Logit Models 1 Alexander Spermann University of Freiburg
Logit Models 1 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 2 1. 2. 3. 4. 5. 6. Logit vs. Probit Models The Multinomial Logit Model Estimation The IIA Assumption Applications (Extensions) Train, K. (2003), Discrete Choice Methods with Simulation (downloadable from http: //elsa. berkeley. edu/books/choice 2. html) Wooldridge, J. M. (2002), Econometric Analysis of Cross Section and Panel Data, Ch. 15 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Binary dependent variable: Let (as in the case of Probit) In the Logit model, F(. ) is given the particular functional form: 3 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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. Estimation: We find the estimated parameters by maximizing the loglikelihood function 4 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models The Logit model is implemented in all major software packages, such as Stata: 5 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Coefficient Magnitudes differ between Logit and Probit: Probit Logit gpa 1, 626 2, 826 tuce 0, 052 0, 095 psi 1, 426 2, 379 This is due to the fact that in binary models, the coefficients are identified only up to a scale parameter 6 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Coefficient magnitudes can be made comparable by standardizing with the variance of the errors: with logarithmic distribution: Var=π2/6 with standard normal distribution: Var=1 approximative conversion of the estimated values using 7 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Estimated coefficients For interpretation we have to calculate the marginal effects of the estimated coefficients (as in the Probit case) (AKA margeff) 8 Interpretation of the marginal effects analogous to the Probit model Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models unordered IIA* yes valid ? no ordered mlogit mprobit nested logit ordered logit/ probit *IIA=independence of irrelevant alternatives (assumption) 9 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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: 10 how do you feel today: very well, fairly well, not too well, miserably Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models A discrete choice underpinning • choice between M alternatives • decision is determined by the utility level Uij, an individual i derives from choosing alternative j • Let: (1) where i=1, …, N individuals; j=0, …, J alternatives 11 The alternative providing the highest level of utility will be chosen. Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 12 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 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 13 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 14 standardization: β 0=0 The special case where J=1 yields the binary Logit model. Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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) 15 It requires a different arrangement of the data (one line per alternative for each i) Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Maximum-Likelihood-Estimation The log likelihood function is globally concave and easy to maximize (Mc. Fadden, 1974) big computational advantage over multinomial probit or nested logit 16 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Interpretation of coefficients The coefficients themselves cannot be interpreted easily but the exponentiated coefficients have an interpretation as the relative risk ratios (RRR) Let “risk ratio“ 17 (for simplicity, only one regressor considered) Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models The relative risk ratio tells us how the probability of choosing j relative to 0 changes if we increase x by one unit: such that “relative risk ratio“ RRR 18 Note: some people also use the term “odds ratio” for the relative risk Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Interpretation: Variable x increases (decreases) the probability that alternative j is chosen instead of the baseline alternative if RRR > (<) 1. 19 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Marginal Effects Elasticities relative change of pij if x increases by 1 per cent 20 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 21 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Ratio of the choice probabilities between two alternatives j and k is independent from any other alternative: 22 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 23 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 24 Not possible according to IIA! Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 25 Problem: probability of walking decreases from 2/3 to 2/4 due to the introduction of blue buses not plausible! Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models : • Reason 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“ 26 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Hausman Test: H 0: IIA is valid („odds ratios” are independent of additional alternatives) Procedure: “omit” a category Do the estimated coefficients change significantly? 27 If they do: reject H 0 cannot apply multinomial logit choose nested logit or multinomial probit instead Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 28 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Observation group: „nonwhite“ 0 = white 1 = black 29 Is the choice of health care insurance determined by the variable “nonwhite”? Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Estimating the M-Logit-Model (with Stata): 30 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 31 customized choice of basic category in Stata: mlogit depvar [indepvars], base (#) Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models Analysing the output: 1) The estimated coefficients are difficult to interpret quantitatively 32 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: Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 2) Calculating the RRR 33 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 34 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models „odds ratio plot“: in Stata: mlogview after mlogit 35 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models • 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. 36 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 3) Marginal Effect Stata computes the marginal effect of “nonwhite“ for each alternative separately. (AKA margeff) 37 Alexander Spermann, University of Freiburg, WS 2007/2008
Logit Models 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 38 (However, none of the coefficients is significant) Alexander Spermann, University of Freiburg, WS 2007/2008
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