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Intentionally Overidentifying Local Projections
\maketitle
\footnotetext[1]{School of Finance, Central University of Finance and Economics, Shahe Higher Education Park, Changping District, Beijing, P.R. China, 102206.
Email: [email removed].}
\footnotetext[2]{Department of Economics, University of Mannheim, L7, 3--5, Mannheim, Germany, 68161.
TEL: +49 621-181-1879; Email: [email removed].}
\footnotetext[3]{Department of Economics, Indiana University, 100 S. Woodlawn Ave., Bloomington, IN 47405, USA. Email: [email removed].}
\begin{bibunit}
\begin{center}
\begin{minipage}{0.88\textwidth}
\small
\noindent\textsc{Abstract.}\quad
Local projections (LPs) have become a standard tool for estimating dynamic effects in macroeconomics and are typically implemented using ordinary least squares (OLS) for a given lag order. We propose an alternative approach that exploits \textit{additional lags} as excluded \textit{instrumental variables} (IVs). Under a finite-order model with martingale-difference shocks, the resulting (optimally weighted) IV estimator is never asymptotically less efficient than OLS and, as we show, can deliver substantial efficiency gains when data are persistent. For the unit-root AR(1) data with i.i.d. errors, our baseline IV estimator (with one overidentifying restriction) has asymptotic variance $2-h^{-1}$, compared with $h$, the asymptotic variance of the OLS lag-augmented estimator in \cite{montiel2021local}, for an integer propagation horizon $h$. We then develop IV-based LP inference that is robust to unknown persistence and moderate local model misspecification, whereas the recursive VAR-based inference is generally not robust to the latter. We also characterize the trade-off between the degree of misspecification and the number of additional lag controls. Through simulations and an empirical application, we demonstrate that the proposed method can yield meaningfully shorter confidence intervals for impulse responses than OLS in finite samples.
\end{minipage}
\end{center}
\vspace{1em}
\textit{Keywords}:
GMM; impulse responses; local projections; misspecification; moment-condition redundancy; persistence; robust inference.
\section{Introduction}
A central task in empirical macroeconomics and finance is to estimate how shocks propagate over time \citep{sims1980macroeconomics,ramey2016macroeconomic}. Since \citet{jorda2005estimation}, local projections (LPs, hereafter) have become a standard tool for this purpose because they estimate horizon-specific impulse responses directly, and can readily accommodate a range of empirically relevant features, including nonlinearities, instability, and rich sets of controls.\footnote{See \cite{jorda2025localprojections,inoue2024unstable,dinh2024rslp,ferreira2025blp, gonccalves2024state}, among many others.}
In recent work, \cite{montiel2021local} and \cite{olea2024double} formally prove that, under the finite-order model, the lag-augmented LP-based inference (i.e. using a sufficient number of controlled lags) is robust to persistence of the data, long horizons, and a nontrivial degree of model misspecification, strengthening the case for LPs relative to other methods of inference on impulse responses. On the other hand, it is well known that the LP estimator tends to have large variance and provide wide confidence intervals.
The current paper extends this line of research by proposing a new LP estimator which exploits the additional information implied by the model. Concretely, instead of using OLS (ordinary least squares), as is common in practice, we intentionally overidentify the local projection by adding orthogonality conditions based on extra lags of the data to construct the optimally weighted IV (via the generalized method of moments (GMM), \cite{hansen1982large}) estimator of the LP regression. While such an estimator is guaranteed to be no less efficient than OLS, we ask whether and when the efficiency gain can be nontrivial and whether the robustness properties established in the earlier literature on local projections can be retained.
Our analysis shows that the new method has considerable potential. Additional orthogonal conditions can yield substantial efficiency gains over LP OLS, arising from the specific structure of the LP regression. As a teaser, additional lags are not necessarily informative. For the autoregression (namely, the horizon-one local projection), lag augmented or not, with conditionally homoskedastic martingale-difference errors (e.g., i.i.d. errors), additional lags are generally redundant excluded instrumental variables; the additional moment conditions do not generate an asymptotically more efficient estimator than OLS. However, such redundancy does not extend to a general LP regression at horizons larger than one, even under conditional homoskedasticity. The optimally weighted GMM can produce a strict efficiency gain over OLS, especially when the data are strongly persistent. The efficiency gain arises primarily from the serial correlation of the GMM (residualized) score process, even when the OLS (residualized) score process can be serially uncorrelated \citep{montiel2021local}. Under conditional heteroskedasticity, as our general inferential theory allows, using additional lags as excluded instruments generally yields non-redundant moment conditions (even for the horizon-one LP regression; see \citet{kuer02ET}).
Regarding robustness, we show that controlling for a sufficient number of lags in the LP regression remains crucial for valid inference with our GMM estimator, robust to both data persistence and moderate model misspecification.
\textbf{Literature}.
The terminology of instruments and GMM has appeared in the VAR/LP literature, but their uses and purposes are all different from ours. The instruments used in the current paper are internal lags that improve efficiency. The method applies to reduced-form impulse responses and, under identification, to structural responses as well. Our use of instruments differs from the recent literature on internal and external instruments for \textit{structural identification} \citep{stock2018identification, lewis2021volatility}. \citet{inoue2026inference} similarly refer to their method as GMM LP, but their approach follows the literature on structural identification. \citet{JordaKoz2011} use GMM and minimum-distance methods to identify and estimate the structural parameter in the solution of a linear rational expectations model.
Instruments used for structural identification can be subject to the well-known weak-IV problem \citep{Mont-SW}, whereas our use of instruments generally is not. Under the standard regularity assumptions, the extra weakly-correlated lags can at most reduce or eliminate the efficiency gain (over OLS) from overidentification, and do not themselves threaten the identification or inference.
\citet{wang2024} use reduced-form VAR innovations as instruments to conduct two-step estimation and associated inference for the generalized impulse responses and multi-horizon Granger causality in the sense of \citet{dufour1998short}. Their motivation is mostly the simplicity of the (persistence-robust) inference (no HAC variance estimator needed, albeit under an additional assumption on innovations) and the ability to handle a larger class of causal parameters. Their instruments \textit{exactly identify} the model, thus the resulting estimator of the impulse response can be more or less efficient than OLS. The current paper focuses exclusively on impulse responses, and uses extra lagged outcomes to construct overidentification restrictions which guarantee no asymptotic efficiency loss relative to OLS.
\citet{lusompa2023local} also exploits the serial correlation in LP regression errors, by constructing a generated outcome variable, which involves estimates of nuisance parameters
(i.e., the full set of impulse responses). Our method is thus simpler to implement:
it relies on a standard GMM estimator with readily available instrumental variables and can
be carried out using off-the-shelf software. We further develop robust inference and allow
for model misspecification, features that are not addressed in Lusompa’s analysis.\footnote{A recent paper by \cite{DeVos2026} warns that GLS (generalized least squares) local projections may achieve efficiency gains at the expense of robustness.}
Relatedly, recently developed smoothing, shrinkage, and Bayesian methods\footnote{See \cite{barnichon2019smooth}, \cite{ferreira2025blp}, \cite{huber2025gsulp}, among others.} can improve precision of the LP by imposing structure across horizons and borrowing information across horizon-specific estimates. By contrast, our approach leaves the local-projection estimand unrestricted and obtains efficiency solely from valid overidentifying moments.
In this paper we consider the finite-order model, following a strand of the literature (surveyed in \cite{olea2024double}) that views the low-order models commonly used in practice as approximations to an underlying infinite-order DGP. Recent work by \citet{xu2023local,xu2026bound} establishes conditions under which LP OLS attains the efficiency bound as the lag order diverges. Thus, under these conditions, we do not expect our GMM estimator to deliver nontrivial efficiency gains.
\textbf{Outline}. Section \ref{sec:implementation} presents the method and its implementation. Section~\ref{sec:eff} derives the actual efficiency gain in a scalar AR(1) model. Section~\ref{sec:uniform} develops uniform inference theory under correct model specification, and Section \ref{sec:local} extends the analysis to the locally misspecified model. Sections \ref{sec:mc} and \ref{sec:application} report Monte Carlo evidence and an empirical application. Technical details are provided in the appendices.
\textbf{Notation}. For two random (column) vectors $y$ and $x$, define $P(y\mid x)=B'x$ as the population linear projection, where $B=[\mathbb E(xx')]^{-1}\mathbb E(xy')$ is the projection coefficient matrix. For covariance-stationary random vectors $x_t$ and $y_t$, define
$\operatorname{LRV}(x_t)=\sum_{k=-\infty}^{\infty}
\operatorname{Cov}(x_t,x_{t-k})$ and
$\operatorname{LRCov}(y_t,x_t)=\sum_{k=-\infty}^{\infty}
\operatorname{Cov}(y_t,x_{t-k})$. For symmetric matrices, $\lambda_{\min}(\cdot)$ and
$\lambda_{\max}(\cdot)$ denote the extreme eigenvalues. For rectangular
matrices, $\lambda_{\min}(\cdot)$ denotes the smallest singular value. The sum $\sum_{\ell=a}^{b}x_{\ell}$ is defined as zero if $a>b$.
\section{Overidentified GMM local projections}
\label{sec:implementation}
In this section we outline the method in a general setting. The formal analysis is provided in Sections \ref{sec:uniform} and \ref{sec:local}.
\subsection{Model and target parameter}
Let $\{y_t: t=1,\cdots,T\}$ be a $K$-dimensional process following the
VAR$(d)$ model,
\begin{equation}
\label{eq:vard}
y_t
=
A_1 y_{t-1}+\cdots+A_d y_{t-d}+u_t,
\end{equation}
where $\{u_t\}$ is a mean-zero white noise innovation sequence. Let
$\Sigma_u:=\mathbb E[u_tu_t']$. Deterministic terms
are omitted in our analysis to maintain the focus. In practice, we can think of $y_t$ as the data after removing deterministic components (e.g., by demeaning or detrending). The finite-order VAR in \eqref{eq:vard} is the maintained model; Section~\ref{sec:local} allows departures from the model \eqref{eq:vard}.
For any $p\ge d$, define the lag vector $
Y_{t,p}:=
(y_t',y_{t-1}',\cdots,y_{t-p+1}')'
\in\mathbb R^{Kp}.$
For a given forecast horizon $h\ge1$, let $y_{1,t+h}:=\nu_1'y_{t+h}$, where $\nu_1$ is the first column of $I_K$. The target is the coefficient $\beta_h$ on $y_t$ in the following horizon-$h$
regression (known as the local projection (LP)),
\begin{align}
\label{eq:vard_lp}
y_{1,t+h}
&=
(\beta_h',\eta_{h,p}')Y_{t,p}+e_{t,h},
\qquad t=p,\cdots,T-h,
\end{align}
where $e_{t,h}$ is the projection error. Under \eqref{eq:vard}, the local projection residual follows $e_{t,h}=\nu_1'\sum_{\ell=0}^{h-1}\Psi_\ell u_{t+h-\ell}$, with $\Psi_\ell=J_d\mathbf A^\ell J_d'$, $\mathbf A$ the $dK\times dK$ companion matrix of $\{A_1,\cdots,A_d\}$,
and $J_d=[I_K,0,\cdots,0]$ being $K
\times dK$. Here $\beta_h\in\mathbb R^K$ collects the
coefficients on $y_t$ and $\eta_{h,p}\in\mathbb R^{K(p-1)}$ those on the included
lag controls. The two leading cases are $p=d$, the usual local projection
\citep{jorda2005estimation}, and $p=d+1$, the lag-augmented local projection (LALP)
\citep{montiel2021local}.
The coefficient $\beta_h$ in population is invariant to the lag blocks included
beyond the VAR order: once $Y_{t,d}$ is included, further lags have no
predictive content for $y_{1,t+h}$ under \eqref{eq:vard}, so their population
coefficients are zero. Importantly, these lags can nonetheless affect the behavior
of the estimator.
\subsection{Optimal weighting and HAC inference}
The LP regression \eqref{eq:vard_lp} is commonly estimated by OLS. OLS is simple but only uses a subset of
moment restrictions available: for any $q\ge p$, the deeper lag stack $Y_{t,q}$ is also
orthogonal to $e_{t,h}$, and OLS leaves these moments unused. An extended set of valid lag moments can therefore enhance asymptotic efficiency. We
treat \eqref{eq:vard_lp} as a linear GMM problem indexed by the integers $(p,q)$.
\vspace{0.5\baselineskip}
\noindent\textbf{Definition}. The $\operatorname{GMM}(p,q)$ estimator of $\beta_h$ (formally defined in \eqref{gmm_iv} below) is the optimally weighted GMM estimator of the regression
\eqref{eq:vard_lp} using $Y_{t,q}$ as the instruments for the regressors $Y_{t,p}$, where
$q\ge p\ge d$.
\vspace{0.5\baselineskip}
To formalize the estimator, define the sample moment function $ \overline g_{h,p,q}(\beta_h,\eta_{h,p})
=\\
T^{-1}\sum_{t=q}^{T-h}
Y_{t,q}
\Bigl(
y_{1,t+h}
-
(\beta_h',\eta_{h,p}')Y_{t,p}
\Bigr)$. For an optimal weighting matrix estimate, let $\widehat e_{t,h}$ be a first-step consistent estimate of the projection
error in \eqref{eq:vard_lp}. We use $\widehat e_{t,h}
=
y_{1,t+h}
-
(\widehat\theta_h^{\operatorname{LALP}})'Y_{t,d+1},$ the OLS LALP
residual, formed by plugging in $\widehat\theta_h^{\operatorname{LALP}}$, the
vector of all coefficients in \eqref{eq:vard_lp} with $p=d+1$.\footnote{Other choices of $\widehat e_{t,h}$ are possible, including OLS
local-projection residuals without lag augmentation or GMM residuals, the
latter leading to an iterated GMM procedure. Any first-step choice, for which
the substituted residual does not affect the probability limit of the
feasible long-run variance estimator, leaves the first-order limit of the GMM
local projection estimator unchanged.} Setting $\widehat s_{t,q,h}=Y_{t,q}\widehat e_{t,h}$, we define the HAC (heteroskedasticity and autocorrelation consistent) score variance estimator
$\widehat{\Omega}_{q,h}
=
\sum_{\ell=-b_T}^{b_T}
k\!\left(\frac{\ell}{b_T}\right)
\widehat{\Gamma}_{q,h}(\ell),$
where $\widehat{\Gamma}_{q,h}(\ell)
=
T^{-1}\sum_{t=\ell+q}^{T-h}
\widehat s_{t,q,h}\widehat s_{t-\ell,q,h}'$
for $\ell\ge0$,
$\widehat{\Gamma}_{q,h}(-\ell)
=
\widehat{\Gamma}_{q,h}(\ell)'$, and $k(\cdot)$ is a bounded symmetric kernel, continuous at zero, with
$k(0)=1$, and $b_T$ is a truncation parameter.
For $q>p$, the population analog of the score process $\widehat s_{t,q,h}$ (or the one after partialling-out (Section \ref{subsec:optimal_iv})) is generally serially correlated (even when the error $u_t$ is assumed to be independent); see Section \ref{subsec:lalp_ar1} for an illustration under the AR(1) model. Thus the HAC-type variance estimator is generally required.\footnote{For the OLS estimator of $\beta_h$, see \citet{montiel2021local} and \citet{xu2023local} for the possibility of avoiding the HAC variance estimator for valid inference.}
The feasible optimally weighted estimator is $\{\widehat{\beta}_h^{\mathrm{GMM}(p,q)},\widehat{\eta}_{h,p}^{\mathrm{GMM}(p,q)}\}
\\=
\arg\min_{\beta_h,\,\eta_{h,p}}
\overline g_{h,p,q}(\beta_h,\eta_{h,p})'
\widehat{\Omega}_{q,h}^{-1}
\overline g_{h,p,q}(\beta_h,\eta_{h,p}).
$
The block of
interest has the closed form
\begin{equation}
\label{gmm_iv}
\widehat{\beta}_h^{\mathrm{GMM}(p,q)}
=
J_p
\left(
\widehat{\Sigma}_{p,q}\widehat{\Omega}_{q,h}^{-1}\widehat{\Sigma}_{p,q}'
\right)^{-1}
\widehat{\Sigma}_{p,q}\widehat{\Omega}_{q,h}^{-1}\widehat m_{q,h},
\end{equation}
where $\widehat m_{q,h}=T^{-1}\sum_{t=q}^{T-h}Y_{t,q}y_{1,t+h}$ and
$\widehat\Sigma_{p,q}=T^{-1}\sum_{t=q}^{T-h}Y_{t,p}Y_{t,q}'$.
When $q=p$, GMM($p,q$) reduces to OLS with $p$ regressors. When $q>p$, the excluded lags generate overidentifying moments,
and $\widehat\Omega_{q,h}^{-1}$ enters the estimator itself, selecting the
efficient combination of IVs; the long-run covariance is then relevant for both
estimation and inference. The variance estimator for $\widehat{\beta}_h^{\mathrm{GMM}(p,q)}$ is defined as
\begin{equation}
\label{eq:Vpq_hat}
\widehat V_h^{\mathrm{GMM}(p,q)}
=
J_p
\left(
\widehat{\Sigma}_{p,q}\widehat{\Omega}_{q,h}^{-1}\widehat{\Sigma}_{p,q}'
\right)^{-1}
J_p'.
\end{equation}
Standard errors are the square roots of the diagonal entries of
$T^{-1}\widehat V_h^{\mathrm{GMM}(p,q)}$. The construction of the variance estimator appears to rely on stationarity. As we show in Section \ref{sec:uniform}, however, the resulting inference remains robust to (near-)unit roots.
\subsection{Exactly-identified optimal IV implementation}
\label{subsec:optimal_iv}
It is useful to have an equivalent form of \eqref{gmm_iv} which separates out the parameter vector of interest. It is obtained as an exactly-identified IV representation after Frisch--Waugh--Lovell residualization.
Write $Y_{t,p}=(y_t',w_{t,p}')'$, where $w_{t,p}=(y_{t-1}',\cdots,y_{t-p+1}')'$ collects the controls. Write the instrument vector $Y_{t,q}=(z_{t,p,q}',w_{t,p}')'$, where $z_{t,p,q}=(y_t',y_{t-p}',\cdots,y_{t-q+1}')'$.
Projecting $y_t$ and $z_{t,p,q}$ on $w_{t,p}$ over
$t=q,\cdots,T-h$, gives residuals $\widehat y_t^{\perp}$ and
$\widehat z_{t,p,q}^{\perp}$, respectively. Let
$\widehat \Sigma_{p,q}^{\perp}=T^{-1}\sum_{t=q}^{T-h}
\widehat y_t^{\perp}(\widehat z_{t,p,q}^{\perp})'$. Let
$\widehat\Omega_{p,q,h}^{\perp}$ be defined like $\widehat{\Omega}_{q,h}$ except that $\widehat z_{t,p,q}^{\perp}\widehat e_{t,h}$ is used in place of $Y_{t,q}\widehat e_{t,h}$. The optimal instrument is the $K$-vector
$\widehat z_{t,p,q}^*
=
\widehat \Sigma_{p,q}^{\perp}
\bigl(\widehat\Omega_{p,q,h}^{\perp}\bigr)^{-1}
\widehat z_{t,p,q}^{\perp}.$
Define
$
\widehat\beta_h^{\operatorname{OIV}(p,q)}
=
\Biggl(
\sum_{t=q}^{T-h}
\widehat z_{t,p,q}^*\,\widehat y_t^{\perp\prime}
\Biggr)^{-1}
\sum_{t=q}^{T-h}
\widehat z_{t,p,q}^*\,y_{1,t+h},
$
obtained by solving the \emph{exactly identified} IV regression, and the variance matrix estimator
$\widehat V_h^{\operatorname{OIV}(p,q)}
=
\Bigl(
\widehat \Sigma_{p,q}^{\perp}
\bigl(\widehat\Omega_{p,q,h}^{\perp}\bigr)^{-1}
\widehat \Sigma_{p,q}^{\perp\prime}
\Bigr)^{-1}.$
We can also define the population quantities under the stationarity assumption of $y_t$, which are useful in the efficiency gain calculation (Section \ref{sec:eff}). Under stationarity, the asymptotic variance
of $\widehat{\beta}_h^{\mathrm{GMM}(p,q)}$ is $V_h^{\mathrm{GMM}(p,q)}
=
J_p
\left(
\Sigma_{p,q}\Omega_{q,h}^{-1}\Sigma_{p,q}'
\right)^{-1}
J_p',
$ where $\Omega_{q,h}
:=
\sum_{\ell=-\infty}^{\infty}
\mathbb E
\left[
Y_{t,q}e_{t,h}e_{t-\ell,h}Y_{t-\ell,q}'
\right]$, $\Sigma_{p,q}:=\mathbb E[Y_{t,p}Y_{t,q}']$ and
$Q_{p,q,h}:=\Sigma_{p,q}\Omega_{q,h}^{-1}\Sigma_{p,q}'$.
To separate out the block of interest, define
$y_t^{\perp}:=y_t-P(y_t\mid w_{t,p})\in\mathbb R^{K}$, the
residualized regressor of interest, and
$z_{t,p,q}^{\perp}:=z_{t,p,q}-P(z_{t,p,q}\mid w_{t,p})\in\mathbb R^{(q-p+1)K}$, the residualized instruments. Both are orthogonal to $w_{t,p}$. Define $\Sigma_{p,q}^{\perp}
:=
\mathbb E\!\left[y_t^{\perp}(z_{t,p,q}^{\perp})'\right],$ and
$
\Omega_{p,q,h}^{\perp}
:=
\sum_{\ell=-\infty}^{\infty}
\mathbb E\!\left[
z_{t,p,q}^{\perp}e_{t,h}\,e_{t-\ell,h}(z_{t-\ell,p,q}^{\perp})'
\right]$.
Write the $K\times K$ matrix
\begin{equation}
\label{eq:Vpq_pop_2}
V_h^{\operatorname{OIV}(p,q)}
=
\left(
\Sigma_{p,q}^{\perp}
\bigl(\Omega_{p,q,h}^{\perp}\bigr)^{-1}
\Sigma_{p,q}^{\perp\prime}
\right)^{-1}.
\end{equation}
\begin{proposition}[Optimal-IV representation of GMM local projections]
\label{prop:optimal_iv_gmm}
(i) Suppose $\widehat{\Omega}_{q,h}$ and $\widehat\Omega_{p,q,h}^{\perp}$ are nonsingular and are built from the same residuals, kernel, and
bandwidth. Then $\widehat\beta_h^{\mathrm{GMM}(p,q)}
=\widehat\beta_h^{\mathrm{OIV}(p,q)}$ and
$\widehat V_h^{\mathrm{GMM}(p,q)}
=
\widehat V_h^{\mathrm{OIV}(p,q)}.$ (ii) Under stationarity, $V_h^{\operatorname{GMM}(p,q)}
=
V_h^{\operatorname{OIV}(p,q)}$.
\end{proposition}
Proposition~\ref{prop:optimal_iv_gmm} is proved in
Appendix~\ref{app:proof_optimal_iv}. Part (i) is a finite-sample identity.\footnote{It extends the finite-sample equivalence result on the residualized-IV representation of \citet{basu2024frisch} to a setting
where the weighting is governed by serial dependence in the moment process.} We will implement the GMM in its optimal IV form; the automatic truncation selection in the HAC variance estimator directly targets the (effective) residualized score for the parameter of interest.
\textbf{\textsc{Lag orders}}. What values of $p$ and $q$ should be used in practice? Our recommendations are $p=d+1$ and $q=d+2$, after the researcher has chosen the model order $d$, typically through some data-dependent information criterion. The resulting GMM$(d+1,d+2)$ is therefore an overidentified version of the LALP of \citet{montiel2021local} with a single set of overidentifying restrictions. This recommendation is based on the fact that the lag-augmented GMM estimator shares the robustness properties of the LALP estimator which have been recently established \citep{montiel2021local,olea2024double}, as we will show in Sections \ref{sec:uniform} and \ref{sec:local} in our setting. Although more overidentifying conditions can theoretically produce larger efficiency gains, too many of them can also lead to poor performance in finite samples, especially when the optimal weighting matrix is estimated. In the next section, we will illustrate in a simple model that even one single set of overidentification conditions can in fact generate a large portion of the efficiency gain that is generated by using all possible lags as instruments.
\subsection{Non-redundancy of the overidentifying restrictions}
\label{subsec:Non-Redundancy}
While the GMM never has a larger asymptotic variance than OLS with the same set of regressors, it is natural to ask when the overidentifying restrictions provide a \textit{strict} efficiency gain. In Appendix \ref{app:gmm_vs_ols}, we provide an analysis of a general time series regression with serially correlated IVs (which accommodates the LP regression \eqref{eq:vard_lp}) to address this question, connecting to the literature on the redundancy of the moment conditions for the GMM estimator (\cite{breusch1999redundancy}, \cite{hall2007information}).
The analysis shows that the redundancy occurs only when two population projection coefficient matrices are equal. The first projection is the ordinary projection of the excluded IVs (i.e. $z_{t,p,q}^{\perp}$, in the context of the LP regression \eqref{eq:vard_lp}, using the notation in Section \ref{subsec:optimal_iv}) on the regressors of interest ($y_t^{\perp}$), after the controls are partialed out. The second projection is the \textit{long-run} projection of the orthogonal moment process ($z_{t,p,q}^{\perp}e_{t,h}$) on the regression score process ($y_t^{\perp}e_{t,h}$), after the controls are partialed out. The two projection coefficient matrices are generally different (thus the added lags carry first-order information), and the main driving factors of the non-equivalence are the serial correlation of the residualized moment process (e.g. the LP with $h>1$), and the conditional heteroskedasticity of the innovation $u_t$. In the case of LALP as we study in the next subsection, the non-redundancy condition for the extra IVs reduces to a particularly simple one.
\subsection{Lag augmentation}
\label{subsec:lag_aug}
We now focus on the LP with the lag-augmented regressor set, i.e. $p=d+1$. We address two questions: the source of the efficiency gain, and the serial correlation in the GMM moment process (why the HAC inference is generally needed).
Assume stationarity in this subsection for illustrative purposes. With lag augmentation we have $y_t^{\perp}=u_t$. Also,
$z_{t,p,q}^{\perp}=(u_t',v_{t-1}')'$, where $v_{t-1}=
\sum_{j=0}^{\infty}C_j u_{t-1-j}$ is the projection residual of $y_{t-d-1}$ on $y_{t-1},\cdots,y_{t-d}$, with the coefficients $C_j$ being square summable in their norms.\footnote{Note that $P(y_{t-d-1}\mid y_{t-1},\cdots,y_{t-d})$ is a reverse projection; it projects on future variables, not on the past, as usual.} Thus $\mathbb{E}[y_t^\perp z_{t,p,q}^{\perp\prime}]=(\Sigma_u,0_K)$.
With lag augmentation, the off-diagonal long-run term in $\operatorname{LRV}\!\left(z_{t,p,q}^\perp e_{t,h}\right)$ is what drives the efficiency gains from overidentification. If we used a diagonal weighting matrix (like the identity matrix), GMM would collapse to LALP; the formula \eqref{eq:Vpq_pop_2} would show that they had the same asymptotic variance. The efficiency gain of the GMM lives entirely in the long run covariance between $u_t e_{t,h}$ and $v_{t-1} e_{t,h}$.
\cite{montiel2021local} show that with the lag-augmented regressor set, the OLS (residualized) LP regression score process for the impulse response can be serially uncorrelated, under appropriate assumptions on the higher-order dependence in innovations (i.e. mean independence of its own past and future; their Assumption 1). Thus the standard Eicker-Huber-White HC inference remains asymptotically valid, without the need to correct for the serial correlation in the score as done by the HAC inference.\footnote{See \citet{xu2023local} for the justification of the Eicker-Huber-White HC inference under the weaker assumption of MDS innovations with a zero fourth-order cumulant condition.} By contrast, the GMM moment process $z_{t,p,q}^\perp e_{t,h}$ is generally serially correlated (even with the mean-independent or i.i.d. innovations), because of the reverse-regression residual $v_{t-1}$. We illustrate these points in the simple model in the next section.
\section{Illustration with AR(1) Data}
\label{sec:eff}
This section shows how overidentification can improve the asymptotic efficiency of the
lag-augmented local projection through a simple example.
Consider the simplest dynamic model for the scalar outcome, AR(1),
\begin{align}
\label{eq:ar1}
y_t = \alpha\, y_{t-1} + u_t, \ \ \ \ \ u_t \overset{\text{i.i.d}}{\sim} (0,\sigma^2),
\end{align}
with $|\alpha|\le 1$, $\mathbb{E}u_t=0$, and $\mathbb{E}u_t^2=
\sigma^2>0$. The GMM method proposed in this paper applies to both the standard LP (using $y_t$ as the only regressor) and the lag-augmented LP (LALP) (using $y_t$ and $y_{t-1}$ as the regressors). We focus on our recommended GMM-LALP regression here, and the corresponding results for the GMM-LP under the AR(1) model are provided in Appendix \ref{app:gmm_lp}.
For $h\ge1$, under the AR(1) model, the LALP regression is $y_{t+h}
=
\beta_h y_{t}+\eta_{h}y_{t-1}+e_{t,h}$, where
$\beta_h=\alpha^h$, $\eta_{h}=0$, and $e_{t,h}=\sum_{i=0}^{h-1}\alpha^i u_{t+h-i}$.
\subsection{The first overidentified estimator}
\label{subsec:lalp_ar1}
We first consider the simplest overidentified estimator, $\operatorname{GMM}(2,3)$. As the benchmark, the LALP estimator of \citet{montiel2021local} partials out $y_{t-1}$ and reduces the regressor effectively to $y_t-\alpha y_{t-1}=u_t$, so the
identifying variation is the innovation in $y_t$. LALP uses this single moment, and has the asymptotic variance
$V_h^{\mathrm{LALP}}
=
\sum_{j=0}^{h-1}\alpha^{2j}.$
With one overidentifying restriction, $\operatorname{GMM}(2,3)$ uses $y_{t-2}$ as the single excluded instrument, besides the regressors $y_{t}$ and $y_{t-1}$.
Specializing the analysis of Section \ref{subsec:lag_aug} to the AR(1) model, and using the notation there, we can show (Appendix~\ref{subsec:hac_gmmlalp}) that if $|\alpha|<1$, the residualized excluded instrument is
\begin{align}
\label{eq:reverse_residual}
v_{t-1}:=y_{t-2}-P(y_{t-2}\mid y_{t-1})
=
-\alpha u_{t-1}
+
(1-\alpha^2)
\sum_{j=0}^{\infty}\alpha^j u_{t-2-j}.
\end{align}
The residualized regressor of interest is $y_t^\perp=y_t-P(y_t\mid y_{t-1})
=u_t$. So the residualized instrument vector is $z_t^\perp=(u_t,v_{t-1})'$. Thus $\mathbb{E}[y_t^\perp z_t^{\perp\prime}]=\sigma^2(1,0)$. Then
\begin{align}
\label{eq:resid_lrv_gmmlalp}
\operatorname{LRV}\!\left(z_t^\perp e_{t,h}\right)
=
\sigma^4
\begin{pmatrix}
\sum_{j=0}^{h-1}\alpha^{2j} & -(h-1)\alpha^{2h-2}\\
-(h-1)\alpha^{2h-2} & \sum_{j=0}^{h-1}\alpha^{2j}
\end{pmatrix},
\end{align}
which is non-diagonal whenever $h>1$ and $\alpha\ne0$.
Equation \eqref{eq:resid_lrv_gmmlalp} also shows the serial correlation in $z_t^\perp e_{t,h}$. The
contemporaneous covariance matrix of $z_t^\perp e_{t,h}$, $\operatorname{Var}\left(z_t^\perp e_{t,h}\right)=\sigma^4\left(\sum_{j=0}^{h-1}\alpha^{2j}\right)I_2$, is diagonal, and its difference from the LRV matrix arises entirely from the off-diagonal elements.
The residualized
GMM variance formula $\left\{
\mathbb{E}(y_t^\perp z_t^{\perp\prime})
\bigl[\operatorname{LRV}\!\left(z_t^\perp e_{t,h}\right)]^{-1}
\mathbb{E}(y_t^\perp z_t^{\perp\prime})'
\right\}^{-1}$ then gives
\begin{align}
\label{eq:vgmm23}
V_h^{\mathrm{GMM}(2,3)}
&=
\sum_{j=0}^{h-1}\alpha^{2j}
-
\frac{(h-1)^2\alpha^{4h-4}}{\sum_{j=0}^{h-1}\alpha^{2j}}.
\end{align}
The asymptotic variance formula \eqref{eq:vgmm23} in fact extends to the unit root case $\alpha=1$, due to the uniform limit theory we establish in Section \ref{sec:uniform}. Comparison with $V_h^{\mathrm{LALP}}
=
\sum_{j=0}^{h-1}\alpha^{2j}$ shows $V_h^{\mathrm{GMM}(2,3)}\leq V_h^{\mathrm{LALP}}$,
and the strict inequality holds for $h>1$, $\alpha\ne0$.
\begin{figure}[t]
\hspace*{-0.5cm}
\makebox[\linewidth][c]{
\includegraphics[width=1.4\textwidth]{Notes/efficiency_IVs_LALP_combine_only_two_kx_update.eps}
}
\caption{Asymptotic standard-error ratio between LALP and GMM$(2,3)$ (solid lines), and between LALP and GMM$(2,\infty)$ (dashed lines).
[AR(1) DGP as in \eqref{eq:ar1}]}
\label{fig:gmm_lalp_ratio}
\begin{minipage}{0.95\linewidth}
\footnotesize
\emph{Note:} The figure plots $\sqrt{V_h^{\mathrm{GMM}(2,2)}/V_h^{\mathrm{GMM}(2,q)}}$
for $q\in\{3,\infty\}$, where $q$ is the number of IVs. $\alpha$ is the slope of the AR(1) DGP. The just-identified $\operatorname{GMM}(2,2)$
coincides with the LALP estimator of \citet{montiel2021local}. We use the formulas in Table~\ref{tab:unit_root_variance_ordering} for the three asymptotic variances for $q\in\{2,3,\infty\}$.
\end{minipage}
\end{figure}
Figure \ref{fig:gmm_lalp_ratio} visualizes the efficiency comparison between LALP and GMM$(2,3)$ by plotting the ratio of the square roots of their asymptotic variances. The figure shows that the efficiency gain of GMM depends on both the persistence level $\alpha$ and the horizon $h$. When the data are only weakly dependent (like $\alpha=0.5$), there is little efficiency gain.
The biggest efficiency gain is achieved at the unit root $\alpha=1$, where
$V_h^{\mathrm{LALP}}=h$, while
$V_h^{\mathrm{GMM}(2,3)}=(2h-1)/h<2$: a single extra lag as the excluded IV caps the variance below
$2$ at every horizon, whereas LALP grows without bound.
As indicated in Section \ref{subsec:lag_aug}, the efficiency gain is driven by the long-run score covariance, which specializes under the AR(1) model \eqref{eq:ar1} as $\operatorname{LRCov}\!\left(u_t e_{t,h},\,v_{t-1}e_{t,h}\right)
=
-(h-1)\sigma^4\alpha^{2h-2}$.
Exact redundancy arises only at $h=1$ (no overlap) or
$\alpha=0$ (no propagation). For $|\alpha|$ sufficiently far from unity, $(h-1)^2\alpha^{4h-4}$ is small for moderate and large $h$, leading to the asymptotic variance of GMM close to that of OLS.
\subsection{More instruments}
\label{sec:theory_ar1}
The asymptotic efficiency of the GMM$(2,3)$ estimator can be further improved if we use further lags to overidentify the LALP. For general $q$, the asymptotic variance of GMM$(2,q)$ under the DGP \eqref{eq:ar1} is given in the following proposition. We also consider the particularly interesting case in which $q$ diverges to infinity, corresponding to the efficiency gain that can be achieved by exploiting information from the infinite past.
\begin{proposition}
\label{prop:ar1_ordering}
Let $\{y_t\}$ follow \eqref{eq:ar1}. For $h\ge1$, write
$\gamma_{h,\ell}=\alpha^\ell\sum_{j=0}^{h-\ell-1}\alpha^{2j}$ for $0\le\ell\le h-1$
and $\gamma_{h,\ell}=0$ otherwise. Then:
\begin{enumerate}[label=(\roman*)]
\item For $q\ge3$,
$V_h^{\mathrm{GMM}(2,q)}=\gamma_{h,0}-r_{h,q}'A_{h,q}^{-1}r_{h,q},$
where the $(q-2)$-vector $r_{h,q}$ has entries
$(r_{h,q})_k=-(h-k)\alpha^{2h-k-1}$ for $1\le k\le\min\{h-1,q-2\}$ and zero
otherwise, and $A_{h,q}$ is the $(q-2)\times(q-2)$ Toeplitz matrix with
$(A_{h,q})_{k\ell}=\gamma_{h,|k-\ell|}$.
\item $\lim_{q\to\infty}\displaystyle V_h^{\mathrm{GMM}(2,q)}=\gamma_{h,0}-\alpha^{2h}\gamma_{h-1,0}
=\sum_{j=0}^{h-1}\alpha^{2j}-\sum_{j=h}^{2h-2}\alpha^{2j}$.
\end{enumerate}
\end{proposition}
The proof is given in
Appendix~\ref{subsec:hac_gmmlalp}.
In Figure \ref{fig:gmm_lalp_ratio}, we plot the ratio of standard deviations between LALP and GMM$(2,\infty)$, together with the ratio of standard deviations between LALP and GMM$(2,3)$. The relative efficiency for other values of $q$ lies between these two curves. At the unit root $\alpha=1$, the IV estimator optimally weighting the infinite past brings the asymptotic variance down to 1, across all horizons $h$, compared with the previous result for GMM(2,3), $2-h^{-1}$. The results also show that a substantial efficiency gain can be obtained with \textit{only one overidentifying restriction}. Additional results (not reported) indicate that unless $q$ is much larger than 3, the further efficiency gain over GMM$(2,3)$ is quite minor (e.g. the relative efficiency gain at $q=6$ is almost indistinguishable from that at $q=3$).
\begin{table}[t]
\centering
\caption{Asymptotic variances of various estimators of the lag augmentation LP regression. [AR(1) DGP as in \eqref{eq:ar1}]}
\label{tab:unit_root_variance_ordering}
\setlength{\tabcolsep}{14pt}
\begin{tabular}{l|l|l}
\hline\hline
Estimator & Variance for $|\alpha|\le1$ & Variance at $\alpha=1$ \\
\hline
&& \\
$\operatorname{GMM}(2,\infty)$ &
$\sum_{j=0}^{h-1}\alpha^{2j}-\sum_{j=h}^{2h-2}\alpha^{2j}$ & $1$ \\
$\operatorname{GMM}(2,3)$
&
$\sum_{j=0}^{h-1}\alpha^{2j}
-
(h-1)^2\alpha^{4h-4}/(\sum_{j=0}^{h-1}\alpha^{2j})$& $2-1/h$ \\
LALP (i.e. GMM(2,2)) &
$\sum_{j=0}^{h-1}\alpha^{2j}$& $h$ \\
LA-AR &
$h^2\alpha^{2(h-1)}$ & $h^2$ \\
\hline\hline
\end{tabular}
\par\vspace{0.8em}
\begin{minipage}{0.9\linewidth}
\footnotesize
\emph{Note:} Asymptotic variances of ordinary LALP, the first overidentified
GMM-LALP estimator, its infinite-instrument limit, and the recursive lag-augmented
AR estimator. The infinite-instrument case is the limit of the optimally
weighted $\operatorname{GMM}(2,q)$ as $q\to\infty$.
\end{minipage}
\end{table}
For convenience, we summarize the comparison in Table \ref{tab:unit_root_variance_ordering}, with particular attention to the unit root case $\alpha=1$.
In Table \ref{tab:unit_root_variance_ordering}, we also include the iterated lag-augmented AR (LA-AR) estimator of \citet{inoue2020uniform}, which is analyzed in \cite{montiel2021local} (p.1817).
Compared with LA-AR, the efficiency advantage of the GMM mainly lies in the high-persistence regime. For example, for $\alpha$ between 0.93 and 1, and for horizons up to 36, the GMM(2,3) always has smaller asymptotic variance than LA-AR. At the unit root, the variance of LA-AR is $h^2$, growing quadratically with the horizon, while that of GMM(2,3) remains uniformly bounded by 2. If the data are only weakly dependent, the LA-AR typically has the smaller variance. Nevertheless, as we show in Section \ref{sec:local}, following the analysis of \citet{olea2024double}, the inference based on the LA-AR estimator is more fragile to model misspecification than our GMM estimator.
\section{Uniform inference}
\label{sec:uniform}
This section establishes that the Wald statistic based on a general GMM($p,q$) LP estimator is asymptotically pivotal \emph{uniformly} over the persistence of the data under the specified conditions. The following assumption is also used in \cite{montiel2021local} (definition 1, p.1804). Following their framework, we assume that the VAR lag order $d$ is fixed.
\begin{assumption}[VAR parameter space]
\label{ass:persistent} Let $A(L)=I_K-A_1L-\cdots-A_dL^d$ denote the VAR polynomial
in \eqref{eq:vard}. Suppose that
$A(L)=B(L)(I_K-DL)$, where
$B(L)=I_K-B_1L-\cdots-B_{d-1}L^{d-1}$ and
$D=\operatorname{diag}(\rho_1,\cdots,\rho_K)$.
For $k=1,\cdots,K$, $\rho_k\in[-1,1]$. The companion matrix
$\mathbf B$ of $B(L)$ satisfies
$\|\mathbf B^\ell\|_2\le C(1-c)^\ell$ for every $\ell\ge1$,
with $C<\infty$ and $c\in(0,1)$.
\end{assumption}
Let $\xi_t=(I_K-DL)y_t$. Then
\begin{equation}
y_t
=
Dy_{t-1}+\xi_t,
\label{eq:persistent-decomposition}
\end{equation}
or, equivalently, $y_{k,t}=\rho_ky_{k,t-1}+\xi_{k,t}$ for each $k$. Thus each variable carries its own potentially persistent root, while $B(L)$ collects the transitory dynamics.
For each $k$, define $\tau_{k,T}=
\sum_{\ell=0}^{T-1}\rho_k^{2\ell}$ and $\Lambda_T=\operatorname{diag}(\tau_{1,T},\cdots,\tau_{K,T})$. In what follows the parameters may depend on $T$. We suppress the triangular-array index for expositional simplicity, and all constants below are uniform over the parameter space and over $T$. Letting the roots vary with $T$ embeds fixed stationary dynamics, shrinking-to-unity sequences, and exact unit roots of either sign in a single parameter space.\footnote{The diagonal form of $D$ is for expository purposes; the same argument applies after a fixed rotation separating the persistent directions from the transitory block.}
\begin{assumption}[Innovations and transitory process]
\label{ass:stable-score}
(i) $\{u_t,\mathcal F_t\}$ is a martingale difference sequence (MDS), where $\mathcal F_t=\sigma(y_0,u_s:s\le t)$;
(ii) $\{u_t\}$ is stationary and $\alpha$-mixing with coefficients $\alpha(j)$ satisfying
$\sum_{j\ge1} j^{2/(r-2)}\alpha(j)^{1-2/r}<\infty$
for some $r>2$, and
$\mathbb E\|u_t\|^{4r+\delta}\le C$
for some $\delta>0$;
(iii) The initial condition satisfies
$\mathbb E\|\Lambda_T^{-1/2}y_0\|^{4r+\delta}\le C$ uniformly over the parameter space.
\end{assumption}
Write $\mathcal P$ for the set of parameters admitted by
Assumption~\ref{ass:persistent}, a generic
element of which is denoted $(D,B)$. Assumption~\ref{ass:stable-score} (i)-(ii) is imposed only on
the primitive innovation process $\{u_t\}$. The moment and dependence
properties of the transitory process $\xi_t$ are instead derived from
its stable-filter representation. Under correct specification (as in this section),
$\xi_t=B(L)^{-1}u_t$; under local misspecification (in the next section), the corresponding
representation is given in \eqref{eq:aux-local-xi-representation}.
\subsection{A rotation of the lags}
\label{subsec:uniform-rotation}
The following rotation separates quasi-differences from levels of the time series, which is especially useful in the persistent regime. We assume $p\ge2$, the rotation thus places $\beta_h$ on a quasi-difference. We emphasize that the rotation is only used in the proof, and does not change any implementation detailed in Section \ref{sec:implementation}.
Let $\Delta_p$ be block upper-bidiagonal such that
\begin{align}
\underset{pK\times pK}{\Delta_p}
&=
\begin{pmatrix}
I_K & -D & 0 & \cdots & 0\\
0 & I_K & -D & \ddots & \vdots\\
\vdots & \ddots & \ddots & \ddots & 0\\
0 & \cdots & 0 & I_K & -D\\
0 & \cdots & \cdots & 0 & I_K
\end{pmatrix}, \quad \underset{pK\times pK}{\Delta_p^{-1}} =
\begin{pmatrix}
I_K & D & D^2 & \cdots & D^{p-1} \\
0 & I_K & D & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & D^2 \\
0 & \cdots & 0 & I_K & D \\
0 & \cdots & \cdots & 0 & I_K
\end{pmatrix}
,
\label{eq:delta-p}
\end{align}
where the inverse follows immediately.
Let $\Upsilon_p=\operatorname{diag}\left(I_{(p-1)K},\Lambda_T^{-1/2}\right)$, and set $H_p=\Upsilon_p\Delta_p$. Then
\begin{align}
H_pY_{t,p}
=
\left(
\xi_t',\,
\xi_{t-1}',\,
\cdots,\,
\xi_{t-p+2}',\,
(\Lambda_T^{-1/2}y_{t-p+1})'
\right)',
\label{eq:rotated-vector}
\end{align}
which stacks quasi-differences above scaled levels whose second moments are uniformly bounded. Their lower-tail nondegeneracy is imposed in Assumption~\ref{ass:rotated-nonsingularity} below.
Rotate the sample objects by $\widecheck\Sigma_{p,q}=H_p\widehat\Sigma_{p,q}H_q'$, $\widecheck\Omega_{q,h}=H_q\widehat\Omega_{q,h}H_q'$, and $\widecheck m_{q,h}=H_q\widehat m_{q,h}$. For $p\ge2$, we have $J_pH_p'
=
J_p,
H_pJ_p'
=
J_p',$
$J_p(H_p')^{-1}=J_p$, and $H_p^{-1}J_p'=J_p'$.
Hence the estimator \eqref{gmm_iv} is numerically identical in the
rotated and original coordinates,
$\widehat\beta_h^{\operatorname{GMM}(p,q)}
=
J_p
\left(
\widecheck\Sigma_{p,q}\widecheck\Omega_{q,h}^{-1}\widecheck\Sigma_{p,q}'
\right)^{-1}
\widecheck\Sigma_{p,q}\widecheck\Omega_{q,h}^{-1}\widecheck m_{q,h},$
and so is its variance \eqref{eq:Vpq_hat},
$\widehat V_h^{\operatorname{GMM}(p,q)}=J_p(\widecheck\Sigma_{p,q}\widecheck\Omega_{q,h}^{-1}\widecheck\Sigma_{p,q}')^{-1}J_p'.$
The rotation therefore reorganizes only the nuisance directions.
\subsection{Wald statistic}
\label{subsec:uniform-main}
For the uniformity result, we impose the following high-level condition. A similar condition is also used in \cite{montiel2021local} and \cite{xu2023local}.
\begin{assumption}[Rotated nonsingularity]
\label{ass:rotated-nonsingularity}
For every $\varepsilon>0$ there exists $\underline c>0$ such that
\begin{align}
\liminf_{T\to\infty}
\inf_{(D,B)\in\mathcal P}
\mathbb P_{D,B}
\left\{
\lambda_{\min}(\widecheck\Sigma_{p,p})\ge\underline c,\;
\lambda_{\min}(\widecheck\Omega_{q,h})\ge\underline c
\right\}
\ge1-\varepsilon.
\label{eq:rotated-nonsingularity}
\end{align}
\end{assumption}
The stochastic formulation in Assumption \ref{ass:rotated-nonsingularity} accommodates every persistence regime. When $\max_k|\rho_k|\le1-c$, it reduces to the usual requirement that the stationary design and long-run covariance limits be positive definite. When some $|\rho_k|\to1$, the corresponding scaled-level block may converge to a random quadratic functional. Assumption~\ref{ass:rotated-nonsingularity} requires the lower tails of this block and of the rotated HAC matrix to remain uniformly away from singularity in probability. Appendix~\ref{app:local-rotated-nonsingularity} provides an illustration. Throughout the paper, the truncation parameter for the HAC estimator is such that $b_T\to\infty$ with $b_T=o(T^{1/2})$.
\begin{theorem}[Uniform GMM inference]
\label{thm:persistence-uniform-gmm}
Let $G$ be a known $g\times K$ matrix of rank $g\le K$. Let $q\ge p\ge\max(d,2)$. Define the Wald statistic
$\\ W_{T,h}
=
T
\bigl(G\widehat\beta_h^{\operatorname{GMM}(p,q)}-G\beta_h\bigr)'
\bigl(
G\widehat V_h^{\operatorname{GMM}(p,q)}G'
\bigr)^{-1}
\bigl(G\widehat\beta_h^{\operatorname{GMM}(p,q)}-G\beta_h\bigr),$
where $\widehat V_h^{\operatorname{GMM}(p,q)}$ is defined in Equation~\eqref{eq:Vpq_hat}. Then under Assumptions~\ref{ass:persistent}--\ref{ass:rotated-nonsingularity},
\begin{align}
\limsup_{T\to\infty}
\;
\sup_{(D,B)\in\mathcal P}
\;
\sup_{x\in\mathbb R}
\left|
\mathbb P_{D,B}(W_{T,h}\le x)
-
F_{\chi_g^2}(x)
\right|
=
0,
\label{eq:uniform-wald-convergence}
\end{align}
where $F_{\chi_g^2}(x)$ is the cumulative distribution function of the chi-square variate with $g$ degrees of freedom.
\end{theorem}
Confidence sets that invert $W_{T,h}$ at the $\chi^2_{g,1-\alpha}$ critical value therefore have asymptotic coverage $1-\alpha$ uniformly over $\mathcal P$. In practice, inference can be improved by the bootstrap, which we implement in our numerical studies in Section \ref{sec:mc} and Section \ref{sec:application}.
The proof of Theorem~\ref{thm:persistence-uniform-gmm}, given in Appendix \ref{sec:uniform-theorem-proof}, splits the parameter space in two. When $\max_k|\rho_k|\le1-c$ the rotated system is stationary and a triangular-array GMM argument applies. When $\max_k|\rho_k|\to1$ the cross blocks between quasi-differences and levels vanish once multiplied by $\Lambda_T^{-1/2}$, so the scaled levels survive only in the nuisance block on the diagonal and affect neither the estimator of $\beta_h$ nor its selected variance. This is the mechanism behind the observation of \citet{sims1990inference} that coefficients on stationary linear combinations of the regressors remain asymptotically normal in the presence of unit roots; the rotation shows that, once $p\ge2$, $\beta_h$ is such a coefficient.
Our proof strategy differs from that used for just-identified local projections. \citet{montiel2021local} partial the lagged controls out of $y_{1,t+h}$ and of $y_t$, reducing inference on $\beta_h$ to a regression on a residualized regressor. Our argument instead follows the classical VAR literature: estimate the whole projection coefficient jointly, rotate so that each block is a quasi-difference or a scaled level, and read the estimator of $\beta_h$ off the first block. An advantage of this method is that we only require $p \ge max(d,2)$, while the residualization-based proof typically needs $p \ge d+1$. Additional lags are therefore allowed but not required for persistence robustness.
A simplifying assumption we make is that $h$ is a fixed horizon. \cite{montiel2021local} and \cite{xu2023local} allow $h$ to grow with the sample size, but both rely on the residualization-based arguments. Moreover, the inference procedure of \cite{montiel2021local} relies on the two-sided MDS assumption which we dispense with. Nevertheless, we find in simulations that the GMM-based LP inference is quite robust over a wide range of horizons.
\section{GMM-LALP under local misspecification}
\label{sec:local}
A finite-order VAR is rarely the true data-generating process. Temporal
aggregation, marginalization of omitted variables, and the equilibrium
dynamics of structural models generally induce moving-average components,
so the reduced form is generically VARMA.\footnote{See
\cite{lutkepohl2005,ravenna2007,fernandez2007}.} This section establishes
the validity of GMM-LALP under local VARMA departures in the spirit of
\citet{olea2024double}, while allowing unit
or near-unit roots in the data.
\subsection{A locally misspecified VAR}
We extend the model \eqref{eq:vard} to the following specification
\begin{equation}
\label{localmis}
y_t
=
A_1y_{t-1}+\cdots+A_dy_{t-d}+\widetilde u_t,
\end{equation}
where the VAR error $\widetilde u_t$ is not exactly a white noise process. Instead, $\widetilde u_t$ is assumed to follow the local VARMA($m,n$) law
\begin{align}
\label{eq:local_arma_error}
\prod_{j=1}^{m}\bigl(I_K-\Phi_{j,T}L\bigr)\widetilde u_t
=
\prod_{k=1}^{n}\bigl(I_K+\Theta_{k,T}L\bigr)u_t,
\end{align}
where the lag orders $m$ and $n$ are fixed but unknown. To regulate the degree of misspecification, assume
$\Phi_{j,T}=T^{-\zeta_{AR}}\Phi_j$ and
$\Theta_{k,T}=T^{-\zeta_{MA}}\Theta_k$, where
$\zeta_{AR}>0$, $\zeta_{MA}>0$, and
$\Phi_j$ and $\Theta_k$ are fixed matrices. Write $\bar\zeta=\min\{\zeta_{AR},\zeta_{MA}\}$.
By \eqref{eq:local_arma_error}, we can write
\begin{align}
\label{eq:local-filter-expansion}
\left\{
\prod_{j=1}^{m}
\bigl(I_K-\Phi_{j,T}L\bigr)
\right\}^{-1}
\prod_{k=1}^{n}
\bigl(I_K+\Theta_{k,T}L\bigr)
=
I_K+\sum_{j\geq1}\Pi_{j,T}L^j.
\end{align}
Each
$\Pi_{j,T}$ is a finite sum of products of exactly $j$ local matrices
and hence $\|\Pi_{j,T}\|=O(T^{-j\bar\zeta})$. Thus, for every $r\geq1$,
$\sum_{j\geq r}\|\Pi_{j,T}\|=O(T^{-r\bar\zeta})$; see Appendix
\ref{subsec:local-aux-lemmas}
(Lemma~\ref{lem:local-filter-approximation}).
\subsection{Lag augmentation and robustness}
\label{subsec:local-asym}
The following proposition gives an exact population representation for the local projection regression under the models \eqref{localmis} and
\eqref{eq:local_arma_error}, using $s$ additional lags as regressors.
\begin{proposition}[Exact lag-augmented representation]
\label{prop:local-lalp-representation}
Let $p=d+s$, for $s\geq0$. Under \eqref{localmis} and
\eqref{eq:local_arma_error}, if $s\geq1$,
\begin{equation}
\label{eq:lalp_local}
y_{1,t+h}
=
(\beta_{h,T}',\eta_{h,p,T}')Y_{t,p}
+
e_{t,h,T},
\end{equation}
for some $\eta_{h,p,T}$, where
\begin{equation}
\label{eq:local_beta_T}
\beta_{h,T}
=
\beta_h
+
\sum_{j=1}^{h}
\Pi_{j,T}'\beta_{h-j}.
\end{equation}
Moreover,
\begin{align}
\label{eq:local-error-decomposition}
e_{t,h,T}
&=
\nu_1'
\sum_{\ell=0}^{h-1}
\Psi_\ell u_{t+h-\ell}
+
\nu_1'
\sum_{j=1}^{\min\{s,h-1\}}
\sum_{\ell=0}^{h-j-1}
\Psi_\ell\Pi_{j,T}
u_{t+h-\ell-j}
+
\mathcal E_{t,h,T},
\end{align}
where
$\mathcal E_{t,h,T}
=
O_p(T^{-(s+1)\bar\zeta})$.
If $s=0$, \eqref{eq:lalp_local} holds with
$(\beta_{h,T},\eta_{h,p,T})=(\beta_h,\eta_{h,d})$, and
\eqref{eq:local-error-decomposition} holds with
$\mathcal E_{t,h,T}=O_p(T^{-\bar\zeta})$.
\end{proposition}
The proof of Proposition~\ref{prop:local-lalp-representation} is given in Appendix \ref{proof:local_mis_lalp_exact}, where
explicit expressions for $\eta_{h,p,T}$ and $\mathcal E_{t,h,T}$ are
also provided. Importantly, the regression form \eqref{eq:lalp_local} shows that under the models \eqref{localmis} and
\eqref{eq:local_arma_error}, the true impulse response is $\beta_{h,T}$. This can be seen from the fact that the error $e_{t,h,T}$ only involves the past ($\mathcal E_{t,h,T}$) and the future (other terms in $e_{t,h,T}$) of $u_t$, but not the contemporaneous innovation $u_t$. Note that $\beta_{h,T}$ generally differs from $\beta_{h}$, the true response under the exact finite-order VAR($d$) as in Section \ref{sec:uniform}, and the difference is of order $T^{-\bar\zeta}$, which can be larger than the order $T^{-1/2}$.
Proposition~\ref{prop:local-lalp-representation} isolates the role of
lag augmentation. With $p=d+s$, the additional $s$ lags successively
absorb the predetermined components through order $s$, leaving terms
of order $T^{-(s+1)\bar\zeta}$. The goal of this section is to study inference based on the same
estimator $\widehat\beta_h^{\operatorname{GMM}(p,q)}$ and its variance
estimator $\widehat V_h^{\operatorname{GMM}(p,q)}$ as in previous
sections under the models \eqref{localmis} and
\eqref{eq:local_arma_error}. The following theorem gives the result.
\begin{theorem}
\label{theorem:local_mis_lalp}
Let $s\geq0$ and
$q\geq p=d+s>1$ be fixed. Suppose $\{y_t\}$ is generated by
\eqref{localmis} and \eqref{eq:local_arma_error}, and
Assumptions~\ref{ass:persistent}--\ref{ass:rotated-nonsingularity}
hold. Assume
\begin{equation}
\bar\zeta=\min\{\zeta_{AR},\zeta_{MA}\}>1/(2s+2).
\end{equation}
Then the Wald statistic
$W_{T,h}^{\operatorname{local}}
=
T
\bigl(G\widehat\beta_h^{\operatorname{GMM}(p,q)}-G\beta_{h,T}\bigr)'
\bigl(G\widehat V_h^{\operatorname{GMM}(p,q)}G'\bigr)^{-1}
\bigl(G\widehat\beta_h^{\operatorname{GMM}(p,q)}-G\beta_{h,T}\bigr)$, where $G$ is as in
Theorem~\ref{thm:persistence-uniform-gmm},
satisfies
\begin{equation}
\label{eq:local_mis_uniform}
\limsup_{T\to\infty}
\;
\sup_{(D,B)\in\mathcal P}
\;
\sup_{x\in\mathbb R}
\left|
\mathbb P_{D,B}
\left(
W_{T,h}^{\operatorname{local}}\le x
\right)
-
F_{\chi_g^2}(x)
\right|
=
0.
\end{equation}
\end{theorem}
Theorem~\ref{theorem:local_mis_lalp} establishes persistence-uniform
inference for GMM-LALP under local VARMA misspecification.
With $p=d+s$, $s\geq0$, valid inference requires the model to be sufficiently close to the VAR($d$) such that
$\bar\zeta>1/(2s+2)$. A larger $s$ allows the inference to remain robust under a greater degree of misspecification. Thus $s=0$ gives the trivial condition
$\bar\zeta>1/2$; without lag augmentation, the inference is robust to misspecification of smaller order than $T^{-1/2}$. $s=1$ recovers the $T^{-1/4}$ boundary
considered by \citet{olea2024double}. With two-lag augmentation ($s=2$), the inference can be robust to misspecification of order arbitrarily close to $T^{-1/6}$.
The proof of Theorem \ref{theorem:local_mis_lalp}, given in Appendix~\ref{proof:local_mis_lalp}, combines
Proposition~\ref{prop:local-lalp-representation} with the
persistence-uniform argument in Section~\ref{sec:uniform}. After
rotation, the moments generated by $\mathcal E_{t,h,T}$ are uniformly
negligible because $T^{1/2-(s+1)\bar\zeta}\to0$. When $s=0$, the same
condition gives
$\sqrt T\|\beta_{h,T}-\beta_h\|=o(1)$.
The same argument establishes first-order equivalence between the
feasible HAC matrix based on $\widehat e_{t,h}$ and its infeasible
counterpart based on $e_{t,h,T}$.
The nonsingularity requirement in Assumption~\ref{ass:rotated-nonsingularity} needed for Theorem \ref{theorem:local_mis_lalp} is similar to the one needed for Theorem \ref{thm:persistence-uniform-gmm}. The local VARMA filter perturbs both
the rotated sample second moments and the corresponding long-run
covariance matrix by asymptotically negligible terms, so that their
smallest eigenvalues inherit the lower bounds of the reference VAR.
Appendix~\ref{app:local-rotated-nonsingularity} illustrates how this
condition can be verified under local misspecification in an AR$(1)$
model with local ARMA$(1,1)$ errors.
We conclude this section by an illustration of the failure of the similarly constructed VAR-based estimator in a simple model. Consider the scalar AR$(1)$ model with local ARMA(1,1) innovations; a special case of \eqref{localmis} and
\eqref{eq:local_arma_error} with $d=m=n=1$. Assume stationarity, $|\alpha|<1$, where $\alpha$ is the AR(1) slope. Let $\zeta_{AR}=\zeta_{MA}=1/2$, so that
$\Pi_{1,T}=T^{-1/2}\pi$, where
$\pi=\Phi_1+\Theta_1$ is scalar. Consider the \emph{lag-augmented} VAR-based impulse response estimator of
\citet{inoue2020uniform}. The estimator fits an AR$(2)$ to obtain two slopes $\widehat\phi_1$ and $\widehat\phi_2$, and reports $\widehat\phi_1^{\,h}$ as the impulse response estimate at horizon $h$. The analysis by \cite{olea2024double} focuses on the \emph{non-augmented} VAR-based response estimator, thus does not cover the lag-augmented estimator.
Since $\beta_h=\alpha^h$, \eqref{eq:local_beta_T} gives
$\beta_{h,T}
=
\alpha^h
+
T^{-1/2}\pi\alpha^{h-1}
+
\sum_{j=2}^{h}
T^{-j/2}\pi\Phi_1^{\,j-1}\alpha^{h-j}$.
The last sum is $O(T^{-1})$ for given $h$.
In Appendix \ref{sec:uniform-lemmas} (Remark \ref{rem:var-illustration}), we show that
\begin{equation}
\sqrt T\bigl(\widehat\phi_1^{\,h}-\beta_{h,T}\bigr)
\xrightarrow{d}
N\Bigl(\alpha^{h-1}\pi(h-1),\;h^2\alpha^{2(h-1)}\Bigr).
\label{eq:var-illustration-aug}
\end{equation}
Compared with the non-augmented estimator, the lag augmentation eliminates the bias (due to misspecification) only at $h=1$ (unless $\alpha=0$).\footnote{The \emph{non-augmented} VAR-based response estimator fits an AR$(1)$ and reports
$\widehat\alpha^{\,h}$ as the impulse response estimate for given $h$. In Appendix \ref{proof:local_mis_lalp}, we show that, in our setting,
\begin{equation}
\sqrt T\bigl(\widehat\alpha^{\,h}-\beta_{h,T}\bigr)
\xrightarrow{d}
N\Bigl(\alpha^{h-1}\pi\bigl[h(1-\alpha^2)-1\bigr],\;
h^2\alpha^{2(h-1)}(1-\alpha^2)\Bigr).
\label{eq:var-illustration-nonaug}
\end{equation}}
The LA-AR estimator is generally asymptotically biased for $h\ge2$.
Because of the bias, the Wald statistic formed from the lag-augmented VAR response estimator $\widehat\phi_1^{\,h}$ (the square of \eqref{eq:var-illustration-aug} divided by a variance estimate) is not asymptotically centered chi-squared, so the inference is asymptotically distorted. If the local
misspecification is larger than order $T^{-1/2}$, equivalently allowing
$\pi\to\infty$, the noncentrality diverges and the VAR-based
Wald statistic explodes. By contrast, under the simple illustrative scalar model with local misspecification, the GMM-LALP Wald statistic (including the one based on OLS LALP as a special case) is asymptotically central $\chi_1^2$ by Theorem~\ref{theorem:local_mis_lalp} even when the local
misspecification is larger than order $T^{-1/2}$.
\section{Monte Carlo simulation}
\label{sec:mc}
Sections \ref{sec:mc} and \ref{sec:application} examine the finite-sample performance of inference based on our recommended GMM-LALP procedure, GMM($d+1,d+2$) (overidentification by one), and compare it with the LALP method of \cite{montiel2021local}. Given the popularity of the LALP and the similar robust properties our GMM approach shares, we focus on the LALP as the comparison target. For practical purposes, we mainly consider the bootstrap implementation of both methods following the algorithm given in \cite{montiel2021local}.
\begin{figure}[t]
\centering
\begin{minipage}{1.05\textwidth}
\noindent\hspace*{-2cm}
\includegraphics[
width=1.2\linewidth,
height=1.99\textheight,
keepaspectratio
]{Notes/figure_coverage_bandwidth_0501_360.png}
\caption{Bootstrap confidence intervals (CIs) based on GMM(2,3) and LALP methods, under the AR(1) data generating process with AR coefficient $\alpha \in \{0.5, 0.99, 1\}$, $T = 360$.}
\label{fig:coverage_bandwidth_theta1_360}
\vspace{-0.0em}
\footnotesize
\textit{Notes:}
Left panels plot the coverage probabilities of bootstrap CIs (Nominal level: 90\%). Right panels plot the ratio of the length of LALP to GMM-LALP CIs; values above one indicate that GMM-LALP CIs are shorter.
\end{minipage}
\end{figure}
In the Monte Carlo simulations, the data are
generated from a scalar ARMA$(1,1)$ process with i.i.d.\ standard normal innovations, the AR coefficient $\alpha \in \{0.5, 0.99, 1\}$ and MA coefficient
$\theta \in \{0, 0.2\}$. For each replication, we demean the series before applying the estimators and bootstrap procedures, and the subsequent local-projection and GMM-LALP regressions are estimated without a constant term. The case $\theta=0$ corresponds to the correctly specified AR
design (with $d=1$), while $\theta=0.2$ introduces misspecification with a moving-average component. We first use the sample
size $T=360$ to match that of our empirical application. We use 5{,}000 Monte Carlo replications and 1{,}000
bootstrap draws. We consider horizons $h = 1, \cdots, 36$.
\begin{figure}[ht]
\centering
\begin{minipage}{1.05\textwidth}
\noindent\hspace*{-2cm}
\includegraphics[
width=1.2\linewidth,
height=1.99\textheight,
keepaspectratio
]{Notes/figure_coverage_bandwidth_0501_360_theta2.png}
\caption{Bootstrap confidence intervals (CIs) based on GMM(2,3) and LALP, under the ARMA(1,1) data generating process with AR coefficient $\alpha \in \{0.5, 0.99, 1\}$ and MA coefficient $\theta=0.2$, $T = 360$.}
\label{fig:coverage_bandwidth_theta2_360}
\vspace{-0.0em}
\footnotesize
\textit{Notes:}
Left panels plot the coverage probabilities of bootstrap CIs (Nominal level: 90\%). Right panels plot the ratio of the length of LALP to GMM-LALP CIs; values above one indicate that GMM-LALP CIs are shorter.
\end{minipage}
\end{figure}
Bootstrap coverage
probabilities and confidence-interval (CI) length ratios for LALP and GMM-LALP are reported in Figure~\ref{fig:coverage_bandwidth_theta1_360} ($\theta=0$) and Figure~\ref{fig:coverage_bandwidth_theta2_360} ($\theta=0.2$). The results show that GMM-LALP CIs generally have similar coverage as LALP CIs, but are remarkably shorter. LALP CIs can be up to 30\% wider than GMM-LALP CIs, and the largest differences appear for the persistent data ($\alpha=0.99$ or 1) and at moderately long horizons. The two CIs perform similarly across all horizons when the data are only weakly dependent ($\alpha=0.5$). We also find that the GMM-LALP can have noticeably better coverage under the misspecified case.
\begin{figure}[ht]
\centering
\begin{minipage}{1.05\textwidth}
\noindent\hspace*{-2cm}
\includegraphics[
width=1.2\linewidth,
height=1.99\textheight,
keepaspectratio
]{Notes/figure_coverage_bandwidth_0501_1200.png}
\caption{Bootstrap confidence intervals (CIs) based on GMM(2,3) and LALP methods, under the AR(1) data generating process with AR coefficient $\alpha \in \{0.5, 0.99, 1\}$, $T = 1200$.}
\label{fig:coverage_bandwidth_theta1_1200}
\vspace{-0.0em}
\footnotesize
\textit{Notes:}
Left panels plot the coverage probabilities of bootstrap CIs (Nominal level: 90\%). Right panels plot the ratio of the length of LALP to GMM-LALP CIs; values above one indicate that GMM-LALP CIs are shorter.
\end{minipage}
\end{figure}
\begin{figure}[t]
\centering
\begin{minipage}{1.05\textwidth}
\noindent\hspace*{-2cm}
\includegraphics[
width=1.2\linewidth,
height=1.99\textheight,
keepaspectratio
]{Notes/figure_coverage_bandwidth_0501_1200_theta2.png}
\caption{Bootstrap confidence intervals (CIs) based on GMM(2,3) and LALP, under the ARMA(1,1) data generating process with AR coefficient $\alpha \in \{0.5, 0.99, 1\}$ and MA coefficient $\theta=0.2$, $T = 1200$.}
\label{fig:coverage_bandwidth_theta2_1200}
\vspace{-0.0em}
\footnotesize
\textit{Notes:}
Left panels plot the coverage probabilities of bootstrap CIs (Nominal level: 90\%). Right panels plot the ratio of the length of LALP to GMM-LALP CIs; values above one indicate that GMM-LALP CIs are shorter.
\end{minipage}
\end{figure}
We also consider a much larger sample size, $T=1200$, albeit one that may be unrealistic in applications, to better approximate the asymptotic setting underlying our theoretical results. The results are reported in Figures \ref{fig:coverage_bandwidth_theta1_1200} and \ref{fig:coverage_bandwidth_theta2_1200}. Both CIs now have the coverage rate quite close to the nominal level across all cases. The differences in CI lengths are even larger; LALP CIs can be up to 60\% wider than GMM-LALP CIs when the data are persistent.
\section{Empirical Application}
\label{sec:application}
We apply the proposed GMM-LALP estimator to the monthly five-variable VAR model of \cite{baker2016measuring}, which has become a standard setting for studying the macroeconomic
effects of uncertainty shocks. The data contain $360$ monthly
observations from January 1985 through December 2014.\footnote{The data are downloaded from
Nicholas Bloom's personal website, \text{https://nbloom.people.stanford.edu/research}.} Table \ref{tab:variable_definitions} shows that all variables except EPU are very highly persistent,
which is precisely the environment in which efficiency losses from standard OLS
local-projection estimation can be empirically relevant.
\begin{table}[b]
\centering
\caption{Variable definitions and first-order autocorrelations}
\label{tab:variable_definitions}
\begin{tabular}{c|c|c}
\hline\hline
Variable & Definition & \begin{tabular}[c]{@{}c@{}}First-order autocorrelation\end{tabular} \\
\midrule
EPU & Economic policy uncertainty index & 0.6707 \\
SP500 & Logarithm of the S\&P 500 index & 0.9984 \\
FFR & Federal funds rate & 0.9973 \\
EMPLOYMENT & Logarithm of employment & 0.9996 \\
IP & Logarithm of industrial production & 0.9995 \\
\hline\hline
\end{tabular}
\end{table}
For each response variable $i$ and each horizon $h=1,\cdots,36$, we estimate a
local projection using the full current vector
$y_t=(\mathrm{EPU}_t,\mathrm{SP500}_t,\mathrm{FFR}_t,\mathrm{EMPLOYMENT}_t,
\mathrm{IP}_t)'$.\footnote{Prior to computing the impulse responses, each series is demeaned. Accordingly, the local-projection regression and GMM-LALP are estimated without the intercept.} The coefficient on the $j$th
element of $y_t$ is interpreted as the horizon-$h$ (reduced-form) response of variable $i$ to
shock variable $j$, conditional on the other current variables and the
lag-augmented controls. We compute both LALP and GMM-LALP estimates and their bootstrap CIs using the same procedures as the previous simulation section. We use 5{,}000 bootstrap replications.
Figure~\ref{fig:application-epu-shock68} reports the responses of all five variables to the EPU shock with 68\% confidence intervals.\footnote{68\% confidence intervals are quite common in empirical macroeconomics, especially for graphical presentation of impulse responses. We also compute 90\% confidence intervals for these responses. See Appendix \ref{app:IRF}.} To examine whether the efficiency gain of GMM-LALP survives a richer dynamic specification, the figure reports two lag choices: the upper panel uses the AIC-selected (Akaike information
criterion) baseline lag order, $d=3$, and the lower panel uses a substantially larger lag order, $d=12$. With lag augmentation, both
LALP and GMM-LALP use $p=d+1$ (thus 4 or 13) monthly lags as regressors, and GMM uses an additional lag as the excluded instrumental variable.
Thus, the comparison is not driven by a parsimonious lag specification alone; it asks whether optimal GMM weighting continues to improve precision even when the lag-augmented projection includes a large number of controls.
Figure~\ref{fig:application-epu-shock68} can be viewed as a non-structural \textit{local-projection} counterpart to the baseline monthly VAR evidence in \cite{baker2016measuring}. Our exercise differs from \cite{baker2016measuring} in an important respect: we do not impose a recursive structural identification scheme (which can be controversial), and hence the responses in Figure~\ref{fig:application-epu-shock68} should not be read as structural IRFs or compared to their estimates in magnitude.\footnote{Their Figure VIII uses the same broad five-variable monthly system---EPU, the stock market, the federal funds rate, employment, and industrial production---with three lags, and reports Cholesky-identified responses of employment and industrial production to an upward EPU innovation.} However, similar to Figure VIII of \cite{baker2016measuring}, the responses of employment and industrial production move downward after an EPU innovation, indicating that the same empirical system points to weaker real activity following increases in policy uncertainty.
Overall, except for a few cases, LALP and GMM-LALP deliver similar point estimates for the impulse responses. The GMM-LALP confidence intervals are systematically shorter than the LALP confidence intervals, reflecting the efficiency gain from using valid lagged instruments and optimally weighting the resulting moment conditions. Using $d=3$, the LALP CI for the response of IP can be up to 55\% wider than the GMM CI (on average 22\% wider). Importantly, this pattern remains visible in the lower panel with a larger lag order, albeit to a lesser extent. Using $d=12$, the LALP CI for the response of IP can be up to almost 30\% wider than the GMM CI (on average 15\% wider).
The tighter CIs by GMM are also economically meaningful. Using $d=3$, for the response of IP, the LALP CIs cover the zero line (meaning the effects are insignificant) across all horizons, while this is not true for GMM CIs. For the response of EMPLOYMENT, GMM-LALP CIs indicate statistical significance at almost half of the horizons, whereas LALP CIs do so at only a couple of horizons.
In summary, the empirical IRFs reinforce the paper's theoretical and simulation results. The optimal GMM weighting provides additional precision conditional on that same lag-augmented specification.
\begin{figure}[t]
\makebox[\linewidth][c]{
\includegraphics[width=1.1\textwidth]{Notes/EPU_shock_response_point_h36_bands_p3_p12_2x5_68_0629.eps}
}
\caption{Responses to an EPU innovation.}
\label{fig:application-epu-shock68}
\begin{minipage}{1\textwidth}
\vspace{2pt}
\footnotesize
\textit{Notes:} The figure plots the impulse responses for EPU, SP500,
FFR, EMPLOYMENT, and IP to an EPU innovation with 68\% confidence intervals. Blue lines are LALP estimates and
bootstrap confidence intervals; orange lines are GMM-LALP estimates and bootstrap confidence intervals. The
model lag order is $d=3$ for the upper panels, and is $d=12$ for the lower panels. The horizons run from 1 to 36 months.
\end{minipage}
\end{figure}
\section{Conclusion}
\label{sec:conclu}
This paper proposes a new method for estimating impulse responses and conducting inference based on local projections. Rather than estimating a local projection by OLS for a given set of lag controls, we exploit additional lags of the data as excluded instruments and optimally combine the resulting moment conditions using GMM. We show that these additional moments can deliver substantial efficiency gains relative to OLS, particularly when the data are persistent. We formally establish the robustness of the proposed inference to data persistence and model misspecification.
\putbib
\end{bibunit}
\newpage