A simple portmanteau test with data-driven truncation point

Abstract

Time series forecasting is an important application of many statistical methods. When it is appropriate to assume that the data may be projected towards the future based on the past history of the dataset, a preliminary examination is usually required to ensure that the data sequence is autocorrelated. This is a quite obvious assumption that has to be made and can be the object of a formal test of hypotheses. The most widely used test is the portmanteau test, i.e., a sum of the squared standardized autocorrelations up to an appropriate maximum lag (the truncation point). The choice of the truncation point is not obvious and may be data-driven exploiting supplementary information, e.g. the largest autocorrelation and the lag where such maximum is found. In this paper, we propose a portmanteau test with a truncation point equal to the lag of the largest (absolute value) estimated autocorrelation. Theoretical and simulation-based comparisons based on size and power are performed with competing portmanteau tests, and encouraging results are obtained.

Appendix. Approximate asymptotic distribution

We derive now the asymptotic distribution of the statistic $Q_{max}$ under Assumptions A and S. It is obtained starting from the joint asymptotic distribution of $(M, Q_d ,p)$ where $M= \max \{N{\hat{r}}(1)^2, N{\hat{r}}(2)^2, \ldots , N{\hat{r}}(L)^2\}$ and $p = {{\mathrm{arg\,max}}_{1 \le j \le L}} \{ {\hat{r}}(j)^2 \}$, obtained in Baragona et al. (2022).

Under Assumptions A and S the asymptotic distribution of M is that of the maximum of L asymptotically independently distributed $\chi ^2_1$ variables: $\Pr \{M<x\}=\varPhi _1(x)^L$.

The joint distribution of M and $Q_d$ for $1 \le d \le L$ is considered in Baragona et al. (2022) (which the reader is referred to for more details) and has a different analytical expression if $d<p$ or $d \ge p$. We only need the expression for $d=p$. In this case, when $p=1$, $Q_{max} \equiv M$, while for $p>1$

$$\begin{aligned} Q_{max} \equiv Q_p= M + \sum _{j \ne p} N {\hat{r}}(j)^2 \end{aligned}$$

is the sum of M plus $(p-1)$ (asymptotically independent) chi squares with one degree of freedom and values limited from 0 to M. Therefore

$$\begin{aligned} \Pr \{Q_{max}<x|p\}= \int _0^x \Pr \{M=y\} \Pr \{Q_d<x|M=y, d=p\} dy \end{aligned}$$

(7)

where $\Pr \{M=y\}$ denotes the density of the maximum: $L \phi _1(y)\varPhi _1(y)^{L-1}$. The distribution function inside the integral equals asymptotically the probability that the sum $S_{p-1}=Z_1+Z_2+\ldots +Z_{p-1}$ is less than $(x-y)$, where the $Z_j$’s are $(p-1)$ independent chi squares with one degree of freedom and truncated at M. The distribution of $S_{p-1}$ may be approximated by a $c \, \chi ^2_\nu$ variate with the same mean and variance, as usual. From the results of Coffrey and Muller (2000):

$$\begin{aligned}{} & {} E(S_{p-1})= (p-1) \varPhi _3(y)/\varPhi _1(y) \\{} & {} Var(S_{p-1})=(p-1)[3\varPhi _5(y)/\varPhi _1(y)-\varPhi _3(y)^2/\varPhi _1(y)^2] \end{aligned}$$

the following values for the constants c and $\nu$ are obtained:

$$\begin{aligned}{} & {} c=Var(S_{p-1})/[2E(S_{p-1})]=\frac{3}{2} \left\{ \frac{\varPhi _5(y)}{\varPhi _3(y)}- \frac{\varPhi _3(y)}{\varPhi _1(y)}\right\} \\{} & {} \nu =2E(S_{p-1})^2/Var(S_{p-1})=\frac{(p-1) \varPhi _3(y)^2}{3\varPhi _3(y)[\varPhi _5(y)-\varPhi _3(y)]}. \end{aligned}$$

With this approximation we obtain

$$\begin{aligned} \Pr \{Q_d<x|M=y, d=p\}= \varPhi _{\nu }[(x-y)/c] / \varPhi _{\nu }[(p-1)y/c] \end{aligned}$$

and finally

$$\begin{aligned} \Pr \{Q_{max}<x|p\}= \int _0^x L \phi _1(y)\varPhi _1(y)^{L-1} \varPhi _{\nu }[(x-y)/c] / \varPhi _{\nu }[(p-1)y/c] dy. \end{aligned}$$