The purpose of this appendix is to show that the standard errors of the estimates obtained from the second-page regression of the 2SLS procedure, using the formula applicable in OLS estimation, are not the "proper" estimates of the "true" standard errors. To see this, we use the income-money supply model given in (20.4.1) and (20.4.2). We estimate the parameters of the overidentified money supply function from the second-stage regression as
Now when we run regression (20.4.6), the standard error of, say, 021 is obtained from the following expression:
But a\ is not the same thing as a22, where the latter is an unbiased estimate of the true variance of u2. This difference can be readily verified from (7). To obtain the true (as defined previously) a^, we proceed as follows:
where /S20 and /S21 are the estimates from the second-stage regression. Hence,
Note the difference between (9) and (10): In (10) we use actual Y1 rather than the estimated Y1 from the first-stage regression.
Having estimated (10), the easiest way to correct the standard errors of coefficients estimated in the second-stage regression is to multiply each one of them by au2 /au>. Note that if Y1t and Y1t are very close, that is, the R2 in the first-stage regression is very high, the correction factor au2 /au, will be close to 1, in which case the estimated standard errors in the second-stage regression may be taken as the true estimates. But in other situations, we shall have to use the preceding correction factor.
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