# Covariance matrix computation VECM, Lutkepohl (2005) p.287

In Lutkepohls book "New introduction to multiple time series analysis" (2005) on p.287 is outlined how to calculate your estimated parameters $$[\hat \Pi,\hat \Gamma]$$ sample covariance matrix $$\hat \Sigma_{co}$$. It is the kronecker product between the two matrices, $$\Sigma_u$$ and another matrix, lets call it $$\Omega$$. A consistent estimate for $$\Sigma_u$$ is its sample counterpart $$\hat \Sigma_u$$. An estimator for $$\Omega$$ can be obtained by $$\hat \Omega = T \begin{bmatrix} Y_{-1}Y_{-1}' & Y_{-1}\Delta X' \\ \Delta X Y_{-1} & \Delta X \Delta X'\\ \end{bmatrix}^{-1}$$

I get very different results when I calculate $$\hat \Sigma_{co}$$ myself, compared to the results obtained by using the python package statsmodels. However, when I calculate $$\Omega$$ without scaling by "T", I get the exact same results as the statsmodels results. This led me to inspect the source code of statsmodels to see how they calculate $$\Omega$$, I notice in the construction of the definition "cov_params_default", that they do not scale $$\Omega$$ by a factor T. (They calculate omega a little different than I do, they use the equation 3 lines below equation 7.2.6 also on p.287, in their construction of omega they should scale by 1/T, point is, they do not use T in any way.)

My question is if anyone could explain why we should not scale $$\hat \Omega$$ by T when calculating the covariance matrix $$\hat \Sigma_{co}$$? My assumption is that the statsmodels people are right ofcourse.

• Check the scaling of sigma_u which might missing the same T so it cancels. Often, matrices like (X'X)^-1 for OLS are not scaled by the number of observations, because they only show up in products where the T, at least partially, cancels. Sep 24, 2020 at 14:49
• My sigma_u and the sigma_u given by the statsmodels package are equivalent. So i dont think that solves my confusion. The sigma_u in statsmodels is calculated by T^-1(u.T@u) as can be seen in the source code. Sep 24, 2020 at 15:07
• Ok, I think the T should not be there for the covariance of the parameter estimates. Equation 7.2.6 is for sqrt(T) times covariance of parameter estimates. This is analogous to OLS (X'X)^-1 * sigma2, where sigma2 is the estimate of the residual variance. The T and sqrt of T are needed for convergence statements in the theory but not for computing cov_params. Sep 24, 2020 at 15:23