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Unformatted text preview: Economics 245A Bivariate Response Models Overview In &rstyear econometrics, you learned results for models in which the depen dent variable is restricted in some way. The &rst result is that not all restrictions require special treatment. For a model with a positive dependent variable, such as wages or income, the log transform is su cient. The second result concerned mod els with binary dependent variables, which we concentrate on today. Recall that a binary dependent variable captures the presence or absence of a nonnumeric quantity. For this reason, binary dependent variable models fall within the class of qualitative (or discrete) dependent variable (response) models. As an example, in a study of work commute choice the dependent variable is Y t = & 1 if individual t drives to work if individual t takes public transit : In binary response models, interest is primarily in the response probability p ( X ) & P ( Y = 1 j X ) = P ( Y = 1 j X 1 ;:::;X K ) ; for various values of X . For a continuous regressor X j , the partial e/ect of X j on the response probability is simply @P ( Y =1 j X ) @X j . When multiplied by & X j (for small & X j ), the partial e/ect yields the approximate change in P ( Y = 1 j X ) when X j increases by & X j holding all other regressors constant. For a discrete regressor X K , the partial e/ect is P ( Y = 1 j x 1 ;:::;x K & 1 ;X K = 1) P ( Y = 1 j x 1 ;:::;x K & 1 ;X K = 0) : Binary response models inherit the features of Bernoulli random variables P ( Y = 0 j X ) = 1 p ( X ) E ( Y j X ) = p ( X ) V ( Y j X ) = p ( X ) [1 p ( X )] : Single Index Models  Linear Probability Model The most widely used binary response models are those for which the regressors enter through a single index p ( X t ) = F ( X t & ) where the scalar X t & is the single index. The leading case is the linear probability model in which F ( z ) = z so p ( X t ) = X t &; which arises from Y t = X t & + U t . The coe cient & k captures the change in p ( X t ) for each one unit change in X k holding the other regressors constant. One can estimate the model via OLS, but the presence of heteroskedasticity in V ( Y j X ) indicates that WLS is e cient. To construct WLS, one must have ^ p ( X t ) = X t b for all t . Because b and X t are not restricted, ^ p may lie outside [0 ; 1] . If the estimated probability lies outside [0 ; 1] for any point in the sample, then WLS cannot be used as the estimated variance for the observation ^ p (1 & ^ p ) is negative. Note that if the model is saturated , in that all interactions between regressor values are included, then the &tted probabilities correspond to cell averages (recall the 140A lecture on functional form) and so lie in [0 ; 1] . The test statistic for H : & 1 = = & K = 0 can be accurately constructed from the OLSE, as under H the error is homoskedastic, EU 2 t = & (1 & & ) . Regardless of how the model is estimated, the R & square is not informative. (Draw a graph with the points...
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.
 Fall '08
 Staff
 Econometrics

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