EconometricsI,Spring,2000MidtermIPleaseanswerallquestions.Pointvaluesaregiveninsquarebracketswitheachquestion.[10]1.Explaincarefullythedifferencebetweenanestimatorthatisunbiasedandonethatisconsistent.Areallunbiasedestimatorsconsistent?Areallconsistentestimatorsunbiased?Again,explain.Anunbiasedestimatorisonethathasexpectationthatisequaltotheparameterbeingestimated.Unbias一dnessrelatesonlytoexpectedvalue,andhasnothingtodowithvarianceorprecision.Aconsistentestimatorisonethatconvergesinprobabilitytotheparameterbeingestimated.Consistencyrelatestothebehavioroftheestimatorasthesamplesizeincreases.Sufficientconditionsforconsistencyarethattheexpectationoftheestimatorconvergetotheparameterasnincreasesandthatthevarianceoftheestimatorconvergetozero.Anunbiasedestimatorcanbeinconsistent,ifitsvariancedoesnotgotozero.Anunbiasedestimatormaybeconsistent.Thesamplemeanisanexample.Aconsistentestimatormightbeunbiased,forexample,thesamplemean,oritmightbebiased,forexample,thesamplemeanplus1/n.[10]2.WhatistheFrisch-Waughtheorem?Explaintheresultinthecontextofthemultipleregressionmodel.TheFrisch-Waughtheoremdescribeshowtocomputethesubvectorsofaleastsquarescoefficientvectorinamultipleregression.Inparticularzonewillgetthesamenumericalanswerforthecoefficientsinamultipleregressionif(1)theyfitthemultipleregressionwithallvariablesincludedor(2)ifeachvariableorsetofvariables,inturn,isregressedontheremainingvariablesandtheresidualsarecomputed,thedependentvariableislikewisetransformed,thenresidualsareregressedonresiduals.IntheFrisch-Waughapplication,theyfoundthatdetrendingproducedthesameresultsastheoriginalregressionifthetimetrendandtheconstanttermwerecontainedinthemultipleregression.[1013.Considerthefollowingexperiment:Wehaven=500observationsonthreevariables,y,xl,andx2.Thevariables,ascanbeseenintheresultsbelow,havemeansequaltozerosoanyregressionsarefitwithoutaconstantterm,astheconstanttermwouldbezeroanyway.Wecomputethefollowingregressions:Regressionofyonx1andx2.TOC\o"1-5"\h\z+ +IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=Y Mean=0.00 ,S.D.= 1.735210951 |IModelsize:Observations= 500,Parameters=2ZDeg.Fr.= 498|IResiduals:Sumofsquares=504.6812512 ,Std.Dev.= 1.00669|IFit: R-squared=.664098,AdjustedR-squared= .66342|IModeltest:F[1, 498]=984?58,Probvalue= .00000|+ + + + + + +IVariable | Coefficient | StandardError |t-ratio |P[|T|>t] | Meanof X|+ + + + + + +XI .9545152414 .043413614 21.987 .0000 0.00X2 .9972568684 .044618141 22.351 .0000 0.00xlisregressedonx2andtheresidualsarecomputed.Thisisvariablexls.x2isregressedonxlandtheresidualsarecomputed.Thisisvariablex2sFinally,yisregressedonxlsandx2s.Theresultsofthesecondregressionaregivenbelow.TOC\o"1-5"\h\z+ 4-IOrdinaryleastsquaresregressionWeightingvariable=noneIIDep.var.=Y Mean=0.00 ,S.D.= 1.735210951 |IModelsize:Observations= 500,Parameters=2,Deg.Fr.= 498|IResiduals:Sumofsquares=504.6812512 ,Std.Dev.= 1.00669IIFit: R-squared=.664098,AdjustedR-squared= .66342|IModeltest:F[1, 498]=984.58,Probvalue= .00000|+ + + + + + +IVariable|CoefficientIStandardError|t-ratio|P[|T|>t]IMeanofX|+ + + + + + +X1S .9560998253 .043413672 22.023 .0000 0.00X2S .9988589528 .044618200 22.387 .0000 0.00Noticethattheleastsquaresregressioncoefficientsinthesecondregressionaredifferentfromthoseinthefirst.(Thisisarealdifference-notroundingerror.)However,theR2andthesumofsquaredresidualsinthetworegressionsareidentical.Thisshouldsuggesttoyouwhatisgoingonhere.Canyougivenanexplanation,intermsofotherresultswehavediscussedinclass?(Hint:thisquestionisnotanapplicationofFrisch-Waugh.)Thiswasthemostdifficultquestiononthetest.Inthefirstregression,weareregressingyonxlandx2.Inthesecond,wefirstcomputexlsbyregressingxlonx2andcomputingtheresiduals.So,xls=xl-cl*x2whereclisthecoefficientinthatregression,cl=xl,x2/x2,x2.Theothervariable,x2siscomputedthesameway,x2s=x2-c2*xlwherec2=xl,x2/xl1xl.So,inthesecondregression,wear一regressingyonalinearcombinationofthevariablesinthefirstregression. Inthenotationofourresultsinclass,thefirstregressionisofyonXwhereXisa2columnmatrix,[xl,x2].Inthesecondregression,weareregressingyonXCwhereCisa2x2matrix,C-1-cl= ?Weknowfromourclassresultsthatwhenweusealinear-Cl1transformationoftheregressors,theR2andsumofsquaredresidualswillbeidentical,buttheregressioncoefficientvectorwillbeC_1timestheoriginalone.4.Theregressionresultsgivenbelowarebasedonadatasetusedinawellknownstudyoflaborsupplyofmarriedwomen.Thedependentvariableisnumberofhoursofworkinthelabormarketinasurveyyear.TheindependentvariablesareWA二wife'sage,WE=wife'seducation(yearsofschool),KL6=numberofchildrenlessthan6inthehousehold,HW=husband'swagerate.TOC\o"1-5"\h\z+ +IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=WHRSMean=1333.066667 ,S.D.= 827.8706386 |IModelsize:Observations= 150,Parameters= 5,Deg.Fr.= 145|IResiduals:Sumofsquares=96999616.91 ,Std.Dev.= 817.90151|IFit: R-squared=.050142zAdjustedR-squared= .02394|IModeltest:F[4, 145]= 1?91,Probvalue= .11126|+ + + + + + +IVariable | Coefficient |Standard Error |t-ratio |P[|T|>t] |Mean of X|+ + + + + + +Constant1128.797382563.205472.004.0469WA3.5721945048.8816910.402.688142.786667WE16.1685363032.946637.491.624312.640000KL6-382.4001782168.57726-2.268.0248.17333333HW-12.3620059421.414083-.577.56467.0102387MatrixCov.Mat.has5rowsand5columns.12 3 4+ 1| .3172004D+062|-3486.12743|-.1252500D+054|-.2547236D+05+ 1| .3172004D+062|-3486.12743|-.1252500D+054|-.2547236D+055| -121.1957-3486.127478.88447.3472647.4375-13.4326-.1252500D+057.3472

1085.4809

-608.8334-200.3219-.2547236D+05-121.1957647.4375-608.8334.2841829D+0577.0989-13.4326-200.321977.0989458.5630TOC\o"1-5"\h\z+ +IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=WHRSMean=1333.066667 ,S.D.= 827.8706386 |IModelsize:Observations= 150zParameters= 3,Deg.Fr.= 147|IResiduals:Sumofsquares=97262475.03 ,Std.Dev.=813.41840|IFit: R-squared=.047568,AdjustedR-squared= .03461|IModeltest:F[2, 147]= 3.67,Probvalue= .02782|+ + + + + + +IVariable| Coefficient | StandardError |t-ratio |P[|T|>t] | Meanof X|+ + + + + + +Constant1464.762583 160.13317 9.147 .0000KL6 -401.9506589 149.81217 -2.683 .0081 .17333333HW -8.847697115 20.375362 -,434 .6648 7.0102387MatrixCov.Mat.has3rowsand3columns.12 3+ 1|.2564263D+05-4337.8222 -2921.40512|-4337.8222 .2244369D+05 63.84713|-2921.4051 63.8471 415.1554[10]a.Testthehypothesisthatthehusband'swageisnotasignificantdeterminantofthewife'shoursworked.Carefullystatethehypothesisandshowexplicitlyhowyoucarryoutthetest.ThehypothesiswouldbethatthecoefficientonHWiszero.Theteststatisticwouldbet(HW)=|(b(hw)-0)|/standarderror.Thiswouldhaveatdistributionwith(150-5)degreesoffreedomundertheassumptionofnormallydistributeddisturbance.Thesamplevalueofthestatisticis-.577.Thecriticalvaluefor95%significance,fromthetableofthetdistributionis1.96.(145isalmostinfinity,sothisisthevaluefromthestandardnormal.)Since.577islessthan1.96,thehypothesisthatthecoefficientiszeroisnotrejected.[10]b.TestthejointhypothesisthatthattheWAandWEvariablesaretogethersignificantdeterminantsofhoursworked.(Thisisnotatestofeacheffectseparately.)Again,showandexplainyourcomputations.Thisistworestrictions,b(WA)andb(WE)equalzero.Thereareseveralwaystocarryoutthistest.Sinceboththeunrestrictedandrestrictedregressionsaregiven,wecanusetheFtest.ThetestcanbebasedontheR2sforthetworegressions.ThestatisticisF[2,145]=[(.050142-.047568)/2]/[(1-.050142)/(150-5)]=0.1964.ThecriticalvaluefromtheF[2,145]tableisslightlylessthan3.00.The.1964isfarlessthan3?00,sothehypothesisthatbothcoefficientsarezeroisnotrejected.[10]c.Aresearchersuggeststhatitisnotthenumberofkidslessthan6thateffectslaborsupplysomuchasthepresenceofanykidslessthan6.Inanattempttofindout,theycomputethetwovariablesK6=1ifthereareanychildrenunder6inthehousehold,0ifnot.MOREKIDS=thenumberofchildrenlessthan6inexcessof1.Thus,ifthereare1or0kidslessthan6,thenMOREKIDS=0,if2ormore,MOREKIDS=KL6-1.Theresultsofregressionofhoursofworkonaconstant,K6,MOREKIDS,WA,WE,andHWareshownbelow.Yourreaction?Dotheregressionresultssupportthetheory?

IOrdinaryleastsquaresregressionWeightingvariable=none|TOC\o"1-5"\h\zIDep.var.=WHRSMean=1333.066667 ,S.D.= 827.8706386 |IModelsize:Observations= 150zParameters= 6,Deg.Fr.= 144|IResiduals:Sumofsquares=95926656.49 ,Std.Dev.= 816.18462|IFit: R-squared=.060649,AdjustedR-squared= .02803IIModeltest:F[5, 144]= 1.86,Probvalue= .10505I+ + + + + + +IVariable | CoefficientI StandardError|t-ratio |P[|T|>t]| MeanofX|+ + + + + + +564.976851.868227.38908-.828452.69997-2.0238.9653893.59032.877488.49121.426761-.670.556342.786667.624312.640000.50407.0102387.0637.4092 .14666667.0449.26666667E-01Constant1055.0637.4092 .14666667.0449.26666667E-01K6 -188.2340788MOREKIDS-915.7912006WA 5.286496276WE 16.13618762HW -14.35508828thetheory.Infact,basedexactlytheopposite.Thenotsignificant,whiletheonekidisextremelylarge-well.Theresultsdefinitelydonotsupportontheresultsabove,onewouldconcludedummyvariable,K6=kidsinthehome,iscoefficientonthenumberofkidsmorethan915hours,andstatisticallysignificantasthetheory.Infact,basedexactlytheopposite.Thenotsignificant,whiletheonekidisextremelylarge-well.[101d.Adifferentresearcher,whoknowsmoreaboutthismarket,hypothesizesthatthelabormarketbehaviorofwomenwithchildrenlessthan6iscompletelydifferentfromthatofwomenwithoutchildrenunder6.Howwouldyouusethedatausedabovetotestthishypothesis?Theregressionresultsaregivenbelow.Testthehypothesis.Showandcarefullyexplainyourcomputations.(Pooleddata-all150observations)+ 4-IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=WHRSMean=1333.066667 ,S.D.= 827.8706386 |IModelsize:Observations= 150zParameters=4,Deg.Fr.= 146|IResiduals:Sumofsquares=100441849.5 ,Std.Dev.=829.43226|IFit: R-squared=.016434rAdjustedR-squared= -.00378|IModeltest:F[3, 146]= .81,Probvalue= .48852|TOC\o"1-5"\h\z+ + + + + + +IVariable| Coefficient | StandardError |t-ratio |P[|T|>t] | Meanof X|+ + + + + + +Constant786.0381147550.206421.429.1552WA12.284195708.12128791.513.132542.786667WE7.97599612233.209770.240.810512.640000HW-11.3245539021.711025-.522.60277.0102387TOC\o"1-5"\h\z(Subsample,nochildrenunder6inthehousehold)+ +IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=WHRSMean=1391.781250 ,S.D.= 820.0404440 |IModelsize:Observations= 128,Parameters= 4,Deg.Fr.= 124|IResiduals:Sumofsquares=84987167.39 ,Std.Dev.= 827.87703|IFit: R-squared=.004872zAdjustedR-squared= 01920|IModeltest:F[3, 124]= .20,Probvalue= .89460|+ + + + + + +IVariable | Coefficient |Standard Error |t-ratio |P[|T|>t] |Mean of X|+ + + + + + +ConstantWAWEConstantWAWEHW935.30553884.73181577420.59718319-1.656453250597.733641.565.12029.2596309.511.610244.37500035.404514.582.561812.53125023.584802-.070.94417.0071797(Subsample,atleastonechildunder6inthehousehold)TOC\o"1-5"\h\z+ +IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=WHRSMean=991.4545455 ,S.D.= 807.9436996 |IModelsize:Observations= 22zParameters=4rDeg.Fr.= 18|IResiduals:Sumofsquares=12584243.26 ,Std.Dev.= 836.13673|IFit: R-squared=.081994,AdjustedR-squared= 07101IIModeltest:F[3, 18]= ?54,Probvalue= .66361|+ + + + + + +

IVariable | Coefficient| StandardError|t-ratio |P[|T|>t]| MeanofX|+ + + + + + +Constant159.9644530 2457.1305 .065 .9488WA34.7443729852.436164.663.516033.545455WE9.339148528112.04441.083.934513.272727HW-65.1649536064.958960-1.003.32917.0280364ThisisanapplicationoftheChowtest.Thethreesetsofregressionsaregiven,sowehavethesumsofsquaresweneed.F[4,150-8]=[100441849.5-(84987167.39+12584243.26)]/4 717609.7775 = =1.044(84987167.39+12584243.26)/(150-4-4) 687122.6102ThecriticalvalueofF[4,142]isabout44ismuchlessthanthis,sothehypothesisthatthetwogroupshavethesameregressionfunctionisnotrejected.TOC\o"1-5"\h\z5.Usingthesamedatausedinpart4,wenowregressthewife'swagerate,WWonaconstantandtheeducationvariable,WE,thenonaconstant,WEandWEsquared.Theresultsappearbelow:+ 4-IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=WWMean=3.792050000 ,S.D.= 2.342414629 |IModelsize:Observations= 150zParameters= 2,Deg.Fr.= 148|IResiduals:Sumofsquares=705.4433955 ,Std.Dev.= 2.18323|IFit: R-squared=.137124,AdjustedR-squared= .13129|IModeltest:F[1, 148]=23?52,Probvalue= .00000|+ + + + + + 4-|Variable | Coefficient |Standard Error |t-ratio |P[|T|>t] | Meanof X|+ + + + + + +Constant-1.338070889 1.0727403 -1.247 .2142WE .4058639944 .83688738E-01 4.850 .0000 12.640000+ +IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=WWMean=3.792050000 ,S.D?= 2.342414629 |IModelsize:Observations= 150,Parameters= 3,Deg.Fr.= 147|IResiduals:Sumofsquares=687.9037819 ,Std.Dev.= 2.16324|IFit: R-squared=.158578,AdjustedR-squared= .14713|IModeltest:F[2, 147]=13?85,Probvalue= .00000I+ + + + + + +IVariable| Coefficient| StandardError|t-ratio |P[|T|>t]|Mean ofX|+ + + + + + +Constant6.656714367 4.2641432 1.561 .1207WE -.8577471274 .65793890 -1.304 .1944 12.640000WE2.4855103864E-01.25078046E-01 1.936 .0548 164.30667MatrixCov.Mat.has3rowsand3columns.12 3+ 1| 18.1829 -2.7822 .10362| -2.7822 .4329 -,01643| .1036 -.0164 .0006[10]a.Basedonthefirstregression,WWobviouslyvariespositively(inthesamedirection)aseducation,WE.Thatis,moreeducation,higherwage.But,thecoefficientonWEinthesecondregressionisnegative.Isn'tthisacontradiction?Explain.It1It1snotagrossregressioncoefficient,whilethesecondisapartialregressioncoefficient.Weknowthatafteraccountingfor

theregression,thereisnothingtopreventthethepresenceofWE2inbeingnegative.Also,sinceWE2isthesquaredirectlyjustbylookingatthecoefficientscoefficientonWE

ofWE,wecannot

thattheeffectfromtellthatcoefficient.Weknowthatafteraccountingfor

theregression,thereisnothingtopreventthethepresenceofWE2inbeingnegative.Also,sinceWE2isthesquaredirectlyjustbylookingatthecoefficientscoefficientonWE

ofWE,wecannot

thatthee