Ljung-Box Test

Definition

---

creation-time: 2026-03-30 14:00 status: in progress parent:

• “forecasting methods” publish: true aliases:
• Ljung-Box Test

---

Definition

The Ljung-Box test is an improved version of theBox-Pierce test for testing whether all autocorrelations up to lag $m$ are jointly zero.

Hypotheses

$H_0:{\rho }_1={\rho }_2=\cdots ={\rho }_m=0\text{(white noise)}$

$H_1:\text{At least one }{\rho }_k\ne 0$

Test Statistic

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

where:

• $n$ = number of observations
• $m$ = maximum lag

Distribution

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

Decision Rule

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

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

Advancement over Box-Pierce

Ljung-Box modifies the Box-Pierce statistic by weighting each $r_k^2$ by $\frac{n+2}{n-k}$, providing better ${\chi }^2$ approximation for small samples.

Comparison with Box-Pierce

Test	Formula	Best For
Box-Pierce	$Q=T{\sum }_{k=1}^m{r}_k^2$	Large $n$
Ljung-Box	$LB=n(n+2){\sum }_{k=1}^m\frac{r_k^2}{n-k}$	Small $n$

Related

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