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Unformatted text preview: Econ 103 UCLA, Fall 2010 Problem Set 5 Solutions by Matthew Baird Part 1: True or False and explain briefly why. 1. In the binary dependent variable model, a predicted value of 0.6 means that the most likely value the dependent variable will take on is 60 percent. FALSE . The dependent variable cannot take on the value of 0.6, as it is binary (it only takes on values of zero or one). The predicted value of 0.6 is the predicted probability that it takes on a value of one, i.e. E ( y | X ) = Pr ( Y = 1 | X ) . 2. The linear probability model is the application of the linear multiple regression model to a binary dependent variable. TRUE . This is by true by definition of the linear probability model (it can be for the linear single regression model as well). 3. In the linear probability model, the interpretation of the slope coefficient is the change in probability that Y=1 associated with a unit change in X, holding other regressors constant. TRUE . Consider the linear regression model Y = β + β 1 X 1 + β 2 X 2 + ··· + β k X k + u Then, Pr ( Y = 1 | X 1 + 1 ,X 2 , ··· X k ) = β + β 1 ( X 1 + 1) + β 2 X 2 + ··· + β k X k = β + β 1 X 1 + β 1 + β 2 X 2 + + ··· β k X k Pr ( Y = 1 | X 1 ,X 2 , ··· X k ) = β + β 1 X 1 + β 2 X 2 + + ··· β k X k From this, Pr ( Y = 1 | X 1 +1 ,X 2 , ··· ,X k )- Pr ( Y = 1 | X 1 ,X 2 , ··· ,X k ) = β 1 . Therefore, the slope coefficient is the change in probability that Y = 1 associated with a unit change in X , holding other regressors constant. 4. For the measure of fit in your regression model with a binary dependent variable, you can meaningfully use the regression R 2 . FALSE . Consider: an R 2 of 1, or a perfect fit, is given when the predicted points are the same as the actual points. However, this is impossible in the case of the binary dependent variable, as the actual points are only zero or one, while the predicted points take on all intermediate values. This suggests the underlying problem, that the closeness of fit cannot be meaningfully derived from the R 2 because of the separation of the actual points and the predicted probabilities. Alternative measures must be considered, such as the pseudo-R 2 or the fraction predicted correctly. 1 Econ 103 UCLA, Fall 2010 5. The probit model forces the predicted values to lie between 0 and 1. TRUE . Recall that the predicted values are given by Pr ( Y = 1 | X ) = Φ( β + β 1 X 1 + ··· + β k X k ) . Φ( · ) is the cumulative distribution function (cdf) for a standard normal random variable. All cdfs are, by definition, bounded by zero and one. Therefore, so are the predicted values. 6. Nonlinear least squares estimators in general are not efficient....
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- Spring '07
- Regression Analysis, Linear Probability Model