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WW p 14 - Nonnacott Ronald J and Thomas H Wonnacott...

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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 = 9-9 u/\)£x? FIGURE 2-11 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 2-ll (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 (2-27) established that the distribution oft? is centered on its target so that B is unbiased. The interpretation of its variance (2-28) 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 (2-28) is large and fl is a comparatively unreliable estimator. To check the intuitive validity of this assertion. consider the scatter diagram in Figure 2-12a. 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 2-12b 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 e-sam 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 (2-11). under moat conditions. Similarly. the normality of a is justified. 14 ...
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