High-Dimensional Covariance Matrix Estimation: Shrinkage Toward a Diagonal Target

High-DimensionalCovarianceMatrixEstimation:ShrinkageTowardaDiagonalTarget SakaiAndoandMingmeiXiao WP/23/257 IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicitcommentsandtoencouragedebate. TheviewsexpressedinIMFWorkingPapersarethoseoftheauthor(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement. 2023 DEC ©2023InternationalMonetaryFundWP/23/257 IMFWorkingPaper ResearchDepartment High-DimensionalCovarianceMatrixEstimation:ShrinkageTowardaDiagonalTargetPreparedbySakaiAndoandMingmeiXiao* AuthorizedfordistributionbyPrachiMishra December2023 IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicitcommentsandtoencouragedebate.TheviewsexpressedinIMFWorkingPapersarethoseoftheauthor(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement. ABSTRACT:Thispaperproposesanovelshrinkageestimatorforhigh-dimensionalcovariancematricesbyextendingtheOracleApproximatingShrinkage(OAS)ofChenetal.(2009)totargetthediagonalelementsofthesamplecovariancematrix.Wederivetheclosed-formsolutionoftheshrinkageparameterandshowbysimulationthat,whenthediagonalelementsofthetruecovariancematrixexhibitsubstantialvariation,ourmethodreducestheMeanSquaredError,comparedwiththeOASthattargetsanaveragevariance.Theimprovementislargerwhenthetruecovariancematrixissparser.OurmethodalsoreducestheMeanSquaredErrorfortheinverseofthecovariancematrix. JELClassificationNumbers: C13,C55 Keywords: High-Dimension;CovarianceMatrix;Shrinkage;DiagonalTarget Author’sE-MailAddress: sando@imf.org;mx235@cam.ac.uk Contents 1Introduction1 2TheoreticalFramework2 2.1SpecialCase:KnownMean5 3Simulation6 3.1Setting7 3.2MainResults9 3.3PerformanceofInverseMatrix12 3.4Alternativemethodbasedonshrinkingcorrelationmatrix14 4Conclusion15 References17 Appendix18 AProofofTheorem118 BProofofTheorem224 CProofofTheorem326 1Introduction EstimatingacovariancematrixΣ:p×panditsinversewhenthedimensionofthematrixpislargerthanthesamplesizeniscentraltomanyempiricalapplications,includingfinancialportfolioselectionandmacroeconomicforecasting((DeMigueletal.(2009),Banetal.(2018),AndoandKim(2022)),andeconometricmethods,suchasGeneralizedMethodofMoments(Hansen(1982))andPrincipalComponentAnalysis(Pearson(1901)).AlthoughLedoitandWolf(2004)developedashrinkageestimatorbasedonanaveragevariancetarget,andChenetal.(2009)improveditsfinitesampleperformanceunderthenormalityassumption,themethodleavesroomforimprovementwhenthediagonalelementsofthetruecovariancema-trixexhibitsubstantialvariation.Forexample,inthesettingofmacroeconomicforecasting,GDPandoutputof,say,thefishingindustrycandifferbyahundredfold,sotheshrink-ageestimatorthattargetstheaveragevariancecanoverestimatethevarianceofthefishingindustry’soutputandunderestimatethatofGDP. Toaccommodatethecasewherethevarianceofrandomvariablesexhibitsubstantialvariation,thispaperproposesashrinkageestimatorthattargetsthediagonalelementsofthesamplecovariancematrix.OurmethodextendstheOracleApproximatingShrinkageestimator(OAS)ofChenetal.(2009)thattargetstheaveragevariance.FollowingEldarandChernoi(2008)andChenetal.(2009),wederivetheoptimalshrinkageparametergiventhetruecovariancematrix(Oracleestimator)andapproximatethisinfeasibleOracleestimatorwithaniterativealgorithm. WeuseasimulationtoshowthatourmethodgeneratesalowerMeanSquaredError(MSE)thanOASwhenthediagonalelementsofthetruecovariancematrixexhibitsub-stantialvariation.Inthespecificationofdecayingoff-diagonalelements,thedegreeofim-provementishigherwhenthetruecovariancematrixissparser.OurmethodalsogeneratesasmallerMSEfortheinverseofthecovariancematrix,whichisoftenanultimategoalofestimatingacovariancematrixinpractice. AsChenetal.(2009),ourmethodisbasedontheoptimalityunderthenormaldistri- bution.ComparedtoSch¨aferandStrimmer(2005)whichalsotargetdiagonalelementsofthecovariancematrixbutwithoutimposingadistributionalassumption,ourmethodper-formsbetterwhenthedistributionisnormal.Inaddition,ourmethodinheritsthedesirablepropertyofOASthattheshrinkageparameterstaysbetween0and1.Thus,theestimatedcovariancematrixispositive-definite,evenwithoutmanuallyrestrictingtheshrinkagepa-rameterasdoneinSch¨aferandStrimmer(2005).Thenormalityassumptionalsoallowsustoderivetheoptimalshrinkageparameterinaclosedform,whichinvolveslesscomputationthanthenon-linearshrinkagemethodofLedoitandWolf(2012). Ourmethod,however,doesnotoutperformexistingmethodsinallcircumstances,andthus,shouldbeconsideredacomplementtothem.Forexample,whenthevariationinthediagonalelementsofthetruecovariancematrixissmall,theOAStendstogeneratealowerMSE.ThisobservationalsosuggestsanalternativemethodofestimatingthecovariancematrixbyapplyingOAStothecorrelationmatrixandscalingitbackbymultiplyingsamplevariances.Toexaminetherobustness,weconductasimulationandshowthatthedifferenceinMSEbetweenOASandourproposedmethodissmallandthatdirectlyshrinkingthesamplecovariancematrixperformsbetterthanapplyingOAStothecorrelationmatrixandscalingitback. Thispaperisorganizedasfollows.Section2describesthetheoretical