Advanced Econometrics Takeshi Amemiya

Iterative Methods for Obtaining the Maximum Likelihood Estimator

The iterative methods we discussed in Section 4.4 can be used to calculate a root of Eq. (9.2.8). For the logit and probit models, iteration is simple because of the global concavity proved in the preceding section. Here we shall only discuss the method-of-scoring iteration and give it an interesting interpreta­tion.

As we noted in Section 4.4, the method-of-scoring iteration is defined by

dlogL

dfi

where fa is an initial estimator of Д, and p2 is the second-round estimator. The iteration is to be repeated until a sequence of estimators thus obtained con­verges. Using (9.2.8) and (9.2.12), we can write (9.2.24) as

where we have defined Ft = F{s.'fix) and fj =/(x'/?1).

An interesting interpretation of the iteration (9.2.25) is possible. From

(9.2.1) we obtain

У1 = Р(х'А) + щ, (9.2.26)

where Eu, = 0 and Vu, = F(x$,)[l - F('P0)]. This is a heteroscedastic non­linear regression model. Expanding F(x')30) in a Taylor series around y?0 = px

and rearranging terms, we obtain

У і ~ Ft -yfiX'Ji =/-x^0 + Uj. (9.2.27)

Thus P2 defined in (9.2.25) can be interpreted as the weighted least squares (WLS) estimator ofp0 applied to (9.2.27) with Vu, estimated by F,(l — Ft). For this reason the method-of-scoring iteration in the QR model is sometimes referred to as the nonlinear weighted least squares (NLWLS) iteration (Walker and Duncan, 1967).