Time Series Analysis: Forecasting and Control, Fourth Edition

CHAPTER SEVEN

Model Estimation

The identification process having led to a tentative formulation for the model, we then need to obtain efficient estimates of the parameters. After the parameters have been estimated, the fitted model will be subjected to diagnostic checks and tests of goodness of fit. As pointed out by R. A. Fisher, for tests of goodness of fit to be relevant, it is necessary that efficient use of the data should have been made in the fitting process. If this is not so, inadequacy of fit may simply arise because of the inefficient fitting and not because the form of the model is inadequate. This chapter contains a general account of likelihood and Bayesian methods for estimation of the parameters in the stochastic model. Throughout the chapter, bold type is used to denote vectors and matrices. Thus, X = {x_ij} is a matrix with x_ij an element in the ith row and jth column and X′ is the transpose of the X.

7.1 STUDY OF THE LIKELIHOOD AND SUM-OF-SQUARES FUNCTIONS

7.1.1 Likelihood Function

Suppose that we have a sample of N observations z with which we associate an N-dimensional random variable, whose known probability distribution p(z|ξ) depends on some unknown parameters ξ. We use the vector ξ to denote a general set of parameters and, in particular, it could refer to the p + q + 1 parameters (ϕ, θ, ) of the ARIMA model.

Before the data are available, p(z|ξ) will associate ...