Aitional Proofs for: Crump, R., V. J. Hotz, G. Imbens an O. Mitnik, Nonparametric Tests for Treatment Effect Heterogeneity Final Draft: December 007
Aitional Proofs Proof of Lemma A. We will generalize the proof in Imbens, Newey an Rier 006. For i we will show [ E ˆΩ Ω ] C ζk K/N so that the result follows by Markov s inequality. [ E ˆΩ Ω ] [ E tr ˆΩ ] tr E [ˆΩ + tr Ω tr E + tr Ω ] tr ˆΩ Ω ] tr E [ˆΩ Ω Note first that E[ˆΩ ] Ω so that, [ˆΩ ] tr Ω. B. The first term of equation B. is ] tr E [ˆΩ N E w W i R kk X i R lk X i N w k l N i k l i j N E [ w W i w W j R kk X i R lk X i R kk X j R lk X j ]. We may partition this epression into terms with i j, N E [ w W i R kk X i R lk X i ] k l i B. an with terms i j, E [ w W i R kk X i R lk X i ] E [ w W j R kk X j R lk X j ]. k l i j B.3 For a ranom variable U with E U < an an event G with Pr G > 0, then E [U G] E [U G] Pr G. Using this we may rewrite equations B. an B.3 as, ] N tr E [ˆΩ π w k l E [ R kk X R lk X W w ] + π w N N N w trω. B.4 []
To eal with the first term of equation B.4 consier, E [ [ R kk X R lk X ] K W w E R kk X k l k l K ζk l K R lk X W w E [ R lk X W w ] ζk tr Ω λ ma Ω ζk K C ζk K. ] B.5 B.6 Equation B.5 follows by, K ζk sup R K sup RkK which then implies that RkK ζk. k k Equation B.6 follows since the maimum eigenvalue of Ω is O see below. Thus, the first term of equation B.4 is π w N N w k l E [ R kk X R lk X ] W w C ζk KN. B.7 To eal with secon term of equation B.4 an the secon term of equation B. we have, trω λ ma Ω K λma Ω K C K. Thus, πw N N trω N + o O K O KN. B.8 Combining the results from equations B.7 an B.8 yiels, [ E ˆΩ Ω ] O ζk KN + O KN O ζk KN. For ii, first note that for any two positive semi-efinite matrices A an B, an conformable vectors a an b, if A B in a positive semi-efinite sense, then for an λ min A min a a a Aa a Aa, λ min B min b b b Bb b Bb, λ ma A ma a a a Aa ā Aā, we have that, λ ma B ma b b b Bb b B b, λ min A λ min B B.9 []
an λ ma A λ ma B. B.0 Now, let f w f X W W w an efine q f 0 /f an note that by Assumptions.3 an 3. we have that 0 < q q q <. Thus we may efine q q + q so that, Ω 0,K E [R K R K W 0] R K R K f 0 R K R K qf R K R K q + q f q R K R K f + R K R K qf q Ω,K + R K R K qf q Ω,K + Q Q is a positive semi-efinite matri, which implies that Ω 0,K q Ω,K in a positive semi-efinite sense. Thus by equation B.9 λ min Ω 0,K q λ min Ω,K q an the minimum eigenvalue of Ω 0,K is boune away from zero. Net, observe that 0 q q q < an so by the above we have that Ω 0,K q Ω,K + R K R K qf q Ω,K + q q R K R K f q Ω,K, in a positive semi-efinite sense. Now by equation B.0 we have λ ma Ω 0,K λ ma q Ω,K q an the maimum eigenvalue of Ω 0,K is boune. Both the minimum an maimum eigenvalue of Ω,K are boune away from zero an boune, respectively, by construction. For iii consier the minimum [3]
eigenvalue of ˆΩ, λ min ˆΩ min min Ω + ˆΩ Ω ˆΩ min Ω + min λ min Ω + λ min ˆΩ Ω λ min Ω ˆΩ Ω ˆΩ Ω B. λ min Ω O p ζkk / N / B. Where B. follows since for a symmetric matri A A tr A λ min A, an since the norm is nonnegative an A λ min A A λ min A for all values of λ min A. Finally, B. follows by part i. Net, consier the maimum eigenvalue of ˆΩ. λ ma ˆΩ ma ma Ω + ˆΩ Ω ˆΩ ma Ω + ma λ ma Ω + λ ma ˆΩ Ω λ ma Ω + ˆΩ Ω ˆΩ Ω B.3 λ ma Ω + O p ζkk / N / B.4 Where B.3 follows by similar arguments as above an B.4 follows by part i. efine For iv let us first Σ R D R N w, D iag { w W i ε w,i; i,..., N }. Net recall that for matrices A an B we have that A + B A + B. [4]
Thus, E ˆΣ Σ E ˆΣ Σ + Σ Σ E ˆΣ Σ + E Σ Σ R ˆD D R E N w + E Σ Σ B.5 B.6 Before we eal with equations B.5 an B.6, we nee to establish conitions for consistency of the estimate errors. Note that, ˆε w,i ε w,i X i Y i ˆµ w X i Y i µ w X i µ w X i ˆµ w X i an so by Lemma A.6 v sup ˆε w,i ε w,i O p ζ K KN + O ζ K K s/. Moreover, ˆε w,i ε w,i ε w,i ˆε w,i ε w,i + O p ζ K KN + O ζ K K s/. an so, for M R ++ Pr ˆε w,i ε w,i > M E ˆε w,i ε w,i M E ε w,i ˆε w,i ε w,i + O p ζ K KN + O ζ K K s/ M O p ζ K KN + O ζ K K s/ an so E ε w,i ˆε w,i ε w,i M C sup + M ˆε w,i ε w,i E ε w,i M + O p ζ K KN + O ζ K K s/ ζ K K s/, O p ζ K KN + O ˆε w,i ε w,i O p ζ K KN + ζ K K s/. B.7 We begin with equation B.6 first. E Σ Σ E [tr Σ ] tr E [ Σ tr tr E ] First note that E [ Σ Σ an so, tr Σ Σ E + tr Σ ] [ Σ ] Σ + tr Σ [ Σ ] tr Σ. B.8 [5]
The first term of equation B.8 is ] tr E [ Σ N w + N w N N w + N k l i j k l i k l i j k l N N N w Equation B.9 may be rewritten as, N k l N N w N N w C C N E [ w W i w W j ε i ε jr kk X i R lk X i R kk X j R lk X j ] N E [ w W i ε 4 w,ir kk X i R lk X i ] E [ w W i ε w,ir kk X i R lk X i ] E [ {Wjw}ε w,jr kk X j R lk X j ] E [ w W ε 4 wr kk X R lk X ] k l E [ w W ε 4 wr kk X R lk X ] k l k l N k l N k l B.9 [ E w W ε wr kk XR lk X ]. B.0 E [ w W E [ ε 4 w X, w W ] R kk X R lk X ] E [ w W E [ ε 4 ] w X RkK X R lk X ] E [ w W R kk X R lk X ] E [ R kk X R lk X ] W w. Thus by equation B.7 we have that equation B.9 is, N E [ w W ε 4 wr kk X R lk X ] C ζk KN. k l B. Equation B.0 an the secon term of equation B.8 are, N N N w k l πw N N π w N w N N N w [ E w W ε wr kk XR lk X ] tr Σ k l [ E σ w X R kk XR lk X ] W w tr Σ tr Σ. B. The first factor is, πw N N N + o. [6]
By Assumption 3. the secon factor is, tr Σ λma Σ K σ4 λ ma Ω K C K B.3 an so equation B. is, N N N w k l [ E w W ε wr kk XR lk X ] tr Σ O KN. Thus, by equations B. an B.3 we have, E Σ Σ O ζk KN + O KN O ζk KN. B.4 Now consier equation B.5. E R N w ˆD D N w N k l i j Then, by equation B.7 we have, N N k l i j R N E [ w W i w W j ˆε i ε i ˆε j ε jr kk X i R lk X i R kk X j R lk X j ]. E [ w W i w W j ˆε i ε i ˆε j ε jr kk X i R lk X i R kk X j R lk X j ] [O p ζ K KN + ζ K K s/] N k l i j [O p ζ K KN + O N E [ w W i w W j R kk X i R lk X i R kk X j R lk X j ] ζ K K s/] ] tr E [ˆΩ. From the proof of i, we have that ] tr E [ˆΩ O ζ K KN + O KN + O K O K, an so we have, E R D D N w R [O p ζ K KN + O ζ K K s/] O K. B.5 [7]
Combining equations B.4 an B.5 yiels, E ˆΣ Σ E Σ Σ + E R D D O ζk KN + [O p ζ K KN + O N w R ζ K K s/] O K O ζk KN + [O p ζ K 4 K N + O ζ K K s/ + O p ζ K 3 KK s/ N ] O K O ζk KN + O p ζ K 4 K 3 N + O ζ K KK s/ + O p ζ K 3 K K s/ N O p ζ K 4 K 3 N + O ζ K KK s/, an the result follows. For v note that, σ Ω Σ σ Ω in a positive semi-efinite sense. Thus, Similarly, λ min Σ λ min σ Ω σ λ min Ω σ min q,. λ ma Σ λ ma σ Ω σ λ ma Ω σ ma q,. For vi consier, λ min ˆΣ min min [ Σ + Σ ˆΣ ˆΣ min Σ + min λ min Σ + λ min Σ ˆΣ λ min Σ Σ ˆΣ ] Σ ˆΣ λ min Σ O p ζ K K 3/ N O p ζ K K / K s/. Net, λ ma ˆΣ ma ma [ Σ + Σ ˆΣ ˆΣ ma Σ + ma λ ma Σ + λ ma Σ ˆΣ λ ma Σ + Σ ˆΣ ] Σ ˆΣ λ ma Σ + O p ζ K K 3/ N + O p ζ K K / K s/. [8]
Proof of Lemma A. For this proof we nee two results. matri an B a conformable matri, then Let A be a symmetric positive efinite λ min B AB λ min A λ min B B, λ ma B AB λ ma A λ ma B B. Using the above result we have, λ min Ω Σ Ω λ min Σ λ min Ω σ min q, [λ ] ma Ω σ min q, [λ ma Ω ] σ min q, [ma q, ], B.6 an, λ ma Ω Σ Ω λ ma Σ λ ma Ω σ ma q, [λ ] min Ω σ ma q, [λ min Ω ] σ ma q, [min q, ]. B.7 Now consier, λ min N V min π 0 Ω 0,K π Σ 0,KΩ 0,K + Ω,K 0 π Σ,KΩ,K min Ω 0,K Σ 0,KΩ π 0 λ min Ω 0,K Σ 0,KΩ 0,K 0,K + min π Ω + λ min Ω,K π Σ,KΩ,K,K Σ,KΩ,K which is boune away from zero by equation B.6 an Assumption.3. λ ma N V ma Ω 0,K π Σ 0,KΩ 0,K + Ω,K 0 π Σ,KΩ,K ma π 0 Ω 0,K Σ 0,KΩ 0,K + ma π π 0 λ ma Ω 0,K Σ 0,KΩ 0,K,,K Σ,KΩ,K Ω + λ ma Ω,K π Σ,KΩ,K, Finally, consier [9]
which is boune by equation B.7 an Assumption.3. For ii we have, ˆΩ λ ˆΣ min ˆΩ λ min ˆΣ ˆΩ λ min [λ min Σ O p ζ K K 3/ N O p ζ K K / K s/] ˆΩ λ min [λ min Σ O p ζ K K 3/ N O p ζ K K / K s/] [λ min O p ζ K K / N /] Ω λ min Σ [λ min Ω ] + Op ζ K K 3/ N + O p ζ K K / K s/. In aition, ˆΩ λ ˆΣ ma ˆΩ λ ma ˆΣ ˆΩ λ ma [λ ma Σ + O p ζ K K 3/ N + O p ζ K K / K s/] ˆΩ λ ma [λ ma Σ + O p ζ K K 3/ N + O p ζ K K / K s/] [λ ma Ω + O p ζ K K / N /] ] λ ma Σ [λ ma Ω + Op ζ K K 3/ N + O p ζ K K / K s/. B.8 B.9 Thus, λ min N ˆV N min N ˆΩ ˆΣ 0,K ˆΩ 0,K 0,K + N 0 N ˆΩ ˆΣ,K ˆΩ,K,K N N 0 min ˆΩ ˆΣ 0,K ˆΩ 0,K 0,K N N 0 λ min ˆΩ 0,K ˆΣ 0,K ˆΩ 0,K + N N min ˆΩ ˆΣ,K ˆΩ,K,K + N N λ min ˆΩ,K ˆΣ,K ˆΩ,K is boune away from zero in probability by equation B.8, Assumption.3 an Assumptions 3. an 3.3. Finally, λ ma N ˆV N ma N ˆΩ ˆΣ 0,K ˆΩ 0,K 0,K + N 0 N ˆΩ ˆΣ,K ˆΩ,K,K N ma ˆΩ N ˆΣ 0 0,K ˆΩ 0,K 0,K + N ma ˆΩ N ˆΣ,K ˆΩ,K,K N N 0 λ ma ˆΩ 0,K ˆΣ 0,K ˆΩ 0,K + N N λ ma ˆΩ,K ˆΣ,K ˆΩ,K which is boune in probability by equation B.9, Assumption.3 an Assumptions 3. an 3.3. Proof of Lemma A.6 For i note that we woul like to provie regressors up to a certain power, say n, incluing cross prouct terms. However, we woul like to relate the number of covariates,, an the uppermost power esire, n, to the number of terms in the series, K. We may o so by, n i + n + K n +. i n i0 [0]
Then note that, n + n +! n n!! n + n + n j + j n +. Thus, we have that K C n +, or equivalently that n C K / an so n s C s K s/ for some C, C R ++. Finally, by Lorentz 986, Theorem 8 we have that sup µw R K γ 0 O n s O K s/, as esire. The proofs of ii an iv may be foun in Imbens, Newey an Rier 006. For iii consier, ˆγK γk 0 N ˆΩ R Y w w N ˆΩ R R γ 0 w N R Yw R γ 0 w λ ma R Yw R γ 0. By Lemma A., the first factor is, λ ma λ ma ˆΩ / Ω / N w ˆΩ / + O p ζ K K / N / O + O p ζ K K / N /. B.30 The secon factor may be broken up into, N ˆΩ / R Yw R γ 0 w N ˆΩ / R Yw µ w X + µ w X R γ 0 w N ˆΩ / R ε w B.3 w + R µw X R γ 0. B.3 N w ˆΩ / []
For equation B.3 we have, E N ˆΩ / R ε w w [ E tr ε wr ˆΩ [ E tr N w N w tr σ E N w σ R ε w ] R R R R ε w ε w ] [ ] E R R R R E [ε w ε w X] N w K C KN, [ ] tr R R R R an so N w ˆΩ / R ε w O p K / N / B.33 by Markov s inequality. For equation B.3 we have, R µw X R γ 0 N w ˆΩ / µw X R γ 0 R R N R R µw X R γ 0 w µw X R γ 0 µw X R γ N 0 w sup µw R K γk 0 O K s/, by i, an so N ˆΩ / R w µw X R γ 0 O K s/ B.34 Note that the thir line of the penultimate isplay follows since R R R R is a projection matri. Combining equations B.30, B.33 an B.34 yiels, [ ˆγK γk 0 O + O p ζ K K / N /] [ O p K / N / + O K s/] O p K / N / + O K s/ + O p ζ K KN + O p ζ K K / K s/ N / O K s/ + O p ζ K KN + O p ζ K K / K s/ N /. However, by Assumption 3.3, we have that, O p ζ K K / K s/ N / o p K s/ o p an so ˆγ K γk 0 Op ζ K KN + O K s/, []
as esire. sup Finally, for v we have, µ w ˆµ sup The first term is O K s/ by i. sup µ 0 w ˆµ sup RK γ 0 ˆγ sup R K γ 0 ˆγ [ ζ K O p ζ K KN + O O p ζ K KN + O µw µ 0 + sup For the secon term we have, K s/] ζ K K s/, µ 0 w ˆµ. where we use the efinition of ζ K an the result from iv. Thus, sup µ w ˆµ O K s/ + O p ζ K KN + O ζ K K s/ O p ζ K KN + O ζ K K s/, as esire. Proof of Lemma A.7 Aitional Details Equation A.4 is / [ˆΩ ˆΣ ] / ˆΩ ˆΩ ε [ˆΩ R ˆΣ ] / ˆΩ ˆΩ R ε w ] / [ˆΣ / ˆΩ ˆΩ / R ε ε w / The first factor is, / [ˆΣ ] ˆΩ ˆΩ / / ˆΣ ˆΣ / λ ˆΩ/ ma ˆΣ / ˆΩ/ λ ma λ ma K / [λ ma Σ / + O p ζ K K 3/ N + O p ζ K K / K s/] [λ ma Ω / + O p ζkk / N /] K / λ ma Σ / λ ma Ω / K / + O p ζ K K N + O ζ K KK s/ O K / + O p ζ K K N + O p ζ K KK s/. [3]
For the secon factor we have, E ˆΩ R ε ε w / [ ε ] E tr N ε w R ˆΩ R ε ε w w [ ε E ε w R R ] R R ε ε w [ ε ] E ε w ε ε w B.35 [ µw E X R γ µw X R γ ] N w sup µw R K γ N w sup µw R K γ 0 + RK γ 0 R K γ N w O K s + O ζkk K s B.36 O N O ζkk K s Thus, ˆΩ / R ε ε w / N / w O p ζkk / K s/ N / by Markov s inequality. Equation B.35 follows by the fact that R R R R is a projection matri an equation B.36 follows from Lemma A.6 i an iv. Then equation A.4 is / [ˆΩ ˆΣ ] / ˆΩ ˆΩ ε [ˆΩ R ˆΣ ] / ˆΩ ˆΩ R ε w [ O K / + O p ζ K K N + O p ζ K KK s/] [ O p ζkk / K s/ N /] O p ζkkk s/ N / + O p ζk 3 K 5/ K s/ N / + O p ζk K 3/ K s/ N / O p ζk K 3/ K s/ N /, by Assumption 3.3. Net, equation A.5 is / [ˆΩ ˆΣ ] / [ˆΩ ] / ˆΩ ˆΩ R ε w Σ Ω ˆΩ R ε w ] / [ ] / [ˆΣ ˆΩ Σ Ω N / w ˆΩ / R ε w The first factor is ] / [ ] / [ˆΣ ˆΩ Σ Ω O p ζ K K N +O p ζ K KN / +O p ζ K KK s/ [4]
by Lemma B.. The secon factor is, E ˆΩ R ε w / N w [ ] E tr ε N wr ˆΩ R ε w w [ E tr ε ] wr R R R ε w [ ] E tr R R R R ε w ε w [ ] tr E R R R R E [ε w ε X] σ w ] tr E [R R R R [ σ w ] E tr R R R R [ R σ w ] E tr R R R σ w tr I K σ w K. Thus, the the secon factor is O p K / by Markov s inequality. Putting this together yiels, / [ˆΩ ˆΣ ] / [ˆΩ ] / ˆΩ ˆΩ R ε w Σ Ω ˆΩ R ε w [O p ζ K K N + O p ζ K KN / + O p ζ K KK s/] O p K / O p ζ K K 3/ N / + O p ζ K K 3/ K s/. Finally, equation A.6 is / [ˆΩ [ Σ Ω Σ Ω ] / ˆΩ / ] / [ ] / ˆΩ R ε w Ω Σ Ω Ω R ε w Ω / / The first factor is, [ / Σ Ω] C I O K /. R ε w [5]
The secon factor is O p ζ K K / N / by Lemma A.. E R ε w / N w [ E tr ] ε N wr R ε w w [ E tr ] R N ε w ε wr w tr E [ R N E [ε w ε ] w X] R w σ w tr E [ R R ] /N w σ w tr Ω σ w K λ ma Ω C K For the thir factor consier, Thus, the thir factor is O p K / by Assumption 3., Lemma A. ii an Markov s inequality. Putting this together yiels, / [ˆΩ ] / [ ] / Σ Ω ˆΩ R ε w Ω Σ Ω Ω R ε w O p ζ K K 3/ N /. Before proving Theorem 3/3. we nee the following lemma. Lemma B. We may partition ˆV an V analogously to the partition of ˆV P an V P in Section 3.4. ˆV00 ˆV0 ˆV ˆV 0 ˆV an V00 V V 0 V 0 V where ˆV 00 an V 00 are scalars, ˆV 0 an V 0 are K vectors, ˆV 0 an V 0 are K vectors an ˆV an V are K K matrices. Then, λ min [N V ] λ min [N V ], λ ma [N V ] λ ma [N V ] an if O p ζ K K 3/ N + O p ζ K K / K s/ o p, [ λ min N ˆV ] [ λ min N ˆV ] [, λ ma N ˆV ] [ λ ma N ˆV ] with probability approaching one. Proof The proof follows by the interlacing theorem, see, for eample, Li an Mathias 00: If A is an n n positive semi-efinite Hermitian matri with eigenvalues λ... λ n, B is a k k principal submatri of A with eigenvalues λ... λ k, then λ i λ i λ i+n k, i,..., k. [6]
In our case, N ˆV an N V are positive semi-efinite, symmetric an thus positive semi-efinite, Hermitian. So then, by the interlacing theorem λ min N V λ min N V λ ma [N V ] λ ma [N V ] an λ ma N V λ ma N V λ min [N V ] λ min [N V ]. Moreover, by Lemma A., N ˆV is nonsingular with probability approaching one so that again by the interlacing theorem we obtain λ min N ˆV λ min N ˆV [ λ ma N ˆV ] [ λ ma N ˆV ] an λ ma N ˆV λ ma N ˆV [ λ min N ˆV ] [ λ min N ˆV ]. Proof of Theorem 3.3 When the conitional average treatment effect is constant we may choose the two approimating sequences, γ0,k 0 an γ0,k, to iffer only by way of the first element the coefficient of the constant term in the approimating sequence. To simplify notation, efine ˆδ K ˆγ,K ˆγ 0,K, δ K γ,k γ 0,K. We may again follow the logic of Lemmas A.3, A.4, an A.5 to conclue that / T ˆδK δk ˆV ˆδK δk K K N 0, We nee only show that T T o p to complete the proof. First, note that γ w,k γw,k 0 γ w,k,i γw,k,i 0 i i γ w,k,i γw,k,i 0 + γ w0,k γw0,k 0 γ γ 0 O KK s/, B.37 by Lemma A.6 ii an ˆγw,K γw,k 0 ˆγw,K,i γw,k,i 0 i i ˆγw,K,i γw,k,i 0 + ˆγw0,K γw0,k 0 ˆγ γ 0 [ O p ζ K KN + O K s/]. B.38 [7]
by Lemma A.6 iii. We may choose the last K elements of the approimating sequence to be equal, γ,k 0 γ0 0,K. This allows us to boun, ˆδ K ˆγ,K ˆγ 0,K ˆγ,K γ,k 0 + γ0,k 0 ˆγ 0,K ˆγ,K γ,k 0 + γ 0 0,K ˆγ 0,K O p ζ K KN + O K s/ B.39 by equation B.38. Also, δk γ,k γ0,k γ,k γ,k 0 + γ0,k 0 γ0,k γ,k γ,k 0 + γ 0 0,K γ0,k O K / K s/ B.40 by equation B.37. T T Net note that, ˆδK δk First consier, δk ˆV δ K tr δk ˆV K δ N δ [ tr K N ˆV ] δ K [ N λ ma N ˆV ] δk / ˆV ˆδK δk ˆδ K ˆV ˆδ K K δk ˆV δ K ˆδ K ˆV K/ δ K B.4 [ N λ ma N ˆV ] δk B.4 N [O + O p ζ K K 3/ N / + O p ζ K K 3/ K s/] O KK s/ O p ζ K K 5/ K s/ N /, B.43 where equation B.4 follows by Lemma B. an equation B.43 follows from Lemma B. an the proof [8]
of Theorem 3.. Now consier, ˆδ K ˆV δ K tr ˆδ K ˆV K δ [ N ˆδ tr K N ˆV ] δ K N λ ma [ N ˆV ] ˆδK δ K N λ ma [ N ˆV ] ˆδK δ K B.44 N [O + O p ζ K K 3/ N / + O p ζ K K 3/ K s/] [ O p ζ K KN + O K s/] [ O K / K s/] O ζ K K K 3s/ N B.45 where equation B.44 follows by Lemma B. an equation B.45 follows from Lemma B. an the proof of Theorem 3.. Thus, we have that equation B.4 is, T T O K / [ O p ζ K K 5/ K s/ N / ] + O p ζ K K K 3s/ N O p ζ K K K s/ N / + O p ζ K K 3/ K 3s/ N which is o p uner Assumptions 3. an 3.3. Proof of Theorem 3.4 First, note that we may partition R K as R R K. R K Net, consier ρ N sup sup µ µ 0 τ X sup X sup X sup X RK γ,k 0 µ + sup RK γ0,k 0 µ 0 X RK ˆγ 0,K R K γ0,k 0 RK ˆγ,K R K γ,k 0 + sup X + sup + sup X R K ˆγ,K R K ˆγ 0,K + R ˆγ 0,K R ˆγ 00,K τ X RK γ0,k 0 µ 0 + sup RK γ,k 0 µ X + sup R K ˆγ0,K γ0,k 0 + sup R K ˆγ,K γ 0,K X X + sup R K ˆγ,K ˆγ 0,K + R ˆγ 0,K R ˆγ 00,K τ X RK γ0,k 0 µ 0 + sup RK γ,k 0 µ X +ζk ˆγ0,K γ0,k 0 + ζk ˆγ,K γ,k 0 + ζk ˆγ,K ˆγ 0,K + R ˆγ 0,K R ˆγ 00,K τ. [9]
Thus, ˆγ,K ˆγ 0,K ζk ρ N sup ζk sup RK γ0,k 0 µ 0 X X ζk sup RK γ,k 0 µ ˆγ0,K γ0,k 0 ˆγ,K γ,k 0 X ζk R ˆγ 0,K R ˆγ 00,K τ. We may follow the steps of the proof of Theorem 3. to obtain, for any M, Pr N / ζ K / K / ˆγ,K ˆγ 0,K > M. Net, we show that this implies that C ˆγ,K ˆγ 0,K V Pr ˆγ,K ˆγ 0,K K > M, K B.46 B.47 for an arbitrary constant C R ++. Denote λ min [N V ] an λ ma [N V ] by λ an λ, respectively an note that by Lemma A. an Lemma B. it follows that λ is boune away from zero an λ is boune. C ˆγ,K ˆγ 0,K V Pr ˆγ,K ˆγ 0,K K > M K C N ˆγ,K ˆγ 0,K [N V ] ˆγ,K ˆγ 0,K K Pr > M K Pr C N ˆγ,K ˆγ 0,K [N V ] ˆγ,K ˆγ 0,K > M K + K Pr λ C N ˆγ,K ˆγ 0,K ˆγ,K ˆγ 0,K > M K + K Pr Nζ K K ˆγ,K ˆγ 0,K ˆγ,K ˆγ 0,K > λ C [ ζ K K M ] K / + K / Pr N / ζ K / K / ˆγ,K ˆγ 0,K > λ C / ζ K M K K / + K / Since for any M, for large enough N, we have / λ C / ζ K M K K / + K / < λ C / it follows that this probability is for large N boune from below by the probability Pr N / ζ K / K / ˆγ,K ˆγ 0,K > λ C / which goes to one by B.46. To conclue we must show that this implies that PrT ˆγ,K ˆγ 0,K ˆV > M Pr ˆγ,K ˆγ 0,K K > M. K Let ˆλ λ min [N ˆV ] for simplicity of notation. Let A enote the event that ˆλ > λ / which satisfies Pr A as N by Lemmas A., A., an B. along with Assumptions 3. an 3.3. Also efine the event A, λ / N ˆγ,K ˆγ 0,K ˆγ,K ˆγ 0,K K K > M. [0]
Note that C ˆγ,K ˆγ 0,K V Pr ˆγ,K ˆγ 0,K K > M K C N ˆγ,K ˆγ 0,K [N V ] ˆγ,K ˆγ 0,K K Pr > M K λ C N ˆγ,K ˆγ 0,K ˆγ,K ˆγ 0,K K Pr > M K which goes to one as N by equation B.47. Since C was arbitrary we may choose C λ / λ an so Pr A as N. Thus, Pr A A as N. Finally, note that the event A A implies that T ˆγ,K ˆγ 0,K ˆV ˆγ,K ˆγ 0,K K K N ˆγ,K ˆγ 0,K [ N ˆV ] ˆγ,K ˆγ 0,K K K ˆλ N ˆγ,K ˆγ 0,K ˆγ,K ˆγ 0,K K K > λ / N ˆγ,K ˆγ 0,K ˆγ,K ˆγ 0,K K K > M. Hence PrT > M. Lemma B. Suppose Assumptions.-.3 an 3.-3. hol. Then i, ˆΣ ˆΩ Σ Ω O p ζ K K N + O p ζ K KN / + O p ζ K KK s/, an ii ˆΩ ˆΣ ˆΩ Ω Σ Ω O p ζ K K 3/ N / + O p ζ K K 3/ K s/. Proof For i note that, ˆΣ ˆΩ Σ Ω ˆΣ ˆΩ Σ ˆΩ + Σ ˆΣ Σ ˆΩ + Σ ˆΩ Ω ˆΩ Σ Ω B.48 B.49 First, consier equation B.48, ˆΣ Σ Op ζ K K 3/ N + O p ζ K K / K s/ by Lemma A.. Net, ˆΩ K / ˆΩ λ ma O K / + O p ζ K KN /. []
For equation B.49 we have that, Σ K / λ ma Σ O K /. Also, we have ˆΩ Ω O p ζ K K / N / B.50 by Lemma A.. Thus, ˆΣ ˆΩ Σ Ω [O p ζ K K 3/ N + O p ζ K K / K s/] [ O K / + O p ζ K KN /] [ + O K /] [ O p ζ K K / N /] O p ζ K K N + O p ζ K KN / + O p ζ K KK s/. For ii note that, ˆΩ ˆΩ ˆΩ + Ω ˆΣ ˆΩ Ω Σ Ω ˆΣ ˆΩ Ω ˆΣ ˆΩ + Ω ˆΣ ˆΩ Ω Σ Ω Ω ˆΣ ˆΩ ˆΣ ˆΩ Σ Ω B.5 B.5 For equation B.5 the first factor is O p ζ K K / N / by equation B.50 an the secon factor is ˆΣ ˆΩ λ ma ˆΣ ˆΩ λ ma ˆΣ λ ma ˆΩ K / λ ma Σ λ ma Ω K / + O p ζ K K N + O p ζ K KK s/ O K / + O p ζ K K N + O p ζ K KK s/. Thus, equation B.5 is ˆΩ Ω ˆΣ ˆΩ For equation B.5 the first factor is, Ω λ ma Ω I O K /. O p ζ K KN / + O p ζ K K 3/ K s/ N /. The secon factor is ˆΣ ˆΩ Σ Ω O p ζ K K N + O p ζ K KN / + O p ζ K KK s/ by i. Thus, equation B.5 is Ω ˆΣ ˆΩ Σ Ω O p ζ K K 3/ N / + O p ζ K K 3/ K s/ []
Finally, ˆΩ ˆΣ ˆΩ Ω Σ Ω O p ζ K KN / + O p ζ K K 3/ K s/ N / + O p ζ K K 3/ N / + O p ζ K K 3/ K s/ O p ζ K K 3/ N / + O p ζ K K 3/ K s/. [3]
Aitional References Imbens, Guio, Whitney Newey an Geert Rier, Mean-square-error Calculations for Average Treatment Effects, unpublishe manuscript, Department of Economics, Harvar University 006. Li, Chi-Kwong, an Roy Mathias, Interlacing Inequalities for Totally Nonnegative Matrices, Linear Algebra an its Applications 34 January 00, 35-44. Lorentz, George, Approimation of Functions New York: Chelsea Publishing Company, 986. [4]