Series-Based HAR Wild Bootstrap Test

• The paper introduces a series-based HAR wild bootstrap test that improves size control over standard two-sample tests under serial dependence.
• It employs orthonormal basis projections to estimate long-run variances and generates dependent multipliers using sine-cosine expansions.
• Empirical results demonstrate that the SHAR-WB method delivers robust inference in both equal and unequal variance scenarios, especially in small samples.

The series-based HAR wild bootstrap test is a two-sample mean inference procedure for serially dependent time series in which HAR means heteroskedasticity and autocorrelation robust, not heterogeneous autoregression. It is designed for settings with two time series,

$$Y_{jt}=\mu_j+u_{jt}, \qquad t=1,\dots,T_j,\quad j=1,2,$$

where the disturbances are independent across the two series but each series may be serially dependent, and the inferential target is

$H_0:\mu_1=\mu_2.$

The method combines a series-based long-run variance estimator, built from orthonormal basis projections, with a dependent wild bootstrap whose multipliers are themselves generated from orthogonal sine-cosine expansions. In the formulation of Hounyo and Kim, the framework covers structural break comparisons with a known break date, treatment-control comparisons, and group-averaged panel or repeated cross-section settings (Hounyo et al., 12 Dec 2025).

1. Inferential setting and basic structure

The starting point is the long-run variance formulation

$$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$

where

$\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$

is the long-run variance of the $j$-th series. The essential complication is that valid two-sample inference depends on $\Omega_1$ and $\Omega_2$, not merely on one-period variances, and the framework explicitly allows

$\Omega_1\neq \Omega_2$

(Hounyo et al., 12 Dec 2025).

The series component enters through a basis expansion. Assumption 1 uses a sequence of piecewise monotonic, continuously differentiable, orthonormal basis functions $\{\phi_\ell(\cdot)\}_{\ell=1}^K$ on $L_2[0,1]$ satisfying

$H_0:\mu_1=\mu_2.$0

The paper uses trigonometric basis functions,

$H_0:\mu_1=\mu_2.$1

Assumption 2 imposes a functional central limit theorem: $H_0:\mu_1=\mu_2.$2 with $H_0:\mu_1=\mu_2.$3 and $H_0:\mu_1=\mu_2.$4 a two-dimensional standard Brownian motion. For increasing-$H_0:\mu_1=\mu_2.$5 asymptotics, Assumption 3 further requires fourth-order stationarity and summability conditions such as

$H_0:\mu_1=\mu_2.$6

together with summability of fourth-order cumulants (Hounyo et al., 12 Dec 2025).

A central conceptual point is that the method is a time-series analogue of Welch-type two-sample inference, with long-run variance heterogeneity replacing unequal i.i.d. variances.

2. Series-HAR long-run variance estimation and test statistics

Let

$H_0:\mu_1=\mu_2.$7

The series-HAR estimator projects these centered observations onto the basis: $H_0:\mu_1=\mu_2.$8 The long-run variance estimator is then

$H_0:\mu_1=\mu_2.$9

Under Assumptions 1 and 2,

$$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$0

and orthonormality yields

$$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$1

Thus the projected coefficient squares behave as asymptotically independent $$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$2-type replicates, and averaging across $$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$3 produces the series-HAR estimator (Hounyo et al., 12 Dec 2025).

The paper distinguishes two asymptotic regimes. With fixed $$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$4, the estimator is not consistent but supports $$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$5-type approximations. With increasing $$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$6 such that $$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$7 and $$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$8, $$\sqrt{T_j}\left(\bar Y_j-\mu_j\right)\to^d N(0,\Omega_j), \qquad j=1,2,$$9 is consistent for $\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$0.

For the equal-long-run-variance case,

$\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$1

the pooled statistic is

$\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$2

Under $\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$3, Theorem 1 gives

$\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$4

for fixed $\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$5 (Hounyo et al., 12 Dec 2025).

For unequal long-run variances, the paper uses the Welch-style statistic

$\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$6

Under increasing $\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$7,

$\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$8

To improve finite-sample performance, the paper also develops a Welch-type degrees-of-freedom correction. Writing

$\Omega_j=\lim_{T_j\to\infty}\Var\!\left(\sqrt{T_j}(\bar Y_j-\mu_j)\right)$9

the feasible adjusted degrees of freedom are

$j$0

with $j$1 any consistent estimator of $j$2, and the paper often sets

$j$3

(Hounyo et al., 12 Dec 2025).

3. Dependent wild bootstrap construction

The distinctive feature of the procedure is the series-based HAR wild bootstrap. The bootstrap sample is generated under the null: $j$4 with bootstrap errors

$j$5

Hence the bootstrap imposes $j$6 through the pooled mean $j$7, while retaining heteroskedasticity and serial dependence through $j$8 (Hounyo et al., 12 Dec 2025).

The multipliers $j$9 are not i.i.d. They satisfy

$\Omega_1$0

and

$\Omega_1$1

where the orthogonal basis functions satisfy

$\Omega_1$2

The paper uses

$\Omega_1$3

for $\Omega_1$4, and generates

$\Omega_1$5

where $\Omega_1$6 and $\Omega_1$7 are i.i.d. with mean $\Omega_1$8 and variance $\Omega_1$9. This yields

$\Omega_2$0

The use of both cosine and sine systems with $\Omega_2$1 i.i.d. variables is motivated by exact unit variance in finite samples (Hounyo et al., 12 Dec 2025).

The induced bootstrap long-run variance is

$\Omega_2$2

or equivalently

$\Omega_2$3

This is the key matching device: the bootstrap covariance is itself a series-averaged projection object (Hounyo et al., 12 Dec 2025).

The bootstrap statistic mirrors the original statistic: $\Omega_2$4 with

$\Omega_2$5

and $\Omega_2$6. The paper emphasizes matched studentization: the same basis $\Omega_2$7 and the same $\Omega_2$8 must be used in the original and bootstrap standardizations (Hounyo et al., 12 Dec 2025).

4. Asymptotic theory and bootstrap validity

The asymptotic theory proceeds in four linked statements. First, under equal long-run variances and fixed $\Omega_2$9, the pooled statistic has a Student limit: $\Omega_1\neq \Omega_2$0 Second, under Assumptions 1–3 and increasing $\Omega_1\neq \Omega_2$1,

$\Omega_1\neq \Omega_2$2

Third, under Assumptions 2–4, finite fourth moments for the external multipliers, and

$\Omega_1\neq \Omega_2$3

the bootstrap consistently reproduces the mean distribution: $\Omega_1\neq \Omega_2$4 A key intermediate lemma is

$\Omega_1\neq \Omega_2$5

Fourth, under

$\Omega_1\neq \Omega_2$6

the bootstrap is valid for the studentized two-sample statistic: $\Omega_1\neq \Omega_2$7 This is the paper’s main bootstrap validity theorem (Hounyo et al., 12 Dec 2025).

The procedure is explicitly positioned as an extension of traditional wild bootstrap methods to time-series settings. Instead of i.i.d. multipliers, serial dependence is induced analytically through the covariance structure of $\Omega_1\neq \Omega_2$8. The paper further notes that the method avoids resampling blocks of observations and is asymptotically related to Shao’s dependent wild bootstrap, with covariance approximately matched to a Daniell/sinc kernel through

$\Omega_1\neq \Omega_2$9

(Hounyo et al., 12 Dec 2025). This situates the method alongside dependent wild bootstrap approaches developed for max-correlation white-noise testing (Hill et al., 2016), while its projection-based long-run variance logic belongs to a different inferential architecture.

5. Tuning, finite-sample behavior, and empirical use

The main tuning parameter is the truncation dimension $\{\phi_\ell(\cdot)\}_{\ell=1}^K$0. The paper adopts Sun’s data-driven rule

$\{\phi_\ell(\cdot)\}_{\ell=1}^K$1

with $\{\phi_\ell(\cdot)\}_{\ell=1}^K$2 estimated from an AR(1)-type approximation. In the bootstrap, the recommended choice is

$\{\phi_\ell(\cdot)\}_{\ell=1}^K$3

The only formal requirement on the multiplier innovations is

$\{\phi_\ell(\cdot)\}_{\ell=1}^K$4

In simulations and empirical work, the paper uses

$\{\phi_\ell(\cdot)\}_{\ell=1}^K$5

bootstrap replications (Hounyo et al., 12 Dec 2025).

The finite-sample evidence shows that the bootstrap is particularly valuable under strong serial dependence, unequal long-run variances, non-Gaussian disturbances, and modest sample sizes. In the AR(1) designs reported in the paper, the SHAR-WB test has the best size control overall. For example, with equal long-run variances, normal errors, $\{\phi_\ell(\cdot)\}_{\ell=1}^K$6, and $\{\phi_\ell(\cdot)\}_{\ell=1}^K$7, the rejection rates are $\{\phi_\ell(\cdot)\}_{\ell=1}^K$8 for the classical $\{\phi_\ell(\cdot)\}_{\ell=1}^K$9, $L_2[0,1]$0 for Welch $L_2[0,1]$1, and $L_2[0,1]$2 for SHAR-WB. With unequal long-run variances in the same setting, the rates are $L_2[0,1]$3, $L_2[0,1]$4, and $L_2[0,1]$5, respectively. With $L_2[0,1]$6 errors, $L_2[0,1]$7, and $L_2[0,1]$8, the classical $L_2[0,1]$9 rejects $H_0:\mu_1=\mu_2.$00 of the time, versus $H_0:\mu_1=\mu_2.$01 for SHAR-WB (Hounyo et al., 12 Dec 2025).

The paper also reports a power trade-off. SHAR-WB typically sacrifices some power relative to less robust procedures, especially under strong dependence, but this accompanies its markedly better size control. In the reported designs, the authors recommend series-HAR $H_0:\mu_1=\mu_2.$02-approximations or SHAR-WB under moderate dependence and moderate sample size, and state a preference for SHAR-WB under strong dependence or small samples (Hounyo et al., 12 Dec 2025).

In empirical applications, the difference is substantive. In the working-from-home experiment, classical and Welch tests produce $H_0:\mu_1=\mu_2.$03 for log calls per second, while $H_0:\mu_1=\mu_2.$04 gives $H_0:\mu_1=\mu_2.$05 and SHAR-WB gives $H_0:\mu_1=\mu_2.$06. In the U.S. macro break application, unemployment yields classical $H_0:\mu_1=\mu_2.$07, Welch $H_0:\mu_1=\mu_2.$08, and SHAR-WB $H_0:\mu_1=\mu_2.$09; inflation yields classical $H_0:\mu_1=\mu_2.$10, Welch $H_0:\mu_1=\mu_2.$11, and SHAR-WB $H_0:\mu_1=\mu_2.$12. These examples show that serial-dependence-robust inference can overturn conclusions obtained from standard two-sample procedures (Hounyo et al., 12 Dec 2025).

6. Scope, interpretation, and relation to adjacent bootstrap methods

A recurring ambiguity concerns the acronym HAR. In (Hounyo et al., 12 Dec 2025), HAR denotes heteroskedasticity and autocorrelation robust. The method is therefore a two-sample mean test with series-based long-run variance estimation and a dependent wild bootstrap; it is not a bootstrap for the heterogeneous autoregressive model of realized volatility. This distinction is essential for correct placement of the method.

Its closest methodological relatives are time-series multiplier and dependent wild bootstrap procedures rather than block-resampling tests. The blockwise multiplier bootstrap for max statistics in weakly dependent high-dimensional time series developed in “Bootstrapping High Dimensional Time Series” (Zhang et al., 2014) provides a general foundation for dependence-aware max-type inference without direct long-run covariance estimation. The dependent wild bootstrap for max-correlation white-noise testing in “A Max-Correlation White Noise Test for Weakly Dependent Time Series” (Hill et al., 2016) similarly shows how serial dependence can be handled through multiplier structures rather than i.i.d. resampling. In nonstationary regression settings, “Nonparametric specification for non-stationary time series regression” (Zhou, 2014) demonstrates that alternative wild bootstrap methods remain consistent when classical residual bootstrap is sensitive to heteroskedasticity, non-stationarity, or temporal dependence. The series-based HAR wild bootstrap test belongs to this broader class of dependence-aware bootstrap procedures, but its distinctive feature is the alignment between basis-projection long-run variance estimation and basis-generated dependent multipliers (Hounyo et al., 12 Dec 2025).

Two limitations delimit its scope. First, the framework assumes independence across the two series; cross-series dependence is not the target setting. Second, performance depends on the truncation choices $H_0:\mu_1=\mu_2.$13 and $H_0:\mu_1=\mu_2.$14, even though the paper supplies a data-driven rule. Within those limits, the method offers a nonparametric, studentized, serial-dependence-robust alternative to conventional two-sample tests, and does so without block resampling of the data themselves (Hounyo et al., 12 Dec 2025).