Least Squares And Instrumental Variables Estimation

The essential results for the classical model with spherical disturbances

are presented in Chapters 2 through 8. To reiterate, we found that the ordinary least squares (OLS) estimator b = (X'X)-1X' y = 0 + (X'X)-1 X'e (10-3)

is best linear unbiased (BLU), consistent and asymptotically normally distributed (CAN), and if the disturbances are normally distributed, like other maximum likelihood estimators considered in Chapter 17, asymptotically efficient among all CAN estimators. We now consider which of these properties continue to hold in the model of (10-1).

To summarize, the least squares, nonlinear least squares, and instrumental variables estimators retain only some of their desirable properties in this model. Least squares remains unbiased, consistent, and asymptotically normally distributed. It will, however, no longer be efficient—this claim remains to be verified—and the usual inference procedures are no longer appropriate. Nonlinear least squares and instrumental variables likewise remain consistent, but once again, the extension of the model brings about some changes in our earlier results concerning the asymptotic distributions. We will consider these cases in detail.

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