Variance-Covariance of disturbances in -xtgls- with autocorrelation within panels

[Sergio Prolo]

I've been using the -xtgls- command with options -panels(heteroskedastic) corr(psar1) igls- to estimate the parameters of a model for my thesis. Now, I'm writing the final document and want to be sure about the estimated variances (sigma) and autocorrelation (rho) parameters.

The documentation states that the variance matrix of disturbances 𝛀 is the result of a Kronecker product of Sigma (a matrix with sigma_i values along the main diagonal, 0 otherwise) and the Identity matrix (in the case where errors are not autocorrelated within panels). Also, it states that '[...] The individual identity matrices along the diagonal of 𝛀 may be replaced with more general structures to allow for serial correlation'. However, the documentation does not detail how these 'more general structures' are represented.

It is my understanding that the variance matrix of disturbances for an AR(1) model has a factor of (1 - rho²)^-1 multiplying all its values. If that is the case, then the main diagonal of 𝛀 would also have its main diagonal multiplied by this factor, giving an actual variance estimation of sigma_i²/(1 - rho_i²) for each panel i. Am I right? If so, are the values stored in e(Sigma) already multiplied by this factor, or are they composed of sigma_i only?

Thank you for your time. Best regards.

[Manh Hoang Ba]

I don't know how to type mathematical formulas in Statalist, so I can't present them directly here, but since xtgls performs the Parks-Kmenta FGLS estimation, you can find the relevant matrix formulas in Kmenta's Elements of Econometrics textbook.