5.2 Maximum likelihood estimation
The log-likelihood is maximised when the following first-order conditions are satisfied:[18]
(21)
Differentiation of (20) gives:[19]
(22)
In considering the terms in (22) it should be remembered that, even if every individual has the same general form of utility function, the individual utilities depend on the personal characteristics in 
.
It is of interest to rewrite the first-order conditions, using (21) and (22) as giving, for all 
:
(23)
This has the simple interpretation that the aim of this method is to make the first derivatives of utility in the observed hours points on average equal to the weighted average of derivatives of utility over all possible hours points. The weights for each individual are equal to the probabilities of each discrete hours level. Although this is an interesting interpretation of the first-order conditions, it does not provide any practical help in trying to solve the highly nonlinear set of equations.
The solution (the set of maximum likelihood estimates for all 
s) can be obtained using numerical methods involving a sequence of iterations which lead efficiently from an arbitrary starting point to the solution. A discussion of Newton’s method, which is often used to maximise functions, can be found in the appendix.
The iterative method involves repeatedly solving the following matrix equation, where 
denotes the vector of parameters in the 
th iteration:
(24)
and the first and second derivatives are evaluated using the parameters . Furthermore, it can be shown that the inverse of the matrix of second derivatives at the final iteration provides an estimate of the variance-covariance matrix of parameter estimates. The application of Newton’s method in the present context therefore requires the second derivatives of the likelihood function. Differentiating (22) again with respect to parameter 
gives:
(25)
where
(26)
An example using this procedure is described in the following section.
