A Bootstrap-Assisted Self-Normalization Approach to Inference in Cointegrating Regressions

abstract{ Traditional inference in cointegrating regressions requires tuning parameter choices to estimate a long-run variance parameter. Even in case these choices are “optimal”, the tests are severely size distorted. We propose a novel self-normalization approach, which leads to a nuisance parameter free limiting distribution without estimating the long-run variance parameter directly. This makes our self-normalized test tuning parameter free and considerably less prone to size distortions at the cost of only small power losses. In combination with an asymptotically justified vector autoregressive sieve bootstrap to construct critical values, the self-normalization approach shows further improvement in small to medium samples when the level of error serial correlation or regressor endogeneity is large. We illustrate the usefulness of the bootstrap-assisted self-normalized test in empirical applications by analyzing the validity of the Fisher effect in Germany and the United States.} \noindentKeywords: Sieve bootstrap, Size distortions, Tuning Parameters \noindentJEL classification: C12, C13, C32

assumptionLet $\{w_t\}_{t \in \ensuremath{{\mathbb Z}}}$ be an $\ensuremath{{\mathbb R}}^{1+m}$-valued, strictly stationary and purely nondeterministic stochastic process of full rank\footnote{The process $\{w_t\}_{t \in \ensuremath{{\mathbb Z}}}$ is of full rank, if the components of its innovation process are linearly independent. For more details we refer to \citeasnoun[p.\,379]{MeKr15}, who study the range of validity of the VAR sieve bootstrap.} with $\ensuremath{{\mathbb E}}(w_t)=0$ and $\ensuremath{{\mathbb E}}(\vert w_t \vert_F^a)<\infty$, for some $a>2$. The autocovariance matrix function $\Gamma(\cdot)$ of $\{w_t\}_{t \in \ensuremath{{\mathbb Z}}}$ fulfills $\sum_{h=-\infty}^\infty (1+\vert h \vert)^k \vert \Gamma (h)\vert_F < \infty$ for some $k\geq 3/2$. For the spectral density matrix $f(\cdot)$ of $\{w_t\}_{t \in \ensuremath{{\mathbb Z}}}$ we assume that there exists a constant $c>0$ such that $\min \sigma(f(\lambda))\geq c$ for all frequencies $\lambda \in (-\pi,\pi]$, where $\sigma(f(\lambda))$ denotes the spectrum of $f(\cdot)$ at frequency $\lambda$.

assumptionThe process $\{w_t\}_{t \in \ensuremath{{\mathbb Z}}}$ has absolutely summable cumulants up to order four. More precisely, we have for all $j=2,\ldots,4$ and $\mathbf{a}=[a_1,\ldots,a_j]'$, with $a_1,\ldots,a_j\in\{1,\ldots,m+1\}$, that \begin{align*} \sum_{h_2,\ldots,h_j=-\infty}^\infty \vert cum_{\mathbf{a}}(0,h_2,\ldots,h_j)\vert <\infty, \end{align*} where $\text{cum}_{\mathbf{a}}(0,h_2,\ldots,h_j)$ denotes the $j$-th joint cumulant of $w_{0,a_1},w_{h_2,a_2},\ldots,w_{h_j,a_j}$ and $w_{t,i}$ denotes the $i$-th element of $w_t$.

assumptionLet $\{w_t\}_{t \in \ensuremath{{\mathbb Z}}}$ fulfill \begin{align} B_T(r)\coloneqq T^{-1/2} \sum_{t=1}^{\lfloor rT \rfloor} w_t \overset{w}{\longrightarrow} B(r) = \Omega^{1/2} W(r),\quad 0\leq r \leq 1, \end{align} as $T\rightarrow \infty$, where $W(r)=[W_{u\cdot v}(r),W_v(r)']'$ is an $(1+m)$-dimensional vector of independent standard Brownian motions and \begin{align*} \Omega^{1/2} =\begin{bmatrix} \Omega_{u \cdot v}^{1/2} & \Omega_{uv}(\Omega_{vv}^{-1/2})'\\ 0 & \Omega_{vv}^{1/2} \end{bmatrix}, \end{align*} where $\Omega_{u\cdot v} \coloneqq \Omega_{uu} - \Omega_{uv} \Omega_{vv}^{-1}\Omega_{vu}$, such that $\Omega^{1/2}(\Omega^{1/2})'=\Omega$. For later usage note that $\Omega_{u\cdot v}$ is a scalar and partition $B(r)=[B_u(r),B_v(r)']'$.

remarkThe IM-OLS estimator is rate-$T$ consistent, but its conditional asymptotic covariance matrix is always larger than (or equal to) the conditional asymptotic covariance matrix of the rate-$T$ consistent D- and FM-OLS estimators \citeaffixed[Proposition 2]{VoWa14}{see}. This stems from the fact that the error in the augmented partial sum regression is integrated rather than stationary. However, this comparison may be misleading, as the asymptotic covariance matrix of the D- and FM-OLS estimators does not account for the finite sample effects of tuning parameter choices required for the D- and FM-OLS estimators but not for the IM-OLS estimator. Simulation results in \citeasnoun{VoWa14} and in Section (ref) in Online Appendix (ref) demonstrate that the IM-OLS estimator performs well relative to the two competitors. The IM-OLS estimator thus serves as a useful starting point for developing a self-normalized test statistic.

theoremLet $(y_t)_{t=1}^T$ and $(x_t)_{t=1}^T$ be generated by (ref) and (ref), respectively and let $\{w_t\}_{t\in \ensuremath{{\mathbb Z}}}$ satisfy Assumption (ref). Then it holds under the null hypothesis given in (ref) that \begin{align} \tau_{\scriptstyleIM}(\hat \eta_T)\overset{w}{\longrightarrow}\mathcal{G}_{\scriptstyleSN}\coloneqq & \frac{\left(R_2\left(\Pi'\right)^{-1}\mathcal{Z}\right)'\left(R_2 V R_2'\right)^{-1}\left(R_2\left(\Pi'\right)^{-1}\mathcal{Z}\right)}{\int_0^1 \left(W_{u\cdot v}(r) - g(r)'\mathcal{Z}\right)^2 dr} \notag\\ =& \frac{\chi_s^2}{\int_0^1 \left(W_{u\cdot v}(r) - g(r)'\mathcal{Z}\right)^2 dr}, \end{align} as $T\rightarrow \infty$.

remarkWe have already mentioned in the introduction that our self-normalization approach is similar in spirit to but different from the approach of \citeasnoun{KVB00} in the stationary time series literature. As \citeasnoun{KiVo02} show that the approach of \citeasnoun{KVB00} is exactly equivalent to using HAC standard errors based on the Bartlett kernel with bandwidth equal to sample size, it is tempting to assume that the traditional kernel estimator of $\Omega_{u \cdot v}$ defined in (ref) with the Bartlett kernel ($\mathcal{K}_\text{\scriptsize Bartlett}$) and $b_T=T$ could also serve as a self-normalizer in our context. However, as $\hat\Omega_{u\cdot v}\left(\mathcal{K}_\text{\scriptsize Bartlett},T\right)$ depends on the OLS residuals in (ref), several nuisance parameters other than $\Omega_{u\cdot v}$ enter the limiting distribution of $\hat\Omega_{u\cdot v}\left(\mathcal{K}_\text{\scriptsize Bartlett},T\right)$. Thus, the resulting test statistic is not nuisance parameter free.\footnote{In particular, this implies that our self-normalization approach is not a special case of standard long-run variance estimation based on the Bartlett kernel with bandwidth equal to sample size. Instead, in Online Appendix (ref), we show that $\hat \eta_T$ is related to a a seemingly natural but inconsistent kernel estimator of $\Omega_{u \cdot v}$ based on the first differences of the residuals in the augmented partial sum regression.} One way around this problem is to replace the OLS residuals in the construction of $\hat\Omega_{u\cdot v}\left(\mathcal{K}_\text{\scriptsize Bartlett},T\right)$ with FM-OLS residuals \citeaffixed{JPS06}{see}. However, the tuning parameter choices required for FM-OLS estimation may have adverse effects on the performance of the corresponding test in finite samples. Moreover, this approach does not fit to the tuning parameter free flavor of self-normalization. The discussion in Online Appendix (ref) reveals some other possible choices of $\kappa$ leading to a nuisance parameter free limiting distribution of $\tau_{\scriptstyle\text{IM}}(\kappa)$. A large scale simulation study comparing all mentioned approaches might be interesting but beyond the scope of this paper.

remarkIn the stationary time series literature local asymptotic power of self-normalized tests is often slightly smaller than local asymptotic power of traditional tests, see, e.\,g.\xspace, \citeasnoun{KVB00} and \citeasnoun{Sh15}. Focusing on the single regressor case, we derive local asymptotic power of the self-normalized test $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$ and compare it with local asymptotic power of the traditional test $\tau_{\scriptstyle\text{IM}}(\hat \Omega_{u\cdot v})$ in Online Appendix (ref). We find that local asymptotic power of the self-normalized test is similar to, but slightly below, local asymptotic power of the traditional test. Interestingly, the corresponding largest relative local asymptotic power loss is smaller than the largest relative local asymptotic power loss of a classical self-normalized test for a restriction on the mean of a stationary time series relative to the corresponding $t$-type test \citeaffixed[p.\,1800]{Sh15}{compare}.

enumerate[wide] • Obtain the bootstrap sample $\left(\ensuremath{\varepsilon}_t^*\right)_{t=1}^T$ by randomly drawing $T$ times with replacement from the centered residuals $\left(\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\right)_{t=q+1}^T$, where $\bar{\hat{\ensuremath{\varepsilon}}}_T(q)\coloneqq (T-q)^{-1}\sum_{t=q+1}^T \hat{\ensuremath{\varepsilon}}_{t}(q)$, and construct $\left(w_t^*\right)_{t=1}^T$ recursively as $w_t^* = \hat\Phi_1(q) w_{t-1}^* + \ldots + \hat\Phi_q(q) w_{t-q}^* + \ensuremath{\varepsilon}_{t}^*$, given initial values $w_{1-q}^*,\ldots,w_0^*$.\footnote{Though irrelevant for developing asymptotic theory, it is advantageous in practical applications to eliminate the dependencies of the results on the initial values of $w_s^*$, $1-q\leq s\leq 0$, to obtain a stationary sample. We suggest to generate a sufficiently large number of $w_t^*$'s and keep the last $T+q$ of them, only.} Partition $w_t^*=[u_t^*,v_t^{*\prime}]'$ analogously to $w_t$ and define $x_t^* \coloneqq \sum_{s=1}^t v_s^*$. • To generate data under the null hypothesis given in (ref), define $y_t^* \coloneqq x_t^{*\prime}\hat\beta_{\scriptstyle\text{IM}}^r + u_t^*$, where the restricted IM-OLS estimator of $\beta$ is given by \begin{align} \hat\beta_{\scriptstyleIM}^r \coloneqq J \left(\hat\theta_{\scriptstyleIM} - \left(\sum_{t=1}^T Z_t Z_t'\right)^{-1} R_2'\left[R_2 \left(\sum_{t=1}^T Z_t Z_t'\right)^{-1} R_2'\right]^{-1} \left(R_2\hat\theta_{\scriptstyleIM}-r_0\right)\right), \end{align} with $J\coloneqq [I_{m},0_{m\times m}]$, such that $R_1\hat\beta_{\scriptstyle\text{IM}}^r = r_0$. • Compute the OLS estimator in the bootstrap augmented partial sum regression \begin{align} S_t^{y^*} = S_t^{x^*\prime}\beta + x_t^{*\prime} \gamma + S_t^{u^*} = Z_t^{*\prime}\theta + S_t^{u^*}, \end{align} where $S_t^{y^*} \coloneqq \sum_{s=1}^t y_s^*$, $S_t^{x^*} \coloneqq \sum_{s=1}^t x_s^*$, $S_t^{u^*} \coloneqq \sum_{s=1}^t u_s^*$ and $Z_t^*\coloneqq [S_t^{x^*\prime},x_t^{*\prime}]'$, to obtain the bootstrap IM-OLS estimator $\hat\theta_{\scriptstyle\text{IM}}^*\coloneqq [\hat\beta_{\scriptstyle\text{IM}}^{*\prime},\hat\gamma_{\scriptstyle\text{IM}}^{*\prime}]'$ of $\theta$ in (ref). Define the corresponding residuals $\hat{S}_t^{u^*} \coloneqq S_t^{y^*} - Z_t^{*\prime}\hat\theta_{\scriptstyle\text{IM}}^*$, $t=1,\ldots,T$, and let $\Delta\hat{S}_t^{u^*}\coloneqq \hat{S}_t^{u^*} - \hat{S}_{t-1}^{u^*}$, $t=2,\ldots,T$, denote their first differences. Define \begin{align*} \hat V_T^*\coloneqq \left(\sum_{t=1}^T Z_t^* Z_t^{*\prime}\right)^{-1}\left(\sum_{t=1}^T c_t^* c_t^{*\prime}\right)\left(\sum_{t=1}^T Z_t^* Z_t^{*\prime}\right)^{-1}, \end{align*} where $c_1^* \coloneqq S_T^{Z^*}$ and $c_t^*\coloneqq S_T^{Z^*} - S_{t-1}^{Z^*}$, with $S_{t}^{Z^*}\coloneqq \sum_{j=1}^t Z_j^*$, for $t=2,\ldots,T$. • Define the bootstrap version of the test statistic $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$ as \begin{align*} \tau_{\scriptstyleIM}^*(\hat \eta_T^*) \coloneqq \left(R_2 \hat\theta_{\scriptstyleIM}^* - r_0\right)'\left[R_2 \hat\eta_T^*\hat V_T^* R_2'\right]^{-1}\left(R_2 \hat\theta_{\scriptstyleIM}^* - r_0\right), \end{align*} where \begin{align*} \hat \eta_T^* \coloneqq T^{-2} \sum_{t=2}^T \left(\sum_{s=2}^t \Delta\hat{S}_s^{u^*} \right)^2. \end{align*} • Let $\alpha$ denote the desired nominal size of the test. Repeat $\text{Steps}\left. 1\right) \text{to} \left. 4\right)$ $B$ times, where $B$ is large and $(B+1)(1-\alpha)$ is an integer, to obtain $B$ realizations of the bootstrap test statistic $\tau_{\scriptstyle\text{IM}}^*(\hat \eta_T^*)$. Reject the null hypothesis in (ref) if the test statistic based on the original observations, $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$, is greater than the $(B+1)(1-\alpha)$-th largest realization of the bootstrap test statistic.

assumptionLet $q_T\rightarrow \infty$ and $q_T = O((T/\ln(T))^{1/3})$ as $T\rightarrow \infty$.

propositionIt holds under Assumptions (ref), (ref) and (ref) that \begin{align*} W_T^*(r)\coloneqq T^{-1/2} \sum_{t=1}^{\lfloor rT \rfloor} \ensuremath{\varepsilon}_t^* \overset{w^*}{\longrightarrow} \Sigma^{1/2} W(r), \quad 0\leq r \leq 1, \quad in $\ensuremath{{\mathbb P}}$, \end{align*} as $T\rightarrow \infty$, with $\Sigma^{1/2}(\Sigma^{1/2})'=\Sigma$.

theoremIt holds under Assumptions (ref), (ref) and (ref) that \begin{align*} B_T^*(r) \coloneqq T^{-1/2} \sum_{t=1}^{\lfloor rT \rfloor} w_t^* \overset{w^*}{\longrightarrow} \Psi(1)\Sigma^{1/2}W(r), \quad 0\leq r \leq 1, \quad in $\ensuremath{{\mathbb P}}$, \end{align*} as $T\rightarrow \infty$. If, in addition, Assumption (ref) is fulfilled, it holds that $\Psi(1)\Sigma^{1/2}W(r) \overset{d}{=} B(r)$, with $B(r)$ introduced in (ref). In particular, $\Psi(1)\Sigma^{1/2}W(r)$ has covariance matrix $\Omega$.

theoremLet the bootstrap quantities be generated as described in Section (ref). Then it holds under Assumptions (ref), (ref), (ref) and (ref) that \begin{align} \begin{bmatrix} T\left(\hat\beta_{\scriptstyleIM}^* - \hat\beta_{\scriptstyleIM}^r\right)\\ \hat\gamma_{\scriptstyleIM}^* - \Omega_{vv}^{-1}\Omega_{vu} \end{bmatrix} \overset{w^*}{\longrightarrow}\Omega_{u \cdot v}^{1/2} \left(\Pi'\right)^{-1}\mathcal{Z}\quad in $\ensuremath{{\mathbb P}}$, \end{align} as $T\rightarrow \infty$, with $\mathcal{Z}$ as defined in (ref). Moreover, it holds under the null hypothesis given in (ref) that \begin{align*} \tau_{\scriptstyleIM}^*(\hat \eta_T^*) &\overset{w^*}{\longrightarrow} \mathcal{G}_{\scriptstyleSN}\quad \text{in $\ensuremath{{\mathbb P}}$}, \end{align*} as $T\rightarrow \infty$, with $\mathcal{G}_{\scriptstyle\text{SN}}$ as defined in (ref).

remarkThe VAR sieve bootstrap scheme allows to capture the second-order dependence structure of $\{w_t\}_{t\in \ensuremath{{\mathbb Z}}}$ in a simple manner. In particular, it only requires the selection of the order of the VAR, which is a straightforward and well understood task in practice. In contrast, choosing the block size for the residual-based block bootstrap proposed in \citeasnoun{PaPo03} for unit root testing seems to be difficult in applications.\footnote{Although \citeasnoun{PoWh04} and \citeasnoun{PPW09} propose estimators of the optimal block size, these are tailor-made for the sample mean of a univariate time series.} The dependent wild bootstrap (DWB), originally proposed in \citeasnoun{Sh10} in a stationary setting and recently extended in \citeasnoun{RhSh19} to univariate unit root testing, might serve as a more non-parametric alternative to the VAR sieve bootstrap. However, it requires the choice of a kernel and a bandwidth parameter to capture dependencies across time. Extending the DWB to our setting and providing a -- possibly data-driven -- rule for selecting the bandwidth parameter for a given kernel seems to be non-trivial. We leave these interesting questions for future research.

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NoHyper\begin{center} \doublespacing { Online Appendix to \\ “A Bootstrap-Assisted Self-Normalization Approach to Inference in Cointegrating Regressions”} { Karsten Reichold and Carsten Jentsch} { \today} Equation, Table and Figure numbers not preceded by a letter refer to the main article. \end{center} \setcounter{page}{1} \setcounter{section}{0} \setcounter{footnote}{29} \setcounter{equation}{0} \setcounter{table}{0} \setcounter{figure}{0} \section{Other Possible Choices for $\kappa$} Our particular choice for $\kappa$ in (ref), the self-normalizer $\hat \eta_T$, is closely related to a seemingly natural kernel estimator of $\Omega_{u \cdot v}$ defined as \begin{align*} \widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right) \coloneqq T^{-1} \sum_{i=2}^T \sum_{j=2}^T \mathcal{K}\left(\frac{\vert i-j\vert}{b_T}\right)\Delta\hat{S}_i^u\Delta\hat{S}_j^u, \end{align*} where, as before, $\mathcal{K}(\cdot)$ is a kernel function and $b_T$ is a bandwidth parameter. In contrast to the traditional estimator $\hat\Omega_{u \cdot v}\left(\mathcal{K},b_T\right)$ defined in (ref), $\widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right)$ is inconsistent for $\Omega_{u \cdot v}$ under common kernel and bandwidth assumptions. Nevertheless, $\widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right)$ serves as a useful starting point to motivate our choice of $\hat \eta_T$ as the self-normalizer. We first note that under traditional kernel and bandwidth assumptions $\widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right)\overset{w}{\longrightarrow} \Omega_{u\cdot v}\left(1+\mathcal{Z}(2)'\mathcal{Z}(2)\right)$, as $T\rightarrow \infty$, where $\mathcal{Z}(2)$ denotes the vector of the last $m$ components of the $2m$-dimensional vector $\mathcal{Z}$ defined in (ref), see \citeasnoun[Proof of Theorem 3]{VoWa14}. That is, the limiting distribution of $\widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right)$ is nuisance parameter free up to its scale-dependence on $\Omega_{u \cdot v}$, which leads to a nuisance parameter free limiting distribution of the test statistic $\tau_{\scriptstyle\text{IM}}(\widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right))$. However, $\widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right)$ is not a self-normalizer in the original sense, as its construction requires tuning parameter choices, which may have adverse effects on the performance of $\tau_{\scriptstyle\text{IM}}(\widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right))$ in finite samples. In particular, unreported preliminary simulation results indicate that $\widetilde \Omega_{u \cdot v}\left(\mathcal{K},b_T\right)$ is less successful in reducing the size distortions of the traditional test relative to $\hat \eta_T$. In the special case where $\mathcal{K}\left(\cdot\right)$ is the Bartlett kernel ($\mathcal{K}_\text{\scriptsize Bartlett}$) and $b_T=T$, it follows from algebraic arguments used in \citeasnoun[Proof of Lemma 1]{CaSh06} in combination with similar arguments as used in the proof of Theorem (ref) that \begin{align*} &\widetilde \Omega_{u \cdot v}\left(\mathcal{K}_\scriptsize Bartlett,T\right) = \hat \eta_T + T^{-2} \sum_{t=2}^T \left(\sum_{s=2}^t \Delta\hat{S}_s^u - \sum_{s=2}^T \Delta\hat{S}_s^u\right)^2\notag \\ & \overset{w}{\longrightarrow} \Omega_{u \cdot v} \left(\int_0^1 \left(W_{u\cdot v}(r) - g(r)'\mathcal{Z}\right)^2 dr \right. \notag\\ & \left. +\int_0^1 \left([W_{u\cdot v}(r) - g(r)'\mathcal{Z}]-[W_{u\cdot v}(1) - g(1)'\mathcal{Z}]\right)^2 dr\right), \end{align*} as $T\rightarrow \infty$. The quantity $\widetilde \Omega_{u \cdot v}\left(\mathcal{K}_\text{\scriptsize Bartlett},T\right)$ is thus closely related to $\hat \eta_T$ and its limiting distribution is again nuisance parameter free up to its scale-dependence on $\Omega_{u \cdot v}$. Unreported preliminary simulation results show that $\tau_{\scriptstyle\text{IM}}(\widetilde \Omega_{u \cdot v}\left(\mathcal{K}_\text{\scriptsize Bartlett},T\right))$ performs similarly to $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$ under the null hypothesis. However, $\tau_{\scriptstyle\text{IM}}(\widetilde \Omega_{u \cdot v}\left(\mathcal{K}_\text{\scriptsize Bartlett},T\right))$ has smaller (local asymptotic) power than $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$ under the alternative. \begin{remark} There is another interesting relation we want to highlight. If we assume for a moment that $\sum_{s=2}^T \Delta\hat{S}_s^u=\hat S_T^u - \hat S_1^u=0$, which does not hold in general, we obtain $\widetilde \Omega_{u \cdot v}\left(\mathcal{K}_\text{\scriptsize Bartlett},T\right) = 2\hat \eta_T$. This relation fits well to the finding of \citeasnoun{KiVo02} in the stationary time series literature. The authors show that the self-normalization approach of \citeasnoun{KVB00} is exactly equivalent to using HAC standard errors based on the Bartlett kernel with bandwidth equal to sample size. From this perspective, choosing $\hat \eta_T$ as the self-normalizer seems to be the natural extension of the approach of \citeasnoun{KVB00} from the stationary time series literature to cointegrating regressions. \end{remark} \section{Local Asymptotic Power} In this section we compare the asymptotic power properties of the self-normalized test $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$ and the traditional test $\tau_{\scriptstyle\text{IM}}(\hat\Omega_{u\cdot v})$ under local alternatives. The results hold under Assumption (ref), with $(y_t)_{t=1}^T$ and $(x_t)_{t=1}^T$ generated by (ref) and (ref), respectively. To ease exposition of the main arguments, we restrict attention to the single regressor case ($m=1$), which suffices to illustrate the main similarities and differences between the two tests. For $m=1$ the regression model becomes $y_t = x_t \beta + u_t$, with $\beta$ being a one-dimensional parameter. We are interested in testing the null hypothesis $\text{H}_0:\beta = \beta_0$. In this case, $\tau_{\scriptstyle\text{IM}}(\kappa)$ defined in (ref) simplifies to \begin{align} \tau_{\scriptstyleIM}(\kappa) = \left(\frac{\left(\hat\beta_{\scriptstyleIM}-\beta_0\right)}{\sqrt{\kappa \hat V_T(1,1)}}\right)^2, \end{align} where $\hat V_T(1,1)$ denotes the upper-left element of the ($2\times 2$)-dimensional matrix $\hat V_T$ defined in (ref). Straightforward calculations reveal that under the local alternative $\text{H}_1:\beta = \beta_0 + c\,T^{-1}$, with $c\in\ensuremath{{\mathbb R}}$, the limiting distribution of $\tau_{\scriptstyle\text{IM}}(\hat \Omega_{u\cdot v})$ is given by \begin{align} \tau_{\text{IM}}(\hat \Omega_{u\cdot v}) \overset{w}{\longrightarrow} \mathcal{G}_{c} \coloneqq \left( \frac{c\,\Omega_{vv}^{1/2}}{\Omega_{u\cdot v}^{1/2}\sqrt{\tilde V(1,1)}} + \frac{\mathcal{Z}(1)}{\sqrt{\tilde V(1,1)}}\right)^2, \end{align} where $\tilde V(1,1)$ denotes the upper-left element of the nuisance parameter free ($2\times 2$)-dimensional matrix $\tilde V$ defined in (ref) and $\mathcal{Z}(1)$ denotes the first element in the two-dimensional vector $\mathcal{Z}$ defined in (ref). Analogously, the limiting distribution of $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$ under the local alternative is given by \begin{align} \tau_{\text{IM}}(\hat \eta_T) & \overset{w}{\longrightarrow} \mathcal{G}_{\text{SN},c} \coloneqq \left( \frac{c\,\Omega_{vv}^{1/2}}{\Omega_{u\cdot v}^{1/2}\sqrt{\int_0^1 \left(W_{u\cdot v}(r) - g(r)'\mathcal{Z}\right)^2 dr}\sqrt{\tilde V(1,1)}}\right.\notag\\ & \left. + \frac{\mathcal{Z}(1)}{\sqrt{\int_0^1 \left(W_{u\cdot v}(r) - g(r)'\mathcal{Z}\right)^2 dr}\sqrt{\tilde V(1,1)}}\right)^2. \end{align} Naturally, for $c=0$ the limiting distributions in (ref) and (ref) coincide with the chi-square distribution with one degree of freedom and the distribution $\mathcal{G}_{\scriptstyle\text{SN}}$ defined in (ref), respectively. It follows that local asymptotic power of $\tau_{\scriptstyle\text{IM}}(\hat \Omega_{u\cdot v})$ at the nominal $5\%$ level is given by $\ensuremath{{\mathbb P}}\left(\mathcal{G}_{c} > \chi^2_{1,0.95}\right)$, where $\chi^2_{1,0.95}$ denotes the $0.95$-quantile of the chi-square distribution with one degree of freedom. Analogously, local asymptotic power of $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$ at the nominal $5\%$ level is given by $\ensuremath{{\mathbb P}}\left(\mathcal{G}_{\scriptstyle\text{SN},c} > q_{1,1,0.95}\right)$, where $q_{1,1,0.95}$ denotes the corresponding $0.95$-quantile for the self-normalized test statistic given in Table (ref) in Online Appendix (ref) ($56.58$; Panel A, $m=1$, $s=1$). Local asymptotic power of both tests depends on $c\,\Omega_{vv}^{1/2}\Omega_{u\cdot v}^{-1/2}$. For a given $c\neq 0$, local asymptotic power of the tests thus depends on both the long-run variance $\Omega_{vv}$ of the first differences $v_t=\Delta x_t$ of the regressor $x_t$ and the long-run variance $\Omega_{u\cdot v}$ of the regression error $u_t$ corrected for its conditional long-run mean given $v_t$. In particular, it follows from the definition of $\Omega_{u\cdot v}$ that local asymptotic power of the tests decreases as the variability in the regression errors increases. To assess the effect of the location parameter $c$, we plot the two power curves as a function of $c$ in Figure (ref) for fixed $\Omega_{vv}$ and $\Omega_{u\cdot v}$ using simulations. In particular, we set $\Omega_{vv}^{1/2}\Omega_{u\cdot v}^{-1/2}=1$ and approximate the stochastic terms using similar methods to those used to generate the critical values in Online Appendix (ref). \begin{figure}[!ht] \begin{center} \end{center} \caption{Asymptotic power of the traditional test and the self-normalized test for $\text{H}_0:\beta = \beta_0$ under the local alternative $\beta = \beta_0 + c\,T^{-1}$. Both tests are carried out at the nominal $5\%$ level.} \end{figure} Figure (ref) shows that both tests have asymptotic power against local alternatives, with local asymptotic power increasing symmetrically as $c$ moves away from zero. Local asymptotic power of the self-normalized test is similar to, but slightly below, local asymptotic power of the traditional Wald-type test. To quantify the local power loss, we consider two measures: the absolute local power loss, \mbox{i.\,e.}\xspace, the difference in local power between the traditional test and the self-normalized test, and the relative local power loss, \mbox{i.\,e.}\xspace, the absolute local power loss divided by local power of the traditional test. The largest absolute local power loss is about $0.06$ and occurs at around $c=\pm 10.6$, whereas the largest relative local power loss is about $10.3\%$ and occurs at around $c=\pm 5.1$. The results are consistent with the findings in the stationary time series literature, where local asymptotic power of self-normalized tests is well known to be often slightly below that of traditional tests, see, \mbox{e.\,g.}\xspace, \citeasnoun{KVB00} and \citeasnoun{Sh15}.\footnote{For comparison, note that \citeasnoun[p.\,1800]{Sh15} analyzes local asymptotic power of a self-normalized test for testing a hypothesis on the mean of a strictly stationary time series $\{X_t\}_{t\in \ensuremath{{\mathbb Z}}}$. He reports that under suitable moment and weak dependence conditions, the largest relative local power loss of the self-normalized test statistic compared to the $t$-type test based on an estimator of the long-run variance of $\{X_t\}_{t\in \ensuremath{{\mathbb Z}}}$ is around $28.5\%$. In this light, the largest relative local power loss of our self-normalized test compared to the traditional test seems to be relatively small.} \begin{remark} Given the results in Theorem (ref), it follows from consistency of the IM-OLS estimator of $\beta$ and imposing the null hypothesis when constructing the bootstrap data $y_t^*$ that the VAR sieve bootstrap is consistent for the limiting null distribution of the self-normalized test statistic even when the null hypothesis is incorrect. In the limit, the bootstrap critical value for the self-normalized test statistic thus coincides with the asymptotic critical value even under deviations from the null hypothesis. Consequently, local asymptotic power of the bootstrap-assisted self-normalized test coincides with local asymptotic power of the asymptotic version of the self-normalized test. \end{remark} \section{Additional Finite Sample Results} \subsection{Estimator Performance} We briefly compare the IM-OLS estimator with the D- and FM-OLS estimators in terms of bias and root mean squared error (RMSE). Implementing the D-OLS estimator requires choosing the numbers of leads and lags of the first differences of the integrated regressors. To this end, we use the Bayesian information criterion (BIC) analyzed in \citeasnoun{ChKu12}, as it appears to be the most successful criterion -- among those considered by the authors -- in reducing the mean squared error of the D-OLS estimator. The FM-OLS estimator is based on an estimator of the long-run covariance matrix of $\{[u_t,v_t']'\}_{t\in\ensuremath{{\mathbb Z}}}$ as defined in (ref). We analyze the results for the Bartlett kernel and the Quadratic Spectral (QS) kernel, together with the corresponding data-dependent bandwidth selection rules of \citeasnoun{An91}. Table (ref) displays results for $\beta_1$.\footnote{Results for $\beta_2$ are similar and therefore not reported.} Although asymptotically unbiased, the D-, FM- and IM-OLS estimators are biased in finite samples. Generally, bias and RMSE increase in $\rho_1,\rho_2$ and $\phi$ and decrease in $T$. In terms of bias the IM-OLS estimator performs similar to the D-OLS estimator and clearly outperforms the FM-OLS estimator. In terms of RMSE the results show that overall the D-OLS estimator performs best, followed by the FM-OLS estimator based on the Bartlett kernel. RMSE of the IM-OLS estimator is comparable to the RMSE of the FM-OLS estimator based on the QS kernel, which is often slightly larger than RMSE of the FM-OLS estimator based on the Bartlett kernel. We conclude that the IM-OLS estimator performs relatively well and serves as a good starting point for self-normalized inference in cointegrating regressions. \begin{table}[!h] \adjustbox{totalheight=\textheight}{\begin{threeparttable} \caption{Bias and RMSE of various estimators of $\beta_1$} \begin{tabular}{ccrrrrrrrrrr} \toprule[1pt]\midrule[0.3pt] &&\multicolumn{5}{c}{$\text{Bias}\times 100$}&\multicolumn{5}{c}{$\text{RMSE}\times 100$}\\ \cmidrule(lr){3-7} \cmidrule(lr){8-12} &&&&\multicolumn{2}{c}{FM-OLS}&&&&\multicolumn{2}{c}{FM-OLS}&\\ \cmidrule(lr){5-6} \cmidrule(lr){10-11} $\phi$&$\rho_1,\rho_2$ & IM-OLS & D-OLS & Bartlett & QS & QML & IM-OLS & D-OLS & Bartlett & QS & QML\\ \midrule \multicolumn{12}{l}{Panel A: $T=75$}\\ \midrule $0$&0 & $-$0.25 & $-$0.27 & 0.04 & 0.11 & $-$0.05 & 5.15 & 3.52 & 3.28 & 3.42 & 15.31 \\ &0.3 & $-$0.25 & $-$0.08 & 0.66 & 0.79 & $-$0.45 & 7.29 & 4.91 & 5.02 & 5.38 & 19.79 \\ &0.6 & 0.88 & 1.65 & 3.64 & 3.70 & $-$6.82 & 12.72 & 11.68 & 11.12 & 11.97 & 374.48 \\ &0.9 & 19.63 & 12.69 & 24.93 & 25.00 & $-$606.33 & 58.08 & 46.22 & 48.09 & 55.54 & 31796.03 \\\midrule $0.3$&0 & $-$0.28 & $-$0.18 & 0.13 & 0.19 & 0.39 & 6.67 & 4.46 & 4.24 & 4.42 & 20.65 \\ &0.3 & $-$0.28 & 0.10 & 0.87 & 1.00 & 0.42 & 9.46 & 7.43 & 6.50 & 6.93 & 23.56 \\ &0.6 & 0.91 & 1.52 & 4.16 & 4.30 & 1.25 & 16.45 & 16.59 & 13.63 & 14.75 & 77.59 \\ &0.9 & 20.44 & 12.89 & 26.44 & 26.37 & 75.50 & 68.99 & 58.15 & 54.10 & 88.03 & 6141.08 \\\midrule $0.9$&0 & $-$0.35 & 0.09 & 0.34 & 0.41 & $-$0.66 & 9.74 & 8.06 & 6.27 & 6.56 & 20.00 \\ &0.3 & $-$0.33 & 0.35 & 1.37 & 1.51 & $-$0.95 & 13.83 & 12.87 & 9.56 & 10.16 & 76.13 \\ &0.6 & 0.96 & 1.60 & 5.14 & 5.59 & $-$3.55 & 23.95 & 26.29 & 18.48 & 19.82 & 354.12 \\ &0.9 & 22.08 & 13.54 & 29.43 & 29.69 & $-$31.68 & 92.30 & 83.42 & 66.97 & 122.62 & 4171.58 \\ \midrule \multicolumn{12}{l}{Panel B: $T=100$}\\ \midrule $0$&0 & $-$0.07 & $-$0.15 & 0.05 & 0.09 & 0.13 & 3.91 & 2.37 & 2.38 & 2.46 & 6.22 \\ &0.3 & $-$0.03 & 0.03 & 0.45 & 0.51 & $-$0.14 & 5.55 & 3.25 & 3.55 & 3.72 & 9.96 \\ &0.6 & 0.74 & 1.53 & 2.48 & 2.44 & $-$2.05 & 9.71 & 6.63 & 7.76 & 8.25 & 115.53 \\ &0.9 & 14.76 & 11.97 & 20.06 & 20.35 & $-$54.65 & 45.30 & 33.92 & 39.03 & 42.27 & 3497.87 \\\midrule $0.3$&0 & $-$0.07 & $-$0.07 & 0.12 & 0.15 & 0.02 & 5.07 & 2.97 & 3.09 & 3.18 & 3.97 \\ &0.3 & $-$0.01 & 0.19 & 0.60 & 0.65 & 0.09 & 7.20 & 4.21 & 4.62 & 4.84 & 11.40 \\ &0.6 & 0.83 & 1.71 & 2.92 & 2.94 & $-$5.09 & 12.56 & 8.85 & 9.67 & 10.28 & 289.60 \\ &0.9 & 15.60 & 12.88 & 21.49 & 22.36 & 12.16 & 53.98 & 42.04 & 44.37 & 50.19 & 903.93 \\\midrule $0.9$&0 & $-$0.06 & 0.12 & 0.30 & 0.31 & 0.24 & 7.39 & 4.42 & 4.56 & 4.71 & 9.54 \\ &0.3 & 0.03 & 0.49 & 1.00 & 1.05 & 0.06 & 10.51 & 6.40 & 6.84 & 7.18 & 21.90 \\ &0.6 & 1.01 & 2.04 & 3.72 & 3.86 & $-$1.24 & 18.27 & 13.33 & 13.42 & 14.37 & 168.09 \\ &0.9 & 17.27 & 14.19 & 24.28 & 25.81 & 123.50 & 72.40 & 58.58 & 55.89 & 66.60 & 4255.25 \\ \midrule \multicolumn{12}{l}{Panel C: $T=250$}\\ \midrule $0$&0 & 0.01 & $-$0.06 & 0.02 & 0.02 & 0.01 & 1.59 & 0.91 & 0.91 & 0.92 & 0.95 \\ &0.3 & 0.03 & 0.02 & 0.12 & 0.12 & 0.01 & 2.27 & 1.25 & 1.33 & 1.35 & 1.38 \\ &0.6 & 0.21 & 0.46 & 0.79 & 0.69 & 0.04 & 3.96 & 2.38 & 2.80 & 2.84 & 2.50 \\ &0.9 & 5.01 & 4.25 & 8.83 & 8.67 & $-$0.17 & 17.71 & 11.78 & 17.39 & 18.36 & 15.07 \\\midrule $0.3$&0 & 0.02 & $-$0.03 & 0.04 & 0.04 & 0.01 & 2.07 & 1.14 & 1.18 & 1.19 & 1.25 \\ &0.3 & 0.05 & 0.10 & 0.17 & 0.16 & $-$0.00 & 2.95 & 1.61 & 1.74 & 1.77 & 1.83 \\ &0.6 & 0.24 & 0.55 & 0.93 & 0.84 & 0.01 & 5.14 & 3.06 & 3.54 & 3.63 & 3.33 \\ &0.9 & 5.32 & 4.98 & 9.64 & 9.82 & $-$1.37 & 21.88 & 14.73 & 19.96 & 21.40 & 36.14 \\\midrule $0.9$&0 & 0.04 & 0.08 & 0.09 & 0.09 & 0.02 & 3.02 & 1.67 & 1.74 & 1.77 & 1.91 \\ &0.3 & 0.07 & 0.25 & 0.30 & 0.29 & 0.07 & 4.30 & 2.41 & 2.58 & 2.65 & 2.87 \\ &0.6 & 0.31 & 0.75 & 1.21 & 1.15 & 0.03 & 7.50 & 4.44 & 5.01 & 5.19 & 6.09 \\ &0.9 & 5.95 & 6.43 & 11.20 & 11.85 & 5.25 & 30.54 & 20.66 & 25.18 & 27.23 & 290.19 \\ \midrule \multicolumn{12}{l}{Panel D: $T=500$}\\ \midrule $0$&0 & 0.01 & $-$0.01 & 0.01 & 0.01 & 0.01 & 0.77 & 0.45 & 0.45 & 0.46 & 0.46 \\ &0.3 & 0.02 & 0.02 & 0.05 & 0.05 & 0.01 & 1.10 & 0.63 & 0.66 & 0.67 & 0.66 \\ &0.6 & 0.08 & 0.21 & 0.30 & 0.25 & 0.03 & 1.93 & 1.16 & 1.33 & 1.33 & 1.17 \\ &0.9 & 1.59 & 1.72 & 3.77 & 3.45 & 0.02 & 8.22 & 5.38 & 8.33 & 8.54 & 5.14 \\\midrule $0.3$&0 & 0.02 & $-$0.00 & 0.02 & 0.02 & 0.01 & 1.00 & 0.58 & 0.59 & 0.59 & 0.60 \\ &0.3 & 0.03 & 0.06 & 0.07 & 0.06 & 0.02 & 1.43 & 0.82 & 0.86 & 0.87 & 0.86 \\ &0.6 & 0.09 & 0.24 & 0.35 & 0.28 & 0.02 & 2.50 & 1.50 & 1.70 & 1.72 & 1.54 \\ &0.9 & 1.71 & 2.05 & 4.11 & 3.92 & $-$0.07 & 10.41 & 6.82 & 9.64 & 10.07 & 6.86 \\\midrule $0.9$&0 & 0.03 & 0.05 & 0.04 & 0.04 & 0.02 & 1.46 & 0.84 & 0.87 & 0.87 & 0.91 \\ &0.3 & 0.04 & 0.13 & 0.12 & 0.10 & $-$0.01 & 2.09 & 1.21 & 1.28 & 1.29 & 1.33 \\ &0.6 & 0.12 & 0.31 & 0.45 & 0.39 & $-$0.01 & 3.65 & 2.18 & 2.43 & 2.47 & 2.34 \\ &0.9 & 1.96 & 2.73 & 4.82 & 4.82 & 0.08 & 14.87 & 9.69 & 12.35 & 13.12 & 10.14 \\ \midrule[0.3pt]\bottomrule[1pt] \end{tabular} \begin{tablenotes} • Notes: The terms “Bartlett” and “QS” signify the kernel used to apply the FM-OLS estimator. QML denotes the quasi maximum likelihood estimator of \citeasnoun{Jo95}, as described in Section (ref). \end{tablenotes} \end{threeparttable}} \end{table} \subsection{Results in the i.i.d. Innovations Case ($a_1=b_1=\rho_3=0$)} \begin{table}[!h] \adjustbox{max width=\textwidth}{\begin{threeparttable} \caption{Empirical sizes of the tests for $\text{H}_0: \beta_1=1,\ \beta_2=1$ at $5\%$ level in the i.i.d. innovations case ($a_1=b_1=\rho_3=0$)} \begin{tabular}{ccccccccccccc} \toprule[1pt]\midrule[0.3pt] \multicolumn{2}{c}&\multicolumn{3}{c}&\multicolumn{8}{c}{Traditional Wald-type tests}\\ \cmidrule(lr){6-13} \multicolumn{3}{c}&\multicolumn{2}{c}{Self-normalized tests}&\multicolumn{4}{c}{Bartlett kernel}&\multicolumn{4}{c}{QS kernel}\\ \cmidrule(lr){4-5} \cmidrule(lr){6-9} \cmidrule(lr){10-13} $\phi$&$\rho_1,\rho_2$ & $\tau_{\tiny \text{IM}}^*(1)$ & $\tau_{\scriptstyle\text{IM}}(\hat{\eta}_{T})$ & $\tau_{\scriptstyle\text{IM}}^*(\hat{\eta}_{T})$ & $\tau_{\scriptstyle\text{D}}(\hat{\Omega}_{u\cdot v})$ & $\tau_{\scriptstyle\text{FM}}(\hat{\Omega}_{u\cdot v})$ & $\tau_{\scriptstyle\text{IM}}(\hat{\Omega}_{u\cdot v})$ & $\tau_{\tiny \text{IM}}^*(\hat{\Omega}_{u\cdot v})$ & $\tau_{\scriptstyle\text{D}}(\hat{\Omega}_{u\cdot v})$ & $\tau_{\scriptstyle\text{FM}}(\hat{\Omega}_{u\cdot v})$ & $\tau_{\scriptstyle\text{IM}}(\hat{\Omega}_{u\cdot v})$ & $\tau_{\tiny \text{IM}}^*(\hat{\Omega}_{u\cdot v})$ \\ \midrule \multicolumn{13}{l}{Panel A: $T=75$}\\ \midrule $0$&0 & 0.09 & 0.03 & 0.07 & 0.16 & 0.16 & 0.11 & 0.07 & 0.19 & 0.20 & 0.14 & 0.06 \\ &0.3 & 0.11 & 0.05 & 0.07 & 0.20 & 0.20 & 0.14 & 0.07 & 0.20 & 0.23 & 0.15 & 0.06 \\ &0.6 & 0.15 & 0.09 & 0.08 & 0.34 & 0.35 & 0.23 & 0.09 & 0.34 & 0.37 & 0.22 & 0.08 \\ &0.9 & 0.48 & 0.35 & 0.19 & 0.72 & 0.79 & 0.66 & 0.25 & 0.79 & 0.84 & 0.73 & 0.26 \\\midrule $0.3$&0 & 0.09 & 0.04 & 0.07 & 0.18 & 0.18 & 0.13 & 0.06 & 0.20 & 0.21 & 0.15 & 0.06 \\ &0.3 & 0.12 & 0.06 & 0.07 & 0.23 & 0.22 & 0.17 & 0.07 & 0.23 & 0.24 & 0.17 & 0.06 \\ &0.6 & 0.18 & 0.09 & 0.09 & 0.40 & 0.36 & 0.25 & 0.09 & 0.41 & 0.40 & 0.27 & 0.08 \\ &0.9 & 0.48 & 0.32 & 0.19 & 0.75 & 0.77 & 0.66 & 0.24 & 0.83 & 0.85 & 0.76 & 0.28 \\\midrule $0.9$&0 & 0.16 & 0.05 & 0.10 & 0.23 & 0.20 & 0.15 & 0.09 & 0.24 & 0.22 & 0.16 & 0.08 \\ &0.3 & 0.19 & 0.06 & 0.10 & 0.32 & 0.26 & 0.19 & 0.11 & 0.33 & 0.29 & 0.21 & 0.09 \\ &0.6 & 0.26 & 0.10 & 0.12 & 0.48 & 0.38 & 0.29 & 0.13 & 0.53 & 0.45 & 0.34 & 0.11 \\ &0.9 & 0.51 & 0.30 & 0.19 & 0.78 & 0.74 & 0.68 & 0.25 & 0.88 & 0.87 & 0.82 & 0.28 \\ \midrule \multicolumn{13}{l}{Panel B: $T=100$}\\ \midrule $0$&0 & 0.08 & 0.03 & 0.07 & 0.13 & 0.14 & 0.10 & 0.06 & 0.15 & 0.17 & 0.12 & 0.06 \\ &0.3 & 0.09 & 0.05 & 0.07 & 0.16 & 0.18 & 0.14 & 0.07 & 0.15 & 0.18 & 0.13 & 0.06 \\ &0.6 & 0.13 & 0.08 & 0.07 & 0.27 & 0.31 & 0.19 & 0.08 & 0.26 & 0.31 & 0.18 & 0.07 \\ &0.9 & 0.39 & 0.27 & 0.14 & 0.66 & 0.73 & 0.57 & 0.18 & 0.72 & 0.79 & 0.64 & 0.20 \\\midrule $0.3$&0 & 0.08 & 0.05 & 0.07 & 0.15 & 0.15 & 0.12 & 0.06 & 0.15 & 0.17 & 0.13 & 0.06 \\ &0.3 & 0.10 & 0.06 & 0.07 & 0.18 & 0.19 & 0.15 & 0.07 & 0.17 & 0.20 & 0.14 & 0.07 \\ &0.6 & 0.14 & 0.09 & 0.08 & 0.29 & 0.32 & 0.22 & 0.09 & 0.29 & 0.33 & 0.22 & 0.08 \\ &0.9 & 0.40 & 0.27 & 0.15 & 0.68 & 0.71 & 0.58 & 0.19 & 0.77 & 0.81 & 0.69 & 0.22 \\\midrule $0.9$&0 & 0.13 & 0.05 & 0.08 & 0.17 & 0.17 & 0.13 & 0.08 & 0.16 & 0.18 & 0.14 & 0.07 \\ &0.3 & 0.15 & 0.06 & 0.09 & 0.21 & 0.22 & 0.17 & 0.09 & 0.21 & 0.24 & 0.17 & 0.09 \\ &0.6 & 0.21 & 0.09 & 0.10 & 0.32 & 0.33 & 0.25 & 0.11 & 0.35 & 0.37 & 0.28 & 0.09 \\ &0.9 & 0.43 & 0.25 & 0.16 & 0.69 & 0.69 & 0.59 & 0.21 & 0.81 & 0.84 & 0.74 & 0.23 \\ \midrule \multicolumn{13}{l}{Panel C: $T=250$}\\ \midrule $0$&0 & 0.06 & 0.04 & 0.06 & 0.08 & 0.09 & 0.07 & 0.06 & 0.09 & 0.10 & 0.08 & 0.05 \\ &0.3 & 0.07 & 0.05 & 0.06 & 0.10 & 0.12 & 0.09 & 0.06 & 0.09 & 0.11 & 0.09 & 0.06 \\ &0.6 & 0.07 & 0.06 & 0.06 & 0.17 & 0.19 & 0.11 & 0.06 & 0.15 & 0.17 & 0.10 & 0.06 \\ &0.9 & 0.18 & 0.13 & 0.07 & 0.38 & 0.52 & 0.28 & 0.09 & 0.40 & 0.55 & 0.31 & 0.09 \\\midrule $0.3$&0 & 0.07 & 0.05 & 0.06 & 0.09 & 0.11 & 0.09 & 0.06 & 0.09 & 0.10 & 0.08 & 0.06 \\ &0.3 & 0.07 & 0.05 & 0.06 & 0.11 & 0.13 & 0.10 & 0.06 & 0.10 & 0.12 & 0.09 & 0.06 \\ &0.6 & 0.08 & 0.06 & 0.06 & 0.17 & 0.20 & 0.13 & 0.06 & 0.16 & 0.19 & 0.12 & 0.06 \\ &0.9 & 0.19 & 0.13 & 0.07 & 0.41 & 0.50 & 0.31 & 0.09 & 0.45 & 0.54 & 0.36 & 0.09 \\\midrule $0.9$&0 & 0.08 & 0.05 & 0.06 & 0.10 & 0.12 & 0.09 & 0.07 & 0.09 & 0.11 & 0.09 & 0.07 \\ &0.3 & 0.10 & 0.05 & 0.07 & 0.12 & 0.15 & 0.11 & 0.07 & 0.11 & 0.14 & 0.10 & 0.07 \\ &0.6 & 0.12 & 0.07 & 0.07 & 0.17 & 0.21 & 0.15 & 0.07 & 0.17 & 0.21 & 0.14 & 0.07 \\ &0.9 & 0.23 & 0.14 & 0.09 & 0.43 & 0.48 & 0.35 & 0.11 & 0.50 & 0.57 & 0.42 & 0.11 \\ \midrule \multicolumn{13}{l}{Panel D: $T=500$}\\ \midrule $0$&0 & 0.05 & 0.04 & 0.05 & 0.07 & 0.07 & 0.07 & 0.05 & 0.07 & 0.07 & 0.07 & 0.05 \\ &0.3 & 0.05 & 0.05 & 0.05 & 0.08 & 0.08 & 0.08 & 0.05 & 0.07 & 0.08 & 0.07 & 0.05 \\ &0.6 & 0.06 & 0.05 & 0.05 & 0.11 & 0.13 & 0.09 & 0.05 & 0.10 & 0.11 & 0.08 & 0.05 \\ &0.9 & 0.08 & 0.06 & 0.05 & 0.20 & 0.34 & 0.13 & 0.05 & 0.20 & 0.35 & 0.14 & 0.05 \\\midrule $0.3$&0 & 0.06 & 0.04 & 0.05 & 0.08 & 0.08 & 0.07 & 0.05 & 0.07 & 0.07 & 0.07 & 0.05 \\ &0.3 & 0.06 & 0.05 & 0.05 & 0.09 & 0.09 & 0.08 & 0.05 & 0.08 & 0.08 & 0.07 & 0.05 \\ &0.6 & 0.06 & 0.05 & 0.05 & 0.12 & 0.13 & 0.10 & 0.05 & 0.11 & 0.12 & 0.09 & 0.05 \\ &0.9 & 0.09 & 0.07 & 0.05 & 0.24 & 0.34 & 0.16 & 0.05 & 0.24 & 0.36 & 0.17 & 0.06 \\\midrule $0.9$&0 & 0.07 & 0.04 & 0.05 & 0.08 & 0.08 & 0.08 & 0.06 & 0.07 & 0.08 & 0.07 & 0.06 \\ &0.3 & 0.07 & 0.05 & 0.05 & 0.09 & 0.10 & 0.09 & 0.06 & 0.08 & 0.10 & 0.08 & 0.05 \\ &0.6 & 0.08 & 0.05 & 0.05 & 0.13 & 0.14 & 0.11 & 0.06 & 0.12 & 0.13 & 0.10 & 0.06 \\ &0.9 & 0.12 & 0.08 & 0.06 & 0.28 & 0.33 & 0.20 & 0.06 & 0.30 & 0.36 & 0.21 & 0.06 \\ \midrule[0.3pt]\bottomrule[1pt] \end{tabular} \begin{tablenotes} • Notes: Superscript “$*$” signifies the use of bootstrap critical values. The asymptotic critical value for the self-normalized test $\tau_{\scriptstyle\text{IM}}(\hat{\eta}_{T})$ is given in Table (ref) in Online Appendix (ref) ($167.23$; Panel A, $m=2$, $s=2$). \end{tablenotes} \end{threeparttable}} \end{table} \begin{figure}[H] \begin{center} \begin{subfigure}{0.4\textwidth} \caption*{$T=100$, $\phi=0$} \end{subfigure}\begin{subfigure}{0.4\textwidth} \caption*{$T=100$, $\phi=0.9$} \end{subfigure} \begin{subfigure}{0.4\textwidth} \caption*{$T=250$, $\phi=0$} \end{subfigure}\begin{subfigure}{0.4\textwidth} \caption*{$T=250$, $\phi=0.9$} \end{subfigure} \end{center} \caption{Size-adjusted power of the tests for $\text{H}_0: \beta_1=1,\ \beta_2=1$ at 5% level for $\rho_1,\rho_2=0.6$ in the i.i.d. innovations case ($a_1=b_1=\rho_3=0$). Notes: Superscript “$*$” signifies the use of bootstrap critical values. Whenever necessary, we use the Bartlett kernel to estimate long-run variance parameters.} \end{figure} \subsection{Comparison with Johansen's Likelihood Ratio Test} We can rewrite the data generating process (DGP) given in (ref) and (ref) as \begin{align*} \underbrace{\begin{bmatrix} 1&-\beta_1&-\beta_2\\ 0&1&0\\ 0&0&1 \end{bmatrix}}_{=:A_0} \begin{bmatrix} y_t\\ x_{1t}\\ x_{2t} \end{bmatrix} = \underbrace{\begin{bmatrix} 0&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}}_{=:A_1} \begin{bmatrix} y_{t-1}\\ x_{1,t-1}\\ x_{2,t-1} \end{bmatrix} + \begin{bmatrix} u_t\\ v_{1t}\\ v_{2t} \end{bmatrix} \Leftrightarrow X_t = \Pi_1 X_{t-1} + \epsilon_t, \end{align*} where $X_t \coloneqq [y_t,x_{1t},x_{2t}]'$, $\epsilon_t\coloneqq A_0^{-1}[u_t,v_{1t},v_{2t}]'$ and $\Pi_1 \coloneqq A_0^{-1}A_1$. While the estimators and tests discussed so far ignore the exact characteristics of the DGP, the reduced rank quasi maximum likelihood (QML) estimator of \citeasnoun{Jo95} and the likelihood ratio (LR) test based upon it are tailor-made for this particular class of DGPs. Under this DGP, the performance of the \textit{parametric} LR test thus serves as a natural benchmark for the performance of the \textit{non-parametric} self-normalized test developed in this paper. To fix ideas, we consider the error correction model of order $k\geq 1$ with one cointegrated relation, \mbox{i.\,e.}\xspace, \begin{align} \Delta X_t = a b' X_{t-1} + \sum_{l=1}^{k-1} \Gamma_l \Delta X_{t-l} + \epsilon_t, \end{align} where $\Delta X_s \coloneqq X_s - X_{s-1}$, $a=[-1,0,0]'$ and $b=[1,-\beta_1,-\beta_2]'$. To test general linear hypotheses on the cointegration vector $b$, we first determine the order $k$ in (ref) similarly as described in the beginning of Section (ref) for the order of the VAR sieve. We then estimate the vector $b$ using the reduced rank QML estimator of \citeasnoun{Jo95}. Testing the null hypothesis $\text{H}_0: \beta_1=1,\ \beta_2=1$ in (ref) corresponds to testing the null hypothesis $\text{H}_0: b = [1,-1,-1]'$ in (ref). Under the null hypothesis, the LR test statistic of \citeasnoun{Jo95} is asymptotically chi-squared distributed with two degrees of freedom. \citeasnoun{CNR15} show how to replace the asymptotic chi-square critical values with bootstrap critical values to reduce the size distortions of the LR test in finite samples. Before we compare the finite sample performance of the self-normalized test based on asymptotic critical values with the LR test based on asymptotic critical values and the bootstrap-assisted self-normalized test with the LR test based on bootstrap critical values generated as described in \citeasnoun[p.\,817]{CNR15}, we should mention one particular drawback of the reduced rank QML estimator. Table (ref) in Online Appendix (ref) shows that the QML estimator of $\beta_1$ has a very large bias for $\rho_1,\rho_2=0.9$ and an extremely large RMSE for $\rho_1,\rho_2\in\{0.6,0.9\}$ in small to medium samples.\footnote{Results for $\beta_2$ are similar.} These results are in line with the findings in related literature, where the QML estimator is observed to occasionally produce estimates that are far away from the true parameter values, see, \mbox{e.\,g.}\xspace, \citeasnoun{BrLu05} and references therein. We now turn to the performance of the tests. Table (ref) displays the empirical sizes of both the LR test and the LR test based on bootstrap critical values ($\text{LR}^*$). Comparing empirical sizes of the LR test with those of the self-normalized test in Table (ref), we note that that the self-normalized test based on asymptotic critical values clearly outperforms the LR test for almost all combinations of $T$, $\rho_1,\rho_2$ and $\phi$. However, in case $\rho_1,\rho_2=0.9$ and $\phi$ is smaller than $0.9$ size distortions of the self-normalized test are slightly larger than those of the LR test. In terms of size-adjusted power we note that the self-normalized test outperforms the LR test in small to medium samples or in case $\rho_1,\rho_2=0.9$, see Figures (ref) and (ref). Replacing asymptotic critical values with bootstrap critical values improves the performance of the LR test in small to medium samples considerably. Comparing the empirical sizes of $\text{LR}^*$ in Table (ref) with those of the bootstrap-assisted self-normalized test in Table (ref) reveals that the two bootstrap tests perform similarly for all sample sizes considered, but $\text{LR}^*$ has slight performance advantages over the bootstrap-assisted self-normalized test in small to medium samples in case $\rho_1,\rho_2=0.9$. However, this performance advantage of $\text{LR}^*$ over the bootstrap-assisted self-normalized test under the null hypothesis comes at the cost of some losses in size-adjusted power, as Figure (ref) reveals. In case $\rho_1,\rho_2$ is large relative to the sample size, we observe that size-adjusted power of $\text{LR}^*$ is considerably smaller than size-adjusted power of the bootstrap-assisted self-normalized test. We also observe that the power difference between these tests can be much larger than the power difference between the bootstrap-assisted self-normalized test and the IM-OLS based Wald-type bootstrap test, compare Figure (ref) in case $T=100$, $\phi=0.9$ and $\rho_1,\rho_2=0.6$ and Figure (ref) in case $T=75$, $\phi\in\{0,0.9\}$ and $\rho_1,\rho_2=0.9$. For larger sample sizes, however, $\text{LR}^*$ has slightly larger size-adjusted power than the bootstrap-assisted self-normalized test and the IM-OLS based Wald-type bootstrap test, compare Figure (ref) in case $T=250$. We conclude that the (bootstrap-assisted) self-normalized test performs well both under the null hypothesis and under the alternative even when compared to the parametric likelihood-ratio (bootstrap) test tailor made for the DGP under consideration. Therefore, in applications focusing on a particular cointegrating relation between the variables, the IM-OLS estimator and the (bootstrap-assisted) self-normalized test are a good alternative to the QML estimator and the likelihood-ration (bootstrap) test, especially when the QML estimator yields implausible estimates of the cointegrating vector. \begin{table}[!t] \adjustbox{max width=\textwidth}{\begin{threeparttable} \caption{Empirical sizes of the LR test and the LR bootstrap test for $\text{H}_0: \beta_1=1,\ \beta_2=1$ at $5\%$ level} \begin{tabular}{cccccccccc} \toprule[1pt]\midrule[0.3pt] \multicolumn{2}{c}&\multicolumn{2}{c}{$T=75$}&\multicolumn{2}{c}{$T=100$}&\multicolumn{2}{c}{$T=250$}&\multicolumn{2}{c}{$T=500$}\\ \cmidrule(lr){3-4} \cmidrule(lr){5-6} \cmidrule(lr){7-8} \cmidrule(lr){9-10} $\phi$&$\rho_1,\rho_2$ & $\text{LR}$ & $\text{LR}^*$ & $\text{LR}$ & $\text{LR}^*$& $\text{LR}$ & $\text{LR}^*$& $\text{LR}$ & $\text{LR}^*$\\ \midrule $0$&0 & 0.13 & 0.06 & 0.10 & 0.06 & 0.07 & 0.06& 0.07 & 0.06\\ &0.3 & 0.14 & 0.06 & 0.11 & 0.06 & 0.08 & 0.06& 0.07 & 0.06\\ &0.6 & 0.16 & 0.07 & 0.13 & 0.06& 0.09 & 0.06& 0.07 & 0.06\\ &0.9 & 0.27 & 0.08 &0.20 & 0.07& 0.11 & 0.07& 0.08 & 0.06\\\midrule $0.3$&0 & 0.14 & 0.07 & 0.11 & 0.06& 0.08 & 0.05& 0.07 & 0.06\\ &0.3 & 0.15 & 0.06 & 0.12 & 0.07& 0.09 & 0.07& 0.07 & 0.06\\ &0.6 & 0.19 & 0.07 & 0.15 & 0.08& 0.09 & 0.07& 0.07 & 0.05\\ &0.9 & 0.32 & 0.10 & 0.25 & 0.09& 0.12 & 0.07& 0.08 & 0.06\\\midrule $0.9$&0 & 0.18 & 0.09 & 0.14 & 0.08& 0.09 & 0.07& 0.07 & 0.06\\ &0.3 & 0.20 & 0.08 & 0.15 & 0.08& 0.10 & 0.06& 0.07 & 0.06\\ &0.6 & 0.23 & 0.09 & 0.18 & 0.07& 0.10 & 0.06& 0.09 & 0.07\\ &0.9 & 0.39 & 0.09 & 0.31 & 0.09& 0.14 & 0.07& 0.12 & 0.08\\ \midrule[0.3pt]\bottomrule[1pt] \end{tabular} \begin{tablenotes} • Note: Superscript “$*$” signifies the use of bootstrap critical values. \end{tablenotes} \end{threeparttable}} \end{table} \begin{figure}[!t] \begin{center} \begin{subfigure}{0.4\textwidth} \caption*{$T=100$, $\phi=0$} \end{subfigure}\begin{subfigure}{0.4\textwidth} \caption*{$T=100$, $\phi=0.9$} \end{subfigure} \begin{subfigure}{0.4\textwidth} \caption*{$T=250$, $\phi=0$} \end{subfigure}\begin{subfigure}{0.4\textwidth} \caption*{$T=250$, $\phi=0.9$} \end{subfigure} \end{center} \caption{Size-adjusted power of the tests for $\text{H}_0: \beta_1=1,\ \beta_2=1$ at 5% level for $\rho_1,\rho_2=0.6$. Notes: Superscript “$*$” signifies the use of bootstrap critical values. Results for the (bootstrap-assisted) self-normalized test and the IM-OLS Wald-type bootstrap tests coincide with those in Figure (ref).} \end{figure} \begin{figure}[!t] \begin{center} \begin{subfigure}{0.4\textwidth} \caption*{$T=75$, $\phi=0$} \end{subfigure}\begin{subfigure}{0.4\textwidth} \caption*{$T=75$, $\phi=0.9$} \end{subfigure} \begin{subfigure}{0.4\textwidth} \caption*{$T=100$, $\phi=0$} \end{subfigure}\begin{subfigure}{0.4\textwidth} \caption*{$T=100$, $\phi=0.9$} \end{subfigure} \end{center} \caption{Size-adjusted power of the tests for $\text{H}_0: \beta_1=1,\ \beta_2=1$ at 5% level for $\rho_1,\rho_2=0.9$. Note: Superscript “$*$” signifies the use of bootstrap critical values. Results for the IM-OLS based Wald-type bootstrap test are based on the Bartlett kernel.} \end{figure} \section{Asymptotic Critical Values} To simulate asymptotically valid critical values, we approximate standard Brownian motions with normalized sums of $10{,}000$ i.i.d. standard normal random variables and approximate the corresponding integrals accordingly. The fact that the numerator of the limiting null distribution of the self-normalized test statistic given in (ref) is invariant to $\Pi$ and only depends on the number of linearly independent restrictions under the null hypothesis justifies to set $\Pi = I_{2m}$ and $R_2=[I_s,0_{s\times (2m-s)}]$.\footnote{Note that the dependence between numerator and denominator in (ref) implies that simply drawing from a chi-square distribution for the numerator is invalid.} We tabulate critical values based on $10{,}000$ replications for various choices of $m$ and $s$ and different deterministic regressors in (ref) in Table (ref). \begin{table}[!ht] \adjustbox{max width=0.85\textwidth}{\begin{threeparttable} \caption{Asymptotic critical values for $\tau_{\scriptstyle\text{IM}}(\hat \eta_T)$} \begin{tabular}{ccccccccccc} \toprule[1pt]\midrule[0.3pt] \multicolumn{1}{c}&\multicolumn{1}{c}{$m=1$}&\multicolumn{2}{c}{$m=2$}&\multicolumn{3}{c}{$m=3$}&\multicolumn{4}{c}{$m=4$}\\ \cmidrule(lr){2-2} \cmidrule(lr){3-4} \cmidrule(lr){5-7} \cmidrule(lr){8-11} $\%$ & $s=1$ & $s=1$ & $s=2$ & $s=1$ & $s=2$ & $s=3$ & $s=1$ & $s=2$ & $s=3$ & $s=4$ \\ \midrule \multicolumn{11}{l}{Panel A: No deterministic regressors}\\ \midrule 90.0 & 36.63 & 66.33 & 122.32 & 94.04 & 172.00 & 240.58 & 131.68 & 232.77 & 318.25 & 402.61 \\ 95.0 & 56.58 & 96.51 & 167.23 & 140.69 & 231.79 & 313.46 & 189.15 & 309.06 & 407.17 & 510.60 \\ 97.5 & 79.24 & 131.79 & 216.99 & 191.68 & 290.47 & 390.38 & 256.38 & 390.07 & 504.08 & 630.19 \\ 99.0 & 120.10 & 189.69 & 286.97 & 266.16 & 375.30 & 494.00 & 355.25 & 505.21 & 645.89 & 767.61 \\ \midrule \multicolumn{11}{l}{Panel B: Intercept ($d_t=1$)}\\ \midrule 90.0 & 64.13 & 94.15 & 168.58 & 126.65 & 221.45 & 305.36 & 162.08 & 278.05 & 382.30 & 481.15 \\ 95.0 & 95.81 & 140.55 & 233.15 & 187.03 & 297.11 & 396.56 & 236.54 & 372.79 & 487.71 & 596.15 \\ 97.5 & 136.10 & 190.23 & 292.64 & 245.93 & 375.55 & 488.35 & 325.56 & 458.37 & 587.85 & 720.31 \\ 99.0 & 187.13 & 263.92 & 381.78 & 338.59 & 474.31 & 602.27 & 421.68 & 582.89 & 719.98 & 872.07 \\ \midrule \multicolumn{11}{l}{Panel C: Intercept and linear time trend ($d_t=[1,t]'$)}\\ \midrule 90.0 & 90.44 & 122.19 & 209.54 & 152.66 & 261.47 & 363.17 & 180.25 & 311.22 & 434.29 & 545.37 \\ 95.0 & 134.19 & 171.46 & 283.33 & 219.51 & 354.08 & 460.37 & 258.75 & 423.39 & 546.31 & 688.21 \\ 97.5 & 183.51 & 231.09 & 357.66 & 294.26 & 433.33 & 569.84 & 342.56 & 524.25 & 686.44 & 810.09 \\ 99.0 & 243.72 & 304.08 & 460.98 & 409.03 & 556.42 & 713.24 & 478.05 & 680.76 & 821.12 & 977.09 \\ \midrule \multicolumn{11}{l}{Panel D: Intercept, linear and square time trend ($d_t=[1,t,t^2]'$)}\\ \midrule 90.0 & 115.13 & 138.49 & 245.91 & 175.40 & 302.95 & 418.77 & 205.72 & 352.96 & 479.57 & 608.35 \\ 95.0 & 166.35 & 200.65 & 331.26 & 255.74 & 402.90 & 530.94 & 303.58 & 465.28 & 621.70 & 764.20 \\ 97.5 & 217.42 & 268.86 & 401.63 & 348.51 & 509.51 & 637.29 & 390.59 & 589.05 & 762.57 & 902.89 \\ 99.0 & 290.63 & 357.58 & 513.85 & 472.49 & 646.48 & 800.81 & 527.20 & 754.26 & 923.59 & 1070.32 \\ \midrule \multicolumn{11}{l}{Panel E: Intercept, linear, square and cubic time trend ($d_t=[1,t,t^2,t^3]'$)}\\ \midrule 90.0 & 137.70 & 166.87 & 292.13 & 197.84 & 340.61 & 465.58 & 229.38 & 392.80 & 533.60 & 680.84 \\ 95.0 & 198.48 & 237.82 & 379.15 & 288.65 & 446.27 & 590.05 & 334.55 & 509.11 & 684.33 & 858.04 \\ 97.5 & 263.30 & 308.64 & 467.71 & 391.70 & 565.65 & 720.19 & 438.56 & 645.41 & 853.07 & 1004.50 \\ 99.0 & 352.56 & 406.48 & 587.03 & 539.71 & 726.07 & 903.53 & 592.44 & 846.82 & 1052.20 & 1222.78 \\ \midrule[0.3pt]\bottomrule[1pt] \end{tabular} \begin{tablenotes} • Notes: $s$ is the number of linearly independent restrictions under the null hypothesis on the coefficients corresponding to the $m\geq s$ integrated regressors. \end{tablenotes} \end{threeparttable}} \end{table} \section{Proofs of Main Results} {0mm} {\bf Proof of Theorem (ref).} \citeasnoun[Lemma 2]{VoWa14} show that \begin{align*} T^{-1/2} \sum_{t=2}^{\lfloor rT \rfloor} \Delta\hat{S}_t^u \overset{w}{\longrightarrow} \Omega_{u \cdot v}^{1/2}\left(W_{u\cdot v}(r) - g(r)'\mathcal{Z}\right),\quad 0\leq r \leq 1, \end{align*} as $T\rightarrow \infty$. The continuous mapping theorem thus yields \begin{align*} \hat \eta_T = T^{-1} \sum_{t=2}^T \left( T^{-1/2} \sum_{s=2}^t \Delta\hat{S}_s^u \right)^2 \overset{w}{\longrightarrow} \Omega_{u \cdot v} \int_0^1 \left(W_{u\cdot v}(r) - g(r)'\mathcal{Z}\right)^2 dr, \end{align*} as $T\rightarrow \infty$. The final result now follows with standard arguments from (ref). $\square$\\ The proof of the remaining results relies on the following lemma. \begin{lemma} Under Assumptions (ref), (ref) and (ref), it holds that \begin{align} \max_{q+1\leq t \leq T} \vert \hat w_t - w_t \vert_F = O_\ensuremath{{\mathbb P}}(T^{-1/2}) \end{align} and \begin{align} q^{1/2}\sum_{j=1}^q \vert \hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F= O_\ensuremath{{\mathbb P}}(q^3/T)=o_\ensuremath{{\mathbb P}}(1), \end{align} where $\tilde \Phi_1(q),\ldots,\tilde \Phi_q(q)$ denote the solution of the sample Yule-Walker equations in the regression of $w_t$ on $w_{t-1},\ldots,w_{t-q}$, $t=q+1,\ldots,T$. \end{lemma} \begin{proof} We provide a proof in Online Appendix (ref). \end{proof} Two key ingredients in the proof of Lemma (ref), which are also useful hereafter, are the following: First, it holds under Assumptions (ref) and (ref) that \begin{align} q^{3/2} \sup_{1\leq j \leq q}\vert \tilde \Phi_j(q) - \Phi_j(q)\vert_F = q^{3/2} O_\ensuremath{{\mathbb P}}((\ln(T)/T)^{1/2}) = O_\ensuremath{{\mathbb P}}(1), \end{align} compare \citeasnoun[Remark 3.3.]{MeKr15}, where $\Phi_1(q),\ldots,\Phi_q(q)$ denote the finite predictor coefficients, \mbox{i.\,e.}\xspace, the solution of the population Yule-Walker equations based on the true moments. Second, under Assumption (ref) with $k\geq 3/2$, there exist constants $q_0\in \ensuremath{{\mathbb N}}$ and $c< \infty$ such that \begin{align} \sum_{j=1}^q (1+j)^k \vert \Phi_j(q) - \Phi_j \vert_F \leq c \sum_{j=q+1}^\infty (1+j)^k \vert \Phi_j\vert_F, \end{align} for all $q\geq q_0$ and the right-hand side converges to zero as $q\rightarrow \infty$, see \citeasnoun[Lemma 3.1]{MeKr15}.\footnote{This result is known as the generalized Baxter's inequality, see \citeasnoun{Ba62} and \citeasnoun[p.\,269]{HaDe88}.}\\ We use Lemma (ref) to prove the following two lemmas. Subsequently, we use the following two lemmas to prove Proposition (ref). \begin{lemma} It holds under Assumptions (ref) and (ref) that \begin{align*} \ensuremath{{\mathbb E}}^*\left(\vert \ensuremath{\varepsilon}_t^*\vert_F^a\right) = (T-q)^{-1}\sum_{t=q+1}^T \vert \hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\vert_F^a = O_\ensuremath{{\mathbb P}}(1), \end{align*} in $\ensuremath{{\mathbb P}}$, for the $a>2$ from Assumption (ref). \end{lemma} \begin{proof} We provide a proof in Online Appendix (ref). \end{proof} \begin{lemma} It holds under Assumptions (ref), (ref) and (ref) that \begin{align*} \ensuremath{{\mathbb E}}^*\left(\ensuremath{\varepsilon}_t^* \ensuremath{\varepsilon}_t^{*\prime}\right) = (T-q)^{-1}\sum_{t=q+1}^T \left(\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\right)\left(\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\right)' = \Sigma + o_\ensuremath{{\mathbb P}}(1), \end{align*} in $\ensuremath{{\mathbb P}}$. \end{lemma} \begin{proof} We provide a proof in Online Appendix (ref). \end{proof} {\bf Proof of Proposition (ref).} Given Lemma (ref) and Lemma (ref), the result follows immediately from \citeasnoun{Ei87}, as in \citeasnoun[p.\,714]{CPS06}.\footnote{For more details we refer to the pre-print of \citeasnoun{PSU10}, which is available on \href{https://www.stephansmeekes.nl/research}{https://www.stephansmeekes.nl/research} (Accessed: March 24, 2022).}$\square$\\ {\bf Proof of Theorem (ref).} Using similar arguments as \citeasnoun[p.\,670]{PSU10}, it follows that \begin{align*} B_T^*(r)= T^{-1/2} \sum_{t=1}^{\lfloor rT \rfloor} w_t^* = \left(I - \sum_{j=1}^q \hat \Phi_j(q)\right)^{-1} W_T^*(r) + T^{-1/2} (\bar w_0^* - \bar w_{\lfloor rT \rfloor}^*), \end{align*} where $\bar w_{t-1}^*\coloneqq \left(I - \sum_{j=1}^q \hat \Phi_j(q)\right)^{-1}\sum_{i=1}^q\left(\sum_{j=i}^q \hat \Phi_j(q)\right) w_{t-i}^*$. It thus remains to show that \begin{align} I - \sum_{j=1}^q \hat \Phi_j(q) \overset{p}{\longrightarrow} \Phi(1) \end{align} and \begin{align} \ensuremath{{\mathbb P}}^*\left( \max_{0\leq t\leq T} \vert T^{-1/2} \bar w_t^* \vert_F > \delta \right)=o_\ensuremath{{\mathbb P}}(1). \end{align} We first show (ref). From Lemma (ref), (ref) and (ref), we obtain \begin{align*} \vert I - \sum_{j=1}^q \hat \Phi_j(q) - \Phi(1)\vert_F &\leq \sum_{j=1}^q \vert \hat \Phi_j(q) - \tilde\Phi_j(q) \vert_F + \sum_{j=1}^q \vert \tilde \Phi_j(q) - \Phi_j(q) \vert_F \\ &\quad + \sum_{j=1}^q \vert \Phi_j(q) - \Phi_j \vert_F + \sum_{j=q+1}^\infty \vert \Phi_j \vert_F\\ & \leq \sum_{j=1}^q \vert \hat \Phi_j(q) - \tilde\Phi_j(q) \vert_F + q\sup_{1\leq j \leq q}\vert \tilde \Phi_j(q)-\Phi_j(q)\vert_F \\ &\quad+ c \sum_{j=q+1}^\infty \vert \Phi_j\vert_F + \sum_{j=q+1}^\infty \vert \Phi_j \vert_F\\ & = o_\ensuremath{{\mathbb P}}(1) + o_\ensuremath{{\mathbb P}}(1) + o(1) + o(1) = o_\ensuremath{{\mathbb P}}(1). \end{align*} To prove (ref), we note that it follows from strict stationarity of $\{\bar w_t^*\}_{t\in \ensuremath{{\mathbb Z}}}$ and Markov's inequality, that \begin{align*} \ensuremath{{\mathbb P}}^*\left( \max_{0\leq t\leq T} \vert T^{-1/2} \bar w_t^* \vert_F > \delta \right) & \leq \sum_{t=0}^T \ensuremath{{\mathbb P}}^*\left( \vert T^{-1/2} \bar w_t^* \vert_F > \delta \right)\\ &\leq (T+1) \ensuremath{{\mathbb P}}^*\left( \vert T^{-1/2} \bar w_t^* \vert_F > \delta \right)\\ &\leq \delta^{-a}(T^{1-a/2}+T^{-a/2}) \ensuremath{{\mathbb E}}^*\left(\vert \bar w_t^* \vert_F^a \right), \end{align*} with the $a>2$ from Assumption (ref), compare \citeasnoun[p.\,486]{Pa02}. Similarly as in \citeasnoun[p.\,671]{PSU10}, we obtain \begin{align*} \ensuremath{{\mathbb E}}^*\left(\vert \bar w_t^* \vert_F^a \right) \leq c\left(m+1\right)^{a/2-1} \left(\sum_{j=0}^\infty \vert \bar{\hat\Psi}_j(q)\vert_F^2\right)^{a/2} \ensuremath{{\mathbb E}}^*\left(\vert \ensuremath{\varepsilon}_t^* \vert_F^a \right), \end{align*} for some constant $c$ and $\bar{\hat\Psi}_j(q) \coloneqq \sum_{i=j+1}^\infty \hat \Psi_i(q)$, where the matrices $\hat \Psi_j$ are determined by the power series expansion of the inverse of $I-\sum_{j=1}^q \hat \Phi_j(q)z^j$. As discussed in \citeasnoun[p.\,671]{PSU10} it follows that $\sum_{j=0}^\infty \vert \bar{\hat\Psi}_j(q)\vert_F^2=O_\ensuremath{{\mathbb P}}(1)$ if we can show that $\sum_{j=1}^q j^{1/2}\vert \hat\Psi_j(q)\vert_F = O_\ensuremath{{\mathbb P}}(1)$, which in turn holds if $\sum_{j=1}^q j^{1/2}\vert \hat\Phi_j(q)\vert_F = O_\ensuremath{{\mathbb P}}(1)$. Using again Lemma (ref), (ref) and (ref), we obtain \begin{align*} \sum_{j=1}^q j^{1/2}\vert \hat\Phi_j(q)\vert_F &\leq \sum_{j=1}^q j^{1/2}\vert \hat\Phi_j(q) - \tilde \Phi_j(q)\vert_F + \sum_{j=1}^q j^{1/2}\vert \tilde\Phi_j(q) - \Phi_j(q)\vert_F \\ & \quad + \sum_{j=1}^q j^{1/2}\vert \Phi_j(q) - \Phi_j\vert_F + \sum_{j=1}^q j^{1/2}\vert \Phi_j\vert_F\\ &\leq q^{1/2} \sum_{j=1}^q \vert \hat\Phi_j(q) - \tilde \Phi_j(q)\vert_F + q^{1/2}\sum_{j=1}^q \vert \tilde\Phi_j(q) - \Phi_j(q)\vert_F\\ &\quad + \sum_{j=1}^q (1+j)\vert \Phi_j(q) - \Phi_j\vert_F + \sum_{j=1}^q j^{1/2}\vert \Phi_j\vert_F\\ &\leq q^{1/2} \sum_{j=1}^q \vert \hat\Phi_j(q) - \tilde \Phi_j(q)\vert_F + q^{3/2} \sup_{1\leq j \leq q} \vert \tilde\Phi_j(q) - \Phi_j(q)\vert_F\\ &\quad + \sum_{j=1}^q (1+j)\vert \Phi_j(q) - \Phi_j\vert_F + \sum_{j=1}^q j^{1/2}\vert \Phi_j\vert_F\\ & = o_\ensuremath{{\mathbb P}}(1) + O_\ensuremath{{\mathbb P}}(1) + o(1) + O(1) = O_\ensuremath{{\mathbb P}}(1). \end{align*} This completes the proof, since $\ensuremath{{\mathbb E}}^*\left(\vert \ensuremath{\varepsilon}_t^* \vert_F^a \right)= O_\ensuremath{{\mathbb P}}(1)$ by Lemma (ref) for the $a>2$ from Assumption (ref). $\square$\\ {\bf Proof of Theorem (ref).} The first result in Theorem (ref) follows from the bootstrap invariance principle result in Theorem (ref) and similar arguments as used in \citeasnoun[Proof of Theorem 2]{VoWa14}. The second result then follows from the bootstrap invariance principle and similar arguments as used in the proof of Theorem (ref). $\square$ \section{Proofs of Auxiliary Results} {0mm} {\bf Proof of Lemma (ref).} The solution of the sample Yule-Walker equations in the regression of $w_t$ on $w_{t-1},\ldots,w_{t-q}$, $t=q+1,\ldots,T$, can be written in compact form as a ($(m+1)\times q(m+1)$)-dimensional matrix \begin{align*} \boldsymbol{\tilde \Phi}(q)\coloneqq [\tilde \Phi_1(q),\ldots,\tilde \Phi_q(q)] = \tilde \Gamma \tilde G^{-1}, \end{align*} with the ($(m+1)\times q(m+1)$)-dimensional matrix $\tilde \Gamma \coloneqq \left[\tilde \Gamma(1),\ldots,\tilde\Gamma(q)\right]$, the ($q(m+1)\times q(m+1)$)-dimensional matrix $\tilde G \coloneqq \left(\tilde \Gamma(s-r) \right)_{r,s=1,\ldots,q}$ and the ($(m+1)\times (m+1)$)-dimensional empirical autocovariance matrix of $w_1,\ldots,w_T$ at lag $-q+1\leq h \leq q$, given by \begin{align*} \tilde \Gamma(h) \coloneqq T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(w_{t+h} - \bar{w}_T)(w_t - \bar{w}_T)', \end{align*} where $\bar{w}_T\coloneqq T^{-1}\sum_{t=1}^T w_t$. Analogously, the solution of the sample Yule-Walker equations in the regression of $\hat w_t$ on $\hat w_{t-1},\ldots,\hat w_{t-q}$, $t=q+1,\ldots,T$, can be written in compact form as \begin{align*} \boldsymbol{\hat \Phi}(q)\coloneqq [\hat \Phi_1(q),\ldots,\hat \Phi_q(q)] = \hat \Gamma \hat G^{-1}, \end{align*} with $\hat \Gamma \coloneqq \left[\hat \Gamma(1),\ldots,\hat\Gamma(q)\right]$, $\hat G \coloneqq \left(\hat \Gamma(s-r) \right)_{r,s=1,\ldots,q}$ and $\hat \Gamma(h)$ the empirical autocovariance matrix of $\hat w_1,\ldots,\hat w_T$ at lag $-q+1\leq h \leq q$. Taking the difference of $\boldsymbol{\tilde \Phi}(q)$ and $\boldsymbol{\hat \Phi}(q)$, adding and subtracting $\hat \Gamma \tilde G^{-1}$ and using \begin{align*} \tilde G^{-1} - \hat G^{-1}=\tilde G^{-1}(\hat G - \tilde G)\hat G^{-1} \end{align*} leads to \begin{align*} \boldsymbol{\hat\Phi}(q) - \boldsymbol{\tilde\Phi}(q) = \hat \Gamma \tilde G^{-1}(\tilde G - \hat G)\hat G^{-1} - (\tilde \Gamma - \hat \Gamma) \tilde G^{-1} \end{align*} Hence, we have to consider $\tilde G - \hat G$ in more detail ($\tilde \Gamma - \hat \Gamma$ works similarly). A typical block element of $\tilde G - \hat G$ is \begin{align*} \tilde \Gamma (h) - \hat \Gamma(h) & = T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(w_{t+h} - \bar{w}_T)(w_t - \bar{w}_T)'\\ &\quad -T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(\hat w_{t+h} - \bar{\hat w}_T)(\hat w_t - \bar{\hat w}_T)' \\ & = T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(w_{t+h} - \bar{w}_T)(w_t - \hat w_t - (\bar{w}_T- \bar{\hat w}_T))' \\ & \quad - T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(\hat w_{t+h} - w_{t+h} - (\bar{\hat w}_T - \bar{w}_T))(\hat w_t - \bar{\hat w}_T)' \\ &= A_1(h)-A_2(h), \end{align*} with an obvious definition for $A_1(h)$ and $A_2(h)$. Let us consider $A_1(h)$ in more detail ($A_2(h)$ works similarly). Using $\hat w_t=[\hat u_t,v_t']'$ and $\hat u_t = y_t - x_t'\hat\beta_{\scriptstyle\text{IM}}$ together with the model equations (ref) and (ref), we get \begin{align} w_t-\hat w_t=\begin{bmatrix} u_t-\hat u_t \\ 0_{m\times 1} \end{bmatrix}=\begin{bmatrix} y_t - x_t'\beta-(y_t - x_t'\hat\beta_{\text{IM}}) \\ 0_{m\times 1} \end{bmatrix}=\begin{bmatrix} x_t'(\hat\beta_{\text{IM}}-\beta) \\ 0_{m\times 1} \end{bmatrix}. \end{align} Before we continue, note that the last equality implies \begin{align*} \max_{1\leq t \leq T}\vert \hat w_t - w_t \vert_F \leq T^{-1/2}\max_{1\leq t \leq T}\vert T^{-1/2} x_t \vert_F\vert T\left(\hat\beta_{\text{IM}} - \beta\right)\vert_F = O_\ensuremath{{\mathbb P}}(T^{-1/2}), \end{align*} since $\max_{1\leq t \leq T}\vert T^{-1/2} x_t \vert_F=\sup_{0\leq r \leq 1}\vert T^{-1/2} x_{\lfloor rT \rfloor}\vert_{F}$ converges by Assumption (ref) and the continuous mapping theorem to $\sup_{0\leq r \leq 1}\vert B_v(r)\vert_{F}=O_\ensuremath{{\mathbb P}}(1)$ and $\hat\beta_{\scriptstyle\text{IM}}$ is rate-$T$ consistent. This proves (ref). We now proceed with the proof of (ref). From (ref) we obtain \begin{align*} A_1(h) & = T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(w_{t+h} - \bar{w}_T)\begin{bmatrix} (x_t-\bar{x}_T)'(\hat\beta_{\text{IM}}-\beta) \\ 0_{m\times 1} \end{bmatrix}' \\ & = T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(w_{t+h} - \bar{w}_T)[(x_t-\bar{x}_T)'(\hat\beta_{\text{IM}}-\beta),0_{1\times m}] \\ & = \tilde\Gamma_{w,x}(h)(\hat\beta_{\text{IM}}-\beta)e_1', \end{align*} where $\tilde\Gamma_{w,x}(h)\coloneqq T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(w_{t+h} - \bar{w}_T)(x_t-\bar{x}_T)'$ is ($(m+1)\times m$)-dimensional and $e_1\coloneqq(1,0_{1\times m})'$ is the first $(m+1)$-dimensional unit vector. Denoting the part of $\tilde G - \hat G$ that consists of block-entries $A_1(h)$ by $\tilde G_1 - \hat G_1$, we get \begin{align*} \tilde G_1 - \hat G_1 = \begin{bmatrix} \tilde\Gamma_{w,x}(s-r)(\hat\beta_{\text{IM}}-\beta)e_1' \end{bmatrix}_{r,s=1,\ldots,q} = \tilde\Gamma_{w,x} (I_q\otimes ((\hat\beta_{\text{IM}}-\beta)e_1')), \end{align*} where $\tilde\Gamma_{w,x}\coloneqq (\tilde\Gamma_{w,x}(s-r))_{r,s=1,\ldots,q}$ is ($q(m+1)\times qm$)-dimensional. For the second factor we have \begin{align*} \vert I_q\otimes ((\hat\beta_{\text{IM}}-\beta)e_1')\vert_F & = \sqrt{\text{tr}\left((I_q\otimes ((\hat\beta_{\text{IM}}-\beta)e_1'))'I_q\otimes ((\hat\beta_{\text{IM}}-\beta)e_1')\right)} \\ & = \sqrt{\text{tr}\left(I_q\otimes (e_1(\hat\beta_{\text{IM}}-\beta)'(\hat\beta_{\text{IM}}-\beta)e_1'\right))} \\ & = \sqrt{\text{tr}\left(I_q\otimes \text{diag}\left(\sum_{i=1}^m (\hat\beta_{\text{IM},i}-\beta_i)^2,0,\ldots,0\right)\right)} \\ & = \sqrt{q\sum_{i=1}^m (\hat\beta_{\text{IM},i}-\beta_i)^2} \\ & = q^{1/2}T^{-1} \vert T(\hat\beta_{\text{IM}}-\beta)\vert_F \\ & = O_P(q^{1/2}T^{-1}). \end{align*} Next, let us consider $\tilde\Gamma_{w,x}$ in more detail. Recall that $w_t=(u_t,v_t')'$ and $x_t=\sum_{k=1}^tv_k$. To avoid lengthy index notation, w.l.o.g. we can assume that $m=1$ and consider the second element of $w_t$ only (the first element works similarly). We thus consider the scalar quantity \begin{align*} \tilde\Gamma_{w,x}(h)=T^{-1} \sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}(v_{t+h} - \bar{v}_T)(\sum_{k=1}^t v_k-T^{-1}\sum_{i=1}^T\sum_{j=1}^iv_j). \end{align*} Taking the expectation of the squared Frobenius norm of the corresponding ($q\times q$)-dimensional matrix $\tilde\Gamma_{w,x}$ and combining the sums over $r$ and $s$, leads to \begin{align*} & \ensuremath{{\mathbb E}}(\vert\tilde\Gamma_{w,x}\vert_F^2) \\ &= \sum_{r,s=1}^q T^{-2} \sum_{t_1,t_2=\max\{1,1-(s-r)\}}^{\min\{T,T-(s-r)\}}\ensuremath{{\mathbb E}}\left[(v_{t_1+s-r} - \bar{v}_T)(\sum_{k_1=1}^{t_1} v_{k_1}-T^{-1}\sum_{i_1=1}^T\sum_{j_1=1}^{i_1}v_{j_1})\right. \\ & \qquad\qquad\qquad\qquad\qquad\times\left.(v_{t_2+s-r} - \bar{v}_T)(\sum_{k_2=1}^{t_2} v_{k_2}-T^{-1}\sum_{i_2=1}^T\sum_{j_2=1}^{i_2}v_{j_2})\right] \\ &= T^{-2}\sum_{h=-q+1}^{q-1}(q-\vert h\vert)\sum_{t_1,t_2=\max\{1,1-h\}}^{\min\{T,T-h\}}\ensuremath{{\mathbb E}}\left[(v_{t_1+h} - \bar{v}_T)(\sum_{k_1=1}^{t_1} v_{k_1}-T^{-1}\sum_{i_1=1}^T\sum_{j_1=1}^{i_1}v_{j_1})\right. \\ & \qquad\qquad\qquad\qquad\qquad\times\left.(v_{t_2+h} - \bar{v}_T)(\sum_{k_2=1}^{t_2} v_{k_2}-T^{-1}\sum_{i_2=1}^T\sum_{j_2=1}^{i_2}v_{j_2})\right]. \end{align*} Note that the last expectation is of the form $\ensuremath{{\mathbb E}}(ABCD)$ with $\ensuremath{{\mathbb E}}(A)=\ensuremath{{\mathbb E}}(B)=\ensuremath{{\mathbb E}}(C)=\ensuremath{{\mathbb E}}(D)=0$. Hence, by using common rules for joint cumulants of centered random variables \citeaffixed{Br81}{see, \mbox{e.\,g.}\xspace,}, we get \begin{align*} & \ensuremath{{\mathbb E}}(ABCD) \\ & = \text{cum}(A,B,C,D)+\ensuremath{{\mathbb E}}(AB)\ensuremath{{\mathbb E}}(CD)+\ensuremath{{\mathbb E}}(AC)\ensuremath{{\mathbb E}}(BD)+\ensuremath{{\mathbb E}}(AD)\ensuremath{{\mathbb E}}(BC) \\ & = \text{cum}(A,B,C,D)+\text{Cov}(A,B)\text{Cov}(C,D)+\text{Cov}(A,C)\text{Cov}(B,D)\\ & \quad +\text{Cov}(A,D)\text{Cov}(B,C), \end{align*} where $\text{Cov}(\cdot,\cdot)$ denotes the covariance of two random variables. Hence, the first term corresponding to the fourth-order cumulant becomes \begin{align*} & T^{-2}\sum_{h=-q+1}^{q-1}(q-\vert h\vert) \sum_{t_1,t_2=\max\{1,1-h\}}^{\min\{T,T-h\}}\text{cum}\left(v_{t_1+h} - \bar{v}_T,\sum_{k_1=1}^{t_1} v_{k_1}-T^{-1}\sum_{i_1=1}^T\sum_{j_1=1}^{i_1}v_{j_1},\right. \\ & \qquad\qquad\qquad\qquad\qquad\left.v_{t_2+h} - \bar{v}_T,\sum_{k_2=1}^{t_2} v_{k_2}-T^{-1}\sum_{i_2=1}^T\sum_{j_2=1}^{i_2}v_{j_2}\right) \end{align*} leading to $2^4=16$ terms when exapnding the cumulant. Exemplarily, for the absolute value of the first one (the others work similarly), we get from the common calculation rules for cumulants \begin{align*} & \vert T^{-2}\sum_{h=-q+1}^{q-1}(q-\vert h\vert) \sum_{t_1,t_2=\max\{1,1-h\}}^{\min\{T,T-h\}}\text{cum}\left(v_{t_1+h},\sum_{k_1=1}^{t_1} v_{k_1},v_{t_2+h},\sum_{k_2=1}^{t_2} v_{k_2}\right)\vert \\ &\leq T^{-2}\sum_{h=-q+1}^{q-1}\vert q-\vert h\vert\vert \sum_{t_1,t_2=\max\{1,1-h\}}^{\min\{T,T-h\}}\sum_{k_1=1}^{t_1}\sum_{k_2=1}^{t_2} \vert \text{cum}\left(v_{t_1+h},v_{k_1},v_{t_2+h},v_{k_2}\right)\vert \\ &\leq \frac{q}{T^2}\sum_{h=-q+1}^{q-1} \sum_{t_1,t_2=1}^{T}\sum_{k_1,k_2=1}^T \vert \text{cum}\left(v_{t_1+h},v_{k_1},v_{t_2+h},v_{k_2}\right)\vert. \end{align*} By combining the sums over $t_1$ and $t_2$ and those over $k_1$ and $k_2$, respectively, the above term becomes \begin{align*} & \frac{q}{T^2}\sum_{h=-q+1}^{q-1} \sum_{l=-(T-1)}^{T-1}\sum_{s=\max\{1,1-l\}}^{\min\{T,T-l\}}\sum_{i=-(T-1)}^{T-1}\sum_{j=\max\{1,1-i\}}^{\min\{T,T-i\}}\vert \text{cum}\left(v_{s+l+h},v_{j+i},v_{s+h},v_{j}\right)\vert \\ &\leq \frac{q}{T^2}\sum_{h=-q+1}^{q-1} \sum_{i,l=-(T-1)}^{T-1}\sum_{s,j=1}^T\vert \text{cum}\left(v_{s+l+h},v_{j+i},v_{s+h},v_{j}\right)\vert \\ &\leq \frac{q}{T^2}\sum_{h=-q+1}^{q-1} \sum_{i,l=-(T-1)}^{T-1} \sum_{k=-(T-1)}^{T-1}\sum_{r=\max\{1,1-k\}}^{\min\{T,T-k\}}\vert \text{cum}\left(v_{r+k+l+h},v_{r+i},v_{r+k+h},v_{r}\right)\vert \\ &\leq \frac{q}{T}\sum_{h=-q+1}^{q-1} \sum_{i,l=-(T-1)}^{T-1} \sum_{k=-(T-1)}^{T-1}\vert \text{cum}\left(v_{k+l+h},v_{i},v_{k+h},v_{0}\right)\vert, \end{align*} where we also combined the sums over $s$ and $j$ and made use of the (strict) stationarity of $\{v_t\}_{t\in\mathbb{Z}}$. Finally, combining the sums over $h$ and $k$, we get the bound \begin{align*} & \frac{q}{T}\sum_{r=-(T+q-2)}^{T+q-2} (2q-1) \sum_{i,l=-(T-1)}^{T-1} \vert \text{cum}\left(v_{r+l},v_{i},v_{r},v_{0}\right)\vert \\ &\leq 2\frac{q^2}{T} \sum_{j=-(2T+q-3)}^{2T+q-3}\sum_{r=-(T+q-2)}^{T+q-2} \sum_{i=-(T-1)}^{T-1} \vert \text{cum}\left(v_{j},v_{i},v_{r},v_{0}\right)\vert\\ &\leq 2\frac{q^2}{T}\sum_{j,i,r=-\infty}^{\infty} \vert \text{cum}\left(v_{j},v_{i},v_{r},v_{0}\right)\vert=O(q^2 T^{-1}) \end{align*} due to the summability condition imposed on the fourth order cumulants in Assumption (ref) and, hence, vanishes for $T\rightarrow \infty$. However, the leading term is the term corresponding to $\text{Cov}(A,B)\text{Cov}(C,D)$. That is, we have to consider \begin{align*} & T^{-2}\sum_{h=-q+1}^{q-1}(q-\vert h\vert)\sum_{t_1,t_2=\max\{1,1-h\}}^{\min\{T,T-h\}}\text{Cov}(v_{t_1+h} - \bar{v}_T,\sum_{k_1=1}^{t_1} v_{k_1}-T^{-1}\sum_{i_1=1}^T\sum_{j_1=1}^{i_1}v_{j_1}) \\ & \qquad\qquad\qquad\qquad\qquad\times \text{Cov}(v_{t_2+h} - \bar{v}_T,\sum_{k_2=1}^{t_2} v_{k_2}-T^{-1}\sum_{i_2=1}^T\sum_{j_2=1}^{i_2}v_{j_2}) \\ &= \sum_{h=-q+1}^{q-1}(q-\vert h\vert) \left(T^{-1}\sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}\text{Cov}(v_{t+h} - \bar{v}_T,\sum_{k=1}^{t} v_{k}-T^{-1}\sum_{i=1}^T\sum_{j=1}^{i}v_{j})\right)^2. \end{align*} Hence, we have to compute \begin{align*} T^{-1}\sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}\text{Cov}(v_{t+h} - \bar{v}_T,\sum_{k=1}^{t} v_{k}-T^{-1}\sum_{i=1}^T\sum_{j=1}^{i}v_{j}). \end{align*} This leads to four terms to consider, which can be treated with similar arguments. For the first term we get \begin{align*} T^{-1}\sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}\text{Cov}(v_{t+h},\sum_{k=1}^{t} v_{k}) = T^{-1}\sum_{t=\max\{1,1-h\}}^{\min\{T,T-h\}}\sum_{k=1}^{t}\gamma_v(t+h-k), \end{align*} where $\gamma_v(h)$ is the covariance of the one-dimensional process (still assumed for notational brevity) $\{v_t\}_{t\in\ensuremath{{\mathbb Z}}}$ at lag $h$. \mbox{E.\,g.}\xspace, for $h\geq 0$, this can be exactly computed to equal \begin{align*} T^{-1}\sum_{t=1}^{T-h}\sum_{k=1}^{t}\gamma_v(t+h-k)=T^{-1}\sum_{j=h}^{T-1}(T-j)\gamma_v(j)\leq \sum_{j=h}^{T-1}\gamma_v(j) \end{align*} and its absolute vale can be bounded by $\sum_{j=-\infty}^{\infty}\vert \gamma_v(j)\vert <\infty$ due to the second-order cumulant condition imposed in Assumption (ref). Similar arguments yield the same bound for $h<0$ and for the other three terms. Hence, the term of $\ensuremath{{\mathbb E}}(\vert \tilde\Gamma_{w,x}\vert_F^2)$ that corresponds to $\text{Cov}(A,B)\text{Cov}(C,D)$ is of order $O(q^2)$. In total we thus have $\ensuremath{{\mathbb E}}(\vert \tilde\Gamma_{w,x}\vert_F^2)=O(q^2)$. Note that we have proven this result for $m=1$. However, as $m$ is fixed, the result also holds for $m>1$. Therefore, we obtain for the ($q(m+1)\times qm$)-dimensional matrix $\tilde\Gamma_{w,x}$ that $\vert \tilde\Gamma_{w,x}\vert_F=O_\ensuremath{{\mathbb P}}(q)$. It follows that $\vert \tilde G_1-\hat G_1\vert_F=O_\ensuremath{{\mathbb P}}(q)O_P(q^{1/2}T^{-1})=O_\ensuremath{{\mathbb P}}(q^{3/2}T^{-1})$ and similarly also $\vert \tilde G-\hat G\vert_F=O_\ensuremath{{\mathbb P}}(q)O_\ensuremath{{\mathbb P}}(q^{1/2}T^{-1})=O_\ensuremath{{\mathbb P}}(q^{3/2}T^{-1})$. Further, we have to consider $\tilde G^{-1}$. In the following, let $\mu_{\min}(A)$ and $\mu_{\max}(A)$ denote the smallest and largest eigenvalue of a matrix $A$, respectively and define $G \coloneqq \left( \Gamma(s-r) \right)_{r,s=1,\ldots,q}\in\ensuremath{{\mathbb R}}^{q(m+1)\times q(m+1)}$. Similar to the above, using the fourth-order cumulant condition from Assumption (ref), we can show that $\vert \tilde G-G\vert_F=O_\ensuremath{{\mathbb P}}(qT^{-1/2})=o_\ensuremath{{\mathbb P}}(1)$. Then, to show boundedness in probability of $\tilde G^{-1}$ (similar for $\hat G^{-1}$), for all $\epsilon>0$, we have to find a $K<\infty$ and a $T_0<\infty$ both large enough such that for all $T>T_0$, it holds that \begin{align*} \ensuremath{{\mathbb P}}(\vert \tilde G^{-1}\vert_2>K)<\epsilon, \end{align*} where $\vert A \vert_2$ denotes the spectral norm of a matrix $A$. Let $\epsilon>0$. Then, due to positive semi-definiteness of $\tilde G$ by construction and invertibility, see \citeasnoun[Lemma 3.4 and Remark 3.2]{MeKr15}, we have positive definiteness of $\tilde G$ and, consequently, of $\tilde G^{-1}$. Hence, we get \begin{align*} \ensuremath{{\mathbb P}}(\vert \tilde G^{-1}\vert_2>K) =&\ensuremath{{\mathbb P}}(\mu_{\max}(\tilde G^{-1})>K)=\ensuremath{{\mathbb P}}(\mu_{\min}^{-1}(\tilde G)>K) \\ =& \ensuremath{{\mathbb P}}(\mu_{\min}(\tilde G)<\frac{1}{K},\vert \tilde G-G\vert_2\geq \delta) \\ &+ \ensuremath{{\mathbb P}}(\mu_{\min}(\tilde G)<\frac{1}{K},\vert \tilde G-G\vert_2<\delta). \end{align*} Further, as $\vert \tilde G-G\vert_2 \leq \vert \tilde G-G\vert_F=o_\ensuremath{{\mathbb P}}(1)$, for any $\delta>0$, we can choose $T_0$ large enough to have $\ensuremath{{\mathbb P}}(\vert \tilde G-G\vert_2\geq \delta)\leq \epsilon$. Then the first term on the last right-hand side can be bounded by $\ensuremath{{\mathbb P}}(\vert \tilde G-G\vert_2\geq \delta)\leq\epsilon$. For the second term, as $\tilde G$, $G$ and hence also $\tilde G-G$ are symmetric with real-valued entries, these matrices are Hermitian such that Weyl's theorem \citeaffixed{HoJo12}{see, \mbox{e.\,g.}\xspace, Theorem 4.3.1 in} applies, leading to the inequality $\mu_{\min}(G)+\mu_{\min}(\tilde G-G)\leq\mu_{\min}(\tilde G)$. It follows that \begin{align*} & \ensuremath{{\mathbb P}}(\mu_{\min}(\tilde G)<\frac{1}{K},\vert \tilde G-G\vert_2<\delta) \\ &\leq \ensuremath{{\mathbb P}}(\mu_{\min}(G)+\mu_{\min}(\tilde G-G)<\frac{1}{K},\vert \tilde G-G\vert_2<\delta) \\ &= \ensuremath{{\mathbb P}}(\mu_{\min}(G)<\frac{1}{K}-\mu_{\min}(\tilde G-G),\vert \tilde G-G\vert_2<\delta). \end{align*} From symmetry of $\tilde G-G$ we get that the eigenvalues of $(\tilde G-G)'(\tilde G-G)$ are exactly the squared eigenvalues of $\tilde G-G$. Hence, the bound \begin{align*} \vert \tilde G-G\vert_2=\sqrt{\mu_{\max}((\tilde G-G)'(\tilde G-G))}<\delta \end{align*} implies also $\mu_{\min}(\tilde G-G)\geq -\delta$ such that the last right-hand side can be bounded by \begin{align} \ensuremath{{\mathbb P}}(\mu_{\min}(G)<\frac{1}{K}+\delta,\vert \tilde G-G\vert_2<\delta)\leq \ensuremath{{\mathbb P}}(\mu_{\min}(G)<\frac{1}{K}+\delta). \end{align} Next, note that $\mu_{\min}(G)\geq \tilde c$ for some constant $\tilde c>0$ by Assumption (ref). Therefore, the right-hand side in (ref) becomes zero if we choose $\delta<\tilde c/2$ small enough and $K>2/\tilde c$ large enough, such that $\frac{1}{K}+\delta<\tilde c$. This completes the proof of $\vert \tilde G^{-1}\vert_2=O_\ensuremath{{\mathbb P}}(1)$. Furthermore, from Assumption (ref) we also get $\vert \tilde G\vert_2=O_\ensuremath{{\mathbb P}}(1)$ and similarly $\vert \hat G\vert_2=O_\ensuremath{{\mathbb P}}(1)$ and $\vert \hat G\vert_2=O_\ensuremath{{\mathbb P}}(1)$. Altogether, we get \begin{align*} &\vert \boldsymbol{\tilde \Phi}(q) - \boldsymbol{\hat \Phi}(q)\vert_2 \leq \vert \tilde \Gamma - \hat \Gamma\vert_2 \vert \tilde G^{-1}\vert_2 + \vert \hat \Gamma\vert_2 \vert \tilde G^{-1}\vert_2\vert \tilde G - \hat G\vert_2\vert \hat G^{-1}\vert_2 \\ &\leq \vert \tilde G - \hat G\vert_2 \left(\vert \tilde G^{-1}\vert_2 + \vert \hat G\vert_2 \vert \tilde G^{-1}\vert_2\vert \hat G^{-1}\vert_2\right) \\ &\leq \vert \tilde G - \hat G\vert _{F} \left(\vert \tilde G^{-1}\vert_2 + \vert \hat G\vert_2 \vert \tilde G^{-1}\vert_2\vert \hat G^{-1}\vert_2\right) \\ &= O_\ensuremath{{\mathbb P}}(q^{3/2}T^{-1})\left(O_\ensuremath{{\mathbb P}}(1)+O_\ensuremath{{\mathbb P}}(1)O_\ensuremath{{\mathbb P}}(1)O_\ensuremath{{\mathbb P}}(1)\right) \\ &= O_\ensuremath{{\mathbb P}}(q^{3/2}T^{-1}). \end{align*} Since $\boldsymbol{\tilde \Phi}(q) - \boldsymbol{\hat \Phi}(q)$ is ($(m+1)\times q(m+1)$)-dimensional and $m$ is fixed, it holds that \citeaffixed{Ge07}{see, \mbox{e.\,g.}\xspace} \begin{align*} \vert \boldsymbol{\tilde \Phi}(q) - \boldsymbol{\hat \Phi}(q)\vert _{F} \leq \sqrt{m+1}\ \vert \boldsymbol{\tilde \Phi}(q) - \boldsymbol{\hat \Phi}(q)\vert_2 = O_\ensuremath{{\mathbb P}}(q^{3/2}T^{-1}). \end{align*} This implies that \begin{align*} q^{1/2}\sum_{j=1}^q \vert \hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F \leq q^{3/2} \vert \boldsymbol{\tilde \Phi}(q) - \boldsymbol{\hat \Phi}(q)\vert _{F} = O_\ensuremath{{\mathbb P}}(q^3T^{-1}), \end{align*} which is $o_\ensuremath{{\mathbb P}}(1)$ since $q^{3}T^{-1}= o(1)$ by Assumption (ref). $\square$ In the proofs of Lemma (ref) and (ref) we repeatedly use the fact that by convexity, $\vert \sum_{i=1}^k z_i \vert^a \leq k^{a-1} \sum_{i=1}^k \vert z_i\vert^a$, for all $a,k\geq 1$.\\ {\bf Proof of Lemma (ref).} Let $\tilde \ensuremath{\varepsilon}_t(q) \coloneqq w_t - \sum_{j=1}^q \tilde \Phi_j(q) w_{t-j}$, $t=q+1,\ldots,T$, denote the Yule-Walker residuals in the regression of $w_t$ on $w_{t-1},\ldots,w_{t-q}$, $t=q+1,\ldots,T$ and define $\bar{\tilde{\ensuremath{\varepsilon}}}_T(q)\coloneqq (T-q)^{-1}\sum_{t=q+1}^T \tilde{\ensuremath{\varepsilon}}_{t}(q)$.\footnote{In the following, $a$ denotes the fixed $a>2$ from Assumption (ref). However, the results also hold for $1\leq \tilde a < a$.} For $q+1\leq t \leq T$ we have \begin{align*} &\vert \hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\vert_F^a\\ &\leq \left(\vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q)\vert_F + \vert \tilde \ensuremath{\varepsilon}_t(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F + \vert \bar{\hat{\ensuremath{\varepsilon}}}_T(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F \right)^a\\ & \leq \left(\vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q)\vert_F + \vert \tilde \ensuremath{\varepsilon}_t(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F + (T-q)^{-1} \sum_{t=q+1}^T \vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) \vert_F \right)^a\\ & \leq 3^{a-1} \left(\vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q)\vert_F^a + \vert \tilde \ensuremath{\varepsilon}_t(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a + \left((T-q)^{-1} \sum_{t=q+1}^T \vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) \vert_F\right)^a \right). \end{align*} Hence, \begin{align*} &(T-q)^{-1} \sum_{t=q+1}^T \vert \hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\vert_F^a\\ &\leq 3^{a-1}\left((T-q)^{-1} \sum_{t=q+1}^T \vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) \vert_F^a + (T-q)^{-1} \sum_{t=q+1}^T \vert \tilde \ensuremath{\varepsilon}_t(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a\right. \\ & + \left.\left((T-q)^{-1} \sum_{t=q+1}^T \vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) \vert_F\right)^a\right)\\ & = 3^{a-1}\left(F_{T,a} + (T-q)^{-1} \sum_{t=q+1}^T \vert \tilde \ensuremath{\varepsilon}_t(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a + (F_{T,1})^a\right), \end{align*} where $F_{T,a}\coloneqq (T-q)^{-1} \sum_{t=q+1}^T \vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) \vert_F^a$. We now consider $F_{T,a}$ in more detail. For $q+1\leq t \leq T$ we have \begin{align*} &\vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) \vert_F\\ &= \vert\hat w_t - \sum_{j=1}^q \hat \Phi_j(q) \hat w_{t-j} - (w_t - \sum_{j=1}^q \tilde \Phi_j(q) w_{t-j})\vert_F\\ & = \vert\hat w_t - w_t - \sum_{j=1}^q \left( \hat \Phi_j(q) - \tilde \Phi_j(q) + \tilde \Phi_j(q)\right)\left(\hat w_{t-j} - w_{t-j} + w_{t-j}\right) + \sum_{j=1}^q \tilde \Phi_j(q) w_{t-j}\vert_F\\ & = \vert\hat w_t - w_t - \sum_{j=1}^q \left( \hat \Phi_j(q) - \tilde \Phi_j(q)\right)\left(\hat w_{t-j} - w_{t-j} + w_{t-j}\right) - \sum_{j=1}^q \tilde \Phi_j(q) \left(\hat w_{t-j} - w_{t-j}\right)\vert_F\\ & \leq \max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F + \sum_{j=1}^q \vert \hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F \vert \hat w_{t-j} - w_{t-j}\vert_F \\ & + \sum_{j=1}^q \vert\tilde \Phi_j(q)\vert_F \vert\hat w_{t-j} - w_{t-j}\vert_F + \sum_{j=1}^q \vert\hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F \vert w_{t-j}\vert_F\\ & \leq \max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F + \max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F \sum_{j=1}^q \vert \hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F\\ & + \max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F \sum_{j=1}^q \vert \tilde \Phi_j(q)\vert_F + \sqrt{m+1} q \vert \boldsymbol{\hat \Phi}(q) - \boldsymbol{\tilde \Phi}(q)\vert_F\ q^{-1} \sum_{j=1}^q \vert w_{t-j}\vert_F, \end{align*} where we have used that \begin{align*} \vert \hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F &\leq \vert \boldsymbol{\hat \Phi}(q) - \boldsymbol{\tilde \Phi}(q)\vert_F \vert [0,I_{m+1},0]'\vert_F\\ &= \sqrt{m+1}\vert \boldsymbol{\hat \Phi}(q) - \boldsymbol{\tilde \Phi}(q)\vert_{F}. \end{align*} Therefore, \begin{align*} &\vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) \vert_F^a\\ &\leq 4^{a-1} \left( \left(\max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F\right)^a + \left(\max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F\right)^a \left( \sum_{j=1}^q \vert\hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F \right)^a\right.\\ &\left. + \left(\max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F\right)^a \left(\sum_{j=1}^q \vert \tilde \Phi_j(q)\vert_F\right)^a \right.\\ & \left.+ \left(\sqrt{m+1} q \vert \boldsymbol{\hat \Phi}(q) - \boldsymbol{\tilde \Phi}(q)\vert_F\right)^a \left(q^{-1} \sum_{j=1}^q \vert w_{t-j}\vert_F\right)^a\right). \end{align*} It follows that \begin{align*} F_{T,a}& \leq 4^{a-1} \left( \left(\max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F\right)^a + \left(\max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F\right)^a \left( \sum_{j=1}^q \vert\hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F \right)^a\right.\\ &\left. + \left(\max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F\right)^a \left(\sum_{j=1}^q \vert \tilde \Phi_j(q)\vert_F\right)^a \right.\\ & \left. + \left(\sqrt{m+1} q \vert \boldsymbol{\hat \Phi}(q) - \boldsymbol{\tilde \Phi}(q)\vert_{F}\right)^a (T-q)^{-1} \sum_{t=q+1}^T \left( q^{-1} \sum_{j=1}^q \vert w_{t-j}\vert_F\right)^a\right). \end{align*} From Lemma (ref) we have $\max_{1\leq t \leq T} \vert\hat w_t - w_t\vert_F = O_\ensuremath{{\mathbb P}}(T^{-1/2})$ and $\sum_{j=1}^q \vert \hat \Phi_j(q) - \tilde \Phi_j(q)\vert_F= O_\ensuremath{{\mathbb P}}(q^{5/2}T^{-1})$. Moreover, the proof of Lemma (ref) shows that $q \vert \boldsymbol{\hat \Phi}(q) - \boldsymbol{\tilde \Phi}(q)\vert_{F}=O_\ensuremath{{\mathbb P}}(q^{5/2} T^{-1})$. Further, by (ref), (ref) and Assumption (ref) we have \begin{align*} \sum_{j=1}^q \vert \tilde \Phi_j(q)\vert_F &\leq \sum_{j=1}^q \vert \tilde \Phi_j(q) - \Phi_j(q)\vert_F + \sum_{j=1}^q \vert \Phi_j(q) - \Phi_j\vert_F + \sum_{j=1}^q \vert \Phi_j\vert_F\\ & \leq q \sup_{1\leq j \leq q} \vert \tilde \Phi_j(q) - \Phi_j(q)\vert_F + c \sum_{j=q+1}^\infty \vert \Phi_j\vert_F + \sum_{j=1}^\infty \vert \Phi_j\vert_F\\ & = O_\ensuremath{{\mathbb P}}(1). \end{align*} Finally, note that \begin{align*} (T-q)^{-1} \sum_{t=q+1}^T \left( q^{-1} \sum_{j=1}^q \vert w_{t-j}\vert_F\right)^a&\leq (T-q)^{-1} \sum_{t=q+1}^T q^{-a} q^{a-1} \sum_{j=1}^q \vert w_{t-j}\vert_F^a\\ & = (T-q)^{-1} q^{-1} \sum_{t=q+1}^T \sum_{j=1}^q \vert w_{t-j}\vert_F^a\\ & \leq (T-q)^{-1} \sum_{t=1}^{T-1} \vert w_t \vert_F^a, \end{align*} where the last inequality follows from the fact that each element in the double sum occurs at most $q$ times, \mbox{i.\,e.}\xspace, $\sum_{t=q+1}^T \sum_{j=1}^q \vert w_{t-j} \vert_F^a \leq q \sum_{t=1}^{T-1} \vert w_t \vert_F^a$. From $\sup_{t\in \ensuremath{{\mathbb Z}}}\ensuremath{{\mathbb E}}\left( \vert w_t \vert_F^a \right)< \infty$ (by stationarity of $\{w_t\}_{t\in \ensuremath{{\mathbb Z}}}$) and Markov's inequality, it follows that $(T-q)^{-1} \sum_{t=1}^{T-1} \vert w_t \vert_F^a=O_\ensuremath{{\mathbb P}}(1)$. In total, we thus have $F_{T,a} = O_{\ensuremath{{\mathbb P}}}((q^{5/2}T^{-1})^a)=o_{\ensuremath{{\mathbb P}}}(1)$. Therefore, \begin{align*} (T-q)^{-1} \sum_{t=q+1}^T \vert \hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\vert_F^a \leq 3^{a-1} \left((T-q)^{-1} \sum_{t=q+1}^T \vert \tilde \ensuremath{\varepsilon}_t(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a + o_\ensuremath{{\mathbb P}}(1)\right). \end{align*} It thus remains to show that $(T-q)^{-1} \sum_{t=q+1}^T \vert \tilde \ensuremath{\varepsilon}_t(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a = O_\ensuremath{{\mathbb P}}(1)$. We now follow \citeasnoun[Proof of Lemma 3.2]{Pa02} and \citeasnoun[Proof of Lemma 2]{PSU10}. Define $\ensuremath{\varepsilon}_t(q)\coloneqq \ensuremath{\varepsilon}_t + \sum_{j=q+1}^\infty \Phi_j w_{t-j}$ and note that \begin{align*} &(T-q)^{-1} \sum_{t=q+1}^T \vert \tilde \ensuremath{\varepsilon}_t(q) - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a \&= (T-q)^{-1} \sum_{t=q+1}^T \vert \tilde \ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t(q) + \ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t + \ensuremath{\varepsilon}_t - \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a\\ &\leq 4^{a-1}\left(A_{T,a}+B_{T,a}+C_{T,a}+D_{T,a}\right), \end{align*} where \begin{align*} A_{T,a} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T \vert \ensuremath{\varepsilon}_t \vert_F^a,\\ B_{T,a} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T \vert \ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t \vert_F^a = (T-q)^{-1} \sum_{t=q+1}^T \vert \sum_{j=q+1}^\infty \Phi_j w_{t-j} \vert_F^a,\\ C_{T,a} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T \vert \tilde\ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t(q) \vert_F^a,\\ D_{T,a} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T \vert \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a = \vert \bar{\tilde{\ensuremath{\varepsilon}}}_T(q) \vert_F^a = \vert (T-q)^{-1} \sum_{t=q+1}^T \tilde \ensuremath{\varepsilon}_t(q) \vert_F^a. \end{align*} We first consider $B_{T,a}$. Note that $\ensuremath{{\mathbb E}}\left(\vert B_{T,a}\vert_F\right) \leq \sup_{t\in\ensuremath{{\mathbb Z}}} \ensuremath{{\mathbb E}}\left(\vert \ensuremath{\varepsilon}_t(q)-\ensuremath{\varepsilon}_t\vert_F^a\right)$. Using Minkowski's inequality, we have \begin{align*} \ensuremath{{\mathbb E}}\left(\vert \ensuremath{\varepsilon}_t(q)-\ensuremath{\varepsilon}_t\vert_F^a\right) &= \ensuremath{{\mathbb E}}\left(\vert \sum_{j=q+1}^\infty \Phi_j w_{t-j} \vert_F^a\right) = \left(\left[ \ensuremath{{\mathbb E}}\left(\vert \sum_{j=q+1}^\infty \Phi_j w_{t-j} \vert_F^a\right) \right]^{1/a}\right)^a\\ &\leq \left( \sum_{j=q+1}^\infty \left[\ensuremath{{\mathbb E}}\left( \vert \Phi_j w_{t-j} \vert_F^a \right)\right]^{1/a} \right)^a \leq \left( \sum_{j=q+1}^\infty \left[\ensuremath{{\mathbb E}}\left( \vert \Phi_j \vert_F^a\vert w_{t-j} \vert_F^a \right)\right]^{1/a} \right)^a\\ &\leq \left( \sum_{j=q+1}^\infty \vert \Phi_j \vert_F \left[\ensuremath{{\mathbb E}}\left( \vert w_{t-j} \vert_F^a \right)\right]^{1/a} \right)^a \leq \left( \sum_{j=q+1}^\infty \vert \Phi_j \vert_F \left[\sup_{t\in \ensuremath{{\mathbb Z}}}\ensuremath{{\mathbb E}}\left( \vert w_t \vert_F^a \right)\right]^{1/a} \right)^a\\ & \leq \sup_{t\in \ensuremath{{\mathbb Z}}}\ensuremath{{\mathbb E}}\left( \vert w_t \vert_F^a \right) \left( \sum_{j=q+1}^\infty \vert \Phi_j \vert_F \right)^a. \end{align*} From Assumption (ref) we have $\sup_{t\in \ensuremath{{\mathbb Z}}}\ensuremath{{\mathbb E}}\left( \vert w_t \vert_F^a \right)<\infty$ and $\sum_{j=q+1}^\infty \vert \Phi_j \vert_F = o(1)$. Markov's inequality thus yields $B_{T,a}=o_\ensuremath{{\mathbb P}}(1)$. Analogously, $\ensuremath{{\mathbb E}}\left(\vert A_{T,a}\vert_F\right) \leq \sup_{t\in\ensuremath{{\mathbb Z}}} \ensuremath{{\mathbb E}}\left(\vert \ensuremath{\varepsilon}_t\vert_F^a\right)$. Using Minkowski's inequality, we have as above \begin{align*} \ensuremath{{\mathbb E}}\left(\vert \ensuremath{\varepsilon}_t\vert_F^a\right) &= \ensuremath{{\mathbb E}}\left(\vert w_t - \sum_{j=1}^\infty \Phi_j w_{t-j}\vert_F^a\right) \leq 2^{a-1}\left(\ensuremath{{\mathbb E}}\left(\vert w_t \vert_F^a \right)+ \ensuremath{{\mathbb E}}\left(\vert \sum_{j=1}^\infty \Phi_j w_{t-j}\vert_F^a\right)\right)\\ &\leq 2^{a-1}\left(\sup_{t\in\ensuremath{{\mathbb Z}}} \ensuremath{{\mathbb E}}\left(\vert w_t\vert_F^a\right)+ \ensuremath{{\mathbb E}}\left(\vert \sum_{j=1}^\infty \Phi_j w_{t-j}\vert_F^a\right)\right)\\ & \leq 2^{a-1} \sup_{t\in\ensuremath{{\mathbb Z}}} \ensuremath{{\mathbb E}}\left(\vert w_t\vert_F^a\right) \left(1+\left( \sum_{j=1}^\infty \vert \Phi_j \vert_F\right)^a\right) < \infty. \end{align*} Using Markov's inequality we conclude that $A_{T,a}=O_\ensuremath{{\mathbb P}}(1)$. We now turn to $C_{T,a}$. By definition, \begin{align*} \tilde \ensuremath{\varepsilon}_t(q) &= w_t - \sum_{j=1}^q \tilde \Phi_j(q) w_{t-j} = \ensuremath{\varepsilon}_t(q) - \sum_{j=1}^q \left(\tilde \Phi_j(q) - \Phi_j \right) w_{t-j}\\ &= \ensuremath{\varepsilon}_t(q) - \sum_{j=1}^q \left(\tilde \Phi_j(q) - \Phi_j(q) \right) w_{t-j} - \sum_{j=1}^q \left(\Phi_j(q) - \Phi_j \right) w_{t-j}. \end{align*} Hence, \begin{align*} \vert \tilde \ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t(q) \vert_F^a \leq 2^{a-1} \left( \vert \sum_{j=1}^q \left(\tilde \Phi_j(q) - \Phi_j(q) \right) w_{t-j} \vert_F^a + \vert \sum_{j=1}^q \left(\Phi_j(q) - \Phi_j \right) w_{t-j} \vert_F^a \right). \end{align*} It follows that $C_{T,a} = 2^{a-1} \left( C_{1T,a} + C_{2T,a}\right)$, where \begin{align*} C_{1T,a} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T \vert \sum_{j=1}^q \left(\tilde \Phi_j(q) - \Phi_j(q) \right) w_{t-j} \vert_F^a,\\ C_{2T,a} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T \vert \sum_{j=1}^q \left(\Phi_j(q) - \Phi_j \right) w_{t-j} \vert_F^a. \end{align*} We consider both terms separately. First note that \begin{align*} C_{1T,a} &\leq q^{a-1} (T-q)^{-1} \sum_{t=q+1}^T \sum_{j=1}^q \vert \tilde \Phi_j(q) - \Phi_j(q) \vert_F^a \vert w_{t-j} \vert_F^a\\ & \leq q^{a-1} \left( \sup_{1\leq j \leq q} \vert \tilde \Phi_j(q) - \Phi_j(q) \vert_F \right)^a (T-q)^{-1} \sum_{t=q+1}^T \sum_{j=1}^q \vert w_{t-j} \vert_F^a\\ & \leq \left( q \sup_{1\leq j \leq q} \vert \tilde \Phi_j(q) - \Phi_j(q) \vert_F \right)^a (T-q)^{-1} \sum_{t=1}^{T-1} \vert w_t \vert_F^a, \end{align*} where the third inequality follows again from the fact that $\sum_{t=q+1}^T \sum_{j=1}^q \vert w_{t-j} \vert_F^a \leq q \sum_{t=1}^{T-1} \vert w_t \vert_F^a$. As $(T-q)^{-1} \sum_{t=1}^{T-1} \vert w_t \vert_F^a=O_\ensuremath{{\mathbb P}}(1)$ it follows from (ref) that $C_{1T,a} = o_\ensuremath{{\mathbb P}}(1)$. Moreover, using Minkowski's inequality, we obtain \begin{align*} \ensuremath{{\mathbb E}}\left(\vert C_{2T,a} \vert_F\right) &= (T-q)^{-1} \sum_{t=q+1}^T \ensuremath{{\mathbb E}}\left( \vert \sum_{j=1}^q \left(\Phi_j(q) - \Phi_j \right) w_{t-j} \vert_F^a \right)\\ & = (T-q)^{-1} \sum_{t=q+1}^T \left(\left[\ensuremath{{\mathbb E}}\left( \vert \sum_{j=1}^q \left(\Phi_j(q) - \Phi_j \right) w_{t-j} \vert_F^a \right)\right]^{1/a}\right)^a\\ & \leq (T-q)^{-1} \sum_{t=q+1}^T \left(\sum_{j=1}^q \left[ \ensuremath{{\mathbb E}}\left( \vert \left(\Phi_j(q) - \Phi_j \right) w_{t-j} \vert_F^a \right) \right]^{1/a}\right)^a\\ & \leq (T-q)^{-1} \sum_{t=q+1}^T \sup_{t\in \ensuremath{{\mathbb Z}}}\ensuremath{{\mathbb E}}\left( \vert w_t \vert_F^a \right) \left( \sum_{j=1}^q \vert \Phi_j(q) - \Phi_j \vert_F \right)^a\\ & = \sup_{t\in \ensuremath{{\mathbb Z}}}\ensuremath{{\mathbb E}}\left( \vert w_t \vert_F^a \right) \left( \sum_{j=1}^q \vert \Phi_j(q) - \Phi_j \vert_F \right)^a. \end{align*} From (ref) and Markov's inequality it follows that $C_{2T,a} = o_\ensuremath{{\mathbb P}}(1)$. In total we thus have $C_{T,a} = o_\ensuremath{{\mathbb P}}(1)$. Finally, we consider $D_{T,a}$. It holds that $(T-q)^{-1} \sum_{t=q+1}^T \tilde \ensuremath{\varepsilon}_t(q) = D_{1T} + D_{2T} + D_{3T}$, where \begin{align*} D_{1T} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T \ensuremath{\varepsilon}_t,\\ D_{2T} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T (\ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t),\\ D_{3T} &\coloneqq (T-q)^{-1} \sum_{t=q+1}^T (\tilde\ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t(q)). \end{align*} By Chebyshev's weak law of large numbers Wh01, $D_{1T} \overset{p}{\longrightarrow} \ensuremath{{\mathbb E}}\left(\ensuremath{\varepsilon}_t\right) = 0$, \mbox{i.\,e.}\xspace, $D_{1T}=o_\ensuremath{{\mathbb P}}(1)$. Moreover, $\vert D_{2T}\vert_F\leq B_{T,1} = o_\ensuremath{{\mathbb P}}(1)$ and $\vert D_{3T} \vert_F \leq C_{T,1} = o_{\ensuremath{{\mathbb P}}}(1)$. By the continuous mapping theorem we thus have $D_T=o_\ensuremath{{\mathbb P}}(1)$. This completes the proof. $\square$\\ {\bf Proof of Lemma (ref).} It follows from Assumption (ref) that \begin{align*} \vert(T-q)^{-1} \sum_{t=q+1}^T \ensuremath{\varepsilon}_t\ensuremath{\varepsilon}_t' - \Sigma\vert_F = o_\ensuremath{{\mathbb P}}(1). \end{align*} Therefore, \begin{align*} \vert \ensuremath{{\mathbb E}}^*\left(\ensuremath{\varepsilon}_t^* \ensuremath{\varepsilon}_t^{*\prime}\right) - \Sigma\vert_F \leq \vert \ensuremath{{\mathbb E}}^*\left(\ensuremath{\varepsilon}_t^* \ensuremath{\varepsilon}_t^{*\prime}\right) - (T-q)^{-1} \sum_{t=q+1}^T \ensuremath{\varepsilon}_t\ensuremath{\varepsilon}_t'\vert_F + o_\ensuremath{{\mathbb P}}(1). \end{align*} Moreover, \begin{align*} &\vert \ensuremath{{\mathbb E}}^*\left(\ensuremath{\varepsilon}_t^* \ensuremath{\varepsilon}_t^{*\prime}\right) - (T-q)^{-1} \sum_{t=q+1}^T \ensuremath{\varepsilon}_t\ensuremath{\varepsilon}_t'\vert_F \&= \vert (T-q)^{-1} \sum_{t=q+1}^T \left(\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\right)\left(\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\right)' - \ensuremath{\varepsilon}_t\ensuremath{\varepsilon}_t'\vert_F \\ & = \vert(T-q)^{-1} \sum_{t=q+1}^T \left(\left[\left(\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\right) - \ensuremath{\varepsilon}_t\right]\left(\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\right)'\right. \\ & \left.+ \ensuremath{\varepsilon}_t \left[\left(\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\right)-\ensuremath{\varepsilon}_t\right]'\right)\vert_F\\ & \leq E_{1T} + E_{2T}, \end{align*} where \begin{align*} E_{1T} & = (T-q)^{-1} \sum_{t=q+1}^T \vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q) - \ensuremath{\varepsilon}_t\vert_F\vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\vert_F,\\ E_{2T} & = (T-q)^{-1} \sum_{t=q+1}^T \vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)-\ensuremath{\varepsilon}_t\vert_F\vert\ensuremath{\varepsilon}_t\vert_F. \end{align*} The Cauchy-Schwarz inequality yields \begin{align*} E_{1T} &\leq \left((T-q)^{-1} \sum_{t=q+1}^T \vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q) - \ensuremath{\varepsilon}_t\vert_F^2 (T-q)^{-1} \sum_{t=q+1}^T \vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\vert_F^2\right)^{1/2},\\ E_{2T} &\leq \left((T-q)^{-1} \sum_{t=q+1}^T \vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)-\ensuremath{\varepsilon}_t\vert_F^2 (T-q)^{-1} \sum_{t=q+1}^T \vert\ensuremath{\varepsilon}_t\vert_F^2\right)^{1/2}. \end{align*} From the proof of Lemma (ref) we have $(T-q)^{-1} \sum_{t=q+1}^T \vert\ensuremath{\varepsilon}_t\vert_F^2 = A_{T,2} = O_\ensuremath{{\mathbb P}}(1)$ and $(T-q)^{-1} \sum_{t=q+1}^T \vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\vert_F^2 = O_\ensuremath{{\mathbb P}}(1)$. It thus remains to show that $(T-q)^{-1} \sum_{t=q+1}^T \vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q) - \ensuremath{\varepsilon}_t\vert_F^2 = o_\ensuremath{{\mathbb P}}(1)$. To this end note that \begin{align*} &\vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q) - \ensuremath{\varepsilon}_t\vert_F\\ &\leq \vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q)\vert_F + \vert \tilde \ensuremath{\varepsilon}_t(q) - (w_t - \sum_{j=1}^q \Phi_j(q)w_{t-j})\vert_F\\ &\quad + \vert w_t - \sum_{j=1}^q \Phi_j(q)w_{t-j} - \ensuremath{\varepsilon}_t \vert_F + \vert \bar{\hat{\ensuremath{\varepsilon}}}_T(q) \vert_F\\ & = \vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q)\vert_F + \vert \sum_{j=1}^q \left(\tilde\Phi_j(q) - \Phi_j(q)\right)w_{t-j}\vert_F\\ &\quad + \vert \sum_{j=1}^q \left(\Phi_j(q) - \Phi_j\right)w_{t-j}\vert_F + \vert \sum_{j=q+1}^\infty \Phi_j w_{t-j}\vert_F + \vert \bar{\hat{\ensuremath{\varepsilon}}}_T(q)\vert_F. \end{align*} Hence, \begin{align*} &\vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q) - \ensuremath{\varepsilon}_t\vert_F^2\\ &\leq 5\left( \vert \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q)\vert_F^2 + \vert \sum_{j=1}^q \left(\tilde\Phi_j(q) - \Phi_j(q)\right)w_{t-j}\vert_F^2\right.\\ &\left. + \vert \sum_{j=1}^q \left(\Phi_j(q) - \Phi_j\right)w_{t-j}\vert_F^2 + \vert \sum_{j=q+1}^\infty \Phi_j w_{t-j}\vert_F^2 + \vert \bar{\hat{\ensuremath{\varepsilon}}}_T(q) \vert_F^2 \right). \end{align*} In the notation of the proof of Lemma (ref), we obtain \begin{align*} (T-q)^{-1} \sum_{t=q+1}^T \vert\hat \ensuremath{\varepsilon}_t(q) - \bar{\hat{\ensuremath{\varepsilon}}}_T(q) - \ensuremath{\varepsilon}_t\vert_F^2 &\leq 5\left( F_{T,2} + C_{1T,2} + C_{2T,2} + B_{T,2} + \vert \bar{\hat{\ensuremath{\varepsilon}}}_T(q) \vert_F^2 \right)\\ & = 5\vert \bar{\hat{\ensuremath{\varepsilon}}}_T(q) \vert_F^2 + o_\ensuremath{{\mathbb P}}(1). \end{align*} From $\hat \ensuremath{\varepsilon}_t(q) = \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) + \tilde \ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t(q) + \ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t + \ensuremath{\varepsilon}_t$, with $\ensuremath{\varepsilon}_t(q)$ as defined in the proof of Lemma (ref), it follows that \begin{align*} \vert \bar{\hat{\ensuremath{\varepsilon}}}_T(q) \vert_F &\leq \vert (T-q)^{-1} \sum_{t=q+1}^T \hat \ensuremath{\varepsilon}_t(q) - \tilde \ensuremath{\varepsilon}_t(q) \vert_F + \vert (T-q)^{-1} \sum_{t=q+1}^T \tilde \ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t(q) \vert_F\\ &\quad+ \vert (T-q)^{-1} \sum_{t=q+1}^T \ensuremath{\varepsilon}_t(q) - \ensuremath{\varepsilon}_t \vert_F + \vert (T-q)^{-1} \sum_{t=q+1}^T \ensuremath{\varepsilon}_t \vert_F\\ & \leq F_{T,1} + C_{T,1} + B_{T,1} + \vert D_{1T}\vert_F = o_\ensuremath{{\mathbb P}}(1). \end{align*} This completes the proof. $\square$ \begin{thebibliography}{xx} \harvarditem[Andrews]{Andrews}{1991}{An91} Andrews, D.W.K. 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