Linear Least Squares

[beta v, r] = gls (y, x, o)

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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.

[beta sigma, r] = ols (y, x)

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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.