If I am convinced that a series is mostly trend+season, what is it I should check about the residuals? 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.


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*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?)
What else?  
 A: The reason we examine the residuals is to possibly extract more signal thus making the noise less noisy. Furthermore examining the residuals can give us a clue as to potential over-modelling as model simplification may be in order.
modelling is an iterative process ...... eyeballs are sometimes useful AND sometimes useless leading to obfuscation rather than clarity ... it all depends on the time of the day and the age/experience of the analyst. Recall beauty is in the eye of the beer-holder !
after each step perform the remedy suggested by the step. Following is an example of 1 sequence of model revision strategies.


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*If this is CAUSAL MODEL examine the residuals to see if they can be predicted by lags of the causal series. Your reflection implies that this is not a causal problem.

*examine the residuals for pulses

*examine the residuals for seasonal pulses

*examine the residuals for step/level shifts &  for time trends

*examine the model coefficients to see if they significantly changed at any point in time via the CHOW test

*examine the acf of the residuals

*examine the error variance to see if it significantly changed at any point in time

*examine the error variance to see if it is correlated to the expected value of Y via the Box-Cox test

*If this is CAUSAL MODEL examine the residuals to see if they can be predicted by lags of the causal series.


these are the steps that AUTOBOX http://autobox.com/cms/ (a piece of software that I helped to develop) takes in it's Tour de Force to form a reasonable model. As you might perceive the ORDER of these tests and remedies are important so results may differ depending upon the order/sequence . AUTOBOX evaluates alternative pathing using using proprietary AI and concludes with a dominant strategy/order.
