5.5 Yitzhaki Weights as a Version of Theil Weights (1950)
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1 5.5 Yitzhaki Weights as a Version of Theil Weights (195) Another intuition behind the Yitzhaki weights (Yitzhaki, 1989). Sample Size (finite sample) Take Case () =() (propensity score is the instrument). 15
2 Recall the Theil (195) formula for OLS = + ( ) = OLS is weighted average of all pairwise OLS slopes. P 1 ˆ = ( )( ) P1 ( ) Form pairwise slopes (Theil) = 1 [ 6= ] ˆ = X 1 = ( ) 16
3 Weights are obviously positive on each if 6= Yitzhaki orders the and produces a pairwise representation of OLS 1 (neglect ties) Concomitants 1 17
4 Slopes for ordered data = [ 6= 1 ] Substitute into formula for OLS and collect terms on the X μ ˆ = = =1 ( ) is proportion of bigger than Obviously weights are positive. They place more weight on the center of the distribution of the. 18
5 5.6 Derivation of the IV Weight = = () () ( () ) ( () ) = 1 + (1 ) Under assumptions ( 1 ) (we keep implicit) =( 1 ) + 19
6 Let ( ) =. ( () ) = [( 1 ) + ] () ª = ( 1 ) () = ( 1 ) () () Pr ( () ) 1 = 1 ( 1 ) = ( ) 11
7 = = ( () ) Z 1 Z Z Z [ 1 ( 1 ) ( )] () ( 1 ) ( ) 1 Z 1 Z ( 1 ) () ( {z } ) ( ) ( )=1 1 Reverse order of integrals = Z 1 ( 1 ) Z 1 () ( ) = () =1 Pr ( =1 ) 111
8 Look at shape of weight: ( () ) =( () ) 11
9 Why does the weight integrate to 1? Z 1 Z 1 () ( ()) () Integrate by parts R = R = + () ( ()) () Z 1 Z 1 = ( () ) () () ( ) weights integrate to 1 since the denominator is just this
10 5.7 Deriving the General IV Weights on MTE We consider instrumental variables conditional on = using a general function of as an instrument. To simplify the notation, we keep the conditioning on implicit. Let () be any function of such that (()) 6=. Consider the population analog of the IV estimator, ( () ) ( () ). 114
11 First consider the numerator of this expression, ( () ) = ([ () ( ())] ) = (( () ( ())) ( + ( 1 ))) = (( () ( ())) ( 1 )) where the second equality comes from substituting in the definition of and the third equality follows from conditional independence assumption (A). 115
12 Define () () (()). Then 116
13 ( () ) ³ = () 1[ ()] ( 1 ) ³ = () 1[ ()] ( 1 ) ³ = () 1[ ()] ( 1 ) μ = () 1[ ()] ( 1 ) Z ( () () )Pr(() ) = ( 1 = ) {z } MTE Z = MTE ( )( () () )Pr(() ) 117
14 where the first equality follows from plugging in the model for ; the second equality follows from the law of iterated expectations with the inside expectation conditional on ( );the third equality follows from conditional independence assumption (A-); the fourth equality follows from Fubini s Theorem and the law of iterated expectations with the inside expectation conditional on ( = ); (and implicitly on ); this allows to reverse the order of integration in a multiple integral; the fifth equality follows from the normalization that is distributed unit uniform conditional on ; and the final equality follows from plugging in the definition of MTE.Nextconsider the denominator of the IV estimand. By iterated expectations ( () )= ( () ()). 118
15 Thus, the population analog of the IV estimator is given by Z MTE ( ) ( ) (5.1) where ( )= ( () () )Pr( () ) ( () ()) (5.11) where by assumption ( () ()) 6=. 119
16 If () and () are continuous random variables then an interpretation of the weight can be derived from (5.11) by noting that Z Z 1 ( ( ())) ( ) Z Z 1 = ( ( ())) () ( () =) 1
17 Write Z 1 ( () =) = 1 ( () =) = () ( () =) where ( () =) is the probability of ( () ) given () = (and implicitly = ). Likewise, Pr[ () ()] = ( ()). Using these results, we may write the weight as ( )= () () () (). 11
18 For fixed and evaluation points, ( ()) is a function of the random variable (). The numerator of the preceding expression is the covariance between () and the probability that the random variable () is greater than the evaluation point conditional on (). 1
19 ( ()) is a function of the random variables and (). The denominator of the above expression is the covariance between () and the probability that the random variable () is greater than the random variable conditional on (). Thus, it is clear that if the covariance between () and the conditional probability that ( () ) given () is positive for all, then the weights are positive. The conditioning is trivially satisfied if () = () so the weights are positive and IV estimates a gross treatment eect. 13
20 5.8 Understanding the Structure of the IV Weights IV ( )= Z Z 1 ( ( ())) ( ) ( () ) (5.1) Theweightsarealwayspositiveif () is monotonic in the scalar. In this case () and () havethesamedistribution and ( ) collapses to a single distribution. 14
21 Weighting function for () = () is maximal for = ( () = ) and minimal for = 1. For () =(), weights are symmetric with respect to is the density of is symmetric. IV weights MTE more where density of () is higher. 15
22 If the instrument is () (so () = ()) then the weights are everywhere non-negative because from (5.9) ( () () ) ( ()) In this case the density of ( () ()) collapses to the density of (). Observethattheweightscanbeconstructedfromdata on ( ) Data on ( () ()) pairs and ( () ) pairs (for each value) are all that is required. Recall Tables 1A and 1B: weights on MTE ( ) generating IV are dierent from the weights on TT ( ) 16
23 Discrete Instruments () Support of the distribution of () contains a finite number of values 1 Support of the instrument () is also discrete, taking distinct values. (() () ) is constant in for within any ( +1 ) interval, and Pr( () ) is constant in for within any ( +1 ) interval. Let denote the weight on the LATE for the interval ( +1 ). 17
24 Under monotonicity, or uniformity Z IV = ( 1 = ) IV ( ) (5.13) = = 1 X =1 Z +1 ( 1 = ) 1 X LATE ( +1 ) =1 1 ( +1 ) In general, this formula is true, under index suciency even if monotonicity is violated. It s certainly true under (A-1) (A-5). True where ( +1 ) is replaced by the Wald estimator. 18
25 Let be the smallest value of the support of (). = P =1 P ( ()) ( ( )) ( () ) ( +1 ) (5.14) 19
26 Generalizes the expression presented by Imbens and Angrist (1994) and Yitzhaki (1989, 1996) Their analysis of the case of vector only considers the case where () and () are perfectly dependent because () is a monotonic function of (). More generally, the weights can be positive or negative for any but they must sum to 1 over the. 13
27 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.1 w(x); =, =5, / =.1 w(x); =, =1, / =.1 w(x); =, =, / = x, g(x) =.1*x, X ~ (, ); 77
28 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.1718 w(x); =, =5, / =.18 w(x); =, =1, / = w(x); =, =, / = x, g(x) =.1*x +.1*exp(x), X ~ (, ); 78
29 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.7183 w(x); =, =5, / =.118 w(x); =, =1, / = w(x); =, =, / = x, g(x) =.1*exp(x), X ~ (, ); 79
30 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.386 w(x); =, =5, / =1. w(x); =, =1, / = w(x); =, =, / = x, g(x) =.1*x +.1*exp(x), X ~ (, ); 8
31 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.86 w(x); =, =5, / =.917 w(x); =, =1, / = w(x); =, =, / = x, g(x) =.1*x +.1*exp(x), X ~ (, ); 81
32 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.1 w(x); =, =5, / =.1 w(x); =, =1, / =.1 w(x); =, =, / = x, g(x) =.1*x, X ~ (, ); 8
33 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.1 w(x); =, =5, / =.1 w(x); =, =1, / =.1 w(x); =, =, / = x, g(x) =.1*x +.1*x, X ~ (, ); 83
34 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / = w(x); =, =5, / = w(x); =, =1, / = w(x); =, =, / = x, g(x) =.1*x, X ~ (, ); 84
35 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.14 w(x); =, =5, / =.14 w(x); =, =1, / =.14 w(x); =, =, / = x, g(x) =.1*x +.1*x, X ~ (, ); 85
36 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.4 w(x); =, =5, / =.4 w(x); =, =1, / =.4 w(x); =, =, / = x, g(x) =.1*x, X ~ (, ); 86
37 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.1 w(x); =, =5, / =.1 w(x); =, =1, / =.1 w(x); =, =, / = x, g(x) =.1*x, X ~ (, ); 87
38 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / =.1 w(x); =, =5, / =.1 w(x); =, =1, / =.1 w(x); =, =, / = x, g(x) =.1*x +.1*cos(x), X ~ (, ); 88
39 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / = w(x); =, =5, / = w(x); =, =1, / = w(x); =, =, / = x, g(x) =.1*cos(x), X ~ (, ); 89
40 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / = w(x); =, =5, / =.9536 w(x); =, =1, / = w(x); =, =, / = x, g(x) =.1*x +.1*cos(x), X ~ (, ); 9
41 Yitzhaki, s Weights d[g(x)]/dx w(x); =, =, / = w(x); =, =5, / = w(x); =, =1, / = w(x); =, =, / = e x, g(x) =.1*x +.1*cos(x), X ~ (, ); 91
42 Yitzhaki, s Weights 1.8 d[g(x)]/dx; b =.5, c = w(x); X ~ U(,1), X,Y / X = x, g(x) = b*x + c*x, X ~ U(,1). 9
43 Yitzhaki, s Weights 1.8 d[g(x)]/dx; b =.5, c =.5 w(x); X ~ U(,1), X,Y / X = x, g(x) = b*x + c*x, X ~ U(,1). 93
44 Yitzhaki, s Weights 1.8 d[g(x)]/dx; b =, c =.5 w(x); X ~ U(,1), X,Y / X = x, g(x) = b*x + c*x, X ~ U(,1). 94
45 Yitzhaki, s Weights 5 4 d[g(x)]/dx; b =.5, c =.5 w(x); X ~ U(,1), X,Y / X = x, g(x) = b*x + c*cos( *x), X ~ U(,1). 95
46 Yitzhaki, s Weights 5 4 d[g(x)]/dx; b =, c =.5 w(x); X ~ U(,1), X,Y / X = x, g(x) = b*x + c*cos( *x), X ~ U(,1). 96
47 Beta PDF: BetaPDF(x,,) = 1 = = 5 = x of BetaPDF(x,,), =5 97
48 Yitzhaki, s Weights d[g(x)]/dx; b =.5, c = w(x); =1, / =.5 w(x); =, / =.5 w(x); =5, / =.5 w(x); =1, / = x, g(x) = b*x + c*x, X ~ BetaPDF(x,, ); =5 98
49 Yitzhaki, s Weights d[g(x)]/dx; b =.5, c =.5 w(x); =1, / =1.556 w(x); =, / = w(x); =5, / =1.4 w(x); =1, / = x, g(x) = b*x + c*x, X ~ BetaPDF(x,, ); =5 99
50 Yitzhaki, s Weights d[g(x)]/dx; b =, c =.5 w(x); =1, / =.7559 w(x); =, / =.6673 w(x); =5, / =.515 w(x); =1, / = x, g(x) = b*x + c*x, X ~ BetaPDF(x,, ); =5 1
51 Yitzhaki, s Weights d[g(x)]/dx; b =, c =.5 w(x); =1, / = w(x); =, / =.871 w(x); =5, / =1.15 w(x); =1, / = x, g(x) = b*x + c*log(x), X ~ BetaPDF(x,, ); =5 11
52 Yitzhaki, s Weights d[g(x)]/dx; b =.5, c =.5 w(x); =1, / =1.48 w(x); =, / =1.375 w(x); =5, / =1.655 w(x); =1, / = x, g(x) = b*x + c*log(x), X ~ BetaPDF(x,, ); =5 1
53 Yitzhaki, s Weights d[g(x)]/dx; b =.5, c =.5 w(x); =1, / =.5864 w(x); =, / =.988 w(x); =5, / = w(x); =1, / = x, g(x) = b*x + c*x, X ~ BetaPDF(x,, ); =5 13
54 Yitzhaki, s Weights d[g(x)]/dx; b =, c =.5 w(x); =1, / =.834 w(x); =, / = w(x); =5, / = w(x); =1, / = x, g(x) = b*x + c*x, X ~ BetaPDF(x,, ); =5 14
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