FIGURE 2.2 Population regression line (data of Table 2.1).
4I am indebted to James Davidson on this perspective. See James Davidson, Econometric Theory, Blackwell Publishers, Oxford, U.K., 2000, p. 11.
5In the present example the PRL is a straight line, but it could be a curve (see Figure 2.3).
CHAPTER TWO: TWO-VARIABLE REGRESSION ANALYSIS: SOME BASIC IDEAS 41
This figure shows that for each X (i.e., income level) there is a population of Y values (weekly consumption expenditures) that are spread around the (conditional) mean of those Y values. For simplicity, we are assuming that these Y values are distributed symmetrically around their respective (conditional) mean values. And the regression line (or curve) passes through these (conditional) mean values.
With this background, the reader may find it instructive to reread the definition of regression given in Section 1.2.
2.2 THE CONCEPT OF POPULATION REGRESSION FUNCTION (PRF)
From the preceding discussion and Figures. 2.1 and 2.2, it is clear that each conditional mean E(Y | Xi) is a function of Xi, where Xi is a given value of X. Symbolically,
where f (Xi) denotes some function of the explanatory variable X. In our example, E(Y | Xi) is a linear function of Xj-. Equation (2.2.1) is known as the conditional expectation function (CEF) or population regression function (PRF) or population regression (PR) for short. It states merely that the expected value of the distribution of Y given Xi is functionally related to Xi. In simple terms, it tells how the mean or average response of Y varies with X.
What form does the function f (Xi) assume? This is an important question because in real situations we do not have the entire population available for examination. The functional form of the PRF is therefore an empirical question, although in specific cases theory may have something to say. For example, an economist might posit that consumption expenditure is linearly related to income. Therefore, as a first approximation or a working hypothesis, we may assume that the PRF E(Y | Xi) is a linear function of Xi, say, of the type
where fa and fa are unknown but fixed parameters known as the regression coefficients;fa and fa are also known as intercept and slope coefficients, respectively. Equation (2.2.1) itself is known as the linear population regression function. Some alternative expressions used in the literature are linear population regression model or simply linear population regression. In the sequel, the terms regression, regression equation, and regression model will be used synonymously.
In regression analysis our interest is in estimating the PRFs like (2.2.2), that is, estimating the values of the unknowns fa and fa on the basis of observations on Y and X. This topic will be studied in detail in Chapter 3.
42 PART ONE: SINGLE-EQUATION REGRESSION MODELS
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