--- title: Series-Based HAR Wild Bootstrap Test url: https://www.emergentmind.com/topics/series-based-har-wild-bootstrap-test type: topic --- # Series-Based HAR Wild Bootstrap Test 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 [2512.11259]. ## 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 \[ \Omega_1\neq \Omega_2 \] [2512.11259]. 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 \[ \int_0^1 \phi_\ell(x)\,dx=0. \] The paper uses trigonometric basis functions, \[ \phi_{\ell}(x)= \begin{cases} \sqrt{2}\cos(2\pi \ell x), & \text{if } \ell \text{ is odd},\\[4pt] \sqrt{2}\sin(2\pi \ell x), & \text{if } \ell \text{ is even}, \end{cases} \qquad \ell=1,\dots,K. \] Assumption 2 imposes a functional central limit theorem: \[ \frac{1}{\sqrt{T}}\sum_{t=1}^{\lfloor rT\rfloor}u_t \to^d \begin{pmatrix} \sqrt{\Omega_1}&0\\ 0&\sqrt{\Omega_2} \end{pmatrix} W_2(r), \] with \(u_t=(u_{1t},u_{2t})'\) and \(W_2(r)\) a two-dimensional standard Brownian motion. For increasing-\(K\) asymptotics, Assumption 3 further requires fourth-order stationarity and summability conditions such as \[ \sum_{h=-\infty}^{\infty}|h|^3|\gamma_j(h)|<\infty, \qquad \gamma_j(h)=E(u_{jt}u_{j,t+h}), \] together with summability of fourth-order cumulants [2512.11259]. 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 \[ \hat u_{jt}=Y_{jt}-\bar Y_j. \] The series-HAR estimator projects these centered observations onto the basis: \[ \hat z_{j,\ell} := \frac{1}{\sqrt{T_j}}\sum_{t=1}^{T_j}\phi_\ell\!\left(\frac{t}{T_j}\right)\hat u_{jt}, \qquad \hat\Omega_{j,\ell}:=\hat z_{j,\ell}\hat z_{j,\ell}'. \] The long-run variance estimator is then \[ \hat\Omega_j=\frac{1}{K_j}\sum_{\ell=1}^{K_j}\hat\Omega_{j,\ell}. \] Under Assumptions 1 and 2, \[ \hat\Omega_{j,\ell}\to^d \Omega_j\chi^2(1), \] and orthonormality yields \[ \Cov\!\left(\int_0^1 \phi_k(r)\,dW_{2j}(r),\int_0^1 \phi_\ell(r)\,dW_{2j}(r)\right)=0, \qquad k\neq \ell. \] Thus the projected coefficient squares behave as asymptotically independent \(\Omega_j\chi^2(1)\)-type replicates, and averaging across \(\ell\) produces the series-HAR estimator [2512.11259]. The paper distinguishes two asymptotic regimes. With **fixed \(K_j\)**, the estimator is not consistent but supports \(t\)-type approximations. With **increasing \(K_j\)** such that \(K_j\to\infty\) and \(K_j/T_j\to0\), \(\hat\Omega_j\) is consistent for \(\Omega_j\). For the equal-long-run-variance case, \[ \Omega_1=\Omega_2=:\Omega, \] the pooled statistic is \[ t_{0,\text{HAR}}= \frac{\bar Y_1-\bar Y_2} {s_{p,\text{HAR}}\sqrt{\frac{1}{T_1}+\frac{1}{T_2}}}, \qquad s_{p,\text{HAR}}= \sqrt{\frac{K_1\hat\Omega_1+K_2\hat\Omega_2}{K_1+K_2}}. \] Under \(H_0\), Theorem 1 gives \[ t_{0,\text{HAR}}\to^d t(K_1+K_2) \] for fixed \(K_1,K_2\) [2512.11259]. For unequal long-run variances, the paper uses the Welch-style statistic \[ t_{1,\text{HAR}}= \frac{\bar Y_1-\bar Y_2} {\sqrt{\hat\Omega_1/T_1+\hat\Omega_2/T_2}}. \] Under increasing \(K_j\), \[ t_{1,\text{HAR}}\to^d N(0,1). \] To improve finite-sample performance, the paper also develops a Welch-type degrees-of-freedom correction. Writing \[ \rho_T=T_2/T_1, \] the feasible adjusted degrees of freedom are \[ \hat K_{\text{adf}}= \frac{\left(\rho_T^{1/2}\tilde\Omega_1+\rho_T^{-1/2}\tilde\Omega_2\right)^2} {\rho_T\tilde\Omega_1^2/K_1+\rho_T^{-1}\tilde\Omega_2^2/K_2}, \] with \(\tilde\Omega_j\) any consistent estimator of \(\Omega_j\), and the paper often sets \[ \tilde\Omega_j=\hat\Omega_j \] [2512.11259]. ## 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: \[ Y_{jt}^*=\mu^*+u_{jt}^*, \qquad \mu^*=\frac{T_1\bar Y_1+T_2\bar Y_2}{T_1+T_2}, \] with bootstrap errors \[ u_{jt}^*=\hat u_{jt}\eta_{jt}. \] Hence the bootstrap imposes \(H_0:\mu_1=\mu_2\) through the pooled mean \(\mu^*\), while retaining heteroskedasticity and serial dependence through \(\hat u_{jt}\eta_{jt}\) [2512.11259]. The multipliers \(\eta_{jt}\) are not i.i.d. They satisfy \[ E^*(\eta_{jt})=0,\qquad \Var^*(\eta_{jt})=1, \] and \[ \Cov^*(\eta_{jt},\eta_{js}) = \frac{1}{K_j^*}\sum_{\ell=1}^{2K_j^*} \psi_\ell\!\left(\frac{t}{T_j}\right)\psi_\ell\!\left(\frac{s}{T_j}\right), \] where the orthogonal basis functions satisfy \[ \int_0^1 \psi_\ell^2(x)\,dx=\frac12, \qquad \int_0^1 \psi_\ell(x)\,dx=0. \] The paper uses \[ \psi_{2\ell-1}(x)=\cos(2\pi \ell x)=:\psi_{1,\ell}(x), \qquad \psi_{2\ell}(x)=\sin(2\pi \ell x)=:\psi_{2,\ell}(x), \] for \(\ell=1,\dots,K_j^*\), and generates \[ \eta_{jt} = \frac{1}{\sqrt{K_j^*}} \sum_{\ell=1}^{K_j^*} \left[ \psi_{1,\ell}\!\left(\frac{t}{T_j}\right)v_{j,1\ell} + \psi_{2,\ell}\!\left(\frac{t}{T_j}\right)v_{j,2\ell} \right], \] where \(v_{j,1\ell}\) and \(v_{j,2\ell}\) are i.i.d. with mean \(0\) and variance \(1\). This yields \[ \Cov^*(\eta_{jt},\eta_{js}) = \frac{1}{K_j^*}\sum_{\ell=1}^{K_j^*} \cos\!\left(2\pi \ell \frac{t-s}{T_j}\right), \qquad \Var^*(\eta_{jt})=1. \] The use of both cosine and sine systems with \(2K_j^*\) i.i.d. variables is motivated by exact unit variance in finite samples [2512.11259]. The induced bootstrap long-run variance is \[ \hat\Omega_{\text{boot},T_j} = \Var^*\left(\frac{1}{\sqrt{T_j}}\sum_{t=1}^{T_j}u_{jt}^*\right) = \frac{1}{T_j}\sum_{t=1}^{T_j}\sum_{s=1}^{T_j} \hat u_{jt}\hat u_{js}\Cov^*(\eta_{jt},\eta_{js}), \] or equivalently \[ \hat\Omega_{\text{boot},T_j} = \frac{1}{T_j}\sum_{t=1}^{T_j}\sum_{s=1}^{T_j} \frac{1}{K_j^*}\sum_{\ell=1}^{K_j^*} \Bigg[ \psi_{1,\ell}\!\left(\frac{t}{T_j}\right)\psi_{1,\ell}\!\left(\frac{s}{T_j}\right) + \psi_{2,\ell}\!\left(\frac{t}{T_j}\right)\psi_{2,\ell}\!\left(\frac{s}{T_j}\right) \Bigg]\hat u_{jt}\hat u_{js}. \] This is the key matching device: the bootstrap covariance is itself a series-averaged projection object [2512.11259]. The bootstrap statistic mirrors the original statistic: \[ t_{1,\text{HAR}}^*= \frac{\bar Y_1^*-\bar Y_2^*} {\sqrt{\hat\Omega_1^*/T_1+\hat\Omega_2^*/T_2}}, \] with \[ \hat\Omega_j^* = \frac{1}{K_j}\sum_{\ell=1}^{K_j}\hat\Omega_{j,\ell}^*, \qquad \hat z_{j,\ell}^* = \frac{1}{\sqrt{T_j}}\sum_{t=1}^{T_j}\phi_\ell\!\left(\frac{t}{T_j}\right)\hat u_{jt}^*, \qquad \hat\Omega_{j,\ell}^* = \hat z_{j,\ell}^*\hat z_{j,\ell}^{*\prime}, \] and \(\hat u_{jt}^*=Y_{jt}^*-\bar Y_j^*\). The paper emphasizes **matched studentization**: the same basis \(\phi_\ell\) and the same \(K_j\) must be used in the original and bootstrap standardizations [2512.11259]. ## 4. Asymptotic theory and bootstrap validity The asymptotic theory proceeds in four linked statements. First, under equal long-run variances and fixed \(K\), the pooled statistic has a Student limit: \[ t_{0,\text{HAR}}\to^d t(K_1+K_2). \] Second, under Assumptions 1–3 and increasing \(K_j\), \[ t_{1,\text{HAR}}\to^d N(0,1). \] Third, under Assumptions 2–4, finite fourth moments for the external multipliers, and \[ K_j^*\to\infty,\qquad K_j^*/T_j\to0, \] the bootstrap consistently reproduces the mean distribution: \[ \sup_{x\in\mathbb R} \left| P^*\!\left(\sqrt{T_j}(\bar Y_j^*-\mu^*)