计量经济学_多分定类因变量.ppt

1、Applied MicroeconometricsChapter 3Multinomial and ordered models,The Multinomial Logit Model (MNL) Estimation The IIA Assumption Applications Extensions to MNL Ordered Probit,Train, K. (2003), Discrete Choice Methods with Simulation (downloadable from /books/choice2.html) Wool

2、dridge, J.M. (2002), Econometric Analysis of Cross Section and Panel Data, Ch. 15,unordered,ordered,ordered logit/ probit,IIA* valid ?,yes,no,mlogit,mprobit nested logit,*IIA=independence of irrelevant alternatives (assumption),Making the right decision between alternative models,Multiple alternativ

3、es without obvious ordering, Choice of a single alternative out of a number of distinct alternatives,e.g.: which means of transportation do you use to get to work?,bus, car, bicycle etc., Example for ordered structure: how do you feel today: very well, fairly well, not too well, miserably,Multinomia

4、l models,Choice between M alternatives Decision is determined by the utility level Uij, an individual i derives from choosing alternative j Let where i=1,N individuals; j=0,J alternatives The alternative providing the highest level of utility will be chosen.,(1),Derivation of the Multinomial Logit m

5、odel,Probability that alternative j will be chosen: In order to calculate this probability, the maximum of a number of random variables has to be determined. In general, this requires solving multidimensional integrals analytical solutions do not exist,Derivation of the Multinomial Logit model,Excep

6、tion: If the error terms ij in (1) are assumed to be independently & identically standard extreme value distributed, then an analytical solution exists. In this case, similar to binary logit, it can be shown that the choice probabilities are,Derivation of the Multinomial Logit model,Standardization:

7、 0=0,The special case where J=1 yields the binary Logit model.,Derivation of the Multinomial Logit model,Different kinds of independent variables Characteristics that do not vary over alternatives (e.g., socio-demographic characteristics, time effects) Characteristics that vary over alternatives (e.

8、g., prices, travel distances etc.),In the latter case, the multinomial logit is often called “conditional logit” (CLOGIT in Stata) It requires a different arrangement of the data (one line per alternative for each i),Independent variables,Estimation is by Maximum Likelihood The log likelihood functi

9、on is globally concave and easy to maximize (McFadden, 1974) big computational advantage over multinomial probit or nested logit,Estimation of the MNL,The coefficients themselves cannot be interpreted easily but the exponentiated coefficients have an interpretation as the relative risk ratios (RRR),

10、Let then,(for simplicity, only one regressor considered),“risk ratio“,Interpretation of coefficients,The relative risk ratio tells us how the probability of choosing j relative to 0 changes if we increase x by one unit:,“relative risk ratio“RRR,Note: some people also use the term “odds ratio” for th

11、e relative risk,such that,Interpretation of coefficients,Variable x increases (decreases) the probability that alternative j is chosen instead of the baseline alternative if RRR () 1.,Interpretation:,Interpretation of coefficients,Marginal Effects,Marginal effects in the MNL,Elasticities,relative ch

12、ange of pij if x increases by 1 per cent,Important assumption of the multinomial Logit-Model it implies that the decision between two alternatives is independent from the existence of more alternatives,Independence of Irrelevant Alternatives (IIA),Ratio of the choice probabilities between two altern

13、atives j and k is independent from any other alternative:,Independence of Irrelevant Alternatives (IIA),Problem: This assumption is invalid in many situations.,Example: red bus - blue bus“ - problem,initial situation: only red buses an individual chooses to walk with probability 2/3 - probability of

14、 taking a red bus is 1/3 probability ratio: 2:1,Independence of Irrelevant Alternatives (IIA),Introduction of blue buses:,It is rational to believe that that the probability of walking will not change. If the number of red buses = number of blue buses: Person walks with P=4/6 Person takes a red bus

15、with P=1/6 Person takes a blue bus with P=1/6,New probability ratio for walking vs. red bus = 4:1,Not possible according to IIA!,Independence of Irrelevant Alternatives (IIA),The following probabilities result from the IIA-assumption: P(by foot)=2/4 P(red bus)=1/4 P(blue bus)=1/4, such that Problem:

16、 probability of walking decreases from 2/3 to 2/4 due to the introduction of blue buses not plausible!,Independence of Irrelevant Alternatives (IIA),:,Reason of IIA property: assumption that error termns are independently distributed over all alternatives. The IIA property causes no problems if all

17、alternatives considered differ in almost the same way.,e.g., probability of taking a red bus is highly correlated with the probability of taking a blue bus “substitution patterns“,Independence of Irrelevant Alternatives (IIA),H0: IIA is valid (odds ratios” are independent of additional alternatives)

18、 Procedure: “omit” a category Do the estimated coefficients change significantly? If they do: reject H0 cannot apply multinomial logit choose nested logit or multinomial probit instead,Hausman Test for validity of IIA,Often one would like to know whether certain alternatives can be merged into one:

19、e.g., do employment states such as “unemployment” and “nonemployment” need to be distinguished? The Cramer-Ridder tests the null hypothesis that the alternatives can be merged. It has the form of a LR test: 2(logLU-logLR),Cramer-Ridder Test,Derive the log likelihood value of the restricted model whe

20、re two alternatives (here, A and N) have been merged:,where log,is the log likelihood of the,of the pooled model, and nA and nN are the number of times A and N have been chosen,restricted model, log,is the log likelihood,Cramer-Ridder Test,Data:,616 observations of choice of a particular health insu

21、rance,3 alternatives:,indemnity plan“: deductible has to be paid before the benefits of the policy can apply prepaid plan“: prepayment and unlimited usage of benefits uninsured“: no health insurance,Application,Observation group: nonwhite“,0 = white,1 = black,Is the choice of health care insurance d

22、etermined by the variable “nonwhite”?,1 = black,Application,Stata estimation output for the MNL,Application,If one does not choose a category as baseline, Stata uses the alternative with the highest frequency.,here: indemnity is used as the baseline category used for comparison,customized choice of

23、basic category in Stata: mlogit depvar indepvars, base (#),Application,The estimated coefficients are difficult to interpret quantitatively,The coefficient indicates how the logarithmized probability of choosing the alternative prepaid“ instead of indemnity“ changes if nonwhite“ changes from 0 to 1.

24、 More intuitive to exponentiate coeffs and form RRRs:,Interpreting the output,Calculation of RRR,Probability of choosing “prepaid“ over “indemnity“ is 1.9 times higher for black individuals “uninsure“ over “indemnity“ is 1.5 times higher for black individuals,Calculation of RRR,Stata computes the ma

25、rginal effect of “nonwhite“ for each alternative separately.,(AKA margeff),Marginal effects,Interpretation: If the variable “nonwhite“ changes from 0 to 1 the probability of choosing alternative “indemnity“ decreases by 15.2 per cent. the probability of choosing alternative “prepaid“ increases by 15

26、.0 per cent. the probability of choosing alternative “uninsure“ rises by 0.2 per cent (However, none of the coefficients is significant),Marginal effects,unordered,ordered,ordered logit/ probit,IIA* valid ?,yes,no,mlogit,mprobit nested logit,*IIA=independence of irrelevant alternatives (assumption),

27、Making the right decision between alternative models,Example for ordered structure Observed dependent variable yi contains m categories such as educational outcomes (basic schooling, completed apprenticeship, A-levels, university degree etc.) Latent model is linear, e.g. where yi* is the amount of e

28、ducation sought and xi could be variables such as abilities, parents education and income, distance to city etc. Note that xi does not contain an intercept (well see shortly why not).,Ordered models,Define a series of threshold values 1,m-1 Category j is chosen if so that in analogy to the binary mo

29、del,Observed and latent outcomes,Suppose that i is distributed standard normal. Then the log likelihood function is where The log likelihod function is then maximised with respect to 1,m-1 and .,Estimation by ML,Choice of voting alternatives by ILO delegates coding: yes = 2, abstention = 1, no = 0 e

30、xplanatory variables: GDP, Govt CEOs party affiliation,Example,. oprob newvote ceo_left ceo_cent ceo_righ income2 income3 income4 income5 Iteration 0: log likelihood = -2090.4973 Iteration 1: log likelihood = -2040.3852 Iteration 2: log likelihood = -2040.3438 Iteration 3: log likelihood = -2040.343

31、8 Ordered probit regression Number of obs = 2103 LR chi2(7) = 100.31 Prob chi2 = 0.0000 Log likelihood = -2040.3438 Pseudo R2 = 0.0240 - newvote | Coef. Std. Err. z P|z| 95% Conf. Interval -+- ceo_left | .2443465 .0660788 3.70 0.000 .1148343 .3738586 ceo_cent | .038681 .1267729 0.31 0.760 -.2097893

32、.2871514 ceo_righ | -.361733 .0716722 -5.05 0.000 -.502208 -.221258 income2 | .2484576 .0818912 3.03 0.002 .0879538 .4089615 income3 | .2016928 .0810596 2.49 0.013 .0428189 .3605666 income4 | .2825082 .0778242 3.63 0.000 .1299755 .4350408 income5 | -.1632577 .1101756 -1.48 0.138 -.3791979 .0526826 -+- /cut1 | -.0628592 .0603017 -.1810484 .0553301 /cut2 | .28