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Fumio Hayashi

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eBook - ePub


Fumio Hayashi

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The most authoritative and comprehensive synthesis of modern econometrics available Econometrics provides first-year graduate students with a thoroughly modern introduction to the subject, covering all the standard material necessary for understanding the principal techniques of econometrics, from ordinary least squares through cointegration. The book is distinctive in developing both time-series and cross-section analysis fully, giving readers a unified framework for understanding and integrating results. Econometrics covers all the important topics in a succinct manner. All the estimation techniques that could possibly be taught in a first-year graduate course, except maximum likelihood, are treated as special cases of GMM (generalized methods of moments). Maximum likelihood estimators for a variety of models, such as probit and tobit, are collected in a separate chapter. This arrangement enables students to learn various estimation techniques in an efficient way. Virtually all the chapters include empirical applications drawn from labor economics, industrial organization, domestic and international finance, and macroeconomics. These empirical exercises provide students with hands-on experience applying the techniques covered. The exposition is rigorous yet accessible, requiring a working knowledge of very basic linear algebra and probability theory. All the results are stated as propositions so that students can see the points of the discussion and also the conditions under which those results hold. Most propositions are proved in the text.For students who intend to write a thesis on applied topics, the empirical applications in Econometrics are an excellent way to learn how to conduct empirical research. For theoretically inclined students, the no-compromise treatment of basic techniques is an ideal preparation for more advanced theory courses.

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Finite-Sample Properties of OLS


The Ordinary Least Squares (OLS) estimator is the most basic estimation procedure in econometrics. This chapter covers the finite- or small-sample properties of the OLS estimator, that is, the statistical properties of the OLS estimator that are valid for any given sample size. The materials covered in this chapter are entirely standard. The exposition here differs from that of most other textbooks in its emphasis on the role played by the assumption that the regressors are “strictly exogenous.”
In the final section, we apply the finite-sample theory to the estimation of the cost function using cross-section data on individual firms. The question posed in Nerlove’s (1963) study is of great practical importance: are there increasing returns to scale in electricity supply? If yes, microeconomics tells us that the industry should be regulated. Besides providing you with a hands-on experience of using the techniques to test interesting hypotheses, Nerlove’s paper has a careful discussion of why the OLS is an appropriate estimation procedure in this particular application.

1.1 The Classical Linear Regression Model

In this section we present the assumptions that comprise the classical linear regression model. In the model, the variable in question (called the dependent variable, the regressand, or more generically the left-hand [-side] variable) is related to several other variables (called the regressors, the explanatory variables, or the right-hand [-side] variables). Suppose we observe n values for those variables. Let yi be the i-th observation of the dependent variable in question and let (xi1, xi2, . . . , xiK) be the i-th observation of the K regressors. The sample or data is a collection of those n observations.
The data in economics cannot be generated by experiments (except in experimental economics), so both the dependent and independent variables have to be treated as random variables, variables whose values are subject to chance. A model is a set of restrictions on the joint distribution of the dependent and independent variables. That is, a model is a set of joint distributions satisfying a set of assumptions. The classical regression model is a set of joint distributions satisfying Assumptions 1.1–1.4 stated below.

The Linearity Assumption

The first assumption is that the relationship between the dependent variable and the regressors is linear.
Assumption 1.1 (linearity):
where ÎČ’s are unknown parameters to be estimated, and Δi is the unobserved error term with certain properties to be specified below.
The part of the right-hand side involving the regressors, ÎČ1xi1 + ÎČ2xi2 + · · · + ÎČKxiK, is called the regression or the regression function, and the coefficients (ÎČ’s) are called the regression coefficients. They represent the marginal and separate effects of the regressors. For example, ÎČ2 represents the change in the dependent variable when the second regressor increases by one unit while other regressors are held constant. In the language of calculus, this can be expressed as ∂yi/∂xi2 = ÎČ2. The linearity implies that the marginal effect does not depend on the level of regressors. The error term represents the part of the dependent variable left unexplained by the regressors.
Example 1.1 (consumption function): The simple consumption function familiar from introductory economics is
where CON is consumption and YD is disposable income. If the data are annual aggrega...