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1、EconometricTheory,4,1988,468-497.PrintedintheUnitedStatesofAmerica.STATISTICALINFERENCEINREGRESSIONSWITHINTEGRATEDPROCESSES:PART1JOONY.PARKANDPETERC.B.PHILLIPSCowlesFoundation,YaleUniversityThispaperdevelopsamultivariateregressiontheoryforintegratedprocesseswhichsimplifiesandextendsmuchearlierwo
2、rk.Ourframeworkallowsforbothstochasticandcertaindeterministicregressors,vectorautoregressions,andregressorswithdrift.Themainfocusofthepaperisstatisticalinference.ThepresenceofnuisanceparametersintheasymptoticdistributionsofregressionFtestsisexploredandnewtransformationsareintroducedtodealwiththe
3、sedependencies.Somespecializationsofourtheoryareconsideredindetail.Inmodelswithstrictlyexogenousregressors,wedemonstratethevalidityofcon-ventionalasymptotictheoryforappropriatelyconstructedWaldtests.Thesetestsprovideasimpleandconvenientbasisforspecificationrobustinferencesinthiscontext.Singleequ
4、ationregressiontestsarealsostudiedindetail.HereitisshownthattheasymptoticdistributionoftheWaldtestisamixtureofthechisquareofconventionalregressiontheoryandthestandardunit-rootthe-ory.Thenewresultaccommodatesbothextremesandintermediatecases.1.INTRODUCTIONRecently,therehasbeengrowinginterestinthet
5、heoryofregressionamongtimeseriesthatareindividuallywell-explainedbymodelsoftheARIMAtype.Suchmodelsgenerateasimpleclassofnonstationarytimeserieswhichwegenericallydescribeasintegratedprocesses.Morespecifically,wecallatimeseriesIX,Janintegratedprocessoforderk(inshort,anI(k)process)ifthetimeseriesof
6、kthorderdifferencesIkX,Jisstationary(anI(0)pro-cess).I(1)processesbehavelikeaccumulatedsumsofstationaryinnovationsandtheypossessasingleunitroot.Whenwerunregressionswithsuchtimeseries,theasymptoticpropertiesoftheregressioncoefficients,statisticaltests,andregressiondiagnosticsareverydifferentfromt
7、hoseofregressionswithstationarytimeseries.Someofthesedifferenceshavebecomeappar-entinrecentworkbyPhillips[15,16],PhillipsandDurlauf[19],PhillipsandOuliaris[20],Stock[23],andStockandWatson[24].Theaimofthepresentpaperanditsseq
