Appendix J

Random Intercept Model

(Sources: Cameron and Trivedi [1], Hsiao [2])

The random effects model can be written as

(J.1)

or

(J.2)

where . The variance–covariance matrix of is

(J.3)

Its inverse is

(J.4)

The individual-specific effects are assumed to be realizations of i.i.d. random variables with distribution and the error is i.i.d. . The model can be re-expressed as , where the error term has two components . For this reason the random effects model is also called the error components model or random intercept model. The random intercept model can be estimated by the maximum likelihood method.

When and are random and normally distributed, the logarithm of the likelihood function is

(J.5)

where

(J.6)

The maximum likelihood estimator (MLE) of is obtained by solving the following first-order conditions simultaneously:

(J.7)

(J.8)

(J.9)

(J.10)

Simultaneous solution of Equations J.7–J.10 is complicated. The Newton–Raphson iterative procedure can be used to solve for the MLE. The procedure uses an initial trial value of to start the iteration by substituting it into the formula

(J.11)

to obtain a revised estimate of . The process is repeated until the jth iterative solution is close to the th iterative solution .