"Note, though, a\ is still biased, which can be seen intuitively as follows: We know that ¡1 = Y — ¡2XX2 — ¡3X3, whereas a\ = Y — ¿2X2, and even if ¿2 = ¡2, the two intercept estimators will not be the same.
12For details, see Adrian C. Darnell, A Dictionary of Econometrics, Edward Elgar Publisher, 1994, pp. 371-372.
514 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
Since 0 < r|3 < 1, it follows that var (<52) > var (92); that is, the variance of <52 is generally greater than the variance of /32 even though, on average, a2 = p2 [i.e., E(a2) = &].
The implication of this finding is that the inclusion of the unnecessary variable X3 makes the variance of <52 larger than necessary, thereby making <52 less precise. This is also true of <51.
Notice the asymmetry in the two types of specification biases we have considered. If we exclude a relevant variable, the coefficients of the variables retained in the model are generally biased as well as inconsistent, the error variance is incorrectly estimated, and the usual hypothesis-testing procedures become invalid. On the other hand, including an irrelevant variable in the model still gives us unbiased and consistent estimates of the coefficients in the true model, the error variance is correctly estimated, and the conventional hypothesis-testing methods are still valid; the only penalty we pay for the inclusion of the superfluous variable is that the estimated variances of the coefficients are larger, and as a result our probability inferences about the parameters are less precise. An unwanted conclusion here would be that it is better to include irrelevant variables than to omit the relevant ones. But this philosophy is not to be espoused because addition of unnecessary variables will lead to loss in efficiency of the estimators and may also lead to the problem of multicollinearity (why?), not to mention the loss of degrees of freedom. Therefore,
In general, the best approach is to include only explanatory variables that, on theoretical grounds, directly influence the dependent variable and that are not accounted for by other included variables.13
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