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Unformatted text preview: LIMIT THEOREMS FOR REGRESSIONS WITH UNEQUAL AND DEPENDENT ERRORS FRIEDHELM EICKER UNIVERSITY OF FREIBURG IM BREISGAU and COLUMBIA UNIVERSITY, NEW YORK 1. Summary Thispaperdealswiththeasymptoticdistributionofthevectorialleastsquares estimators (LSE) fortheparameters inmultiplelinearregressionsystems. The regressionconstantsareassumed tobeknown; theerrorsareassumed (a)tobe independentbutnotnecessarilyidenticallyornormally distributed (section3), or (b)toconstitutea generalizedlineardiscretestochasticprocess (section4). The latterpart includesthe case ofregressionfortime series.Conditionsare studied under which the joint distribution functions (d.f.'s)of the vectorial LSE's tend to a multivariatenormal d.f.as the sample sizeincreases. In the proofacentrallimittheorem (CLT) forweightedaveragesofindependentran domvariablesisused.Incase(a),atheoremforlargeclassesoflinearregressions isproved (theorem3.2),whose conditionsareinacertainsensealsonecessary. The theorem simultaneously permits consistent estimationofthe limitingco variancematrixoftheLSE's. The resultsincase(b)arecontainedintheorems 4.2,4.3,4.4,4.5,4.6. Theyarenotnaturallyofasclosedaformasthosepertinent tocase (a)becauseofthe more complicatednatureoftheproblem.Someuseof spectraltheory ismade. Severalexamplesare discussed (section3.3).The as sumptionsmade inthispaperareweakerthanthoseofresults published earlier in the literature. (Fora more recent survey, compare [6].)Their structure is quitesimplesothattheyoughttobeusefulinapplications.Section4.4contains some remarks on multivariateregressionequations. 2. Introduction (notations) There existsa considerablenumber ofpublicationsdealingwith theasymp totic normality ofparameter estimatesfor linear regressions, many ofwhich deal with specific cases, however, or are unnecessarily narrow in the as sumptions made. The most generalpaper among these,and theone closestto Research supported in part by National Science Foundation Grant NSFGP3694 at ColumbiaUniversity. 59 60 FIFTH BERKELEY SYMPOSIUM: EICKER thepresentnote,seems tobe [9]. Inthatpapervectorialregressionequations are consideredwhilewe are predominantly interestedinscalarones. For that case,however, the assumptions of [9]aremore restrictivethan those ofthe presentnote. For theindividualcomponentsofthevectorialLSE theasymptoticnormality was alreadyprovedundergeneralassumptionsin[1]forthecaseofindependent errors. Thesystemof(scalar)linearregressionequationsisdenotedby (2.1) yt = Xtil + ***+ Xtq3q + Et, t = 1, * ,n, n > qbeingthesamplesize.Inmatrixnotationthisbecomes (2.2) Y(n) = Xn3 + e (n), where y(n) isthe (column)vector ofobservations (ndimensional),X. = (xtj), the (nX q)matrixofknown regressionconstants assumed to be offullrank throughout, ,B = (i,, **, #3,)' isthevectorofunknown regressionparameters ('denotes the transpose), and e(n) = (el, **, en)' isthenvectoroferrorrandom variables(r.v.'s)aboutwhichwe assume throughoutthat (2.3) Eet = O, O < EJ, < a> , forall t....
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.
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