In many Kaggle competitions where linear regression has been applied, I see people plot the y distribution and then take the log of (or other transformation of) the dependent variable to make y normal distributed. What would happen if we don't do this step: transform the dependent variable into a normal distribution?
I have a few thoughts but wasn't sure which one is more correct and which one is wrong.
- if y is skewed, after running the linear regression model, we could have the heteroscedasticity problem (variance is not constant along the predicted y)
- we know the linear regression coefficients are still unbiased even if y is skewed; however, the t-test of the coefficient wouldn't make sense anymore because y is not a normal distribution.
- Because X is usually standardized, y's distribution should match X's normal distribution.