Large Sample Properties of Matching Estimators for Average Treatment Effects by Alberto Abadie & Guido Imbens

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1 Addtonal Proofs for: Large Sample Propertes of atchng stmators for Average Treatment ffects by Alberto Abade & Gudo Imbens Remnder of Proof of Lemma : To get the result for U m U m, notce that U m U m B m m, where B m uu fz + u Pr X z u m Pr X z u m du R Boundedness of X mples that B m converges unformly Transformng to polar coordnates agan leads to B m where, as before br log r S r ωω fz + rωλ S dω r r m s fz + sωλ S dω ds dr S e br ãr dr, r s S fz + sωλ S dω ds, m s S fz + sωλ S dω ds and ãr r + ωω fz + rωλ S dω S r m s S fz + sωλ S dω ds r s S fz + sωλ S dω m ds That s, ãr qrpr, qr r + cr, and, as before, pr gr m, where ωω fz + rωλ S dω S cr r, s S fz + sωλ S dω ds Clearly, r gr r s S fz + sωλ S dω ds s S fz + sωλ S dω ds lm r qrr + lm cr r fz λ S dωi S

2 Hence, lm r ãrr m+ lm prr m lm qrr + r r m fz λ S dω S fz λ S dωi S m fz λ S dω I S Therefore, the condtons of Lemma A hold for α m +, β m a fz λ S dω I S b fz S λ S dω Applyng Lemma A, we get m + B m Γ a b m+/ + o m+/ m+/ / m + fz Γ λ S dω S I m+/ + o m + Γ fz Hence, usng the fact that lm we have that m /m!, m U m U m m B m π / Γ + / / I m+/ + o m+/ m+/ / m + π / Γ fz m! Γ + I + o / / Usng the same technques as for the frst two moments, U m 3 C m m, where C m br log e br ār dr, r s S fz + sωλ S dω ds,

3 and ār r + fz + rωλ S dω S r m s S fz + sωλ S dω ds r s S fz + sωλ S dω m, ds and C m converges unformly Let ār qrpr, qr r + cr, and pr gr m, where fz + rωλ S dω S cr r, s S fz + sωλ S dω ds ow, Hence, r gr r s S fz + sωλ S dω ds s S fz + sωλ S dω lm r qrr + lm cr fz λ S dω r S lm r ārr m+ ds lm prr m lm qrr + r r m fz λ S dω fz λ S dω S S m fz λ S dω S Therefore, the condtons of Lemma A hold for α m + 3, β m a fz λ S dω S b fz S λ S dω Applyng Lemma A, we get m + 3 C m Γ a b m+3/ fz + o m+3/ m+3/ 3/ m + 3 Γ λ S dω S + o m+3/ m + 3 π / 3/ Γ fz Γ + / + o m+3/ Hence, usng the fact that lm m /m!, m 3 m+3/ m+3/

4 we have that U m 3 m C m m + 3 π / 3/ Γ fz m! Γ + / + o 3/ m + 3 U m 3 π / 3/ Γ fz m! Γ + / + o 3/ Therefore U m 3 O 3/ 3/ 3/ Proof of Theorem 5: Frst, assume wthout loss of generalty that the support of X s, If the support of X s x, x, we can always wor wth the transform X x/x x and obtan the same estmator The results wll not change because they are functons of ratos of denstes, so the Jacobans of the transformaton cancel Pr j J X j, W, W j, ι W Pr j j m X j, W, W j, ι W m Pr j j m X j, W, W j, ι W Pr j j m X x, X j, W, W j, ι W f x dx m m + m m + m { x x x } m f xdx { x x x } m f xdx f x dx f x dx x m x m f x dx m f x dx u m +v f x dx where u x, v x otce that m f x dx u u f x dx x f x dx x u +v m f x dx f x dx m m m f x dx f u du f x dx f x dx f u du m f + v dv, au exp bu du, 4

5 where f x dx u au f u u m f x dx m, and bu log u f x dx It s easy to see that lm u au/u m f f m, and lm u bu/u f Applyng Lemma A we obtan: ow, because, lm m f x dx u m /m!, m u f x dx m f u du m! f f m + o m we obtan: m m f x dx u u f x dx m f u du f + o f Smlarly, m m +v f x dx +v f x dx m f + v dv f + o f Therefore: Pr j j m X j, W, W j, ι f W + o f, and Pr j J X j, W, W j, ι f W + o f 5

6 ow, let us calculate the jont probablty of two matches: Pr j J, j J X j, W W, W j, ι W Pr j J, j J X j, W W, W j, ι W, X X x Pr j J, j J X j, W W, W j, ι W, X x, X x f x f x / dx dx Let P, j Pr j J, j J X j, W W, W j, ι W, X x, X x ow, let us calculate P, j for x x Three cases: x x : otce that Let P, j P m m x m x f x dx m f m x dx x m f x dx m x f x dx m ow let u x Then, where f x dx x f x dx m x f x dx m x f x dx m m x f x dx f x dx m m f x dx f x dx f x dx f x dx x x x m m f x dx f x dx u u f x dx f u du a u exp b u du, u a u f u b u log f x dx u u f x dx f x dx u m, f x dx u 6 m

7 It s easy to see that and a u lm u u m f f m, b u lm f u u As a result, Lemma A holds wth α m+, β, a f f m, and b f : m! m f x dx x f f m f m+ f m! f m+ + o ow, because, m+ m+ f x dx x + o m+ m x f x dx f x dx lm m m+ /m! m, we obtan: P f m f m + f f + o + o x x : otce that Let P, j P m m! f x dx m!m! m m +! x m x x f x dx f x dx x m m m + m! f x dx m m m!m! m m +! x m x m x x m + j f x dx f x dx f x f x dx dx x 7

8 Let u x, u x otce that for x x, sup u, u Also, let S ++ {ω ω, ω R : ω, ω, ω } m m x f x dx f x dx x where x x S ++ a r +u u f x dx m m + f x f x dx dx m +u f x dx f x dx u f x dx m m + f + u f u du du m mn{ /ω,/ω } m +rω r f x dx f x dx rω +rω rω b r log f x dx m m + S ++ rf + rω f rω +rω rω f + rω f rω dr λ S dω m mn{ /ω,/ω } a r exp b r dr λ S dω, f x dx rω +rω rω It can be easly seen that for ω > and ω > a r lm r r m+m f ω f m b r lm f ω + ω r r Therefore, Lemma A holds wth α m + m, β, m +rω f x dx f x dx f x dx ω f m, m m+m, a f f m+m ω m ω m, and b f ω + ω mn{ /ω,/ω } +rω rω m +rω r f x dx f x dx rω f x dx m m + m + m! f + rω f rω dr f ω m ω m f ω + ω m+m m+m m + o m+m 8

9 Therefore, mn{ /ω,/ω } +rω rω! m m +! r f x dx m m + rω m +rω f x dx f x dx f + rω f rω dr m f ω m ω m m + m! + o m+m f ω + ω In addton, t can be shown that last ntegral s bounded by a constant To see that, notce that rω f x dx m frω m, In addton, for r /ω we have +rω f x dx m frω m, and +rω f x dx m m + +rω f x dx m m + rω rω frω + ω m m + Let s fω + ω r mn{ /ω,/ω } +rω rω! m m +! r f x dx f f m+m ω m ω m mn{ /ω,/ω } f f m+m f m+m m m + rω m +rω f x dx f x dx f + rω f rω dr m! m m +! rm+m fω + ω r m m + dr ω m ω m ω + ω m+m! m m +! fω+ω mn{ /ω,/ω } s m+m s m m+ ds It s easy to see that mn{ /ω, /ω } ω + ω If /ω /ω, then ω ω ω so ω ω + ω An analogous argument apples for the case /ω /ω In addton, because the support of X has length one, t has to be the case that f 9

10 Therefore, the upper lmt of the ntegral s smaller than one We obtan mn{ /ω,/ω } +rω rω f m+m f f m+m! m m +! r f x dx m m + ω m ω m ω + ω m+m rω m +rω f x dx f x dx f + rω f rω dr! m m +! m s m+m s m m+ ds The ntegral n the rght hand sde s a Beta functon wth parameters m +m and m m +, whch s equal to m + m! m m +!/ +! Therefore, mn{ /ω,/ω } +rω rω f m+m f f m+m! m m +! r f x dx m m + rω m +rω f x dx f x dx f + rω f rω dr ω m ω m ω + ω m m+m + m! whch s ntegrable over ω, ω S ++, because ω m ω m S ++ f m+m f f m+m ω + ω λ m+m S dω m! m!, m + m see, eg, Gradshteyn and Ryzh,, eq m ω m ω m ω + ω m m+m + m!, Then, applyng Lebesgue s Domnated Convergence Theorem, and usng agan the result n last equaton we obtan: P f f + o 3 x x : Ths case s analogous to the case wth x x and contrbutes the same amount to the ntegral: Hence, P 3 + f f + o Pr j J, j J X j, W W, W j, ι W P + P + P 3 + f f + o

11 As a result the frst two moments of K are gven by: K W, X x, ι W f x + o f x K W, X x, ι W f x f x Let p PrW + K σw X o f x f x + K σw X W PrW + + K σw X W PrW We focus on the frst term The second term can be calculated analogously Frst, notce that / p/ p almost surely From the proof of Lemma 3, we obtan: q n K q W, X x, ι W C Sq, n, n for some C > and all q Usng Chernoff s Inequalty, t s easy to show that all postve moments of / exst, condtonal of X x and W and are bounded unformly n arov s Inequalty mples that / n s asymptotcally unformly ntegrable, whch, n turn, mples convergence of moments: K W, X x f x p f x p + o K W, X x f x p f x p + + f x f x p p + o Therefore, + K W, X x + K + K W, X x + f x p f x p + f x p f x p + + f x p f x p + o + p f x + p f x p f x p f x + p f x p f x + o Because ex pf x/pf x + pf x, and thus ex/ex pf x/pf x, ths can be wrtten as + K W, X x ex + ex ex + ex ex + o

12 Because the bound on the condtonal moments of K, and because the moments of / n condtonal on X x and W do not depend on x, and are bounded unformly n, we obtan: + K σw X W ex + ex ex + ex ex σx W + o After some algebra t can be shown that: ex + ex ex + ex ex σx W p σ X + ex ex ex σx The analogous result holds condtonng on W, so the result follows Although t s not necessary for the proof of Theorem 5, t can be seen that V converges to ts expectaton Before provng that result we gve a couple of prelmnary lemmas As before, wthout loss of generalty assume that the support of X s, Defne f xdx f W, A P f xdx f W, A Let X,, for,, be the covarates for the control observatons, wth P, and K, the correspondng versons of P and K Also defne X, to be the order statstcs so that X, < X,+, and let A be the correspondng catchment areas: A X, + X, /, X, + X,+ / for, and P, the correspondng probabltes that s, based on the orderng of the covarates, not based on the orderng of the probabltes themselves: X, +X,+ / P, f xdx f xdx Defne A X, +X, / P, f X, f X, F X,+ F X, / Lemma A6: For all δ >, and all fxed K and maxx,+k X, o p +δ, max P, P, o p +δ max P, o p +δ,

13 Proof of Lemma A6: Frst, consder part Because F X,+ F X, f x X,+K X, for some x X,, X,+K, t follows that max X,+K X, sup x f x max F X,+K F X, Wth the support of X compact and the densty bounded away from zero the frst factor s bounded By the fact that the dstrbuton functon of a contnuous random varable has a unform dstrbuton, and by the fact that the order statstcs of unform random varables can be wrtten as ratos of sums of d unt exponentals the second factor has the same dstrbuton as max +K,, K m / ɛ m m ɛ m, K for d unt exponental ɛ m Hence t wll be suffcent to show that max,, K m ɛ +m/ j ɛ j o p +δ To show ths we frst show that K / K max ɛ +m ɛ j ɛ +m / + o p +δ, A6 m and second that max K m j m ɛ +m / + o p +δ A7 To show that A6 holds, consder K / δ max ɛ +m ɛ j m j K ɛ +m / + δ max m + / j ɛ j/ δ max ɛ K ɛ +m + m j ɛ j A8 The second factor n A8, δ max ɛ s o p because all moments of ɛ m exsts because t s unt exponental The frst factor n A8 converges to zero, so A8 s o p and A6 holds To show that A7 holds, consder Pr δ max Pr K ɛ +m / + > C m δ max K ɛ +m > C m K Pr δ ɛ +m > C m Pr ɛ m C δ m 3

14 Pc > /δ Then the rght hand sde s equal to Pr ɛ m C δ m By Chebyshev s nequalty ths can be bounded from above by m ɛ m C δ ɛ m C δ, m whch converges to zero because fnshes part of the Lemma ext, consder part Because max m ɛ m P, P, max P, P,, t s suffcent to show that max P, P, o p +δ By the trangle nequalty s fnte gven that the ɛ are d unt exponental Ths δ max P, P, δ max X, +X,+ / X, +X, / f xdx f X, X,+ X, / A9 + δ max Consder the frst term, A9: δ max f X, X,+ X, / f X, f X, F X,+ F X, / X, +X,+ / X, +X, / δ max δ max f xdx f X, X,+ X, / X,+ X, / max X,+ X, max δ max X,+ X, sup f x x sup x X,,X,+ X,+ X, sup x f x f X, f x A The last factor s bounded Because for all δ > we have δ max X,+ X, op, t follows that for δ > δ / δ max X,+ X, δ δ δ max X,+ X, op 4

15 Hence A9 s o p To show that the second term, A, s o p t s suffcent to show that δ max X,+ X, f X, F X,+ F X, o p, because f x s bounded By a mean value theorem t follows that X,+ X, f x F X,+ F X, for some x X,, X,+ So, for ths value of x, δ max X,+ X, f X, F X,+ F X, δ max X,+ X, f x f X, δ max X,+ X, f X, + f x x X, f X, δ/ max X,+ X, δ/ x X f, x f X, δ/ max X,+ X, δ/ x X, f sup x x,x f x Because x X,, X,+ and X,+ X, o p δ/, t follows that δ/ max X,+ X, δ/ and x X, are op Because the last factor s bounded the product s o p and thus A s o p Ths fnshed part of the Lemma Fnally, consder part of the Lemma By part of the Lemma t s suffcent to show that P, o p δ Because f x and f x are bounded and bounded away from zero t s suffcent to show that max F X,+ F X, o p δ Ths follows by the same argument as n part of the Lemma Lemma A7: Suppose that g s contnuously dfferentable on the nterval, If ɛ,, ɛ unt-exponental, then for fxed, as, / / g ɛ j ɛ l ɛ +j ɛ l p gxdx j l and 4 / g ɛ j j l ɛ l j + j + l ɛ +j / l ɛ l p + gxdx Proof: We show that for arbtrary fxed ntegers K, K, and K 3 mnk, K we have K / / p g ɛ j ɛ +K3 ɛ l gxdx K j l ɛ l l are d A A A3 Applyng to each of the terms n A then gves the desred result We prove that A3 holds for K K K 3 The dfference wth terms wth other values of K, K, and K 3 s of order o p By the trangle nequalty we have / / g ɛ j ɛ l ɛ ɛ l gxdx j l l 5

16 / / / g ɛ j ɛ l ɛ ɛ l g / + ɛ ɛ l j l l l / ɛ + g / + ɛ ɛ l g / + + l + g/ + ɛ g/ g/ + gxdx + We wll show for each of the terms A4-A7 that they are of order o p Frst note that max,, ɛ + + o p j A4 A5 A6 A7 A8 Ths follows because max,, + ɛ + j max,, + ɛ The expectaton of ɛ s zero, wth second moment equal to µ and fourth moment fnte Then defne Z + j ɛ The fourth moment of Z s bounded by C / 4 for some fnte C Then, for < α < /4, Prmax Z C α PrZ C α j PrZ 4 C 4 α Z 4 /C 4 4α C / 4 4α /C 4 C 3+4α 4 /C 4 ext, note that / max,, ɛ ɛ l j l ɛ j / + o p j A9 Ths follows from / max,, ɛ ɛ l j + l ɛ j / + max + j ɛ l ɛ l/ + o p j ɛ l ɛ l/ + j 6

17 Combned these two results mply that / max,, ɛ ɛ l / + o p j l A3 ow consder A4 / g ɛ j j max x max x l ɛ l g x max g x max / / ɛ ɛ l g / + ɛ ɛ l l l / ɛ j ɛ l / + j l / ɛ ɛ l l / ɛ j ɛ l / + o p j ext, consder A5 / ɛ g / + ɛ ɛ l g / l l g / + ɛ l ɛ l/ + o p ext, consder A6 Because ɛ, t follows that the expectaton of ths term s zero Its varance s g/ / + max x, gx / + whch converges to zero, mplyng that ths term s o p Fnally, consder A7 By contnuty of g, ths non-stochastc term converges to zero Ths fnshes the proof for part The same argument shows that 4 / g ɛ j j l ɛ l j + ɛ +j / l ɛ l 4 g / j + ɛ +j / The expected value for j + ɛ +j s 4 + Usng the exstence of fourth moments one can then show that 4 g / j + ɛ +j / 4 g / 4 + / p, whch n turn mples the second part of the Lemma Lemma A8: p W σx K W σ X P + P P p 7

18 Proof of Lemma A8: We wll show that W σx K P P P σ X, K, P, P, P, p We frst prove ths for the sngle match case wth Wth, condtonal on,, and X W X,, W X the -vector K, wth th element K, has a multnomal dstrbuton wth parameters P,,, P, and The moment generatng functon of the frst components of the vector K s t,, t exp We need the followng moments: µ, K, X P,, K t µ, K, X P, + P, P,, µ 4, K, 4 X P, + 7 P, P, expt + P, +4 P, P,, 4 µ,,,j K, K, j X P, P,j + P, P,j + P, P,j + 3 P, P,j, where the last expectaton s for j Then σx, K, P, + P, P, σx, σx,j K, µ K, j µ j σx 4, µ 4, + σx, σx,j µ,,,j µ, µ,j j σ 4 X, P, + 7 P, +4 P, P, 4 + σ X, σx,j P, P,j P, P,j j 8

19 + 4 + P, P,j P, P,j Usng the fact that for all δ >, max P, o p +δ, all sums n ths expresson can be shown to be convergng to zero n probablty For example, choosng δ < / σ 4 X, 3 P, 4 sup σx 4 x 3 max P, o p 4+4δ o p ow consder the case wth general The margnal dstrbuton of K, remans bnomal wth parameters and P,, and so the moments µ,, µ,, and µ 4, are as before The dfference s n the moment µ,,,j There are two possbltes Frst, A A j In that case µ,,,j s as before Second, A A j In that case µ,,,j maxµ 4,, µ 4,j Frst we shall show that that out of the pars, j the number of pars wth A A j s less than Consder the catchment area for unt, A X, + X, /, X, + X,+ / Ths overlaps only wth the catchment areas for unts + to +, a total of unts Hence the total number of pars, j wth overlap n the catchment areas s less than or equal to For the unts wth overlappng catchment areas the correlaton between K, and K, j s hgher than for unts wth non-overlappng catchment areas Hence we can bound the second cross moment for such unts from above by µ 4, µ 4,j Hence the dfference wth the expresson for s bounded by max µ 4, Ths converges to zero usng the expresson for µ 4, gven above Lemma A9: W σx P + P P p σx f X f X W + + σ X f X f X W Proof of Lemma A9: To prove ths result we show frst that and second that W P P p σx fx f X W, A3 W P p p + σ X the combnaton of whch gves the desred result For A3 consder: W σx P σx, P, 9 f X f X W A3

20 Ths can be bounded n absolute value by sup x σx max P, By Lemma A6 max P, o p +δ for any δ >, so max P, o p +δ, and sup x σx max P, o p +δ for all δ > Choose δ < /, so that sup x σx max P, o p and thus σ X, P, o p Hence n order to show that A3 holds t remans to show that We can wrte W σx p P σx fx f X W A33 W σx P σx, P, σx, P, σx, P, + o p, σx, fx, f X, F X,+ F X, + o p where the second to last equalty follows from Lemma A6 Wth X d wth cdf F x, the vector F X,, F X has the same dstrbuton as the vector ɛ / j ɛ j,, ɛ / j ɛ j, wth the ɛ j d unt exponental Hence to show A33 t suffces to show, for d unt exponental ɛ j, that σf ff j ɛ j / ɛ l ɛ j/ l ɛ l f F j ɛ j/ l ɛ l ɛ +j / ɛ l j l p σx fx f X W j By Lemma A7, for g contnuously dfferentable on, and d unt-exponental ɛ we have / / g ɛ j ɛ l ɛ +j ɛ l p gzdz j l j Wth gz σf z f F z/f F z, ths equals gzdz σf ff z z f F z dz σz f z f z df z σx fx f X W Thus A33 holds, and therefore A3 For A3 we have: l l W σx P σx, P, σx, P, + o p σf ɛ j / ɛ l j l f F j ɛ j/ f F j ɛ j/ l ɛ l l ɛ l m + ɛ +m j ɛ +o p j

21 g F ɛ j / ɛ l m + ɛ +m j ɛ + o p, j j l wth gz σ zf z /f z Then usng part of Lemma A7 shows that A3 holds Proof of Theorem 5: Condtonal on X and W the varance of the smple matchng estmator wth matches s V + + K σw X W σ X + W σ X + W K W K σ X + σ X + K W σ X K W σ X Frst we shall show that ths converges to V p σx W + p σx p f X p f X W σ +p X fx f X + σ X + f X p f X p W +p σx W + p σx p f X p f X W σ + p X fx f X + σ X + f X p f X p W We wll loo at convergence of one of the terms n V to the correspondng term n V Specfcally, we wll show that K W σ X p p σ X The others follow by the same argument By Lemma A8 fx f X + σ X + f X f X p p W A34 W σx K P P P p By Lemma A9 W σx P + P P p σx p f X p f X W + p + p,

22 so that W σx P + P P p p σx p f X p f X W + + p p The fnal step conssts of showng that V s equal to σ X + ex ex ex σ + X + ex ex ex σx σx Ths follows from smple algebra For example, collectng the terms wth σx and no factor / n V : p σx W + p σx p f X p f X W +p σx f X p f X p W p σx p f X + p f X + p f X p f X W p σx + ex ex + p ex ex W p σx + ex ex + ex ex W σ p X ex W σ X ex Remander of proof of Lemma A5: Let µ 4 be a bound to the fourth centered condtonal moments of ε Because µ 4 <, t s easy to show that the varance of the frst term s o The varance of the second term multpled tmes J 4 s: K q J j ε l j σ W X lj J K q j + J K q K t q t> j K q J µ 4 + ε l j σ W X lj ε l j σ W X lj J j max K q J L µ 4 J µ 4 K q + J L µ 4 ε l jt σ W X ljt max K q

23 Usng Bonferron s Inequalty: max K q max K 4q n Pr K 4q > n n Pr max K 4q > n K 4q, whch s unformly bounded for all Because the frst moment of a random varable s bounded by the square-root of the second moment we obtan that / max K q s unformly bounded for all, and the second term on the rght hand sde of equaton A s o p Let σ be a unform bound for the condtonal varance of ε, and σ 4 σ The varance of the thrd term on the rght hand sde of equaton A dvded by 4/J 4 s: K q + t> J j h>j ε ljε lh J K q K t q JJ σ4 j h>j ε ljε lh J K q J j h>j K q + J LJ σ 4 j h>j ε ljtε lh t ε ljε lh max K q o The last nequalty holds because wthn the same treatment group, each observaton s used as a match at most J L tmes; and because there are J possble combnatons of that observaton and the other J matches The varance of the fourth term on the rght hand sde of equaton A dvded by 4/J s: K q ε + J j ε lj K q K t q ε t> J j ε lj K q ε ε t J σ4 K q + J j ε ljt J j 4 J L σ ε lj max K q o 3

24 The varance of the ffth term on the rght hand sde of equaton A dvded by 4/J s: K q ε + J µ W X µ W X lj j K q ε J µ W X µ W X lj j K q K t q ε t> J µ W X µ W X lj ε t j J µ Wt X t µ Wt X ljt j σ J K q µ W X µ W X lj o The varance of the sxth term on the rght hand sde of equaton A dvded by 4/J 4 s: K q + J J ε lj j j J K q ε ljt j j J σ + j j J µ W X µ W X lj ε lj J µ W X µ W X lj j J K q K t q t> j J µ Wt X t µ Wt X ljt K q J L σ max ε lj j J µ W X µ W X lj J µ W X µ W X lj j max K q max J µ W X µ W X lj o As a result, the varance of each term on the rght hand sde of equaton A s o, so equaton A9 holds Ths result, along wth the result n equaton A8 guarantees that the result of the lemma holds Here we also prove the second part of the lemma The analyss from the begnnng of ths proof to equaton A7 apples here wthout change Recall also from the proof of Lemma 3 that for any q, / K q W s unformly bounded As a result, we obtan: W K q σ W X X, W σw X max σ W X X, W σw X W j K q W o p A35 4

25 To obtan the second result of the lemma, t s left to be proven that for q : otce that: J + W K q σ W X X, W σ W X o p A36 J + J W W W K q J K q ε K q σ W X X, W σ W X J j J j ε l j σ W X lj + J ε lj + J J W W K q ε K q J j J W W K q K q ε σw X J j h>j ε ljε lh µ W X µ W X lj J ε lj j j µ W X µ W X lj A37 The means of the terms on the rght-hand sde of equaton A37 are zero It s left to be shown that the varances are o The varance of the frst term on the rght hand sde of equaton A37 s K q ε σw X K q ε σw X W µ 4 The varance of the second term multpled tmes J 4 s: W K q J j J K q W j + K q J µ 4 W JJ L µ 4 JJ L µ 4 ε l j σ X lj W t,t> K q W W W ε l j σ X lj K t q J j ε l j σ X lj + JJ L µ 4 K q W J W, K K q max W t K t q max W K t q W, K JJ L µ 4 j o p ε l jt σ X ljt K q max K t q W t + o + o max K t q + o, W 5

26 where A s the ndcator functon of the set A that s A x f x A, zero otherwse Bonferron s Inequalty: max K q max W K 4q W n n Pr max K 4q > n W Pr K 4q > n W K 4q W, Usng whch s unformly bounded Because the frst moment of a random varable s bounded by the square-root of the second moment we obtan that / max W K q s unformly bounded, and the second term on the rght hand sde of equaton A37 s o p The varance of the thrd term on the rght hand sde of equaton A37 dvded by 4/J 4 s: W + JJ σ4 K q W K q J j h>j JJ J L σ 4 JJ J L σ 4 ε ljε lh W t,t> J K t q K q W W j h>j J K q W ε ljε lh J j h>j + JJ J L σ 4, K K q max W t K t q max K q, K W W JJ J L σ 4 j h>j ε ljtε lh t W + o + o ε ljε lh K q max W t K t q max K q + o o W The varance of the fourth term on the rght hand sde of equaton A37 dvded by 4/J s: W + J σ4 K q ε W J j K q ε lj W t,t> K t q ε J j W ε lj K q ε ε t K q W + J 4 L σ J j W ε ljt J j ε lj K q max W t K q o 6

27 The varance of the ffth term on the rght hand sde of equaton A37 dvded by 4/J s: W K q ε W J µ W X µ W X lj j K q ε J µ W X µ W X lj j + K q K t q ε W t> σ J µ W X µ W X lj ε t j W J µ Wt X t µ Wt X ljt j J K q µ W X µ W X lj o The varance of the sxth term on the rght hand sde of equaton A37 dvded by 4/J 4 s: W + J K q W W J ε lj j j J K q ε ljt j j J σ K q j j J µ W X µ W X lj ε lj J µ W X µ W X lj j J K t q W t,t> j J µ Wt X t µ Wt X ljt W + JJ L σ W t ε lj j J K q max µ Wt X t µ Wt X ljt W j J µ W X µ W X lj J K q max K t q max µ Wt X t µ Wt X W t W ljt o t As a result, the varance of each term on the rght hand sde of equaton A37 s o, so equaton A36 holds Ths result, along wth the result n equaton A35 guarantees that the result of the lemma holds j Remnder of proof of Theorem 8: For the varance of τ sm,t W Y Ŷ τ sm,t W W we obtan: Y Ŷ τ t τ sm,t τ t Y Ŷ τ t + op A38 7

28 In addton, Y Ŷ τ t W µ X µ X jm τ t + ε W m W m + µ X µ X jm τ t ε W m ε jm m ε jm A39 Because the sample maxmum of the norms of the matchng dscrepances, X X jm, s o p, and the regresson functon µ, s Lpschtz, we obtan W µ X µ X jm o p A4 m Consder the frst term on the rght hand sde of equaton A39: W W µ X µ X jm τ t m µ X µ X τ t + µ X µ X jm W m µ X µ X τ t + µ X µ X jm + µ X µ X τ t W W m µ X µ X jm m µ X µ X τ t + o p, A4 by Hölder s Inequalty and equaton A4 ext, consder the second term on the rght hand sde of equaton A39: W ε m ε jm W ε + m W ε j + m m n>m ε jmε jn ε ε jm m Therefore, W ε m ε jm + W W + W K σ X, W W + W K m n>m ε jmε jn ε σ X, W W ε ε jm m A4 8

29 The expectatons condtonal on X and W of each of the three terms on the rght hand sde of last expresson are zero, so the uncondtonal expectatons are also zero Because the fourth condtonal moments of ε are unformly bounded, and because / K W s unformly bounded see Proof of Lemma 3, we obtan: W + W K ε σ X, W W + W K The varance of the second term dvded by 4/ 4 s ε jmε jn W m n>m + + W 4 m n>m ε σ X, W o ε jmε jn W W j,j> m n>m W W W σ 4 ε jmε jn m n>m K K σ 4 o The varance of the thrd term dvded by 4/ s ε ε jm ε ε jm W m W m + jmj ε ε jm ε j ε W W,j> m m σ 4 o As a result, we obtan W ε m ε jm W W + W K σ X, W o p Fnally consder the last term on the rght hand sde of equaton A39 Let Ψ t, µ X µ X jm τ t m otce that there s a fnte bound Ψ t, such that Ψ t, Ψt for all The condtonal expectaton of the last term of equaton A39 s zero, so the uncondtonal expectaton s also zero The condtonal varance jnj ε jmjε 9

30 of ths term dvded by 4 s: jmj Ψ t,ψ t,j ε ε jm ε j ε W W j m m Ψ t,ψ t,jε ε j W W j + Ψ t,ψ t,jε W W j + Ψ t,ψ t,j W W j otce that: Ψ t,ψ t,jε ε j W W j Ψ t, ε Ψt W Ψ t,ψ t,jε ε jmj, W W j W + m Ψ t,ψ t,j ε jm ε jmj W j m m Ψ t,ψ t,j ε jm W W j,j> m m + Ψt ε jmj m ε jm ε jmj m m ε W o, Ψ t, ε jm W m ε jmj Ψt σ W K K ε o W As a result, we obtan: Ŷ Ŷ τ sm,t µ X µ X τ t W Applyng the prevous lemma: Therefore, W + W K V τx,t W Y Ŷ τ sm,t W + σ X, W W + W K σ X, W + o p Addtonal References W + W K σ X, W o p W + W K σ X, W p V τx,t Gradshteyn, IS and I Ryzh,, Tables of Integrals, Seres, and Products, 6th edton, Academc Press, San Dego 3