The Local Projection Residual Bootstrap for AR(1) Models

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spacing{1.2} \begin{abstract} This paper proposes a local projection residual bootstrap method to construct confidence intervals for impulse response coefficients of AR(1) models. Our bootstrap method is based on the local projection (LP) approach and involves a residual bootstrap procedure applied to AR(1) models. We present theoretical results for our bootstrap method and proposed confidence intervals. First, we prove the uniform consistency of the LP-residual bootstrap over a large class of AR(1) models that allow for a unit root, conditional heteroskedasticity of unknown form, and martingale difference shocks. Then, we prove the asymptotic validity of our confidence intervals over the same class of AR(1) models. Finally, we show that the LP-residual bootstrap provides asymptotic refinements for confidence intervals on a restricted class of AR(1) models relative to those required for the uniform consistency of our bootstrap. \end{abstract}

KEYWORDS: Bootstrap, Local Projection, Uniform Inference, Asymptotic Refinements.

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Introduction

This paper contributes to a growing literature on confidence interval construction for impulse response coefficients based on the local projection (LP) approach (Jorda2005). In this literature, the LP approach estimates an impulse response coefficient as one of the slope coefficients in a linear regression of a future outcome on current or lag-augmented covariates (ramey2016macroeconomic; NakamuraSteinsson2018; MO-PM2020). Recent theoretical results exist for the asymptotic validity of the confidence intervals constructed around the LP estimator, which hold over a large class of vector autoregressive models (xu2022). Since these confidence intervals have small-sample coverage distortions (e.g., coverage probability is lower than expected), their bootstrap versions are recommended for practical use. However, theoretical results for these bootstrap versions are unknown, even for the AR(1) model. This paper proposes a different bootstrap method to construct LP confidence intervals with theoretical guarantees for a class of AR(1) models that allow for a unit root, conditional heteroskedasticity of unknown form, and martingale difference shocks.

We propose an LP-residual bootstrap method to construct confidence intervals for impulse response coefficients of AR(1) models. Our bootstrap method is based on the LP approach and involves a residual bootstrap procedure applied specifically to AR(1) models.\footnote{Section (ref) presents the LP-residual bootstrap for VAR(p) models, but its theoretical properties are unknown and left for future research; see Remarks (ref) and (ref) for further discussion. Appendix (ref) reports a Monte Carlo simulation for the LP-residual bootstrap for vector autoregressive models.} Our bootstrap confidence intervals are centered at the LP estimator and use heteroskedasticity-consistent (HC) standard errors and a bootstrap critical value. Section (ref) presents the details.

We rely on the asymptotic distribution theory initially developed in MO-PM2020 and generalized in xu2022. In their framework, a root $R_n(h)$ based on the LP approach can be defined for a given horizon $h$ and a sample size $n$. Here, by a root we refer to a real-valued function depending on the data and an impulse response coefficient. Their results guarantee the root $R_n(h)$ is asymptotically distributed as a standard normal distribution for a class of VAR models that allow for multiple unit roots and conditional heteroskedasticity of unknown form\textcolor{black}{, and even at intermediate horizons, i.e., horizons $h$ that are allowed to grow with $n$, e.g., $h = h_n \propto n^{\zeta}$, $\zeta \in [0,1)$}. As a result, the root $R_n(h)$ can be used to construct a confidence interval $C_n(h, 1-\alpha)$ for an impulse response coefficient using a normal critical value (quantile of the asymptotic distribution). Furthermore, $C_n(h, 1-\alpha)$ has asymptotic coverage equal to the nominal level $1-\alpha$ uniformly over the parameter space (VAR model coefficients) and \textcolor{black}{a wide range of} intermediate horizons (e.g., \textcolor{black}{uniform over $h \le h_n$, where $h_n$ is any fixed sequence such that $h_n = o(n)$}). Nevertheless, Monte Carlo simulations report \textcolor{black}{that} $C_n(h, 1-\alpha)$ has a lower coverage probability than expected.

We propose the LP-residual bootstrap method to approximate the distribution of the root $R_n(h)$ as an alternative to the asymptotic distribution. We use our approximation to calculate bootstrap-based critical values; see Section (ref) for the step-by-step procedure. Specifically, we construct a confidence interval $C_n^*(h,1-\alpha)$ for an impulse response coefficient using the root $R_n(h)$ and a bootstrap critical value; see Section (ref) for details.

Our first result proves the uniform consistency of the LP-residual bootstrap. More concretely, we demonstrate in Section (ref) that the distribution of the root $R_n(h)$ can be approximated by its bootstrap version uniformly over the parameter space (e.g., $\rho \in [-1,1]$) and \textcolor{black}{a wide range of} intermediate horizons (e.g., \textcolor{black}{uniform over $h \le h_n$, where $h_n$ is any fixed sequence such that $h_n = o(n)$}). Our result applies to a large class of AR(1) models that allow for a unit root, conditional heteroskedasticity of unknown form as in gonccalves2004bootstrapping, which includes ARCH and GARCH shocks, and a sequence of shocks that satisfy the martingale difference assumption. To obtain this result, we prove the root $R_n(h)$ is asymptotically distributed as a standard normal distribution for sequences of AR(1) models with i.i.d. shocks (Theorem (ref)). In particular, we prove that a high-level assumption (Assumption 3 in MO-PM2020corrigendum and Assumption 4 in xu2022) necessary for the theoretical properties of $C_n(h,1-\alpha)$ can be verified for sequences of AR(1) models with i.i.d. shocks (Proposition (ref)).

Our first result implies that the LP-residual bootstrap method provides asymptotically valid confidence intervals over a large class of AR(1) models that allow for a unit root, conditional heteroskedasticity of unknown form (e.g., GARCH shocks), and martingale difference shocks. Moreover, our confidence interval $C_n^*(h,1-\alpha)$ has an asymptotic coverage equal to the nominal level $1-\alpha$ uniformly over $\rho \in [-1,1]$ and \textcolor{black}{a wide range of} intermediate horizons.

Our second set of results shows that the LP-residual bootstrap provides asymptotic refinements to the confidence intervals on a more restricted class of AR(1) models (e.g., \textcolor{black}{$|\rho|\le 1-a$, where $a \in (0,1)$,} \textcolor{black}{and} i.i.d. shocks \textcolor{black}{with} positive continuous density), that is, the size of the error in coverage probability (ECP) of $C_n^*(h,1-\alpha)$ is $o(n^{-1})$, whereas the size of the ECP of $C_n(h,1-\alpha)$ is $O(n^{-1})$. More concretely, Theorem (ref) shows the ECP of $C_n^*(h,1-\alpha)$ is $o(n^{-(1+\epsilon)})$ for some $\epsilon \in (0,1/2)$. To obtain these results, we derive Edgeworth expansions for the distribution of the root $R_n(h)$ and its bootstrap version for a fixed $h$ and \textcolor{black}{$|\rho| \le 1-a$, where $a \in (0,1)$; that is, the Edgeworth expansions are obtained for stationary AR(1) models and fixed horizons}. An informal discussion to calculate the size of the ECP using Edgeworth expansions appears in Section (ref), while the formal results are established in Section (ref).

Other bootstrap methods to construct confidence intervals for the impulse response coefficients have been considered and recommended based on simulation studies in the growing literature on LP inference. MO-PM2020 use a wild bootstrap procedure to generate new samples and compute critical values, but the theoretical results for their bootstrap method are unknown. KilianKim2011 present a simulation study including a block-bootstrap method to construct confidence intervals based on the LP approach, but the theory of their block-bootstrap method is unknown; see Remarks (ref) and (ref) for alternative block-bootstrap procedures with theoretical guarantees. Recently, lusompa2021local proposes a block wild bootstrap method for confidence interval construction that is point-wise valid for a class of stationary data-generating processes; however, his bootstrap method is not applicable for an AR(1) model with a unit root. In contrast, we present a bootstrap method based on the LP approach with theoretical guarantees for a class of AR(1) models that allow for a unit root, conditional heteroskedasticity of unknown form, and martingale difference shocks.

More broadly, we contribute to the literature on confidence interval construction for impulse response coefficients. For short horizons (fixed $h$), the problem of confidence interval construction has been studied by andrews1993exactly, hansen1999grid, inoue2002bootstrapping, Jorda2005, Mikusheva2007,mikusheva2015second, among others. For long horizons ($\textcolor{black}{h} \textcolor{black}{=} h_n \textcolor{black}{\propto} (1-b)n$, $b \in (0,1)$), the problem of confidence interval construction was \textcolor{black}{discussed} and revised by \textcolor{black}{phillips1998impulse}, gospodinov2004asymptotic, pesavento2006small, and Mikusheva2012 since the \textcolor{black}{standard} methods for short horizons \textcolor{black}{may} produce invalid confidence intervals when the data-generating process allows for unit roots. Recently, the problem of confidence interval construction for intermediate horizons ($h_n = o\left(n\right)$) was addressed in MO-PM2020 and xu2022, which was a case not covered in the literature. In this paper, we propose bootstrap confidence intervals that are asymptotically valid at short and intermediate horizons.

\textcolor{black}{ We also contribute to the literature on uniform inference in autoregressive models, where the confidence intervals for impulse response coefficients are uniformly valid, that is, they have an asymptotic coverage equal to the nominal level uniformly over the parameter space (e.g., uniformly over $ \rho \in [-1,1]$ for the AR(1) model). Mikusheva2007,Mikusheva2012 shows that the grid bootstrap proposed by hansen1999grid provides confidence sets that are uniformly valid for the impulse responses when the sequence of shocks is a martingale difference sequence with constant conditional variance. However, it is unknown if the grid bootstrap is uniformly valid for AR(1) models with GARCH shocks; we report simulations for the grid bootstrap in Section (ref) and Online Supplemental Appendix (ref). InoueKillian2020 show that confidence intervals based on a lag-augmented autoregressive method are uniformly valid for impulse response coefficients when the sequence of shocks is i.i.d. It is unknown if their results hold for martingale difference shocks. MO-PM2020 and xu2022 show that confidence intervals based on (lag-augmented) local projections are uniformly valid for impulse response coefficients; nevertheless, Monte Carlo simulations report lower coverage probability than expected. In contrast, our bootstrap method produces confidence intervals that are uniformly valid for a larger class of martingale difference shocks with conditional heteroskedasticity of unknown form (allowing for GARCH shocks). }

The remainder of the paper is organized as follows. In Section (ref), we describe the setup and previous results. In Section (ref), we introduce our bootstrap confidence interval and the LP-residual bootstrap. In Sections (ref) and (ref), we study the theoretical properties of the LP-residual bootstrap: uniform consistency and asymptotic refinements. In Section (ref), we investigate the numerical performance of the LP-residual bootstrap using a small simulation study. In Section (ref), we describe how to implement the LP-residual bootstrap for VAR models. Finally, in Section (ref), we present concluding remarks. All the proofs are presented in Appendices (ref) and (ref), and \textcolor{black}{Online} Supplemental Appendices (ref) and (ref). Additional simulation results appear in \textcolor{black}{Online} Supplemental Appendix (ref).

Setup and Previous Results on Local Projection

Consider an AR(1) model,

equation[equation omitted — 104 chars of source]

Denote the impulse response coefficient at horizon $h \in \mathbf{N}$ by

equation[equation omitted — 64 chars of source]

An estimator for $\beta(\rho,h)$ based on the LP approach is obtained as the slope coefficient of $y_t$ in the linear regression of $y_{t+h}$ on $y_t$ and $y_{t-1}$,

equation[equation omitted — 135 chars of source]

where $(\hat{\beta}_n(h),~\hat{\gamma}_n(h))$ and $\{\hat{\xi}_t(h): 1 \le t \le n-h\}$ are the coefficient vector and residuals of the linear regression (ref), respectively. This lag-augmented LP approach was developed in MO-PM2020, where they give conditions under which the coefficient $\hat{\beta}_n(h)$ consistently estimates $\beta(\rho,h)$. Equation (ref) is a lag-augmented LP regression since the coefficient on $y_{t-1}$ is known to be zero under (ref); see Remark (ref) for additional details on this LP approach.

Let $\hat{s}_n(h)$ be the heteroskedasticity-consistent (HC) standard error of $\hat{\beta}_n(h)$ in the lag-augmented LP regression (ref), which can be computed as follows

equation[equation omitted — 236 chars of source]

where $\hat{u}_t(h) \equiv y_t- \hat{\rho}_n(h) y_{t-1}$ and

equation[equation omitted — 154 chars of source]

For a given $h \in \textbf{N}$, we consider the following real-valued root for the parameter $\beta(\rho,h)$:

equation[equation omitted — 104 chars of source]

where $\beta(\rho,h)$ is as in (ref), $\hat{\beta}_n(h)$ is computed as in (ref), and $\hat{s}_n(h)$ is as in (ref). We denote the distribution of the root $R_n(h)$ by

equation[equation omitted — 100 chars of source]

where $x \in \mathbf{R}$, $h \in \textbf{N}$, $P$ is the distribution of the shocks $\{u_t: t \ge 1 \}$, $\rho \in \textbf{R}$, and $P_{\rho}$ denote the probability distribution of the sequence $\{y_t: t \ge 1\}$, which is defined jointly by the distribution $P$ and the parameter $\rho$ in (ref).

Let $c_n(h,1-\alpha)$ be the $1-\alpha$ quantile of $|R_n(h)|$ under the distribution $P_{\rho}$,

equation[equation omitted — 160 chars of source]

Ideally, we would use the root $R_n(h)$ and the critical value $c_n(h,1-\alpha)$ to construct confidence sets for $\beta(\rho,h)$ with a coverage probability of 1-$\alpha$. That is collecting all the parameters $\beta(\rho,h)$ such that $|R_n(h)| \le c_n(h,1-\alpha)$, which is equivalent to defining the next confidence interval

align*[align* omitted — 164 chars of source]

However, the critical value $c_n(h,1-\alpha)$ is unknown since the distribution of the root is unknown in general. As a result, the confidence interval $\tilde{C}_n(h,1-\alpha) $ is infeasible. For this reason, it is common to approximate the distribution of the root $R_n(h)$ relying on asymptotic distribution theory or bootstrap methods to approximate the infeasible $c_n(h,1-\alpha)$.

Previous Results

The asymptotic distribution theory developed in MO-PM2020 and xu2022 implies that the distribution $J_n(x,h,P,\rho)$ converges to the standard normal distribution $\Phi(x)$ whenever certain assumptions on the distribution of the shocks $P$ hold. Moreover, this convergence is uniform over the values of $\rho \in [-1,1]$ and \textcolor{black}{a wide range of} intermediate horizons, that is

equation[equation omitted — 178 chars of source]

where \textcolor{black}{$h_n$ is any fixed sequence such that} $h_n \le n$ and $h_n = o\left(n\right)$. Assumptions (ref) and (ref) in Section (ref) are sufficient conditions on the distribution $P$ to obtain (ref) due to Theorem 2 in xu2022.

The confidence interval for $\beta(\rho,h)$ based on asymptotic distribution theory is defined as

equation[equation omitted — 177 chars of source]

where $z_{1-\alpha/2} \equiv \Phi^{-1}(1-\alpha/2)$ is the $1-\alpha/2$ quantile of the standard normal distribution. The result in (ref) implies that \textcolor{black}{the} confidence interval $C_n(h,1-\alpha)$ is uniformly asymptotically valid in the sense that its asymptotic coverage probability is equal to the nominal level $1-\alpha$ uniformly over $\rho$ and \textcolor{black}{a wide range of intermediate horizons} $h$,

equation*[equation* omitted — 193 chars of source]

where \textcolor{black}{$h_n$ is any fixed sequence such that} $h_n \le n$ and $h_n = o\left(n\right)$. Three features of $C_n(h,1-\alpha)$ deserve further discussion. First, it is simpler to compute than \textcolor{black}{the} available alternatives in the sense that it does not require any tuning parameter. It is common to use heteroskedasticity- and autocorrelation-robust (HAR) standard errors for inference whenever we have dependent data. The major complication of HAR standard errors is the choice of the (truncation) tuning parameter; see lazarus2018har. In contrast, the HC standard errors $\hat{s}_n(h)$ defined in (ref) are simple to compute and sufficient for inference under certain conditions on the distribution $P$; see Remark (ref) for further explanation. Second, the uniform asymptotic validity of the confidence interval $C_n(h,1-\alpha)$ avoids pre-testing procedures about the nature of the data-generating process ($|\rho|<1$ vs $\rho=1$) that can distort inference; see Mikusheva2007. In particular, inference using $C_n(h,1-\alpha)$ holds regardless of the value of $\rho \in [-1,1]$. Third, the confidence interval $C_n(h,1-\alpha)$ has theoretical guarantees at intermediate horizons (e.g., \textcolor{black}{$h=h_n \propto n^{\zeta}$, $\zeta \in (0,1)$}). This is an important feature for inference on impulse response coefficients at intermediate horizons. Other methods to construct confidence intervals that work at short horizons ($h$ fixed) may have problems at long and intermediate horizons; see \textcolor{black}{phillips1998impulse}, gospodinov2004asymptotic, pesavento2006small, Mikusheva2012, and MO-PM2020 for additional discussion.

remarkThe HC standard errors $\hat{s}_n(h)$ defined in (ref) are sufficient for the construction of valid confidence intervals under certain conditions on the distribution $P$. In particular, as it was pointed out by xu2022, it is sufficient and necessary that the scores $\{ \xi_t(\rho,h) u_t : 1 \le t \le n-h \}$ be serially uncorrelated, where $\xi_t(\rho,h) \equiv \sum_{\ell=1}^{h} \rho^{h-\ell} u_{t+\ell}$. To explain the sufficiency of this condition, we use the derivations presented on page 1811 in MO-PM2020 that imply that the root $R_n(h)$ defined in (ref) can be written as follows $$ \frac{\left((n-h)^{-1/2} \sum_{t=1}^{n-h} \xi_t(\rho,h) u_t \right)}{ E\left[ \xi_t(\rho,h)^2 u_t^2\right]^{1/2}} \times \frac{ \left[(n-h)^{-1} \sum_{t=1}^{n-h} \hat{\xi}_t(h)^2 \hat{u}_t(h)^2\right]^{-1/2} }{E\left[ \xi_t(\rho,h)^2 u_t^2\right]^{-1/2}} + \varepsilon_n(\rho,h)~,$$ where $\varepsilon_n(\rho,h)$ is a remainder error term. We derive three implications under Assumptions (ref) and (ref), presented in Section (ref). First, the term in parentheses converges to a normal distribution with variance correctly scaled by the denominator when the scores are serially uncorrelated. This condition is guaranteed by part (ii) of Assumption (ref). Second, the term between brackets converges in probability to its denominator due to serially uncorrelated scores. Third, the remainder error term $\varepsilon_n(\rho,h)$ converges in probability to zero. Importantly, xu2022 proposed alternative standard errors for the construction of confidence intervals under serially correlated scores.

remarkThe lag-augmented LP regression has the purpose of making the effective regressor of interest stationary. To see this, let us use the AR(1) model in (ref) to obtain $ y_{t+h} = \beta(\rho,h) y_t + \xi_t(\rho,h)$, where $\xi_t(\rho,h) = \sum_{\ell=1}^h \rho^{h-\ell} u_{t+h}$, which can be rewritten as $$y_{t+h} = \beta(\rho,h) u_t + \rho \beta(\rho,h) y_{t-1} + \xi_t(\rho,h)~. $$ Based on the previous equality, an estimator for $\beta(\rho,h)$ is defined as the slope coefficient of $u_t$ in the linear regression of $y_{t+h}$ on $u_t$ and $y_{t-1}$. This estimator is ideal since the effective regressor is stationary (by assumption). However, this regression is unfeasible since $u_t$ is not observed. Nevertheless, the estimator can also be obtained in the lag-augmented LP regression of $y_{t+h}$ on $y_t$ and $y_{t-1}$ since $y_t$ is a linear combination of $u_t$ and $y_{t-1}$ due to (ref).

The LP-Residual Bootstrap

This paper proposes an LP-residual bootstrap for confidence interval construction. Our confidence interval for the impulse response coefficient $\beta(\rho,h)$ is defined as

equation[equation omitted — 188 chars of source]

where $\hat{\beta}_n(h)$ is an estimator for $\beta(\rho,h)$ defined in (ref), $\hat{s}_n(h)$ is its heteroskedasticity-consistent (HC) standard error defined in (ref), and $c_n^*(h, 1-\alpha)$ is a bootstrap critical value defined in (ref).

Bootstrap Critical Value

Let $Y^{(n)} \equiv \{y_t : 1 \le t \le n\}$ be data generated by (ref). Let $c_n^*(h, 1-\alpha)$ be the bootstrap critical value involving the following steps:

itemize• Estimate $\rho$ in the AR(1) model defined in (ref) with the data $Y^{(n)}$ using linear regression, denoted by \begin{equation} \hat{\rho}_n \equiv \left(\sum_{t=1}^n y_{t-1}^2\right)^{-1}\left(\sum_{t=1}^n y_{t-1}y_t\right) , \end{equation} and compute the centered residuals \begin{equation} \{ \Tilde{u}_t \equiv \hat{u}_t - n^{-1}\sum_{t=1}^n \hat{u}_t : 1 \le t \le n \} , \end{equation} where $\hat{u}_t \equiv y_{t} - \hat{\rho}_n y_{t-1}$. • Generate a new sample of size $n$ using (ref), (ref), and (ref). Define the sample as \begin{equation*} y_{b,t}^* = \hat{\rho}_n y_{b,t-1}^* + u_{b,t}^* , \quad y_{b,0}^* = 0 , t=1,\ldots,n , \end{equation*} where $\{u_{b,t}^* : 1 \le t \le n\}$ is a random sample from the empirical distribution of the centered residuals defined in (ref). The new sample $\{ y_{b,t}^* : 1 \le t \le n \} $ is called the bootstrap sample. • Compute $\hat{\beta}_{b,n}^*(h)$ and $\hat{s}_{b,n}^*(h)$ as in (ref) and (ref) using the lag-augmented LP regression and the bootstrap sample $\{ y_{b,t}^* : 1 \le t \le n \} $. Define the bootstrap version of the root \begin{equation} R_{b,n}^*(h) = \frac{\hat{\beta}_{b,n}^*(h) - \beta(\hat{\rho}_n,h)}{\hat{s}_{b,n}^*(h)} , \end{equation} where $\beta(\rho,h)$ and $\hat{\rho}_n$ are as in (ref) and (ref), respectively. • Define the bootstrap critical value as the $1-\alpha$ quantile of $|R_{b,n}^*(h)|$ conditional on the data $Y^{(n)}$, denoted by \begin{equation} c_n^{*}(h, 1-\alpha) \equiv \inf \left \{ u \in R : P_{\rho}\left ( |R_{b,n}^*(h)| \le u \mid Y^{(n)} \right) \ge 1-\alpha \right\} . \end{equation}

We named this procedure the LP-residual bootstrap due to steps 2 and 3. Step 2 generates bootstrap samples based on the estimated model and a residual bootstrap procedure. Step 3 computes the bootstrap version of the root based on the lag-augmented LP regression. To our knowledge, this bootstrap procedure is new; see Remark (ref) and (ref) for other bootstrap procedures involving roots based on LP estimators.

We use the bootstrap critical value $c_n^{*}(h, 1-\alpha)$ in the construction of the confidence interval defined in (ref). The explicit formula in (ref) has two implications. First, the bootstrap critical value $c_n^{*}(h, 1-\alpha)$ depends on the data, the sample size $n$, and the horizon $h$. Second, we can compute $c_n^{*}(h, 1-\alpha)$ with perfect accuracy whenever we use the exact empirical distribution of the centered residuals defined in (ref). However, the computation of an exact distribution can be computationally demanding; therefore, it is common to approximate it using Monte Carlo procedures as we describe in Remark (ref), which has a theoretical justification due to the Glivenko–Cantelli theorem.

remarkIt is a common practice to approximate the bootstrap critical value $c_n^{*}(h, 1-\alpha)$ using a Monte Carlo procedure (Horowitz2001, Horowitz2019). We generate $B$ bootstrap samples of size \textcolor{black}{$n$}, where each $b$-th bootstrap sample $\{y_{b,t}^*: 1 \le t \le n\}$ is generated as in step 2. We then obtain $\{ |R_{b,n}^*(h)| : 1 \le b \le B \}$, where each $R_{b,n}^*(h)$ is computed as in step 3. Finally, we approximate the bootstrap critical value $c_n^{*}(h, 1-\alpha)$ by the $1-\alpha$ quantile of $\{ |R_{b,n}^*(h)| : 1 \le b \le B\}$, denoted by \begin{equation*} c_{b,n}^*(h, 1-\alpha) \equiv \inf \left \{ u \in R : \frac{1}{B} \sum_{b=1}^B I\left\{ |R_{b,n}^*(h)| \le u \right\} \ge 1-\alpha \right\} . \end{equation*} The accuracy of the approximation improves as the number of bootstrap samples $B$ increases. We use $B=1000$ in our simulation study presented in Section (ref).

remarkAnother bootstrap procedure to approximate the infeasible critical value $c_n(h,1-\alpha)$ is presented in Section 5 of MO-PM2020. They use the wild bootstrap procedure described in gonccalves2004bootstrapping. For this reason, we name their procedure the LP-wild bootstrap. The only difference with respect to the LP-residual bootstrap is in Step 2. The LP-wild bootstrap defines the shocks as follows: $ u_{b,t}^* = \tilde{u}_t z_{b,t}$ for all $t = 1, \ldots, n$, where $\{\tilde{u}_t: 1 \le t \le n\}$ are the centered residuals defined in (ref) and $\{z_{b,t} : 1 \le t \le n \}$ is an i.i.d. sequence of standard normal random variables independent of the data $Y^{(n)}$. To our knowledge, the theoretical properties of the LP-wild bootstrap are unknown. We include the LP-wild bootstrap in our simulation study presented in Section (ref).

remarkAn alternative to the symmetric percentile-t confidence interval defined in (ref) is the equal-tailed percentile-t confidence interval. The latter is proposed and recommended in Section 5 of MO-PM2020, while the former has been found to perform better in simulations reported by gonccalves2004bootstrapping. Furthermore, symmetric confidence intervals are known to perform better asymptotically in terms of coverage error in the case of i.i.d. data; see Section 3.6 in Hall1992TheBA. For these reasons, we focus on and study the properties of the symmetric percentile-t confidence interval in the next sections. Remark (ref) presents additional discussion of the equal-tailed percentile-t confidence interval based on the LP-residual bootstrap. We include equal-tailed percentile-t confidence intervals based on both LP-residual and LP-wild bootstrap in our simulation study in Section (ref).

remarkWe propose the LP-residual bootstrap method for constructing confidence intervals, aiming to provide a more accurate asymptotic approximation than the first-order asymptotic distribution for conducting inference. In Sections (ref) and (ref), we study the validity of this bootstrap method and its theoretical properties under assumptions on the distribution of the shocks and under the assumption of correct specification, i.e., the data are generated from the AR(1) model in (ref). To our knowledge, the theoretical properties of the root $R_n(h)$ for general forms of misspecification are unknown. Recent work by olea2024double imply that $R_n(h)$ is still asymptotically pivotal under a specific form of local misspecification. The analysis of the theoretical properties of the LP-residual bootstrap under misspecification is \textcolor{black}{outside} the scope of this paper.

Uniform Consistency

We show the uniform consistency of the LP-residual bootstrap (Theorem (ref)) and that our proposed bootstrap confidence interval $C_n^*(h,1-\alpha)$ defined in (ref) is uniformly asymptotically valid (Theorem (ref)). In what follows, we first present and discuss the assumptions, and we then establish the results.

The following assumption imposes restrictions on the distributions of the shocks $P$. These assumptions are based on the general framework developed by xu2022 that generalized the work of MO-PM2020.

assumption\begin{enumerate} • $\{u_t: 1 \le t \le n\}$ is covariance-stationary and satisfies $E[u_t \mid \{u_s \}_{s < t}] = 0$ almost surely. • $E[u_t^2u_{t-s}u_{t-r}] = 0$ for all $s \neq r$, for all $t,r,s \ge 1$. • $\{u_t: 1 \le t \le n\}$ is strong mixing with mixing numbers $\{ \alpha(j): j \ge 1\}$. There exists $\zeta > 2$, $\epsilon>1$, and $C_{\alpha}<\infty$, such that $\alpha(j) \le C_{\alpha}j^{-2\zeta \epsilon/(\zeta-2)}$, for all $j$. • For $\zeta$ defined in (iii), $E[u_t^{8\zeta}] \le C_8 < \infty$, and $E[u_t^2 \mid \{ u_s \}_{s<t}] \ge C_\sigma $ almost surely. \end{enumerate}

Part (i) of Assumption (ref) assumes the shocks are a martingale difference sequence. This assumption allows for uncorrelated dependent shocks and implies that the shock $u_t$ is uncorrelated with $y_{t-1}$. Part (ii) in Assumption (ref) includes a large class of conditional heteroskedastic autoregressive models (e.g., ARCH and GARCH shocks), and it has been common in the literature; for instance, gonccalves2004bootstrapping use a similar assumption (Assumption A') to prove the asymptotic consistency of the wild bootstrap for autoregressive processes. Moreover, this assumption implies that the process $\{ \xi_t(\rho,h) u_t : 1 \le t \le n-h \}$ is serially uncorrelated, where $\xi_t(\rho,h) \equiv \sum_{\ell=1}^{h} \rho^{h-\ell} u_{t+\ell}$, which is important for the use of HC standard errors as we discussed in Remark (ref). Part (iii) and (iv) of Assumption (ref) are mild regularity conditions on the distribution of the shocks $P$ to establish uniform bounds of approximation errors, which can be relaxed if stronger assumptions are imposed over the serial dependence of the shocks; see Assumption (ref) in Appendix (ref).

The next assumption is a high-level assumption and imposes additional restrictions on the distributions of the shocks $P$.

assumption\begin{equation*} \lim_{M \to \infty} \lim_{n \to \infty} \inf_{ |\rho| \le 1 } P_{\rho} \left( g(\rho,n)^{-2} n^{-1}\sum_{t=1}^{n}y_{t-1}^2 \ge 1/M \right) = 1 , \end{equation*} where $ g(\rho,k) = \left(\sum_{\ell=0}^{k-1} ~ \rho^{2 \ell}\right)^{1/2}~.$

This assumption implies that the estimator $\hat{\rho}_n(h)$ defined in (ref) is well-behaved in the sense that its denominator after scaled by the factor $g(\rho,n-h)$ converges to a strictly positive limit. As a result, we can replace the residual $\hat{u}_t(h) \equiv y_t - \hat{\rho}_n(h)y_{t-1}$ by the shock $u_t$, which implies the second and third implication discussed in Remark (ref). We show in Proposition (ref) that Assumption (ref) can be verified if the shocks are i.i.d. and satisfied mild regularity conditions (Assumption (ref)). In Appendix C of MO-PM2020, this assumption is verified for AR(1) models whenever a contiguity condition holds.

Assumptions (ref) and (ref) guarantee that the distribution $J_n(\cdot,h,P,\rho)$ defined in (ref) can be approximated by the standard normal distribution $\Phi(\cdot)$ uniformly on $\rho \in [-1,1]$ and \textcolor{black}{a wide range of horizons} $h$ as in (ref). Let $\hat{P}_n$ be the empirical distribution of the centered residuals defined in (ref) and let $\hat{\rho}_n$ be the estimator of $\rho$ defined in (ref). Using this notation $J_n(\cdot,h,\hat{P}_n,\hat{\rho}_n)$ is the distribution of the bootstrap root $R_{b,n}^*(h)$ defined in (ref) conditional on the data $Y^{(n)}$. The next theorem shows that the distribution $J_n(\cdot,h,P,\rho)$ can be approximated by the bootstrap distribution $J_n(\cdot,h,\hat{P}_n,\hat{\rho}_n)$ uniformly on $\rho \in [-1,1]$ and \textcolor{black}{a wide range of} intermediate horizons (e.g., \textcolor{black}{uniform over $h \le h_n$, where $h_n$ is any fixed sequence such that $h_n = o(n)$}), i.e., the LP-residual bootstrap is uniformly consistent.

theoremSuppose Assumptions (ref) and (ref) hold. Then, for any $\epsilon>0$ and for any sequence $h_n$ \textcolor{black}{such that $h_n \le n$ and} $h_n = o\left(n\right)$, we have \begin{equation} \sup_{|\rho|\le 1} P_{\rho}\left( \sup_{h \le h_n} \sup_{x\in \mathbf{R}} | J_n(x,h,P,\rho) - J_n(x,h,\hat{P}_n,\hat{\rho}_n) | > \epsilon \right) \to 0 \quad as \quad n \to \infty , \end{equation} where $J_n(x,h,\cdot,\cdot)$ is as in (ref), $\hat{P}_n$ is the empirical distribution of the centered residuals defined in (ref), and $\hat{\rho}_n$ is as in (ref).

Theorem (ref) shows that the LP-residual bootstrap is uniformly consistent, i.e., the bootstrap distribution $J_n(\cdot,h,\hat{P}_n,\hat{\rho}_n)$ approximates the distribution $J_n(\cdot,h,P,\rho)$ uniformly over the parameter space ($\rho \in [-1,1]$) and \textcolor{black}{a wide range of} intermediate horizons ($h \le h_n)$. Two features of this uniform approximation result deserve further discussion. First, uniform consistency of bootstrap methods over the parameter spaces of autoregressive models is not just a technical detail but a crucial property to guarantee reliable inference methods; see Mikusheva2007. Otherwise, it is possible to obtain for any sample size $n$ a parameter $\rho_n$ such that the distance between the distributions $J_n(\cdot,h,\hat{P}_n,\hat{\rho}_n)$ and $J_n(\cdot,h,P,\rho)$ is far from zero. Second, the uniform approximation over the horizons is necessary for inference purposes at intermediate horizons. Other valid methods for a fixed $h$ do not necessarily work for $h$ growing with the sample size.

The proof of Theorem (ref) is presented in Appendix (ref). It has two main ideas. First, we show that the approximation result presented in (ref) also holds for sequences of AR(1) models with i.i.d. shocks (Theorem (ref)),

equation*[equation* omitted — 207 chars of source]

where $\mathbf{P}_{n,0}$ denotes the set of all distributions that satisfy Assumption (ref) in Appendix (ref), $h_n$ is as in Theorem (ref), $J_n(\cdot,h,P,\rho)$ is as in (ref) and $\Phi(\cdot)$ is the standard normal distribution. Assumption (ref) imposes stronger restrictions on the dependence of the shocks (i.i.d.) and some mild regularity conditions. The formal result is presented in Appendix (ref) as Theorem (ref). Second, we show that Assumptions (ref) and (ref) imply the existence of a sequence of events $E_n$ with probability approaching 1 such that the empirical distributions $\hat{P}_n $ conditional on the event $E_n$ verify Assumption (ref). In other words, we show that $\hat{P}_n \in \mathbf{P}_{n,0}$ holds with a probability approaching 1. The construction of the events $E_n$ relies on Lemma (ref) in Appendix (ref). We use the previous two ideas to approximate the distribution $J_n(\cdot,h,\hat{P}_n,\hat{\rho}_n)$ by the standard normal distribution $\Phi(\cdot)$ conditional on the event $E_n$. Finally, we conclude that the distributions $J_n(\cdot,h,\hat{P}_n,\hat{\rho}_n)$ and $J_n(\cdot,h,P,\rho)$ are asymptotically close since both have the same asymptotic limit.

The next result shows that the confidence interval $C_n^*(h,1-\alpha)$ defined in (ref) is uniformly asymptotically valid in the sense that its asymptotic coverage probability is equal to $1-\alpha$ uniformly over $\rho$ and \textcolor{black}{a wide range of horizons} $h$.

theoremSuppose Assumptions (ref) and (ref) hold. Then, for any sequence $h_n$ \textcolor{black}{such that $h_n \le n$ and} $h_n = o\left(n\right)$, we have \begin{equation} \sup_{|\rho|\le 1} \sup_{h \le h_n} \left| P_{\rho} \left( \beta(\rho,h) \in C_n^*(h,1-\alpha) \right) - (1-\alpha) \right| \to 0 \quad as \quad n \to \infty , \end{equation} where $\beta(\rho,h)$ and $C_n^*(h,1-\alpha)$ are as in (ref) and (ref), respectively.

Theorem (ref) provides the theoretical justification to conduct inference on the impulse response coefficient $\beta(\rho,h)$ using our bootstrap confidence interval $C_n^*(h,1-\alpha)$. Note that the only difference with respect to the confidence interval $C_n(h,1-\alpha)$ defined in (ref) is the critical value, which was equal to $z_{1-\alpha/2}$. The critical value $z_{1-\alpha/2}$ was the same for different sample sizes $n$ and horizons $h$. Instead, we now use a critical value $c_n^*(h,1-\alpha)$ that depends on the data, the sample size, and the horizon. We evaluate the difference in coverage probability between the confidence intervals $C_n(h,1-\alpha)$ and $C_n^*(h,1-\alpha)$ using simulations in Section (ref). The simulation results provide evidence that the coverage probability of our proposed confidence interval $C_n^*(h,1-\alpha)$ is closer to $1-\alpha$ than that of $C_n(h,1-\alpha)$.

The proof of Theorem (ref) is presented in Appendix (ref). It only relies on the uniform consistency of the bootstrap procedure. We next sketch the main arguments of the proof. We first note that (ref) is equivalent to $$ \sup_{|\rho|\le 1} ~ \sup_{h \le h_n} \left| P_{\rho} \left( |R_n(h)| \le c_n^*(h,1-\alpha) \right) - (1-\alpha) \right| \to 0 \quad \text{as} \quad n \to \infty~. $$ We then use that the bootstrap critical value $c_n^*(h,1-\alpha)$ is included in $[z_{1-\alpha/2 -\epsilon},~z_{1-\alpha/2 +\epsilon}]$ with a probability approaching 1 for arbitrary $\epsilon>0$; see Lemma (ref) in Appendix (ref). This result is possible because the root $R_n(h)$ is asymptotically normal and the LP-residual bootstrap is uniformly consistent. Third, we can conclude using algebra manipulation and the asymptotic normality of the root $R_n(h)$ that $$ \limsup_{n \to \infty} \sup_{|\rho|\le 1} \sup_{h \le h_n} \left| P_{\rho} \left( |R_n(h)| \le c_n^*(h,1-\alpha) \right) - (1-\alpha) \right| \le 2\epsilon ~,$$ which implies (ref) since $\epsilon>0$ was arbitrary.

remarkWe can use the LP-residual bootstrap to construct equal-tailed percentile-$t$ confidence intervals denoted by $C_{per-t,n}^{*}(h,1-\alpha)$. That is \begin{equation} C_{per-t,n}^{*}(h,1-\alpha) \equiv \left[ \hat{\beta}_n(h) - q_n^*(h,1-\alpha/2) \hat{s}_n(h), \hat{\beta}_n(h) - q_n^*(h,\alpha/2) \hat{s}_n(h)\right] , \end{equation} where $\hat{\beta}_n(h)$ is as in (ref), $\hat{s}_n(h)$ is as in (ref), and $q_n^*(h,\alpha_0)$ is the $\alpha_0$-quantile of the bootstrap root $R_{b,n}^*(h)$ defined in (ref). Three features of $C_{per-t,n}^{*}(h,1-\alpha)$ deserve further discussion. First, the bootstrap quantiles $q_n^*(h,\alpha_0)$ can be approximated using Monte Carlo procedures in a similar way as we discussed in Remark (ref). Second, the confidence interval $C_{per-t,n}^{*}(h,1-\alpha)$ can be asymmetric around $\hat{\beta}_n(h)$ by construction, which is not the case of $C_n^{*}(h,1-\alpha)$ that is a symmetric one. Third, $C_{per-t,n}^{*}(h,1-\alpha)$ is uniformly asymptotically valid, $$ \sup_{|\rho|\le 1} \sup_{h \le h_n} \left| P_{\rho} \left( \beta(\rho,h) \in C_{per-t,n}^{*}(h,1-\alpha) \right) - (1-\alpha) \right| \to 0 \quad \text{as} \quad n \to \infty~,$$ where \textcolor{black}{$h_n$ is any fixed sequence such that} $h_n \le n$ and $h_n = o\left(n\right)$. The proof of this claim follows directly by Theorem (ref), Lemma (ref), and the proof of Theorem (ref). We include $C_{per-t,n}^{*}(h,1-\alpha)$ in our simulation study in Section (ref).

remarkFor short horizons (fixed $h$), the available grid bootstrap (hansen1999grid,Mikusheva2012) is a valid alternative to our bootstrap confidence interval $C_n^*(h,1-\alpha)$ when the conditional variance of the shocks is constant. The grid bootstrap is a method to construct confidence intervals for the parameter $\beta(\rho,h)$ defined in (ref) based on test inversion. Mikusheva2007,Mikusheva2012 shows that the grid bootstrap provides confidence intervals that are uniformly asymptotically valid in the sense that its asymptotic coverage probability is equal to $1-\alpha$ uniformly on $\rho \in [-1,1]$. Nevertheless, when the conditional variance of the shocks is not constant (e.g., GARCH shocks), it is unknown if the confidence intervals based on the grid bootstrap are valid. In contrast, $C_n^*(h,1-\alpha)$ remains valid for a larger class of AR(1) models. We include the grid bootstrap presented in Mikusheva2012 in our simulation study presented in Section (ref).

remarkIf we restrict our analysis to data-generating processes with weak dependence (e.g., $|\rho| \le 1-a$ for some $a \in (0,1)$) and consider stronger assumptions in the distribution of the shocks $\{u_t: 1 \le t \le n\}$, then both claims in (ref) and (ref) can hold for long horizons (e.g., $h_n \le (1-b)n$ for some $b \in (0,1)$). In other words, the confidence interval $C_n^*(h,1-\alpha)$ has theoretical guarantees for long horizons under certain conditions. Assumptions 1-2 in MO-PM2020 are sufficient to guarantee this claim; a formal proof can be derived following the same strategy presented in Appendix (ref) to prove Theorem (ref) and (ref). In particular, the proof of Theorem (ref) can be adapted for long horizons since $|\rho| \le 1-a$ implies that $g(\rho,h_n)^2/(n-h_n) \to 0$ as $n \to \infty$ for any $h_n \le (1-b) n$, where $g(\rho,h) = \{ \sum_{\ell=1}^{h} \rho^{2(\ell-1)} \}^{1/2}$. This technical condition was satisfied when $|\rho| \le 1$ and $h_n = o(n)$.

remarkFor strictly stationary data, the results in Theorems (ref) and (ref) can be extended to vector autoregressive (VAR) models considered in MO-PM2020 that satisfy their Assumptions 1 and 2. A proof of these extensions may be done using the finite sample inequalities presented in their online appendix and following the approach we presented in Appendixes (ref) and (ref). We leave the details of a formal proof to future research. For non-stationary data, it is an open question whether the LP-residual bootstrap is consistent for VAR models. Our approach relies on verifying Assumption (ref) for an appropriate sequence of AR(1) models; therefore, an analogous approach may require a similar step for VAR models, which is \textcolor{black}{outside} the scope of this paper.

remarkWe can use Theorem (ref) to show the uniform validity of alternative methods to construct confidence intervals for $\beta(\rho,h)$; however, some alternative confidence intervals can be impractical at the intermediate horizon. For instance, a confidence interval $C_{la-ar}^*(h,1-\alpha)$ for $\beta(\rho,h)$ can be obtained by first constructing a confidence interval for $\rho$ using Theorem (ref) (taking $h=1$) and then by using $\beta(\rho,h) = \rho^h$ (monotone transformation). Unfortunately, the confidence interval $C_{la-ar}^*(h,1-\alpha)$ can be very wide asymptotically for certain data-generation processes and intermediate horizons. More concretely, for any $L>1$ it can be shown $P_{\rho}\left( [1/L,L] \subseteq C_{la-ar}^*(h,1-\alpha) \right) \to 1 $ as $n \to \infty$ when $\rho = 1-c_1/n$ (local-to-unit models) and $h \sim \sqrt{n}$. We formally establish this result in Proposition (ref) in Appendix (ref). This result is similar to the ones presented in Appendix B.2.2 in MO-PM2020 for the lag-augmented AR bootstrap confidence interval of InoueKillian2020, which is a bootstrap confidence interval related but different to $C_{la-ar}^*(h,1-\alpha)$.

Asymptotic Refinements

\textcolor{black}{This section will impose conditions on the data-generating process that further restrict the class of AR(1) models relative to that considered in Section (ref), ruling out local-to-unity and unit-root models. These conditions are explicit in Theorems (ref) and (ref), where we calculate the sizes of the error in coverage probability (ECP) for the confidence intervals $C_n(h,1-\alpha)$ and $C_n^*(h,1-\alpha)$ defined in (ref) and (ref), respectively. The results of these theorems show that the LP-residual bootstrap can provide asymptotic refinements for confidence intervals, that is, the ECP of $C_n^{*}(h,1-\alpha)$ is $o(n^{-1})$, whereas the ECP of $C_n(h,1-\alpha)$ is $O(n^{-1})$.}

\textcolor{black}{Section (ref) first provides an informal discussion of the elements and challenges involved in obtaining asymptotic refinements for confidence intervals with the LP-residual bootstrap. Section (ref) then formalizes the discussion by giving conditions on the data-generating process (Assumption (ref) and $\rho \in [-1+a, 1-a] $ for a given $a \in (0,1)$) that are sufficient to establish these asymptotic refinements (Theorems (ref) and (ref)).}

Informal Discussion on Asymptotic Refinements

\textcolor{black}{This section gives an informal exposition on how a bootstrap method can provide asymptotic refinements for confidence intervals when the root is asymptotically pivotal, i.e., the asymptotic distribution of the root does not depend on any unknown parameters.} The explanation below is not new; see hall1996bootstrap, Horowitz2001,Horowitz2019, and lahiri2003resampling. It has the purpose of introducing the main elements and challenges that arise to obtain asymptotic refinements \textcolor{black}{in the context of dependent data generated from an AR(1) model. It also describes the approach considered in this paper; see Remark (ref) for alternative methods. }

\textcolor{black}{Main elements: For the sake of exposition, suppose the root $R_n(h)$ has an Edgeworth expansion up to an error of size $o(n^{-1})$, that is, the distribution of the root $R_n(h)$ has an asymptotic expansion,

equation[equation omitted — 153 chars of source]

where $q_j(x,h,P,\rho)$ are polynomials in $x \in \textbf{R}$ such that (i) their coefficients are continuous function of moments of $P$ and $\rho$ and (ii) $q_j(x,h,P,\rho) = (-1)^{j+1} q_j(-x,h,P,\rho)$ for $j=1,2$. Similarly, suppose the bootstrap root $R_n^*(h)$ has an Edgeworth expansion,

equation[equation omitted — 192 chars of source]

where $J_n(x,h,\cdot,\cdot)$ is as in (ref), $\hat{P}_n$ is the empirical distribution of the centered residuals defined in (ref), and $\hat{\rho}_n$ is the estimator of $\rho$ defined in (ref).}

\textcolor{black}{ The approximations in (ref) and (ref) are commonly used to show that the bootstrap methods provide more accurate approximations than the asymptotic distribution theory; see Hall1992TheBA for a textbook reference for the case of i.i.d. data. We next sketch an informal calculation of the sizes of the ECP of the confidence intervals $C_n(h,1-\alpha)$ and $C_n^{*}(h,1-\alpha)$.}

\textcolor{black}{The coverage probability of $C_n(h,1-\alpha)$ is equal to $P_{\rho} \left( |R_n(h)| \le z_{1-\alpha/2} \right)$ by the definitions of $C_n(h,1-\alpha)$ and $R_n(h)$ in (ref) and (ref), respectively. Note that (ref) and the properties of $q_j(\cdot,h,P,\rho)$ imply that for any $x>0$, we have

equation[equation omitted — 153 chars of source]

Taking $x = z_{1-\alpha/2}$, we conclude the size of the ECP of $C_n(h,1-\alpha)$ is $O(n^{-1})$.}

\textcolor{black}{Similarly, the coverage probability of $C_n^*(h,1-\alpha)$ is equal to $P_{\rho} \left( |R_n(h)| \le c_n^*(h,1-\alpha) \right)$ by the definitions in (ref) and (ref). Now, we will argue that

align[align omitted — 180 chars of source]

where $c_n(h,1-\alpha)$ is as in (ref). This is sufficient to conclude that the size of the ECP of $C_n^*(h,1-\alpha)$ is $o(n^{-1})$ since $P_{\rho} \left( |R_n(h)| \le c_n(h,1-\alpha)\right) = 1-\alpha$ by definition. Using the properties of $q_j(\cdot,h,\hat{P}_n,\hat{\rho}_n)$ and (ref), we obtain

align[align omitted — 275 chars of source]

where the last equality uses $q_2(x,h,\hat{P}_n,\hat{\rho}_n) = q_2(x,h,P,\rho) + o_p\left(1\right) $. Note that (ref) and (ref) looks similar. Taking $x = c_n(h,1-\alpha)$ in (ref) and $x = c_n^*(h,1-\alpha)$ in (ref), it can be conclude that $c_n^*(h,1-\alpha) = c_n(h,1-\alpha) + o_p\left(n^{-1}\right)$, which will imply (ref). }

\textcolor{black}{ The informal explanation presented above suggests that the LP-residual bootstrap can provide asymptotic refinements when there exist valid Edgeworth expansions as in (ref)-(ref). We present in Section (ref) conditions (Assumption (ref) and $\rho \in [-1+a, 1-a] $, where $a \in (0,1)$) under which the previous informal discussion can be formalized. }

\textcolor{black}{ The challenges: Edgeworth expansions as in (ref)-(ref) are not always available or valid in the context of AR(1) models. For instance, in the case of the local-to-unity and unit-root models, the Edgeworth expansion for the least-squares estimate of the AR(1) model defined in (ref) is no longer valid; see phillips2023estimation. In this case, alternative asymptotic approximations were developed to prove asymptotic refinements of the bootstrap, e.g., park2003bootstrap,park2006bootstrap and mikusheva2015second. To our knowledge, there are no available theoretical results about valid Edgeworth expansions for the root $R_n(h)$ defined in (ref) that can be applied directly.}

\textcolor{black}{ Nevertheless, for stationary AR(1) models (when $\rho \in [-1+a,1-a]$, $a \in (0,1)$), asymptotically valid Edgeworth expansions were obtained; see phillips1977approximations, phillips1977general, bose1988, among others. Therefore, we will restrict our analysis to stationary AR(1) models to obtain valid Edgeworth expansions for the root $R_n(h)$ and its bootstrap version $R_n^*(h)$ when $\rho \in [-1+a,1-a]$, $a \in (0,1)$, and $h$ is fixed.}

Formal Conditions and Results

This section presents conditions under which the LP-residual bootstrap provides asymptotic refinements to the confidence interval. Under these conditions, we calculate the sizes of the ECP for $C_n(h,1-\alpha)$ and $C_n^*(h,1-\alpha)$ in Theorems (ref) and (ref), respectively.

The following assumption imposes stronger conditions on the distribution of the shocks $P$ than the ones presented in Assumption (ref). We use this assumption \textcolor{black}{and $\rho \in [-1+a, 1-a]$ for some $a \in (0,1)$} to formalize the informal explanation about asymptotic refinements presented in Section (ref).

assumption\begin{enumerate} • $\{u_t: 1 \le t \le n\}$ is a sequence of i.i.d. random variables with $E[u_t] =0$. • $u_t$ has a positive continuous density. • $E[e^{x u_t}] \le e^{x^2 c_u^2}$ for all $|x| \le 1/c_u$ and $E[u_t^2] \ge C_{\sigma} $ for some constants $c_u, C_\sigma>0$. \end{enumerate}

Part (i) of Assumption (ref) imposes stronger conditions over the serial dependence of the shocks. This assumption is common for theoretical analysis of the asymptotic refinement of the bootstrap method in autoregressive models. An incomplete list of previous research that uses this assumption includes bose1988, park2003bootstrap,park2006bootstrap, and mikusheva2015second. Parts (ii) and (iii) of Assumption (ref) are sufficient technical conditions on the distribution of the shocks $P$ to establish the existence of the Edgeworth expansions presented in (ref)-(ref). Part (ii) implies that the distribution $J_n(\cdot,h,P,\rho)$ defined in (ref) is continuous and guarantees that a data-dependent version of the Cram\'er condition holds, which is a common condition to guarantee the existence of Edgeworth expansions; see Remark (ref) for further discussion. Part (iii) implies that any sufficiently large number of moments exist and are uniformly bounded by a function of the constant $c_u$, which is important to guarantee the Edgeworth expansion for the bootstrap distribution $J_n(\cdot,h,\hat{P}_n,\hat{\rho}_n)$. Although this condition is strong, it is not atypical in the literature of the asymptotic refinement of the bootstrap method with dependent data; for instance, hall1996bootstrap and inoue2006bootstrapping assume the existence of 33rd and 36th moments, respectively, while andrews2002higher assumes that all the \textcolor{black}{moments exist}.

We rely on Assumption (ref), the approach and results presented in bhattacharya1978validity and bhattacharya1987some, and the general framework developed by GotzeHipp1983 to prove the existence of Edgeworth expansions with dependent data. The framework of GotzeHipp1983 requires weakly dependent data and verifying stronger regularity conditions than the ones needed in the case of i.i.d. data; see Hall1992TheBA and lahiri2003resampling for textbook references. Therefore, we restrict our analysis to data-generating processes with weak dependence (e.g., $|\rho| \le 1-a$ for some $a \in (0,1)$) in a similar way to previous research on asymptotic refinements involving dependent data that includes bose1988, hall1996bootstrap, lahiri1996edgeworth, andrews2002higher, andrews2004block, and inoue2006bootstrapping. It is an open question whether there exist Edgeworth expansions as in (ref)-(ref) for the case \textcolor{black}{of local-to-unity or unit-root models}. See Remark (ref) for further discussion on alternative methods and available results.

theoremSuppose Assumption (ref) \textcolor{black}{holds}. Fix a given $h \in \textbf{N}$ and $a \in (0,1)$. Then, for any $\rho \in [-1+a,1-a]$, we have \begin{equation} | P_{\rho} \left( \beta(\rho,h) \in C_n(h,1-\alpha) \right) - (1-\alpha) | = O(n^{-1}) \end{equation} where $\beta(\rho,h)$ is as in (ref) and $C_n(h,1-\alpha)$ is as in (ref).

The ECP of $C_n(h,1-\alpha)$ has a similar size as the one derived in our informal explanation in Section (ref). Similar sizes of the ECP were obtained for symmetrical confidence intervals in the i.i.d. data case; see Hall1992TheBA and Horowitz2001, Horowitz2019.

The proof of Theorem (ref) is presented in Appendix (ref). It uses two main ideas developed previously in the literature. First, we approximate the distribution $J_n(\cdot,h,P,\rho)$ by another distribution $\Tilde{J}_n(\cdot,h,P,\rho)$ up to an error of size $O\left(n^{-1-\epsilon}\right)$ for a fixed $\epsilon \in (0,1/2)$; similar approach has been used in hall1996bootstrap and andrews2002higher, andrews2004block. Second, we use that the distribution $\Tilde{J}_n(\cdot,h,P,\rho)$ admits an Edgeworth expansion up to an error of size $O\left(n^{-3/2}\right)$ based on the results of bhattacharya1978validity and GotzeHipp1983,gotze1994asymptotic; see Theorem (ref) in Appendix (ref). These two ideas guarantee the existence of the Edgeworth expansion presented in (ref). We then conclude the proof by standard derivations similar to the one derived in our informal explanation presented in Section (ref).

The next theorem shows that the LP-residual bootstrap provides asymptotic refinements to the confidence intervals. In other words, the size of the ECP of our bootstrap confidence interval defined in (ref) for $\beta(\rho,h)$ is $o(n^{-1})$.

theoremSuppose Assumption (ref) \textcolor{black}{holds}. Fix a given $h \in \textbf{N}$ and $a \in (0,1)$. Then, for any $\rho \in [-1+a,1-a]$ and $\epsilon \in (0,1/2)$, we have \begin{equation} | P_{\rho} \left( \beta(\rho,h) \in C_n^*(h,1-\alpha) \right) - (1-\alpha) | = o \left( n^{-(1+\epsilon)} \right) , \end{equation} where $\beta(\rho,h)$ is as in (ref) and $C_n^*(h,1-\alpha)$ is as in (ref).

Theorem (ref) presents the size of the ECP of the confidence interval $C_n^*(h,1-\alpha)$ in (ref). This is similar to the one derived in our informal explanation in Section (ref), but it is typically larger than those obtained for the ECP of symmetrical confidence intervals using bootstrap methods in the i.i.d. data case; see Hall1992TheBA and Horowitz2001, Horowitz2019.

The proof of Theorem (ref) is presented in Appendix (ref). It relies on two claims: the existence of the Edgeworth expansion for the distribution $J_n(\cdot,h,P,\rho)$ and the existence of constants $C_1$ and $C_2$ such that $P_{\rho}(|\Delta_n| > C_1 n^{-(1+\epsilon)} ) \le C_2 n^{-(1+\epsilon)} $, where $\Delta_n = c_n^*(h,1-\alpha)-c_n(h,1-\alpha)$, and $c_n(h,1-\alpha)$ and $c_n^*(h,1-\alpha)$ are defined in (ref) and (ref), respectively. We next sketch the proof based on those two claims. We can derive

align*[align* omitted — 357 chars of source]

where the last equality follows from the existence of the Edgeworth expansion for the distribution $J_n(\cdot,h,P,\rho)$ (our first claim), which implies $$ P_{\rho} \left( |R_n(h)| \le c_n(h,1-\alpha) + O\left( n^{-(1+\epsilon)} \right) \right) = 1-\alpha + O\left( n^{-(1+\epsilon)} \right)~.$$ Note that the first claim follows from Theorem (ref). To prove our second claim, we first show that there is an event $E_n$ such that (i) $J_n(\cdot,h,\hat{P}_n,\hat{\rho}_n)$ has an Edgeworth expansion as in (ref) conditional on $E_n$ and (ii) the probability of the complement of $E_n$ is equal to $O\left(n^{-(1+\epsilon)}\right)$ for any $\epsilon \in (0,1/2)$; see Lemma (ref) in Appendix (ref). We then follow standard arguments in the literature to prove this claim. Finally, note that $O(n^{-(1+\epsilon)})$ for any $\epsilon \in (0,1/2)$ is equivalent to $o(n^{-(1+\epsilon)})$ for any $\epsilon \in (0,1/2)$, which is the error stated in Theorem (ref).

remarkThe bootstrap methods proposed in hall1996bootstrap and andrews2002higher can be adapted for the construction of confidence intervals for the impulse response $\beta(h,\rho)$ defined in (ref). Four points based on their framework and results deserve further discussion. First, their bootstrap method consists of the nonoverlapping block bootstrap scheme (carlstein1986use) and overlapping block bootstrap (kunsch1989jackknife). Second, they show that their bootstrap methods provide asymptotic refinements to the critical values of $t$-tests based on generalized method of moments (GMM) estimators $\hat{\theta}_T$ and weakly dependent data $\{Z_t:1 \le t \le n\}$. One of their main conditions is that the series of moment functions $\{g(Z_t,\theta): t \ge 1 \}$ are uncorrelated beyond some finite lags, i.e. for some $\kappa >0$ we have $E[g(Z_t,\theta)g(Z_s,\theta)'] = 0$ for any $t,s\ge 1$ such that $|t-s|>\kappa$. Third, the LP estimator $\hat{\beta}_n(h)$ defined in (ref) can be presented as a GMM estimator using the following dependent data $\{Z_t = (y_{t-1}, y_t, y_{t+h}) : 1 \le t \le n\}$ and moment function: $g(y_{t+h},x_t,\theta)=(y_{t+h}-\theta x_t)x_t$, where $x_t = (y_t, y_{t-1})'$. Then, we can invoke their results and use their bootstrap methods but only for the case of $|\rho|<1$ and under additional assumptions. Note that their main condition can be verified with $\kappa = h$. Fourth, we can construct confidence intervals for $\beta(\rho,h)$ based on their asymptotic distribution theory.

remarkAs we mentioned in Remark (ref), we can use the bootstrap methods presented in hall1996bootstrap and andrews2002higher, andrews2004block to construct confidence intervals for $\beta(\rho,h)$ since the LP estimator $\hat{\beta}_n(h)$ defined in (ref) can be presented as a GMM estimator. Their results provide sizes of the ECP of these confidence intervals that are qualitatively similar to the one found in Theorem (ref).

remarkThe size of the ECP of $C_{per-t,n}^{*}(h,1-\alpha)$ is $O(n^{-1})$. We presented and discussed the equal-tailed percentile-t confidence interval $C_{per-t,n}^{*}(h,1-\alpha)$ in Remark (ref). To compute the size of its ECP, we can use the existence of the Edgeworth expansions presented in (ref)-(ref) and Theorem 5.2 in Hall1992TheBA. The size of the ECP of $C_{per-t,n}^{*}(h,1-\alpha)$ is similar to the one obtained in (ref) for the ECP of $C_n(h,1-\alpha)$; therefore, the LP-residual bootstrap does not provide asymptotic refinement for equal-tailed percentile-t confidence intervals. Similar conclusions were obtained for the case of i.i.d. data; see Hall1992TheBA and Horowitz2001, Horowitz2019.

remarkWe use part (ii) of Assumption (ref) to verify that a dependent-data version of the Cramer condition required in GotzeHipp1983 holds, which is an important condition for the existence of the Edgeworth expansion in the dependent-data case. However, verifying that condition is quite difficult in general, as pointed out by hall1996bootstrap and gotze1994asymptotic, among others. Therefore, we proceed in two steps based on the results by gotze1994asymptotic that propose simple and verifiable conditions to guarantee the conditions required by GotzeHipp1983, including the dependent-data version of the Cramer condition. We first approximate the distribution $J_n(\cdot,h,P,\rho)$ by a distribution $\Tilde{J}_n(\cdot,h,P,\rho)$. We then use part (ii) of Assumption 5.1 to verify the conditions required in Theorem 1.2 of gotze1994asymptotic, which guarantee the existence of Edgeworth expansion for the distribution $\Tilde{J}_n(\cdot,h,P,\rho)$.

remarkFor strictly stationary data-generating processes, the results in Theorems (ref) and (ref) can be extended to the family of vector autoregressive (VAR) models that satisfy similar assumptions to the ones presented in Assumption (ref), which are stronger than Assumptions 1 and 2 in MO-PM2020. These extensions can be shown by verifying the conditions required in gotze1994asymptotic. We leave the details of a formal proof for the VAR models for future research.

remarkAn alternative method for asymptotically approximating a finite sample distribution is the stochastic embedding and strong approximation principle used in park2003bootstrap,park2006bootstrap and mikusheva2015second. Using this method in the local-to-unit asymptotic framework for the AR(1) model, mikusheva2015second showed that the grid bootstrap version of the t-statistic approximates its finite sample distribution up to an error of size $o(n^{-1/2})$. It is an open question whether these techniques can be adapted to show that LP-residual bootstrap provides asymptotic refinements to the confidence intervals when $\rho=1$.

Simulation Study

We examine the finite sample performance of $C_n^*(h,1-\alpha)$ defined in (ref) using different data-generating processes. We consider a sample size $n = 95$, which is the median sample size based on 71 papers that have utilized the LP approach; see herbst2024bias. Additionally, we examine other confidence intervals presented in the paper.

Monte Carlo Design

We use four designs for the distribution of the shocks $\{u_t:1 \le t \le n\}$ and two values for the parameter $\rho \in \{0.95, 1\}$ in our Monte Carlo simulation. The shocks are defined according to the GARCH(1,1) model:

equation*[equation* omitted — 143 chars of source]

where the distribution of $v_t$ and the parameter vector $(\omega_0,\omega_1,\omega_2)$ are specified as follows:

itemize• $v_t \sim N(0,1)$, $\omega_0=1$, and $\omega_1=\omega_2=0$. • $v_t \sim N(0,1)$, $\omega_0=0.05$, $\omega_1= 0.3$, and $\omega_2=0.65$. • $v_t \sim t_4/\sqrt{2}$, $\omega_0=1$, and $\omega_1=\omega_2=0$. • $v_t|B_t=j \sim N(m_j,\sigma_j^2)$, where $B_t \in \{0,1\}$, $B_t = 1$ with probability $p = 0.25$, $m_0 = 2/\sigma_2$, $m_1 =-6/\sigma_2$, $\sigma_0 = 0.5/\sigma_2$, $\sigma_1 = 2/\sigma_2$, and $\sigma_2^2 = p(m_1^2+\sigma_1)+(1-p)(m_0^2+\sigma_0)$, $\omega_0=0.05$, $\omega_1= 0.3$, and $\omega_2=0.65$.

We consider nine different confidence intervals for each design and each value of $\rho$. All our confidence intervals use the HC standard errors $\hat{s}_n(h)$ defined in (ref). Additionally, we consider alternative HC standard errors $\hat{s}_{j,n}(h)$ defined as

equation*[equation* omitted — 231 chars of source]

for $j=2,3$, where $\hat{\xi}_{2,t}(h)^2 = \hat{\xi}_{t}(h)^2/(1-\mathbb{P}_{h,tt})$ and $\hat{\xi}_{3,t}(h)^2 = \hat{\xi}_{t}(h)^2/(1-\mathbb{P}_{h,tt})^2$. We use the projection matrix $\mathbb{P}_h = \mathbb{X}_h (\mathbb{X}_h'\mathbb{X}_h)^{-1}\mathbb{X}_h'$, where $\mathbb{X}_h$ is a matrix with row elements equal to $(\hat{u}_t(h),~ y_{t-1})$ for $t=1,\ldots,n-h$. The confidence intervals that we use are listed below\textcolor{black}{.}

enumerate• RB: confidence interval as in (ref) based on the LP-residual bootstrap. • $\text{RB}_{per-t}$: equal-tailed percentile-t confidence interval as in (ref). It is based on the LP-residual bootstrap and discussed in Remark (ref). • $\text{RB}_{hc3}$: confidence interval as in (ref) but using $\hat{s}_{3,n}(h)$ and $c_{3,n}^*(h,1-\alpha)$ instead of $\hat{s}_{n}(h)$ and $c_n^*(h,1-\alpha)$, where $c_{3,n}^*(h,1-\alpha)$ is computed as in Section (ref) but using $\hat{s}_{3,n}^*(h)$ instead of $\hat{s}_{n}^*(h)$. • WB: confidence interval as in (ref) but using $c_n^{wb,*}(h,1-\alpha)$ instead of $c_n^*(h,1-\alpha)$, where $c_n^{wb,*}(h,1-\alpha)$ is based on the LP-wild bootstrap; see Remark (ref). • $\text{WB}_{per-t}$: equal-tailed percentile-t confidence interval as in (ref) but using $q_n^{wb,*}(h,\alpha_0)$ instead of $q_n^*(h,\alpha_0)$, where $q_n^{wb,*}(h,\alpha_0)$ is based on the LP-wild bootstrap discussed in Remark (ref). • $\text{GB}_{LR}$: confidence interval based on the grid bootstrap presented in Section 3.3 in Mikusheva2012. It uses the LR statistic. • \textbf{AA:} standard confidence interval as in (ref). • \textbf{$\text{AA}_{hc2}$:} standard confidence interval as in (ref) but using $\hat{s}_{2,n}(h)$ instead of $\hat{s}_{n}(h)$. • \textbf{$\text{AA}_{hc3}$:} standard confidence interval as in (ref) but using $\hat{s}_{3,n}(h)$ instead of $\hat{s}_{n}(h)$.

Discussion and Results

In all the designs, the shocks have zero mean and variance one. Designs 1-2 verify Assumption (ref) presented in Section (ref). Design 1 also verifies Assumption (ref) due to Proposition (ref) in Appendix (ref). Assumption (ref) can be tedious to verify in general since it involves computing a probability for all the parameters $\rho$ in the parameter space and taking their infimum. In contrast, designs 3-4 do not verify all the parts of Assumption (ref). Design 3 considers shocks without a fourth moment, i.e., it does not verify part (iv) of Assumption (ref), which was a regularity condition. Design 4 considers a distribution of the shocks (GARCH errors with asymmetric $v$ and nonzero skewness) that lie outside the class of conditional heteroskedastic processes that we consider in this paper, i.e., it does not verify part (ii) of Assumption (ref). As we discussed in Remark (ref), part (ii) of Assumption (ref) was a sufficient condition for the validity of the HC standard errors $\hat{s}_n(h)$ in the construction of confidence intervals.

table[table omitted — 2,064 chars of source]

Tables (ref) and (ref) report the coverage probabilities (in %) of our simulations. Columns are labeled as the confidence intervals we specified in Section (ref). For all the designs on the distribution of the shock and values of $\rho$, we use 5000 simulations to generate data with a sample size $n = 95$ based on the AR(1) model (ref). In each simulation, we compute the nine confidence intervals described above for horizons $h \in \{1,6,12,18\}$. The confidence intervals have a nominal level equal to $1-\alpha = 90\%$. The bootstrap critical values are computed using $B=1000$ as described in Remark (ref). We summarize our findings from the simulations below.

table[table omitted — 2,080 chars of source]

Five features of Table (ref) deserve discussion. First, it shows that our recommended confidence interval RB has a coverage probability closer to 90% than the confidence intervals AA, $\text{AA}_{hc2}$, and $\text{AA}_{hc3}$ for all the designs 1-2, values of $\rho$, and horizons $h$, with some few exceptions. The lowest coverage probability of RB, AA, \textbf{$\text{AA}_{hc2}$}, and \textbf{$\text{AA}_{hc3}$} are $85\%$, $77\%$, $78\%$, and $79\%$, respectively, and occur when $\rho=1$ and horizon $h=18$. Second, \textbf{RB} and \textbf{$\text{RB}_{hc3}$} have better performance than \textbf{$\text{RB}_{per-t}$}, especially when $\rho = 1$ and the horizon is a significant fraction of the sample size ($h \in \{12,18\}$). Third, \textbf{WB} and \textbf{$\text{WB}_{per-t}$} have larger coverage probability than \textbf{RB} for all the designs 1-2, values of $\rho$, and horizons $h$, with some few exceptions. The larger coverage of \textbf{WB} and \textbf{$\text{WB}_{per-t}$} is associated with a larger median length of their confidence intervals, as we reported in Table (ref) in \textcolor{black}{Online Supplemental} Appendix (ref). Fourth, \textbf{$\text{AA}_{hc3}$} presents a coverage probability closer to $90\%$ and larger than \textbf{AA} and \textbf{$\text{AA}_{hc2}$} for all the designs 1-2, values of $\rho$, and horizons $h$. This finding suggests that using $\hat{s}_{3,n}(h)$ instead of $\hat{s}_n(h)$ can improve the coverage probability of the confidence interval; however, confidence intervals based on bootstrap methods (e.g., \textbf{RB} and \textbf{$\text{WB}_{per-t}$}) report coverage probability closer to $90\%$. Fifth, \textbf{$\text{GB}_{LR}$} has a coverage probability close to 90% on design 1 (i.i.d. shocks), while it has some distortions on design 2 that are larger on $\rho = 0.95$. As we mentioned in Remark (ref), it is unknown if the grid bootstrap is valid for design 2. The coverage probability of \textbf{$\text{GB}_{LR}$} is constant across horizons because the LR statistic is invariant to monotonic transformations; see Section 4.3 and footnote 6 on Mikusheva2012 for more details.

Table (ref) presents results for designs 3-4. Our findings for design 3 are qualitatively similar to Table (ref), which was discussed above. This suggests that failing part (iv) of Assumption (ref) (a regularity condition) does not have a major effect on the coverage probability of the confidence intervals that we considered. In contrast, design 4 shows that some of our qualitative findings can change if we fail to verify part (ii) of Assumption (ref). This result is consistent with existing theory since this assumption was a sufficient condition for the validity of confidence intervals that use HC standard errors $\hat{s}_n(h)$; see Remark (ref). In particular, $\text{RB}_{per-t}$ has a coverage probability closer to $90\%$ and larger than RB and $\text{RB}_{hc3}$. The small sample size ($n=95$) does not explain the findings for design 4. We obtain similar results for a sample size $n=240$ in Table (ref) in \textcolor{black}{Online Supplemental} Appendix (ref).

Finally, Table (ref) in \textcolor{black}{Online Supplemental} Appendix (ref) reports the statistical power of the confidence intervals specified in Section (ref). Here, we refer by statistical power to the coverage probabilities (in %) of (size-adjusted) confidence intervals for parameters different than the true one. In this sense, a low coverage probability of a confidence interval is desirable. We find all the confidence intervals have coverage probability around $80\%$ on horizon $h=1$ and designs 1, 2, and 3, which suggests they have statistical power at $h=1$. We also notice that $\text{RB}_{per-t}$, $\text{WB}_{per-t}$ and $\text{GB}_{LR}$ have a coverage probability strictly lower than $90\%$ for horizon $h=6$ and designs 1, 2, and 3. Moreover, they have a lower coverage \textcolor{black}{probability} than all the other confidence intervals. Finally, all the confidence intervals have coverage \textcolor{black}{probability} above $90\%$ on design 4, with the exception of $\text{GB}_{LR}$ for horizon $h=1$.

LP-Residual Bootstrap for VAR Models

This section describes the LP-residual bootstrap method to construct confidence intervals for a scalar function of impulse responses of VAR(p) models, where $p$ denotes the number of lags. More concretely, we propose the confidence interval in (ref) for $\nu' \beta_{h,i}$, where $\beta_{h,i} \in \mathbf{R}^k$ is the vector containing all the impulse response coefficients of the reduced-form shocks in the variable $i$ at $h$ periods in the future. Here, $\nu \in \mathbf{R}^k\setminus \{ 0 \}$ is a user-specified vector, e.g., $\nu = e_j$ (the j-th unit vector) implies $\nu' \beta_{h,i}$ is the impact of the $j$-th reduced-form shock in the variable $i$ at $h$ periods in the future.

The confidence interval for $\nu' \beta_{h,i}$ is defined as

equation[equation omitted — 225 chars of source]

where $\hat{\beta}_{i,n}(h)$, $\hat{s}_{i,n}(h,\nu)$, and $c_n^*(h,1-\alpha)$ are defined in (ref), (ref), and (ref), respectively.

Let $\{y_t \in \mathbf{R}^k: 1 \le t \le n\}$ be the available time-series data. Suppose the data have been demeaned. Denote $X_t = (y_{t-1}',\ldots,y_{t-p}')'$ for all $t=p+1,\ldots, n$. Let $\hat{\beta}_{i,n}(h)$ be obtained from an OLS regression between $y_{i,t+h}$ and $(y_t',X_t')$,

equation[equation omitted — 128 chars of source]

Let $ \hat{s}_{i,n}(h,\nu)$ be the standard error for $\nu'\hat{\beta}_{i,n}(h)$ defined by

equation[equation omitted — 259 chars of source]

where $$ \hat{u}_t(h) = y_t - \hat{A}(h) X_t~, \quad \hat{A}(h) = \left( \sum_{t=p+1}^{n-h} y_t X_t' \right) \left( \sum_{t=p+1}^{n-h} X_t X_t' \right)^{-1} $$ and $$ \hat{\Sigma}(h) = \frac{1}{n-h-p} \sum_{t=p+1}^{n-h} \hat{u}_t(h) \hat{u}_t(h)'~ \textcolor{black}{.}$$ Finally, let $c_n^*(h, 1-\alpha)$ be the bootstrap critical value involving the following steps:

itemize• Estimate a VAR(p) model with the data $Y^{(n)}$ using linear regression, $$ y_t = \hat{A}_n X_t + \hat{u}_t~, ~ t = p+1,\ldots, n$$ where \begin{equation} \hat{A}_n = \left( \sum_{t=p+1}^{n} y_t X_t' \right) \left( \sum_{t=p+1}^{n} X_t X_t' \right)^{-1} , \end{equation} and compute the centered residuals \begin{equation} \left\{ \Tilde{u}_t \equiv \hat{u}_t - \frac{1}{n-p}\sum_{t=p+1}^n \hat{u}_t : p+1 \le t \le n \right \} . \end{equation} • Generate $B$ new samples of size $n$ using (ref) and (ref). Define the sample as \begin{equation*} y_{b,t}^* = \sum_{\ell=1}^p \hat{A}_{n,\ell} y_{b,t-\ell}^* + u_{b,t}^* , \quad t= p + 1,\ldots,n , \end{equation*} where the initial p observations $(y_{b,1}^*,\ldots,y_{b,p}^*)$ are \textcolor{black}{drawn} at random from the $n-p+1$ blocks of $p$ consecutive observations in the original data. Here, $\hat{A}_n = (\hat{A}_{n,1},\ldots, \hat{A}_{n,p})$ are matrices estimated in (ref) and $\{u_{b,t}^* : 1 \le t \le n\}$ is a random sample from the empirical distribution of the centered residuals defined in (ref). The new sample $\{ y_{b,t}^* : 1 \le t \le n \} $ is called the bootstrap sample. • Compute $\hat{\beta}_{b,i,n}^*(h)$ and $\hat{s}_{b,i,n}^*(h)$ as in (ref) and (ref) using the lag-augmented LP regression and the bootstrap sample $\{ y_{b,t}^* : 1 \le t \le n \} $ for each $b=1,\ldots, B$. Define \begin{equation*} R_{b,n}^*(h,\nu) = \frac{\nu'\hat{\beta}_{b,i,n}^*(h) - \nu'\beta_i(\hat{A}_n,h)}{\hat{s}_{b,i,n}^*(h,\nu)} , b=1,\ldots, B \end{equation*} where $\beta_i(A,h) \in \mathbf{R}^k$ is the impulse response of all \textcolor{black}{reduced-form} shocks in the variable $i$ at horizon $h$ implied by the VAR(p) model with coefficients $A= (A_1,\ldots,A_p)$. Here, $\hat{A}_n$ is as in (ref). • Compute the $1-\alpha$ quantile of the $B$ draws of $R_{b,n}^*(h,\nu) $. Denote this by \begin{equation} c_n^{*}(h, 1-\alpha) \equiv \inf \left \{ u \in R : \frac{1}{B} \sum_{b=1}^B I\{ |R_{b,n}^*(h,\nu)| \le u \} \ge 1-\alpha \right\} . \end{equation}

The theoretical properties of the bootstrap confidence interval defined in (ref) are unknown for general VAR models. However, Monte Carlo simulations presented in \textcolor{black}{Online Supplemental} Appendix (ref) suggest that confidence intervals based on the LP-residual bootstrap perform better in terms of coverage probability than those based on first-order asymptotic theory. Remarks (ref) and (ref) provide further discussion on how to extend some of the results presented in this paper to general VAR models.

remarkMO-PM2020 proposed a different bootstrap confidence interval for the impulse response coefficients of VAR(p) models. As we discussed in Remark (ref), they use a wild bootstrap procedure ---which we refer to as the LP wild bootstrap--- to define the bootstrap shocks used to generate the bootstrap sample with an estimated VAR model (similar to Step 2 above). They use the LP wild bootstrap to construct equal-tailed percentile-t confidence intervals that differ from the symmetric percentile-t confidence intervals defined in (ref), which we recommend for the same reasons presented in Remark (ref) and based on our theoretical results for the AR(1) model. To our knowledge, the theoretical properties of the LP wild bootstrap procedure and the confidence intervals proposed by MO-PM2020 remain unknown. We include their recommended confidence intervals in the simulations presented in \textcolor{black}{Online Supplemental} Appendix (ref).

Concluding Remarks

This paper contributes to a growing literature on confidence interval construction for impulse response coefficients based on the local projection approach. Specifically, we propose the LP-residual bootstrap method to construct confidence intervals for the impulse response coefficients of AR(1) models at intermediate horizons. We prove two theoretical properties of this method: uniform consistency and asymptotic refinements. For a large class of AR(1) models that allow for a unit root, conditional heteroskedasticity of unknown form, and martingale difference shocks, we show that the proposed confidence interval $C_n^*(h,1-\alpha)$ defined in (ref) has an asymptotic coverage probability equal to its nominal level $1-\alpha$ uniformly over the parameter space (e.g., $\rho \in [-1,1]$) and \textcolor{black}{a wide range of} intermediate horizons. For a restricted class of AR(1) models (e.g., $|\rho|\le \textcolor{black}{1-a}$ \textcolor{black}{where $a \in (0,1)$} and i.i.d. shocks with positive continuous density), we demonstrate that the error in coverage probability of $C_n^*(h,1-\alpha)$ has size $o(n^{-1})$, that is, the LP-residual bootstrap provides asymptotic refinements to the confidence intervals.

This paper considered the AR(1) model as the first step in understanding the theoretical properties of the LP-residual bootstrap. Three possible directions exist for future research. First, the uniform consistency of the LP-residual bootstrap method is an open question for the general vector auto-regressive (VAR) model. This bootstrap method is described in Section (ref). Second, the asymptotic refinement property of this method is unknown for the unit-root model ($\rho=1$) or general VAR models. Third, future work is needed to prove the uniform consistency of the LP-wild bootstrap discussed in Remark (ref).