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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?

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Notwithstanding IrishStat's comments, you could use a Breusch-Godfrey test. It is used to test for a lack of correlation among the residuals of a regression model.

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).

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A case in point is where the residuals are perceived to be independent via the tests you define BUT are not normally distributed is when the mean of the errors is not-constant. Including a constant in the model guarantees that the overall mean of the errors is zero BUT not necessarily for all time intervals. If you have one anomaly in the residuals this will inflate the variance of the errors thus providing a downward-bias to the correlation coefficient. If you have an error process that has a mean shift at a particular point in time you again will have indflated error variance and a (severe) downward bias ("Alice in Wonderland") in the acf of the errors. In summary the tests you are relying on assume that there is no mean bias in the errors. Simply use Intervention Detection procedures to identify omitted Pulses, Level Shifts , Seasonal Pulses and/or Local Time Trends and then incorporate any and all of these statistically significant variables into your Transfer Function. The fixup will then allow you to proceed with your standard tests. You then might find that the error variance might be related to the level of Y suggesting the need fpr a power transform ( logs/reciprovals/square root etc )/ Alternatively the error variance may have changed at fixed points over time suggesting GLS or stochastically suggesting the need for a GARCH augmentation.

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