Logit Models Alexander Spermann, University of Freiburg, SS Logit Models

Alexander Spermann, University of Freiburg, SS Logit vs. Probit Models 2.The Multinomial Logit Model 3.Estimation 4.The IIA Assumption 5.Applications 6.(Extensions) Train, K. (2003), Discrete Choice Methods with Simulation (downloadable from Wooldridge, J.M. (2002), Econometric Analysis of Cross Section and Panel Data, Ch. 15

Logit Models Alexander Spermann, University of Freiburg, SS In the Logit model, F(.) is given the particular functional form: Binary dependent variable: Let (as in the case of Probit)

Logit Models Alexander Spermann, University of Freiburg, SS Estimation: We find the estimated parameters by maximizing the log- likelihood function The model is called Logit because the residuals of the latent model are assumed to be extreme value distributed. The difference between two extreme value distributed random variables εik-εij is distributed logistic.

Logit Models Alexander Spermann, University of Freiburg, SS The Logit model is implemented in all major software packages, such as Stata:

Logit Models Alexander Spermann, University of Freiburg, SS This is due to the fact that in binary models, the coefficients are identified only up to a scale parameter ProbitLogit gpa1,6262,826 tuce0,0520,095 psi1,4262,379 Coefficient Magnitudes differ between Logit and Probit:

Logit Models Alexander Spermann, University of Freiburg, SS Coefficient magnitudes can be made comparable by standardizing with the variance of the errors: with logarithmic distribution : Var=π 2 /6 with standard normal distribution: Var=1  approximative conversion of the estimated values using

Logit Models Alexander Spermann, University of Freiburg, SS Estimated coefficients For interpretation we have to calculate the marginal effects of the estimated coefficients (as in the Probit case) Interpretation of the marginal effects analogous to the Probit model (AKA margeff)

Logit Models Alexander Spermann, University of Freiburg, SS unordered ordered ordered logit/ probit IIA* valid ? yes no mlogit mprobit nested logit *IIA=independence of irrelevant alternatives (assumption)

Logit Models Alexander Spermann, University of Freiburg, SS Multiple alternatives 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

Logit Models Alexander Spermann, University of Freiburg, SS A discrete choice underpinning choice between M alternatives decision is determined by the utility level U ij, an individual i derives from choosing alternative j Let: where i=1,…,N individuals; j=0,…,J alternatives (1) The alternative providing the highest level of utility will be chosen.

Logit Models Alexander Spermann, University of Freiburg, SS The probability that alternative j will be chosen is: 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

Logit Models Alexander Spermann, University of Freiburg, SS Exception: If the error terms εij 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

Logit Models Alexander Spermann, University of Freiburg, SS standardization : β 0 =0 The special case where J=1 yields the binary Logit model.

Logit Models Alexander Spermann, University of Freiburg, SS Different kinds of independent variables 1)Characteristics that do not vary over alternatives (e.g., socio-demographic characteristics, time effects) 2)Characteristics that vary over alternatives (e.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)

Logit Models Alexander Spermann, University of Freiburg, SS Maximum-Likelihood-Estimation The log likelihood function is globally concave and easy to maximize (McFadden, 1974)  big computational advantage over multinomial probit or nested logit

Logit Models Alexander Spermann, University of Freiburg, SS The coefficients themselves cannot be interpreted easily but the exponentiated coefficients have an interpretation as the relative risk ratios (RRR) Let Interpretation of coefficients (for simplicity, only one regressor considered) “ risk ratio“

Logit Models Alexander Spermann, University of Freiburg, SS 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 the relative risk such that

Logit Models Alexander Spermann, University of Freiburg, SS Variable x increases (decreases) the probability that alternative j is chosen instead of the baseline alternative if RRR > (<) 1. Interpretation:

Logit Models Alexander Spermann, University of Freiburg, SS Marginal Effects Elasticities  relative change of p ij if x increases by 1 per cent

Logit Models Alexander Spermann, University of Freiburg, SS Independence of Irrelevant Alternatives (IIA) : Important assumption of the multinomial Logit-Model  it implies that the decision between two alternatives is independent from the existence of more alternatives

Logit Models Alexander Spermann, University of Freiburg, SS Ratio of the choice probabilities between two alternatives j and k is independent from any other alternative:

Logit Models Alexander Spermann, University of Freiburg, SS Problem: This assumption is invalid in many situations. Example: „red bus - blue bus“ - problem initial situation: -an individual chooses to walk with probability 2/3 -- probability of taking the bus is 1/3 probability ratio: 2:1

Logit Models Alexander Spermann, University of Freiburg, SS Introduction of blue buses: It is rational to believe 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 with P=1/6 Person takes a blue bus with P=1/6 New probability ratio: 4:1 Not possible according to IIA!

Logit Models Alexander Spermann, University of Freiburg, SS 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: probability of walking decreases from 2/3 to 2/4 due to the introduction of blue buses  not plausible!

Logit Models Alexander Spermann, University of Freiburg, SS : R eason of IIA property: assumption that error termns are independently distributed over all alternatives. The IIA property causes no problems if all 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“

Logit Models Alexander Spermann, University of Freiburg, SS Hausman Test: H0: IIA is valid („odds ratios” are independent of additional alternatives) 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

Logit Models Alexander Spermann, University of Freiburg, SS Cramer-Ridder Test Often you want to know whether certain alternatives can be merged into one: e.g., do you have to distinguish between employment states such as “unemployment” and “nonemployment” The Cramer-Ridder tests the null hypothesis that the alternatives can be merged. It has the form of a LR test: 2(logL U -logL R )~χ²

Logit Models Alexander Spermann, University of Freiburg, SS Derive the log likelihood value of the restricted model where two alternatives (here, A and N) have been merged: where log is the log likelihood of the of the pooled model, and n A and n N are the number of times A and N have been chosen restricted model, log is the log likelihood

Logit Models Alexander Spermann, University of Freiburg, SS Data: 616 observations of choice of a particular health insurance 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

Logit Models Alexander Spermann, University of Freiburg, SS Observation group: „nonwhite“ 0 = white 1 = black Is the choice of health care insurance determined by the variable “nonwhite”?

Logit Models Alexander Spermann, University of Freiburg, SS Estimating the M-Logit-Model (with Stata):

Logit Models Alexander Spermann, University of Freiburg, SS 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 basic category in Stata: mlogit depvar [indepvars], base (#)

Logit Models Alexander Spermann, University of Freiburg, SS Analysing the output: 1)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. More intuitive to exponentiate coeffs and form RRRs:

Logit Models Alexander Spermann, University of Freiburg, SS ) Calculating the RRR

Logit Models Alexander Spermann, University of Freiburg, SS 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

Logit Models Alexander Spermann, University of Freiburg, SS „odds ratio plot“: in Stata: mlogview after mlogit

Logit Models Alexander Spermann, University of Freiburg, SS Alternatives U und P are located on the right of baseline category I i.e. compared to I there is a higher probability for them to be chosen if “nonwhite“ has the value 1 Distance of the two alternatives measures the magnitude of this effect: the gap between U and I is smaller than the gap between P and I.

Logit Models Alexander Spermann, University of Freiburg, SS ) Marginal Effect Stata computes the marginal effect of “nonwhite“ for each alternative separately. (AKA margeff)

Logit Models Alexander Spermann, University of Freiburg, SS 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 15.0 per cent. the probability of choosing alternative “uninsure“ rises by 0.2 per cent (However, none of the coefficients is significant)