第9章-Multiple-Regression-教学讲解课件

MultipleRegressionAnalysisP289

多元回归分析之模型设定和数据问题

y=b0+b1x1+b2x2+...bkxk+uSpecificationandDataProblems模型设定和数据问题1MultipleRegressionAnalysisChapterOutline 本章大纲FunctionalFormmisspecification函数形式误设-讨论模型误设的结果-P289UsingProxyvariablesforunobservedexplanatoryvariables对观测不到的变量使用代理变量-讨论用代理变量来减轻有偏性PropertiesoftheOLSUnderMeasurementError有测量误差的OLS性质-推导和解释MissingData,NonrandomSamples,andoutliers数据缺失、非随机样本和离群点-讨论额外的数据问题2ChapterOutline 本章大纲FunctionaFunctionalForm

函数形式Howdoweknowifwe’vegottentherightfunctionalformforourmodel?我们如何知道模型是否得到正确的函数形式呢？P289:异方差的出现可以看成是模型的错误设定，但不影响有偏性和一致性，还可以通过WLS来减轻；本章讨论u与xi的相关性，如果相关，称xi为外生变量，为什么？当被忽略的自变量为其他变量的函数时，将产生函数形式误设这一问题。何谓函数形式误设？3FunctionalForm

函数形式HowdoweFunctionalForm(continued)

函数形式（续）

First,useeconomictheorytoguideyou首先，用经济理论的指导Thinkabouttheinterpretation考虑它的解释Doesitmakemoresenseforxtoaffectyinpercentage(uselogs)orabsoluteterms?x影响y的更合理的方式是百分比的形式（用log形式），还是绝对量的形式？Doesitmakemoresenseforthederivativeofx1tovarywithx1(quadratic)orwithx2(interactions)ortobefixed?x1的系数更合理的形式是随x1变化（二次形式），随x2变化（交互作用），还是固定不变？P290:2个误设案例，一个是忽略了二次项，一个是忽略了交叉项。也可能是没有用LOG形式；回顾第三章P85假设3不成立的几种情况，函数形式误设的后果P290EXP.9.1-阅读4FunctionalForm(continued)

函数FunctionalFormMisspecification

函数形式误设Amultipleregressionmodelsuffersfromfunctionalformmisspecificationwhenitdoesnotproperlyaccountfortherelationshipbetweenthedependentandtheobservedexplanatoryvariables.

当一个多元回归模型不能正确地说明被解释变量和观察到的解释变量之间的关系时，此模型存在函数形式误设问题。5FunctionalFormMisspecificatiFunctionalFormMisspecification

函数形式误设Misspecifyingthefunctionalformofamodelcanhaveseriousconsequences.Wemayobtainbiasedorinconsistentestimatorsofthepartialeffects.误设一个模型的函数形式可能产生严重的后果。我们得到的局部效应的估计量可能有偏或不一致。Onewayout:toaddquadratictermsofanysignificantvariablestoamodelandtoperformajointtestofsignificance.

一种方法：向模型加入任何重要变量的二次项，进行一个联合显著性检验。-加入二次项，对二次项系数联合显著性F检验通过时，显示的症状往往是误设，如误将对数模型为水平模型。另外经济数据中，二次项可以解决大部分非线性问题-P2906FunctionalFormMisspecificatiExample:ModelingCrime

例子：对犯罪建模-P292Dependentvariable:被解释变量：Narr86,#timesarrested,1986（1986年被捕次数）ExplanatoryVariables：解释变量：pcnvproportionofpriorconvictions以前被定罪比例avgsen avgsentencelength,mos.平均判刑期限，单位：月tottime timeinprisonsince18,mos.18岁以来的服刑时间，单位：月Ptime86mos.inprisonduring19861986年的服刑时间，单位：月解读：1.为什么加入二次项，因为水平项T检验很显著；2.加入变量的二次项后，原先的水平变量系数变化很大；同时二次项联合F显著；3.二次项加入，模型的解读变得困难，可能有更深刻的实际意义7Example:ModelingCrime

例子：对犯罪Example:ModelingCrime

例子：对犯罪建模Explanatoryvariables解释变量Qemp86#quartersemployed,19861986年被雇佣季度数inc86 legalincome,1986,$100s1986年合法收入，单位：百美元black =1ifblack如果是黑人，black=1hispan =1ifHispanic如果是西班牙裔，hispan=1First,weregressthedependentvariablesontheindependentvariables,withoutanysquareterms.首先，我们将被解释变量向解释变量回归，不包含任何平方项。8Example:ModelingCrime

例子：对犯罪

regnarr86pcnvavgsentottimeptime86qemp86inc86blackhispanSource|SSdfMSNumberofobs=2725-------------+------------------------------F(8,2716)=26.47Model|145.390104818.173763Prob>F=0.0000Residual|1864.957052716.686655763R-squared=0.0723-------------+------------------------------AdjR-squared=0.0696Total|2010.347162724.738012906RootMSE=.82865------------------------------------------------------------------------------narr86|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------

pcnv|-.1332344.0403502-3.300.001-.2123546-.0541141avgsen|-.0113177.0122401-0.920.355-.0353185.0126831tottime|.0120224.00943521.270.203-.0064785.0305233

ptime86|-.0408417.008812-4.630.000-.0581206-.0235627qemp86|-.0505398.0144397-3.500.000-.0788538-.0222258

inc86|-.0014887.0003406-4.370.000-.0021566-.0008207black|.3265035.04541567.190.000.2374508.4155561hispan|.1939144.03971134.880.000.1160469.2717818_cons|.5686855.036046115.780.000.4980048.6393661------------------------------------------------------------------------------99Plottingnarr86againstpncv

绘图：narr86关于pncv10Plottingnarr86againstpncv

绘Plottingnarr86againstinc86

绘图：narr86关于pncv11Plottingnarr86againstinc86Plottingnarr86againstptime86

绘图：narr86关于pncv12Plottingnarr86againstptime8

narr86Coef.Std.Err.tP>|t|[95%Conf.Interval]

pcnv.5525236.15423723.580.000.2500892.8549579

pcnvsq-.7302119.1561177-4.680.000-1.036333-.4240903avgsen-.0170216.0120539-1.410.158-.0406574.0066142tottime.011954.00928251.290.198-.0062474.0301554

ptime86.2874334.04425826.490.000.2006501.3742166

pt86sq-.0296076.0038634-7.660.000-.037183-.0220321qemp86-.0140941.0173612-0.810.417-.0481366.0199485

inc86-.0034152.0008037-4.250.000-.0049912-.0018392

inc86sq7.19e-062.56e-062.810.0052.17e-06.0000122black.292296.044836.520.000.2043916.3802004hispan.1636175.03945074.150.000.0862609.240974_cons.5046065.036835313.700.000.4323784.5768347AddingQuadratictermstosignificantVariables加入重要变量的平方项13narr86Coef.Drawbacksofaddingsquaretermstodetectfunctionalformmisspecification

取消加入平方项以检测函数形式误设

Theoretically,wecantestjointexclusionrestrictionstoseeifhigherordertermsorinteractionsbelongtothemodel理论上，我们作排除性约束的联合检验，来看高阶项和交叉项是否属于模型。Itcanbetedioustoaddandtestextraterms.Manydegreesoffreedomsmaybeused. 加入和检验另外的项过程会很单调乏味且冗长。当原模型解释变量多时可能会消耗掉许多自由度。14DrawbacksofaddingsquareterDrawbacksofaddingsquaretermstodetectfunctionalformmisspecification

取消加入平方项以检测函数形式误设Somenonlinearitiescannotbepickedupbyaddingquadraticterms.Forexample,wemayfindasquaretermmatterswhenusinglogsismoreappropriate. 一些非线性关系不能通过加入二次项捕捉。例如，当我们发现平方项重要时，可能对数形式更加适合。15DrawbacksofaddingsquareterRamsey’sRESETP292

Ramsey回归设定误差检验AtestoffunctionalformisRamsey’sregressionspecificationerrortest(RESET)一种函数形式的检验是Ramsey回归设定误差检验（RESET）。RESETaddspolynomialsintheOLSfittedvaluestotheoriginalregression.RESET在原回归中加入OLS拟合值的多项式-没有明确的原理指出到底要加入多少个高次方的项，但是平方和立方一般是有用的。16Ramsey’sRESETP292Ramsey’sRESET

Ramsey回归设定误差检验

Insteadofaddingfunctionsofthex’sdirectly,weaddandtestfunctionsofŷ我们加入并检验ŷ的多次项函数，而不是直接加入x的函数。注意：如何加入函数项的？P293So,estimatey=b0+b1x1+…+bkxk+d1ŷ2+d1ŷ3+errorandtest所以，估计y=b0+b1x1+…+bkxk+d1ŷ2+d1ŷ3+error，并检验。H0:d1=0,d2=0,usingFstatisticorLMstatistic.H0:d1=0,d2=0，用F统计量或LM统计量。17Ramsey’sRESET

Ramsey回归设定误差检验Ramsey’sRESET

Ramsey回归设定误差检验AsignificantFstatisticsuggestssomesortoffunctionalformproblem.一个显著的F统计量说明函数形式可能存在问题。ThedistributionofFisapproximatelyF2,n-k-3inlargesamplesunderthenullhypothesisandtheG-Massumptions.在零假设和G-M假定下，F的分布大样本近似为F2,n-k-3分布。自由度的说明：减少了2个自由度P29318Ramsey’sRESET

Ramsey回归设定误差检验ImplementingRESETinStata

在STATA中实施RESETSTATAusescommandovtestafterregcommand.STATA在reg命令后，使用ovtest命令。ŷ2,ŷ3,andŷ4

areusedinstata.STATA使用ŷ2,ŷ3和ŷ4。regnarr86pcnvavgsentottimeptime86qemp86inc86blackhispan

ovtest

RamseyRESETtestusingpowersofthefittedvaluesofnarr86RESET检验使用narr86拟合值的幂函数项 Ho:modelhasnoomittedvariablesF(3,2713)=4.19,Prob>F=0.005819ImplementingRESETinStata

在SImplementingRESETinStata

在STATA中实施RESETAnalternativeistospecifytheoption,rhs.一个替代的方法是指定选择，rhsInthiscasethepowertermsofalltheexplanatoryvariablesinsteadofthefittedvaluesareusedinthetest.在这种情况下，检验中使用所有解释变量的幂函数项，而不是拟合值的相应项。ovtest,rhs RamseyRESETtestusingpowersoftheindependentvariablesRESET检验使用解释变量的幂函数项Ho:modelhasnoomittedvariablesF(18,2698)=9.73Prob>F=0.000020ImplementingRESETinStata

在SCautionsinUsingRESET

使用RESET的注意事项RESETisgoodatdetectingmisspecificationsintheformofnonlinearities,notgeneralomittedvariables. RESET在探测非线性形式的函数误设时很好用，而不是用于检测一般的遗漏变量。Wooldridge(1995)showsthatRESEThasnopowerfordetectingomittedvariableswhenevertheyhaveexpectationsthatarelinearintheincludedindependentvariables. Wooldridge在1995年证明：当被遗漏变量的期望值时所包含自变量的线性函数时，RESET无法探测出遗漏变量问题。P294：对RESET作用的正确评价：1.有的认为可以检测遗漏变量和异方差，但是Wooldridge不这样认为21CautionsinUsingRESET

使用RESECautioninusingofRESET

使用RESET的注意事项However,iftheomittedvariablehavenonlinearexpectationsinthedependentvariables,asignificantRESETcanindicateomitted-variableproblem. 尽管如此，如果被遗漏变量的期望是自变量的非线性形式时，一个显著的RESET可以指出遗漏变量问题。AlsonoticethatthedrawbackoftheRESETtestiswhenthenullisrejected,RESETdoesnotsuggestwhattodointhenextstep. 也要注意到，RESET检验的一个缺陷是，当零假设被拒绝后，它并不能建议我们下一步怎么做。22CautioninusingofRESET

使用REHousingPriceExample

住房价格的例子Thisexampleisusedfortwopurposes. 使用这个例子有两个目的。First,logformscanbebetterindealingwithnonlinearitiesthenusingthelevelvariables. 首先，处理非线性问题时，log形式可能比变量原形式更好。Second,asignificantRESETmayindicatenonlineareffectofomittedvariables,likethevariable“assess”addedinlater. 其次，一个显著的RESET可能指出被遗漏变量的非线性效应，比如稍后加入的变量“assess”。23HousingPriceExample

住房价格的例子THousingPriceExample

住房价格的例子Dataused:hprice1.dta,variables使用数据：hprice1.dta,变量assessassessedvalue,$1000s（评估价，单位：千美元）pricehouseprice,$1000s（房价，单位：千美元）lotsizesizeoflotinsquarefeet（土地的面积，单位：平方英尺）sqrftsizeofhouseinsquarefeet（房屋的面积，单位：平方英尺）bdrmsnumberofbedrooms（卧室数）24HousingPriceExample

住房价格的例子DHousingPriceExample

住房价格的例子

P293阅读

regpricelotsizesqrftbdrmsovtest

RamseyRESETtestusingpowersofthefittedvaluesofprice（RESET检验用拟合价格的幂函数项）Ho:modelhasnoomittedvariablesF(3,81)=4.26Prob>F=0.007625HousingPriceExample

住房价格的例子HousingPriceExample:thelogforms

住房价格的例子：log形式Thelogformdonotrejectthenullofnomisspecificationat5%significancelevel.Log形式的回归在5％水平上没有拒绝零假设：没有函数形式误设。--结论：第二个模型即对数模型更好一些。-P293reglpricellotsizelsqrftbdrmsovtestRamseyRESETtestusingpowersofthefittedvaluesoflprice（RESET检验用lprice拟合值的幂函数项）Ho:modelhasnoomittedvariablesF(3,81)=2.45Prob>F=0.069226HousingPriceExample:thelogHousingPriceExample:thelogforms

住房价格的例子：log形式reglpricelassessllotsizelsqrftbdrmsInthisstepvariablelassessisasignificantvariablewitht=6.89.这一步中，变量lassess显著，t=6.89ovtest

RamseyRESETtestusingpowersofthefittedvaluesoflprice（RESET检验使用lprice拟合值的幂函数项）Ho:modelhasnoomittedvariablesF(3,80)=1.11Prob>F=0.350927HousingPriceExample:thelogHousingPriceExample:thelogforms

住房价格的例子：log形式Noticetheresultsaredifferentfromthetextbooksinceŷ2,ŷ3,andŷ4

areusedinstata,insteadofŷ2,ŷ3

asinthetextbook

. 注意这里的结果和课本上不同，因为课本上使用ŷ2,ŷ3

，这里stata用的是ŷ2,ŷ3,和

ŷ4

。Youcanreplicatethetextbookresultbyputtingŷ2,ŷ3

intothemainequation,anduseFtesttotesttheirjointsignificances.

你可以通过以下方法得到课本的结果：向主方程加入ŷ2,ŷ3

，使用F检验检验它们的联合显著性。28HousingPriceExample:thelogNonnestedAlternativeTests:MR

非嵌套替代模型的检验：MRP294

-如何检验非嵌套模型？二种方法：MR方法、DM方法

Whichofthefollowingmodelisbetter?下面哪一个模型更好？MizonandRichard(1986):Constructacomprehensivemodelthatcontainseachmodelasaspecialcaseandthentotesttherestrictionsthatledtoeachofthemodels.

MizonandRichard(1986):

构造一个综合模型，将每个模型都作为一个特殊情况包含其中，然后检验导致每个模型变的约束。注意：第6章P199曾提出用拟合优度监测29NonnestedAlternativeTests:MNonnestedAlternativeTests

非嵌套替代模型的检验Intheaboveexample,thecomprehensivemodelis在上例中，综合模型是：

andtest

30NonnestedAlternativeTests

非嵌NonnestedAlternativeTests:DM

嵌套替代模型的检验：DMDavidsonandMacKinnon(1981):if(9.6)istrue,thenthefittedvaluesfrom(9.7),shouldbeinsignificantin(9.6).DavidsonandMacKinnon(1981):如果（9.6）正确，那么从（9.7）得到的拟合值在（9.6）中应当不显著。注意：D-M检验的思路，是一个t检验P29431NonnestedAlternativeTests:DNonnestedAlternativeTests:DM

嵌套替代模型的检验：DMTotest(9.6),wefirstestimatemodel(9.7)byOLStoobtainthefittedvalues.为了检验（9.6），我们首先通过OLS估计模型（9.7）以得到拟合值。Putthisfittedvalueasanadditionalexplanatoryvariablein(9.6),usetstatistictotestitssignificance.将得到的拟合值作为另外的解释变量放到（9.6）中，用t统计量检验其显著性。32NonnestedAlternativeTests:DTheHousingPriceExample:MR

住房价格的例子：MRThecompetingmodels:竞争模型是：

(1)

reglpricebdrmscolonialassesslotsizesqrft(2)reglpricebdrmscoloniallassessllotsizelsqrft

Thecombinedregression:组合的回归：

reglpricecolonialbdrmsassesslotsizesqrftlassessllotsizelsqrft

33TheHousingPriceExample:MR

TheHousingPriceExample:MR

住房价格的例子：MRTestingwhether(2)istherightone:检验（2）是否正确：testassesslotsizesqrft

F(3,79)=2.92,Prob>F=0.0392Testingwhether(1)istherightone:检验（1）是否正确： testlassessllotsizelsqrftF(3,79)=3.97,Prob>F=0.0108Inclusive.34TheHousingPriceExample:MR

TheHousingPriceExample:DM

住房价格的例子：DMTestingwhether(2)istherightone:检验（2）是否正确：reglpriceassessbdrmslotsizesqrftcolonial

predictyl,xbreglpricelassessllotsizelsqrftbdrmscolonialylThetablebelowshowthatylisaninsignificantvariable.下表显示yl不是一个显著的变量。35TheHousingPriceExample:DM

Source|SSdfMSNumberofobs=88-------------+------------------------------F(6,81)=48.11Model|6.2607657361.04346095Prob>F=0.0000Residual|1.7568377981.021689355R-squared=0.7809-------------+------------------------------AdjR-squared=0.7646Total|8.0176035287.092156362RootMSE=.14727

-----------------------------------------------------------------------------lprice|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+---------------------------------------------------------------lassess|.6762505.33745562.000.048.00481971.347681llotsize|-.0119247.0419541-0.280.777-.0954003.0715508lsqrft|-.1258866.1407801-0.890.374-.4059949.1542216bdrms|.0152289.0245180.620.536-.0335542.0640121colonial|.0243595.0397240.610.541-.0546788.1033977

yl|.4346309.36462431.190.237-.2908571.160119_cons|.3062863.57372220.530.595-.83524091.447813-----------------------------------------------------------------------------36Source|SS

Source|SSdfMSNumberofobs=88-------------+------------------------------F(6,81)=48.27Model|6.2654426361.04424044Prob>F=0.0000Residual|1.7521608981.021631616R-squared=0.7815-------------+------------------------------AdjR-squared=0.7653Total|8.0176035287.092156362RootMSE=.14708

----------------------------------------------------------------------------lprice|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+--------------------------------------------------------------assess|.0004822.00099150.490.628-.0014906.002455bdrms|-.0032415.0236591-0.140.891-.0503157.0438326lotsize|1.48e-061.68e-060.880.381-1.86e-064.83e-06sqrft|.0000404.00005820.690.489-.0000753.0001562colonial|.0207546.04268410.490.628-.0641735.1056826

ys|.7382357.3435822.150.035.05461531.421856_cons|1.2247571.6193960.760.452-1.9973334.446848----------------------------------------------------------------------------Testingwhether(1)istherightone检验（1）是否正确：37Source|SSNonnestedAlternativeTests:Comments

嵌套替代模型的检验：注释Theaboveexamplefavorsthelogmodel,butitisoftenpossibletoseebothmodelsberejected,orneithermodelberejected.上面的例子偏好log模型，但可能经常看到两个模型都被拒绝，或，没有一个被拒绝。38NonnestedAlternativeTests:CNonnestedAlternativeTests:Comments

嵌套替代模型的检验：注释WhenbotharerejectedMoreworkonspecificationneedstobedone.However,iftheeffectsofkeyindependentvariablesonyarenotverydifferent,thenitdoesnotreallymatterwhichmodelisused.

当两个都被拒绝需要在模型设定上花更多功夫尽管如此，如果关键解释变量对y的效应差别不是很大，那么用哪个模型关系不是很大。WhenbotharenotrejectedWecanusetheadjustedR-squaredtochoosebetweenthem.当两个都未被拒绝我们可以用调整过的R2在它们之间选择。39NonnestedAlternativeTests:CProxyVariablesP295

代理变量

Whatifmodelismisspecifiedbecausenodataisavailableonanimportantxvariable?如果模型误设是因为得不到一个重要解释变量的数据，怎么办？比如人的能力，是一个模糊变量，很难衡量Itmaybepossibletoavoidormitigateomittedvariablebiasbyusingaproxyvariable.可能通过使用一个代理变量避免或减轻遗漏变量偏误。Aproxyvariableissomethingthatisrelatedtotheunobservedvariablethatwewouldliketocontrolforinouranalysis. 代理变量就是与我们在分析中试图控制而又观测不到的变量相关的变量。注意：引入代理变量的目的是什么？不是检测beta3，而是为了正确获取beta1和beta240ProxyVariablesP295ProxyVariables

代理变量-代理变量要与原始变量相关-P29641ProxyVariables

代理变量-代理变量要与原始变ProxyVariables

代理变量42ProxyVariables

代理变量42ProxyVariables

代理变量43ProxyVariables

代理变量43ProxyVariables

代理变量

P296

引入代理变量需要怎样的条件呢？44ProxyVariables

代理变量P296ProxyVariables

代理变量P296

45ProxyVariables

代理变量P296ProxyVariables(continued)

代理变量（续）Whenthesetwoassumptionsaresatisfied,wearerunningregressionsy=(b0+b3d0)+b1x1+b2x2+b3d3x3+(u+b3v3)andhavejustredefinedintercept,errortermx3coefficient.当这两个假设被满足，我们作回归y=(b0+b3d0)+b1x1+b2x2+b3d3x3+(u+b3v3)，只要重新定义截距项，误差项和x3系数。46ProxyVariables(continued)

代理TheIQExample.reglwageeducexpertenuremarriedsouthurbanblack

Source|SSdfMSNumberofobs=935-------------+------------------------------F(7,927)=44.75Model|41.837761975.97682312Prob>F=0.0000Residual|123.818521927.133569063R-squared=0.2526-------------+------------------------------AdjR-squared=0.2469Total|165.656283934.177362188RootMSE=.36547

----------------------------------------------------------------------------lwage|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+---------------------------------------------------------------

educ|.0654307.006250410.470.000.0531642.0776973exper|.014043.00318524.410.000.007792.020294tenure|.0117473.0024534.790.000.0069333.0165613married|.1994171.03905025.110.000.1227801.276054south|-.0909036.0262485-3.460.001-.142417-.0393903urban|.1839121.02695836.820.000.1310056.2368185

black|-.1883499.0376666-5.000.000-.2622717-.1144281_cons|5.395497.11322547.650.0005.173295.617704--------------------------------------------------------------------------

47TheIQExample.reglwageeduPlottingstandardizedIQagainstStandardizedWage

绘图：标准化的IQ关于标准化的工资48PlottingstandardizedIQagain4949TheRegressionAddingIQ

加入IQ的回归.reglwageeducexpertenuremarriedsouthurbanblacksdIQ

Source|SSdfMSNumberofobs=935-------------+------------------------------F(8,926)=41.27Model|43.536016185.44200202Prob>F=0.0000Residual|122.120267926.131879338R-squared=0.2628-------------+------------------------------AdjR-squared=0.2564Total|165.656283934.177362188RootMSE=.36315----------------------------------------------------------------------------lwage|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+--------------------------------------------------------------

educ|.0544106.00692857.850.000.0408133.068008exper|.0141458.00316514.470.000.0079342.0203575tenure|.0113951.00243944.670.000.0066077.0161825married|.1997644.03880255.150.000.1236134.2759154south|-.0801695.0262529-3.050.002-.1316916-.0286473urban|.1819463.02679296.790.000.1293645.2345281

black|-.1431253.0394925-3.620.000-.2206304-.0656202

sdIQ|.0535739.01492933.590.000.0242747.0828731_cons|5.536914.119208846.450.0005.3029635.770864----------------------------------------------------------------------------50TheRegressionAddingIQ

加入IQ的CautionsinUsingProxyVariables

使用代理变量注意事项

Whenassumptionsarenotsatisfiedwecannotgetconsistentestimators.Sayx3*=d0+d1x1+d2x2+d3x3+v3

Thenweareactuallyestimatingy=(b0+b3d0)+(b1+b3d1)x1+(b2+b3d2)x2+b3d3x3+(u+b3v3)Biaswilldependonsignsofb3anddj当假设不满足时，我们不能得到无偏、一致的估计量比如x3*=d0+d1x1+d2x2+d3x3+v3实际上，我们可以估计y=(b0+b3d0)+(b1+b3d1)x1+(b2+b3d2)x2+b3d3x3+(u+b3v3)。偏误方向将依赖于b3

和dj的符号。51CautionsinUsingProxyVariabLaggedDependentVariables

滞后的被解释变量

P302

Whatifthereareunobservedvariables,andyoucan’tfindreasonableproxyvariables?如果存在不可观测的变量，并且你又找不到合理的解释变量，怎么办？Maybepossibletoincludealaggeddependentvariabletoaccountforomittedvariablesthatcontributetobothpastandcurrentlevelsofy 可以包含一个滞后的被解释变量，说明同时影响过去和当前y水平的被遗漏变量。Obviously,youmustthinkpastandcurrentyarerelatedforthistomakesense.很显然的,我们必须认为过去和当前的y相关，才有意义。52LaggedDependentVariables

滞后的TheCrimeExample

犯罪的例子Variables:变量lcrmrtelog(crimerateper1000persons)（log（以1000人为单位的犯罪率））llawexpclog(lawexpenditure)（log（诉讼费用））lcrmrt_1lcrmrtelagged（滞后的lcrmrte）unemunemploymentrate（失业率）53TheCrimeExample

犯罪的例子VariablTheCrimeExample:WithoutLaggedDependentVariable

犯罪的例子：不包含滞后的被解释变量.reglcrmrtellawexpcunemifyear==87Source|SSdfMSNumberofobs=46-------------+------------------------------F(2,43)=1.30Model|.2719871992.1359936Prob>F=0.2824Residual|4.4899821443.104418189R-squared=0.0571-------------+------------------------------AdjR-squared=0.0133Total|4.7619693445.105821541RootMSE=.32314

----------------------------------------------------------------------------lcrmrte|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+--------------------------------------------------------------llawexpc|.2033652.17265341.180.245-.1448236.5515539unem|-.0290032.0323387-0.900.375-.0942205.0362141_cons|3.3428991.2505272.670.011.82097215.864826----------------------------------------------------------------------------54TheCrimeExample:WithoutLagTheCrimeExample:WithLaggedDependentVariable

犯罪的例子：包含滞后的被解释变量.reglcrmrtellawexpclcrmrt_1unem

Source|SSdfMSNumberofobs=46-------------+------------------------------F(3,42)=29.73Model|3.2373284631.07910949Prob>F=0.0000Residual|1.5246408842.036300973R-squared=0.6798-------------+------------------------------AdjR-squared=0.6570Total|4.7619693445.105821541RootMSE=.19053

----------------------------------------------------------------------------lcrmrte|Coef.Std.Err.tP>|t|[95%Conf.Interval]-----------+----------------------------------------------------------------llawexpc|-.1395764.1086412-1.280.206-.3588231.0796704lcrmrt_1|1.193923.13209859.040.000.92733711.460508unem|.008621.01951660.440.661-.0307652.0480072_cons|.0764511.82114330.090.926-1.5806831.733585----------------------------------------------------------------------------55TheCrimeExample:WithLaggedMeasurementError

测量误差

P392

Sometimeswehavethevariablewewant,butwethinkitismeasuredwitherror有时，我们有需要的变量，但我们认为它的测量存在误差。Examples:Asurveyaskshow