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Unformatted text preview: Economics 245A Bivariate Response Models Overview In &rst-year 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 non-numeric 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