# Testing normality and independence of time series residuals

The simplest form of a white noise process is where its observations are uncorrelated. We can check this by applying e.g. a portmanteau test such as Lung - Box or Box - Pierce. The series might be Gaussian white noise where the observations are uncorrelated and also normally distributed and hence independent. We can test this with a normality test and a portmanteau test. As far as I know there is a third case where the observations are uncorrelated and independent without being normally distributed. In that case how can we test whether the observations are independent? Is there a statistical test for this?

First, you perform your regression. Get the residuals. Run a regression of the residuals on all the variables from your regression of interest from step 1 plus some number of lagged residuals. You can guess how many lags you should include by looking at the autocorrelation function. You can test for a lack of serial correlation by testing that the coefficients on the lags of the residuals are jointly 0 by using an F test or a version of a Lagrange multiplier test (the test statistic is the number of observations in the second, auxiliary regression times the $R^2$ from that regression; the test statistic is distributed as a $\chi^2_l$, where $l$ is the number of lags, under the null of no serial correlation).