Consider the time series to be estimated with OLS:

$Y_t = a + \bf{X}_t\bf{b} + e_t$

where $Y_t$ is skewed, $\bf{X}_t$ is a vector of regressor values at time $t$, $\bf{b}$ is a vector of coefficient estimates.

Scenario (1): The design matrix is such that the skewness in the response is perfectly accounted for by the design matrix - i.e., there is no non-normality in the residuals. Are our coefficient estimates minimum variance?

Scenario (2): The design matrix is such that the skewness in the response flows into the residuals (suppose for this scenario that the dependent has major skew and the residuals has smaller skew but still comes up on a hypothesis test of non-normality over the estimates of the residuals). Are our coefficient estimates minimum variance? What estimator should I use (I only know of 1: a variant of the Wild bootstrap that accounts for skew)?


This is answered by Gauss-Markov theorem. If the residuals are mean zero, homoscedastic and uncorrelated the OLS estimates have minimum variance among all unbiased estimates. Since there are distributions which are asymmetrical, but with zero mean, as long as you only care about minimum variance and not normality, the answer to both of your question is yes.

A side note. Skewness is a property of single distribution. In time series the sample is a draw from stochastic process. Unless you are working with strictly stationary variables each $Y_t$ might have different distribution.

  • $\begingroup$ +1 but I think it may also be worth mentioning that in scenario (1) the usual inference you'd use for least squares models is still valid whereas it is not in scenario (2). $\endgroup$
    – Macro
    Aug 22 '12 at 12:54
  • $\begingroup$ @Macro, well asymptotically it is valid in scenario (2) also. Of course it depends then on sample size and the third moment of errors. $\endgroup$
    – mpiktas
    Aug 22 '12 at 13:52
  • $\begingroup$ @mpiktas If the residuals have mean 0, homoscedasticity and uncorrelated but are skewed $\textbf{and are minimum variance}$, then why does my bootstrap book advertise a benefit of one of the variants of the Wild is that it "accounts for skewness"? $\endgroup$
    – user13253
    Aug 22 '12 at 13:57
  • 2
    $\begingroup$ @user1125946 - The bootstrap can be used to estimate the sampling distribution of the coefficient estimates, which distribution may itself be skewed. This does not change the minimum variance property of OLS under the Gauss-Markov assumptions, however. It just avoids the perhaps-inappropriate use of the (symmetric) Normal or t distributions when calculating, e.g., confidence intervals for the coefficients. $\endgroup$
    – jbowman
    Aug 22 '12 at 17:17

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