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 m – g 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).
[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.