some thoughts on t test and residual normality in reg

[Bo Chen]

Hi all,

So this might be a widely known issue, and a recent post on this can be found here (https://www.statalist.org/forums/for...-residual-term). In summary,

1. in a regression, normality of the residuals is needed if one wants to use the t-test results of the estimated coefficients;
2. with a large sample size, law of large number kicks in, and the estimated coefficients follow a normal distribution, regardless of the distribution of the residuals.

The question is: most of the time, the normality assumption fails (and most people do not really test it), and one relies on the large sample properties of the coefficients to do inference. That's to say, the coefficients are asymptotically normally distributed, and the t-tests reported in the regression tables should not be used. So I am wondering why Stata (and many other statistical packages) doesn't throw a warning to the users about this. Or one additional option can be added to let the user decide whether they want the t-test or the z-score test, instead of reporting the t-test by default.

Thanks!

[Joao Santos Silva]

Dear Bo,

If the number of degrees of freedom is large, the t and normal distributions are indistinguishable. So, with large samples, it does not make any difference.

Best wishes,

Joao

[Carlo Lazzaro]

Bo:
the following table gives some example on what you're interested in http://www.sjsu.edu/faculty/gerstman...er/t-table.pdf.
If I remeber correctly, I was taught that after 120 degrees of freedom using t or normal distribution is actually immaterial.
I fail to get why one should be worried about the normality of coefficients and related ttests when the only prerequisite about normality in OLS refers to residuals.

[Bo Chen]

Thank you, Joao and Carlo, for reminding me that t distribution converges to normal distribution as df increases!