Test to show a regressor is non stochastic?

[Peter Jakobsen]

Other than producing and eyeballing a scatter diagram, is there a standard way to prove/discover that an independent variable is non-stochastic so that it complies with the assumptions of simple OLS?

Thank you,

Peter J.

[Clyde Schechter]

The only distributional requirement for simple OLS is that the residuals of the regression be normally distributed. And even that only applies in small samples. Much too much time is wasted by people trying to prove that they fulfill this requirement when it is not actually required.

But more importantly, you cannot test this requirement by doing anything with the dependent variable itself. You must first run the regression and then examine the distribution of the residuals. There is absolutely no requirement that the dependent variable be normal, nor any other distributional requirement on it.

If you want to test the normality of the residuals, that is problematic to do because most of the tests for normality will, in small samples, lack enough power to detect departure from normality that might actually matter. And in large sample, since normality of residuals isn't necessary, any departure from normality detected by such tests is irrelevant. Probably the least harmful way to check the normality of residuals in a small sample is graphically.

regress whatever
predict resid, resid
qnorm resid

If the scatter of the points departs in a visually large way far from the diagonal, then, with a small sample, you cannot rely on the t-tests and F-tests (nor do the confidence intervals have an assurance of correct coverage). Again, with a large sample, this is all irrelevant anyway. I should also emphasize that even with a small sample, the coefficient estimates are unbiased and the standard errors are correct, regardless of the distribution of the residuals (or of anything else).

[Peter Jakobsen]

Thank you for your generous answer. I do find it odd when ever I see vertical dots so neatly arranged for data that are simply not discrete in actual fact. For example, you are only 20 years old for a split second, after that you are 20.001 years old and very quickly 20.1 years old, etc etc.