Model Specification and
A DAPM MAY: (1) provide a complete characterization of the joint distribution of all of the variables being studied; or (2) imply restrictions on some moments of these variables, but not reveal the form of their joint distribution. A third possibility is that there is not a well-developed theory for the joint distribution of the variables being studied. Which of these cases obtains for the particular DAPM being studied determines the feasible estimation strategies; that is, the feasible choices of D in the definition of an estimation strategy. This chapter introduces the maximum likelihood (ML), generalized method of moments (GMM), and linear least-squares projection (LLP) estimators and begins our development of the interplay between model formulation and the choice of an estimation strategy discussed in Chapter 1.
Suppose that a DAPM yields a complete characterization of the joint distribution of a sample of size T on a vector of variablesdenote the family of joint density functions of implied by the DAPM and indexed by the K-dimensional parameter vector β. Suppose further that the admissible parameter space associated with this DAPM is Θ ⊆ ℝK and that there is a unique β0 ∈ Θ that describes the true probability model generating the asset price data.
In this case, we can take LT(β) to be our sample criterion function— called the likelihood function of the data—and obtain the maximum likelihood (ML) estimatorby maximizing LT(β). In ML estimation, we start with the joint density function of evaluate the random variable at the realization comprising the observed historical sample, and then maximize the value of this density over the choice of β ∈ Θ. This amounts to maximizing,