To use OLS for inference, is it necessary in all cases that the premise of homoscedasticity is met?

I need to check the influence of some features (eg age, income...) on a variable y (whether or not I bought a property) in a real case. I thought about using linear regression interpreting the coefficients and I know that if there is heteroscedasticity it affects the reliability of the significance tests and confidence intervals are incorrect. But I'm not sure it would affect the regression coefficients as well.


1 Answer 1


Heteroscedasticity makes it so that the OLS estimator is not the best linear unbiased estimator of the regression slopes and makes it so that the usual standard errors (and the quantities based on them, such as p-values and confidence intervals) are incorrect. It doesn't affect the interpretation of the regression coefficients, which depends not the on estimation procedure or assumption but on the structure of the model. That is, if you specify $$E[Y|X] = \beta_0 + \beta_1 x_1 + ... + \beta_k x_k$$ the usual regression model, it doesn't matter how you estimate the coefficients or whether the assumptions are valid. The model itself tells you that for two observations that differ in $x_1$ by one unit but are the same on the other predictors, the expected difference in their outcomes is $\beta_1$. You might use a terrible method for estimating the coefficients, but that doesn't change their interpretation.

If you have heteroscedasticity, just use a robust standard error and carry on.

  • 5
    $\begingroup$ Either use robust standard errors and carry on or find a model specification that accounts for heteroskedasticity. $\endgroup$ May 22, 2022 at 19:42
  • $\begingroup$ So, under heteroskedasticity the coefficients are wrong too ? $\endgroup$
    – Alysson
    May 23, 2022 at 0:24
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    $\begingroup$ @Alysson they are not wrong. They are unbiased and consistent but not the best (most precise) estimates possible, unlike under homoscedasticity, in which case they are unbiased, consistent, and asymptotically efficient. This is a minor technical point that shouldn't mean much for most practical applications. $\endgroup$
    – Noah
    May 23, 2022 at 0:41
  • 1
    $\begingroup$ I found a quick demonstration here: stats.stackexchange.com/questions/249714/… $\endgroup$ May 23, 2022 at 5:21

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