Lecture 3 Discrete Choice Models

1 RS Lecture 171 Lecture 3 Discrete Choice ModelsLimited Dependent VariablesDiscrete Dependent Variable Continuous dependent variableTruncated/Censored Regr. ModelsDiscrete Choice Models (DCM)Duration (Hazard)ModelsTruncated,Censored To date we have implicitly assumed that the variable yiis a continuous random variable. But, the CLM does not require this assumption! The dependent variable can have discontinuities. For example, it can be Discrete or follow counts. In these cases, linearity of the conditional expectations is unusual. Different types of discontinuities generate different Models :RS Lecture 17 From Frances and Paap (2001)With limited dependent variables, the conditional mean is rarelylinear. We need to use adjusted Models . Limited Dependent VariablesLimdep: Discrete Choice Models (DCM) We usually study Discrete data that represent a decision, a Choice . Sometimes, there is a single Choice .

2 Then, the data come in binary form with a 1 representing a decision to do something and a 0 being a decision not to do > Single Choice (binary Choice Models ): Binary Data Data: yi= 1 (yes/accept) or 0 (no/reject)- Examples: Trade a stock or not, do an MBA or not, etc. Or we can have several choices. Then, the data may come as 1, 2,.., J, where J represents the number of > Multiple Choice (multinomial Choice Models )Data:yi= 1(opt. 1), 2 (opt. 2), .., J (opt. J) - Examples: CEO candidates, transportation modes, Lecture 17 Limdep: DCM Binary Choice - ExampleFlyGroundLimdep: DCM Mulinomial Choice - ExampleRS Lecture 17 Limdep: Truncated/Censored Models Truncated variables:We only sample from (observe/use) a subset of the population. The variable is observed only beyond a certain threshold level ( truncation point )- store expenditures, labor force participation , income below poverty line.

3 Censored variables:Values in a certain range are all transformed to/grouped into (or reported as) a single hours worked, exchange rates under Central Bank : Censoring is essentially a defect in the sample data. Presumably, if they were not censored, the data would be a representative sample from the population of = Not Healthy1 = HealthyLimdep: Censored Health Satisfaction DataRS Lecture 17 Limdep: Duration/Hazard Models We model the time between two : Time between two trades Time between cash flows withdrawals from a fund Time until a consumer becomes inactive/cancels a subscription Time until a consumer responds to direct mail/ a questionnaire Consumers Maximize Utility. Fundamental Choice Problem: Max U(x1,x2,..) subject to prices and budget constraints A Crucial Result for the Classical Problem: Indirect Utility Function: V = V(p,I) Demand System of Continuous Choices The Integrability Problem: Utility is not revealed by demandsMicroeconomics behind Discrete Choice *( , I) /( , I) / IjjVpxV = ppRS Lecture 17 Theory is silent about Discrete choices Translation to Discrete Choice - Existence of well defined utility indexes: Completeness of rankings- Rationality: Utility maximization- Axioms of revealed preferences Choice and consideration sets: Consumers simplify Choice situations Implication for Choice among a set of Discrete alternatives Commonalities and uniqueness Does this allow us to build Models ?

4 What common elements can be assumed? How can we account for heterogeneity? Revealed choices do not reveal utility, only rankings which are scale for Discrete Choice We will model Discrete Choice . We observe a Discrete variable yiand a set of variables connected with the decision xi, usually called covariates. We want to model the relation between yiand xi. It is common to distinguish between covariates zithat vary by units (individuals or firms), and covariates that vary by Choice (and possibly by individual), wij. Example of zi s: individual characteristics, such as age or education. Example of wij: the cost associated with the Choice , for example the cost of investing in bonds/stocks/cash, or the price of a product. This distinction is important for the interpretation of these Models using utility maximizing Choice behavior. We may put restrictions on the way covariates affect utilities: the characteristics of Choice ishould affect the utility of Choice i, but not the utility of Choice Choice Models (DCM)RS Lecture 17 The modern literature goes back to the work by Daniel McFadden in the seventies and eighties (McFadden 1973, 1981, 1982, 1984).

5 Usual Notation:n= decision makeri,j= Choice optionsy = decision outcomex = explanatory variables/covariates = parameters = error termI[.] = indicator function: equal to 1 if expression within brackets is true, 0 otherwise. Example: I[y=j|x] = 1 if jwas selected (given x)= 0 otherwiseDiscrete Choice Models (DCM) Q: Are the characteristics of the consumers relevant? Predicting behavior- Individual for example, will a person buy the add-on insurance?- Aggregate for example, what proportion of the population will buy the add-on insurance? Analyze changes in behavior when attributes change. For example, how will changes in education change the proportion of who buy the insurance?DCM What Can we Learn from the Data? RS Lecture 17 Application: Health Care Usage (Greene)German Health Care Usage Data, N = 7,293, Varying Numbers of PeriodsData downloaded from Journal of Applied Econometrics Archive.

6 This is an unbalanced panel with 7,293 individuals. This is a large data set. There are altogether 27,326 observations. The number of observations ranges from 1 to 7. (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987). (Downloaded from the JAE Archive)Variables in the file areDOCTOR = 1(Number of doctor visits > 0)HOSPITAL = 1(Number of hospital visits > 0)HSAT = health satisfaction, coded 0 (low) - 10 (high) DOCVIS = number of doctor visits in last three monthsHOSPVIS = number of hospital visits in last calendar yearPUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by add-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net income in German marks / 10000.(4 observations with income=0 were dropped)HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in yearsFEMALE = 1 for female headed household, 0 for maleEDUC = years of educationApplication: Binary Choice Data (Greene)RS Lecture 17Q: Does income affect doctor s visits?

7 What is the effect of age on doctor s visits? Is gender relevant?27,326 Observations 1 to 7 years, panel 7,293 households observed We use the 1994 year => 3,337 household observationsDescriptive Statistics============================== ===========================Variable Mean Minimum Maximum--------+------------------------ ------------------------DOCTOR| .657980 .474456 .000000 | | .444764 .216586 .340000E-01 | .463429 .498735 .000000 : Health Care Usage (Greene)DCM: Setup Choice Set1. Characteristics of the Choice set- Alternatives must be mutually exclusive: No combination of Choice alternatives. For example, no combination of different investments types (bonds, stocks, real estate, etc.). - Choice set must be exhaustive: all relevant alternatives included. If we are considering types of investments, we should include all: bonds; stocks; real estate; hedge funds; exchange rates; commodities, etc.

8 If relevant, we should include international and domestic financial Finite (countable) number of Lecture 17 DCM: Setup RUM2. Random utility maximization (RUM)Assumption: Revealed preference. The decision maker selects the alternative that provides the highest utility. That is,Decision maker nselects Choice iif Uni> Unj j iDecomposition of utility: A deterministic (observed), Vnj, and random (unobserved) part, nj:Unj= Vnj+ nj- The deterministic part, Vnj, is a function of some observed variables, xnj(age, income, sex, price, etc.):Vnj= + 1 Agen+ 2 Incomenj+ 3 Sexn+ 4 Pricenj- The random part, nj, follows a distribution. For example, a : Setup RUM2. Random utility maximization (RUM) We think of an individual s utility as an unobservable variable, with an observable component, V, and an unobservable (tastes?) randomcomponent, . The deterministic part is usually intrinsic linear in the + 1 Agen+ 2 Incomenj+ 3 Sexn+ 4 Pricenj- In this formulation, the parameters, , are the same for all individuals.

9 There is no heterogeneity. This is a useful assumption for estimation. It can be Lecture 172. Random utility maximization (continuation)Probability Model: Since both U s are random, the Choice is random. Then, n selects i over j if:=> Pnj= F(X, ) is a CDF. Vnj- Vnj= h(X, ). h(.) is usually referred as the index function. To evaluate the CDF, F(X, ),f( n) needs to be specified. nnnjnininjnidfijVVIP )()( < = )(Prob)(Prob)(ProbijVVijVVijUUPnjnininjnjnjnininjnini < = +> += >=DCM: Setup - RUM01()[],Pr1iiF h xy == DCM: Setup - RUMh(X, ).RS Lecture 17 To evaluate the integral, f( n) needs to be specified. Many possibilities: Normal: Probit Model, natural for behavior. Logistic: Logit Model, allows thicker tails. Gompertz: Extreme Value Model, asymmetric distribution. We can use non-parametric or semiparametric methods to estimate the CDF F(X, ).

10 These methods impose weaker assumptions than the fully parametric model described above. In general, there is a trade-off: Less assumptions, weaker conclusions, but likely more robust : Setup - RUMDCM: Setup RUM Different f( n) RS Lecture 17 DCM: Setup - RUM Note: Probit? Logit?A one standard deviation change in the argument of a standard Normal distribution function is usually called a Probability Unit or Probitfor short. Probit graph papers have a normal probability scales on one axis. The Normal qualitative Choice model became known as the Probitmodel. The it was transmitted to the Logistic Model (Logit) and the Gompertz Model (Gompit).DCM: Setup - Distributions Many candidates for CDF , Pn(x n ) = F(Zn),: Normal (Probit Model)= (Zn) Logistic (Logit Model)= 1/[1+exp(-Zn)] Gompertz (Gompit Model) = 1 exp[-exp(Zn)] Suppose we have binary (0,1) data.