2.1 Previous Results

The asymptotic distribution theory developed in Montiel Olea and Plagborg-Møller (Reference Montiel Olea and Plagborg-Møller2021) and Xu (Reference Xu2023) 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 a wide range of intermediate horizons, that is,

(9) $$ \begin{align} \sup_{|\rho|\le 1} ~ \sup_{h \le h_n} ~ \sup_{x\in \mathbf{R}} |J_n(x,h,P,\rho) - \Phi(x)| \to 0 \quad \text{as} \quad n \to \infty, \end{align} $$

where $h_n$ is any fixed sequence such that $h_n \le n$ and $h_n = o\left (n\right )$ . Assumptions 4.1 and 4.2 in Section 4 are sufficient conditions on the distribution P to obtain (9) due to Theorem 2 in Xu (Reference Xu2023).

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

(10) $$ \begin{align} C_n(h,1-\alpha) \equiv \left[ \hat{\beta}_n(h) - z_{1-\alpha/2}~\hat{s}_n(h),~\hat{\beta}_n(h) + z_{1-\alpha/2}~ \hat{s}_n(h)\right]\!, \end{align} $$

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 (9) implies that 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 a wide range of intermediate horizons h,

$$ \begin{align*} \sup_{|\rho|\le 1} ~ \sup_{h \le h_n} \left| P_{\rho} \left( \beta(\rho,h) \in C_n(h,1-\alpha) \right) - (1-\alpha) \right| \to 0 \quad \text{as} \quad n \to \infty, \end{align*} $$

where $h_n$ 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 the available alternatives in the sense that it does not require any tuning parameters. 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 Lazarus et al., Reference Lazarus, Lewis, Stock and Watson2018). In contrast, the HC standard errors $\hat {s}_n(h)$ defined in (4) are simple to compute and sufficient for inference under certain conditions on the distribution P (see Remark 2.1 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 Mikusheva, Reference 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., $h=h_n \propto n^{\zeta }$ , $\zeta \in (0,1)$ ). 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 Phillips (Reference Phillips1998), Gospodinov (Reference Gospodinov2004), Pesavento and Rossi (Reference Pesavento and Rossi2006), Mikusheva (Reference Mikusheva2012), and Montiel Olea and Plagborg-Møller (Reference Montiel Olea and Plagborg-Møller2021) for additional discussion).

Remark 2.1. The HC standard errors $\hat {s}_n(h)$ defined in (4) are sufficient for the construction of valid confidence intervals under certain conditions on the distribution P. In particular, as it was pointed out by Xu (Reference Xu2023), 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 Montiel Olea and Plagborg-Møller (Reference Montiel Olea and Plagborg-Møller2021) that imply that the root $R_n(h)$ defined in (6) can be written as follows:

$$ \begin{align*}\frac{\left((n-h)^{-1/2} \sum_{t=1}^{n-h} \xi_t(\rho,h) u_t \right)}{ E\left[ \xi_t(\rho,h)^2 u_t^2\right]^{1/2}} \times \frac{ \left[(n-h)^{-1} \sum_{t=1}^{n-h} \hat{\xi}_t(h)^2 \hat{u}_t(h)^2\right]^{-1/2} }{E\left[ \xi_t(\rho,h)^2 u_t^2\right]^{-1/2}} + \varepsilon_n(\rho,h),\end{align*} $$

where $\varepsilon _n(\rho ,h)$ is a remainder error term. We derive three implications under Assumptions 4.1 and 4.2, presented in Section 4. 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 4.1. 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, Xu (Reference Xu2023) proposed alternative standard errors for the construction of confidence intervals under serially correlated scores.

Remark 2.2. The 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 (1) 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

$$ \begin{align*}y_{t+h} = \beta(\rho,h) u_t + \rho \beta(\rho,h) y_{t-1} + \xi_t(\rho,h).\end{align*} $$

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 (1).

3.1 Bootstrap Critical Value

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

1. Step 1: Estimate $\rho $ in the AR(1) model defined in (1) with the data $Y^{(n)}$ using linear regression, denoted by

   (12) $$ \begin{align} \hat{\rho}_n \equiv \left(\sum_{t=1}^n y_{t-1}^2\right)^{-1}\left(\sum_{t=1}^n y_{t-1}y_t\right)\!, \end{align} $$

   and compute the centered residuals

   (13) $$ \begin{align} \{ \tilde{u}_t \equiv \hat{u}_t - n^{-1}\sum_{t=1}^n \hat{u}_t : 1 \le t \le n \}, \end{align} $$

   where $\hat {u}_t \equiv y_{t} - \hat {\rho }_n y_{t-1}$ .

2. Step 2: Generate a new sample of size n using (1), (12), and (13). Define the sample as

   $$ \begin{align*} y_{b,t}^* = \hat{\rho}_n y_{b,t-1}^* + u_{b,t}^*, \quad y_{b,0}^* = 0,~ t=1,\ldots,n, \end{align*} $$

   where $\{u_{b,t}^* : 1 \le t \le n\}$ is a random sample from the empirical distribution of the centered residuals defined in (13). The new sample $\{ y_{b,t}^* : 1 \le t \le n \} $ is called the bootstrap sample.

3. Step 3: Compute $\hat {\beta }_{b,n}^*(h)$ and $\hat {s}_{b,n}^*(h)$ as in (3) and (4) 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

   (14) $$ \begin{align} R_{b,n}^*(h) = \frac{\hat{\beta}_{b,n}^*(h) - \beta(\hat{\rho}_n,h)}{\hat{s}_{b,n}^*(h)}, \end{align} $$

   where $\beta (\rho ,h)$ and $\hat {\rho }_n$ are as in (2) and (12), respectively.

4. Step 4: Define the bootstrap critical value as the $1-\alpha $ quantile of $|R_{b,n}^*(h)|$ conditional on the data $Y^{(n)}$ , denoted by

   (15) $$ \begin{align} c_n^{*}(h, 1-\alpha) \equiv \inf \left \{ u \in \textbf{R} : P_{\rho}\left ( |R_{b,n}^*(h)| \le u \mid Y^{(n)} \right) \ge 1-\alpha \right\}\!. \end{align} $$

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 3.2 and 5.1 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 (11). The explicit formula in (15) 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 (13). 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 3.1, which has a theoretical justification due to the Glivenko–Cantelli theorem.

Remark 3.1. It is a common practice to approximate the bootstrap critical value $c_n^{*}(h, 1-\alpha )$ using a Monte Carlo procedure (Horowitz, Reference Horowitz2001, Reference Horowitz2019). We generate B bootstrap samples of size 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

$$ \begin{align*} c_{b,n}^*(h, 1-\alpha) \equiv \inf \left \{ u \in \boldsymbol{R} : \frac{1}{B} \sum_{b=1}^B I\left\{ |R_{b,n}^*(h)| \le u \right\} \ge 1-\alpha \right\}\!. \end{align*} $$

The accuracy of the approximation improves as the number of bootstrap samples B increases. We use $B=1,000$ in our simulation study presented in Section 6.

5.1 Informal Discussion on Asymptotic Refinements

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 Hall and Horowitz, Reference Hall and Horowitz1996; Horowitz, Reference Horowitz2001, Reference Horowitz2019; Lahiri, Reference Lahiri2003). It has the purpose of introducing the main elements and challenges that arise to obtain asymptotic refinements in the context of dependent data generated from an AR(1) model. It also describes the approach considered in this article (see Remark 5.6 for alternative methods).

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,

(19) $$ \begin{align} J_n(x,h,P,\rho) = \Phi(x) + \sum_{j=1}^2 n^{-j/2} q_j(x,h,P,\rho)\phi(x) + o\left(n^{-1}\right)\!, \end{align} $$

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,

(20) $$ \begin{align} J_n(x,h,\hat{P}_n,\hat{\rho}_n) = \Phi(x) + \sum_{j=1}^2 n^{-j/2} q_j(x,h,\hat{P}_n,\hat{\rho}_n)\phi(x) + o_p\left(n^{-1}\right)\!, \end{align} $$

where $J_n(x,h,\cdot ,\cdot )$ is as in (7), $\hat {P}_n$ is the empirical distribution of the centered residuals defined in (13), and $\hat {\rho }_n$ is the estimator of $\rho $ defined in (12).

The approximations in (19) and (20) are commonly used to show that the bootstrap methods provide more accurate approximations than the asymptotic distribution theory (see Hall (Reference Hall1992) 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 $C_n^{*}(h,1-\alpha )$ .

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 (10) and (6), respectively. Note that (19) and the properties of $q_j(\cdot ,h,P,\rho )$ imply that for any $x>0$ , we have

(21) $$ \begin{align} P_{\rho} \left( |R_n(h)| \le x \right) = 2\Phi(x) - 1 + n^{-1} 2q_2(x,h,P,\rho) \phi(x) + o\left(n^{-1}\right)\!. \end{align} $$

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

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 (11) and (6). Now, we will argue that

(22) $$ \begin{align} P_{\rho} \left( |R_n(h)| \le c_n^*(h,1-\alpha) \right) = P_{\rho} \left( |R_n(h)| \le c_n(h,1-\alpha)\right) + o\left(n^{-1}\right)\!, \end{align} $$

where $c_n(h,1-\alpha )$ is as in (8). 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 (20), we obtain

(23) $$ \begin{align} P_{\rho} \left( |R_{b,n}^*(h)| \le x \mid Y^{(n)}\right) &= 2\Phi(x) - 1 + n^{-1} 2q_2(x,h,\hat{P}_n,\hat{\rho}_n) \phi(x) + o_p\left(n^{-1}\right)~\notag\\ &= 2\Phi(x) - 1 + n^{-1} 2q_2(x,h,P,\rho) \phi(x) + o_p\left(n^{-1}\right)\!, \end{align} $$

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 (20) and (21) look similar. Taking $x = c_n(h,1-\alpha )$ in (21) and $x = c_n^*(h,1-\alpha )$ in (23), it can be concluded that $c_n^*(h,1-\alpha ) = c_n(h,1-\alpha ) + o_p\left (n^{-1}\right )$ , which will imply (22).

The informal explanation presented above suggests that the LP-residual bootstrap can provide asymptotic refinements when there exist valid Edgeworth expansions as in (19) and (20). We present in Section 5.2 conditions (Assumption 5.1 and $\rho \in [-1+a, 1-a] $ , where $a \in (0,1)$ ) under which the previous informal discussion can be formalized.

The Challenges: Edgeworth expansions as in (19) and (20) 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 (12) is no longer valid (see Phillips, Reference Phillips2023). In this case, alternative asymptotic approximations were developed to prove asymptotic refinements of the bootstrap, e.g., Park (Reference Park2003, Reference Park2006) and Mikusheva (Reference Mikusheva2015). To our knowledge, there are no available theoretical results about valid Edgeworth expansions for the root $R_n(h)$ defined in (6) that can be applied directly.

Nevertheless, for stationary AR(1) models (when $\rho \in [-1+a,1-a]$ , $a \in (0,1)$ ), asymptotically valid Edgeworth expansions were obtained (see Phillips, Reference Phillips1977a, Reference Phillips1977b; Bose, Reference 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.

5.2 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 5.1 and 5.2, respectively.

The following assumption imposes stronger conditions on the distribution of the shocks P than the ones presented in Assumption 4.1. We use this assumption and $\rho \in [-1+a, 1-a]$ for some $a \in (0,1)$ to formalize the informal explanation about asymptotic refinements presented in Section 5.1.

Part (i) of Assumption 5.1 imposes stronger conditions over the serial dependence of the shocks. This assumption is common for the theoretical analysis of the asymptotic refinement of the bootstrap method in autoregressive models. An incomplete list of previous research that uses this assumption includes Bose (Reference Bose1988), Park (Reference Park2003, Reference Park2006), and Mikusheva (Reference Mikusheva2015). Parts (ii) and (iii) of Assumption 5.1 are sufficient technical conditions on the distribution of the shocks P to establish the existence of the Edgeworth expansions presented in (19) and (20). Part (ii) implies that the distribution $J_n(\cdot ,h,P,\rho )$ defined in (7) is continuous and guarantees that a data-dependent version of the Cramér condition holds, which is a common condition to guarantee the existence of Edgeworth expansions (see Remark 5.4 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, Hall and Horowitz (Reference Hall and Horowitz1996) and Inoue and Shintani (Reference Inoue and Shintani2006) assume the existence of 33rd and 36th moments, respectively, while Andrews (Reference Andrews2002) assumes that all the moments exist.

We rely on Assumption 5.1, the approach and results presented in Bhattacharya and Ghosh (Reference Bhattacharya and Ghosh1978) and Bhattacharya (Reference Bhattacharya1987), and the general framework developed by Götze and Hipp (Reference Götze and Hipp1983) to prove the existence of Edgeworth expansions with dependent data. The framework of Götze and Hipp (Reference Götze and Hipp1983) requires weakly dependent data and verifying stronger regularity conditions than the ones needed in the case of i.i.d. data (see Hall (Reference Hall1992) and Lahiri (Reference Lahiri2003) 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 Bose (Reference Bose1988), Hall and Horowitz (Reference Hall and Horowitz1996), Lahiri (Reference Lahiri1996), Andrews (Reference Andrews2002, Reference Andrews2004), and Inoue and Shintani (Reference Inoue and Shintani2006). It is an open question whether there exist Edgeworth expansions as in (19) and (20) for the case of local-to-unity or unit-root models. See Remark 5.6 for further discussion on alternative methods and available results.

Theorem 5.1. Suppose Assumption 5.1 holds. Fix a given $h \in \textbf {N}$ and $a \in (0,1)$ . Then, for any $\rho \in [-1+a,1-a]$ , we have

(24) $$ \begin{align} | P_{\rho} \left( \beta(\rho,h) \in C_n(h,1-\alpha) \right) - (1-\alpha) | = O(n^{-1}), \end{align} $$

where $\beta (\rho ,h)$ is as in (2) and $C_n(h,1-\alpha )$ is as in (10).

The ECP of $C_n(h,1-\alpha )$ has a similar size as the one derived in our informal explanation in Section 5.1. Similar sizes of the ECP were obtained for symmetrical confidence intervals in the i.i.d. data case (see Hall, Reference Hall1992; Horowitz, Reference Horowitz2001, Reference Horowitz2019).

The proof of Theorem 5.1 is presented in Appendix A.3. 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 Hall and Horowitz (Reference Hall and Horowitz1996) and Andrews (Reference Andrews2002, Reference Andrews2004). 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 Bhattacharya and Ghosh (Reference Bhattacharya and Ghosh1978) and Götze and Hipp (Reference Götze and Hipp1983, Reference Götze and Hipp1994) (see Theorem B.2 in Appendix B.3). These two ideas guarantee the existence of the Edgeworth expansion presented in (19). We then conclude the proof by standard derivations similar to the one derived in our informal explanation presented in Section 5.1.

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 (11) for $\beta (\rho ,h)$ is $o(n^{-1})$ .

Theorem 5.2. Suppose Assumption 5.1 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

(25) $$ \begin{align} | P_{\rho} \left( \beta(\rho,h) \in C_n^*(h,1-\alpha) \right) - (1-\alpha) | = o \left( n^{-(1+\epsilon)} \right)\!, \end{align} $$

where $\beta (\rho ,h)$ is as in (2) and $C_n^*(h,1-\alpha )$ is as in (11).

Theorem 5.2 presents the size of the ECP of the confidence interval $C_n^*(h,1-\alpha )$ in (25). This is similar to the one derived in our informal explanation in Section 5.1, 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 Hall, Reference Hall1992; Horowitz, Reference Horowitz2001, Reference Horowitz2019).

The proof of Theorem 5.2 is presented in Appendix A.4. 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 (8) and (15), respectively. We next sketch the proof based on those two claims. We can derive

$$ \begin{align*} P_{\rho} \left( \beta(\rho,h) \in C_n^*(h,1-\alpha) \right) &= P_{\rho} \left( |R_n(h)| \le c_n^*(h,1-\alpha) \right) \\&= P_{\rho} \left( |R_n(h)| \le c_n(h,1-\alpha) + \Delta_n, \left| \Delta_n \right|\le C_1 n^{-(1+\epsilon)} \right)\\& \quad + O\left( n^{-(1+\epsilon)} \right)\\&= 1-\alpha + O\left( n^{-(1+\epsilon)} \right)\!, \end{align*} $$

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

$$ \begin{align*}P_{\rho} \left( |R_n(h)| \le c_n(h,1-\alpha) + O\left( n^{-(1+\epsilon)} \right) \right) = 1-\alpha + O\left( n^{-(1+\epsilon)} \right)\!.\end{align*} $$

Note that the first claim follows from Theorem 5.1. 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 (20) 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 B.5 in Appendix B.1). 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 5.2.

6.1 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:

$$ \begin{align*} u_t = \tau_t v_t, \quad \tau_t^2 = \omega_0 + \omega_1 u_{t-1}^2 + \omega_2 \tau_{t-1}^2, \quad v_t \text{ are } i.i.d., \end{align*} $$

where the distribution of $v_t$ and the parameter vector $(\omega _0,\omega _1,\omega _2)$ are specified as follows:

1. Design 1: $v_t \sim N(0,1)$ , $\omega _0=1$ , and $\omega _1=\omega _2=0$ .

2. Design 2: $v_t \sim N(0,1)$ , $\omega _0=0.05$ , $\omega _1= 0.3$ , and $\omega _2=0.65$ .

3. Design 3: $v_t \sim t_4/\sqrt {2}$ , $\omega _0=1$ , and $\omega _1=\omega _2=0$ .

4. Design 4: $v_t|B_t=j \sim N(m_j,\sigma _j^2)$ , 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 (4). Additionally, we consider alternative HC standard errors $\hat {s}_{j,n}(h)$ defined as

$$ \begin{align*} \hat{s}_{j,n}(h) \equiv \left( \sum_{t=1}^{n-h} \hat{u}_t(h)^2 \right)^{-1/2} \left(\sum_{t=1}^{n-h} \hat{\xi}_{j,t}(h)^2 \hat{u}_t(h)^2 \right)^{1/2} \left( \sum_{t=1}^{n-h} \hat{u}_t(h)^2 \right)^{-1/2}, \end{align*} $$

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.

1. 1. RB: confidence interval as in (11) based on the LP-residual bootstrap.

2. 2. RB ${}_{\boldsymbol {per-t}}$ : equal-tailed percentile-t confidence interval as in (18). It is based on the LP-residual bootstrap and discussed in Remark 4.1.

3. 3. RB ${}_{\boldsymbol {hc3}}$ : confidence interval as in (11) 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 3.1 but using $\hat {s}_{3,n}^*(h)$ instead of $\hat {s}_{n}^*(h)$ .

4. 4. WB: confidence interval as in (11) 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 3.2).

5. 5. WB ${}_{\boldsymbol {per-t}}$ : equal-tailed percentile-t confidence interval as in (18) 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 3.2.

6. 6. GB ${}_{\boldsymbol {LR}}$ : confidence interval based on the grid bootstrap presented in Section 3.3 in Mikusheva (Reference Mikusheva2012). It uses the LR statistic.

7. 7. AA: standard confidence interval as in (10).

8. 8. AA ${}_{\boldsymbol {hc2}}$ : standard confidence interval as in (10) but using $\hat {s}_{2,n}(h)$ instead of $\hat {s}_{n}(h)$ .

9. 9. AA ${}_{\boldsymbol {hc3}}$ : standard confidence interval as in (10) but using $\hat {s}_{3,n}(h)$ instead of $\hat {s}_{n}(h)$ .

6.2 Discussion and Results

In all the designs, the shocks have zero mean and variance one. Designs 1 and 2 verify Assumption 4.1 presented in Section 4. Design 1 also verifies Assumption 4.2 due to Proposition B.1 in Appendix B.2. Assumption 4.2 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 and 4 do not verify all the parts of Assumption 4.1. Design 3 considers shocks without a fourth moment, i.e., it does not verify part (iv) of Assumption 4.1, 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 article, i.e., it does not verify part (ii) of Assumption 4.1. As we discussed in Remark 2.1, part (ii) of Assumption 4.1 was a sufficient condition for the validity of the HC standard errors $\hat {s}_n(h)$ in the construction of confidence intervals.

Tables 1 and 2 report the coverage probabilities (in %) of our simulations. Columns are labeled as the confidence intervals we specified in Section 6.1. For all the designs on the distribution of the shock and values of $\rho $ , we use 5,000 simulations to generate data with a sample size $n = 95$ based on the AR(1) model (1). 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 $B=1,000$ as described in Remark 3.1. We summarize our findings from the simulations below.

Table 1 Coverage probability (in %) of confidence intervals for $\beta (\rho ,h)$ with a nominal level of 90% and $n = 95$

Note: In total, 5,000 simulations and 1,000 bootstrap iterations.

Table 2 Coverage probability (in %) of confidence intervals for $\beta (\rho ,h)$ with a nominal level of 90% and $n = 95$

Note: In total, 5,000 simulations and 1,000 bootstrap iterations.

Five features of Table 1 deserve discussion. First, it shows that our recommended confidence interval RB has a coverage probability closer to 90% than the confidence intervals AA, AA ${}_{\boldsymbol {hc2}}$ , and AA ${}_{\boldsymbol {hc3}}$ for all designs 1 and 2, values of $\rho $ , and horizons h, with some few exceptions. The lowest coverage probabilities of RB, AA, AA ${}_{\boldsymbol {hc2}}$ , and AA ${}_{\boldsymbol {hc3}}$ are $85\%$ , $77\%$ , $78\%$ , and $79\%$ , respectively, and occur when $\rho =1$ and horizon $h=18$ . Second, RB and RB ${}_{\boldsymbol {hc3}}$ have better performance than RB ${}_{\boldsymbol {per-t}}$ , especially when $\rho = 1$ and the horizon is a significant fraction of the sample size ( $h \in \{12,18\}$ ). Third, WB and WB ${}_{\boldsymbol {per-t}}$ have larger coverage probability than RB for all designs 1 and 2, values of $\rho $ , and horizons h, with some few exceptions. The larger coverage of WB and WB ${}_{per-t}$ is associated with a larger median length of their confidence intervals, as we reported in Table E.1 in the Supplementary Material. Fourth, AA ${}_{hc3}$ presents a coverage probability closer to $90\%$ and larger than AA and AA ${}_{hc2}$ for all designs 1 and 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., RB and WB ${}_{per-t}$ ) report coverage probability closer to $90\%$ . Fifth, 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 4.2, it is unknown if the grid bootstrap is valid for design 2. The coverage probability of GB ${}_{LR}$ is constant across horizons because the LR statistic is invariant to monotonic transformations (see Section 4.3 and footnote 6 in Mikusheva (Reference Mikusheva2012) for more details).

Table 2 presents results for designs 3 and 4. Our findings for design 3 are qualitatively similar to Table 1, which was discussed above. This suggests that failing part (iv) of Assumption 4.1 (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 4.1. 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 2.1). In particular, RB ${}_{per-t}$ has a coverage probability closer to $90\%$ and larger than RB and 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 E.3 in the Supplementary Material.

Finally, Table E.2 in the Supplementary Material reports the statistical power of the confidence intervals specified in Section 6.1. 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 1–3, which suggests they have statistical power at $h=1$ . We also notice that RB ${}_{per-t}$ , WB ${}_{per-t}$ , and GB ${}_{LR}$ have a coverage probability strictly lower than $90\%$ for horizon $h=6$ and designs 1–3. Moreover, they have a lower coverage probability than all the other confidence intervals. Finally, all the confidence intervals have coverage probability above $90\%$ on design 4, with the exception of GB ${}_{LR}$ for horizon $h=1$ .