第9章 Multiple Regression Analysis 多元回歸分析之模型設(shè)定和數(shù)據(jù)問(wèn)題

MultipleRegressionAnalysisP289

多元回歸分析之模型設(shè)定和數(shù)據(jù)問(wèn)題

y=b0+b1x1+b2x2+...bkxk+uSpecificationandDataProblems模型設(shè)定和數(shù)據(jù)問(wèn)題1ChapterOutline 本章大綱FunctionalFormmisspecification

函數(shù)形式誤設(shè)-討論模型誤設(shè)的結(jié)果-P289UsingProxyvariablesforunobservedexplanatoryvariables

對(duì)觀測(cè)不到的變量使用代理變量-討論用代理變量來(lái)減輕有偏性PropertiesoftheOLSUnderMeasurementError

有測(cè)量誤差的OLS性質(zhì)-推導(dǎo)和解釋MissingData,NonrandomSamples,andoutliers

數(shù)據(jù)缺失、非隨機(jī)樣本和離群點(diǎn)-討論額外的數(shù)據(jù)問(wèn)題2FunctionalForm

函數(shù)形式Howdoweknowifwe’vegottentherightfunctionalformforourmodel?我們?nèi)绾沃滥Ｐ褪欠竦玫秸_的函數(shù)形式呢？P289:異方差的出現(xiàn)可以看成是模型的錯(cuò)誤設(shè)定，但不影響有偏性和一致性，還可以通過(guò)WLS來(lái)減輕；本章討論u與xi的相關(guān)性，如果相關(guān)，稱xi為外生變量，為什么？當(dāng)被忽略的自變量為其他變量的函數(shù)時(shí)，將產(chǎn)生函數(shù)形式誤設(shè)這一問(wèn)題。何謂函數(shù)形式誤設(shè)？3FunctionalForm(continued)

函數(shù)形式（續(xù)）

First,useeconomictheorytoguideyou首先，用經(jīng)濟(jì)理論的指導(dǎo)Thinkabouttheinterpretation考慮它的解釋Doesitmakemoresenseforxtoaffectyinpercentage(uselogs)orabsoluteterms?x影響y的更合理的方式是百分比的形式（用log形式），還是絕對(duì)量的形式？Doesitmakemoresenseforthederivativeofx1tovarywithx1(quadratic)orwithx2(interactions)ortobefixed?x1的系數(shù)更合理的形式是隨x1變化（二次形式），隨x2變化（交互作用），還是固定不變？P290:2個(gè)誤設(shè)案例，一個(gè)是忽略了二次項(xiàng)，一個(gè)是忽略了交叉項(xiàng)。也可能是沒(méi)有用LOG形式；回顧第三章P85假設(shè)3不成立的幾種情況，函數(shù)形式誤設(shè)的后果P290EXP.9.1-閱讀4FunctionalFormMisspecification

函數(shù)形式誤設(shè)Amultipleregressionmodelsuffersfromfunctionalformmisspecificationwhenitdoesnotproperlyaccountfortherelationshipbetweenthedependentandtheobservedexplanatoryvariables.

當(dāng)一個(gè)多元回歸模型不能正確地說(shuō)明被解釋變量和觀察到的解釋變量之間的關(guān)系時(shí)，此模型存在函數(shù)形式誤設(shè)問(wèn)題。5FunctionalFormMisspecification

函數(shù)形式誤設(shè)Misspecifyingthefunctionalformofamodelcanhaveseriousconsequences.Wemayobtainbiasedorinconsistentestimatorsofthepartialeffects.誤設(shè)一個(gè)模型的函數(shù)形式可能產(chǎn)生嚴(yán)重的后果。我們得到的局部效應(yīng)的估計(jì)量可能有偏或不一致。Onewayout:toaddquadratictermsofanysignificantvariablestoamodelandtoperformajointtestofsignificance.

一種方法：向模型加入任何重要變量的二次項(xiàng)，進(jìn)行一個(gè)聯(lián)合顯著性檢驗(yàn)。-加入二次項(xiàng)，對(duì)二次項(xiàng)系數(shù)聯(lián)合顯著性F檢驗(yàn)通過(guò)時(shí)，顯示的癥狀往往是誤設(shè)，如誤將對(duì)數(shù)模型為水平模型。另外經(jīng)濟(jì)數(shù)據(jù)中，二次項(xiàng)可以解決大部分非線性問(wèn)題-P2906Example:ModelingCrime

例子：對(duì)犯罪建模-P292Dependentvariable:被解釋變量：Narr86,#timesarrested,1986（1986年被捕次數(shù)）ExplanatoryVariables：解釋變量：pcnvproportionofpriorconvictions以前被定罪比例avgsen

avgsentencelength,mos.平均判刑期限，單位：月tottime timeinprisonsince18,mos.18歲以來(lái)的服刑時(shí)間，單位：月Ptime86mos.inprisonduring19861986年的服刑時(shí)間，單位：月解讀：1.為什么加入二次項(xiàng)，因?yàn)樗巾?xiàng)T檢驗(yàn)很顯著；2.加入變量的二次項(xiàng)后，原先的水平變量系數(shù)變化很大；同時(shí)二次項(xiàng)聯(lián)合F顯著；3.二次項(xiàng)加入，模型的解讀變得困難，可能有更深刻的實(shí)際意義7Example:ModelingCrime

例子：對(duì)犯罪建模Explanatoryvariables解釋變量Qemp86#quartersemployed,19861986年被雇傭季度數(shù)inc86 legalincome,1986,$100s1986年合法收入，單位：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，我們將被解釋變量向解釋變量回歸，不包含任何平方項(xiàng)。8

regnarr86pcnv

avgsen

tottimeptime86qemp86inc86blackhispanSource|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-.0541141

avgsen|-.0113177.0122401-0.920.355-.0353185.0126831

tottime|.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|.3265035.04541567.190.000.2374508.4155561

hispan|.1939144.03971134.880.000.1160469.2717818_cons|.5686855.036046115.780.000.4980048.6393661------------------------------------------------------------------------------9Plottingnarr86againstpncv

繪圖：narr86關(guān)于pncv10Plottingnarr86againstinc86

繪圖：narr86關(guān)于pncv11Plottingnarr86againstptime86

繪圖：narr86關(guān)于pncv12

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

pcnv.5525236.15423723.580.000.2500892.8549579

pcnvsq

-.7302119.1561177-4.680.000-1.036333-.4240903

avgsen-.0170216.0120539-1.410.158-.0406574.0066142

tottime.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.292296.044836.520.000.2043916.3802004

hispan.1636175.03945074.150.000.0862609.240974_cons.5046065.036835313.700.000.4323784.5768347AddingQuadratictermstosignificantVariables加入重要變量的平方項(xiàng)13Drawbacksofaddingsquaretermstodetectfunctionalformmisspecification

取消加入平方項(xiàng)以檢測(cè)函數(shù)形式誤設(shè)

Theoretically,wecantestjointexclusionrestrictionstoseeifhigherordertermsorinteractionsbelongtothemodel理論上，我們作排除性約束的聯(lián)合檢驗(yàn)，來(lái)看高階項(xiàng)和交叉項(xiàng)是否屬于模型。Itcanbetedioustoaddandtestextraterms.Manydegreesoffreedomsmaybeused. 加入和檢驗(yàn)另外的項(xiàng)過(guò)程會(huì)很單調(diào)乏味且冗長(zhǎng)。當(dāng)原模型解釋變量多時(shí)可能會(huì)消耗掉許多自由度。14Drawbacksofaddingsquaretermstodetectfunctionalformmisspecification

取消加入平方項(xiàng)以檢測(cè)函數(shù)形式誤設(shè)Somenonlinearitiescannotbepickedupbyaddingquadraticterms.Forexample,wemayfindasquaretermmatterswhenusinglogsismoreappropriate. 一些非線性關(guān)系不能通過(guò)加入二次項(xiàng)捕捉。例如，當(dāng)我們發(fā)現(xiàn)平方項(xiàng)重要時(shí)，可能對(duì)數(shù)形式更加適合。15Ramsey’sRESETP292

Ramsey回歸設(shè)定誤差檢驗(yàn)AtestoffunctionalformisRamsey’sregressionspecificationerrortest(RESET)一種函數(shù)形式的檢驗(yàn)是Ramsey回歸設(shè)定誤差檢驗(yàn)（RESET）。RESETaddspolynomialsintheOLSfittedvaluestotheoriginalregression.RESET在原回歸中加入OLS擬合值的多項(xiàng)式-沒(méi)有明確的原理指出到底要加入多少個(gè)高次方的項(xiàng)，但是平方和立方一般是有用的。16Ramsey’sRESET

Ramsey回歸設(shè)定誤差檢驗(yàn)

Insteadofaddingfunctionsofthex’sdirectly,weaddandtestfunctionsof?我們加入并檢驗(yàn)?的多次項(xiàng)函數(shù)，而不是直接加入x的函數(shù)。注意：如何加入函數(shù)項(xiàng)的？P293So,estimatey=b0+b1x1+…+bkxk+d1?2+d1?3+errorandtest所以，估計(jì)y=b0+b1x1+…+bkxk+d1?2+d1?3+error，并檢驗(yàn)。H0:d1=0,d2=0,usingFstatisticorLMstatistic.H0:d1=0,d2=0，用F統(tǒng)計(jì)量或LM統(tǒng)計(jì)量。17Ramsey’sRESET

Ramsey回歸設(shè)定誤差檢驗(yàn)AsignificantFstatisticsuggestssomesortoffunctionalformproblem.一個(gè)顯著的F統(tǒng)計(jì)量說(shuō)明函數(shù)形式可能存在問(wèn)題。ThedistributionofFisapproximatelyF2,n-k-3inlargesamplesunderthenullhypothesisandtheG-Massumptions.在零假設(shè)和G-M假定下，F(xiàn)的分布大樣本近似為F2,n-k-3分布。自由度的說(shuō)明：減少了2個(gè)自由度P29318ImplementingRESETinStata

在STATA中實(shí)施RESETSTATAusescommandovtestafterregcommand.STATA在reg命令后，使用ovtest命令。?2,?3,and?4

areusedinstata.STATA使用?2,?3和?4

。regnarr86pcnv

avgsen

tottimeptime86qemp86inc86blackhispan

ovtest

RamseyRESETtestusingpowersofthefittedvaluesofnarr86RESET檢驗(yàn)使用narr86擬合值的冪函數(shù)項(xiàng) Ho:modelhasnoomittedvariablesF(3,2713)=4.19,Prob>F=0.005819ImplementingRESETinStata

在STATA中實(shí)施RESETAnalternativeistospecifytheoption,rhs.一個(gè)替代的方法是指定選擇，rhsInthiscasethepowertermsofalltheexplanatoryvariablesinsteadofthefittedvaluesareusedinthetest.在這種情況下，檢驗(yàn)中使用所有解釋變量的冪函數(shù)項(xiàng)，而不是擬合值的相應(yīng)項(xiàng)。ovtest,rhs RamseyRESETtestusingpowersoftheindependentvariablesRESET檢驗(yàn)使用解釋變量的冪函數(shù)項(xiàng)Ho:modelhasnoomittedvariablesF(18,2698)=9.73Prob>F=0.000020CautionsinUsingRESET

使用RESET的注意事項(xiàng)RESETisgoodatdetectingmisspecificationsintheformofnonlinearities,notgeneralomittedvariables. RESET在探測(cè)非線性形式的函數(shù)誤設(shè)時(shí)很好用，而不是用于檢測(cè)一般的遺漏變量。Wooldridge(1995)showsthatRESEThasnopowerfordetectingomittedvariableswhenevertheyhaveexpectationsthatarelinearintheincludedindependentvariables. Wooldridge在1995年證明：當(dāng)被遺漏變量的期望值時(shí)所包含自變量的線性函數(shù)時(shí)，RESET無(wú)法探測(cè)出遺漏變量問(wèn)題。P294：對(duì)RESET作用的正確評(píng)價(jià)：1.有的認(rèn)為可以檢測(cè)遺漏變量和異方差，但是Wooldridge不這樣認(rèn)為21CautioninusingofRESET

使用RESET的注意事項(xiàng)However,iftheomittedvariablehavenonlinearexpectationsinthedependentvariables,asignificantRESETcanindicateomitted-variableproblem. 盡管如此，如果被遺漏變量的期望是自變量的非線性形式時(shí)，一個(gè)顯著的RESET可以指出遺漏變量問(wèn)題。AlsonoticethatthedrawbackoftheRESETtestiswhenthenullisrejected,RESETdoesnotsuggestwhattodointhenextstep. 也要注意到，RESET檢驗(yàn)的一個(gè)缺陷是，當(dāng)零假設(shè)被拒絕后，它并不能建議我們下一步怎么做。22HousingPriceExample

住房?jī)r(jià)格的例子Thisexampleisusedfortwopurposes. 使用這個(gè)例子有兩個(gè)目的。First,logformscanbebetterindealingwithnonlinearitiesthenusingthelevelvariables. 首先，處理非線性問(wèn)題時(shí)，log形式可能比變量原形式更好。Second,asignificantRESETmayindicatenonlineareffectofomittedvariables,likethevariable“assess”addedinlater. 其次，一個(gè)顯著的RESET可能指出被遺漏變量的非線性效應(yīng)，比如稍后加入的變量“assess”。23HousingPriceExample

住房?jī)r(jià)格的例子Dataused:hprice1.dta,variables使用數(shù)據(jù)：hprice1.dta,變量assessassessedvalue,$1000s（評(píng)估價(jià)，單位：千美元）pricehouseprice,$1000s（房?jī)r(jià)，單位：千美元）lotsizesizeoflotinsquarefeet（土地的面積，單位：平方英尺）sqrftsizeofhouseinsquarefeet（房屋的面積，單位：平方英尺）bdrmsnumberofbedrooms（臥室數(shù)）24HousingPriceExample

住房?jī)r(jià)格的例子

P293閱讀

regpricelotsize

sqrft

bdrms

ovtest

RamseyRESETtestusingpowersofthefittedvaluesofprice（RESET檢驗(yàn)用擬合價(jià)格的冪函數(shù)項(xiàng)）Ho:modelhasnoomittedvariablesF(3,81)=4.26

Prob>F=0.007625HousingPriceExample:thelogforms

住房?jī)r(jià)格的例子：log形式Thelogformdonotrejectthenullofnomisspecificationat5%significancelevel.Log形式的回歸在5％水平上沒(méi)有拒絕零假設(shè)：沒(méi)有函數(shù)形式誤設(shè)。--結(jié)論：第二個(gè)模型即對(duì)數(shù)模型更好一些。-P293reg

lprice

llotsize

lsqrft

bdrmsovtestRamseyRESETtestusingpowersofthefittedvaluesoflprice

（RESET檢驗(yàn)用lprice擬合值的冪函數(shù)項(xiàng)）Ho:modelhasnoomittedvariablesF(3,81)=2.45

Prob>F=0.069226HousingPriceExample:thelogforms

住房?jī)r(jià)格的例子：log形式reg

lprice

lassess

llotsize

lsqrft

bdrmsInthisstepvariablelassessisasignificantvariablewitht=6.89.這一步中，變量lassess顯著，t=6.89ovtest

RamseyRESETtestusingpowersofthefittedvaluesoflprice

（RESET檢驗(yàn)使用lprice擬合值的冪函數(shù)項(xiàng)）Ho:modelhasnoomittedvariablesF(3,80)=1.11

Prob>F=0.350927HousingPriceExample:thelogforms

住房?jī)r(jià)格的例子：log形式Noticetheresultsaredifferentfromthetextbooksince?2,?3,and?4

areusedinstata,insteadof?2,?3

asinthetextbook

. 注意這里的結(jié)果和課本上不同，因?yàn)檎n本上使用?2,?3

，這里stata用的是?2,?3,和

?4

。Youcanreplicatethetextbookresultbyputting?2,?3

intothemainequation,anduseFtesttotesttheirjointsignificances.

你可以通過(guò)以下方法得到課本的結(jié)果：向主方程加入?2,?3

，使用F檢驗(yàn)檢驗(yàn)它們的聯(lián)合顯著性。28NonnestedAlternativeTests:MR

非嵌套替代模型的檢驗(yàn)：MRP294

-如何檢驗(yàn)非嵌套模型？二種方法：MR方法、DM方法

Whichofthefollowingmodelisbetter?下面哪一個(gè)模型更好？MizonandRichard(1986):Constructacomprehensivemodelthatcontainseachmodelasaspecialcaseandthentotesttherestrictionsthatledtoeachofthemodels.

MizonandRichard(1986):

構(gòu)造一個(gè)綜合模型，將每個(gè)模型都作為一個(gè)特殊情況包含其中，然后檢驗(yàn)導(dǎo)致每個(gè)模型變的約束。注意：第6章P199曾提出用擬合優(yōu)度監(jiān)測(cè)29NonnestedAlternativeTests

非嵌套替代模型的檢驗(yàn)Intheaboveexample,thecomprehensivemodelis在上例中，綜合模型是：

andtest

30NonnestedAlternativeTests:DM

嵌套替代模型的檢驗(yàn)：DMDavidsonandMacKinnon(1981):if(9.6)istrue,thenthefittedvaluesfrom(9.7),shouldbeinsignificantin(9.6).DavidsonandMacKinnon(1981):如果（9.6）正確，那么從（9.7）得到的擬合值在（9.6）中應(yīng)當(dāng)不顯著。注意：D-M檢驗(yàn)的思路，是一個(gè)t檢驗(yàn)P29431NonnestedAlternativeTests:DM

嵌套替代模型的檢驗(yàn)：DMTotest(9.6),wefirstestimatemodel(9.7)byOLStoobtainthefittedvalues.為了檢驗(yàn)（9.6），我們首先通過(guò)OLS估計(jì)模型（9.7）以得到擬合值。Putthisfittedvalueasanadditionalexplanatoryvariablein(9.6),usetstatistictotestitssignificance.將得到的擬合值作為另外的解釋變量放到（9.6）中，用t統(tǒng)計(jì)量檢驗(yàn)其顯著性。32TheHousingPriceExample:MR

住房?jī)r(jià)格的例子：MRThecompetingmodels:競(jìng)爭(zhēng)模型是：

(1)

reg

lprice

bdrmscolonialassesslotsize

sqrft(2)reg

lprice

bdrmscoloniallassess

llotsize

lsqrft

Thecombinedregression:組合的回歸：

reg

lpricecolonialbdrmsassesslotsize

sqrft

lassess

llotsize

lsqrft

33TheHousingPriceExample:MR

住房?jī)r(jià)格的例子：MRTestingwhether(2)istherightone:檢驗(yàn)（2）是否正確：testassesslotsize

sqrft

F(3,79)=2.92,Prob>F=0.0392Testingwhether(1)istherightone:檢驗(yàn)（1）是否正確： testlassess

llotsize

lsqrftF(3,79)=3.97,Prob>F=0.0108Inclusive.34TheHousingPriceExample:DM

住房?jī)r(jià)格的例子：DMTestingwhether(2)istherightone:檢驗(yàn)（2）是否正確：

reg

lpriceassessbdrms

lotsize

sqrftcolonial

predictyl,xb

reg

lprice

lassess

llotsize

lsqrft

bdrmscolonialylThetablebelowshowthatylisaninsignificantvariable.下表顯示yl

不是一個(gè)顯著的變量。35

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|.6762505.33745562.000.048.00481971.347681

llotsize|-.0119247.0419541-0.280.777-.0954003.0715508

lsqrft|-.1258866.1407801-0.890.374-.4059949.1542216

bdrms|.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-----------------------------------------------------------------------------36

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-.0014906.002455

bdrms|-.0032415.0236591-0.140.891-.0503157.0438326

lotsize|1.48e-061.68e-060.880.381-1.86e-064.83e-06

sqrft|.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檢驗(yàn)（1）是否正確：37NonnestedAlternativeTests:Comments

嵌套替代模型的檢驗(yàn)：注釋Theaboveexamplefavorsthelogmodel,butitisoftenpossibletoseebothmodelsberejected,orneithermodelberejected.上面的例子偏好log模型，但可能經(jīng)常看到兩個(gè)模型都被拒絕，或，沒(méi)有一個(gè)被拒絕。38NonnestedAlternativeTests:Comments

嵌套替代模型的檢驗(yàn)：注釋W(xué)henbotharerejectedMoreworkonspecificationneedstobedone.However,iftheeffectsofkeyindependentvariablesonyarenotverydifferent,thenitdoesnotreallymatterwhichmodelisused.

當(dāng)兩個(gè)都被拒絕需要在模型設(shè)定上花更多功夫盡管如此，如果關(guān)鍵解釋變量對(duì)y的效應(yīng)差別不是很大，那么用哪個(gè)模型關(guān)系不是很大。WhenbotharenotrejectedWecanusetheadjustedR-squaredtochoosebetweenthem.當(dāng)兩個(gè)都未被拒絕我們可以用調(diào)整過(guò)的R2在它們之間選擇。39ProxyVariablesP295

代理變量

Whatifmodelismisspecifiedbecausenodataisavailableonanimportantxvariable?如果模型誤設(shè)是因?yàn)榈貌坏揭粋€(gè)重要解釋變量的數(shù)據(jù)，怎么辦？比如人的能力，是一個(gè)模糊變量，很難衡量Itmaybepossibletoavoidormitigateomittedvariablebiasbyusingaproxyvariable.可能通過(guò)使用一個(gè)代理變量避免或減輕遺漏變量偏誤。Aproxyvariableissomethingthatisrelatedtotheunobservedvariablethatwewouldliketocontrolforinouranalysis. 代理變量就是與我們?cè)诜治鲋性噲D控制而又觀測(cè)不到的變量相關(guān)的變量。注意：引入代理變量的目的是什么？不是檢測(cè)beta3，而是為了正確獲取beta1和beta240ProxyVariables

代理變量-代理變量要與原始變量相關(guān)-P29641ProxyVariables

代理變量42ProxyVariables

代理變量43ProxyVariables

代理變量

P296

引入代理變量需要怎樣的條件呢？44ProxyVariables

代理變量P296

45ProxyVariables(continued)

代理變量（續(xù)）Whenthesetwoassumptionsaresatisfied,wearerunningregressionsy=(b0+b3d0)+b1x1+b2x2+b3d3x3+(u+b3v3)andhavejustredefinedintercept,errortermx3coefficient.當(dāng)這兩個(gè)假設(shè)被滿足，我們作回歸y=(b0+b3d0)+b1x1+b2x2+b3d3x3+(u+b3v3)，只要重新定義截距項(xiàng)，誤差項(xiàng)和x3系數(shù)。46TheIQExample.reg

lwage

educ

expertenuremarriedsouthurbanblack

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|.0654307.006250410.470.000.0531642.0776973

exper|.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--------------------------------------------------------------------------

47PlottingstandardizedIQagainstStandardizedWage

繪圖：標(biāo)準(zhǔn)化的IQ關(guān)于標(biāo)準(zhǔn)化的工資4849TheRegressionAddingIQ

加入IQ的回歸.reg

lwage

educ

expertenuremarriedsouthurbanblacksdIQ

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|.0544106.00692857.850.000.0408133.068008

exper|.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----------------------------------------------------------------------------50CautionsinUsingProxyVariables

使用代理變量注意事項(xiàng)

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

Thenweareactuallyestimatingy=(b0+b3d0)+(b1+b3d1)x1+(b2+b3d2)x2+b3d3x3+(u+b3v3)Biaswilldependonsignsofb3anddj當(dāng)假設(shè)不滿足時(shí)，我們不能得到無(wú)偏、一致的估計(jì)量比如x3*=d0+d1x1+d2x2+d3x3+v3實(shí)際上，我們可以估計(jì)y=(b0+b3d0)+(b1+b3d1)x1+(b2+b3d2)x2+b3d3x3+(u+b3v3)。偏誤方向?qū)⒁蕾囉赽3

和dj的符號(hào)。51LaggedDependentVariables

滯后的被解釋變量

P302

Whatifthereareunobservedvariables,andyoucan’tfindreasonableproxyvariables?如果存在不可觀測(cè)的變量，并且你又找不到合理的解釋變量，怎么辦？Maybepossibletoincludealaggeddependentvariabletoaccountforomittedvariablesthatcontributetobothpastandcurrentlevelsofy 可以包含一個(gè)滯后的被解釋變量，說(shuō)明同時(shí)影響過(guò)去和當(dāng)前y水平的被遺漏變量。Obviously,youmustthinkpastandcurrentyarerelatedforthistomakesense.很顯然的,我們必須認(rèn)為過(guò)去和當(dāng)前的y相關(guān)，才有意義。52TheCrimeExample

犯罪的例子Variables:變量lcrmrtelog(crimerateper1000persons)（log（以1000人為單位的犯罪率））llawexpclog(lawexpenditure)（log（訴訟費(fèi)用））lcrmrt_1lcrmrtelagged（滯后的lcrmrte

）unemunemploymentrate（失業(yè)率）53TheCrimeExample:WithoutLaggedDependentVariable

犯罪的例子：不包含滯后的被解釋變量.reg

lcrmrte

llawexpc

unemifyear==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-.1448236.5515539

unem|-.0290032.0323387-0.900.375-.0942205.0362141_cons|3.3428991.2505272.670.011.82097215.864826----------------------------------------------------------------------------54TheCrimeExample:WithLaggedDependentVariable

犯罪的例子：包含滯后的被解釋變量.reg

lcrmrte

llawexpclcrmrt_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|1.193923.13209859.040.000.92733711.460508

unem|.008621.01951660.440.661-.0307652.0480072_cons|.0764511.82114330.090.926-1.5806831.733585----------------------------------------------------------------------------55MeasurementError

測(cè)量誤差

P392

Sometimeswehavethevariablewewant,butwethinkitismeasuredwitherror有時(shí)，我們有需要的變量，但我們認(rèn)為它的測(cè)量存在誤差。Examples:Asurveyaskshowmanyhoursdidyouworkoverthelastyear,orhowmanyweeksyouusedchildcarewhenyourchildwasyoung例子：一個(gè)調(diào)查問(wèn)你在過(guò)去的一年中工作了多少小時(shí)，或當(dāng)你的孩子小時(shí)，你照看孩子用了多少周。Measurementerrorinydifferentfrommeasurementerrorinxy的測(cè)量誤差與x的測(cè)量誤差不同。56MeasurementErrorinaDependentVariable

被解釋變量的測(cè)量誤差

Lety*bethevariableofourinterest,butyisitsreportedvalue.Definemeasurementerrorase0=y–y*令y*為我們感興趣的變量，但y是它的報(bào)告值。定義測(cè)量誤差為e0=y–y*

。Themodely=b0+b1x1+…+bkxk+u+e0isestimated.估計(jì)的模型y=b0+b1x1+…+bkxk+u+e0

WhenwillOLSproduceunbiasedresults?什么時(shí)候OLS產(chǎn)生有偏的結(jié)果？57MeasurementErrorinaDependentVariable

被解釋變量的測(cè)量誤差I(lǐng)fe0andxj,uareuncorrelated，theresultsisunbiased.如果e0和xj,u不相關(guān)，結(jié)果無(wú)偏。IfE(e0)≠0thenb0willbebiased,though如果E(e0)≠0，那么b0有偏。Whileunbiased,wefacelargervariancesthanwithnomeasurementerror當(dāng)無(wú)偏時(shí)，我們要比沒(méi)有測(cè)量誤差時(shí)面臨更大的方差。見(jiàn)P303公式結(jié)論：1、當(dāng)e與資本了相關(guān)時(shí)，導(dǎo)致有偏性，2、無(wú)關(guān)時(shí)，只增大方差，模型還是合適的58MeasurementErrorinanExplanatoryVariable

解釋變量的測(cè)量誤差

Wewishtoestimatey=b0+b1x1*+u.

我們希望估計(jì)y=b0+b1x1*+u。Definemeasurementerrorase1=x1–x1*.定義測(cè)量誤差為e1=x1–x