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.

  • $\begingroup$ 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. $\endgroup$ – Josef Sep 24 at 14:49
  • $\begingroup$ 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. $\endgroup$ – confused student Sep 24 at 15:07
  • $\begingroup$ 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. $\endgroup$ – Josef Sep 24 at 15:23

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