On Computing Maximum-Likelihood Estimates of the Unbalanced Two-Way Random-Effects Model

Abstract

This article applies the ECME algorithm to derive an easily implemented iterative feasible generalized least squares procedure for calculating maximum-likelihood estimates of the parameters of the unbalanced two-way random-effects model. The algorithm increases the log-likelihood monotonically and the fitted variance components are guaranteed to be non negative. This article applies the algorithm in an example.

Keywords:

• EM algorithm
• Error components
• Incomplete panel
• Maximum likelihood

2000 Mathematics Subject Classification:

• 62p20

Notes

Liu and Rubin (1994) described the ECME algorithm's advantages and investigate its properties.

For the purposes of deriving the imputed log-likelihood, the variables in Z are treated as fixed.

Variations on this ECME algorithm are possible. For example, instead of calculating a new fit for during the first CM step, one could calculate the new fit during the second CM step as , where u ⁺ =  y  −  Z γ⁺. In this case, however, the matrix (Ω ⁻¹)⁺, which is used in the second CM step, is necessarily evaluated at , , and —that is, unlike in the algorithm described in the text, (Ω ⁻¹)⁺ cannot be calculated based on a new fit for , for that fit is calculated during the second CM step. Moreover, an extra calculation of Ω⁻¹ is required because after is calculated in the second CM step, we set , , and for the matrix (Ω ⁻¹) ^c that is used when the first CM step is repeated. On the other hand, for the algorithm described in the text, the matrix (Ω ⁻¹) ^c , which is used when the first CM step is repeated, is the matrix (Ω ⁻¹)⁺ used in the prior second CM step.

Meng and van Dyk (1998) noted that, without extra computational effort, Newton–Raphson iterations can produce negative variance estimates.

Baltagi and Li (1992) did not report an estimate for β₀. I obtained an estimate of −58.33.

The value of the log-likelihood in (1), including the constant term, was −1095.249.