Estimation and Inference of Discontinuity in Density

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Also, by the change of variables,1h 2 E [g i ()] = Z 1(1; u) K (u) ff (c + uh) exp (a l + b l uh)g du;h u
w.p.a.1. On the other hand, an expansion of (15) around ^; ^(^) = ( 0 ; 0) yieldswhere H =1nhP ni=10 = 1nh0B@g i (~)g i (~) 01+ ~ 0 g i (~)nXi=11 0 00 h 00 0 hg i ( 0 ) + 1nh1CA~; , ~ nXi=111 + ~ 0 g i (~)is a point on the line joining@g i (^)@ 0 H 1 !H (^ 0 ) ^V3^ (^) ; (18)^; ^(^)and ( 0 ; 0), and ^V 3 = 2 is implicitly de…ned. Combining (17) multiplied H 1 from left and (18),0 =01nhP ni=1 g i ( 0 )!+ ^M H (^ 0)^ (^)!; where ^M =!0 ^G0 1; (19)^G 2^V3where^G 1 = 1nhnXi=111 + ^ (^) 0 g i (^)@g i (^)@ 0 H 1 ; ; ^G 2 = 1nhnXi=111 + ~ 0 g i (~)@g i (^)@ 0 H 1 ;which satisfy ^G 1 ; ^Gp2 ! G by a similar argument to Lemma 1 and the consistency of ^. Also note thatp^V 3 ! V . Since G is full column rank and V is positive de…nite (Assumption 2), ^M is invertible w.p.a.1.By solving (19) for p nhH (^ 0 ), we havepnhH (^ 0 ) = G 0 V 1 G 1G 0 V 1 1pnhFrom this and an expansion of1 pnhP ni=1 g i (^) around ^ = 0 ,1pnhnXg i (^) =i=1nXg i ( 0 ) + o p (1) :i=1hI G G 0 V 1 G 1G 0 V 1i 1pnhnXg i ( 0 ) + o p (1) : (20)i=11PCombining (16), (20), p nnh i=1 g i ( 0 ) ! d N (0; V ) (Lemma 1), and 2 ^V 11Lemma 1 and 2 with the consistency of ^), we have^V 11 1 ^V 2 ^V 1p! V 1 (by` (^)d! 0 V 1=2 h I G G 0 V 1 G 1G 0 V 1i 0V1 h I G G 0 V 1 G 1G 0 V 1i V 1=2 = 0 h I A A 0 A 1 A0 i = 2 (1) ;where N (0; I) and A = V1=2 G. Therefore, the conclusion is obtained.17
Page 3 and 4: the empirical likelihood con…denc
Page 7 and 8: whereKlj G = K Xj;hG ( JX1 IkGk=1K
Page 9 and 10: the binned data, the computational
Page 11: type of derivation. Second, the emp
Page 15: AMathematical AppendixHereafter “
Page 19 and 20: Let g i;1 ( 0 ) be the …rst eleme
Page 21 and 22: References[1] Angrist, J. D. and V.
Page 23 and 24: Table 1: Finite-sample biases, stan
Page 25 and 26: Table 3: Finite-sample sizes, quant
Page 27 and 28: Frequencies0.0160.0140.0120.010.008
Page 29: Densities0.015Binning EstimatorLoca

Estimation and Inference of Discontinuity in Density

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