Imagine I have a time-series. I eyeball it and it looks mostly like a trend plus a seasonal component plus noise. I take the series, subtract from it the trend and subtract from it the seasons. What's left is the residuals. So in the question we have a bunch of series: the series, series minus trend, series minus seasonals, series minus trend minus seasonals equals residuals.
What things should I look at?
Below are some ideas. Please point out the ones that are good ideas and the ones that don't make sense.
normality test on residuals (residuals should follow a normal distribution)
Dickey-Fuller test on residuals (residuals should not have a unit root?)
look at the Durbin-Watson statistic? (residuals should not be auto-correlated?)
some other stationary or trend-stationarity tests? Should these tests be performed on the residuals or on the original series?
KPSS test on series minus seasonals (to test trend-stationarity?)