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.
78,031 characters · 11 sections · 108 citation commands
Local Projection Inference is Simpler and More Robust Than You Think
Keywords: impulse response, local projection, long horizon, uniform inference.
Impulse response functions are key objects of interest in empirical macroeconomic analysis. It is increasingly popular to estimate these parameters using the method of local projections Jorda2005: simple linear regressions of a future outcome on current covariates Ramey2016,Angrist2018,Nakamura2018,Stock2018. Since local projection estimators are regression coefficients, they have a simple and intuitive interpretation. Moreover, inference can be carried out using textbook standard error formulae, adjusting for serial correlation in the (multi-step forecast) regression residuals.
Despite its popularity, there exist no theoretical results justifying the use of local projection inference over autoregressive procedures. From an identification and estimation standpoint, Kilian2017 and PlagborgMoller2019 argue that neither local projections nor Vector Autoregressions (VARs) dominate the other in terms of mean squared error in finite samples, and in population the two methods are equivalent. However, from an inference perspective, the only available guidance on the relative performance of local projections comes in the form of a small number of simulation studies, which by necessity cannot cover the entire range of empirically relevant data generating processes.
In this paper we show that---in addition to its intuitive appeal---frequentist local projection inference is robust to two common features of macroeconomic applications: highly persistent data and the estimation of impulse responses at long horizons. Key to our result is that we consider lag-augmented local projections, which use lags of the variables in the regression as controls. Formally, we prove that standard confidence intervals based on such lag-augmented local projections have correct asymptotic coverage uniformly over the persistence in the data generating process and over a wide range of horizons.\footnote{We focus on marginal inference on individual impulse responses, not simultaneous inference on a vector of several response horizons IK2016,MontielOlea2019.} This means that confidence intervals remain valid even if the data exhibits unit roots, and even at horizons $h$ that are allowed to grow with the sample size $T$, e.g., $h=h_T \propto T^{\eta}$, $\eta \in [0,1)$. In fact, when persistence is not an issue, and the data is known to be stationary, local projection inference is also valid at long horizons; i.e., horizons that are a non-negligible fraction of the sample size ($h_T \propto T$).
Lag-augmenting local projections not only robustifies inference, it also simplifies the computation of standard errors by obviating the adjustment for serial correlation in the residuals. It is common practice in the local projections literature to compute Heteroskedasticity and Autocorrelation Consistent/Robust (HAC/HAR) standard errors Jorda2005,Ramey2016,Kilian2017,Stock2018. Instead, we prove that the usual Eicker-Huber-White heteroskedasticity-robust standard errors suffice for lag-augmented local projections. The reason is that, although the regression residuals are serially correlated, the regression scores (the product of the residuals and residualized regressor of interest) are serially uncorrelated under weak assumptions. This finding further simplifies local projection inference, as it side-steps the delicate choice of HAR procedure and associated difficult-to-interpret tuning parameters Lazarus2018.
The robustness properties of lag-augmented local projection inference stand in contrast to the well-known fragility of standard autoregressive procedures. Textbook autoregressive inference methods for impulse responses (such as the delta method) are invalid in some cases with near-unit roots or medium-long to long horizons (e.g., $h_T \propto \sqrt{T})$, as discussed further below. We show that lag-augmented local projection inference is valid when the data has near-unit roots and the horizon sequence satisfies $h_{T}/T \rightarrow 0$. Though the method fails in the case of unit roots and very long horizons $h_T \propto T$, existing VAR-based methods that achieve correct coverage in this case are either highly computationally demanding or result in impractically wide confidence intervals. When the data is stationary and interest centers on short horizons, local projection inference is valid but less efficient than textbook AR inference. Thus, the robustness afforded by our recommended procedure is not a free lunch. We provide a detailed comparison with alternative inference procedures in (ref) below.
Our results rely on assumptions that are similar to those used in the literature on autoregressive inference. In particular, we assume that the true model is a VAR($p$) with possibly conditionally heteroskedastic innovations and known lag length. We discuss the choice of lag length $p$ in (ref). The key assumption that we require on the innovations is that they are conditionally mean independent of both past and future innovations (which is trivially satisfied for i.i.d. innovations). Our strengthening of the usual martingale difference assumption is crucial to avoid HAC inference, but we show that the assumption is satisfied for a large class of conditionally heteroskedastic innovation processes. The robustness property of local projection inference only obtains asymptotically if the researcher controls for all $p$ lags of all of the variables in the VAR system. Thus, our paper highlights the advantages of multivariate modeling even when using single-equation local projections.
To illustrate our theoretical results, we present a small-scale simulation study suggesting that lag-augmented local projection confidence intervals achieve a favorable trade-off between coverage and length. Since local projection estimation is subject to small-sample biases just like VAR estimation Herbst2020, we consider a simple and computationally convenient bootstrap implementation of local projection. The simulations suggest that non-augmented autoregressive procedures with delta method standard errors have more severe under-coverage problems than local projection inference, especially at moderate and long horizons. Autoregressive confidence intervals can be meaningfully shorter than lag-augmented local projection intervals in relative terms, but in absolute terms the difference in length is surprisingly modest. Our simulations also indicate that lag-augmented local projections with heteroskedasticity-robust standard errors have better coverage/length properties than more standard non-augmented local projections with off-the-shelf HAR standard errors. Finally, although the lag-augmented autoregressive bootstrap procedure of Inoue2020 achieves good coverage, it yields prohibitively wide confidence intervals at longer horizons when the data is persistent.
\paragraph{Related Literature.} It is well known that standard autoregressive (AR) inference on impulse responses requires an auxiliary rank condition to rule out super-consistent limit distributions, thus yielding a $\sqrt{T}$-normal limit with strictly positive variance, see Assumption B of Inoue2020. When this rank condition holds, the textbook AR impulse response estimator is asymptotically normal even in the presence of (near-)unit roots Inoue2002. However, there are two common features of the data that lead to violations of the rank condition. First, the condition can fail when some linear combinations of the variables exhibit no persistence Benkwitz2000. Second, in the presence of (near-)unit roots, certain linear combinations of the autoregressive coefficients are necessarily super-consistent Sims1990. This compromises textbook AR inference for certain combinations of impulse response horizons and parameter values that typically cannot be ruled out a priori, especially in AR(1) or VAR(1) models, but also in higher-order autoregressions (Phillips1998; Inoue2020). In an important paper, Inoue2020 show that lag-augmented autoregressive inference solves the rank problem caused by (near-)unit roots, but data generating processes that lack persistence still need to be ruled out a priori. We build on their ideas, which in turn are based on Toda1995 and Dolado1996. As we show, the validity of lag-augmented local projection (LP) inference does not hinge on auxiliary rank conditions.
Moreover, the validity of textbook AR inference is also compromised when the length of the impulse response horizon is large Pesavento2007,Mikusheva2012. Standard bootstrap methods rectify some of these problems, but not all. Several papers have proposed AR-based methods for impulse response inference at long horizons $h=h_T \propto T$ Wright2000,Gospodinov2004,Pesavento2007,Mikusheva2012,Inoue2020. With the exception of Mikusheva2012, the literature on long-horizon inference has exclusively focused on near-unit root processes as opposed to devising uniformly valid procedures. The Hansen1999 grid bootstrap analyzed by Mikusheva2012 is asymptotically valid at short and long horizons. However, it is not valid at intermediate horizons (e.g., $h_T \propto \sqrt{T}$), unlike the LP procedure we analyze. Mikusheva2012 argues, though, that the grid bootstrap is close to being valid at intermediate horizons, although it is much more computationally demanding than our recommended procedure, especially in VAR models with several parameters. Inoue2020 show that a version of the Efron bootstrap confidence interval, when applied to lag-augmented AR estimators, is valid at long horizons. We show that this procedure delivers impractically wide confidence intervals at moderately long horizons when the data is persistent, unlike lag-augmented LP.
We appear to be the first to prove the uniform validity of lag-augmented LP inference. Mikusheva2007,Mikusheva2012 and Inoue2020 derive the uniform coverage properties of various AR inference procedures, but they do not consider LP. The pointwise properties of LP procedures have been discussed by Jorda2005, Kilian2017, and Stock2018, among others. Kilian2011 and Brugnolini2018 present simulation studies comparing AR inference and LP inference. Brugnolini2018 finds that the lag length in the LP matters, which is consistent with our theoretical results.
Though the theoretical results in this paper appear to be novel, Dufour2006 and Breitung2019 have discussed some of the main ideas presented herein. First, both these papers state that lag augmentation in LP avoids unit root asymptotics, but neither paper considers inference at long horizons or derives uniform inference properties. Second, Breitung2019 further argue that HAC inference in LP can be avoided if the true model is a VAR($p$), although it is not clear from their discussion what are the assumptions needed for this to be true. Neither of these papers provide results concerning the efficiency of lag-augmented LP inference relative to other lag-augmented or non-augmented inference procedures, as we do in (ref).
Local projections are closely related to multi-step forecasts. Richardson1989 and Valkanov2003 develop a non-standard limit distribution theory for long-horizon forecasts. Chevillon2017 proves a robustness property of direct multi-step inference that involves non-normal asymptotics due to the lack of lag augmentation. Phillips2013 test the null hypothesis of no long-horizon predictability using a novel approach that requires a choice of tuning parameters, but yields uniformly-over-persistence normal asymptotics. This test is based on an estimator with a faster convergence rate than ours in the non-stationary case. However, to the best of our knowledge, their approach does not carry over immediately to impulse response inference, and it is not obvious whether the procedure is uniformly valid over both short and long horizons.
\paragraph{Outline.} (ref) provides a non-technical overview of our results in the context of a simple AR(1) model, including an illustrative simulation study. (ref) provides an in-depth comparison of lag-augmented LP with other inference procedures. (ref) presents the formal uniformity result for a general VAR($p$) model. (ref) describes a simple bootstrap implementation of lag-augmented local projection that we recommend for practical use. (ref) concludes. Proofs are relegated to (ref) and the Online Supplement. (ref) contain further simulation and theoretical results. The supplement and a full Matlab code repository are available online.\footnote{\url{https://github.com/jm4474/Lag-augmented_LocalProjections} }
This section provides an overview of our results in the context of a simple univariate AR(1) model. The discussion here merely intends to illustrate our main points. (ref) presents general results for VAR($p$) models.
\paragraph{Model.} Consider the AR(1) model for the data $\lbrace y_t \rbrace$:
The parameter of interest is a nonlinear transformation of $\rho$, namely the impulse response coefficient at horizon $h \in \mathbb{N}$. We denote this parameter by $\beta(\rho,h) \equiv \rho^h$. In (ref) below we argue that the zero initial condition $y_0=0$ is not needed for our results to go through. Our main assumption in the univariate model is:
The assumption requires the innovations to be mean independent relative to past and future innovations. This is a slight strengthening of the usual martingale difference assumption on $u_t$. (ref) is trivially satisfied if $\lbrace u_t \rbrace$ is i.i.d., but it also allows for stochastic volatility and GARCH-type innovation processes.\footnote{For example, consider processes $u_t = \tau_t \varepsilon_t$, where $\varepsilon_t$ is i.i.d. with $E(\varepsilon_t)=0$, and for which one of the following two sets of conditions hold: (a) $\lbrace \tau_t \rbrace$ and $\lbrace \varepsilon_t \rbrace$ are independent processes; or (b) $\tau_t$ is a function of lagged values of $\varepsilon_t^2$, and the distribution of $\varepsilon_t$ is symmetric. (ref) is in principle testable, but that is outside the scope of this paper.}
\paragraph{Local Projections With and Without Lag Augmentation.} We consider the local projection (LP) approach of Jorda2005 for conducting inference about the impulse response $\beta(\rho, h)$. A common motivation for this approach is that the AR(1) model (ref) implies
where the regression residual (or multi-step forecast error), \[\xi_t(\rho,h) \equiv \sum_{\ell=1}^h \rho^{h-\ell}u_{t+\ell},\] is generally serially correlated, even if the innovation $u_t$ is i.i.d.
The most straight-forward LP impulse response estimator simply regresses $y_{t+h}$ on $y_t$, as suggested by equation (ref), but the validity of this approach is sensitive to the persistence of the data. Specifically, this standard approach leads to a non-normal limiting distribution for the impulse response estimator when $\rho \approx 1$, since the regressor $y_t$ exhibits near-unit-root behavior in this case. Hence, inference based on normal critical values will not be valid uniformly over all values of $\rho \in [-1,1]$ even for fixed forecast horizons $h$. If $\rho$ is safely within the stationary region, then the LP estimator is asymptotically normal, but inference generally requires the use of Heteroskedasticity and Autocorrelation Robust (HAR) standard errors to account for serial correlation in the residual $\xi_t(\rho,h)$.
To robustify and simplify inference, we will instead consider a lag-augmented local projection, which uses $y_{t-1}$ as an additional control variable. In the autoregressive literature, “lag augmentation” refers to the practice of using more lags for estimation than suggested by the true autoregressive model. Define the covariate vector $x_t \equiv (y_t,y_{t-1})'$. Given any horizon $h \in \mathbb{N}$, the lag-augmented LP estimator $\hat{\beta}(h)$ of $\beta(\rho,h)$ is given by the coefficient on $y_t$ in a regression of $y_{t+h}$ on $y_t$ and $y_{t-1}$:
Here $\hat{\beta}(h)$ is the impulse response estimator of interest, while $\hat{\gamma}(h)$ is a nuisance coefficient.
The purpose of the lag augmentation is to make the effective regressor of interest stationary even when the data $y_t$ has a unit root. Note that equations (ref)--(ref) imply
If $u_t$ were observed, the above equation suggests regressing $y_{t+h}$ on $u_t$, while controlling for $y_{t-1}$. Intuitively, this will lead to an asymptotically normal estimator of $\beta(\rho,h)$, since the regressor of interest $u_t$ is stationary by (ref), and we control for the term that involves the possibly non-stationary regressor $y_{t-1}$. Fortunately, due to the linear relationship $y_t=\rho y_{t-1}+u_t$, the coefficient $\hat{\beta}(h)$ on $y_t$ in the feasible lag-augmented regression (ref) on $(y_t,y_{t-1})$ precisely equals the coefficient on $u_t$ in the desired regression on $(u_t,y_{t-1})$. This argument for why lag-augmented LP can be expected to have a uniformly normal limit distribution even when $\rho \approx 1$ is completely analogous to the reasoning for using lag augmentation in AR inference Sims1990,Toda1995,Dolado1996,Inoue2002,Inoue2020. In the LP case, lag augmentation has the additional benefit of simplifying the computation of standard errors, as we now discuss.
\paragraph{Standard Errors.} We now define the standard errors for the lag-augmented LP estimator. We will show that, contrary to conventional wisdom (e.g., Jorda2005; Ramey2016), HAR standard errors are not needed to conduct inference on lag-augmented LP, despite the fact that the regression residual $\xi_t(\rho,h)$ is serially correlated. Instead, it suffices to use the usual heteroskedasticity-robust Eicker-Huber-White standard error of $\hat{\beta}(h)$:\footnote{This is computed by the {\tt regress, robust} command in Stata, for example. The usual homoskedastic standard error formula suffices if $u_t$ is assumed to be i.i.d.}
where we define the lag-augmented LP residuals
and the residualized regressor of interest \[\hat{u}_t(h) \equiv y_t - \hat{\rho}(h)y_{t-1},\quad t=1,2,\dots,T-h,\] \[\hat{\rho}(h) \equiv \frac{\sum_{t=1}^{T-h} y_t y_{t-1}}{\sum_{t=1}^{T-h} y_{t-1}^2}.\] As mentioned in the introduction, the fact that we may avoid HAR inference simplifies the implementation of LP inference, as there is no need to choose amongst alternative HAR procedures or specify tuning parameters such as bandwidths Lazarus2018.
Why is it not necessary to adjust for serial correlation in the residuals? Since lag-augmented LP controls for $y_{t-1}$, equation (ref) suggests that the estimator $\hat{\beta}(h)$ is asymptotically equivalent with the coefficient in a linear regression of the (population) residualized outcome $y_{t+h} - \beta(\rho,h+1)y_{t-1}$ on the (population) residualized regressor $u_t = y_t - \rho y_{t-1}$:
The second term in the decomposition above determines the sampling distribution of the lag-augmented local projection. Although the multi-step regression residual $\xi_t(\rho,h)$ is serially correlated on its own, the regression score $\xi_t(\rho,h)u_t$ is serially uncorrelated under (ref).\footnote{Breitung2019 make this same observation, but they appear to claim that it is sufficient to assume that $\lbrace u_t \rbrace$ is white noise, which is incorrect.} For any $s < t$,
Thus, the heteroskedasticity-robust (but not autocorrelation-robust) standard error $\hat{s}(h)$ suffices for doing inference on $\hat{\beta}(h)$.\footnote{Stock2018 mention a similar conclusion for the distinct case of LP with an instrumental variable, under some conditions on the instrument.} Notice that this result crucially relies on (i) lag-augmenting the local projections and (ii) the strengthening in (ref) of the usual martingale difference assumption on $\lbrace u_t \rbrace$ (as remarked above, the strengthening still allows for conditional heteroskedasticity and other plausible features of economic shocks).\footnote{The nuisance coefficient $\hat{\gamma}(h)$ is not interesting per se, but note that inference on this coefficient would generally require HAR standard errors, and its limit distribution is in fact non-standard when $\rho \approx 1$.}
Though lag augmentation robustifies and simplifies local projection inference, it is not necessarily a free lunch. We show in (ref) that the relative efficiency of non-augmented and lag-augmented local projection estimators depends on $\rho$ and $h$.
\paragraph{Lag-Augmented Local Projection Inference.} Define the nominal $100 (1-\alpha)\%$ lag-augmented LP confidence interval for the impulse response at horizon $h$ based on the standard error $\hat{s}(h)$: \[\hat{C}(h,\alpha) \equiv \left[ \hat{\beta}(h)-z_{1-\alpha/2}\: \hat{s}(h)\:,\: \hat{\beta}(h)+z_{1-\alpha/2}\: \hat{s}(h) \right],\] where $z_{1-\alpha/2}$ is the $(1-\alpha/2)$ quantile of the standard normal distribution.
Our main result shows that the lag-augmented LP confidence interval above is valid regardless of the persistence of the data, i.e., whether or not the data has a unit root. Crucially, the result does not break down at moderately long horizons $h$. We provide a formal result for VAR($p$) models in (ref) and for now just discuss heuristics. Consider any upper bound $\bar{h}_T$ on the horizon which satisfies $\bar{h}_T/T \to 0$. Then (ref) below implies that
where $P_\rho$ denotes the distribution of the data $\lbrace y_t \rbrace$ under the AR(1) model (ref) with parameter $\rho$. In words, the result states that, for sufficiently large sample sizes, LP inference is valid even under the worst-case choices of parameter $\rho \in [-1,1]$ and horizon $h \in [1,\bar{h}_T]$. As is well known, such uniform validity is a much stronger result than pointwise validity for fixed $\rho$ and $h$. In fact, if we restrict attention to only the stationary region $\rho \in [-1+a,1-a]$, $a \in (0,1)$, then the statement (ref) is true with the upper bound $\bar{h}_T = (1-a)T$ on the horizon. That is, if we know the time series is not close to a unit root, then local projection inference is valid even at long horizons $h$ that are non-negligible fractions of the sample size $T$.
We now present a small simulation study to show that lag-augmented LP achieves a favorable trade-off between robustness and efficiency relative to other procedures. For clarity, we continue to assume the simple AR(1) model (ref) with known lag length. Our baseline design considers homoskedastic innovations $u_t \stackrel{i.i.d.}{\sim} N(0,1)$. In (ref) we present results for ARCH innovations.
We stress that, although we use the AR(1) model for illustration here, the central goal of this paper is to develop a procedure that is feasible even in realistic VAR($p$) models. Thus, we avoid computationally demanding procedures, such as the AR grid bootstrap, which are difficult to implement in applied settings. We provide an extensive theoretical comparison of various inference procedures in (ref).
(ref) displays the coverage and median length of impulse response confidence intervals at various horizons. We consider several versions of AR inference and LP inference, either implemented using the bootstrap or using delta method standard errors. “LP” denotes local projection and “AR” autoregressive inference. “LA” denotes lag augmentation. The subscript “$b$” denotes bootstrap confidence intervals constructed from a wild recursive bootstrap design Goncalves2004, as described in (ref) (for LP we use the percentile-t confidence interval). Columns without the “$b$” subscript use delta method standard errors. For LA-LP, we always use Eicker-Huber-White standard errors as discussed in (ref), whereas non-augmented LP always uses HAR standard errors.\footnote{As an off-the-shelf, state-of-the-art HAR procedure, we choose the Equally Weighted Cosine (EWC) estimator with degrees of freedom as recommended by Lazarus2018. The degrees of freedom depend on the effective sample size $T-h$ and thus differ across horizons $h$.} The column “AR-LA” is the Efron bootstrap confidence interval for lag-augmented AR estimates developed by Inoue2020 and discussed further in (ref).\footnote{We use the Pope1990 bias-corrected AR estimates to generate the bootstrap samples, as recommended by Inoue2020.} All estimation procedures include an intercept. The sample size is $T=240$. We consider data generating processes (DGPs) $\rho \in \lbrace 0,.5,.95,1\rbrace $ and horizons $h$ up to 60 periods (25% of the sample size, which is not unusual in applied work). The nominal confidence level is 90%. We use 5,000 Monte Carlo repetitions, with 2,000 bootstrap draws per repetition.
\afterpage{
}
Consistent with our theoretical results, the bootstrap version of lag-augmented local projection (column 1) achieves coverage close to the nominal level in almost all cases, whereas the competing procedures either under-cover or return impractically wide confidence intervals. In contrast, non-augmented LP (columns 3 and 4) exhibits larger coverage distortions in almost all cases. As is well known, textbook AR delta method confidence intervals (column 6) severely under-cover when $\rho>0$ and the horizon is even moderately large.
It is only when both $\rho=1$ and $h \geq 36$ that lag-augmented local projection exhibits serious coverage distortions, again consistent with our theory. However, even in these cases, the coverage distortions are similar to or less pronounced than those for non-augmented LP and for delta method AR inference.
Although the Inoue2020 lag-augmented AR bootstrap confidence interval (column 5) achieves correct coverage for $\rho>0$ at all horizons, this interval is extremely wide in the problematic cases where $\rho$ is close to 1 and the horizon $h$ is intermediate or long. We explain this fact theoretically in (ref). Confidence intervals with median width greater than 1 would appear to be of little practical use, since the true impulse response parameter is bounded above by 1 in the AR(1) model.\footnote{In the AR(1) model, we could intersect all confidence intervals with the interval $[-1,1]$. In this case, the median length of the Inoue2020 confidence interval is close to 1, cf. (ref).} Note also that the Inoue2020 interval severely under-covers when $\rho=0$ at all even (but not odd) horizons $h$, as explained theoretically in (ref).\footnote{Inoue2020 assume $\rho \neq 0$ and discuss why this restriction is necessary in their case.}
Although outperformed by bootstrap procedures, the lag-augmented local projection delta method interval (column 2) performs well among the group of delta method procedures. Its coverage distortions are much less severe than textbook AR delta method inference (column 4) and non-augmented LP inference with HAR standard errors (column 6). Recall that the lag-augmented LP confidence interval is at least as easy to compute as these other delta method confidence intervals. The reason why the bootstrap improves on the coverage properties of the delta method procedures is related to the well-known finite-sample bias of AR and LP estimators Kilian1998, Herbst2020.\footnote{Our bootstrap implementation of non-augmented LP also appears to be quite effective at correcting the most severe coverage distortions of the delta method procedure.}
(ref) illustrates the fact that the robustness of lag-augmented local projection inference entails an efficiency loss relative to AR inference when $\rho$ is well below 1, although this loss is not large in absolute terms. In percentage terms, local projection confidence intervals are much wider than AR-based confidence intervals when $\rho \ll 1$ and the horizon $h$ is intermediate or long, since AR procedures mechanically impose that the impulse response function tends to 0 geometrically fast with the horizon. Yet, in absolute terms, the median length of the LP confidence intervals is not so large as to be a major impediment to applied research. The relative efficiency of lag-augmented LP vs. non-augmented LP cannot be ranked and depends on the DGP and on the horizon. When $\rho$ is close to 1, lag-augmented LP intervals are sometimes (much) narrower than lag-augmented AR intervals. We analytically characterize the various efficiency trade-offs in (ref).
(ref) shows that the preceding qualitative conclusions extend to richer models. There we consider a bivariate VAR(4) model with varying degrees of persistence, as well as two empirically calibrated VAR(12) models with four or five observables.
The simulations and theoretical results in this paper suggest that lag-augmented local projection is the only known confidence interval procedure that achieves uniformly valid coverage over the DGP and over a wide range of horizons, while preserving reasonable average length and remaining computationally feasible in realistic settings. However, the simulations also suggest that lag-augmented local projection inference is less efficient than standard AR inference when the data is stationary. In this section we discuss in more detail the coverage and length properties of alternative confidence interval procedures for impulse responses. We review the well-known drawbacks of textbook AR inference, provide new results on the relative length of lag-augmented LP vs. non-augmented LP and lag-augmented AR, and discuss the computational challenges of the AR grid bootstrap. We refer the reader back to the small-scale simulation study in (ref) for illustrations of the following arguments.
\paragraph{Textbook Autoregressive Inference.} The uniformity result ((ref)) for lag-augmented LP stands in stark contrast to textbook AR inference on impulse responses, which suffers from several well-known issues. First, for the standard OLS AR estimator, the usual asymptotic normal limiting theory is invalid when the derivative of the impulse response parameter with respect to the AR coefficients has a singular Jacobian matrix. In the AR(1) model, this occurs at all horizons $h\geq 2$ in the white noise case $\rho=0$ Benkwitz2000. Second, as with non-augmented LP, textbook AR inference is not uniformly valid when the data is nearly non-stationary, unless one further restricts the parameter space (Phillips1998; Inoue2002; Inoue2020).\footnote{This is well known in the AR(1) model. In the AR(2) model, a non-normal limit arises at $h=2$ when there is a unit root and the autoregressive coefficients are equal Inoue2020.} Third, pre-testing for the presence of a unit root does not yield uniformly valid inference and can lead to poor finite sample performance Mikusheva2007. Fourth, plug-in AR inference with normal critical values must necessarily break down at medium-long horizons $h=h_T \propto T^{1/2}$ and at long horizons $h_T \propto T$, due to the severe nonlinearity of the impulse response transformation at such horizons Mikusheva2012. Wright2000 and Pesavento2006,Pesavento2007 construct confidence intervals for persistent processes at long horizons $h=h_T \propto T$ by inverting the non-standard AR limit distribution, but these tailored procedures do not work uniformly over the parameter space or over the horizon.
The severe under-coverage of delta method AR inference is starkly illustrated in (ref) (see column 6 of (ref)). As discussed in detail by Inoue2020, standard bootstrap approaches to AR inference do not solve all the uniformity issues.
We must emphasize, however, that if we restrict attention to stationary processes and short-horizon impulse responses, the standard OLS AR impulse response estimator is more efficient than lag-augmented LP. Hence, there is a trade-off between efficiency in benign settings and robustness to persistence and longer horizons, as is also clear in the simulation results in (ref). We expand upon the efficiency properties of the standard AR estimator in (ref).
\paragraph{Lag-Augmented AR Inference.} The above-mentioned non-uniformity of the textbook AR inference method in the case of near-non-stationary data can be remedied by lag augmentation Inoue2020. In the case of an AR(1) model, the lag-augmented AR estimator $\hat{\beta}_\text{ARLA}(h)$ is given by $\hat{\rho}_1^h$, where $(\hat{\rho}_1,\hat{\rho}_2)$ are the OLS coefficients from a regression of $y_t$ on $(y_{t-1},y_{t-2})$ (i.e., we estimate an AR(2) model). The intuition why this guarantees a normal limiting distribution even in the unit root case is the same as in (ref). Lag-augmented AR and lag-augmented LP coincide at horizon $h=1$, but not at longer horizons. Lag augmentation involves a loss of efficiency: The lag-augmented AR estimator is strictly less efficient than the non-augmented AR estimator except when the true process is white noise (see (ref)). Note that lag augmentation by itself does not solve the above-mentioned issues that occur when the Jacobian of the impulse response transformation is singular, or when doing inference at medium-long or long horizons.\footnote{The AR(1) simulations in (ref) show that the coverage of the Inoue2020 confidence interval is 0 at all even horizons when $\rho=0$. This is because the true impulse response is 0, but the bootstrap samples of $\hat{\rho}_1^h$ are all strictly positive. Their procedure achieves uniformly correct coverage at odd horizons.}
The bootstrap confidence interval for lag-augmented AR proposed by Inoue2020 has valid coverage even at long horizons. Specifically, Inoue2020 show that the Efron bootstrap confidence interval---applied to recursive AR bootstrap samples of $\hat{\beta}_\text{ARLA}(h)$---has valid coverage even at long horizons $h=h_T \propto T$, as long as the largest autoregressive root is bounded away from 0.\footnote{For intuition, consider the AR(1) case. The Efron bootstrap preserves monotonic transformations, and the bootstrap transformation $\beta(\rho,h)=\rho^h$ is monotonic (if we restrict attention to $\rho \in (0,1]$ or $\rho \in [-1,0)$). Hence, the Efron confidence interval is valid for $\rho^h$ if it is valid for $\rho$ itself. In more general VAR($p$) models, the same argument can be applied at long horizons, since here only the largest autoregressive root matters for impulse responses (if the roots are well-separated).}
Unfortunately, we show in (ref) that the expected length of the lag-augmented AR interval is prohibitively large when the data is persistent and the horizon is long. Precisely, in the case of an AR(1) model, $\hat{\beta}_\text{ARLA}(h)=\hat{\rho}_1^h$ is inconsistent for sequences of DGPs $\rho=\rho_T$ and horizons $h=h_T$ such that $h_{T} \propto T^{\eta}, \eta \in [1/2,1]$, and $h_T(1-\rho_T) \to a \in [0,\infty)$. The reason is that the lag-augmented coefficient estimator $\hat{\rho}_1$ converges at rate $T^{-1/2}$ even in the unit root case, implying that the estimation error in $\hat{\rho}_1$ is not negligible when raising the estimator to a power of $h=h_T$. This implies that the Efron bootstrap confidence interval is inconsistent (i.e., its length does not shrink to 0 in probability) for such sequences $\rho_T$ and $h_T$. In fact, when $\eta>1/2$, the width of the confidence interval for the $h_{T}$ impulse response is almost equal to the entire positive part of the parameter space $[0,1]$ with probability equal to the nominal confidence level. This contrasts with the lag-augmented LP confidence interval, which is consistent for any sequence $\rho_T \in [-1,1]$ and any sequence $h_T$ such that $h_T/T\to 0$. The large width of the Inoue2020 interval is illustrated in the simulations in (ref) (see the second-to-last column in (ref)).
Interestingly, if we restrict attention to stationary processes and short horizons, the relative efficiency of lag-augmented AR and lag-augmented LP inference is ambiguous. In the context of a stationary, homoskedastic AR(1) model with a fixed horizon $h$ of interest, (ref) shows that lag-augmented AR is more efficient than lag-augmented LP when $\rho$ is small or when the horizon $h$ is large, and vice versa. For any horizon $h$, there exists some cut-off value for $\rho \in (0,1)$, above which lag-augmented LP is more efficient. Intuitively, the nonlinear impulse response transformation $\rho \mapsto \rho^h$ is highly sensitive to values of $\rho$ near 1 whenever $h$ is large, which compounds the effects of estimation error in $\hat{\rho}$, whereas LP is a purely linear procedure.
\paragraph{AR Grid Bootstrap and Projection.} The grid bootstrap of Hansen1999 represents a computationally intensive approach to doing valid inference at fixed and long horizons, regardless of persistence, but it is invalid at intermediate horizons, as shown by Mikusheva2012. The grid bootstrap is based on test inversion, so it requires running an autoregressive bootstrap on each point in a fine grid of potential values for the impulse response parameter of interest. It also requires estimating a constrained OLS estimator that imposes the hypothesized null on the impulse response at each point in the grid. Recall that lag-augmented LP inference is computationally simple and valid at any horizon $h=h_T$ satisfying $h_T/T \to 0$. However, in the case of unit roots and long horizons $h_T \propto T$, lag-augmented LP inference with normal critical values is not valid, while the grid bootstrap is valid Mikusheva2012.
Another computationally intensive approach is to form a uniformly valid confidence set for the AR parameters and then map it into a confidence interval for impulse responses by projection. Although doable in the AR(1) model, this approach would appear to be computationally infeasible and possibly highly conservative in realistic VAR($p$) settings, unlike lag-augmented LP (see (ref)).
\paragraph{Other Local Projection Approaches.} Non-augmented LP is not robust to non-stationarity, as already discussed in (ref). If the data is stationary and the horizon $h$ is fixed, the relative efficiency of non-augmented LP and lag-augmented LP is generally ambiguous, as shown in (ref) in the case of a homoskedastic AR(1) model. There are two competing forces. On the one hand, as shown in (ref), non-augmented LP uses the regressor $y_t$, which has higher variance than the effective regressor $u_t$ in the lag-augmented case. By itself, this suggests that non-augmented LP should be more efficient. On the other hand, absent lag augmentation, the LP regression scores are serially correlated and thus have a larger long-run variance. On balance, (ref) shows that lag-augmented LP is relatively more efficient the smaller is $\rho$ and the larger is $h$.
In some empirical settings, the researcher may directly observe the autoregressive innovation, or some component of the innovation, for example by constructing narrative measures of economic shocks Ramey2016. For concreteness, consider the AR(1) model (ref) and assume we observe the innovation $u_t$. In this case, it is common in empirical practice to simply regress $y_{t+h}$ on $u_t$, without controls. Although this strategy provides consistent impulse response estimates when the data is stationary, it is inefficient relative to lag-augmented LP, since the latter approach additionally controls for the variable $y_{t-1}$, which would otherwise show up in the error term in the representation (ref). Thus, lag augmentation is desirable on robustness and efficiency grounds even if some shocks are directly observed.
\paragraph{Summary.} Existing and new theoretical results confirm the main message of our simulations in (ref): Lag-augmented LP is the only known procedure that is computationally feasible in realistic problems and can be shown to have valid coverage under a wide range of DGPs and horizon lengths, without achieving such valid coverage by returning a confidence interval that is impractically wide. This robustness does come at the cost of a loss of efficiency relative to non-robust AR methods. However, the efficiency loss is large in relative terms only in stationary, short-horizon cases, where lag-augmented LP confidence intervals do well in absolute terms, as illustrated in (ref). Based on these results, we believe that it is only in the case of highly persistent data and very long horizons $h=h_T \propto T$ that the use of alternative robust procedures should be considered, such as the computationally demanding AR grid bootstrap.
This section presents the inference procedure and theoretical uniformity result for a general VAR($p$) model. In this case, the lag-augmented LP procedure controls for $p$ lags of all the time series that enter into the VAR model. We follow Mikusheva2012 and Inoue2020 in assuming that the lag length $p$ is finite and known. We also assume that the VAR process has no deterministic dynamics for simplicity. See (ref) for further discussion of these assumptions.
Consider an $n$-dimensional VAR($p$) model for the data $y_t=(y_{1,t},\dots,y_{n,t})'$:
Let $A \equiv (A_1, \ldots, A_{p})$ denote the $n \times np$ matrix collecting all the autoregressive coefficients. The assumption of zero pre-sample initial conditions $y_0=\dots=y_{1-p}=0$ is made for notational simplicity and can be relaxed, as discussed below in the remarks after (ref). As in the AR(1) case, we assume that the $n$-dimensional innovation process $\lbrace u_{t} \rbrace$ satisfies the strengthening of the martingale difference condition in (ref) (which from now on will refer to the vector process $\lbrace u_t \rbrace$).
We seek to do inference on a scalar function of the reduced-form impulse responses of the VAR model. Generalizations to structural impulse responses and joint inference require more notation but are otherwise straight-forward, see (ref). Let $\beta_{i}(A,h)$ denote the $n \times 1$ vector containing each of variable $i$'s reduced-form impulse responses at horizon $h \geq 0$. Without loss of generality, we focus on the impulse responses of the first variable $y_{1,t}$. Thus, we seek a confidence interval for the scalar parameter $\nu'\beta_{1}(A,h)$, where $\nu \in \mathbb{R}^n \backslash \lbrace 0 \rbrace$ is a user-specified vector. For example, the choice $\nu = e_j$ (the $j$-th unit vector) selects the horizon-$h$ response of $y_{1,t}$ with respect to the $j$-th reduced-form innovation $u_{j,t}$.
Local projection estimators of impulse responses are motivated by the representation
see Jorda2005 and Kilian2017. Here $\delta_{1,\ell}(A,h)$ is an $n \times 1$ vector of regression coefficients that can be obtained by iterating on the VAR model (ref). The model-implied multi-step forecast error in this regression is
\paragraph{Multivariate Lag-Augmented Local Projection.} The lag-augmented LP estimator corresponding to the VAR model (ref) is motivated by (ref). We regress $y_{1,t+h}$ on the $n$ variables $y_{t}$, using the $np$ variables $(y'_{t-1}, \ldots, y'_{t-p})$ as additional controls. According to equation (ref), the population regression coefficients on the last $n$ control variables $y_{t-p}$ equal zero. Thus, we are including one additional lag in the estimation of the impulse response coefficients. Given any horizon $h \in \mathbb{N}$, the lag-augmented LP estimator $\hat{\beta}_1(h)$ of $\beta_1(A,h)$ is given by the vector of coefficients on $y_{t}$ in the regression of $y_{1,t+h}$ on $x_{t} \equiv (y_{t}', y_{t-1}', \ldots, y_{t-p}')'$:
where $\hat{\beta}_{1}(h)$ is a vector of dimension $n \times 1$.
The usual (Eicker-Huber-White) heteroskedasticity-robust standard error for $\nu'\hat{\beta}_1(h)$ is defined as \[\hat{s}_1(h,\nu) \equiv \frac{1}{T-h}\left\lbrace \nu'\hat{\Sigma}(h)^{-1} \left(\sum_{t=1}^{T-h} \hat{\xi}_{1,t}(h)^2\hat{u}_t(h)\hat{u}_t(h)' \right) \hat{\Sigma}(h)^{-1}\nu\right\rbrace^{1/2},\] where \[\hat{\xi}_{1,t}(h) \equiv y_{1,t+h} - \hat{\beta}_1(h)' y_{t} - \hat{\gamma}_{1}(h)' X_{t}, \quad X_{t} \equiv (y_{t-1}' , \ldots, y_{t-p}')', \] \[ \hat{u}_{t}(h) \equiv y_{t} -\hat{A}(h)X_{t}, \quad \hat{A}(h) \equiv \left( \sum_{t=1}^{T-h} y_{t}X_{t}' \right) \left ( \sum_{t=1}^{T-h} X_{t} X_{t}' \right)^{-1}, \] and \[\hat{\Sigma}(h) \equiv \frac{1}{T-h}\sum_{t=1}^{T-h} \hat{u}_t(h)\hat{u}_t(h)'.\] The $1-\alpha$ confidence interval for $\nu'\beta_1(A,h)$ is defined as \[\hat{C}_1(h,\nu,\alpha) \equiv \left[ \nu'\hat{\beta}_1(h)-z_{1-\alpha/2}\: \hat{s}_1(h,\nu)\:,\: \nu'\hat{\beta}_1(h)+z_{1-\alpha/2}\: \hat{s}_1(h,\nu) \right].\]
\paragraph{Parameter Space.} We consider a class of VAR processes with possibly multiple unit roots combined with arbitrary stationary dynamics. Specifically, we will prove that the confidence interval $\hat{C}_1(h,\nu,\alpha)$ has uniformly valid coverage over the following parameter space. Let $\|M\| \equiv \sqrt{\operatorname*{trace}(M'M)}$ denote the Frobenius matrix norm, and let $I_n$ denote the $n \times n$ identity matrix.
This parameter space contains any stationary VAR process (for sufficiently small $a,\epsilon$ and sufficiently large $C$) as well as many---but not all---non-stationary processes. Lag polynomials $A(L)$ in this parameter space imply that the process $\lbrace y_t \rbrace$ can be written in the form $y_t = \operatorname*{diag}(\rho_1,\dots,\rho_n)y_{t-1} + \tilde{y}_t$, where $\tilde{y}_t \equiv B(L)^{-1}u_t$ is a stationary process whose impulse responses at horizon $\ell$ decay at the geometric rate $(1-\epsilon)^\ell$. We allow all the roots $\rho_1,\dots,\rho_n$ to be potentially close to or equal to 1. Mikusheva2012 considers the same class of processes but with $\rho_2=\dots=\rho_n=0$. We are not aware of other uniform inference results that allow multiple near-unit roots. Although the parameter space in (ref) appears more restrictive than the local-to-unity framework of Phillips1988, we argue below that our uniform coverage result applied to the parameter space $\mathcal{A}(a,C,\epsilon)$ immediately implies an extended result that also covers processes with cointegration among the control variables $y_{2,t},\dots,y_{n,t}$. However, we do impose the restriction that the response variable of interest $y_{1,t}$ has at most one root near unity, as in Wright2000, Pesavento2006, Mikusheva2012, and Inoue2020.
Our main result requires two further technical assumptions in addition to (ref). Let $\lambda_{\min}(M)$ denote the smallest eigenvalue of a symmetric positive semidefinite matrix $M$.
Part ((ref)) of (ref) is a common requirement for consistent estimation of regression standard errors with possibly heteroskedastic residuals. Part ((ref)) is a standard weak dependence restriction on the second moments of $u_t$ Brillinger2001.
We will write $\rho(A)=(\rho_1(A),\dots,\rho_n(A))'$ to represent any of the possible vectors of roots $\rho_1,\dots,\rho_n$ corresponding to a collection of autoregressive coefficients $A=(A_1, \ldots, A_{p}) \in \mathcal{A}(0,C,\epsilon)$. This is a slight abuse of notation, since the mapping from $A(L)$ to $\rho_i$'s is one-to-many. Define $g(\rho,h)^2 \equiv \min\lbrace \frac{1}{1-|\rho|},h\rbrace$ and $\rho_i^*(A,\epsilon) \equiv \max\lbrace |\rho_{i}(A)|, 1-\epsilon/2 \rbrace$. Define also the $np \times np$ diagonal matrix $G(A,h,\epsilon) \equiv I_p \otimes \operatorname*{diag}(g(\rho_1^*(A,\epsilon) ,h),\dots,g(\rho_n^*(A,\epsilon),h))$.
This high-level assumption ensures that the properly scaled (matrix) “denominator” in the VAR OLS estimator $\hat{A}(h)$ is uniformly non-singular asymptotically, so the estimator is uniformly well-defined with high probability in the limit. Hence, the assumption is essentially necessary for our result.
How can (ref) be verified? $G(A_T,T,\epsilon)^{-1}\left[\frac{1}{T}\sum_{t=1}^T X_tX_t'\right]G(A_T,T,\epsilon)^{-1}$ is known to converge in distribution in a pointwise sense to an almost surely positive definite (perhaps stochastically degenerate) random matrix under stationary, local-to-unity, or unit root sequences $\lbrace A_T \rbrace$ Phillips1988,Hamilton1994.\footnote{Note that the diagonal entries of $G(A,T,\epsilon)^{-1}$ are constants for stationary VAR coefficient matrices $A$, whereas these diagonal entries are proportional to $T^{-1/2}$ under local-to-unity or unit root sequences.} (ref) requires that such convergence obtains for all possible sequences $\lbrace A_T \rbrace$. In (ref) we illustrate how the assumption can be verified in the AR(1) model under an additional weak condition on the innovation process.
We now state the result that the LP estimator $\nu'\hat{\beta}_1(h)$ is asymptotically normally distributed uniformly over the parameter space in (ref), even at long horizons $h$. Let $P_A$ denote the probability measure of the data $\lbrace y_t \rbrace$ when it is generated by the VAR($p$) model (ref) with coefficients $A \in \mathcal{A}(a,C,\epsilon)$. The distribution of the innovations $\lbrace u_t \rbrace$ is fixed.
The uniform asymptotic normality established above immediately implies that the confidence interval $\hat{C}_1(h,\nu,\alpha)$ has uniformly valid coverage asymptotically. Part ((ref)) considers stationary VAR processes whose largest roots are bounded away from 1; then inference is valid even at long horizons $h=h_T \propto T$. Part ((ref)) allows all or some of the $n$ roots $\rho_1,\dots,\rho_n$ to be near or equal to 1, but then we require $h_T/T \to 0$.
\paragraph{Remarks.}
In this section we describe the bootstrap implementation of lag-augmented local projection that we recommend for practical use. We find in simulations that the bootstrap procedure is effective at correcting small-sample coverage distortions. These distortions arise primarily due to the small-sample bias of local projection, which Herbst2020 show is analogous to the well-known bias of the VAR OLS estimator Kilian1998.
Our baseline algorithm is based on a wild autoregressive bootstrap design, which allows for heteroskedastic VAR innovations Goncalves2004 as in our theoretical results. Guided by simulation evidence, we construct the bootstrap confidence interval using the equal-tailed percentile-t method, which has a built-in bias correction (Kilian1998; Kilian2017).
The bootstrap procedure for computing a $1-\alpha$ confidence interval proceeds as follows, assuming a VAR($p$) model:
Instead of the above recursive VAR design, it is also possible to use the standard fixed-design pairs bootstrap, as in any linear regression with serially uncorrelated scores.\footnote{This is the bootstrap carried out by Stata's {\tt bootstrap} command with standard settings.} In this case, the usual Efron bootstrap confidence interval is valid, like the percentile-t interval. However, simulations suggest that the pairs bootstrap procedure is less accurate in small samples than the above recursive bootstrap design, mirroring the results in Goncalves2004 for autoregressive inference.
Our online code repository implements the above recommended bootstrap procedure, as well as several alternative LP- and VAR-based procedures, see (ref).
Local projection inference is already popular in the applied macroeconomics literature. The simple nature of local projections has allowed the methods of causal analysis in macroeconomics to connect with the rich toolkit for program evaluation in applied microeconomics; see for example Angrist2018, Nakamura2018, Stock2018, and rambachan2019econometric. We hope the novel results in this paper on the statistical properties of local projections may further this convergence.
\paragraph{Recommendations for Applied Practice.} The simplicity and statistical robustness of lag-augmented local projection inference makes it an attractive option relative to existing inference procedures. We recommend that applied researchers conduct inference based on lag-augmented local projections with heteroskedasticity-robust (Eicker-Huber-White) standard errors. This procedure can be implemented using any regression software and has desirable theoretical properties relative to textbook delta method autoregressive inference and to non-augmented local projection methods. In particular, we showed that confidence intervals based on lag-augmented local projections that use robust standard errors with standard normal critical values are uniformly valid over the persistence in the data and for a wide range of horizons. We also suggested a simple bootstrap implementation in (ref), which seems to achieve even better finite-sample performance.
Conventional VAR-based procedures deliver smaller standard errors than local projections in many cases, but this comes at the cost of fragile coverage, especially at longer horizons. In our opinion, there are only two cases in which the lag-augmented local projection inference method is inferior to competitors: (i) If the data is known to be at most moderately persistent and interest centers on very short impulse response horizons, in which case textbook VAR inference is valid and efficient. (ii) When the data has (near-)unit roots and interest centers on horizons that are a substantial fraction of the sample size, in which case the computationally demanding AR grid bootstrap may be deployed if feasible Hansen1999,Mikusheva2012. In all other cases, lag-augmented local projection inference appears to achieve a competitive trade-off between robustness and efficiency.
How should the VAR lag length $p$ be chosen in practice? Naive pre-testing for $p$ causes uniformity issues for subsequent inference Leeb2005. Though we leave the development of a formal procedure for future research (see below), our theoretical analysis yields three insights. First, users of local projection should worry about the choice of $p$ in order to obtain robust inference, just as users of VAR methods do. Second, $p$ should be chosen conservatively, as is conventional in VAR analysis Kilian2017. In our framework there is no asymptotic efficiency cost of controlling for more than $p_0$ lags if the true model is a VAR($p_0$), and the simulation results in (ref) confirm that the cost is also small in finite samples. Third, the logic of (ref) suggests that in realistic models where the higher-lag VAR coefficients are relatively small, it is not crucial to get $p$ exactly right: What matters is that we include enough control variables so that the effective regressor of interest approximately satisfies the conditional mean independence condition ((ref)).
\paragraph{Directions for Future Research.} It would be interesting to relax the assumption of a finite lag length $p$ by adopting a VAR($\infty$) framework. We are not aware of existing work on uniform inference in such settings. One possibility would be to base inference on a sieve VAR framework that lets the lag length used for estimation tend to infinity at an appropriate rate as in Goncalves2007. A second possibility is to impose a priori bounds on the rate of decay of the VAR coefficients, and then take the resulting worst-case bias of finite-$p$ local projection estimators into account when constructing confidence intervals Armstrong_Kolesar:2018.
Due to space constraints, we leave a proof of the validity of the suggested bootstrap strategy to future work. It appears straight-forward, albeit tedious, to prove its pointwise validity. Proving uniform validity requires extending the already lengthy proof of (ref).
Several extensions of the results in this paper could be pursued by adopting techniques from the VAR literature. First, the results of PlagborgMoller2019 suggest straight-forward ways to generalize our results on reduced-form impulse response inference to structural inference. Second, our assumption of no deterministic dynamics in the VAR model could presumably be relaxed using standard arguments. Third, by considering linear system estimators rather than single-equation OLS, our results on scalar inference could be extended to simultaneous inference on several impulses IK2016,MontielOlea2019. Finally, whereas we adopt a frequentist perspective in this paper, it remains an open question whether local projection inference is relevant from a Bayesian perspective.