1、個體計量模型在財務上的應用個體計量模型在財務上的應用余士迪余士迪元智大學財金系教授Econometric modelqSingle equation modelqSystem of equations model qSimultaneous equation ModelSingle equation modelikikiiuxxy.110iiiuxyCan be written asSystem of equations modelmummxmy2u22x2y1u11x1y)(0)(uuEuESimultaneous equationsmummxmymm1y1mmy2u22xmym21y212
2、y1u11xmym12y121yMicroeconometricsqDiscrete choice modelsqSample selection models qDuration models Probit Model&Logit Model Multinomial Choice Model Multinomial Logit Model Nested Logit Model Mixed Logit Model Multinomial Probit Model Bivariated Probit Model Multivariate Probit Model Sequential Choic
3、e Model Ordered Probit Model Count DataDiscrete Choice ModelSample Selection ModeloCensored modeloSample selection modelDuration ModeloDuration modeloSplit population modelBinary Choice Model Choose ADont Choose AIndividual i051015202501人數*2*10*2*1122*211*1iUiUifiUiUifiyiuixiUiuixiUBinary Choice Mod
4、el*2*1*iUiUiy(Unobserved variable)0*iyif00*iyif1iyiuix*iyiu N（0,1）Probit ModelAssume:Binary Choice ModeloBoczar(1978,J.of Finance)BankFinance CompanyPersonal loan debtor0*00*1*iyifiyifiyiuixiyBinary choice modelObtain a credit from a bankObtain a credit from a financial company)(1)()0()0*()1(xFxiuPi
5、uxPiyPiyPThe probability of choosing alternative 1 is given by Probit Model)()()0()0()0(*xFxuPuxPyPyPiiiiThe probability of choosing alternative 0 is given byProbit ModelXzdzeXFXYP21222121)()|1(Probit model)0()1()0()0(*iiiiyPyPyPyPniiiiiiyPyyPyL1)0(ln)1()1(lnlnThe loglikelihood for this model is giv
6、en byProperties of Maximum Likilihood Estimator0.2.4.6.81Probabiltity P=1-5051 01 52 0P re d ic te d d e s ire d H o u rslo g is ticn o rm a lu n ifo r mProbit,logit vs.OLSPr(y=1)X-2-10120.51Red Line:ProbitBlue Line:LogitModeling DecisionoThis yes or no type decision leads to a dummy variable.oThe d
7、ependent variable of our model is a dummy variable.oWe will be modeling the probability function,P(Y=1).Simplest ModelLinear Probability Models.y variableexplanator theof functiona is andnobservatio tonobservatio from varieswhich)1()var(easticity.heterosced from suffers Modelyyymodel regression basi
8、cour For i221iiiiiikkiiiPPXXPEeEPicture of LPMX10X0X1y Problems of LPMoPredictions outside 0-1 range.oHeteroscedasticitynThis can be solved and a estimated GLS estimator developed.oCoefficient Determination has little meaning.oConstant marginal effect.Interpreting the Probit Model0)()()(iijiijiiijii
9、jixIwhenoccurschangeMaximumxfxxxFxxFxPThe logit modeljiijjiiiFFeefeefeiiiii2xxijiij2xxxx1x1xxPstill is in x change a ofeffect marginal The1x isfunction density The11)xF(model.our for on distributi normal theofinsteadon distributi logistic theuses modellogit The niiiXFYXFYL1)(ln()1()(1(ln(lnThe Log-L
10、ikelihood function:LIMDEP Command Read;NVAR=7;Nobs=200;file=;names=.$Regress;LHS=y1;RHS=one,x1,x2,.$Probit;LHS=y1;RHS=one,x1,x2,.$Logit;LHS=y1;RHS=one,x1,x2,.$More often cited are R-square values based on likelihood ratios.Maddala R2=1 -(LR/LUR)2/nMcFadden R-square:R2=1 -(log(LUR)/log(LR)PROBIT,LOGI
11、T Goodness of Fit Measures?Jacobson and Roszbach(2003,Journal of Banking&Finance)-Bivariate Probit ModelProviding a loan?YesNoLoan defaults?YesNo,111*1iuixiy,0*100*111iyifiyifiy0*i2yif00*i2yif1i2y)11,0(21Niuiui2ui2x2*i2yBivariate Probit Model(if loan granted)(if loan not granted)(if loan does not de
12、fault)(if loan defaults)LIMDEP CommandBivariated Probit Model Read;NVAR=7;Nobs=200;file=;names=.$Bivariate Probit;LHS=Y1,Y2;RHS=one,x1,x2,.;RH2=one,z1,z2,.$JuJJxJyuxyuxy*22221111.*0*00*1jiyifjiyifjiyMultivariate Probit ModelEXAMPLE:CigaretteYesNoAlcoholMarijuanaCocaineYesYesYesNoNoNo444*433322221111
13、*3*uxyuxyuxyuxy0*i1yif00*i1yif1i1y0*200*212iyifiyifiy0*400*441iyifiyifiy0*300*331iyifiyifiy),0(243424142323132212214321Nuuuuiiii11223344),(),()0,0,0,0()1,1,1,1(432144433322211143214321XXXXuuuudFXuXuXuXuUUUUPYYYYPHausman and Wise(1978,Econometrica)Individual iAlternatives JMultinomial Choice Model:Ex
14、ample:Credit Card231Multinomial Choice Model Example:The classic RUM model,where individual is utility from choosing alternative j is given by:ijijijUXu The probability of choosing alternative 1 is given by11111,1iijijiPuXjdFuwhere11ijijiuuu11ijiijXXXand*3*4*2*4*1*44*4*3*2*3*1*33*4*2*3*2*1*22*4*1*3*
15、1*2*11iUiUiUiUiUiUifiUiUiUiUiUiUifiUiUiUiUiUiUifiUiUiUiUiUiUifiyMultinomial Choice ModeliuixiUiuixiUiuixiUiuixiU44*433*322*211*1Multinomial Logit ModelLet m21P,.P,PXFPPP1m11XFPPP2m22be the probabilities associated these m categoriesXFPPP1mm1m1mxGxF1xFPPjjjmj(j=1,2,.m-1)If xxGjjexp)(JkjjjxxP2exp1expJ
16、kjmxP2exp11McFadden 1973JjzzeijikikijijikijijijijeejiobeFJjkzzobjkUUobjiob1)chooses(Pr then)(ondistributi Weibull with theiid are serror term theiff shown that becan it)(Pr )(Pr )chooses(Pr estimatingin interested reWeMultinomial Logit Model Jkxj*jj*jJkxxikikijeiobqqJjeejiob20211)0 chooses(Pr as res
17、ults same thegives ,any vectorfor because ion,normalizat of kind some have tonecessary isit-parameters heidentify t tonecessary-thisdo and 0,one call es,alternativ relabel always can you-0 with0 ealternativ an is thereassumes implicilty this-,.,1 1)chooses(Pr1Yij0YijIf the ith individual falls in th
18、e jth categoryotherwiseim2i1iYimY2in1IY1 jP.PPL Independence of Irrelevant Alternatives(IIA)ikjikijikijijikijijxikijikijzzzzzzzzikijikijePPzzeeeeeePPPP:logit lMultinomia,on dependsjust :logit lConditionaesalternativother oft independen are ratios odds the models,logit lconditiona and lmultinomia In
19、theOrdered Probit Model AAAAAABBBExample:Blume,Lim,Mackinlay(1998,Jornal of Finance)：Corporate bond rating(債券評等)*33*22*11*4321*iiiiyifyifyifyifiyiuixiyiu N（0,1）Ordered Probit Model)*()1(1iyPiyP)*()2(21iyPiyP)*()3(32iyPiyP)*()4(43iyPiyP是否核準否是不良債權還款正常將來還款正常Never Fail將來不正常Eventually FailAuctionNoYesNoY
20、esNoYesSequential Choice Model:Example:法拍屋iuixiyiuixiyiuixiy33*322*211*1Sequential Response ModelN(0,1)i.i.d.3,2,1iuiuiuFirst Auction No=1-F(1x)Yes=F(2x)No=1-F(2x)Yes=F(3x)No=1-F(3x)Yes=F(1x)Sequential Choice Model Then the probabilities can be written as XFP11XFXF1P212XFXF1XF1P3213XF1XF1XF1P3214Biv
21、ariate Probit Model?Multinomial Choice Model?Ordered Probit Model?Sequential Choice Model?EXAMPLE:Model Selection:Joint decision vs.Sequential decisionEXAMPLE:Model SelectionIoannides and Rosenthal(1994,The Review of Economics and Statistics):“Estimating the consumption and investment demand for hou
22、sing and their effect on housing tenure status”Multinomial Choice Model?(喜歡、購買)(不喜歡、購買)(喜歡、不購買)(不喜歡、不購買)Ordered Probit Model?(喜歡、購買)(不喜歡、購買)(喜歡、不購買)(不喜歡、不購買)Intensity of UtilitySequential Choice Model?購買不購買不喜歡喜歡Bivariate Probit Model?喜歡不喜歡購買不購買Count regressionAppropriate when the dependent variable
23、is a non-negative integer(0,1,2,3,)Distributions and ModelsnPoisson ModelnNegative Binomial ModelnZero-inflated Poisson Modeln Zero-inflated Negative Binomial ModelPoisson Regression!r)()exp()rY(obPrriiip1jijj0ixln1101!exp)!)(exp(iiiipjiiijjiiniiYiiYYxYYLiWhy not use linear regression?oTypical count
24、 data in health care:nLarge number of 0 values and small valuesnDiscrete nature of dataoResult:nUnusual distribution Normal distribution vsPoisson distribution2)()(yVaryE2)()(yVaryEBell shaped curveNormal distributionPoisson distributionNot bell shapednext slide“Intensity of process”Poisson with =0.
25、50102030405060708002468101214161820222426283032Relative FrequencyWhen Count Data Cannot be Treated NormallyDistribution of PH 01234567036912151821242730333639424548515457606366697275Number of Scripts FilledRelative Frequency56.1279.15704.182ssxWhen they probably can.024681012141234 5678 9 10 11 12 1
26、3 14 15 16 17 18 19 20 21 22 23 2422.337.104.102ssxWhat happens when mean variance?oOverdispersion:when variance meanSometimes called“unobserved heterogeneity”oZero-Inflated:More zeros than expected by Poisson distributionnEx.If=1(mean=1),then we expect 37%0sOverdispersion:Distribution of ED visits0
27、10203040506070800123456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33Number of visitsRelative frequency21.146.147.02ssx)().exp()(110yVarXXyEkkPoisson Regression modelsNegative Binomial Regression models2)()(yVaryE).exp()(110uXXyEkku is Weibull distributionOverdispersion
28、and Zero InflationDistribution of PH 01234567036912151821242730333639424548515457606366697275Number of Scripts FilledRelative Frequency56.1279.15704.182ssxZero-inflated Poisson)|0*Pr()1Pr()0Pr()0Pr(PoissonYzzY)|*Pr()1Pr()0Pr(PoissonjYzjYExampleBao article:Predicting the use of outpatient mental heal
29、th services:do modeling approaches make a difference?Inquiry.2002 Summer;39(2):168-83.Observed dataPoisson and Zero-Inflated PoissonNegative Binomial ModelZero-Inflated Negative Binomial ModelTOBIT ModelTOBIT MODELiiiuxy*0*iiyy00*iiyy if 2,0Nui TOBIT MODEL 0000*iiiiiiiyyEyPyyEyPyE)0()0(uxuxEuxPiiii)
30、(*iiiiiixuuExxupTOBIT MODELauuEii|22121211112iuaiiiaiiiiaiudeaduauPuuduauufuiTOBIT MODELaaeaeaaaui12101211222221TOBIT MODEL0*iiyyE)0(uxuxEii)(iiiixuuEx)(iixxTOBIT MODEL0*iiyyiiixx)(22)|()(iiiiiiixuuExuuEVarTOBIT MODEL222)(iiiixxxVarTOBIT MODELwhere ixaiiiixxxuuE2222p.f.iiiixixiiiiduxuPufuduxuufuxuuE
31、ii2222TOBIT MODELletixa222ZuuZdudZ aFduufua12dZeZdZeZdueu分子aZZaua22222222222222121TOBIT MODELNOTE dxxfxgxgxfdxxgxfdxxgxfdxxgxfxgxfdxdxgdxxdfdxxdgxfxgxfdxdTOBIT MODELlet ZZf 22222222ZZZeZdedZzexg aaadZeeadZeeaeZZdeeZ分子aZaaZaZzaZaZ1212102222222222222222222222limTOBIT MODEL aaaaaaaaaaa全部11111112222iiii
32、ixxxuuE222TOBIT MODEL aaaaaaaaaaa全部11111112222iiiiixxxuuE222The log-likelihood function 10)1(iifFL122012/12)(21)2(1ln)1ln(lniiixyFLiuixiy111*10*100*1*22iyifiyifiyiy)21,0(21Niuiuwhereiuixiy222*2Sample Selection Modeliuixiy111*1*2*1*2*2*1*12iyiyifiyiyiyifiyiy)21,0(21Niuiuiuixiy222*2Self-Selection Mode
33、l)11()11(22)111|2(22)0111|2(22)0*1|2(ixixixixiuiuEixiuixiuEixiyiyESample Selection Modeliixixixiyiy)11()11(220*1|2Heckmans Two-step Estimator(1979)Duration Models:Censored Data Unobserved Heterogeneity Time-Varying CovariatesDDEnd of studyDCCoHazard Rate：t)tTttTt(Plim)t(h0oSurvival Rate)tT(P)t(SHaza
34、rd Rate and Survival Rate)t(S)t(fdt)t(Slnd)t(h=ii)x|S(t)x|t(fLi1n1iiDuration ModelDistributions ParametricnExpoentialnWeibullnLog-normalnLog-logisticnGamma Semi-parametricnCoxs partial likelihood estimator LIMDEP Command-Duration Model Read;NVAR=7;Nobs=200;file=;names=.$Survival;LHS=ln(time),status(
35、exit=1);RHS=one,x1,x2,.;model=Exponential$Survival;LHS=ln(time),status(exit=1);RHS=one,x1,x2,.;model=Weibull$Coxs Semiparametric Estimator:Survival;LHS=ln(time),status(exit=1);RHS=one,x1,x2,.$iuixiy111*1,0*i 1yif00*i 1yif1i 1y0*i2yif00*i2yif1i2yiuixiy222*2Bivariate Probit Model11,0021NiuiuBivariate
36、Probit model)0,0()0,0(2*121iiiiyyPyyPiiiixxiiiiduduuufxuxuPii1221222111),(),(1122Multivariate Probit ModelCigaretteAlcoholMarijuanaCocaineYESYESYESYESNoNoNoNo444*433322221111*3*uxyuxyuxyuxy0*i 1yif00*i 1yif1i 1y0*200*212iyifiyifiy0*400*441iyifiyifiy0*300*331iyifiyifiyMultivariate Normality 134241412
37、3131121,00004321NiuiuiuiuMultivariate Probit ModelqJ=3,Clark(1961)qJ=4,Hausman and Wise(1978,Econometrica)qJ 4McFadden(1989,Econometrica)-Simulation-Based Estimation -high dimensional integralsStern(1997,Journal of Economic Literature)-Simulated Maximum Likelihood Estimator -Simulated Moment Estimat
38、or -GHK simulatorTOBIT MODELiiiuxy*0*iiyy00*iiyy if 2,0Nui TOBIT MODEL 0000*iiiiiiiyyEyPyyEyPyE)0()0(uxuxEuxPiiiiixiiiiiixuuExxup(*TOBIT MODELauuEii22121211112iuaiiiaiiiiaiudeaduauPuuduauufuiTOBIT MODELaaeaeaaaui12101211222221TOBIT MODELwhere ixaiiiixxxuuE2222p.f.iiiixixiiiiduxuPufuduxuufuxuuEii2222
39、TOBIT MODELletixa222ZuuZdudZ aFduufua12dZeZdZeZdueu分子aZZaua22222222222222121TOBIT MODELNOTE dxxfxgxgxfdxxgxfdxxgxfdxxgxfxgxfdxdxgdxxdfdxxdgxfxgxfdxdTOBIT MODELlet ZZf 22222222ZZZeZdedZzexg aaadZeeadZeeaeZZdeeZ分子aZaaZaZzaZaZ1212102222222222222222222222limTOBIT MODEL aaaaaaaaaaa全部11111112222iiiiixxxuu
40、E222TOBIT MODELNOTE2222limlimZZZZeZeZby LHopital rule 011lim22ZZZeDDEnd of studyDCCDuration ModelDuration modelqCensored dataqUnobserved heterogeneity qTime-varying covariates2.2 Hazard Analysist)tTttTt(Plim)t(h0Survival rate and Hazard rate)tT(P)t(S)t(S)t(fdt)t(Slnd)t(h2.2 Nonparametric Hazard Anal
41、ysisKaplan-Meier estimatorLife table estimatorFigure2Kaplan-Meier Estimates of Survival Function00.10.20.30.40.50.60.70.80.9101020304050607080Age of the first useSurvival functiontobacco usealcoholmarijuana usecocaine useFigure 2:Kaplan-Meier Estimates of Survival FunctionFigure3 Life Table Estimaat
42、es of Survival Function00.10.20.30.40.50.60.70.80.9101020304050607080Age at the first useSurvival functiontobacco usealcohol usemarijuana usecocaine useFigure 3:Life Table Estimates of Survival FunctionThe density and survival functions);()0();()1(ixiTSiYPixiTfiYP f (Ti;wi):the probability density f
43、unction of the failure timeS (ti;wi):the probability of survivalqExponentialqWeibullqLog-logisticqLog-normalThe specifications for f (Ti;xi)and S (ti;xi):00.020.040.060.080.10.120.140.160.180.20.220.240.260.280.30.320.340.360.380.40.420.4401020304050607080Age at the first useHazard functiontobacco u
44、sealcohol usemarijuana usecocaine useFigure 4:Life Table Estimates of Hazard FunctionsEventually fail assumptionii)x|S(t)x|t(fLi1n1ii是否核準否是不良債權還款正常Eventually Fail Assumption是否核準否是不良債權還款正常將來還款正常Never Fail將來不正常Eventually FailSchmidt and Witte(1989)-Split population duration model);()()(1)0();()()1(iwi
45、TSixGixGiYPiwiTfixGiYPG(xi):the probability of eventual failure f(Ti;wi):the probability density function of the failure timeS(Ti;wi):the probability of survivalSchmidt and Witte(1989)-Likelihood function,)iw;it(S)ix(G0i)ix(G1ln)iw;iT(f1i)ix(GlnLln4444333322221111*uxyuxyuxyuxy8888777766665555*uxyuxy
46、uxyuxy4.3 Multivariate Split Population Duration ModelMultivariate probit model41444313332122211111*xyxyxyxyMultivariate duration model82888727776266652555*xyxyxyxyUnobserved heterogeneityThe frailty (m=1,2)is assumed to follow a gamma distribution with mean 1 and variance :.1111),(mmmmvemvmvmm =ind
47、ividuals probability of eventual failure for a type k event(k=1,2,3,4).follows a Weibull distribution)x(Giki)x(Gjkithi.11kixkeizekiGWhether Parti2ZkiAssume the survival function is log-logistic.The second frailty enters the hazard function as:,)1(2kiwekiweizkiwhere kixkkitkiw/)ln(,and is the failure
48、 time or thecensored time,whichever is earlier.kitDuration PartThe cumulative hazard,the survival function,and the density function are:,)1ln(2)1(2)1(2,)1ln(2),1ln(2)1(20kiweizekiwekiweizkiSkiwekiweizkifkiweizekiHekiSkiweizkidwkiwekiweizkiH The likelihood function is given by idzizidzizkidkiSkiGkiGkidkifkkiGkiL2)2,2(1)1,1(1)1()1(00B.3 Simultaneous Equations ModelsoM.J.Lee(1995,Journal of Applied Econometrics)mummxmymmymmyuxmymyyuxmymyy*11,*11,222*,2*11,22111*,1*22,11*謝謝指教
