Box-Pierce vs Ljung-Box Comparison

Comparison

Both tests check whether all autocorrelations up to lag $m$ are jointly zero (white noise test).

Box-Pierce Test

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

Ljung-Box Test

$LB=n(n+2){\sum }_{k=1}^m\frac{r_k^2}{n-k}$

Key Differences

Aspect	Box-Pierce	Ljung-Box
Formula	$Q=T\sum r_k^2$	$LB=n(n+2)\sum \frac{r_k^2}{n-k}$
Weighting	Equal weights	Weighted by $\frac{n+2}{n-k}$
Best for	Large $n$	Small $n$
${\chi }^2$ approximation	Poor for $n<100$	Better for all $n$

Recommendations

Sample Size	Test to Use
$n<100$	Ljung-Box
$n\ge 100$	Either (Ljung-Box still preferred)

Why Ljung-Box is Better

The weighting $\frac{n+2}{n-k}$ gives more weight to early lags where we have more data points, improving the ${\chi }^2$ approximation.

Related

• Box-Pierce Test
• Ljung-Box Test
• White Noise