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## Ljung-Box Q-Test

The sample autocorrelation function (ACF) and partial autocorrelation function (PACF) are useful qualitative tools to assess the presence of autocorrelation at individual lags. The Ljung-Box Q-test is a more quantitative way to test for autocorrelation at multiple lags jointly [1]. The null hypothesis for this test is that the first m autocorrelations are jointly zero,

`${H}_{0}:{\rho }_{1}={\rho }_{2}=\dots ={\rho }_{m}=0.$`

The choice of m affects test performance. If N is the length of your observed time series, choosing$m\approx \mathrm{ln}\left(N\right)$ is recommended for power [2]. You can test at multiple values of m. If seasonal autocorrelation is possible, you might consider testing at larger values of m, such as 10 or 15.

The Ljung-Box test statistic is given by

`$Q\left(m\right)=N\left(N+2\right){\sum }_{h=1}^{m}\frac{{\stackrel{^}{\rho }}_{h}^{2}}{N-h}.$`

This is a modification of the Box-Pierce Portmanteau “Q” statistic [3]. Under the null hypothesis, Q(m) follows a ${\chi }_{m}^{2}$ distribution.

You can use the Ljung-Box Q-test to assess autocorrelation in any series with a constant mean. This includes residual series, which can be tested for autocorrelation during model diagnostic checks. If the residuals result from fitting a model with g parameters, you should compare the test statistic to a ${\chi }^{2}$ distribution with mg degrees of freedom. Optional input arguments to `lbqtest` let you modify the degrees of freedom of the null distribution.

You can also test for conditional heteroscedasticity by conducting a Ljung-Box Q-test on a squared residual series. An alternative test for conditional heteroscedasticity is Engle’s ARCH test (`archtest`).

## References

[1] Ljung, G. and G. E. P. Box. “On a Measure of Lack of Fit in Time Series Models.” Biometrika. Vol. 66, 1978, pp. 67–72.

[2] Tsay, R. S. Analysis of Financial Time Series. 3rd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2010.

[3] Box, G. E. P. and D. Pierce. “Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models.” Journal of the American Statistical Association. Vol. 65, 1970, pp. 1509–1526.