Multivariate portmanteau tests for weak multiplicative seasonal VARMA models

Abstract

Numerous multivariate time series encountered in real applications display seasonal behavior. In this paper we consider portmanteau tests for testing the adequacy of structural multiplicative seasonal vector autoregressive moving-average (SVARMA) models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak SVARMA). We study the asymptotic distributions of residual autocorrelations at seasonal lags of multiple of the length of the seasonal period under weak assumptions on the noise. We deduce the asymptotic distribution of the proposed multivariate portmanteau statistics, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. A set of Monte Carlo experiments and an application of U.S. monthly housing starts and housing sold are presented.

Appendix: Proofs

Proof of Proposition 1

The proof of Proposition 1 is similar to that given by Boubacar Mainassara and Francq (2011) for weak VARMA models. $\square$

Proof of Proposition 2

The proof is a straightforward extension of those obtained in Box and Pierce (1970), Ljung and Box (1978), Chitturi (1974), Hosking (1980). $\square$

Proof of Theorem 3

Let $\tilde{\ell }_n(\theta ,\varSigma _e)=-2{n}^{-1}\log \tilde{\mathrm {L}}_n(\theta ,\varSigma _e)$. As in Boubacar Mainassara and Francq (2011), it can be shown that ${\ell }_n(\theta ,\varSigma _e)=\tilde{\ell }_n(\theta ,\varSigma _e)+\mathrm {o}(1)$ a.s, where

$$\begin{aligned} \ell _n(\theta ,\varSigma _e):=-\frac{2}{n}\log {\mathrm {L}}_n(\theta ,\varSigma _e)=\frac{1}{n}\sum _{t=1}^n \left\{ d\log (2\pi )+\log \det \varSigma _e+e_t'(\theta )\varSigma _e^{-1}e_t(\theta )\right\} , \end{aligned}$$

and where $\left( e_t(\theta )\right)$ is given by

$$\begin{aligned} \begin{aligned} {e}_{t}(\theta )=&X_t-\sum _{i=1}^p A_{0}^{-1}A_{i} X_{t-i} - \sum _{j=1}^{p_s} A_{0}^{-1}\varLambda _{0}^{-1}\varLambda _{j}A_{0} X_{t-sj}\\&+\sum _{i=1}^p \sum _{j=1}^{p_s} A_{0}^{-1}\varLambda _{0}^{-1}\varLambda _{j} A_{i} X_{t-i-sj} \\&+\sum _{i=1}^q A_{0}^{-1}\varLambda _{0}^{-1}\varPsi _{0}B_{i}B_{0}^{-1}\varPsi _{0}^{-1}\varLambda _{0}A_{0} {e}_{t-i}(\theta )\\&+\sum _{j=1}^{q_s} A_{0}^{-1}\varLambda _{0}^{-1}\varPsi _{j}\varPsi _{0}^{-1}\varLambda _{0}A_{0} {e}_{t-sj}(\theta ) \\&-\sum _{i=1}^q \sum _{j=1}^{q_s} A_{0}^{-1}\varLambda _{0}^{-1} \varPsi _{j}B_{i}B_{0}^{-1} \varPsi _{0}^{-1}\varLambda _{0}A_{0}{e}_{t-i-sj}(\theta ). \end{aligned} \end{aligned}$$

(15)

It can be also shown uniformly in $\theta \in \varTheta$ that

$$\begin{aligned} \frac{\partial \ell _n(\theta ,\varSigma _e)}{\partial \theta }=\frac{\partial \tilde{\ell }_n(\theta ,\varSigma _e)}{\partial \theta }+\mathrm {o}(1)\quad a.s. \end{aligned}$$

The same equality holds for the second-order derivatives of $\tilde{\ell }_n(\theta ,\varSigma _e)$. Under A6, we have almost surely $\hat{\theta }_n\rightarrow \theta _0\in {\mathop {\varTheta }\limits ^{\circ }}$. Thus $\partial \tilde{\ell }_n(\hat{\theta }_n,\hat{\varSigma }_e)/\partial \theta =0$ for sufficiently large n, and a standard Taylor expansion of the derivative of $\tilde{\ell }_n$ about $(\theta _0,\varSigma _{e0}),$ taken at $(\hat{\theta }_n,\hat{\varSigma }_e),$ yields

$$\begin{aligned} 0&=\sqrt{n}\frac{\partial \tilde{\ell }_n(\hat{\theta }_n,\hat{\varSigma }_e)}{\partial \theta } =\sqrt{n}\frac{\partial \tilde{\ell }_n(\theta _0,\varSigma _{e0})}{\partial \theta }+\frac{\partial ^2 \tilde{\ell }_n(\theta ^*,\varSigma _e^*)}{\partial \theta \partial \theta '}\sqrt{n}\left( \hat{\theta }_n-\theta _0\right)&\nonumber \\&=\sqrt{n}\frac{\partial {\ell }_n(\theta _0,\varSigma _{e0})}{\partial \theta }+\frac{\partial ^2 {\ell }_n(\theta _0,\varSigma _{e0})}{\partial \theta \partial \theta '}\sqrt{n}\left( \hat{\theta }_n-\theta _0\right) +\mathrm {o}_\mathbb {P}(1),&\end{aligned}$$

(16)

where $\theta ^*$ is between $\theta _0$ and $\hat{\theta }_n$, and $\varSigma _e^*$ is between $\varSigma _{e0}$ and $\hat{\varSigma }_e$, with $\hat{\varSigma }_e=n^{-1}\sum _{t=1}^n \tilde{e}_t(\hat{\theta }_n)\tilde{e}'_t(\hat{\theta }_n)$. Thus, by standard arguments, we have from (16):

$$\begin{aligned} \sqrt{n}\left( \hat{\theta }_n-\theta _0\right)= & {} -J ^{-1}\sqrt{n}\frac{\partial {\ell }_n(\theta _0,\varSigma _{e0})}{\partial \theta }+\mathrm {o}_\mathbb {P}(1)\\ {}= & {} J ^{-1}\sqrt{n}Y_n+\mathrm {o}_\mathbb {P}(1) \end{aligned}$$

where

$$\begin{aligned} Y_n= & {} -\frac{\partial {\ell }_n(\theta _0,\varSigma _{e0})}{\partial \theta } \nonumber \\= & {} -\frac{1}{n}\sum _{t=1}^n \frac{\partial }{\partial \theta }\left\{ d\log (2\pi )+\log \det \varSigma _{e0}+e_t'(\theta _0)\varSigma _{e0}^{-1}e_t(\theta _0)\right\} . \end{aligned}$$

(17)

Using well-known results on matrix derivatives [see (5) of Appendix A.13 in Lütkepohl (2005)], we have

$$\begin{aligned} Y_n=-\frac{2}{n} \sum _{t=1}^n \frac{\partial e'_{t}(\theta _0)}{\partial \theta }\varSigma _{e0}^{-1}e_{t}(\theta _0). \end{aligned}$$

Now, using the elementary relation $\mathrm{vec}(ABC)=(C'\otimes A)\mathrm{vec}(B)$ [see (4) of Appendix A.12 in Lütkepohl (2005)], we have $\mathrm{vec}\gamma (s\ell )={n}^{-1} \sum _{t=s\ell +1}^{n}e_{t-s\ell }\otimes e_t$. It is easily shown that for $\ell ,\ell '\ge 1,$

$$\begin{aligned} \text{ Cov }(\sqrt{n}\mathrm{vec}\gamma (s\ell ),\sqrt{n}\mathrm{vec}\gamma (s\ell '))= & {} \frac{1}{n} \sum _{t=s\ell +1}^{n}\sum _{t'=s\ell '+1}^{n}\mathbb {E}\left( \left\{ e_{t-s\ell }\otimes e_t \right\} \left\{ e_{t'-s\ell '}\otimes e_{t'}\right\} '\right) \\ {}\rightarrow & {} \varGamma (s\ell ,s\ell ') \quad \text{ as } \quad n\rightarrow \infty . \end{aligned}$$

Then, we have

$$\begin{aligned} \varSigma _{\gamma _m(s)}=\left\{ \varGamma (s\ell ,s\ell ')\right\} _{1\le \ell ,\ell '\le m} \end{aligned}$$

By stationarity of $(e_t)$ and $\left( Y_t\right)$ and the dominated convergence Theorem, we have

$$\begin{aligned}&\text{ Cov }(\sqrt{n}J^{-1}Y_n,\sqrt{n}\mathrm{vec}\gamma (s\ell ))\\&\quad = - \frac{2}{n} \sum _{t=1}^n\sum _{t=s\ell +1}^{n}J^{-1}\text{ Cov }\left( \frac{\partial e'_{t}(\theta _0)}{\partial \theta }\varSigma _{e0}^{-1}e_{t},e_{t-s\ell }\otimes e_{t}\right) \\&\quad =-\frac{2}{n}\sum _{h=-n+1}^{n-1}(n-|h|)J^{-1}\text{ Cov }\left( \frac{\partial e'_{t}(\theta _0)}{\partial \theta }\varSigma _{e0}^{-1}e_{t},e_{t-h-s\ell }\otimes e_{t-h}\right) \\&\qquad \rightarrow -\sum _{h=-\infty }^{+\infty }2J^{-1}\mathbb {E}\left( \frac{\partial e'_{t}(\theta _0)}{\partial \theta }\varSigma _{e0}^{-1}e_{t}\left\{ e_{t-s\ell -h}\otimes e_{t-h}\right\} '\right) . \end{aligned}$$

Then we have

$$\begin{aligned} \varSigma '_{\gamma _m(s),\hat{\theta }_n}=-2J^{-1}\sum _{h=-\infty }^{+\infty }\mathbb {E}\left( \frac{\partial e'_{t}(\theta _0)}{\partial \theta }\varSigma _{e0}^{-1}e_{t}\left\{ \left( \begin{array}{c} {e}_{t-1s-h}\\ \vdots \\ {e}_{t-ms-h}\end{array}\right) \otimes e_{t-h}\right\} '\right) . \end{aligned}$$

Applying the central limit Theorem (CLT) for mixing processes [see Herrndorf (1984)] we directly obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\text{ Var }(\sqrt{n}J^{-1}Y_n)= J^{-1}IJ^{-1} \end{aligned}$$

which gives the asymptotic covariance matrix given in (7). The existence of these matrices is ensured by the Davydov inequality [see Davydov (1968)] and A7. The proof is then complete. $\square$

Proof of Theorem 4

Considering $\hat{\varGamma }(sh)$ and $\gamma (sh)$ as values of the same function at the points $\hat{\theta }_n$ and $\theta _0$, a Taylor expansion about $\theta _0$ gives

$$\begin{aligned} \mathrm{vec}\hat{\varGamma }_e(sh)= & {} \mathrm{vec}\gamma (sh)+\frac{1}{n} \sum _{t=sh+1}^{n}\left\{ e_{t-sh}(\theta )\otimes \frac{\partial e_{t}(\theta )}{\partial \theta '} \right. \\&\left. + \frac{\partial e_{t-sh}(\theta )}{\partial \theta '}\otimes e_{t}(\theta )\right\} _{\theta =\theta _n^*} (\hat{\theta }_n-\theta _0)+\mathrm {O}_\mathbb {P}(1/n)\\= & {} \mathrm{vec}\gamma (sh)+\mathbb {E}\left( e_{t-sh}(\theta _0)\otimes \frac{\partial e_{t}(\theta _0)}{\partial \theta '}\right) (\hat{\theta }_n-\theta _0)+\mathrm {O}_\mathbb {P}(1/n), \end{aligned}$$

where $\theta _n^*$ is between $\hat{\theta }_n$ and $\theta _0.$ The last equality follows from the consistency of $\hat{\theta }_n$ and the fact that $\left( \partial e_{t-sh}/\partial \theta '\right) (\theta _0)$ is not correlated with $e_t$ when $h\ge 0.$ Then for $h=1,\dots ,m,$

$$\begin{aligned} {\hat{\varGamma }}_m(s):= & {} \left( \left\{ \text{ vec }\hat{\varGamma }_e (1s)\right\} ',\dots ,\left\{ \text{ vec }\hat{\varGamma }_e(ms)\right\} ' \right) '\nonumber \\= & {} \gamma _m(s)+\varPhi _m(s)(\hat{\theta }_n-\theta _0)+\mathrm {O}_\mathbb {P}(1/n), \end{aligned}$$

(18)

where $\varPhi _m(s)$ is defined in (8). From Theorem 3, we have obtained the asymptotic joint distribution of $\gamma _m(s)$ and $\hat{\theta }_n-\theta _0$. Using (18) and the TCL of Herrndorf (1984), we obtain that the asymptotic distribution of $\sqrt{n}\hat{\varGamma }_m(s),$ is normal, with mean zero and covariance matrix

$$\begin{aligned} \lim _{n\rightarrow \infty }\text{ Var }(\sqrt{n}\hat{\varGamma }_m(s))= & {} \lim _{n\rightarrow \infty }\text{ Var }(\sqrt{n}\gamma _m(s)) +\varPhi _m(s)\lim _{n\rightarrow \infty }\text{ Var }(\sqrt{n}(\hat{\theta }_n-\theta _0)) \varPhi '_m(s)\\ {}&+\varPhi _m(s)\lim _{n\rightarrow \infty }\text{ Cov } (\sqrt{n}(\hat{\theta }_n-\theta _0),\sqrt{n}\gamma _m(s)) \\&+\lim _{n\rightarrow \infty }\text{ Cov } (\sqrt{n}\gamma _m(s),\sqrt{n}(\hat{\theta }_n-\theta _0))\varPhi '_m(s) \\ {}= & {} \varSigma _{\gamma _m(s)}+\varPhi _m(s)\varOmega \varPhi '_m(s)+\varPhi _m(s)\varSigma _{\hat{\theta }_n,\gamma _m(s)} +\varSigma '_{\hat{\theta }_n,\gamma _m(s)}\varPhi '_m(s). \end{aligned}$$

From a Taylor expansion about $\theta _0$ of $\mathrm{vec}\hat{\varGamma }_e(0)$ we have, $\mathrm{vec}\hat{\varGamma }_e(0)=\mathrm{vec}\gamma (0)+\mathrm {O}_\mathbb {P}(n^{-1/2}).$ Moreover, $\sqrt{n}(\mathrm{vec}\gamma (0)-\mathbb {E}\mathrm{vec}\gamma (0))=\mathrm {O}_\mathbb {P}(1)$ by the CLT for mixing processes [see Herrndorf (1984)]. Thus $\sqrt{n}(\hat{S}_e\otimes \hat{S}_e- S_e\otimes S_e)=\mathrm {O}_\mathbb {P}(1)$ and, using (9) and the ergodic Theorem, we obtain

$$\begin{aligned} n\left\{ \mathrm{vec}(\hat{S}_e^{-1}\hat{\varGamma }_e(sh)\hat{S}_e^{-1})- \mathrm{vec}( S_e^{-1}\hat{\varGamma }_e(sh) S_e^{-1})\right\} =\mathrm {O}_\mathbb {P}(1). \end{aligned}$$

In the previous equalities, we also use $\mathrm{vec}(ABC)=(C'\otimes A)\mathrm{vec}(B)$ and $(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}$ when A and B are invertible. It follows that

$$\begin{aligned}\hat{\rho }_m(s)= & {} \left( \left\{ \text{ vec }\hat{R}_e (1s)\right\} ',\dots ,\left\{ \text{ vec }\hat{R}_e(ms)\right\} ' \right) '\\ {}= & {} \left( \left\{ (\hat{S}_e\otimes \hat{S}_e)^{-1}\text{ vec }\hat{\varGamma }_e(1s)\right\} ',\dots ,\left\{ (\hat{S}_e\otimes \hat{S}_e)^{-1}\text{ vec }\hat{\varGamma }_e(ms)\right\} ' \right) '\\ {}= & {} \left\{ I_m\otimes (\hat{S}_e\otimes \hat{S}_e)^{-1}\right\} \hat{\varGamma }_m(s) =\left\{ I_m\otimes (S_e\otimes S_e)^{-1}\right\} \hat{\varGamma }_m(s)+\mathrm {O}_\mathbb {P}(n^{-1}). \end{aligned}$$

We now obtain (10) from (9). Hence, we have

$$\begin{aligned} \text{ Var }(\sqrt{n}\hat{\rho }_m(s))=\left\{ I_m\otimes (S_e\otimes S_e)^{-1}\right\} \varSigma _{\hat{\varGamma }_m(s)}\left\{ I_m\otimes (S_e\otimes S_e)^{-1}\right\} . \end{aligned}$$

This completes the proof. $\square$

Proof of Theorem 6

The proof is similar to that given by Francq et al. (2005) for Theorem 3. $\square$