计量经济分析（第六版）答案 Notes-25.docx

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1、Econometrics IProfessor William GreeneNotes 25. Discrete Choice, Duration and Censoring1. Discrete Choice Among Multiple AlternativesA. Random Utility ModelU(i,j) = Px(iJ) + 8(ij)B. The Independent Utilities Extreme Value - Multinomial Logit Models(i,j)exp(-exp(8(i,j), all independentLeads to Prob i

2、ndividual i makes choice j = Prob U(i,j) u(i,k) for all k not jexppx(z;y)exptpCz,；)expp,x(z,y)1)=-expP*x(zJ) exp logexpBx(i,攵)exp/VIV 二 inclusive value = LogSum” Used as a measure of consumer surplusC. Estimated by maximum likelihood: LogL = S, log Prob actual outcomesD. Application: Survey of 210 p

3、eople traveling between Sydney and MelbourneMode = 0/1 for four alternatives: l=Airz 2=Train, 3=Bus, 4=Car, Ttme = Terminal waiting time, 0 for Car Invc = Invehicle cost for all stages, Invt = Invehicle time for all stages, Gc = Generalized cost measure = Invc + Invt * value of time, Chair = Dummy v

4、ariable for chosen mode is Air, Hine = Household income in thousands, Psize = Traveling party size.+I Discrete choice (multinomial logit) model |I Maximum Likelihood EstimatesIIDependent variableChoice|IWeighting variableONE|INumber of observations210|IIterations completed6|ILog likelihood function-

5、182.3383|ILog-L for Choicemodel =-182.3383|IR2=l-LogL/LogL*Log-L fncnR-sqrdRsqAdj|INo coefficients-291.1218.37367.36561|IConstants only-283.7588.35742.34915IIChi-squared 5=202.84091|ISignificance forchi-squared= 1.00000|IResponse data aregiven asind. choice.|I Number of obs.=210z skipped 0 bad obs.

6、|+I Variable | Coefficient | Standard Error |b/St.Er.|P|Z|z | +GC.7559792205E-01.18249800E-014.142.0000TTME-.1029031876.10987031E-01-9.366.0000INVT-.1434742416E-01.26519082E-02-5.410.0000INVC-.8952256682E-01.19952710E-01-4.487.0000HINCA.2363930104E-01.11552924E-012.046.0407A_AIR4.0657414501.05260373

7、.863.0001A二TRAIN4.273933648.512142808.345.0000A 二 BUS3.714449873.508562107.304.0000(Note:E+nn or E-nn meansmultiply by 10to + or -nnpower.)+Derivative (times100) Averagedoverobservations.Attribute is GCin choiceAIREffects on probabilities of allchoices in themodel:* indicates directDerivative effect

8、of the attribute.Decompositionof EffectTotal |TrunkLimbBranchChoiceEffect|Trunk=Trunk1Limb=Lmb1:1Branch=B(1:1,1)*Choice=AIR.000.000.000.828.828|Choice=TRAIN.000.000.000-.257-.257|Choice=BUS.000.000.000-.168-.168|Choice=CAR.000.000.000-.403-.403|+E. Problems and questions about the model:1. The IIA p

9、roblem - Red Bus/Blue Bus problem:The ratio of any two probabilities is independent of all other alternatives.If you throw in an irrelevant alternative, all probabilities adjust to preserve ratios.2. Independent utilities? Seem simplausible. 1. above is a consequence.F. An alternative model: There a

10、re many1. A nested logit model: Allow some heteroscedasticity and correlation:a. Partition the choice set.b. Within a partition, equal variances, not necessarily uncorrelatedc. Across partitions, allows heteroscedasticity:2. Approach to estimation:Prob actual choice = Prob group Prob choice | group3

11、. Application: Public (Bus, Train) and Private (Air,Car)+FIML: Nested Multinomial Logit Model Maximum Likelihood EstimatesDependent variableMODEWeighting variableONENumber of observations840Iterations completed21Log likelihood function-171.5440Restricted log likelihood-291.1218Chi-squared239.1556Deg

12、rees of freedom10Significance level.0000000R2=l-LogL/LogL*Log-L fncnR-sqrdRsqAdjNo coefficients-291.1218.41075.40124Constants only-283.7588.39546.38571At start values-182.3383.05920.04403Response data aregiven asind. choice.+Tree Structure Specified for the Nested Logit Model Sample proportions are

13、marginal, not conditional.Choicesmarked with * areexcluded for theIIA test.Trunk(prop.)|Limb(prop.)|Branch(prop.)|Choice11(prop.)|Weight|IIA 11Trunkl1.00000|Lmbl|1111-L1.00000IPUBLIC1I PRIVATE1 +.44286ITRAIN IBUS.55714|AIR ICAR +.30000| .142861 .276191 .280951+-1.00011.00011.00011.0001 +I Variable |

14、 Coefficient I Standard Error |b/St.Er.|P|Z|z |+Attributes in the Utility Functions (beta)GC.6319944270E-01.17808876E-013.549.0004TTME-.7130802994E-01.11497764E-01-6.202.0000INVT-.1307742437E-01.25655552E-02-5.097.0000IN VC-.7208563816E-01.19890552E-01-3.624.0003HINCA.1787182506E-01.80413475E-022.22

15、2.0263A_AIR2.229411274.554629394.020.0001ATRAIN1.664083959.567903702.930.0034abus1.833564375.916116412.001.0453IV parameters, tau(jIi,1)z sigma(i11)A phi(1)PUBLIC1.820251751.377035754.828.0000PRIVATE2.494503616.508123764.909.0000Derivative (times 100) Averaged over observations.Attribute is GCin cho

16、ice AIREffects on probabilities of all choices in the model: * indicates direct Derivative effect of the attribute.Decomposition of EffectTrunk Limb Branch ChoiceTrunk=Trunk1Limb=Lmb1|1Branch=PUBLICTotalEffect*Choice=AIR.000.000.608.377.985Choice=TRAINBranch=PRIVATE.000.000.238-.377-.139Choice=BUS.0

17、00.000.296-.539-.242Choice=CAR.000.000-.550.000-.550+G. A Random Parameters Model: For individual i,。=0 + v(i) where v(i) is a random vector.1. Model is estimated by random simulation2. Application| Random Parameters Logit ModelDependent variable Number of observations Iterations completed Log likel

18、ihood function Significance levelMODE 840 16 -160.1339 .0000000R2=l-LogL/LogL* No coefficients Constants only At start values Replications forLog-L fncn -291.1218 -283.7588 -182.3383R-sqrd .44994 .43567 .12178simulated probs.RsqAdj | .43561 | .42096 | .09889 | 30 II Hessian was not PD. Using BHHH es

19、timator.|I Number of obs.=210z skipped 0 bad obs. |+A_AIR5.0254023743.65829741.374.1695ATRAIN10.962655183.82426292.867.0041A-BUS9.3478542703.26839242.860.0042GC.2491490021.100179192.487.0129TTME-.2342273626.77743691E-01-3.013.0026INVT-.5045120364E-01.18800678E-01-2.683.0073IN VC-.3001815521.10769456

20、-2.787.0053HINCA.6481668668E-01.66740609E-01.971.3315Derived standarddeviations of parameterdistributionsNsA_AIR1.9158456922.3111342.829.4071NsATRA工.61913790981.5176036.408.6833nsabus2.5701674381.27234892.020.0434NsGC.2254800113E-01.37784221E-01.597.5507NsTTME.8710050978E-01.40302704E-012.161.0307Ns

21、INVT.1879565624E-02.44580028E-02.422.6733NsINVC.9103876676E-01.41057987E-012.217.0266NsHINCA.7388416523E-01.44265121E-011.669.0951+|Variable | Coefficient I Standard Error |b/St.Er.|P|Z|z | +Random parameters in utility functionsIL Models for Duration - Hazard Function ModelingA. What is the hazard

22、rate?B. Modeling hazard rates: What is the probability of a transition?C. Duration dependence what are positive and negative duration dependenceD. Modeling Duration with parametric models1. Standard approaches2. Dealing with censoring in duration dataE. Estimation and interpretation1. Maximum likelihood estimation2. Interpreting results