class10 - Discrete Dependent Variables We are now going to...

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Discrete Dependent Variables We are now going to deal with discrete dependent variables. An example is a binary variable y i = & 1 0 if person i works otherwise The objective of regression models involving discrete dependent variables is to understand what variables ( X ) determine whether a given person i works or not (in the example above). Discrete dependent variables can be classi&ed in many possible ways. These classi&cations typically also determine the estima- tion method that is used to estimate the e/ect of regressors on the expected value of the discrete dependent variable given value of the regressors. A useful classi&cation is as follows: y i Categorical Non-categorical Unordered Ordered Sequential 1. Non-categorical d.d.v. These are d.d.v. that have limited support and typically take integer values. Examples are count data (the number of patents, the number of mosquito bites, the number of children in a class, etc.), or duration data (the number of months one is being unemployed, the number of years between two major wars, etc.). 2. Categorical, unordered d.d.v. These are d.d.v. that are categorical (de&ne "categories" or "choices"), but cannot be ordered in any meaningful way. Examples include d.d.v. that de&ne mode of transport to work, y i = 8 > > < > > : 1 2 3 4 if i takes car if i takes bus if i takes helicopter if i takes bike or occupation 1
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y i = 8 > > < > > : 1 2 3 4 if i is a doctor if i is an actor if i is a judge of American Idol if i is a plumber Note: The order does not matter because I could shu› e the values assigned to the four possible modes of transport (i.e., instead of assigning the value y i = 1 to "car", I could assign it to "bike" - the choice is completely arbitrary). Things would be di/erent if I was to "order" the modes of transport by their environmental impact, or their cost, say. In this case I would be dealing with the next type of d.d.v. . 3 . Categorical, ordered d.d.v. These are d.d.v. that are categorical and can be ordered in some meaningful way. Examples include cases in which you are given data on income in classes, y i = 8 > > < > > : 1 2 3 4 if i has earnings between 0 and 10k if i has earnings between 10 and 100k if i has earnings between 100 and 1m if i has earnings above 1m Note: here the values assigned to the d.d.v. are assumed to pick up the order with which the variable moves. 3 . Categorical, sequential d.d.v. These are d.d.v. that are categorical and follow a sequence. Examples include classifying individuals by their level of schooling, y i = 8 > > < > > : 1 2 3 4 if i has not completed HS if i has completed HS but not college if i has completed college but not a PhD if i has completed a PhD 1 Linear Probability Model This is the simplest possible model that can be used. In practice, it is just OLS (or GLS) applied to a case in which the dependent variable is a binary variable y i = f 0 ; 1 g . The model is y = X& + U for N individuals. The vector y is just a bunch of zeros or ones. We are making the usual assumption E ( U j X ) = 0 (we&ll check it later). The LPM estimates are as usual b & LPM = ( X 0 X ) & 1 X 0 y Note that since y is a {0,1} variable, E ( y ) = Pr( y = 1) and E ( y j
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This note was uploaded on 07/28/2011 for the course ECON 102C taught by Professor Pistaferri,l during the Spring '11 term at Stanford.

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class10 - Discrete Dependent Variables We are now going to...

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