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Limited Dependent Variables: Binary Models

Limited Dependent Variables: Binary Models. Erik Nesson Ball State University MBSW 2013. Outline. Overview of LDVs Binary Outcome Models Linear Probability Model Logit and Probit Interpretation of Coefficients Odds ratios vs. marginal effects Implementation in Stata.

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Limited Dependent Variables: Binary Models

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  1. Limited Dependent Variables:Binary Models Erik NessonBall State University MBSW 2013

  2. Outline • Overview of LDVs • Binary Outcome Models • Linear Probability Model • Logit and Probit • Interpretation of Coefficients • Odds ratios vs. marginal effects • Implementation in Stata

  3. Binary Outcome Models • Dependent variable takes values of 0 or 1 • Model of interest is probability that y=1 conditional on independent variables • Here represents matrix of covariates and represents vector of coefficients • Examples • Is a person obese? Does a person smoke? Does a person contract a disease?

  4. Example: Secondhand Smoke • Half of adult non-smokers are exposed to environmental tobacco smoke (ETS) • Main point of public policies is to reduce ETS exposure in specific areas • Ex: smokefree air laws are meant to reduce ETS exposure at work • But most research about tobacco control policies focuses on reducing smoking • Main question: • Do tobacco control policies reduce ETS exposure in the workplace?

  5. How to measure ETS exposure? • NHANES Dataset • Repeated cross-section dataset covering 1988-1994 and 1999-2004 • Roughly 10k individuals interviewed every year • All individuals complete extensive survey AND receive physical • Contains extensive demographic and health history information • Sample • Non-smoking, employed individuals age 18 to 65 • 8,554 individuals

  6. Dependent and Independent Variables • Indicators of ETS Exposure • Q: “… how many hours per day can you smell the smoke from other people’s cigarettes, cigars, and/or pipes?” • Serum cotinine levels • Cotinine is major metabolite of nicotine with 8-16 hr half life • Very low levels can be detected • Tobacco Control Policies • Cigarette Taxes measured in real $2009 • The percent of each individual’s state living under a workplace SFA law (from 0 to 100)

  7. Summary Statistics

  8. Self-Reported ETS Exposure at Work by Job Category

  9. Tabulation of Observable Cotinine Levels by Job Category

  10. Basic Model • Basic model estimates exposure to ETS as a function of tobacco control policies, individual characteristics, other geographic characteristics: • TC: tobacco control policies • X: individual characteristics • Z: geographic characteristics • and : state and time fixed effects • Main coefficients of interest:

  11. Linear Probability Model • Assume conditional probability is linear in • Upsides to LPM • Easy to run: run a linear regression: • Interpretation is easy! • In words, “Every unit increase in is associated with a percentage point change in the probability that y=1.” • Prediction is easy!

  12. Binary Outcome Models • Downsides to LPM • may not be linear • Predicted probability does not need to be between 0 and 1 • Constant marginal effect may not be reasonable!

  13. Linear Probability Results • Table shows marginal effects with standard errors in parentheses

  14. Logit or Probit • Assume that is a function such that for any value of • As and as • For Logit • For Probit • Note: is the standard normal CDF

  15. Interpretation of Coefficients • Continuous variable: • If then what is marginal effect, i.e. ? • Using some calculus: • Where is the density function associated with • Some notes: • doesn’t only depend on • What values of should we use? • Marginal effect isn’t constant like in linear probability model

  16. Interpretation of Coefficients • Discrete variable: • If then marginal effect of increasing from 0 to 1 = • Some notes: • Again, marginal effect doesn’t only depend on • What values of should we use? • Marginal effect isn’t constant like in linear probability model

  17. Calculating Marginal Effects • For both continuous and discrete variables, other coefficients and variable values enter into marginal effects calculation • Three common approaches: • Marginal effect at the mean: Use mean values of other variables • Average marginal effect: Calculate marginal effect for each observation • Marginal effect at some other value of coefficients

  18. Calculating Marginal Effects • Calculating marginal effect at the mean for • Continuous coefficient • Find by plugging in mean values for • Multiply by • Discrete coefficient • Find when by plugging in mean values for and 1 for • Find when by plugging in mean values forand 0 for • Subtract two values

  19. Calculating Marginal Effects • Calculating average marginal effect for • Continuous coefficient • Find for each observation and multiply by • Find mean value for all observations • Discrete coefficient • Find when for each observation • Findwhen for each observation • Subtract two values for each observation • Find mean value for all observations

  20. Implementation in Stata • Code for estimating marginal effect at the mean in Stata: • logit y x • margins, dydx(varlist) atmeans • Code for estimating average marginal effect in Stata: • logit y x • margins, dydx(varlist) • Usually Stata is smart enough to determine which independent variables are binary

  21. Odds Ratios • Common in other fields to run a logit model and report an odds ratio • What are odds? • Odds of an event = • The odds ratio • Odds ratio>1: event is more likely to happen • Odds ratio<1: event is less likely to happen

  22. Odds Ratios and Logit • How do Odds Ratios work in Logit? • Odds given : • Note: • Similarly, odds given : • Note:

  23. Odds Ratios and Logit • Then Odds Ratio = • Plugging in: • Quick notes about the odds ratio • Odds ratio does not depend on other coefficients or independent variables • May be difficult to translate into policy • Odds ratios are very easy to compute! Simply exponentiate coefficients

  24. Odds Ratios and Logit • What does an odds ratio of 2 mean? • Odds of y=1 are 2x greater when x=1 than when x=0 • Could be that • Odds that y=1|x=1 = 4 and odds that y=1|x=0 = 2 • Odds that y=1|x=1 = 3 and odds that y=1|x=0 = 1.5 • Odds that y=1|x=1 = 2 and odds that y=1|x=0 = 1 • So odds ratios are not equal to marginal effects • Do not tell us about differences in probability

  25. Odds Ratios and Marginal Effects • Some notation: • Odds that

  26. Logit Results • Table shows odds ratios, standard errors in parentheses, and marginal effects in brackets

  27. Other Issues • Standard errors are complicated • Be wary of canned programs (like Stata!) which allow calculation of robust variance/covariance matrices. • Interaction terms are also complicated • Odds ratios can be difficult to interpret • Marginal effects are better!

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