Goodness-of-Fit and Hypothesis Testing
IN THIS CHAPTER we explore in more depth the testing of hypotheses implied by a DAPM. The types of hypotheses considered are separated into two categories: (1) overall goodness-of-fit tests of whether a model is consistent with the data, in a sense that we make precise, and (2) tests of constraints on the parameters in a DAPM. For the purposes of this chapter we focus on classical testing in a stationary and ergodic statistical environment. The new problems induced by time trends and applications of Bayesian methods are discussed in the context of specific empirical issues.
In the context of GMM estimation, the sample criterion function is constructed from sample counterparts to the M moment conditions E[h(zt,θ0)] = 0, namely HT(θT) = (1/T) Σt h(zt,θT) = 0. When M > K, there are more moment conditions than unknown parameters and, consequently, estimation proceeds by minimizing the GMM criterion functionwhere the distance matrix has been chosen optimally as discussed in Chapter 3. In minimizing QT(θ) over the choice of θ ∈ Θ, the GMM estimator is chosen to set K linear combinations of the M sample moment conditions HT to zero (the K first-order conditions):
Yet, if the model is correctly specified, all M sample moment equations HT(θT) should be close to zero. This observation suggests that we can construct a goodness-of-fit test of the model by examining whether linear combinations of HT(θT) that are not set to zero in estimation are in fact close to zero.
Conveniently, it turns out that the minimized value of the GMM criterion function, scaled by sample size, TQT(θT) is a goodness-of-fit test based