# Correlated error restrictions and OLS

I have a VAR model of the form

$$Y_t = \beta Y_{t-1} + \varepsilon_t$$

Where $$Y_t$$ and $$\varepsilon_t$$ are $$n\times 1$$ vectors, and $$\beta$$ is an $$n \times n$$ matrix.

The residuals $$\varepsilon_{t,i}$$ ($$i=1,2,...,n$$) are each normally distributed with a mean of zero, but they may be correlated with one-another (i.e. the variance-covariance matrix $$\Sigma$$ is non-diagonal). There is no serial correlation, and $$\Sigma$$ does not change over time.

I have a matrix of data $$D$$, which has $$m$$ observation (i.e. $$D$$ is a $$n \times m$$ matrix). I wish to estimate the elements of $$\beta$$ and $$\Sigma$$ using OLS estimation (and/or maximum likelihood, if that would be different).

How I estimate this model, when errors are known to be correlated?

I've looked at other posts on OLS with correlated errors, but these don't match the constraints of my problem (i.e. errors correlated with one-another, but not serially correlated), and tend to focus on heteroscedasticity.

• You can consider generalized least squares, or maximum likelihood model that accounts for the dependence. Feb 28, 2022 at 18:12
• @DifferentialPleiometry, equation-by-equation OLS is the efficient estimator for VAR where the right-hand-side variables are the same in all equations. In that case, GLS collapses to equation-by-equation OLS. MLE collapses to equation-by-equation OLS in case the errors are normal, and they are here. So equation-by-equation OLS seems to be the way to go. Feb 28, 2022 at 20:16
• @RichardHardy Purportedly SUR estimation is more efficient than equation-by-equation estimation. Feb 28, 2022 at 20:31
• @DifferentialPleiometry, generally, yes, but in this case, no. This is a well known result that applies not only to VAR models but also to any SUR system whenever the right-hand-side variables are the same across equations. Many econometric textbooks contain this result. Feb 28, 2022 at 20:39
• @RichardHardy Are you saying the equation-by-equation OLS dominate these generalizations, or that these generalizations do not dominate equation-by-equation OLS? Feb 28, 2022 at 20:46

Estimate your model equation-by-equation using OLS. This is the efficient estimator for a VAR where the right-hand-side variables are the same in all equations. In that case, GLS collapses to equation-by-equation OLS. MLE collapses to equation-by-equation OLS in case the errors are normal (and you say they are). A more detailed explanation could be found e.g. in Chapter 3 of Lütkepohl "New Introduction to Multiple Time Series Analysis" (2005).

• Can you cite a source that isn't so heavily pay-walled? Feb 28, 2022 at 20:57
• @DifferentialPleiometry, Wikipedia shows the first equivalence. The second one follows from the equivalence between OLS and MLE under normality. Feb 28, 2022 at 21:00
• I would appreciate a comment from the downvoter on what may be wrong with my answer. Constructive criticism can be a great thing. Feb 28, 2022 at 21:01
• Thanks @RichardHardy! Can you point me to the relevant chapter/section of Lütkepohl's book? Feb 28, 2022 at 21:52
• @Mich55, it has been a while since I read the book, but the table of contents suggests it is chapter 3. Mar 1, 2022 at 14:01

The correlation structure $$\Sigma$$ of the error terms has an impact on the standard error estimate of the regression coefficients $$\beta$$s. That is, you can still use OLS to estimate $$\beta$$s but the estimate of covariance of $$\hat\beta$$s underestimates the true covariance because of the non-diagonal $$\Sigma$$, and hence you will draw a wrong inference. The easiest solution would be to use a sandwich-estimator of the covariance of $$\hat\beta$$s, where $$\hat\beta$$s are the OLS estimates of $$\beta$$s (under the assumption of independent errors $$\epsilon_i$$).

• Something like that would be the case under serial correlation but not in the setting of the OP. Feb 28, 2022 at 20:50
• I strongly disagree with this comment because the sandwich-estimator does not make any assumption about the correlation structure, and by definition, it should work even with the absence of serial correlation. However, I noticed that my answer does not help for the second part of your problem where you want to estimate the correlation using your data matrix D. Feb 28, 2022 at 23:37
• The problem with sandwich is that in absence of serial correlation, it is an inefficient estimator (since it is ignores what is known and estimating things that are known) while OLS provides an efficient one. Mar 1, 2022 at 14:04