lecture12 - Marginal eects Suppose our model for the latent...

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Marginal e/ects Suppose our model for the latent variable y is: y = + ±x + u We have studied a number of cases of interest: A. The econometrician observes y = y . Then @E ( y j x ) @x = ± B. Probit model: y = 1 ( y > 0) . The marginal e/ect of interest is: @E ( y j x ) @x = @ Pr( y = 1 j x ) @x = @ Pr( y > 0 j x ) @x = @ Pr( + ±x ) @x = @ + ±x ) @x = ±² ( + ±x ) C. Tobit model: y = + ±x + u y = y & 0 if y > 0 otherwise We may be interested in two marginal e/ects @E ( y j x;y > 0) @x and @E ( y j x ) @x x among "participants"; the second is the e/ect in the population at large. Recall that E ( y j x ) = E ( y j x;y > 0)Pr( y > 0 j x ) + E ( y j x;y = 0)Pr( y = 0 j x ) = E ( y j x;y > 0)Pr( y > 0 j x ) 1
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Now in our case Pr( y > 0 j x + ±x ) E ( y j x;y > 0) = E ( y j x;y > 0) = + ±x + E ( u j u > ( + ±x )) = + ±x + ² ³ + ±x ² ± + ±x ² ± @E ( y j x;y > 0) @x = ± + ² @ h ³ + ±x ² ± = + ±x ² ±i @x = ± 8 < : 1 ³ + ±x ² ± + ±x ² ± 2 4 ² + ±x ² ³ + ³ + ±x ² ± + ±x ² ± 3 5 9 = ; The second marginal e/ect is @E ( y j x ) @x = @E ( y j x;y > 0) @x Pr( y > 0 j x ) + E ( y j x;y > 0) @ Pr( y > 0 j x ) @x = ± 8 < : 1 ³ + ±x ² ± + ±x ² ± 2 4 ² + ±x ² ³ + ³ + ±x ² ± + ±x ² ± 3 5 9 = ; ² + ±x ² ³ + ² 0 @ + ±x ² + ³ + ±x ² ± + ±x ² ± 1 A ± ² ³ ² + ±x ² ³ = ± ±³ + ±x ² ± ³ 2 + ±³ + ±x ² + ± ³ 2 = ± ² + ±x ² ³ the marginal e/ect can be decomposed into two parts: a) @E ( y j x;y> 0) @x Pr( y > 0 j x ) : This is the change in y among participants weighted by the proportion of participants in the population. This is known as change in the intensive margin. b) E ( y j x;y > 0) @ Pr( y> 0 j x ) @x : A change in x changes also the probability of participating, for example shifting some people±s decision to participate. This is known as change in the extensive margin. 2
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Program Evaluation 2 The best use of the Heckman sample selection correction is in the program evaluation literature (see one of the previous lectures). 1 Restating the problem Consider an individual who is faced with the problem of choosing whether to take or not a given "treatment". Assume that d i = 1 if the individual takes the treatment, and 0 otherwise. Associated to each possible alternative there is an outcome, usually an eco- nomic return such as earnings. Assume that earnings in each possible state (0,1) are given by: y 0 i = & 0 + u 0 i (1) y 1 i = 1 + u 1 i (2) respectively in state 0 (no treatment) and 1(treatment). People select into regime 1 if ± 0 + ± 1 z i + " i > 0 . Hence, z i is an "instrument". This is a very simple model in which the outcome is equal to an e/ect
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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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lecture12 - Marginal eects Suppose our model for the latent...

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