Hi, it is said that t=estimate / standard error. If we have Heteroskedasticity, error will be lower and t will be too high.
I do not get it why error is lower under Heteroskedasticity!? Please explain. My logic:
If variable X is getting bigger, error is going be bigger as well (as it growth with X) and t test will be smaller?
Heteroskedasticity - your model shows small errors in earlier periods and larger errors in later periods.
Think: you want no errors in your model (ideally). The small erros in the earlier periods will average with the larger errors in the outter period and produce a lower error.
Lower error - lower standard error - higher t - statistic (artificially). Since your t statistic is big (your alpha is small), you will reduce your chances of a type 1 (rejecting the null when when it is true).
my two cents!
… The small erros in the earlier periods will average with the larger errors in the outter period and produce a lower error…
this is not clear, Could you add few more sentences :)?
Your regression model will produce an estimate. That estimate is compared to the actual number that occur, the difference is your error (alpha).
The differential (error) could be smaller at some point and wider…so your model with average small errors (good) with wider erros(bad) and give you a smaller standard deviation.
To OP and Rasec:
Heteroscedasticity doesn’t necessarily mean small to large variation in the errors. The spread could go from large to small or from small to large to small (football-shaped), for example. Heteroscedasticity can take many forms, but is simply described as non-constant error variance across all settings (combinations) of the independent variables. The rationale for understatement is also incorrect. It’s not because you’re averaging big and small errors. In a general sense, it’s because the traditional OLS variance and standard errors are derived under the assumption of homoscedasticity. When this assumption is violated, the traditional formulas do not account for the non-constant variance (which is why we need heteroscedasticity robust standard errors). In other words, the true variance and standard errors are larger than the calculated (traditional) OLS standard errors/variances. Lastly, your inference that a larger magnitude t-stat implies a smaller alpha is incorrect. Alpha is independent of a calculated test statistic. Alpha is a chosen level of significance, rather than an observed level. The observed level of significance (p-value) will decrease as the t-statistic increases in magnitude. In this case, heteroscedasticity will actually increase the chance of making a Type I error. This is because the t-stats are inflated, leading to a more likely rejection of the null hypothesis. Hope this helps.
All i’m saying about the T-stat and alpha is this: as the violation occurs…your standard error will be smaller and higher t-stat.
Higher t-stat --> the rejection region gets smaller.
But this is fundamentally incorrect. The rejection region is only linked to the selected alpha and therefore, the selected critical values (same as selected significance level). As you increase the confidence level, and similarly, the magnitude of the critical values, (decrease alpha) you decrease the rejection region (area beyond the critical values). The opposite is also true. The calculated t-statistic has absolutely nothing to do with the size of the rejection region. The rejection region size is based completely on the chosen significance level as denoted by alpha and the corresponding critical values. To be clear, I do agree that the violation inflates the t-tests via understated standard errors.
Semantics! LOL Thanks for clearing it up!
It is not semantics lol, what you saying is not totally accurate. Look at the alpha as a benchmark or a threshold, if you pass it, you reject the null, if not you fail to reject the null. Do not confuse alpha with the p-values, they always shown at the regression table, so you must compare them with the alpha (“the benchmark”).
I agree with your idea; alpha is a threshold (benchmark) probability.
Also, I think many people confuse alpha and the p-value because of the way certain sources choose to define a p-value. Yes, it’s true, the p-value is the smallest level that you could set alpha to and still reject the null (I doubt this phrase increases most readers’ intuitions about a p-value, though). Following this idea can be bad practice and can cause cherry picking results (adjusting alpha up or down after seeing p-values just to find a “result” that fits your objective or only “conducting” tests that you know will give you significant results). Again, this definition (in my mind) isn’t very good.
A better definition (again, my opinion) for a p-value: Given that the null hypothesis is true, a p-value is the probability of seeing a result that is at least as contradictory to the null hypothesis.
This is part of the reason why we reject the null for (relatively) small p-values. For example, given a level of alpha (0.05), this p-value (0.01) is small (in fact, it is below our threshold for making a Type I error). So, if the null hypothesis is true, then there is only a 1% chance that we will observe something at least as contradictory to the null. Therefore, it is (very) unlikely that the null hypothesis is true (reject Ho).
I know this was a little bit off topic, but maybe it’ll be useful to someone.
Edit: I changed a few words to make it a little more precise.