A monotonic property for iterative GLS in the two-way random effects model

This paper shows that maximum likelihood estimation for the two-way random effects model can be obtained as an iterated GLS procedure based on two subsets of the parameters: The first subset contains the regression coefficients β, and the second subset contains two variance components ratios, θ₁ and θ₂. Fixing θ_i (i=1, 2) and iterating between β and θ_j (j=1, 2 and j≠i), the sequence of θ_j's generated by this algorithm form a monotonic sequence. This result is an extension of Breusch's (1987) 'remarkable property' for iterative GLS from the one-way to the two-way model. Since the θ_i's are both between zero and one, a search over θ_i while iterating on the other θ_j and β will guard against the possibility of multiple local maxima of the likelihood function. However, such a search procedure can be relatively costly. This paper suggests an alternative computationally more efficient algorithm which makes use of Deaton's (1975) ridge-walking algorithm and Breusch's (1987) monotonic property. The proposed algorithm is shown to converge rapidly for the investment equation considered by Grunfeld (1958).