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Unformatted text preview: I EcollOmetrica. Vol. 38, No. 2 (~arch. 1970) TESTS OF EQUALITY BEIWEEN SETS OF COEFFICIENTS IN j .··1WO LINEAR REGRESSIONS: AN EXPOSITORY NOTE . I By FRANKLIN M.FlSHER 1 . 1. lNTRODUCIlON Tmi PROBU::M OF testing the equality of ~ts of ;egr~i~n coeffi.ci~nts in two ~r moreregr~ions arises with considerable frequency in econometrics. While the tests involved can all be found' in the statistical literature. the derivation thereof is usually very' difficult for students in econometrics courses to follow. 1 Accordingly. the present note attempts to derive the relevant results in a unified and relatively simple way and to show the close relation ofthe tests involved to the standard F test of the hypothesis that a subset of the regression coefficients in a single regI:~ion has all its elements zero. Since the purpose is to give a self-contained exposition useful to students, some of the material is well known, and, as indicated, nearly all or it can be found in the literature. I have tried to spell out points that students may not recognize. 2. LEMMAS In what follows, 13 is a vector of normal variableS with mean zero and variance-covariance matrix ql I. LEMMA 2.1: Let M be any symmetric and idempotent matrix (i.e., Ml = M). Let u Me. Then (U'U/q2) is distributed as ./ with tr M degrees offreedom. PROOF: Since M is symmetric, there exists a matrix P such that (2.1) M P'AP, P'P I, where A is a diagonal matrix with diagonal elements the characteristic roots or M. Since M 3 is idempotent, all its characteristic roots must be either zero or one, so we can write A in the form = [Ir OJ (2.2) A 0 0 where r is the rank of M. Note that r tr Ai.4 Now define v = Pe. Then v is a vector of normal variables, since each clement is a linear combination of normal variables. Further, ._- (2.3) E(v) = PE(c) = 0, Thus the' clements of I' arc mutually uncorrelated and, since they are normal, they are mutually independent. Clearly. however, , (2.4) u'u/a! = c'M'Mc/a 2 c'Me/q2 = c'P'APe/a 2 = l"Al~/ql = I (t'Ja)2. i= 1 1 am indebted to E. Kuh for discussion but retain responsibility for error. 2 The paper in this area best known to economists is Chow [I], but even Ihis is oncn quite difficuh for students. The tests \Vere in large part first derived by Kullback and Rosenblatt [4J. Chow's own presentation of m:my of the tests follows Kempthorne (3, pp. 54-66] and a treatment of some of the material similar 10 that of the present note can be found in Graybill [2. p. li6 and Section 6-4]' See also Rao [5, pp. 153-157]. 1 am indebted to referees for some of these references. 1 If M is idempotent. !d" = At. so any characteristic root of M must be equal to its own square. • Because the trace of any matrix equals the sum of its characteristic roots and all the nonzero characteristic roots of M arc unity, 361 362 FRANKLIN M. FISHER The last tenn is evidently the sum of squares of r (= tr M) independently distributed standard normal deviates which is 9istributed as stated. Q.E.D....
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This note was uploaded on 12/08/2011 for the course ECON 703 taught by Professor Bernardmorzuch during the Fall '10 term at UMass (Amherst).
- Fall '10