Wild bootstrap tests for autocorrelation in vector autoregressive models

Abstract

Tests for error autocorrelation (AC) are derived under the assumption of independent and identically distributed errors. The tests are not asymptotically valid if the errors are conditionally heteroskedastic. In this article we propose wild bootstrap (WB) Lagrange multiplier tests for error AC in vector autoregressive (VAR) models. We show that the WB tests are asymptotically valid under conditional heteroskedasticity of unknown form. WB tests based on a version of the heteroskedasticity-consistent covariance matrix estimator are found to have the smallest error in rejection probability under the null and high power under the alternative. We apply the tests to VAR models for credit default swap prices and Euribor interest rates. An important result that we find is that the WB tests lead to parsimonious models while the asymptotic tests suggest that a long lag length is required to get white noise residuals.

Appendix

Proof of Proposition 1 To prove the asymptotic validity of the recursive-design WB $Q_{ \text {LM}}^{*}$ test, we need to show that $\sqrt{T} \text{ vec }(\widehat{\mathbf {B}}-\mathbf {B})$ and $\sqrt{T}\text{ vec }( \widehat{\mathbf {B}}^{*}-\widehat{\mathbf {B}})$ have the same asymptotic multivariate normal distribution, $\sqrt{T}\mathbf {c}_{h}$ and $\sqrt{T} \mathbf {c}_{h}^{*}$ have the same asymptotic multivariate normal distribution, and

$$\begin{aligned} \text{ plim }\frac{1}{T}\mathbf {ZF}(\mathbf {I}_{h}\otimes \mathbf {U}^{\prime })=\text{ plim }\frac{1}{T}\mathbf {Z}^{*}\mathbf {F}(\mathbf {I}_{h}\otimes \mathbf {U}^{*\prime })=\widetilde{\mathbf {G}}, \end{aligned}$$

(15)

where $\widetilde{\mathbf {G}}$ is defined in (10).

To show that $\sqrt{T}\text{ vec }(\widehat{\mathbf {B}}-\mathbf {B})$ and $\sqrt{T}\text{ vec }(\widehat{\mathbf {B}}^{*}-\widehat{\mathbf {B}})$ have the same asymptotic multivariate normal distribution, we use Theorem 5.1 of Brüggemann et al. (2014). Brüggemann et al. prove consistency of the moving block bootstrap in vector autoregressive models with conditionally heteroscedastric errors. Their result also holds for the WB estimates of the mean parameters $\mathbf {B}$ (see Brüggemann et al. 2014, p. 8). Hence, we can conclude that

$$\begin{aligned} \sup _{\mathbf {z}}\left| \text{ P }\left( \sqrt{T}\text{ vec }(\widehat{ \mathbf {B}}-\mathbf {B})\le \mathbf {z}\right) -\text{ P }^{*}\left( \sqrt{T }\text{ vec }(\widehat{\mathbf {B}}^{*}-\widehat{\mathbf {B}})\le \mathbf {z} \right) \right| \rightarrow \mathbf {0} \end{aligned}$$

in probability.

The asymptotic multivariate normal distribution of $\sqrt{T}\mathbf {c}_{h}$ is obtained from Lemma A.1 of Brüggemann et al. (2014):

$$\begin{aligned} \sqrt{T}\mathbf {c}_{h}\overset{D}{\rightarrow }\text{ N }(\mathbf {0}, \varvec{\Omega }_{h}), \end{aligned}$$

where $\varvec{\Omega }_{h}=(\tau _{0,i,0,j})$ is a $(K^{2}h\times K^{2}h)$ block matrix and $\tau _{0,i,0,j}$ is defined in (3). The proof that $\sqrt{T}\mathbf {c}_{h}^{*}$ has the same asymptotic multivariate normal distribution is entirely analogous to that in the proof of Lemma A.3 of Goncalves and Kilian. Define $S_{t}=\varvec{\mathbf {\lambda }} ^{\prime }\mathbf {c}_{h}^{*}$ for arbitrary $\varvec{\mathbf {\lambda }}\in \mathbf {R}^{m}$, $m=K^{2}h$, $\varvec{\mathbf {\lambda }}^{\prime }\varvec{\mathbf {\lambda }}=1$, and apply Theorem A.1 and the techniques of the proof of Lemma A.3 of Gonçalves and Kilian (2004) to $S_{t}$.

The result (15) follows from the multivariate analogue of Lemma A.2 of Gonçalves and Kilian (2004).

Proof of Proposition 2 The proof of Proposition 2 is similar to the proof of Proposition 1, and is omitted. The required convergence results for the fixed-design WB were proved by Hafner and Herwartz (2009).