Modified Schwarz and Hannan–Quinn information criteria for weak VARMA models

Abstract

Numerous multivariate time series admit weak vector autoregressive moving-average (VARMA) representations, in which the errors are uncorrelated but not necessarily independent nor martingale differences. These models are called weak VARMA by opposition to the standard VARMA models, also called strong VARMA models, in which the error terms are supposed to be independent and identically distributed (iid). This article considers the problem of order selection of the weak VARMA models by using the information criteria. It is shown that the use of the standard information criteria are often not justified when the iid assumption on the noise is relaxed. As a consequence, we propose the modified versions of the Schwarz or Bayesian information criterion and of the Hannan and Quinn criterion for identifying the orders of weak VARMA models. Monte Carlo experiments show that the proposed modified criteria estimate the model orders more accurately than the standard ones. An illustrative application using the squared daily returns of financial series is presented.

Appendix

Proof of Proposition 1

Since the selected model with $k_1^{\prime }$ parameters is overfitted, we have $s_0$ overfitted parameters whose true values are zeros.

To test $s_0$ linear constraints on the elements of $\theta _0$ , we thus consider a null hypothesis of the form

$$\begin{aligned} H_0:R_0\theta _0={\mathfrak {r}}_0 \end{aligned}$$

where $R_0$ is a known $s_0\times k_0$ matrix of rank $s_0$ and ${\mathfrak {r}}_0$ is a known $s_0$-dimensional vector. The Wald, Lagrange multiplier (LM) and likelihood ratio (LR) principles are employed frequently for testing $H_0$. For instance, consider the LM test. Let $\hat{\theta }_n^c$ be the restricted QMLE of the parameter under $H_0$. Define the Lagrangean

$$\begin{aligned} {\mathscr {L}}(\theta ,\lambda )=\tilde{\ell }_n(\theta )-\lambda ^{\prime }(R_0\theta -{\mathfrak {r}}_0), \end{aligned}$$

where $\lambda$ denotes a $s_0$-dimensional vector of Lagrange multipliers. The first-order conditions yield

$$\begin{aligned} \frac{\partial \tilde{\ell }_n}{\partial \theta }(\hat{\theta }_n^c)=R_0^{\prime }\hat{\lambda },\quad R_0\hat{\theta }_n^c={\mathfrak {r}}_0. \end{aligned}$$

It will be convenient to write $a\mathop {=}\limits ^{c}b$ to signify $a=b+c$. A Taylor expansion gives under $H_0$

$$\begin{aligned} 0= & {} \sqrt{n}\frac{\partial \tilde{\ell }_n(\hat{\theta }_n)}{\partial \theta }\mathop {=}\limits ^{o_P(1)}\sqrt{n}\frac{\partial \tilde{\ell }_n(\hat{\theta }_n^c)}{\partial \theta }-J\sqrt{n}\left( \hat{\theta }_n-\hat{\theta }_n^c\right) . \end{aligned}$$

We deduce that

$$\begin{aligned} \sqrt{n}(R_0\hat{\theta }_n-{\mathfrak {r}}_0)= & {} R_0\sqrt{n}(\hat{\theta }_n-\hat{\theta }_n^c)\mathop {=}\limits ^{o_P(1)}R_0J^{-1}\sqrt{n}\frac{\partial \tilde{\ell }_n(\hat{\theta }_n^c)}{\partial \theta } = R_0J^{-1}R_0^{\prime }\sqrt{n}\hat{\lambda }. \end{aligned}$$

Thus under $H_0$ and the previous assumptions,

$$\begin{aligned} \sqrt{n}\hat{\lambda }\mathop {\rightarrow }\limits ^{{\mathscr {L}}}{\mathscr {N}}\left\{ 0,(R_0J^{-1}R_0^{\prime })^{-1}R_0\varOmega R_0^{\prime }(R_0J^{-1}R_0^{\prime })^{-1}\right\} . \end{aligned}$$

(20)

Standard Taylor expansions show that

$$\begin{aligned} \sqrt{n}(\hat{\theta }_n-\hat{\theta }_n^c)\mathop {=}\limits ^{o_P(1)}-\sqrt{n}J^{-1}R_0^{\prime }\hat{\lambda }, \end{aligned}$$

and that the LR statistic satisfies

Using the previous computations and standard results on quadratic forms of normal vectors [see e.g. Lemma 17.1 in van der Vaart (1998)], we find that the $\mathrm {\mathbf {LR}}_n$ statistic is asymptotically distributed as $\sum _{i=1}^{s_0}\mathrm {\lambda }_i Z_i^2$ where the $Z_i^{\prime }$s are iid ${\mathscr {N}}(0,1)$ and $\mathrm {\lambda }_1,\dots , \mathrm {\lambda }_{s_0}$ are the eigenvalues of

$$\begin{aligned} \varSigma _{\mathrm {\mathbf {LR}}}=J^{-1/2}S_{\mathrm {\mathbf {LR}}}J^{-1/2},\quad S_{\mathrm {\mathbf {LR}}}=\frac{1}{2}R_0^{\prime }(R_0J^{-1}R_0^{\prime })^{-1}R_0\varOmega R_0^{\prime }(R_0J^{-1}R_0^{\prime })^{-1}R_0. \end{aligned}$$

When $p^{\prime }\ge p_0$ and $q^{\prime }\ge q_0$ with either $p^{\prime }$ or $q^{\prime }$ greater then their true value, even though the model might not be identified in this case. So eventually

$$\begin{aligned} \hbox {BIC}_c(k_1)-\hbox {BIC}_c(k^{\prime }_1)= & {} 2\left\{ \log \tilde{\mathrm {L}}_n(\hat{\theta }_{n,k^{\prime }_1})-\log \tilde{\mathrm {L}}_n(\hat{\theta }_{n,k_1})\right\} -(c_{k^{\prime }_1}-c_{k_1})\log (n)\\ {}= & {} \mathrm {\mathbf {LR}}_n- (c_{k^{\prime }_1}-c_{k_1})\log (n), \end{aligned}$$

where $\hat{\theta }_{n,k}$ is the QMLE of the true k-dimensional parameter $\theta _{0,k}$. Combining these results, the asymptotic probability that the $\hbox {BIC}_c$ criterion selects the overfitted model is

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathbb {P}}\left\{ \hbox {BIC}_c(k_1)\ge \hbox {BIC}_c(k^{\prime }_1)\right\}= & {} \lim _{n\rightarrow \infty }{\mathbb {P}}\left\{ \sum _{i=1}^{s_0}\mathrm {\lambda }_i Z_i^2\ge (c_{k^{\prime }_1}-c_{k_1})\log (n) \right\} =0. \end{aligned}$$

The proof is complete. $\square$

Proof of Proposition 2

It is similar to that given for Proposition 1 and we omit it. $\square$

Proof of Proposition 3

Let $\hat{\theta }_{n,k_1}$ the QMLE of the true parameter $\theta _{0,k_1}$ with dimensional $k_1$. The QMLE satisfies $\log {\mathrm {L}}_n(\hat{\theta }_{n,k_1})\ge \log {\mathrm {L}}_n(\theta _{0,k_1})$ almost surely.

The $\hbox {BIC}_c$ criterion underfits if $\hbox {BIC}_c(k_1^{\prime \prime })\le \hbox {BIC}_c(k_1)$. In this case, since the selected model with $k_1^{\prime \prime }$ parameters is misspecified, as n grows to infinity, eventually

$$\begin{aligned} \log \tilde{\mathrm {L}}_n(\hat{\theta }_{n,k^{\prime \prime }_1})> \log \tilde{\mathrm {L}}_n(\hat{\theta }_{n,k_1}). \end{aligned}$$

(21)

Thus, the asymptotic probability that the $\hbox {BIC}_c$ criterion selects the underfitted model is

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathbb {P}}\left\{ \hbox {BIC}_c(k^{\prime \prime }_1)\le \hbox {BIC}_c(k_1)\right\}= & {} \lim _{n\rightarrow \infty }{\mathbb {P}}\left\{ -2\log \tilde{\mathrm {L}}_n(\hat{\theta }_{n,k^{\prime \prime }_1})+c_{k^{\prime \prime }_1}\log (n)\right. \\\le & {} \left. -2 \log \tilde{\mathrm {L}}_n(\hat{\theta }_{n,k_1})+c_{k_1}\log (n)\right\} \\= & {} \lim _{n\rightarrow \infty }{\mathbb {P}}\left\{ -2\left\{ \log \tilde{\mathrm {L}}_n(\hat{\theta }_{n,k^{\prime \prime }_1})- \log \tilde{\mathrm {L}}_n(\hat{\theta }_{n,k_1}) \right\} \right. \\\le & {} \left. (c_{k_1}-c_{k^{\prime \prime }_1})\log (n)\right\} . \end{aligned}$$

From (21), the fact that $\log (n)/n\rightarrow 0$ and $\mathrm {\mathbf {LR}}_n/n$ tends to a strictly positive constant, the conclusion follows. $\square$

Proof of Proposition 4

It is similar to that given for Proposition 3 and we omit it. $\square$