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1、INTRODUCTIONTOVECTORANDMATRIXDIFFERENTIATIONEconometrics2HeinoBohnNielsenSeptember21,2005hisnoteexpandsonappendixA.7inVerbeek(2004)onmatrixdifferenti-ation.WefirstpresenttheconventionsforderivativesofscalarandvectorTfunctions;thenwepresentthederivativesofanumberofspecialfunctionsparticularlyusefulinec
2、onometrics,and,finally,weapplytheideastoderivetheordinaryleastsquares(OLS)estimatorinthelinearregressionmodel.Weshouldemphasizethatthisnoteiscursoryreading;therulesforspecificfunctionsneededinthiscourseareindicatedwitha(∗).1ConventionsforScalarFunctionsLetβ=(β1,...,βk)0beak×1vectorandletf(β)=f(β1,...,
3、βk)beareal-valuedfunctionthatdependsonβ,i.e.f(·):Rk7−→Rmapsthevectorβintoasinglenumber,f(β).Thenthederivativeoff(·)withrespecttoβisdefinedas⎛⎞∂f(β)⎜∂β1⎟∂f(β)=⎜..⎟.(1)∂β⎝.⎠∂f(β)∂βk∂f(β)Thisisak×1columnvectorwithtypicalelementsgivenbythepartialderivative.∂βiSometimesthisvectorisreferredtoasthegradient.
4、Itisusefultorememberthatthederivativeofascalarfunctionwithrespecttoacolumnvectorgivesacolumnvectorastheresult1.1∂f(β)WecannotethatWooldridge(2003,p.783)doesnotfollowthisconvention,andletbea1×k∂βrowvector.1Similarly,thederivativeofascalarfunctionwithrespecttoarowvectoryieldsthe1×krowvector³´∂f(β)∂f(β
5、)∂f(β)0=∂β···∂β.∂β1k2ConventionsforVectorFunctionsNowlet⎛⎞g1(β)⎜.⎟g(β)=⎜..⎟⎝⎠gn(β)beavectorfunctiondependingonβ=(β1,...,βk)0,i.e.g(·):Rk7−→Rnmapsthek×1vectorintoan×1vector,wheregi(β)=gi(β1,...,βk),i=1,2,...,n,isareal-valuedfunction.Sinceg(·)isacolumnvectoritisnaturaltoconsiderthederivativeswithrespe
6、cttoarowvector,β0,i.e.⎛⎞∂g1(β)∂g1(β)···⎜∂β1∂βk⎟∂g(β)=⎜......⎟,(2)∂β0⎝...⎠∂gn(β)∂gn(β)···∂β1∂βkwhereeachrow,i=1,2,...,n,containsthederivativeofthescalarfunctiongi(·)withrespecttotheelementsinβ.Theresultisthereforean×kmatrixofderivativeswith∂gi(β)typicalelement(i,j)givenby.Ifthevectorfunctionisdefineda
7、sarowvector,it∂βjisnaturaltotakethederivativewithrespecttothecolumnvector,β.Wecannotethatitholdsingeneralthat0µ¶0∂(g(β))∂g(β)=,(3)∂β∂β0whichinthecaseaboveisak×nmatrix.Applyingtheconventionsin(1)and(2)
