lecture11

# lecture11 - Other Discrete Dependent Variable Models Our...

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Unformatted text preview: Other Discrete Dependent Variable Models Our model is still y &amp; i = &amp; + x i + u i where y &amp; i is a latent dependent variable (for example, net utility associated to making a certain choice), and the only thing we observe is the index: y i = &amp; 1 if y &amp; i &amp; if y &amp; i &lt; 1 Logit model In the logit model, instead of making the assumption that u is standard normally distributed (which would give the probit model), one makes the assumption that it follows a logistic distribution. A logistic distribution has c.d.f. given by F ( z ) = e z 1 + e z Note that this is a symmetric distribution because F ( z ) = 1 F ( z ) . The p.d.f. is given by f ( z ) = @F ( z ) @z = e z (1 + e z ) 2 = F ( z ) F ( z ) = F ( z )[1 F ( z )] Hence Pr( y i = 1) = Pr( y &amp; i &amp; 0) = Pr( u i &amp; ( &amp; + x i )) = Pr( u i ( &amp; + x i )) = F ( &amp; + x i ) = e &amp; + x i 1 + e &amp; + x i This model is also estimated by ML. The individual contribution to the likelihood function is l i = e &amp; + x i 1 + e &amp; + x i y i 1 1 + e &amp; + x i (1 y i ) and the log-likelihood is ln L = n X i =1 ln l i 1 The FOC is @ ln L @&amp; = 0 ! n X i =1 &amp; y i &amp; e &amp; + x i 1 + e &amp; + x i x i = 0 and again a numerical method must be used to obtain the estimates and &amp; . Logit and probit will often give very similar results as the shape of their c.d.f.&amp;s is very similar except at the tails. Since E ( y i j x i ) = e &amp; + x i 1+ e &amp; + x i , the logit marginal eect is @E ( y i j x i ) @x i = @F ( + &amp;x i ) @x i = &amp;f ( + &amp;x i ) = &amp; e &amp; + x i (1 + e &amp; + x i ) 2 2 Ordered Probit Suppose that instead of having a binary index, we have a multiple index for an ordered discrete dependent variable. For example, suppose that y &amp; i is the utility from working, and the only thing we observe are categories of hours of work y i = 8 &gt; &gt; &gt; &gt; &lt; &gt; &gt; &gt; &gt; : 1 2 3 ::: k if y &amp; i 1 if 1 &lt; y &amp; i 2 if 2 &lt; y &amp; i 3 ::: if y &amp; i &gt; k 1 with 1 2 3 ::: k 1 being &quot;thresholds&quot;. Here y i = 1 could corre- spond to 0 hours, y i = 2 to 1 to 1-10 hours, and so forth. Figure 1 gives an illustration (for k = 6 ). 1 2 3 4 5 y* y i =1 y i =2 y i =3 y i =4 y i =5 y i =6 1 2 3 4 5 y* 1 2 3 4 5 y* y i =1 y i =2 y i =3 y i =4 y i =5 y i =6 Note that the assumption of normality about u i is all we need to estimate the model with ML. In particular: 2 Pr( y i = j j x i ) = Pr( &amp; j &amp; 1 &amp; y i &amp; &amp; j j x i ) = Pr( &amp; j &amp; 1 ( + x i ) &amp; y i &amp; &amp; j ( + x i )) = &amp;( &amp; j ( + x i )) &amp;( &amp; j &amp; 1 ( + x i )) with &amp; = 1 and &amp; k = 1 . It follows that the individual contribution to the log likelihood is ln l i = k X j =1 1 f y i = j g ln[&amp;( &amp; j ( + x i )) &amp;( &amp; j &amp; 1 ( + x i ))] where 1 f y i = j g is an indicator function, i.e., a variable that equals 1 if the statement in brackets is true, and 0 otherwise. Note that cannot be identi&amp;ed....
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## lecture11 - Other Discrete Dependent Variable Models Our...

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