I want to prove the following statement:

Under $H_{0}$ the test statistic $Q=n(n+2)$ $sum limits_{k=1}^h frac{hat{p}_{k}^2}{n-k}$ follows a $chi ^2(h)$ chi-squared distribution with $h$ degrees of freedom. $H_{0}$ is the hypothesis that all data points are independently distributed.

$n$: sample size

$hat{p}_{k}$: sample autocorrelation at lag k

$h$: number of lags being tested

I know that if $H_{0}$ is valid it follows that $Q=0$ because $hat{p}_{k}=0$ for all k.

But how do I conclude that $Q$ is chi-squared distributed?

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#### Best Answer

First, note that $$Q=nsum_{k=1}^h frac{(n+2)hat{p}_{k}^2}{n-k}$$ and that $frac{n+2}{n-k}to1$ as $ntoinfty$, so that $Q$ will behave like $$tilde Q=nsum_{k=1}^hhat{p}_{k}^2$$ asymptotically.

To show that $tilde Q$ is $chi^2(h)$ under $H_0$, consider the following intermediate result adapted from Brockwell and Davis (1991), Theorem 7.2.1: Let $hat{p}=(hat p_1,ldots,hat p_h)^top$. For a white noise process (i.e., one for which the null is true) $$Y_t=mu+epsilon_t$$ with $E|epsilon_t|^4<infty$ it holds that $$ sqrt{n}hat{p}to_d N(0,I_h) $$ Thus, the first $h$ sample autocorrelations are multivariate normal with expected value 0 each (the true autocorrelation of any order for a white noise process) and asymptotic covariance matrix equal to the identity matrix. Hence, they are asymptotically independent. This also implies that each autocorrelation is asymptotically standard normal. Next, observe that $$ tilde Q=sqrt{n}hat{p}^topsqrt{n}hat{p}$$ Now, we know that the sum of $h$ independent squared standard normal random variables is $chi^2(h)$.

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