ and define it as:
Contrary to $\mathcal{R}(h)$, $\mathcal{R}(h)^*$ explicitly controls for future realizations of $x_t$ so that it describes dynamic responses that do not incorporate the effect of persistence (regardless of the value of $\gamma$), i.e., the responses are observationally equivalent to those that would arise from a data generating process with $\gamma=0$:\footnote{The definition of $\mathcal{R}(h)^*$ is not new. When $x_t$ is the shock variable of interest, this impulse response is referred to as the “traditional impulse response function” by koop1996impulse: $\mathcal{R}(h)^* = \operatorname{\mathbb{E}} \left[ y_{t+h} | x_t=1, x_{t+1}=0,...,x_{t+h}=0, \Omega_{t-1} \right] - \operatorname{\mathbb{E}} \left[ y_{t+h} | x_t=0, x_{t+1}=0,...,x_{t+h}=0, \Omega_{t-1} \right]$. It provides an answer to the question “what is the effect of a shock of size 1 hitting the system at time $t$ on the state of the system at time $t+h$ given that no other shocks hit the system?”.}
Note that, if $\gamma=0$ (the shock is not persistent), then $\mathcal{R}(h)=\mathcal{R}(h)^*$ $\forall$ $h$. By contrast, if $\gamma \neq 0$, then $\mathcal{R}(h) \neq \mathcal{R}(h)^*$ $\forall$ $h>0$.
We now consider the two most frequently used methods to estimate impulse responses when a shock is independently identified, DLMs and LPs, and compare the objects that they identify when the shock is persistent. We first consider the case of DLMs. The use of these models is widespread in applied macroeconomics.\footnote{See, for example, romer2004measure, cerra2008myth, romer2010macroeconomic, alesina2015output, arezki2017news, coibion2018cyclical for interesting applications based on DLM methods, or baek2019abcs for a discussion of their properties. As mentioned in the introduction, these methods are also a special case of more general specifications such as VARs with exogenous variables (or VAR-X). We develop this point further in Appendix (ref), when generalizing some of the results of the paper.} In the case of system (ref), note that we can recover the response function $\mathcal{R}(h)^{DLM}$ using the following regression:\footnote{This regression should include as many lags as the response horizon $h=0,1,\ldots,H$.}
and it follows that $\mathcal{R}(h)^{DLM} = \frac{\partial y_{t+h}} {\partial x_{t}} =\theta_h$ $\forall$ $h$.
The second main method to compute impulse responses is LPs, proposed by jorda2005estimation. LPs are more robust to certain sources of misspecification and for this reason, their use has increased in recent times (see ramey2016macroeconomic for examples). LPs compute impulse responses by estimating an equation for each response horizon $h=0,1,\ldots,H$:
where the sequence of coefficients $\{\delta_{h}\}_{h=0} ^{H}$ determines the response of the variable of interest $\mathcal{R}(h)^{LP}=\delta_h$ for each horizon $h$.\footnote{Unrelated to our case at hand, note that the structure of the LPs induce serial correlation in the residuals $\xi_{t+h}$. This is usually corrected by computing autocorrelation-robust standard errors (jorda2005estimation). See olea2020local for a recent contribution on inference in LPs.}
We now consider under which conditions both methods identify the same objects.
Following the above proposition, when $\gamma\neq0$, LPs recover a dynamic response that includes three dynamic effects: (i) the effect that $x_t$ has directly on $y_{t+h}$ (due to a lagged impact of the shock), (ii) the effect that $x_t$ has through the persistence of $y_t$, and (iii) the effect that $x_t$ has on $y_{t+h}$ through $x_{t+h}$ (since $cov(x_t,x_{t+h}) \neq 0$ when $\gamma\neq0$). The first two effects are independent of $\gamma$ and are shut down in our simple specification of system (ref) (we will incorporate them in our simulation exercises in the next subsection). The last effect (the persistence effect of $x_t$) drives the difference between $\mathcal{R}(h)^{DLM}$ and $\mathcal{R}(h)^{LP}$. In particular, $\mathcal{R}(h)^{LP} = \mathcal{R}(h)^{} = \delta \gamma^h$, while $\mathcal{R}(h)^{DLM} =\mathcal{R}(h)^{*} = 0$ for all $h \geq 1$.
To understand why LPs, unlike DLMs, incorporate this third effect, consider the LPs when $h=1$:
where $\delta_{1}=\mathcal{R}(1)^{LP}$. The direct effect of $x_t$ on $y_{t+1}$ is 0. If $x_t$ had no persistence, then $\delta_1$ would be 0. However, when $\gamma \neq 0$, we can use system (ref) to express $y_{t+1}$ as a function of $x_t$:
where $u^*_{t+1}=\delta \varepsilon_{t+1}+u_{t+1}$. This shows that the coefficient $\delta_1$ in equation (ref) will also recover the persistence effect of $x_t$: $\delta_1 = \delta\gamma$. The intuition is that between period $t$ and period $t+1$, $x_t$ affects $x_{t+1}$ when $\gamma \neq 0$. Since $x_{t+1}$ is not a regressor in equation (ref), then this effect is absorbed by $\delta_1$.\footnote{This omitted variables problem is also briefly mentioned in alesina2015output in the particular context of fiscal consolidation plans.}
When impulse responses are identified using DLMs, the treatment of the persistence of $x_t$ is different. Consider a version of equation (ref) expressed in terms of $t+1$:
As noted earlier, the sequence of coefficients $\theta_h$ determines the response function. Consider the response when $h=1$, i.e., $\mathcal{R}(1)^{DLM} = \theta_1$. Note that, while we know from system (ref) that $\frac{\partial y_{t+1}} {\partial x_{t}} = \delta \gamma$, the coefficient recovered by $\theta_1$ is indeed $\left. \frac{\partial y_{t+1}} {\partial x_{t}} \right | _{x_{t+1}} = 0$. That is, since the DLM controls for $x_{t+1}$, the persistence effect of $x_t$ is accounted for.
In other words, DLMs identify:
while LPs identify:
Note that the difference between $\mathcal{R}^{LP}$ and $\mathcal{R}^{DLM}$ is positive (negative) when $\gamma >0$ ($\gamma <0$). In empirical applications, $\gamma$ may be positive or negative.\footnote{For example, $\gamma$ seems to be positive in ramey2018government, and negative in romer2004measure.}
In this subsection we lay out two methods that can render the responses from DLMs and LPs identical, even under the presence of persistence.
A researcher may be interested in recovering responses as if the shock were serially uncorrelated ($\mathcal{R}(h)^{*}$). (We discuss in Section (ref) when the object of interest may be $\mathcal{R}(h)^{*}$, or $\mathcal{R}(h)$ instead.) However, we have shown that $\mathcal{R}^{LP} (h) \neq \mathcal{R}(h)^{*}$ if $\gamma \neq 0$ and $h\geq1$.
Two apparent methods to avoid LPs picking up the effect of persistence in $x_t$ are: (i) to include lags in the regression (ref), or (ii) to replace $x_t$ with the error term that purges out the persistence:
However, neither of these methods yields $\mathcal{R}^{*} (h)$. The reason is that replacing $x_t$ with ${\varepsilon}_t$ does not include any further information between $t$ and $t+h$, so the responses of the dependent variable will still be affected by $x_{t+h}$. This point is further developed in Appendix (ref).
A third potential method to exclude the effect of persistence would be recasting system (ref) as a VAR that includes the shock as an endogenous variable. However, since in this case LPs and a VAR would identify the same impulse responses (see plagborg2018local) the VAR responses would also include an effect due to the persistence of the shock---we explore this in more detail in Appendix (ref).
Instead, we propose a method based on the inclusion of leads of the persistent shock variable. In particular, given that the DGP of system (ref) poses an AR(1) for $x_t$, one should regress:
where $\delta_{h,0}$ is the $h$-horizon response identified by LPs that include leads of the shock $x_t$, which we denote as $\mathcal{R}^{F} (h)$. In more general processes, in which the autocorrelation of the shock may be of an order larger than one, the optimal choice of leads can be derived adapting the procedure from choi2012model.\footnote{See also lee2020lag for lag order selection in LPs.} The most conservative procedure would be to include $h$ leads of the shock in each period $h$. This is the choice implemented in Section (ref), when considering empirical applications.
Intuitively, leads of $x_t$ in equation (ref) act as controls for the persistence of the shock throughout the response horizon, so that the parameter $\delta_{h,0}$ reflects the dynamic response to a counterfactual serially-uncorrelated shock, that is, controlling for the effect due to $\frac{\partial x_{t+1}}{\partial x_t}\neq0$ built in system (ref) when $\gamma \neq 0$.
As noted earlier, $\mathcal{R}(h)^{DLM} = \mathcal{R}(h)^{*}$ regardless of the value of $\gamma$. However, in some instances, the researcher may be interested in the response that includes the effect of persistence ($\mathcal{R}(h)^{} $). In this subsection, we show how to adapt DLMs to recover these responses. Intuitively, the idea is to compute the impulse responses in system (ref) with respect to $\varepsilon_t$ instead of $x_t$.
Consider a recursive substitution of $x_t$ in system (ref):
The responses of $y_t$ to $\varepsilon_t$, which we denote by $\mathcal{R}(h)^{DLM-per}$, can be obtained from the coefficients $\tilde{\theta}_h$ in:
Proposition (ref) establishes a direct equivalence between the coefficients obtained from equation (ref) and those obtained from LPs in equation (ref): $\tilde{\theta}_h = \delta_h$ $\forall$ $h$. The former are also related to the coefficients estimated from the DLM in terms of $x_t$, as in equation (ref): $\theta_0= \tilde{\theta}_0 = \delta$, $\theta_1= \tilde{\theta}_1 -{\gamma} \tilde{\theta}_0, \ldots, \theta_h= \tilde{\theta}_h -{\gamma} \tilde{\theta}_{h-1}$. Intuitively, the response of $y_{t+1}$ to $x_t$ has an overall effect of $\delta_1=\tilde{\theta}_1$, 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 ${\gamma} \delta$). The standard DLM estimation from equation (ref), since it accounts for the evolution of $x_t$ over the response horizon, 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 impulse response functions using both LPs and DLMs. Our goal is twofold. First, to evaluate quantitatively the conclusions reached in the previous subsection using a plausible calibration of 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 frequently present in real empirical applications. In particular, we consider the following process:
where $\operatorname{\mathbb{E}}(\varepsilon_{t-s} u_{t-r})=0$ $\forall s,r \gtreqless 0$, and $u_t$ and $\varepsilon_t$ follow $\mathcal{N}(0,1)$ distributions. We set $B_0 = 1.5$, $B_1 = 1$ and $\rho = 0.9$.
Compared to system (ref), the new DGP described in system (ref) includes persistence in the outcome variable through $\rho$, and allows the shock $x_t$ to have lagged effects on $y_t$ through $B_1$.\footnote{We introduce this extra lag of the shock to make explicit the distinction between the effect due to the persistence of the shock and the effect of lagged values of the shock on current outcomes.}
We simulate system (ref) for 100 million periods and recover the dynamic responses of $y_t$ to the shock $x_t$ using LPs:
We consider three cases: (i) no persistence ($\gamma= 0$), without including leads in the estimation (i.e., setting $\beta_{h,f}=0$); (ii) some persistence ($\gamma = 0.2$) and still $\beta_{h,f}=0$; (iii) some persistence ($\gamma = 0.2$) and including a lead of the explanatory variable (i.e., allowing $\beta_{h,f}\neq0$).\footnote{The choice of $\gamma=0.2$ is based on an empirical application that we will present in Section (ref). Of course, larger values of $\rho$ would yield higher biases due to the persistence of the process.}
Note that equation (ref) must include a lag of shock $x_{t}$ to capture the effect of $B_1$ in system (ref). However, this does not control for the potential persistence of shock $x_{t}$, as will be apparent in the simulations.
Figure (ref) shows the results of our simulations. In case (i) (dark-blue solid line), the response has a contemporaneous effect of $\hat{\beta}_{1,0}=1.5$ and peaks at the following period due to the the fact that both $\rho$ and $B_1$ have positive values. Using the language of the previous section, the impulse response function estimated by LPs with no persistence is asymptotically equivalent to the one obtained directly from equation (ref), that is, $\hat{\mathcal{R}}(h)^{LP} \rightarrow \mathcal{R}(h)^{*}$.
In case (ii) (red solid line), the introduction of persistence in the shock $x_t$ results in a larger 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 general equilibrium models), but naively estimates equation (ref), implicitly setting $\beta_{h,f}=0$, then the dynamic response is upwardly biased due to the persistence of the shock, i.e., $\hat{\mathcal{R}}(h)^{LP} > \mathcal{R}(h)^{*}$ 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 persistence parameters $\gamma$ and $\rho$ would increase the difference between both responses (blue and red lines in Figure (ref)).
In case (iii) (dashed grey line in Figure (ref)), we see that the inclusion of leads of $x_t$ renders the response of the outcome variable to a persistent shock identical to the one obtained when considering a shock without persistence, i.e., $\hat{\mathcal{R}}(h)^{F} \rightarrow \mathcal{R}(h)^{*}$. In Appendix (ref) we provide an alternative simulation where the shock $x_t$ in (ref) 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 using DLMs always yields the same estimates regardless of the persistence in $x_t$, that is, $\hat{\mathcal{R}}^{DLM} (h) \rightarrow \mathcal{R}^{*} (h)$ for any value of $\gamma$.
First, note that, since $\rho<1$, system (ref) can be inverted and re-written as:
where $L$ represents the lag operator.
Given the independence of $u_t$ and $x_t$, the representation from equation (ref) suggests that the dynamic responses of $y_t$ from $x_t$ can be obtained from the coefficients $\vartheta_h$ in the following regression:
where $H$ is the response horizon.\footnote{baek2019abcs show that for autoregressive distributed lag models, setting the lag order to H is a necessary condition to achieve consistency. }
We estimate equation (ref) fo three different cases: (i) assuming that $\gamma=0$ in the data generating process described in system (ref), (ii) assuming that $\gamma=0.2$ and (iii) replacing $x_t$ with $\hat{\varepsilon_t}$ in equation (ref) (i.e., following equation (ref)).
The results are shown in Figure (ref). Cases (i) and (ii) are displayed in blue and dashed grey lines, respectively. As argued earlier, since equation (ref) controls for all potential dynamic effects of $x_t$, including its persistence, the coefficients $\vartheta_h$ reflect the responses to a shock as if the variable $x_t$ showed no persistence, regardless of the value of $\gamma$. Hence, we have that $\hat{\mathcal{R}}(h)^{DLM} \rightarrow \mathcal{R}(h)^{*}$ for any $\gamma$. Note that these impulse response functions are the same as those obtained with LPs ($\hat{\mathcal{R}}(h)^{LP}$) when $\gamma = 0$, or when we include leads in the LPs ($\hat{\mathcal{R}}(h)^{F}$).
Case (iii) is shown in the red line in Figure (ref). As argued in the previous subsection, when computing the impulse response with respect to $\varepsilon_t$, we are allowing the DLMs to pick up the effect that is due to the persistence in $x_t$. In other words, since we do not implicitly control for the leads of $x_t$ but for those of $\varepsilon_t$ in the DLM, we are not taking into account the persistence of $x_t$. In this case, the responses are equal to those obtained from LPs when $\gamma \neq 0$: $\hat{\mathcal{R}}(h)^{DLM-per} = \hat{\mathcal{R}}(h)^{LP} \rightarrow \mathcal{R}(h)$.
In the presence of a persistent shock, a researcher needs to determine what object to identify. Table (ref) summarizes the adjustments required in LPs and DLMs depending on the choice of the object of interest.
The researcher faces two options: to identify the response as if the shock were uncorrelated ($\mathcal{R}(h)^{*}$) or the response that includes the effect of persistence ($\mathcal{R}(h)$). There are arguments in favor of both. Ultimately, deciding for one or the other may depend on what specific question the researcher is trying to address.
Since $\mathcal{R}(h)^{*}$ can be understood as the IRF resulting from a standardized shock (so that it becomes serially uncorrelated), it should be the desired object when the researcher wants to establish comparisons across dynamic responses. There are at least three instances when $\mathcal{R}(h)^{*}$ can facilitate comparisons. First, a shock identified from within a model (say, a structural VAR) or the innovation to a stochastic process in a DSGE model are, by construction, a m.d.s. (they are non-persistent). Given the absence of serial correlation, the thought experiment carried out in such cases is equivalent to constructing and IRF such as the shock takes the value of 1 on impact and 0 afterwards. Contrary to VAR-identified shocks or innovations in a DSGE model, narratively-identified shocks may display serial correlation. If this is the case, $\mathcal{R}(h)$ (resulting, for example, from standard LP) will identify a different object, since the effect of serial correlation is included in the IRFs. In this instance, $\mathcal{R}(h)^{*}$ will provide the same macroeconomic experiment as, for example, a DSGE model.\footnote{ As mentioned earlier, when $x_t$ is the shock of interest, $\mathcal{R}(h)^{*}$ is defined as the “traditional impulse response” in koop1996impulse.}
Second, $\mathcal{R}(h)^{*}$ can also be an object of interest when the researcher wants to compare the effects of different shocks, e.g., whether fiscal or monetary policy is more effective in stimulating output. For example, it may be the case that fiscal shocks tend to show more persistence or that a given identification procedure tends to generate shocks with different degree of serial correlation. If the effect of persistence amounts to a non-negligible amount of the dynamic response, this could wrongly lead to the conclusion that one shock is more effective than the other when the true underlying cause is that the DGP of both shocks is different. Since $\mathcal{R}(h)^{*}$ effectively standardizes the dynamic responses to shocks with different data generating processes, this would facilitate such comparison.
Third, in a similar vein, $\mathcal{R}(h)^{*}$ can be useful when the researcher wants to compare the effects of the same shock using data from different countries. This is because $\mathcal{R}(h)^{*}$ provides a standardization of the data generating processes of the shocks, which may be heterogeneous across countries.\footnote{Consider the following example: we want to compare the effects of fiscal policy in the US (using a news variable) and in another country (where we have availability of an alternative news variables). Consider the case that the news variables have different amounts of serial correlation and we obtain estimates of the government spending multipliers in both countries. Could we conclude that government spending is more effective in one country versus the other? Potentially, both policies could be equally effective but their sources of identification (news variables) may have different DGPs (one with more serial correlation than other), what leads to different multipliers.}
On the other hand, $\mathcal{R}(h)$ should be the object of interest when the researcher is interested in estimating the most likely dynamic response of a variable to a shock according to the historical data. This argument is similar to the one posed by fisher2010stock and ramey2018government to support the use of the cumulative multiplier (the ratio of the integral of the output response to that of the government spending response) to evaluate the effectiveness of fiscal policies. If we consider the effects of a monetary policy shock that cuts the policy rate by one percentage point, it is important to note that, if that shock displays persistence, then the total monetary policy action (the evolution of the nominal interest following the initial tightening) may be different to what would occur if the shock were non-persistent.
Importantly, and regardless of the experiment that one wants to run, looking at the difference between $\mathcal{R}(h)$ and $\mathcal{R}(h)^{*}$ is informative by itself, as it speaks about how much of the dynamic response is due to the implied DGP of the shock variable. Put differently, it informs the researcher of a propagation mechanism: $\mathcal{R}(h)$ includes the propagation through the persistence of $x_t$ while $\mathcal{R}(h)^{*}$ does not.
Further to this, the methods that underlie the construction of $\mathcal{R}(h)^{*}$ when using local projections can be exported to more general uses. Hence the inclusion of leads of different variables can help in decomposing an IRF in different channels of propagation (where serial correlation is just one of them). This avenue could be particularly informative in highlighting what economics models can bring the dynamic responses closer to the data.
In this section we use the empirical work of ramey2018government to show the quantitative relevance of serial correlation in an actual example. We do so by computing two types of IRFs, $\mathcal{R}(h)$ and $\mathcal{R}(h)^{*}$, as described above.\footnote{In the appendix, we consider additional applications, based on guajardo2014expansionary, romer2004measure, gertler2015monetary, and romer2010macroeconomic.}
ramey2018government, building on previous work by ramey2011identifying and owyang2013government, produce a series of announces about future defense spending between 1890q1-2014q1, scaled by previous quarter trend real GDP.\footnote{ramey2018government estimate trend GDP as sixth degree polynomial for the logarithm of GDP and multiplier by the GDP deflator. In fact, it is the use of the GDP deflator and trend GDP as a way to scale the shocks what seems to induce the persistence. The persistence is also present when the shock is scaled by previous-quarter GDP, as in owyang2013government.} This series, plotted in panel D of Figure (ref), has a positive autocorrelation of $18.4\%$ (47.0% in the subsample after WWII).\footnote{This positive autocorrelation is significant at a confidence level of 90% when considering standard errors that are robust to the presence of heteroskedasticity and persistence (with more than one lag) for the whole sample. For the subsample starting after WWII, the autocorrelation is significant at any level. }
ramey2018government use LPs to estimate the response of output and government spending to a shock in future defense spending. We follow their same approach and sample and estimate the following equations for output ($y_t$) and government spending ($g_t$):
where $z_{t}$ includes $P$ lags of $y_t$, $g_t$ and $shock_t$. Note that, following the discussion in previous sections, we include $h$ leads of the variable $shock_t$. In particular, for each horizon $h$ we include $h$ leads.
To replicate ramey2018government's estimates, we set $\gamma_{f,h}=0$, $\forall$ $f,h$. The black, solid line in Figure (ref) represents the estimated responses of output (left panel) and government spending (right panel) to the shock.\footnote{Figure (ref) also replicates the original 95% confidence intervals computed using the Newey-West correction.} As noted in Section (ref), these dynamic responses are the equivalent to the $\mathcal{R}(h)$ as defined in equation (ref) (with the only difference being that $\Omega_{t-1}$ includes now the past history of $z_t$). The results closely resemble those in ramey2018government (Figure 5 of their paper).\footnote{We drop the last $h$ observations of the sample, so that the specifications with and without leads can be fully comparable. This does not have any discernible effect when replicating the original results from ramey2018government.}
Next, we allow $\gamma_{f,h}\neq0$. As discussed in Section (ref), this amounts to estimating $\mathcal{R}(h)^*$ as defined in equation (ref). In the red lines in Figure (ref), we observe that the dynamic responses change considerably when the leads are included. For example, after two years, output and government spending are 40% lower than in ramey2018government's estimates. The large observed difference between $\mathcal{R}(h)$ and $\mathcal{R}(h)^*$ suggests that the persistence of the news variable plays a non-negligible role in explaining the dynamic transmission of the fiscal shock to output and government spending.
Whether to include leads or not also has implications for inference. The 95% confidence intervals when leads are included (shown in dashed lines in Figure (ref)) are substantially narrower than when they are not (grey areas in Figure (ref)). The latter are around 50% broader after two years, and more than twice as big after three years.
The dynamic responses of output and government spending are informative about the expected path of these variables after a shock. To obtain a measure of the efficiency of fiscal policy (i.e., the increase of output per each dollar increase in government spending), ramey2018government use the cumulative multiplier, computed as:\footnote{ramey2018government show that the cumulative multiplier can be obtained in one step yielding identical results to those obtained combining equations (ref) and (ref).}
We find that this statistic is not substantially affected by persistence of the shock (Figure (ref)). Given that both output and government spending react similarly when including leads of the shock, taking the ratio of the two variables attenuates the differences between both specifications.\footnote{Even though the multiplier does not change much when accounting for persistence, the fact that the expected 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 important variables such as public debt or future changes in tax liabilities.}
\paragraph{Non-linear effects.} We now investigate whether the effect of persistence in the shock can affect the responses in a non-linear setting, i.e., if government spending multipliers are different in expansions and recessions.\footnote{See ramey2018tenyears for a recent summary of this debate. For example, an influential study by auerbach2012measuring finds that government spending multipliers are higher during recessions using a non-linear VAR. alloza2017fiscal highlights the role of the information used to define a period of recession, and finds that output responds negatively to government spending shocks in a post-WWII sample under different identification and estimation approaches.} For this, we follow ramey2018government and estimate a series of non-linear LPs:
where $x_t$ is either output or government spending and $S_t$ is a binary variable indicating the state of the economy. When $S_t=1$, the economy is booming and, when $S_t=0$, the economy is 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 effects during expansions and recessions.
We first replicate the non-linear responses of output and government spending during booms and recessions obtained by ramey2018government. Hence, we estimate equation (ref) setting $\delta_{A,f,h}=\delta_{B,f,h}=0$ $\forall f,h$, which identifies $\mathcal{R}(h)$. Our results, shown in Figure (ref) in black lines, resemble very closely those from the authors. Next, we repeat the experiment accounting for potential persistence, that is, including leads of the shock (identifying $\mathcal{R}(h)^*$). The results are shown in red lines in Figure (ref). While relatively similar in the case of expansions, the responses are quantitatively different during recessions. The estimates that include leads lie outside of the 95% confidence bands during much of the response horizon. The results suggest that ignoring the effect of persistence could yield responses during recessions that, after 2--3 years, are twice as large as the responses that account for the effect of persistence. Or, in other words, persistence in the shock is responsible for up to 50% of the dynamic transmission of the shock during recessions.
In Figure (ref), we show how these responses map into estimates of non-linear fiscal multipliers. In the case of expansions, the results do not change much depending on whether the persistence is accounted for (red solid line, $\mathcal{R}(h)^*$) or not (black solid line, $\mathcal{R}(h)$). In either case, they resemble those in ramey2018government (see Figure 6 of their paper). In recessions, however, the results change substantially depending on whether the persistence is controlled for or not. If it is not (black solid line), the multiplier has a negative value upon impact and substantially falls in the following quarter to a value of -2. It becomes positive before the end of the first year, and fully converges to the value of the multiplier during expansions after six quarters. If the persistence is excluded from the dynamic responses (red dashed line), the cumulative multiplier is -1 (instead of -2) and becomes positive after the first year. Furthermore, the multiplier during recessions remains lower than the multiplier during expansions for a much longer period. When the persistence is not accounted for, this convergence is achieved after 6 quarters, as mentioned above. However, when including leads of 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 multiplier could be lower during recessions than during expansions, and part of this difference may 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, as those in equation (ref). This is particularly useful since, contrary to threshold VARs, LPs do not impose any restriction on the evolution of state $S_t$ (while non-linear VARs that interact the 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 macroeconomic experiments that can help understand how restrictive this condition is. In particular, by including leads of the state $S_t$ in equation (ref) we are identifying the counterfactual response to a fiscal shock when the underlying state of the economy is not allowed to change (as in threshold VARs). We perform this experiment and report the multipliers during booms in recessions in green lines in Figure (ref). We observe that, when the state is not allowed to change, the multiplier during recessions is slightly higher in the short run, but essentially unchanged at medium and longer horizons. This exercise allows us to illustrate how the use of leads of variables in conjunction with LPs can help understand interesting counterfactual exercises and shed light on the dynamic transmission of shocks.
We have shown that persistence results in the estimation of different responses when using LPs 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 be adapted. 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, the difference between both types of responses is informative about how much of the dynamic transmission 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 down channels of transmission. For example, one may be interested in the effects of monetary policy 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., tax changes) will deliver responses holding that variable constant. This methodology allows to separate 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 of leads achieves a similar goal in LPs, hence allowing to construct interesting macroeconomic experiments. We leave these questions for future research.
\setcounter{page}{1}