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ChapterILINEARALGEBRAANDMATRIXMETHODSINECONOMETRICSHENRITHEIL*UniversityofFloridaContents1.Introduction2.Whyarematrixmethodsusefulineconometrics?2.1.Linearsystemsandquadraticforms2.2.Vectorsandmatricesinstatisticaltheory2.3.Leastsquaresinthestandardlinearmodel2.4.Vectorsandmatricesinconsumptiontheory3.Partitionedmatrices3.I,Thealgebraofpartitionedmatrices3.2.Block-recursivesystems3.3.Incomeandpricederivativesrevisited4.Kroneckerproductsandthevectorizationofmatrices4.I.ThealgebraofKroneckerproducts4.2.Jointgeneralizedleast-squaresestimationofseveralequations4.3.Vectorizationofmatrices5.Differentialdemandandsupplysystems5.1.Adifferentialconsumerdemandsystem5.2.Acomparisonwithsimultaneousequationsystems5.3.Anextensiontotheinputsofafirm:Asingularityproblem5.4.Adifferentialinputdemandsystem5.5.Allocationsystems5.6.Extensions6.Definiteandsemidefinitesquarematrices6.I.CovariancematricesandGauss-Markovfurtherconsidered6.2.Maximaandminima6.3.Block-diagonaldefinitematrices7.Diagonalizations7.1.nestandarddiagonalizationofasquarematrix5:7*::;::1616171920;;2:29303”:*ResearchsupportedinpartbyNSFGrantSOC76-82718.TheauthorisindebtedtoKennethClements(ReserveBankofAustralia,Sydney)andMichaelIntriligator(UniversityofCalifornia,LosAngeles)forcommentsonanearlierdraftofthischapter.HundhookofEconometrics,VolumeI,EditedbyZ.GrilichesandM.D.Intriligator0North-HollandPublishingCompany,I983H.Theil1.2.Specialcases7.3.Aitken’stheorem7.4.TheCholeskydecomposition7.5.Vectorswrittenasdiagonalmatrices7.6.Asimultaneousdiagonalizationoftwosquarematrices7.7.Latentrootsofanasymmetricmatrix8.Principalcomponentsandextensions8.1.Principalcomponents8.2.Derivations8.3.Furtherdiscussionofprincipalcomponents8.4.Theindependencetransformationinmicroeconomictheory8.5.Anexample8.6.Aprincipalcomponentinterpretation9.Themodelingofadisturbancecovariancematrix9.1.Rationalrandombehavior9.2.Theasymptoticsofrationalrandombehavior9.3.Applicationstodemandandsupply10.TheMoore-Penroseinverse10.1.Proofoftheexistenceanduniqueness10.2.Specialcases10.3.AgeneralizationofAitken’stheorem10.4.DeletinganequationfromanallocationmodelAppendixA:LinearindependenceandrelatedtopicsAppendixB:TheindependencetransformationAppendixC:RationalrandombehaviorReferences::535657586164Ch.1:LinearAlgebraandMatrixMethoak1.IntroductionVectorsandmatricesplayedaminorroleintheeconometricliteraturepublishedbeforeWorldWarII,buttheyhavebecomeanindispensabletoolinthelastseveraldecades.Partofthisdevelopmentresultsfromtheimportanceofmatrixtoolsforthestatisticalcomponentofeconometrics;anotherreasonisthein-creaseduseofmatrixalgebraintheeconomictheoryunderlyingeconometricrelations.Theobjectiveofthischapteristoprovideaselectivesurveyofbothareas.Elementarypropertiesofmatricesanddeterminantsareassumedtobeknown,includingsummation,multiplication,inversion,andtransposition,buttheconceptsoflineardependenceandorthogonalityofvectorsandtherankofamatrixarebrieflyreviewedinAppendixA.ReferenceismadetoDhrymes(1978),Graybill(1969),orHadley(1961)forelementarypropertiesnotcoveredinthischapter.Matricesareindicatedbyboldfaceitalicuppercaseletters(suchasA),columnvectorsbyboldfaceitaliclowercaseletters(a),androwvectorsbyboldfaceitaliclowercaseletterswithaprimeadded(a’)toindicatethattheyareobtainedfromthecorrespondingcolumnvectorbytransposition.Thefollowingabbreviationsareused:LS=leastsquares,GLS=generalizedleastsquares,ML=maximumlikelihood,6ij=Kroneckerdelta(=lifi=j,0ifi*j).2.Whyarematrixmethodsusefulineconometrics?2.1.LinearsystemsandquadraticformsAmajorreasonwhymatrixmethodsareusefulisthatmanytopicsineconomet-ricshaveamultivariatecharacter.Forexample,considerasystemofLsimulta-neouslinearequationsinLendogenousandKexogenousvariables.Wewritey,,andx,~forthe&hobservationonthelthendogenousandthekthexogenousvariable.Thenthejthequationforobservation(Ytakestheformk=l(2.1)tively:rYIIY12-.*YILPI1Pl2-.-PILY21Y22...Y2LP21P22...P2Lr=...,B=.........YLIYL2..YLL_P’P,,...P,L_KIWhentherearenobservations((Y=1,...,n),thereareLnequationsoftheform(2.1)andnequationsoftheform(2.2).WecancombinetheseequationscompactlyintoE=6H.Theilwhere&ajisarandomdisturbanceandthey’sandp’sarecoefficients.Wecanwrite(2.1)forj=l,...,Lintheformy;I’+x&B=E&,(2.2)whereyL=[yal...yaL]andx&=[xal...xaK]areobservationvectorsontheendog-enousandtheexogenousvariables,respectively,E&=[E,~...caL]isadisturbancevector,andrandBarecoefficientmatricesoforderLXLandKXL,respec-Yr+XB=E,(2.3)whereYandXareobservationmatricesofthetwosetsofvariablesofordernXLandnXK,respectively:YllYl,...YlLXIIX12...XlKY21Y22.-Y2Lx21X22-.-X2Ky=...3x=...3........_YnlYtlZ...Y?lL_Xnlxn2.-.nKXandEisannXLdisturbancematrix:-%IEl2...ElLE2lE22...&2L......Enl%2...nLENotethatrissquare(LXL).Ifrisalsonon-singular,wecanpostmultipy(2.3)byr-t:Y=-XBr-'+Er-'.(2.4)Ch.I:LinearAlgebraandMatrixMethodsIThisisthereducedformforallnobservationsonallLendogenousvariables,eachofwhichisdescribedlinearlyintermsofexogenousvaluesanddisturbances.Bycontrast,theequations(2.1)or(2.2)or(2.3)fromwhich(2.4)isderivedconstitutethestructuralformoftheequati