By Thomas B. Fomby, R. Carter Hill, Stanley R. Johnson
This publication had its notion in 1975in a pleasant tavern close to the college of Businessand PublicAdministration on the UniversityofMissouri-Columbia. of the authors (Fomby and Hill) have been graduate scholars of the 3rd (Johnson), and have been (and are) fascinated by educating econometrics successfully on the graduate point. We determined then to write down a booklet to function a finished textual content for graduate econometrics. quite often, the fabric integrated within the bookand itsorganization were ruled via the query, " Howcould the topic be most sensible awarded in a graduate class?" For content material, this has intended that we have got attempted to hide " the entire bases " and but haven't tried to be encyclopedic. The meant goal has additionally affected the levelofmathematical rigor. now we have tended to turn out in basic terms these effects which are simple and/or quite ordinary. Proofs that may call for inordinant quantities of sophistication time have easily been referenced. The publication is meant for a two-semester path and paced to confess extra huge remedy of parts of particular curiosity to the teacher and scholars. now we have nice self belief within the skill, undefined, and patience of graduate scholars in ferreting out and knowing the passed over proofs and effects. finally, this can be how one earnings adulthood and a fuller appreciation for the topic as a minimum. it really is assumed that the readers of the booklet could have had an econometric equipment direction, utilizing texts like J. Johnston's Econometric equipment, 2d ed.
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This follows directly from Aitken's theorem. Although unbiasedness and consistency in the estimation of p are retained when ordinary least squares is applied to a generalized least squares model, serious implications arise with respect to hypothesis testing . Importantly, incorrect use of ordinary least squares leads to invalid hypothesis testing because of biased and inconsistent estimation of the true ordinary least squares covariance matrix a 2 (X' X) -1 X'QX(X' X) - 1 by O' 2(X' X) -1 . From the developments in (ii) and (iii) above, the bias in using O' 2 (X 'X) - 1 arises because (X'X)-1X'QX(X'X)-1 =f (X'X)-1 and EO' 2 =f 0'2.
It is important to know the properties of ordinary least squares estimators when the y are based on mistaken ass umptions about the covariance stru cture of the errors. In particular, con sider the generalized least squa res model y = Xp + e, where Eee' = a 2 0 and 0 is a positi ve definite and symmetric matrix but not the identity matrix. Th en the following result s ar e ava ilable. P (i) The ordinar y least squa res estimator = (X' X) -l X'y is unbiased. If limT _ lX,(X'X /T) is finite and nonsingular and limT _ oo(X'OX /T ) is finite, pis consistent.
The intuitive reason for this is that in ajoint test of several hypotheses any single hypothesis is "affected" by the information in the other hypotheses. 5 when confidence intervals are discussed. The method of test construction will be the likelihood ratio method proposed by Neyman and Pearson (1928). Kendall and Stuart (1973) note that likelihood ratio methods have played a role in the theory of tests analogous to that of maximum likelihood methods in the theory of estimation. The method of maximum likelihood provides a means of obtaining point estimates while,the likelihood ratio method provides a means of obtaining test statistics.