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FIGURE 2.4 Regression lines based on two different samples.

to "fit" the scatters reasonably well: SRFi is based on the first sample, and SRF2 is based on the second sample. Which of the two regression lines represents the "true" population regression line? If we avoid the temptation of looking at Figure 2.1, which purportedly represents the PR, there is no way we can be absolutely sure that either of the regression lines shown in Figure 2.4 represents the true population regression line (or curve). The regression lines in Figure 2.4 are known as the sample regression lines. Sup-

CHAPTER TWO: TWO-VARIABLE REGRESSION ANALYSIS: SOME BASIC IDEAS 49

posedly they represent the population regression line, but because of sampling fluctuations they are at best an approximation of the true PR. In general, we would get N different SRFs for N different samples, and these SRFs are not likely to be the same.

Now, analogously to the PRF that underlies the population regression line, we can develop the concept of the sample regression function (SRF) to represent the sample regression line. The sample counterpart of (2.2.2) may be written as

where Y is read as "Y-hat" or "Y-cap" Yj = estimator of E(Y | Xj) fa = estimator of fa fa = estimator of fa

Note that an estimator, also known as a (sample) statistic, is simply a rule or formula or method that tells how to estimate the population parameter from the information provided by the sample at hand. A particular numerical value obtained by the estimator in an application is known as an estimate.13

Now just as we expressed the PRF in two equivalent forms, (2.2.2) and (2.4.2), we can express the SRF (2.6.1) in its stochastic form as follows:

where, in addition to the symbols already defined, ui denotes the (sample) residual term. Conceptually Ui is analogous to ui and can be regarded as an estimate of ui. It is introduced in the SRF for the same reasons as ui was introduced in the PRF.

To sum up, then, we find our primary objective in regression analysis is to estimate the PRF

on the basis of the SRF

because more often than not our analysis is based upon a single sample from some population. But because of sampling fluctuations our estimate of

13As noted in the Introduction, a hat above a variable will signify an estimator of the relevant population value.

50 PART ONE: SINGLE-EQUATION REGRESSION MODELS

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