We have shown that, in the framework of the classical linear regression model, the least-squares estimators are unbiased (and efficient) in any sample size, small or large. But sometimes, as discussed in Appendix A, an estimator may not satisfy one or more desirable statistical properties in small samples. But as the sample size increases indefinitely, the estimators possess several desirable statistical properties. These properties are known as the large sample, or asymptotic, properties. In this appendix, we will discuss one large sample property, namely, the property of consistency, which is discussed more fully in Appendix A. For the two-variable model we have already shown that the OLS estimator fa is an unbiased estimator of the true fa. Now we show that fa is also a consistent estimator of fa. As shown in Appendix A, a sufficient condition for consistency is that fa2 is unbiased and that its variance tends to zero as the sample size n tends to infinity.

Since we have already proved the unbiasedness property, we need only show that the variance of fa2 tends to zero as n increases indefinitely. We know that var (fa) = = J^XPr (26)

By dividing the numerator and denominator by n, we do not change the equality.

XYxf/nJ

106 PART ONE: SINGLE-EQUATION REGRESSION MODELS

where use is made of the facts that (1) the limit of a ratio quantity is the limit of the quantity in the numerator to the limit of the quantity in the denominator (refer to any calculus book); (2) as n tends to infinity, a2/n tends to zero because a2 is a finite number; and [(J] x2)/n] = 0 because the variance of X has a finite limit because of Assumption 8 of CLRM.

The upshot of the preceding discussion is that the OLS estimator fa is a consistent estimator of true fa. In like fashion, we can establish that fa is also a consistent estimator. Thus, in repeated (small) samples, the OLS estimators are unbiased and as the sample size increases indefinitely the OLS estimators are consistent. As we shall see later, even if some of the assumptions of CLRM are not satisfied, we may be able to obtain consistent estimators of the regression coefficients in several situations.

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