Binary Response Lecture 22 Lecture 22

Today’s plan Three models: Linear probability model Probit model Logit model L22.xls provides an example of a linear probability model and a logit model Lecture 22

Discrete choice variable Defining variables: Yi = 1 if individual : Yi = 0 if individual: The discrete choice variable Yi is a function of individual characteristics: Yi = a + bXi + ei Takes BART Buys a car Joins a union Does not take BART Does not buy a car Does not join a union Lecture 22

Graphical representation X = years of labor market experience Y = 1 [if person joins union] = 0 [if person doesn’t join union] X Y 1 Observed data with OLS regression line Lecture 22

Linear probability model The OLS regression line in the previous slide is called the linear probability model predicting the probability that an individual will join a union given their years of labor market experience Using the linear probability model, we estimate the equation: using we can predict the probability Lecture 22

Linear probability model (2) Problems with the linear probability model 1) Predicted probabilities don’t necessarily lie within the 0 to 1 range 2) We get a very specific form of heteroskedasticity errors for this model are note: values are along the continuous OLS line, but Yi values jump between 0 and 1 - this creates large variation in errors 3) Errors are non-normal We can use the linear probability model as a first guess can be used for start values in a maximum likelihood problem Lecture 22

McFadden’s Contribution Suggestion: curve that runs strictly between 0 and 1 and tails off at the boundaries like so: Y 1 Lecture 22

McFadden’s Contribution Recall the probability distribution function and cumulative distribution function for a standard normal: 1 PDF CDF Lecture 22

Probit model For the standard normal, we have the probit model using the PDF The density function for the normal is: where Z = a + bX For the probit model, we want to find Lecture 22

Probit model (2) The probit model imposes the distributional form of the CDF in order to estimate a and b The values have to be estimated as part of the maximum likelihood procedure Lecture 22

Logit model The logit model uses the logistic distribution Density: Cumulative: 1 Standard normal F(Z) Logistic G(Z) Lecture 22

Maximum likelihood Alternative estimation that assumes you know the form of the population Using maximum likelihood, we will be specifying the model as part of the distribution Lecture 22

Maximum likelihood (2) For example: Bernoulli distribution where: (with a parameter ) We have an outcome 1 1 1 0 0 0 0 1 0 0 The probability expression is: We pick a sample of Y1….Yn Lecture 22

Maximum likelihood (3) Probability of getting observed Yi is based on the form we’ve assumed: If we multiply across the observed sample: Given we think that an outcome of one occurs r times: Lecture 22

Maximum likelihood (3) If we take logs, we get This is the log-likelihood We can differentiate this and obtain a solution for Lecture 22

L(a, b) = Si [Yi log(Gi) + (1 - Yi) log(1 - Gi)] Maximum likelihood (4) In a more complex example, the logit model gives Instead of looking for estimates of we are looking for estimates of a and b Think of G(Zi) as : we get a log-likelihood L(a, b) = Si [Yi log(Gi) + (1 - Yi) log(1 - Gi)] solve for a and b Lecture 22

Example Data on union membership and years of labor market experience (L22.xls) To build the maximum likelihood form, we can think of: intercept: a coefficient on experience : b There are three columns Predicted value Z Estimated probability Estimated likelihood as given by the model The Solver from the Tools menu calculates estimates of a and b Lecture 22

Example (2) How the solver works: Defining a and b using start values Choose start values of a and b equal to zero Define our model: Z = a + bX Define the predictive possibilities: Define the log-likelihood and sum it Can use Solver to change the values on a and b Lecture 22

Comparing parameters How do we compare parameters across these models? The linear probability form is: Y = a + bX where Recall the graphs associated with each model Consequently This is the same for the probit and logit forms Lecture 22

L22.xls example Predicting the linear probability model: If we wanted to predict the probability given 20 years of experience, we’d have: For the logit form: use logit distribution: logit estimated equation is: Lecture 22

L22.xls example (2) At 20 years of experience: Thus the slope at 20 years of experience is: 0.234 x 0.06 = 0.014 Lecture 22