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Why is OLS estimator normally distributed?
The reason this estimator is normally distributed is that it is a linear function of the underlying error vector (as written in the equation you have shown), which is normally distributed under the model assumptions.
Are OLS estimators normally distributed?
However, under non-pathological behaviour for the explanatory variables, and assuming that the error terms are IID with finite variance, this is usually sufficient to allow application of the CLT, which means that the OLS estimator is approximately normally distributed when n is large.
What are the assumptions for linear OLS regression?
In order to present the context for non-normality it may help to review the assumptions for linear OLS regression, which are: Weak exogeneity. This essentially means that the predictor variables, x, can be treated as fixed values, rather than random variables.
Do you need normal distribution of independent variables?
My independent variables are, however, not normally distributed (moderately positively skewed). A square root transformation was successful in normalising the distribution of the IVs. However, after running the logistic regression on the normalised data, I get some very strange results – huge Odds Ratios and Confidence Intervals.
Is the marginal distribution of dependent variables normal?
1) If the distribution of the residuals within each group is normal, and the groups have different means (i.e. in a linear regression there is a slope different from 0) then the marginal distribution of the dependent variable may very well be NOT normal at all.
When do the coefficient estimates are asymptotically distributed?
For example, if the errors follow a t -distribution with 2.01 degrees of freedom (which is not clearly more long-tailed than the errors seen in the OP’s data), the coefficient estimates are asymptotically normally distributed, but it takes much longer to “kick in” than it does for other shorter-tailed distributions.