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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 .)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 NSF-GP-3694 at ColumbiaUniversity. 59 60 FIFTH BERKELEY SYMPOSIUM: EICKER thepresentnote,seems tobe . Inthatpapervectorialregressionequations are consideredwhilewe are predominantly interestedinscalarones. For that case,however, the assumptions of aremore restrictivethan those ofthe presentnote. For theindividualcomponentsofthevectorialLSE theasymptoticnormality was alreadyprovedundergeneralassumptionsinforthecaseofindependent 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 (n-dimensional),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)' isthen-vectoroferrorrandom 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.
- Fall '08