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Unformatted text preview: Economics 140A Qualitative Dependent Variables With each of the classic assumptions covered, we turn our attention to exten- sions of the classic linear regression model. Our &rst extension is to data in which the dependent variable is limited. Dependent variables that are limited in some way are common in economics, but not all require special treatment. We have seen examples with wages, income or consumption as the dependent variables - all of these must be positive. As these strictly positive variables take numerous values, we found the log transform to be su cient. Yet not all restrictions on the dependent variable can be handled so easily. If we model individual choice, the optimal behavior of individuals often results in a sizable fraction of the population at a corner solution. For example, a sizable fraction of working age adults do not work outside the home, so the distribution of hours worked has a sizable pile up at zero. If we &t a linear conditional mean, we will likely predict negative hours worked for some individuals. The log transform used for wages will not work, as the log of zero is unde&ned. Another issues arises with sample selection. It may well be the case that E ( Y j X ) is linear, but nonrandom sampling requires more detailed inference. Finally, a host of other data issues may arise: linear conditional mean functions that switch over regimes, data recorded as counts or analysis of durations between events. As we will see, even if only a &nite number of values are possible, a linear model for E ( Y j X ) may still be appropriate. While all these issues may arise, we focus on perhaps the most common restric- tion in which the dependent variable is qualitative in nature and so takes discrete values. For this reason, such models are also termed discrete dependent variable models or (less frequently) dummy dependent variable models. As we recall from our discussion of qualitative regressors, qualitative variables capture the presence or absence of some non-numeric quantity. For example, in studying home ownership the dependent variable is often Y t = & 1 if household t owns their home otherwise : Many qualitative variables take more than two values. For example, in studies of employment dynamics the dependent variable can take three values Y t = 8 < : 1 if individual t is employed if individual t is unemployed but seeking employment & 1 if individual t is not in the labor force (i.e. not seeking employment) : We focus attention on qualitative dependent variables that take only two values and for ease set these values to 0 and 1. In binary response models, interest is primarily in 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 regressorholding all other regressors constant....
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This note was uploaded on 09/04/2011 for the course ECON 140a taught by Professor Staff during the Fall '08 term at UCSB.
- Fall '08