If the residuals of a model are uncorrelated and/or normally distributed this is indicative of the adequacy of model. Therefore the usual residual analysis using the Q-statistic and the histogram Jarque Bera test of residual normality may be used to test for model adequacy.

For checking model adequacy, there are several methodologies. You can try to Kolmogrov Smirnoff test or Shapiro wilk test to check normality assumption of residuals. Durbin Watson test for autocorrelation. Levene test or white test to check the heteroscedasticity of residuals.

Diebold mariyano (DM test) test to check the performance of model.

Since you're building a predictive model I would definitely do some form of cross validation. Best regards, David

You should refit the model while reserving a part of the series at the end and look at the ex post forecasts. Evaluate a criterion like MAPE or MsymAPE (if they are not adequate because the data are not strictly positive, use RMSE instead) and compare with other, simpler, methods. Look at the forecast intervals. Look at the correlogram of the residuals. But there is no reason to check for normality.