Generalized Method of Moments and Macroeconomics

Article excerpt

We consider the contribution to the analysis of economic time series of the generalized method-of-moments estimator introduced by Hansen. We outline the theoretical contribution, conduct a small-scale literature survey, and discuss some ongoing theoretical research.

KEY WORDS: Bootstrap; Efficiency bounds; HAC estimation; Heteroscedasticity; Serial correlation; Time-series.

1. INTRODUCTION

The three most important developments in time series econometrics in the last 25 years arguably are generalized method-of-moments (GMM) estimation, vector autoregressions (VARs), and the analysis of nonstationary time series (unit roots and cointegration). This article surveys the role of GMM in macroeconomic time series.

The seminal contribution to the literature on GMM was made by Lars Peter Hansen (1982); his work is the focal point of our survey. Hansen's article had important antecedents in the econometrics literature. Two-stage least squares was developed independently by Theil (1953) and Basmann (1957). Basmann (1960) provided an alternative derivation similar to minimum chi-squared estimation (see also Rothenberg 1973). The formulation of two-stage least squares as an optimal instrumental variable (IV) estimator under conditional homoscedasticity and a test for overidentifying restrictions was proposed by Sargan (1958, 1959). These methods were extended to nonlinear models by Amemiya (1974, 1977), Jorgenson and Laffont (1974), Gallant (1977), and Gallant and Jorgenson (1979), Gallant and Jorgenson also proposed a test statistic that ties naturally to Hansen's (1982) test of overidentifying restrictions. In addition, several articles by White (1980, 1982a, 1982b) can be viewed as GMM applied to cross-section linear regression.

The starting point for this article is not this earlier literature, however, but rather Hansen's (1982) contribution. Sections 2-4 exposit the rationale, structure, and impact on applied work of Hansen's article. Section 2 presents a simple rational forecasting example to illustrate why in many applications generalized least squares (GLS) is not an alternative to GMM. Section 3 defines notation and illustrates the use of GMM to estimate a nonlinear time series model (the consumption-based capital asset pricing model), and then outlines a linear dynamic panel model. Section 4 reports a small survey of economics journals, examining the prevalence of GMM and other estimation methods in empirical time series work.

Section 5 turns to subsequent literature that builds on the work of Hansen (1982). This section reviews some current issues of research interest for time series GMM: efficiency bounds, feasible attainment of efficiency bounds, weight matrix estimation, the time series bootstrap, and empirical likelihood methods. Section 6 concludes the article.

2. WHY GENERALIZED METHOD OF MOMENTS?

We use a simple example to motivate use of GMM in time series applications. Suppose that we wish to test the "rationality" of a scalar variable [x.sub.t] as an n period ahead predictor of a variable [q.sub.t+n]; the null is [E.sub.t][q.sub.t+n] = [X.sub.t], where for the moment we leave unspecified the information set used in forming the expectation. The variable [x.sub.t] might be the expectation of [q.sub.t+n] reported by a survey. Alternatively, [x.sub.t] might be a market-determined variable posited by economic theory to be the expectation of [q.sub.t+n] (e.g., [q.sub.t+n] = spot rate, [x.sub.t] = n period-ahead-forward rate). Let [u.sub.t] denote the expectational error, [u.sub.t] = [q.sub.t+n] - [E.sub.t][q.sub.t+n] = [q.sub.t+n] - [x.sub.t]. (The expectational error [u.sub.t] is dated t rather than t + n for consistency with the dating of regression residuals in the main part of this article.) Under the null, [Ex.sub.t][u.sub.t] = O, and [u.sub.t] follows a moving average process of order n - 1. …