2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Chapter 4MultipleRegressionAnalysis:Inference2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Chapter4MultipleRegressionAnalysis:Inference4.2 Testing Hypotheses about a Single Population Parameter: The t Test4.3 Confidence Intervals4.4 Testing Hypotheses about a Single Linear Combination of the Parameters4.5 Testing Multiple Linear Restrictions: The F Test4.1 Sampling Distributions of the OLS Estimators4.6 An application estimation of the weights of CPI components in ChinaAssignments: Promblems 1, 2, 4, 5, 7, 8, 10 Computer Exercises C1, C2, C3, C8, C9 C8: smplifmarr=1andfsize=2(401ksubs.wf1)The End2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Statistical inference in the regression modelHypothesistestsaboutpopulationparametersConstructionofconfidenceintervalsSampling distributions of the OLS estimatorsTheOLSestimatorsarerandomvariablesWealreadyknowtheirexpectedvaluesandtheirvariancesHowever,forhypothesistestsweneedtoknowtheirdistributionInordertoderivetheirdistributionweneedadditionalassumptionsAssumptionaboutdistributionoferrors:normaldistributionChapter4MultipleRegressionAnalysis:Inference4.1 Sampling Distributions of the OLS Estimators (1/5)ChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Assumption MLR.6 (Normality of error terms)independentlyofItisassumedthattheunobservedfactorsarenormallydistributedaroundthepopulationregressionfunction.Theformandthevarianceofthedistributiondoesnotdependonanyoftheexplanatoryvariables.Itfollowsthat:Chapter4MultipleRegressionAnalysis:Inference4.1 Sampling Distributions of the OLS Estimators (2/5)ChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Discussion of the normality assumptionTheerrortermisthesumofmany“differentunobservedfactorsSumsofindependentfactorsarenormallydistributed(CLT)Problems:Howmanydifferentfactors?Numberlargeenough?PossiblyveryheterogenuousdistributionsofindividualfactorsHowindependentarethedifferentfactors?ThenormalityoftheerrortermisanempiricalquestionAtleasttheerrordistributionshouldbeclose“tonormalChapter4MultipleRegressionAnalysis:Inference4.1 Sampling Distributions of the OLS Estimators (3/5)ChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Discussion of the normality assumption (cont.)Exampleswherenormalitycannothold:Wages(nonnegative;also:minimumwage)Numberofarrests(takesonasmallnumberofintegervalues)Unemployment(indicatorvariable,takesononly1or0)Insomecases,normalitycanbeachievedthroughtransformationsofthedependentvariable(e.g.uselog(wage)insteadofwage)Important:Forthepurposesofstatisticalinference,theassumptionofnormalitycanbereplacedbyalargesamplesizeChapter4MultipleRegressionAnalysis:Inference4.1 Sampling Distributions of the OLS Estimators (4/5)ChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.TerminologyTheorem 4.1 (Normal sampling distributions)UnderassumptionsMLR.1MLR.6:TheestimatorsarenormallydistributedaroundthetrueparameterswiththevariancethatwasderivedearlierThestandardizedestimatorsfollowastandardnormaldistributionGauss-Markovassumptions“Classicallinearmodel(CLM)assumptions“Chapter4MultipleRegressionAnalysis:Inference4.1 Sampling Distributions of the OLS Estimators (5/5)ChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.2.1 Theorem 4.2 t Distribution for the Standardized EstimatorsChapter4MultipleRegressionAnalysis:Inference4.2 Testing Hypotheses about a Single Population Parameter: The t Test4.2.3 Two-Sided Alternatives4.2.4 Testing Other Hypotheses about b bj4.2.2 Testing against One-Sided Alternatives4.2.5 Computing p-Values for t Tests4.2.6 A Reminder on the Language of Classical Hypothesis Testing4.2.7 Economic, or Practical, versus Statistical SignificanceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.UnderassumptionsMLR.1MLR.6:Ifthestandardizationisdoneusingtheestimatedstandarddeviation(=standarderror),thenormaldistributionisreplacedbyat-distributionNote:Thet-distributionisclosetothestandardnormaldistributionifn-k-1islarge.Chapter4MultipleRegressionAnalysis:Inference4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators (1/3)Proof:SectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Null hypothesis (for more general hypotheses, see below)t-statistic (or t-ratio)Distribution of the t-statistic if the null hypothesis is trueThet-statisticwillbeusedtotesttheabovenullhypothesis.Thefarthertheestimatedcoefficientisawayfromzero,thelesslikelyitisthatthenullhypothesisholdstrue.Butwhatdoesfar“awayfromzeromean?Thisdependsonthevariabilityoftheestimatedcoefficient,i.e.itsstandarddeviation.Thet-statisticmeasureshowmanyestimatedstandarddeviationstheestimatedcoefficientisawayfromzero.Thepopulationparameterisequaltozero,i.e.aftercontrollingfortheotherindependentvariables,thereisnoeffectofxjonyChapter4MultipleRegressionAnalysis:Inference4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators (2/3)SectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Goal: Define a rejection rule so that, if it is true, H0 is rejected only with a small probability (= significance level, e.g. 5%)Thepreciserejectionruledependsonthealternativehypothesisandthechosensignificancelevelofthetest.Asignificancelevel:theprobabilityofrejectingH0whenitistrue.Chapter4MultipleRegressionAnalysis:Inference4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators (3/3)SectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Testagainst.Testing against one-sided alternatives (greater than zero)4.2.2 Testing against One-Sided Alternatives (1/8)Rejectthenullhypothesisinfavourofthealternativehypothesisiftheestimatedcoefficientistoolarge“(i.e.largerthanacriticalvalue).Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thisisthepointofthet-distributionwith28degreesoffreedomthatisexceededin5%ofthecases.!Rejectift-statisticgreaterthan1.701Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Wage equationTestwhether,aftercontrollingforeducationandtenure,higherworkexperienceleadstohigherhourlywages(1)Testagainst.Onewouldeitherexpectapositiveeffectofexperienceonhourlywageornoeffectatall.Standarderrors4.2.2 Testing against One-Sided Alternatives (2/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Wage equation (cont.)Theeffectofexperienceonhourlywageisstatisticallygreaterthanzeroatthe5%(andevenatthe1%)significancelevel.“Thoughttheestimatedreturnforanotheryearofexperience,holdingtenureandeducationfixed,isnotespeciallylarge,wehavepersuasivelyshownthatthepartialeffectofexperienceispositiveinthepopulation.t-statisticCriticalvaluesforthe5%andthe1%significancelevel(theseareconventionalsignificancelevels).Thenullhypothesisisrejectedbecausethet-statisticexceedsthecriticalvalue.(2)Degreesoffreedom;herethestandardnormalapproximationapplies(3)(4)4.2.2 Testing against One-Sided Alternatives (3/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Testagainst.Testing against one-sided alternatives (less than zero)Rejectthenullhypothesisinfavourofthealternativehypothesisiftheestimatedcoefficientistoosmall“(i.e.smallerthanacriticalvalue).Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thisisthepointofthet-distributionwith18degreesoffreedomsothat5%ofthecasesarebelowthepoint.!Rejectift-statisticlessthan-1.7344.2.2 Testing against One-Sided Alternatives (4/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Student performance and school sizeTestwhethersmallerschoolsizeleadstobetterstudentperformanceTestagainst.Dolargerschoolshamperstudentperformanceoristherenosucheffect?PercentageofstudentspassingmathstestAverageannualtea-chercompensationSchoolenrollment(=schoolsize)Staffperonethousandstudents4.2.2 Testing against One-Sided Alternatives (5/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Student performance and school size (cont.)Onecannotrejectthehypothesisthatthereisnoeffectofschoolsizeonstudentperformance(notevenforalaxsignificancelevelof15%).t-statisticCriticalvaluesforthe5%andthe15%significancelevel.Thenullhypothesisisnotrejectedbecausethet-statisticisnotsmallerthanthecriticalvalue.Degreesoffreedom;herethestandardnormalapproximationapplies4.2.2 Testing against One-Sided Alternatives (6/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Student performance and school size (cont.)Alternativespecificationoffunctionalform:Testagainst.R-squaredslightlyhigher4.2.2 Testing against One-Sided Alternatives (7/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Student performance and school size (cont.)Thehypothesisthatthereisnoeffectofschoolsizeonstudentperformancecanberejectedinfavorofthehypothesisthattheeffectisnegative.t-statisticCriticalvalueforthe5%significancelevel!rejectnullhypothesisHowlargeistheeffect?(smalleffect)+10%enrollment!-0.129percentagepointsstudentspasstest4.2.2 Testing against One-Sided Alternatives (8/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Testing against two-sided alternativesTestagainst.Rejectthenullhypothesisinfavourofthealternativehypothesisiftheabsolutevalueoftheestimatedcoefficientistoolarge.Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thesearethepointsofthet-distributionsothat5%ofthecaseslieinthetwotails.!Rejectifabsolutevalueoft-statisticislessthan-2.06orgreaterthan2.064.2.3 Two-Sided Alternatives (1/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Determinants of college GPALecturesmissedperweekTheeffectsofhsGPAandskippedaresignificantlydifferentfromzeroatthe1%significancelevel.TheeffectofACTisnotsignificantlydifferentfromzero,notevenatthe10%significancelevel.Forcriticalvalues,usestandardnormaldistribution4.2.3 Two-Sided Alternatives (2/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Statistically significant“ variables in a regressionIfaregressioncoefficientisdifferentfromzeroinatwo-sidedtest,thecorrespondingvariableissaidtobestatisticallysignificant“Ifthenumberofdegreesoffreedomislargeenoughsothatthenormalapproximationapplies,thefollowingrulesofthumbapply:statisticallysignificantat10%level“statisticallysignificantat5%level“statisticallysignificantat1%level“4.2.3 Two-Sided Alternatives (3/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Testing more general hypotheses about a regression coefficientNull hypothesist-statisticThe test works exactly as before, except that the hypothesized value is substracted from the estimate when forming the statisticHypothesizedvalueofthecoefficient4.2.4 Testing Other Hypotheses about b bj (1/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Campus crime and enrollmentAninterestinghypothesisiswhethercrimeincreasesbyonepercentifenrollmentisincreasedbyonepercentThehypothesisisrejectedatthe5%levelEstimateisdifferentfromonebutisthisdifferencestatisticallysignificant?4.2.4 Testing Other Hypotheses about b bj (2/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.2.4 Testing Other Hypotheses about b bj (3/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.2.5 Computing p-Values for t Tests (1/2)Computing p-values for t-testsIfthesignificancelevelismadesmallerandsmaller,therewillbeapointwherethenullhypothesiscannotberejectedanymoreThereasonisthat,byloweringthesignificancelevel,onewantstoavoidmoreandmoretomaketheerrorofrejectingacorrectH0Thesmallestsignificancelevelatwhichthenullhypothesisisstillrejected,iscalledthep-valueofthehypothesistestAsmallp-valueisevidenceagainstthenullhypothesisbecauseonewouldrejectthenullhypothesisevenatsmallsignificancelevelsAlargep-valueisevidenceinfavorofthenullhypothesisP-valuesaremoreinformativethantestsatfixedsignificancelevelsChapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.How the p-value is computed (here: two-sided test)Thep-valueisthesignificancelevelatwhichoneisindifferentbetweenrejectingandnotrejectingthenullhypothesis.Inthetwo-sidedcase,thep-valueisthustheprobabilitythatthet-distributedvariabletakesonalargerabsolutevaluethantherealizedvalueoftheteststatistic,e.g.:Fromthis,itisclearthatanullhypothesisisrejectedifandonlyifthecorrespondingp-valueissmallerthanthesignificancelevel.Forexample,forasignificancelevelof5%thet-statisticwouldnotlieintherejectionregion.valueofteststatisticThesewouldbethecriticalvaluesfora5%significancelevel4.2.5 Computing p-Values for t Tests (2/2)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.2.6 A Reminder on the Language of Classical Hypothesis TestingExample 4.5 Housing Prices and Air PollutionWedonotwanttotestthatbnox=0.Instead,H0:bnox=-1t=(-.954+1)/.117=.393Thereislittleevidencethattheelasticityisdifferentfrom-1.wefailtorejectH0atthex%level.H0isacceptedatthex%level.H0:bnox=-.9t=(-.954+.9)/.117=-.462Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.2.7 Economic, or Practical, versus Statistical Significance (1/2)economicsignificance:statisticalsignificance:Example 4.6 Participation Rates in 401(k) Plans Considerbtotemp.Example 4.7 Effect of Job Training on Firm Scrap Rates Considerbhrsemp.Someresearchersinsistonusingsmallersignificancelevelsasthesamplesizeincreases.Mostresearchersarealsowillingtoentertainlargersignificancelevelsinapplicationswithsmallsamplesizes.Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Guidelines:Ifthevariableisstatisticallysignificantattheusuallevels,discussthemagnitudeofthecoefficienttogetanideaofitseconomicimportance.Thefactthatacoefficientisstatisticallysignificantdoesnotnecessarilymeanitiseconomicallyorpracticallysignificant!Ifavariableisstatisticallyandeconomicallyimportantbuthasthewrong“sign,theregressionmodelmightbemisspecified.Ifavariableisstatisticallyinsignificantattheusuallevels(10%,5%,1%),onemaythinkofdroppingitfromtheregression.Ifthesamplesizeissmall,effectsmightbeimpreciselyestimatedsothatthecasefordroppinginsignificantvariablesislessstrong.variableswithsmalltstatisticsthathavethe“wrong”sign.(multicollinearity)4.2.7 Economic, or Practical, versus Statistical Significance (2/2)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Criticalvalueoftwo-sidedtestConfidence intervalsSimple manipulation of the result in Theorem 4.2 implies thatInterpretation of the confidence intervalTheboundsoftheintervalarerandomInrepeatedsamples,theintervalthatisconstructedintheabovewaywillcoverthepopulationregressioncoefficientin95%ofthecasesLowerboundoftheConfidenceintervalUpperboundoftheConfidenceintervalConfidencelevelChapter4MultipleRegressionAnalysis:Inference4.3 Confidence Intervals (1/3)ChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Confidence intervals for typical confidence levelsRelationship between confidence intervals and hypotheses testsrejectinfavorofUserulesofthumbChapter4MultipleRegressionAnalysis:Inference4.3 Confidence Intervals (2/3)ChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Example: Model of firms R&D expendituresSpendingonR&DAnnualsalesProfitsaspercentageofsalesTheeffectofsalesonR&Disrelativelypreciselyestimatedastheintervalisnarrow.Moreover,theeffectissignificantlydifferentfromzerobecausezeroisoutsidetheinterval.Thiseffectisimpreciselyestimatedasthein-tervalisverywide.Itisnotevenstatisticallysignificantbecausezeroliesintheinterval.Chapter4MultipleRegressionAnalysis:Inference4.3 Confidence Intervals (3/3)ChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Testing hypotheses about a linear combination of parametersExample: Return to education at 2 year vs. at 4 year collegesYearsofeducationat2yearcollegesYearsofeducationat4yearcollegesTestagainst.Apossibleteststatisticwouldbe:Thedifferencebetweentheestimatesisnormalizedbytheestimatedstandarddeviationofthedifference.Thenullhypothesiswouldhavetoberejectedifthestatisticistoonegative“tobelievethatthetruedifferencebetweentheparametersisequaltozero.4.4 Testing Hypotheses about a Single Linear Combination of the Parameters (1/4)Chapter4MultipleRegressionAnalysis:InferenceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.InsertintooriginalregressionImpossible to compute with standard regression output becauseAlternative methodUsuallynotavailableinregressionoutputDefineandtestagainst.anewregressor(=totalyearsofcollege)4.4 Testing Hypotheses about a Single Linear Combination of the Parameters (2/4)Chapter4MultipleRegressionAnalysis:InferenceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Estimation resultsThis method works always for single linear hypothesesTotalyearsofcollegeHypothesisisrejectedat10%levelbutnotat5%level4.4 Testing Hypotheses about a Single Linear Combination of the Parameters (3/4)Chapter4MultipleRegressionAnalysis:InferenceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.twoyear.wf1lslwagecjcjc+univexperseriestotcoll=jc+univlslwagecjctotcollexper4.4 Testing Hypotheses about a Single Linear Combination of the Parameters (4/4)Chapter4MultipleRegressionAnalysis:InferenceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.5 Testing Multiple Linear Restrictions: The F Test4.5.2 Relationship between F and t Statistics4.5.3 The R-Squared Form of the F Statistic4.5.1 Testing Exclusion Restrictions4.5.4 Computing p-Values for F Tests4.5.5 The F Statistic for Overall Significance of a Regression4.5.6 Testing General Linear RestrictionsChapter4MultipleRegressionAnalysis:InferenceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Testing multiple linear restrictions: The F-testTesting exclusion restrictions YearsintheleagueAveragenumberofgamesperyearSalaryofmajorlea-guebaseballplayerBattingaverageHomerunsperyearRunsbattedinperyearagainstTestwhetherperformancemeasureshavenoeffect/canbeexludedfromregression.4.5.1 Testing Exclusion Restrictions (1/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Estimation of the unrestricted model (mlb1.wf1)lslog(salary)cyearsgamesyrbavghrunsyrrbisyrNoneofthesevariabelsisstatisticallysignificantwhentestedindividuallyIdea:Howwouldthemodelfitbeifthesevariablesweredroppedfromtheregression?4.5.1 Testing Exclusion Restrictions (2/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Estimation of the restricted modelTest statisticThesumofsquaredresidualsnecessarilyincreases,butistheincreasestatisticallysignificant?NumberofrestrictionsTherelativeincreaseofthesumofsquaredresidualswhengoingfromH1toH0followsaF-distribution(ifthenullhypothesisH0iscorrect)4.5.1 Testing Exclusion Restrictions (3/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Rejection rule (Figure 4.7)AF-distributedvariableonlytakesonpositivevalues.ThiscorrespondstothefactthatthesumofsquaredresidualscanonlyincreaseifonemovesfromH1toH0.Choosethecriticalvaluesothatthenullhypo-thesisisrejectedin,forexample,5%ofthecases,althoughitistrue.4.5.1 Testing Exclusion Restrictions (4/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Test decision in exampleDiscussionThethreevariablesarejointlysignificant“TheywerenotsignificantwhentestedindividuallyThelikelyreasonismulticollinearitybetweenthemNumberofrestrictionstobetestedDegreesoffreedomintheunrestrictedmodelThenullhypothesisisoverwhel-minglyrejected(evenatverysmallsignificancelevels).mlb1.wf1PerformanceinEviews.4.5.1 Testing Exclusion Restrictions (5/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.5.2 Relationship between F and t StatisticsTheFstatisticfortestingexclusionofasinglevariableisequaltothesquareofthecorrespondingtstatistic,whosealternativeistwo-sided.Itispossiblethat,inagroupofseveralexplanatoryvariables,onevariablehasasignificanttstatistic,butthegroupofvariablesisjointlyinsignificantattheusualsignificancelevels.Often,whenavariableisverystatisticallysignificantanditistestedjointlywithanothersetofvariables,thesetwillbejointlysignificant.Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.5.3 The R-Squared Form of the F StatisticChapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.5.4 Computing p-Values for F TestsThep-valueistheprobabilityofobservingavalueofFatleastaslargeaswedid,giventhatthenullhypothesisistrue.Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Test of overall significance of a regressionThe test of overall significance is reported in most regression packages; the null hypothesis is usually overwhelmingly rejectedThenullhypothesisstatesthattheexplanatoryvariablesarenotusefulatallinexplainingthedependentvariable4.5.5 The F Statistic for Overall Significance of a Regression (1/2)Restrictedmodel(regressiononconstant)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.EndSectionChapterExample 4.9 ParentsEducationinaBirthWeightEquation(bwght.wf1)lsbwghtccigsparityfamincmotheducfatheducDependentVariable:BWGHTIncludedobservations:1191VariableCoefficientStd.Errort-StatisticProb.C114.523.72830.7160.0000CIGS-0.5960.110-5.4010.0000PARITY1.7880.6592.7110.0068FAMINC0.0560.0371.5330.1256MOTHEDUC-0.3700.320-1.1580.2470FATHEDUC0.4720.2831.6710.0949R-squared0.038748Meandependentvar119.5298AdjustedR-squared0.034692S.D.dependentvar20.14124F-statistic9.553500Durbin-Watsonstat1.911657Prob(F-statistic)0.0000004.5.5 The F Statistic for Overall Significance of a Regression (2/2)Chapter4MultipleRegressionAnalysis:Inference2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Testing general linear restrictions with the F-testExample: Test whether house price assessments are rationalTheassessedhousingvalue(beforethehousewassold)Sizeoflot(infeet)ActualhousepriceSquarefootageNumberofbedroomsIfhousepriceassessmentsarerational,a1%changeintheassessmentshouldbeassociatedwitha1%changeinprice.4.5.6 Testing General Linear Restrictions (1/4)Inaddition,otherknownfactorsshouldnotinfluencethepriceoncetheassessedvaluehasbeencontrolledfor.Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Unrestricted regressionRestricted regressionTest statisticcannotberejected4.5.6 Testing General Linear Restrictions (2/4)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Performance in Eviews.hprice1.wf1lslog(price)clog(assess)log(lotsize)log(sqrft)log(bdrms)c(2)=1,c(3)=0,c(4)=0,c(5)=04.5.6 Testing General Linear Restrictions (3/4)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Regression output for the unrestricted regressionThe F-test works for general multiple linear hypothesesFor all tests and confidence intervals, validity of assumptions MLR.1 MLR.6 has been assumed. Tests may be invalid otherwise.Whentestedindividually,thereisalsonoevidenceagainsttherationalityofhousepriceassessments4.5.6 Testing General Linear Restrictions (4/4)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.4.6 An application- estimation of the weights of CPI components in China (1/2)Unrestricted model:cpi=w0+w1*food+w2*daily+w3*cloth+w4*equit+w5*health+w6*trans+w7*entertain+w8*dwell+uH0:w0=0,w1+w2+w3+w4+w5+w6+w7+w8=1Restricted model:cpi-food=w2*(daily-food)+w3*(cloth-food)+w4*(equit-food)+w5*(health-food)+w6*(trans-food)+w7*(entertain-food)+w8*(dwell-food)+ucpi-weight.wf1smpl2003m12005m12lscpicfooddailyclothequithealthtransentertaindwelllscpi-fooddaily-foodcloth-foodequit-foodhealth-foodtrans-foodentertain-fooddwell-foodChapter4MultipleRegressionAnalysis:InferenceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.SSRur=0.0358SSRr=0.0372Operate the test in Eviews:Regresstheunrestrictedmodel.IntheEquationwindow,View/CoefficientTests/Wald-CoefficientRestrictionsc(1)=0,c(2)+c(3)+c(4)+c(5)+c(6)+c(7)+c(8)+c(9)=14.6 An application- estimation of the weights of CPI components in China (2/2)Chapter4MultipleRegressionAnalysis:InferenceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.C8401ksubs.wf1smplifmarr=1andfsize=24.6 An application- estimation of the weights of CPI components in China (2/2)Chapter4MultipleRegressionAnalysis:InferenceChapterEnd2013CengageLearning.AllRightsReserved.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart.Chapter4MultipleRegressionAnalysis:InferenceChapter