Introduction

Under certain conditions known as regularity conditions, the maximum likelihood estimator introduced in Chapter 1 possesses a number of important statistical properties and the aim of this chapter is to derive these properties. In large samples, the maximum likelihood estimator is consistent, efficient and normally distributed. In small samples, it satisfies an invariance property, is a function of sufficient statistics and in some, but not all, cases, is unbiased and unique. As the derivation of analytical expressions for the finite-sample distributions of the maximum likelihood estimator is generally complicated, computationally intensive methods based on Monte Carlo simulations or series expansions are used to examine some of these properties.

The maximum likelihood estimator encompasses many other estimators often used in econometrics, including ordinary least squares and instrumental variables (Chapter 5), nonlinear least squares (Chapter 6), the Cochrane-Orcutt method for the autocorrelated regression model (Chapter 7), weighted least squares estimation of heteroskedastic regression models (Chapter 8) and the Johansen procedure for cointegrated nonstationary time series models (Chapter 18).

Preliminaries

Before deriving the formal properties of the maximum likelihood estimator, four important preliminary concepts are reviewed. The first presents some stochastic models of time series and briefly discusses their properties. The second is concerned with the convergence of a sample average to its population mean as T →∞, known as the weak law of large numbers. The third identifies the scaling factor ensuring convergence of scaled random variables to nondegenerate distributions.