X is the univariate time series data (a one dimensional array of cells (e.g. rows or columns)).

Order is the time order in the data series (i.e. the first data point's corresponding date (earliest date=1 (default), latest date=0)).

M is the maximum number of lags to include in the test. If omitted, the default value of log(T) is assumed.

Return_type is a switch to select the return output (1 = P-Value (default), 2 = Test Stats, 3 = Critical Value.

Alpha is the statistical significance of the test (i.e. alpha). If missing or omitted, an alpha value of 5% is assumed.

1. The time series is homogeneous or equally spaced.
2. The time series may include missing values (e.g. #N/A) at either end.
3. The test hypothesis for white-noise:
   
   $$H_{o}: \rho_{1}=\rho_{2}=...=\rho_{m}=0$$
   $$H_{1}: \exists \rho_{k}\neq 0$$
   
   $$1\leq k \leq m$$
   
   Where:
   
   • $H_{o}$ is the null hypothesis.
   • $H_{1}$ is the alternate hypothesis.
   • $\rho_k$ is the population autocorrelation function for lag k
   • $m$ is the maximum number of lags included in the white-noise test.
4. The Ljung-Box modified $Q^*(m)$ statistic is computed as:
   $$Q^* =T(T+2)\sum_{j=1}^{m}\frac{\hat\rho_{j}^2}{T-l}$$
   Where:
   
   • $m$ is the maximum number of lags included in the test.
   • $\hat\rho_j$ is the sample autocorrelation at lag j.
   • $T$ is the number of non-missing values in the data sample.
5. The Ljung-Box modified $Q^*$ statistic has an asymptotic chi-square distribution with $m$ degrees of freedom and can be used to test the null hypothesis that the time series is not serially correlated.
   $$Q^*(m) \sim \chi_{\nu=m}^2()$$
   Where:
   
   • $\chi_{\nu}^2()$ is the Chi-square probability distribution function.
   • $\nu$ is the degrees of freedom for the Chi-square distribution.
6. The Ljung-Box test is a suitable test for all sample sizes including small ones.
7. This is one-side (i.e. one-tail) test, so the computed p-value should be compared with the whole significance level ($\alpha$).

Example 1:

Files Examples

Related Links

• Wikipedia - Ljung–Box test
• Wikipedia - White noise

References

• D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
• James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896
• Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740
• Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848
• Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568