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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,

The choice of m affects test performance. If N is the length of your observed time series, choosing 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

This is a modification of the Box-Pierce Portmanteau "Q" statistic [3]. Under the null hypothesis, Q(m) follows a 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 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.