why statsmodels OLS uses t-test but GLM uses z-test for confidence interval

[Mohsen Vazirizade]

Hi,

As you may now, both smf.ols and mf.glm can be used for fitting a regression model. Theoretically, they should bear the same results if the error is normally distributed. I ran 2 regressions (attached please find the code), once using ols and the other using glm. While, their results are identical, ols uses t-test but glm uses z-test. Consequently, their confidence intervals are different. 

To my knowledge, we always should use t-test since we do not know the actual value for sigma and we estimate that by the samples we have from the population. I was wondering if someone can explain why smf.glm uses z-test.

Thank you

[josef...@gmail.com]

On Wed, Oct 6, 2021 at 3:45 PM Mohsen Vazirizade <s.m.vaz...@gmail.com> wrote:

Hi,

As you may now, both smf.ols and mf.glm can be used for fitting a regression model. Theoretically, they should bear the same results if the error is normally distributed. I ran 2 regressions (attached please find the code), once using ols and the other using glm. While, their results are identical, ols uses t-test but glm uses z-test. Consequently, their confidence intervals are different. 

To my knowledge, we always should use t-test since we do not know the actual value for sigma and we estimate that by the samples we have from the population. I was wondering if someone can explain why smf.glm uses z-test.

Thank you

In GLM the default is the same for all families. So, we only assume that the parameters are asymptotically normal distributed.

Using t-distribution for wald tests will often have better small sample behavior than the normal distribution, but that's mostly from monte carlo simulations and not a theoretical result.

However, you can set `use_t=True` as option in `fit(...)` and the distribution is changed from z to t and chi2 to F.

Josef

 

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