Unformatted text preview: Nonnacott, Ronald J. and Thomas H. Wonnacott. Econometrics. (Second
Edition). New York: John Wiley & Sons, l979. pp. 29—30. 2—7 THE DlSTRlBUTlON 0F [3 Now that we have established the mean and variance off} in (2— 27) and
(2 28) we ask about the shape of the distribution of [3. Let us add (for the
ﬁrst time) the strong assumption that the Y are normal Since [3 IS a linear
combination of the Y“ it follows (from Appendix 2— A) that [3 will also be normal. But even without assuming the Yl are normal as sample size in } Note J Standard deviation of B 6
Ex = l
l Etﬂl=t3 B = 99
u/\)£x? FIGURE 211 The probability distribution of the estimator £ iustiﬁed by a generalized form of the central limit theorem.8 We are now in a position to graph the distribution of 5 in Figure 2ll
(for the time being, the bottom line in this diagram may be disregarded). Our
objective is to develop a clear intuitive idea of how this estimator varies from
sample to sample. First of course we note that (227) established that the
distribution oft? is centered on its target so that B is unbiased. The interpretation of its variance (228) is more subtle, with some inter
esting implications for experimental design. Suppose that the experiment has
been badly designed with the X .s close together. This makes the deviations
x, small hence Z x) small Therefore the variance ofﬁ in (228) is large and
ﬂ is a comparatively unreliable estimator. To check the intuitive validity of
this assertion. consider the scatter diagram in Figure 212a. The bunching of
the X '5 means that the small part of the line being investigated is obscured
by the error e, making the slope estimate 3 very unreliable. In this speciﬁc instance, our estimate has been pulled badly out of line by the errors—in
particular. by the one indicated by the arrow. By contrast, in Figure 212b we show the case where the X ‘s are rea creases the distribution of A will usuall a roach norrnalit ' this can be } Note lar esam le normalit of the tam to mean R. It a lies also to a wet lired sum of random ‘ Sometimes called the " normal approximation theorem.“ the central limit theorem proves the } N0 te
variables web as in (211). under moat conditions. Similarly. the normality of a is justified. 14 ...
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 Fall '10
 BernardMorzuch

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