12.6 Portmanteau Statistics

With the help of the knowledge about the asymptotic distribution of the autocorrelations we can derive a statistic to test the hypothesis of white noise. One can either test the original series $ X_t$ or the residuals of an ARMA($ p,q$) process. The number of estimated parameters is in the first case $ k=0$ and in the second case $ k=p+q$.

Under the null hypothesis it holds for every $ m$

$\displaystyle \rho_1=0, \ldots, \rho_m =0.
$

The alternative hypothesis is accordingly, that at least one $ \rho_i$, $ 1\le i \le m$ is not equal to zero. Under the null hypothesis $ \sqrt{n}\hat{\rho}_{\tau,n}$ is asymptotically standard normally distributed. The statistic

$\displaystyle Q_m = n \sum_{j=1}^m \hat{\rho}_{j,n}^2
$

has an asymptotic $ \chi^2$ distribution with $ m-k$ degrees of freedom. One would reject the null hypothesis at a significance level of $ \alpha$, as long as $ Q_m
> \chi^2_{m-k;\alpha},$ the $ (1-\alpha)$-quantile of the Chi-squared distribution with $ m-k$ degrees of freedom.

Studies show that $ Q_m$ in small samples poorly approximates the asymptotic distribution. This results from the fact that $ \hat{\rho}_{\tau,n}$ is a biased estimator for $ \rho_\tau$. The bias is stronger for small $ \tau$, and thus an asymptotically equivalent statistic can be defined as

$\displaystyle Q_m^* = n(n+2) \sum_{j=1}^m \frac{1}{n-j} \hat{\rho}_{j,n}^2
$

which weights the empirical autocorrelations of smaller order less than those of larger order. The modified Portmanteau statistic $ Q_m^*$ is therefore in small samples frequently closer to the asymptotic $ \chi^2$ distribution. For large $ n$, both statistics performs equally well.