Data-driven portmanteau tests for time series

Abstract

Portmanteau tests and information criteria are widely used for checking the hypothesis of independence in time series. More recently, data-driven versions were proposed, where the tests are calibrated based on the largest estimated autocorrelation. It seems natural to introduce a double test statistic (M, Q) where Q is the portmanteau and M is the largest squared autocorrelation. Both statistics have been investigated at length in the past decades. We computed under reasonable assumptions the bivariate probability distribution of this double statistic, conditional, in addition, to the lag at which the largest autocorrelation is found. Tests of the null hypothesis of independence based on rejection regions in the plane (M, Q) are proposed, and some methods to select the rejection region in order to maximize power when the alternative hypothesis is unknown are suggested. A simulation study and a thorough comparison with some popular tests have been performed to show the advantages of our proposal. Notice that this latter includes some well-known univariate tests, so we could expect not only an optimal choice but also additional information which may turn useful for a better understanding of the time series for both model building and forecasting.

Computational details

Throughout this section, we consider the probability distribution under the null hypothesis $H_0$ of independence, the specification $|H_0$ is omitted for brevity.

1.1 Test O

The critical points $m_\alpha , q_\alpha$ must satisfy $\Pr \{M>m_\alpha \bigcup Q_h>q_\alpha \}= \alpha = 1-\Pr \{M<m_\alpha \bigcap Q_h<q_\alpha \}$ and require computation of the bivariate cumulative distribution function (16). In order to compute $\Pr \{M<u \bigcap Q_h<v\}$ for any u and v it should be considered that the support is limited to $u>0, 0<v<hu$ if $h<p$, and to $u>0, u<v<hu$ if $h \ge p$.

We denote for simplicity

$$\begin{aligned} G(v,x,h, \nu , c)= \varPhi _{h \nu }(\frac{v}{c})/\varPhi _{h \nu }(\frac{hx}{c}). \end{aligned}$$

The complete form of the conditional distribution may be written as follows

$$\begin{aligned}&\Pr \{Q_h<v | M=x, h<p\} = \left\{ \begin{array}{cc} G(v,x,h,\nu ,c) &{} \ 0<v \le hx \\ 1 &{} \ v>hx \\ \end{array} \right. \end{aligned}$$

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$$\begin{aligned}&\Pr \{Q_h<v | M=x, h \ge p\} = \left\{ \begin{array}{ cc} 0 &{} \ 0<v \le x \\ G(v-x,x,h-1,\nu ,c) &{} \ x<v \le hx \\ 1 &{}\ v>hx \\ \end{array} \right. . \end{aligned}$$

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However, for $h=1$ the exact expression is obtained assuming $G(v,x,1,\nu ,c)=G(v,x,1,1,1)$ and $G(v,x,0,\nu ,c)=I_{[v>x]}$ where $I_{[A]}$ denotes the indicator function of the event A. For $h=2$ also the exact expression of the conditional distribution may be easily obtained. When $2 \ge p$ the assumption $G(v,x,1,\nu ,c)=G(v,x,1,1,1)$ makes formula (21) exact, while when $2<p$ the correct expression satisfies (20) on assuming

$$\begin{aligned} G(v,x, 2, \nu ,c) = \int _0^{\max (x,v)} \phi _1(y) \varPhi _1[\max (v-y,x)]/\varPhi _1(x)^2 dy . \end{aligned}$$

For $h \ge 3$ the approximation is satisfying. The quality of the proposed approximation has been checked by simulation. For $3 \le h \le 30$ and $1 \le M \le 20$, 10, 000 replications of $S_h=Z_1+Z_2+ \ldots +Z_h$ were simulated, where each $Z_j$ was generated from the chi-square distribution with one degree of freedom truncated at M⁴. The quantiles at level 0.95 were computed and compared with those of the approximate distribution (13). We found that the differences were uniformly negligible, results for selected values of h and M are shown in Table 2.

Table 2 Simulated and approximate quantiles at level 0.95 for the conditional distribution of $Q_h|M$

On integrating with respect to x and conditioning on p we have:

$$\begin{aligned}&\Pr \{M<u \bigcap Q_h<v | h<p\} =\int _0^u f_M(x)\Pr \{Q_h<v|M=x,h<p\}dx\\&=\left\{ \begin{array}{l l} \int _0^u f_M(x) dx = \varPhi _1(u)^L &{} \, u \le v/h \\ \int _0^{v/h} f_M(x)dx + \int _{v/h}^u f_M(x) G(v,x,h,\nu ,c) dx &{} \, u > v/h \end{array} \right. \end{aligned}$$

that may be written

$$\begin{aligned}&\Pr \{M<u \bigcap Q_h<v | h<p \}\\&\quad = \varPhi _1[\min (u, \frac{v}{h})]^L+ I_{[u>v/h]} \int _{v/h}^u f_M(x)G(v,x,h,\nu ,c)dx . \end{aligned}$$

On the other hand, if $h \ge p$:

$$\begin{aligned}&\Pr \{M<u \bigcap Q_h<v | h \ge p \} =\int _0^u f_M(x)\Pr \{Q_h<v|M=x,h\ge p\}dx \\&= \left\{ \begin{array}{l} \int _0^u f_M(x) dx = \varPhi _1(u)^L \,\,\,\, u \le v/h \\ \int _0^{v/h} f_M(x)dx + \int _{v/h}^u f_M(x) G(v-x,x,h-1,\nu ,c) dx \,\,\,\, v/h< u \le v \\ \int _0^{v/h} f_M(x)dx + \int _{v/h}^v f_M(x) G(v-x,x,h-1,\nu ,c) dx \,\,\,\, u > v \\ \end{array} \right. \end{aligned}$$

that may be written more simply:

$$\begin{aligned}&\Pr \{M<u \bigcap Q_h<v | h \ge p\} = \varPhi _1[\min (u, \frac{v}{h})]^L\\&\quad + I_{[u>v/h]} \int _{v/h}^{\min (u,v)} f_M(x)G(v-x,x,h-1,\nu ,c)dx . \end{aligned}$$

Finally, the required probability results:

$$\begin{aligned} \Pr \{M>u \bigcup Q_h>v\}= & {} 1-\varPhi _1[\min (u,\frac{v}{h})]^L \nonumber \\&- I_{[u>v/h]}\left\{ \frac{L-h}{L}\int _{v/h}^u f_M(x) G(v,x,h,\nu ,c)dx \right. \nonumber \\&\left. +\frac{h}{L}\int _{v/h}^{\min (u,v)} f_M(x) G(v-x,x,h-1,\nu ,c)dx \right\} .\nonumber \\ \end{aligned}$$

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1.2 Test A

For test A the critical points $m_\alpha ,q_\alpha$ must satisfy $\Pr \{M>m_\alpha \bigcap Q_h>q_\alpha \}=\alpha$ and the computation may be based (remembering that $M=\max \{N{\hat{r}}(j)^2\} \le N$) on

$$\begin{aligned} \Pr \{M>u \bigcap Q_h>v\} =\sum _{p=1}^L \frac{1}{L} \int _u^N f_M(x) \Pr \{Q_h>v |M=x, p\}dx . \end{aligned}$$

Thus we have for $h<p$

$$\begin{aligned}&\Pr \{M>u \bigcap Q_h>v | h<p\} \\= & {} \int _u^{N} f_M(x)\left[ 1-\Pr \{Q_h<v |M=x, h<p\} \right] dx \end{aligned}$$

and using (20)

$$\begin{aligned}&\Pr \{M>u \bigcap Q_h>v | h<p\} = \int _{\max (u,v/h)}^{N} f_M(x)[1-G(v,x,h,\nu ,c)]dx \\&\quad = 1-\varPhi _1[\max (u,\frac{v}{h})]^L-\int _{\max (u,v/h)} ^{N}f_M(x) G(v,x,h,\nu ,c)dx . \end{aligned}$$

When $h \ge p$ using (21) we get

$$\begin{aligned} \Pr \{M>u \bigcap Q_h>v | h \ge p\}= & {} \int _u^{N} f_M(x)\left[ 1-\Pr \{Q_h<v |M=x, h \ge p\} \right] dx \\= & {} \int _{\max (u,v/h)} ^v f_M(x)[1-G(v-x,x,h-1,\nu ,c)]dx \\&+ \int _v ^{N} f_M(x) dx = 1- \varPhi _1[\max (u,\frac{v}{h})]^L \\&- I_{[u \le v]} \int _{\max (u,v/h)} ^v f_M(x) G(v-x,x,h-1, \nu ,c) dx. \end{aligned}$$

Thus the required probability may be written

$$\begin{aligned} \Pr \{M>u \bigcap Q_h>v \}= & {} 1- \varPhi _1[\max (u,\frac{v}{h})]^L \nonumber \\&- \frac{L-h}{L} \int _{\max (u,v/h)}^{N} f_M(x) G(v,x,h,\nu ,c)dx \nonumber \\&- I_{[u \le v]} \frac{h}{L} \int _{\max (u,v/h)}^v f_M(x) G(v-x,x,h-1,\nu ,c)dx.\nonumber \\ \end{aligned}$$

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