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Bivariate Response Models

Bivariate Response Models - Economics 245A Bivariate...

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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 0 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 0 t ° )
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where the scalar X 0 t ° is the single index. The leading case is the linear probability model in which F ( z ) = z so p ( X t ) = X 0 t °; which arises from Y t = X 0 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 0 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 0 : ° 1 = ² ² ² = ° K = 0 can be accurately constructed from the OLSE, as under H 0 the error is homoskedastic, EU 2 t = ° 0 (1 ± ° 0 ) . Regardless of how the model is estimated, the R ± square is not informative. (Draw a graph with the points clustered at 0 and 1 on the y-axis and a straight line attempting to °t them.) Example (Married Women±s Labor Force Participation) In a survey of 753 women, 428 report working more than zero hours. Also, 606 have no young children while 118 have exactly one young child.
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