Train_Discrete Choice Methods with Simulation - Chapter 8

8 Numerical Maximization Motivation Most estimation involves maximization of some function, such as the likelihood function, the simulated likelihood function, or squared moment conditions. This chapter describes numerical procedures that are used to maximize a likelihood function. | P1 GEM IKJ P2 GEM IKJ QC GEM ABE T1 GEM August 20 2002 12 39 Char Count 0 CB495-08Drv CB495 Train KEY BOARDED Part II Estimation 187 P1 GEM IKJ CB495-08Drv P2 GEM IKJ QC GEM ABE CB495 Train KEY BOARDED T1 GEM August 20 2002 12 39 Char Count 0 188 T1 GEM P1 GEM IKJ P2 GEM IKJ QC GEM ABE CB495-08Drv CB495 Train KEY BOARDED August 20 2002 12 39 Char Count 0 8 Numerical Maximization Motivation Most estimation involves maximization of some function such as the likelihood function the simulated likelihood function or squared moment conditions. This chapter describes numerical procedures that are used to maximize a likelihood function. Analogous procedures apply when maximizing other functions. Knowing and being able to apply these procedures is critical in our new age of discrete choice modeling. In the past researchers adapted their specifications to the few convenient models that were available. These models were included in commercially available estimation packages so that the researcher could estimate the models without knowing the details of how the estimation was actually performed from a numerical perspective. The thrust of the wave of discrete choice methods is to free the researcher to specify models that are tailor-made to her situation and issues. Exercising this freedom means that the researcher will often find herself specifying a model that is not exactly the same as any in commercial software. The researcher will need to write special code for her special model. The purpose of this chapter is to assist in this exercise. Though not usually taught in econometrics courses the procedures for maximization are fairly straightforward and easy to implement. Once learned the freedom they allow is invaluable. Notation The log-likelihood function takes the form . N 1 ln Pn N where Pn is the probability of the observed outcome for decision maker n N is the sample size and is a K x 1 vector of parameters. In this chapter we divide the log-likelihood .