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I understand that when performing linear regression, one common rule-of-thumb is that for a good 'fit', the residuals should be 1) independently distributed, 2) stationary and 3) not serially correlated. In practical terms, this can be achieved with multiple statistical tools i.e. Augmented Dickey-Fuller (ADF) a Ljung-Box (lag=1) tests on the residuals.

What is the convention, if any, regarding this type of regression analysis i.e. what are the recommended statistical tests to infer if a model fit is appropriate? For instance, is testing for stationarity strictly required when testing for auto-correlation (as non-stationary trends would exhibit autocorrelation)? Or is just testing the residuals for normality (e.g. with a Kolmogorov-Smirnov test) sufficient?

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    $\begingroup$ 1) implies 3), so you do not need both. Also, your post might be a duplicate, so consider taking a look at the existing threads. I think after 10+ years and over 200'000 questions (many of which about regression), the issue of regression model diagnostics has been covered on Cross Validated. $\endgroup$ Jun 9, 2023 at 17:40
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    $\begingroup$ Strictly speaking, (1) is not true, because by construction the residuals with $k$ terms (including an intercept) satisfy up to $k$ linear constraints. (2) and (3) are not usually even defined or meaningful, because the setting does not include any stochastic process assumptions. There are so many things to check and so many ways to do it that we have separate threads for each: see this site search. Using a formal test of Normality is usually not helpful, btw. $\endgroup$
    – whuber
    Jun 9, 2023 at 17:51

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