MODELS OF QUALITATIVE CHOICE by Bambang Juanda

 Models in which the dependent variable involves two ore more qualitative choices.  Valuable for the analysis of survey data in which the behavioral response are qualitative; one uses either the subway, the bus or the automobile; one is either in the labor force or out of the labor force; etc.  Binary-choice models assume that individuals are faced with choice between two alternatives and that their choice depends on their characteristics.

 Although it is reasonable to expect a direct relationship between their characteristics and their choices, we cannot be sure how each and avery individual make a choice.  One purpose of a qualitative choice model is to determine the probability that an individual with a given set of attributes will make one choice rather than the alternative.  We assume that the probability of an individual making a given choice is a linear function of individual attributes.

Overview Continuous Categorical Linear Regression Analysis - ResponseAnalysis -Linear Probability Model -Probit Model

1. Linear Probability Model Y i =  +  X i + ε i (10.1) Where X i = value of attribute, e.g. Income for ith individual, Y i = 1, if first option is chosen (buy a car) 0, if second option is chosen (not buy a car). ε i = independently distributed random variable with 0 mean. To interpret eq. (10.1), we take the expected value of each dependent variable observation Y i : E(Y i ) =  +  X i (10.2) Since Y i can take on only two values (1 dan 0), we can describe the probability distribution of Y by letting: P i = P(Y i =1) dan 1-P i = P(Y i =0), Then, E(Y i ) = 1 (P i ) + 0 (1-P i ) = P i (10.3) Model (10.1)  probability that an ith individual will buy a car, given information about her income. The slope of line measures the effect on the probability of buying a car of a unit change in income.

Estimated Linear Probability Model  +  X i, jika 0<(  +  X i )<1 P i = 1, jika (  +  X i ) ≥ 1 0, jika (  +  X i ) ≤ 0 (10.4)

Probability Distribution of ε i YiYi εiεi Probability 1 1-  -  X i PiPi 0 -  -  X i 1 - P i

E(ε i ) = (1-  -  X i ) P i + (-  -  X i ) (1-P i ) = 0 then P i =  +  X i (1-P i ) = 1 -  -  X i Variance of error term Thus, Variable Y distribute according to Bernouli probability distribution.  error term is heteroscedastic

Difficulties in linear probability model  need to transform the original model in such a way that predictions will lie in the (0;1) interval for all X. One of this transformation forms is the cumulative probability function), F. [1] The resulting probability distribution might be represented as: [1] P i = F(  +  X i ) = F(Z i ) While numerous alternative cumulative probabilitiy functions are possible, we shall consider only two, the normal and the logistic cumulative probabilitiy functions. [1][1] cumulative probability function is F(x i ) = Prob(X≤x i )

2. Probit Model Pi = F(  +  Xi) = F(Zi) Assume there is a theoretical continuous an index Z i, which is determined by an explanatory variable X. Thus, we can write: Z i =  +  X i Assume that Z is a normally distributed random variable, so that the probability that Z is less than (or equal to) Z i can be computed from the cummulative normal probability function. The standardized cumulative normal function is written:: (10.9) Where s is a random variable which is normally distributed with mean zero and unit variance. By construction, the variable P i will lie in the (0;1) interval. P i represents the probability that individual with income X i make a choice (buy a car). Since this probability is measured by the area under the standard normal curve from -  to Z i, the event (buy a car) will be more likely to occur the larger the value of the index Z i. To obtain an estimate of index Z i, we apply the inverse of the cumulative normal function to eq.(10.9) : Z i = F -1 (P i ) =  +  X i

The Relation of Index Z and Cumulative Normal Probability Function Distribution ZF(Z)Z

Linear Probability Model vs Probit Model Linear Model

 While the probit model is more appealing than the linear probability model, it involves nonlinear maximum likelihood estimation.  The theoritical justification for employing the probit model is somewhat limited.  We shall consider a somewhat more appealing model specification, the logit model

It is based on the cumulative logistic probability function and is specified as. Logistic Regression Model (Logit Model ) Simple Logit Model : The Logistic distribution curve is said to be S shaped, so that its interpretation is logic. 0 ≤ E(Y/X) ≤ 1 Interpretation: probability that an ith individual will make a choice (e.g. buy a car), given information about her income X i

Logit Transformation Probability of an event (p i ) is transformed by the form: iindex of all observations (1, 2,..,n). p i probability of an event (e.g. buy a car) occurs for an ith observation. logis natural logarithm (basic number e). Function g(x) is Linear in Parameter, and -~ ≤ g(x) ≤ ~, so that it can be estimated by OLS

Assumption (variable X has an interval scale) P i Predictor (X) Logit Transformation Predictor (X)

Interpretation of Logit Model Coefficients For binary independent variable, e.g. sex (X=1, X=0) X=1X=0 Y=1 Y=0 P(1) : Probability of buying a car for male consumer P(0) : Probability of buying a car for female consumer 11 Total 11

Interpretation of Coefficient  1 = g(X+1) – g(X) For X binary:  1 = g(1) – g(0) Odds Ratio: “How much more likely to buy for male consumer compared to for female consumer” Interpretation of relatif probability approach P(1)/P(0) is applicable when P(x) is small For X continuous, exp(  1) : How much more likely to buy when X increase 1 unit Association measure

Properties of the Odds Ratio -0.5 No Association =x+1=x (1-  ) 100% Confidence Interval of Odds Ratio: exp(c  ± z  /2 c s  ) ^ ^ Note:

Multiple Logistic Regression

Illustration of a model to study the effect of sex (X 1 ), age (X 2 ), and income (X 2 ) on buying a car. logit (p i ) = For an independent continuous variable X, occasionally 1 unit is too small or large to consider  Estimation for the change of “c” unit g(x+c) – g(x) = c  1 Odds Ratio-nya:

Testing a Model with p Independent Variables Testing for the significance of the Model: H 0 :  1 =  2 =…=  p =0 H 1 : ada  j ≠0  Likelihood Ratio Test Statistics (G) ~ Testing for a coefficient partially: H 0 :  j =0 H 1 :  j ≠0  WaldTest Statistics (W) ~ Z  2 (p)

Adjusted Odds Ratio ontrolling for

Types of Logistic Regression Response Variable Yes No Binary Two Categories Type of Logistic Regression Binary Nominal Ordinal Three or More Categories