— Function File: [beta, v, r] = gls (y, x, o)

Generalized least squares estimation for the multivariate model y = x b + e with mean (e) = 0 and cov (vec (e)) = (s^2) o, where y is a t by p matrix, x is a t by k matrix, b is a k by p matrix, e is a t by p matrix, and o is a t p by t p matrix.

Each row of y and x is an observation and each column a variable. The return values beta, v, and r are defined as follows.

beta

The GLS estimator for b.

v

The GLS estimator for s^2.

r

The matrix of GLS residuals, r = y - x beta.

— Function File: [beta, sigma, r] = ols (y, x)

Ordinary least squares estimation for the multivariate model y = x b + e with mean (e) = 0 and cov (vec (e)) = kron (s, I). where y is a t by p matrix, x is a t by k matrix, b is a k by p matrix, and e is a t by p matrix.

Each row of y and x is an observation and each column a variable.

The return values beta, sigma, and r are defined as follows.

beta

The OLS estimator for b, beta = pinv (x) * y, where pinv (x) denotes the pseudoinverse of x.

sigma

The OLS estimator for the matrix s,

               sigma = (y-x*beta)'
                 * (y-x*beta)
                 / (t-rank(x))
          

r

The matrix of OLS residuals, r = y - x * beta.