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Class Notes in Statistics and Econometrics Part 29

CHAPTER 57 Applications of GLS with Nonspherical Covariance Matrix. In most cases in which the covariance matrix is nonspherical, Ψ contains unknown parameters, which must be estimated before formula (26.0.2) can be applied. Of course, if all entries of Ψ are unknown | CHAPTER 57 Applications of GLS with Nonspherical Covariance Matrix In most cases in which the covariance matrix is nonspherical contains unknown parameters which must be estimated before formula 26.0.2 can be applied. Of course if all entries of are unknown such estimation is impossible since one needs n n 1 2 1 parameters to specify a symmetric matrix up to a multiplicative factor but with n observations only n unrelated parameters can be estimated consistently. Only in a few exceptional cases is known and in some even more exceptional cases there are unknown parameters in but 26.0.2 does not depend on them. We will discuss such examples first heteroskedastic disturbances with known relative variances and some examples involving equicorrelated disturbances. 1233 1234 57. APPLICATIONS WITH NONSPHERICAL COVARIANCE 57.1. Cases when OLS and GLS are identical Problem 498. From y Xfi e with e o a21 follows Py PXfi Pe with Pe o a2PPT . Which conditions must P satisfy so that the generalized least squares regression of Py on PX with covariance matrix PPT gives the same result as the original regression PROBLEM 499. We are in the model y Xft e e a2 . As always we assume X has full column rank and is nonsingular. We will discuss the special situation here in which X and are such that X XA for some A. a. 3 points Show that the requirement X XA is equivalent to the requirement that R X R X . Here R B is the range space of a matrix B i.e. it is the vector space consisting of all vectors that can be written in the form Bc for some c. Hint For show first that R X C R X and then show that R X has the same dimension as R X . ANSWER. Clearly R X C R X since SPX XA and every XAc has the form Xd with d Ac. And since SP is nonsingular and the range space is the space spanned by the column vectors and the columns of SPX are the columns of X premultiplied by SP it follows that the range space of SPX has the same dimension as that of X. The ith column of SPX lies in R X i.e. it can be .