Qualitative and Limited Dependent Variable Models ECON 6002 Econometrics Memorial University of Newfoundland Adapted from Vera Tabakova’s notes

 16.1 Models with Binary Dependent Variables  16.2 The Logit Model for Binary Choice  16.3 Multinomial Logit  16.4 Conditional Logit  16.5 Ordered Choice Models  16.6 Models for Count Data  16.7 Limited Dependent Variables

When the dependent variable in a regression model is a count of the number of occurrences of an event, the outcome variable is y = 0, 1, 2, 3, … These numbers are actual counts, and thus different from the ordinal numbers of the previous section. Examples include:  The number of trips to a physician a person makes during a year.  The number of fishing trips taken by a person during the previous year.  The number of children in a household.  The number of automobile accidents at a particular intersection during a month.  The number of televisions in a household.  The number of alcoholic drinks a college student takes in a week.

If Y is a Poisson random variable, then its probability function is This choice defines the Poisson regression model for count data. “rate” Also equal To the variance

If we observe 3 individuals: one faces one event, the other two two events each:

So now you can calculate the predicted probability of a certain number y of events

You may prefer to express this marginal effect as a %:

If there is a dummy Involved, be careful, remember Which would be identical to the effect of a dummy In the log-linear model we saw under OLS

Extensions: overdispersion Under a plain Poisson the mean of the count is assumed to be equal to the average (equidispersion) This will often not hold Real life data are often overdispersed For example: a few women will have many affairs and many women will have few a few travelers will make many trips to a park and many will make few etc.

Extensions: overdispersion use "C:\bbbECONOMETRICS\Rober\GRAD\GROSMORNE.dta", clear

Extensions: negative binomial Under a plain Poisson the mean of the count is assumed to be equal to the average (equidispersion) The Poisson will inflate your t-ratios in this case, making you think that your model works better than it actually does  Or use a Negative Binomial model instead (nbreg) or even a Generalised Negative Binomial (gnbreg), which will allow you to model the overdispersion parameter as a function of covariates of our choice You can also test for overdispersion, to test whether the problem is significant

Extensions: negative binomial sum visits Variable | Obs Mean Std. Dev. Min Max visits |

Extensions: negative binomial

Extensions: excess zeros Often the numbers of zeros in the sample cannot be accommodated properly by a Poisson or Negative Binomial model They would underpredict them too There is said to be an “excess zeros” problem You can then use hurdle models or zero inflated or zero augmented models to accommodate the extra zeros

Extensions: excess zeros Often the numbers of zeros in the sample cannot be accommodated properly by a Poisson or Negative Binomial model They would underpredict them too nbvargr Is a very useful command

Extensions: excess zeros You can then use hurdle models or zero inflated or zero augmented models to accommodate the extra zeros They will also allow you to have a different process driving the value of the strictly positive count and whether the value is zero or strictly positive EXAMPLES: Number of extramarital affairs versus gender Number of children before marriage versus religiosity In the continuous case, we have similar models (e.g. Cragg’s Model) and an example is that of size of Insurance Claims from fires versus the age of the building

Extensions: excess zeros You can then use hurdle models or zero inflated or zero augmented models to accommodate the extra zeros Hurdle Models A hurdle model is a modified count model in which there are two processes, one generating the zeros and one generating the positive values. The two models are not constrained to be the same. In the hurdle model a binomial probability model governs the binary outcome of whether a count variable has a zero or a positive value. If the value is positive, the "hurdle is crossed," and the conditional distribution of the positive values is governed by a zero-truncated count model. Example: smokers versus non-smokers, if you are a smoker you will smoke!

Extensions: excess zeros Hurdle Models In Stata Joseph Hilbe’s downloadable ado HPLOGIT will work, although it does not allow for two different sets of variables, just two different sets of coefficients Example: smokers versus non-smokers, if you are a smoker you will smoke!

Extensions: excess zeros You can then use hurdle models or zero inflated or zero augmented models to accommodate the extra zeros Zero-inflated models (initially suggested by D. Lambert) attempt to account for excess zeros in a subtly different way. In this model there are two kinds of zeros, "true zeros" and excess zeros. Zero-inflated models estimate also two equations, one for the count model and one for the excess zero's. The key difference is that the count model allows zeros now. It is not a truncated count model, but allows for “corner solutions” Example: meat eaters (who sometime just did not eat meat that week) versus vegetarians who never ever do

Extensions: excess zeros webuse fish We want to model how many fish are being caught by fishermen at a state park. Visitors are asked how long they stayed, how many people were in the group, were there children in the group and how many fish were caught. Some visitors do not fish at all, but there is no data on whether a person fished or not. Some visitors who did fish did not catch any fish (and admitted it ) so there are excess zeros in the data because of the people that did not fish.

Extensions: excess zeros. histogram count, discrete freq Lots of zeros!

Extensions: excess zeros Vuong test

Extensions: excess zeros Vuong test

Extensions: truncation Count data can be truncated too (usually at zero) So ztp and ztnb can accommodate that Example: you interview visitors at the recreational site, so they all made at least that one trip In the continuous case we would have to use the truncreg command

Extensions: truncation This model works much better and showcases the bias in the previous estimates: Smaller now estimated Consumer Surplus

Extensions: truncation Now accounting for overdispersion This model works much better and showcases the bias in the previous estimates:

Extensions: truncation and endogenous stratification Example: you interview visitors at the recreational site, so they all made at least that one trip You interview patients at the doctors’ office about how often they visit the doctor You ask people in George St. how often the go to George St… Then you are oversampling “frequent visitors” and biasing your estimates, perhaps substantially

Extensions: truncation and endogenous stratification Then you are oversampling “frequent visitors” and biasing your estimates, perhaps substantially It turns out to be supereasy to deal with a Truncated and Endogenously Stratified Poisson Model (as shown by Shaw, 1988): Simply run a plain Poisson on “Count-1” and that will work (In STATA: poisson on the corrected count) It is more complex if there is overdispersion though 

Extensions: truncation and endogenous stratification Supereasy to deal with a Truncated and Endogenously Stratified Poisson Model Much smaller now estimated Consumer Surplus

Extensions: truncation and endogenous stratification Endogenously Stratified Negative Binomial Model (as shown by Shaw, 1988; Englin and Shonkwiler, 1995): Even after accounting for overdispersion, CS estimate is relatively low

Extensions: truncation and endogenous stratification How do we calculate the pseudo-R2 for this model???

Extensions: truncation and endogenous stratification GNBSTRAT will also allow you to model the overdispersion parameter in this case, just as gnbreg did for the plain case

NOTE: what is the exposure Count models often need to deal with the fact that the counts may be measured over different observation periods, which might be of different length (in terms of time or some other relevant dimension) For example, the number of accidents are recorded for 50 different intersections. However, the number of vehicles that pass through the intersections can vary greatly. Five accidents for 30,000 vehicles is very different from five accidents for 1,500 vehicles. Count models account for these differences by including the log of the exposure variable in model with coefficient constrained to be one. The use of exposure is often superior to analyzing rates as response variables as such, because it makes use of the correct probability distributions

 Censored Data Figure 16.3 Histogram of Wife’s Hours of Work in 1975

Having censored data means that a substantial fraction of the observations on the dependent variable take a limit value. The regression function is no longer given by (16.30). The least squares estimators of the regression parameters obtained by running a regression of y on x are biased and inconsistent—least squares estimation fails.

Having censored data means that a substantial fraction of the observations on the dependent variable take a limit value. The regression function is no longer given by (16.30). The least squares estimators of the regression parameters obtained by running a regression of y on x are biased and inconsistent—least squares estimation fails.

Censoring versus Truncation  With truncation, we only observe the value of the regressors when the dependent variable takes a certain value (usually a positive one instead of zero)  With censoring we observe in principle the value of the regressors for everyone, but not the value of the dependent variable for those whose dependent variable takes a value beyond the limit

We give the parameters the specific values and Assume

 Create N = 200 random values of x i that are spread evenly (or uniformly) over the interval [0, 20]. These we will keep fixed in further simulations.  Obtain N = 200 random values e i from a normal distribution with mean 0 and variance 16.  Create N = 200 values of the latent variable.  Obtain N = 200 values of the observed y i using

Figure 16.4 Uncensored Sample Data and Regression Function

Figure 16.5 Censored Sample Data, and Latent Regression Function and Least Squares Fitted Line

The maximum likelihood procedure is called Tobit in honor of James Tobin, winner of the 1981 Nobel Prize in Economics, who first studied this model. The probit probability that y i = 0 is:

The maximum likelihood estimator is consistent and asymptotically normal, with a known covariance matrix. Using the artificial data the fitted values are:

Because the cdf values are positive, the sign of the coefficient does tell the direction of the marginal effect, just not its magnitude. If β 2 > 0, as x increases the cdf function approaches 1, and the slope of the regression function approaches that of the latent variable model.

Figure 16.6 Censored Sample Data, and Regression Functions for Observed and Positive y values

 Problem: our sample is not a random sample. The data we observe are “selected” by a systematic process for which we do not account.  Solution: a technique called Heckit, named after its developer, Nobel Prize winning econometrician James Heckman.

 The econometric model describing the situation is composed of two equations. The first, is the selection equation that determines whether the variable of interest is observed.

 The second equation is the linear model of interest. It is

 The estimated “Inverse Mills Ratio” is  The estimating equation is

 The maximum likelihood estimated wage equation is The standard errors based on the full information maximum likelihood procedure are smaller than those yielded by the two-step estimation method.

Slide Principles of Econometrics, 3rd Edition  binary choice models  censored data  conditional logit  count data models  feasible generalized least squares  Heckit  identification problem  independence of irrelevant alternatives (IIA)  index models  individual and alternative specific variables  individual specific variables  latent variables  likelihood function  limited dependent variables  linear probability model  logistic random variable  logit  log-likelihood function  marginal effect  maximum likelihood estimation  multinomial choice models  multinomial logit  odds ratio  ordered choice models  ordered probit  ordinal variables  Poisson random variable  Poisson regression model  probit  selection bias  tobit model  truncated data

Further models  Survival analysis (time-to-event data analysis)  Multivariate probit (biprobit, triprobit, mvprobit)

References  Hoffmann, 2004 for all topics  Long, S. and J. Freese for all topics  Cameron and Trivedi’s book for count data  Agresti, A. (2001) Categorical Data Analysis (2nd ed). New York: Wiley.