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We frequently apply various statistical tests to data, including the stationarity test, the t-test, tests for randomness, etc. They are typically used with direct data, or information obtained directly from instruments, subjects, etc. However, there are other types of data which are model implied, i.e., residuals. Even for a model there can be different types of residuals where distribution can be very different e.g. distribution of ordinary residuals can be different from the quantile residuals.

So my question is can I use the same tests directly to such model implied data and use critical values for statistical inference?

Any pointer/textbook where this subject is discussed with practical example will be very helpful.

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  • $\begingroup$ Every textbook explains how to apply tests and describes the circumstances in which they are reliable and meaningful. In light of this, could you please explain what you mean by "use the same tests directly to such model implied data"? $\endgroup$ Commented Jun 29, 2022 at 15:10
  • $\begingroup$ Lets take example of Durbin–Watson statistic. As per the Wikipedia I see that it can be applied in Residual values from a regression model. So 1) can I apply this same test on actual observation 2) When it is applied on the residuals, can it be applied on any type of residuals (i.e. pearson, quantile) and any type of regression model (e.g. lm, glm, gee) without any change? $\endgroup$ Commented Jun 29, 2022 at 17:14

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