9 Maximum Likelihood and Generalised Least Squares

Least squares was motivated by a criterion, minimising the sum of squared residuals, and justified by the Gauss–Markov theorem. Maximum likelihood starts instead from a full probability model for the data and chooses the parameter values under which the observed sample is most probable. For the normal linear model the two coincide, but maximum likelihood extends to models where least squares has nothing to say: limited dependent variables, systems of equations, models with non-spherical disturbances. It also supplies a general theory of efficiency and a general triple of tests. This section develops that theory and applies it to the linear model, first with spherical disturbances and then with the general covariance matrix \(\sigma ^2\Omega \), which leads to generalised least squares.

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