Box-Pierce Test

Definition

TheBox-Pierce test tests whether all autocorrelations up to lag $m$ are jointly zero (i.e., whether the series is white noise).

Hypotheses

$$\begin{array}{rl}{H}_0& :{\rho }_1={\rho }_2=\cdots ={\rho }_m=0\text{(white noise)}\\ H_1& :\text{At least one }{\rho }_k\ne 0\end{array}$$

Test Statistic

$Q=T{\sum }_{k=1}^m{r}_k^2$

where:

• $T$ = number of observations
• $m$ = maximum lag (typically $m\approx T/4$)

Distribution

Under $H_0$: $Q\sim {\chi }_m^2$

Decision Rule

Reject $H_0$ at level $\alpha$ if:

$Q>{\chi }_{m,\alpha }^2$

Limitation

Box-Pierce approximation to ${\chi }^2$ is poor for small $n$ (e.g., $n<100$). Use Ljung-Box Test instead in that case.

Related

• Ljung-Box Test
• Sample Autocorrelation (ACF)
• White Noise