Testing overidentifying restrictions with many instruments and heteroskedasticity

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1 Testng overdentfyng restrctons wth many nstruments and heteroskedastcty The MIT Faculty has made ths artcle oenly avalable. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher Chao, John C. et al. Testng Overdentfyng Restrctons wth Many Instruments and Heteroskedastcty. Journal of Econometrcs 178 (2014): htt://dx.do.org/ /j.jeconom Elsever Verson Orgnal manuscrt Accessed Mon Dec 10 02:15:09 EST 2018 Ctable Lnk Terms of Use Detaled Terms htt://hdl.handle.net/1721.1/ Creatve Commons Attrbuton-NonCommercal-NoDervs Lcense htt://creatvecommons.org/lcenses/by-nc-nd/4.0/

2 Testng Overdentfyng Restrctons wth Many Instruments and Heteroskedastcty John C. Chao, Deartment of Economcs, Unversty of Maryland, Jerry A. Hausman, Deartment of Economcs, MIT, Whtney. Newey, Deartment of Economcs, MIT, Norman R. Swanson, Deartment of Economcs, Rutgers Unversty, Temen Woutersen, Deartment of Economcs, Johns Hokns Unversty, June 2010 JEL classfcaton: C13, C31. eywords: heteroskedastcty, nstrumental varables, jackknfe estmaton, many nstruments, weak nstruments. *Preared for the Test 2010 Conference n amen, Chna. Newey thanks the NSF for fnancal suort.

3 Proosed Runnng Head: JIVE Wth Heteroskedastcty Corresondng Author: Whtney. Newey Deartment of Economcs MIT, E52-262D Cambrdge, MA Abstract Ths aer gves a test of overdentfyng restrctons that s robust to many nstruments and heteroskedastcty. It s based on a jackknfe verson of the Sargan test statstc, havng a numerator that s the objectve functon mnmzed by the JIVE2 estmator of Angrst, Imbens, and rueger (1999). Correct asymtotc crtcal values are derved for ths test when the number of nstruments grows large, at a rate u to the samle sze. It s also shown that the test s vald when the number nstruments s fxed and there s homoskedastcty. Ths test mroves on recently roosed tests by allowng for heteroskedastcty and by avodng assumtons on the nstrument rojecton matrx. The asymtotcs s based on the heteroskedastcty robust many nstrument asymtotcs of Chao et. al. (2010). 1

4 1 Introducton The Sargan (1958) and Hansen (1982) tests of nstrument valdty are known to be qute senstve to the number of restrctons beng tested. Ths aer rooses an alternatve test that s robust to many nstruments and to heteroskedastcty. It s based on subtractng out the dagonal terms n the numerator of the Sargan statstc and normalzng arorately. Ths test has a jackknfe nterretaton, beng based on the objectve functon of the JIVE2 estmator of Angrst, Imbens, and rueger (1999). We show that the test s vald under homoskedastcty wth fxed number of nstruments and wth number of nstruments gong to nfnty at any rate u to the samle sze tself. Recently Anatolyev and Gosodnov (2009) and Lee and Oku (2010) have gven tests that allow for many nstruments but mose homoskedastcty. Our test s vald under ther condtons and also wth heteroskedastcty. Also, we do not mose sde condtons on the nstrument rojecton matrx. The asymtotc theory s based on the results of Chao et. al. (2010) and Hausman et. al. (2010), ncludng a central lmt theorem that moses no sde condtons on the nstrumental varable rojecton matrx. In Secton 2 we descrbe the model and test statstc. In Secton 3 we gve the asymtotc theory. 2 The Model and Test Statstc We adot the same model and notaton as n Hausman et. al. (2010) and Chao et. al. (2010). The model we consder s gven by y n 1 = δ 0 + ε, n G G 1 n 1 = Υ + U, where n s the number of observatons, G s the number of rght-hand sde varables, Υ s the reduced form matrx, and U s the dsturbance matrx. For the asymtotc aroxmatons, the elements of Υ wll be mlctly allowed to deend on n, although we suress deendence of Υ on n, for notatonal convenence. Estmaton of δ 0 wllbebasedonann matrx, Z, of nstrumental varable observatons wth rank(z) =. HerewewlltreatZ and Υ as nonrandom for smlcty 2

5 though t s ossble to do asymtotc theory condtonal on these as n Chao et. al. (2010). We wll assume that E[ε] =0andE[U] =0. Ths model allows for Υ to be a lnear combnaton of Z (.e. Υ = Zπ, for some G matrx π). Furthermore, some columns of may be exogenous, wth the corresondng column of U beng zero. The model also allows for Z to aroxmate the reduced form. For examle, let 0, Υ0, and Z0 denote the th row (observaton) for, Υ, and Z, resectvely. We could let Υ = f 0 (w ) be a vector of unknown functons of a vector w of underlyng nstruments and let Z =( 1 (w ),..., (w )) 0, for aroxmatng functons k (w), such as ower seres or slnes. In ths case, lnear combnatons of Z may aroxmate the unknown reduced form. For estmaton of δ we consder heteroskedastcty robust verson of the Fuller (1977) estmator of Hausman et. al. (2010), referred to as HFUL. Other heteroskedastcty and many nstrument robust estmators could also be used, such as jackknfe nstrumental varable (IV) estmators of Angrst, Imbens, and rueger (1999) or the contnuously udated GMM estmator (CUE). We focus on HFUL because of ts hgh effcency relatve to jackknfe IV, because t has moments, and because t s comutatonally smle relatve to CUE. To descrbe HFUL, let P = Z(Z 0 Z) 1 Z 0, P j denote the j th element of P, and =[y, ]. Let α be the smallest egenvalue of ( 0 ) 1 ( 0 P n P 0 ). =1 Although the matrx n ths exresson s not symmetrcthasrealegenvaluesbecausetsa roduct of symmetrc, ostve sem-defnte matrces. Let ˆα =[ α (1 α)/t ]/[1 (1 α)/t ]. HFUL s gven by Ã! n 1 Ã! n ˆδ = 0 P P 0 ˆα 0 0 Py P y ˆα 0 y. =1 =1 Thus, HFUL can be comuted by fndng the smallest egenvalue of a matrx and then usng ths exlct formulae. Motvaton for HFUL s further dscussed n Hausman et. al. (2010). To descrbe the overdentfcaton statstc, let ˆε = y 0 ˆδ, ˆε =(ˆε 1,...,ˆε n ) 0, ˆε(2) = (ˆε 2 1,...,ˆε2 n) 0, and P (2) be the n-dmensonal square matrx wth j th comonent equal to P 2 j. Also, let P 3

6 denote the double sum over all not equal to j. The test statstc s ˆT = ˆε0 P ˆε P n =1 P ˆε 2 ˆV +, ˆV = ˆε(2)0 P (2)ˆε(2) P P 2 ˆε4 = P ˆε2 P 2 j ˆε2 j Treatng ˆT as f t s ch-squared wth G degrees of freedom wll be asymtotcally correct f no faster than n and when s fxed and ε s homoskedastc. Let q r (τ) betheτ th quantle of the ch-squared dstrbuton wth r degrees of freedom. A test wth asymtotc rejecton frequency α s to reject the null hyothess f ˆT q G (1 α). We wll show that the test wth ths crtcal regon has a robablty of rejecton that converges to α. To exlan the form of ths test statstc, note that the numerator s n ˆε 0 P ˆε P ˆε 2 = ˆε P j ˆε j. =1 Ths object s the numerator of the Sargan (1958) statstc wth the own observaton terms subtracted out. It has a jackknfe form, n the sense that t s the sum of sums where the own observatons have been deleted. JIVE2 estmator of Angrst, Imbens, and rueger (1999). If ˆδ were chosen to mnmze ths exresson, t would be the One effect of removng the own observatons s that P ˆε P j ˆε j wouldbemeanzerofˆε were relaced by ε In fact, P ε P j ε j has a martngale dfference structure that leads to t beng asymtotcally normal as, e.g. as n Lemma A2 of Chao et. al. (2010). The denomnator ncororates a heteroskedastcty consstent estmator of the varance of P ε P j ε j. By drong terms that have zero exectaton, smlarly to Chao et. al. (2010), t follows that for σ 2 = E[ε2 ], E[( ε P j ε j ) 2 ] = E[ P k P jk ε ε 0 jε 2 k + Pjε 2 2 ε 2 j],j k/ {,j} = E[2 Pjε 2 2 ε 2 j]=2 Pjσ 2 2 σj 2 Smlarly to Whte (1980) the varances are relaced by squared resduals to obtan ˆV. Also, 2 s relaced by 1/ and s added to normalze the statstc to be ch-squared wth fxed and ε homoskedastc. Unfortunately, t does not aear ossble to normalze the statstc to be ch-squared f there s heteroskedastcty when s fxed. 4

7 3 Many Instrument Asymtotcs The asymtotc theory we gve combnes the many nstrument asymtotcs of untomo (1980), Mormune (1983), and Bekker (1994) wth the many weak nstrument asymtotcs of Chao and Swanson (2005). Some regularty condtons are mortant for ths theory. Let Z 0,ε,U 0, and Υ0 denote the th row of Z, ε, U, and Υ resectvely. Assumton 1: Z ncludes among ts columns a vector of ones, rank(z) =, and there s a constant C such that P C<1, ( =1,...,n),. The restrcton that rank(z) = s a normalzaton that requres excludng redundant columns from Z. Itcanbeverfed n artcular cases. For nstance, when w s a contnuously dstrbuted scalar, Z = (w ), and k (w) =w k 1, t can be shown that Z 0 Z s nonsngular wth robablty one for <n. 1 The condton P C<1mles that /n C, because/n = P n =1 P /n C. The next condton secfes that the reduced form Υ s a lnear combnaton of a set of varables z havng certan roertes. Assumton 2: Υ = S n z / n where S n = S dag (μ 1n,...,μ Gn )and S s nonsngular. Also, for each j ether μ jn = n or μ jn / n 0, μ n = mn μ jn, and /μ 2 n 0. Also, 1 j G there s C>0such that k P n =1 z z 0/nk C and λ mn ( P n =1 z z 0 /n) 1/C, for n suffcently large. Ths condton s smlar to Assumton 2 of Hansen, Hausman, and Newey (2008). It accommodates lnear models where ncluded nstruments (e.g. a constant) have fxed reduced form coeffcents and excluded nstruments have coeffcents that can shrnk as the samle sze grows, as further dscussed n Hausman et. al. (2010). The μ 2 n can be thought of as a verson of the concentraton arameter, determnng the convergence rate of estmators of δ 0G, just as the concentraton arameter does n other settngs. For μ 2 n = n, the convergence rate wll be n,where Assumtons 1 and 2 ermt to grow as fast as the samle sze, corresondng to a many nstrument asymtotc aroxmaton lke untomo (1980), Mormune (1983), and Bekker (1994). For μ 2 n growng slower than n the convergence rate wll be slower that 1/ n, leadng to an asymtotc aroxmaton lke that of Chao and Swanson (2005). 1 The observatons w 1,...,w n are dstnct wth robablty one and therefore, by <n,cannot all be roots of a th degree olynomal. It follows that for any nonzero a theremustbesome wth a 0 Z = a 0 (w ) 6= 0, mlyng that a 0 Z 0 Za > 0. 5

8 Assumton 3: There s a constant C > 0 such that (ε 1,U 1 ),...,(ε n,u n ) are ndeendent, wth E[ε ]=0,E[U ]=0,E[ε 2 ] <C, E[kU k 2 ] C, Var((ε,U 0)0 )=dag(ω, 0), and λ mn ( P n =1 Ω /n) 1/C. Ths assumton requres second condtonal moments of dsturbances to be bounded. It also moses unform nonsngularty of the varance of the reduced form dsturbances, that s useful n the consstency roof, to hel the denomnator of the objectve functon stay away from zero. Assumton 4: There s a π n such that P n =1 kz π n Z k 2 /n 0. Ths condton and P C<1wll mly that for a large enough samle n n P j Υ Υ 0 j/n = Υ 0 P Υ/n P Υ Υ 0 /n = (1 P )Υ Υ 0 /n Υ 0 (I P )Υ/n =1 =1 n n = (1 P )Υ Υ 0 /n + o(1) (1 C) Υ Υ 0 /n, =1 =1 so that the structurally arameters are dentfed asymtotcally. Also, Assumton 4 s not very restrctve because flexblty s allowed n the secfcaton of Υ.IfwesmlymakeΥ the exectaton of gven the nstrumental varables then Assumton 4 holds automatcally. Assumton 5: There s a constant, C>0, such that wth robablty one, P n =1 kz k 4 /n 2 0, E[ε 4 ] C and E[kU k 4 ] C. It smlfes the asymtotc theory to assume that certan objects converge and to allow for two cases of growth rates of relatve to μ 2 n. These condtons could be relaxed at the exense of further notaton and detal, as n Chao et. al.. Let σ 2 = E[ε 2 ], γ n = P n =1 E[U ε ]/ P n =1 σ 2, Ũ = U εγn, 0 havng th row Ũ 0;andlet Ω = E[ŨŨ 0]. Assumton 6: μ n Sn 1 S 0 and ether I) /μ 2 n α for fnte α or; II) /μ 2 n. Also, each of the followng exst: n n H P = lm (1 P )z z 0 n /n, Σ P = lm (1 P ) 2 z z n σ 0 2 /n, =1 =1 ³ Ψ = lm P 2 n j σ 2 E[ŨjŨ j]+e[ũε 0 ]E[ε j Ũj] 0 /. 6

9 The frst result shows that the ch-square aroxmaton s asymtotcally correct when grows wth n. Theorem 1: If Assumtons 1-6 are satsfed then Pr( ˆT q G (1 α)) α. The next result shows asymtotc valdty of the ch-squared aroxmaton when s fxed. Theorem 2: If E[ε 2 ] = σ2, s fxed, Z 0 Z/n Q nonsngular, Z 0 Υ/n D wth rank(d) =G, E[ε 4 ] C, kυ k C, and Assumton 3 s satsfed, then Pr( ˆT q G (1 α)) α. Ths test should have ower aganst some forms of mssecfcaton. Under mssecfcaton ˆV wll stll be bounded and bounded away from zero. Also, for ε = E[y 0lm(ˆδ)], the normalzed numerator P P j ˆε ˆε j / wll be centered at à ε 0 P ε! P ε 2 /. Assumng a lnear combnaton of Z aroxmates ε ths s close ε 2 (1 P )/ Ths wll ncrease at rate n/ by P bounded away from one. ˆT rovdes secfcaton check for many nstrument estmator ˆδ. Note however that t may not be otmal as a test of the null hyothess that E[ ε ]=0. The magntude of the test statstc under the alternatve grows faster when grows slower. Thus, for hgher ower t would be better to use fewer nstruments. 7

10 4 Aendx A - Proofs of Theorems We wll defne a number of notaton and abbrevatons. C denotes a generc ostve constant that may be dfferent n dfferent uses and let M, CS, and T denote the Markov nequalty, the Cauchy-Schwartz nequalty, and the Trangle nequalty resectvely. Also, for random varables W, Y,andη,let w = E[W ], W = W w, ȳ = E[Y ], Ỹ = Y ȳ, η = E[η ], η = η η, ȳ =(ȳ 1,..., ȳ n ) 0, w =( w 1,..., w n ) 0, μ W = max, μ Y = max, μ η = max, 1 n 1 n 1 n σ W 2 = max ], σ Y 2 =max ], σ η 2 =max ]; n n n The followng Lemmas are secal cases of results n Chao et. al. (2010) but are gven here for exoston: Lemma A1: Suose that the followng condtons hold: ) P s a symmetrc, demotent matrx wth rank(p )=, P C<1; ) (W 1n,U 1,ε 1 ),..., (W nn,u n,ε n ) are ndeendent and D n = P n =1 E [W n Wn 0 ] satsfes kd nk C; ) E [Wn 0 ]=0,E[U ]=0, E[ε ]=0and there exsts a constant C such that E[kU k 4 ] C, E[ε 4 ] C; v) P h n =1 E kw n k 4 0; v) as n. Then for def Σ n = ³ Pj 2 E[U U]E[ε 0 2 j]+e[u ε ]E[ε j Uj] 0 / and any sequences c 1n and c 2n deendng on Z conformable vectors wth kc 1n k C, kc 2n k C, Ξ n = c 0 1n D nc 1n + c 0 2n Σ n c 2n > 1/C, t follows that Y n = Ξ 1/2 n (c 0 1n n W n + c 0 2n U P j ε j / ) =1 d N (0, 1). Proof: Ths s Lemma A2 of Chao et. al. (2010) when Z and Υ are not random. Q.E.D. result, Lemma A2: If Assumtons 1-3 are satsfed then S 1 n P j 0 js 10 n = O (1),S 1 n q P j ε j = O (1 + /r n ). Proof: The second concluson holds by Lemma A5 of Chao et al. (2010), and by that same S 1 n P j 0 js 10 n = z P j zj/n 0 + o (1). 8

11 We also have z P j zj/n 0 = z 0 Pz/n P z z /n and both z 0 Pz/n z 0 z/n and P P z z 0/n z0 z/n are bounded, gvng the frst concluson. Q.E.D. Lemma A3: If ˆδ δ, E[k k 2 ] C, E[ε 4 ] C, ε 1,...,ε n are mutually ndeendent, and ether or max n P 0 then P 2 j ˆε2 ˆε2 j Proof: Hence by ˆδ δ we have Hence w..a.1, for d =3(1+k k 2 ), Also by P,j P 2 j = P P =, P 2 j σ2 σ2 j 0. ˆδ δ 2 ˆδ δ wth robablty aroachng one (w..a.1). ˆε 2 ε 2 2 k k ˆδ δ + k k 2 2 ˆδ δ d ˆδ δ. E[ Pjd 2 d j ]/ C Pj/ 2 C, E[ Pjε 2 2 d j ]/ C. Then by M, Pjd 2 d j / = O (1), Pjε 2 2 d j / = O (1). Therefore, for ˆV n = P Pj 2 ˆε2 ˆε2 j /, Ṽn = P Pj 2 ε2 ε2 j / we have ˆV n Ṽn Pj 2 ˆε 2 ˆε 2 j ε 2 ε 2 j / ˆδ δ 2 Pjd 2 d j / +2 ˆδ δ Let V n = P P 2 j σ2 σ2 j / and v = ε 2 σ2 Note that by P j = P j, Note that E[v 2] E[ε4 ] C, so we have have E[( P 2 jε 2 d j / Pjε 2 2 ε 2 j Pjσ 2 2 σj 2 =2 Pjv 2 σj 2 + Pjv 2 v j. P 2 jv σ 2 j /) 2 ] = 2 C 2 j6= k6= j C 1 max n P 9 P 2 j P 2 jp 2 ke[v 2 ]σ 2 j σ 2 k k P 2 k = C 2 P 2 0. P / C 1 max n P 0.

12 Also, by CS, max,j n P 2 j max n P 2,sothat E[( Then by T and M we have Ṽn V n Pjv 2 v j /) 2 ] = 2 2 PjE[v 4 2 ]E[vj 2 ] C 2,j C 1 max P 2 n,j P 4 j Pj/ 2 = C 1 max P 2 0. n 0. The concluson then follows by T. Q.E.D. ProofofTheorem1:Note that P n ˆε P j ˆε j = h h ε (ˆδ 0 δ) P j ε j j(ˆδ 0 δ) / = P ε P j ε j +(ˆδ δ) 0 S n S 1 n +2(ˆδ δ) 0 S n Sn 1 P j ε j /. P j 0 js 10 n Sn(ˆδ 0 δ)/ If /μ 2 n α< (case I of Assumton 6) then by Theorem 2 of Hausman et. al. (2010) we have Sn(ˆδ 0 δ) =O (1). ThenbyLemmaA2wehave P P n ˆε P j ˆε j = ε P j ε j + o (1). (1) If /μ 2 n (case II of Assumton 6) then by Theorem 2 of Hausman et. al. (2010), (μ n / )S 0 n(ˆδ δ 0 )=O (1), so that by /μ 2 n 0, (ˆδ δ) 0 S n Sn 1 P j js 0 n 10 Sn(ˆδ 0 δ)/ ³ = O (1) /μ 2 n / = o (1), (ˆδ δ) 0 S n S n 1 P j ε j / = O (1)( /μ n )O (1 + /μ n )/ = O (1/μ n + /μ 2 n)=o (1). Therefore, eq. (1) s also satsfed when /μ 2 n. Next, note that σ 2 C by Assumton 3 and P C<1 by Assumton 1, so that V n = P σ 2 P 2 j σ2 j >C ÃP,j P 2 j P P 2! P = C P (1 P ) >C>0. 10

13 Also, E[ε 4 ] C and as shown above, E[P (ε P j ε j ) 2 ]=2V n. Now aly Lemma A1 wth W n =0,c 1n =0,andc 2n = 1. It follows by the concluson of Lemma A1 that P ε P j ε j 2Vn d N(0, 1). Next, by Theorem 1 of Hausman et. al. (2010) we have ˆδ δ, so that by Lemma A3, q ˆV n V n 0. Then by V n bounded and bounded away from zero, V n / ˆV n 1. Therefore by the Slutzky theorem, P P n ˆε P j ˆε j n ε P j ε j = q q2 ˆV n 2 ˆV n Next, note that ˆT q G (1 α) f and only f + o s P (1) Vn n ε P j ε j d = + o (1) N(0, 1). q2 ˆV n ˆV n 2Vn P n ˆε P j ˆε j q 2 ˆV n q G(1 α) 2. It s know that as,[q G (1 α) ( G)] / 2( G) q(1 α), where q(1 α) s the 1 α quantle of the standard normal dstrbuton. Also, we have The concluson now follows. Q.E.D. = s Ã! G q G (1 α) ( G) G q(1 α). 2( G) 2 ProofofTheorem2:It follows n the usual way from the condtons that n ³ˆδ δ0 d N(0,σ 2 (D 0 Q 1 D) 1 ). In addton, t s stratforward to show that Z 0 Z/n Q nonsngular mles that max n P 0; e.g. see McFadden (1982). Furthermore, note for d =3(1+k k 2 ) from the roof of Lemma A3 that E[ P d ] P E[d ] C, so P P d = O (1). Then smlarly to the roof of Lemma A3, by P 0, P (ˆε 2 ε 2 ) 2 P ˆε ε 2 P d ˆδ δ = O (1)o (1) 0. 11

14 Also, we have E[( P ε 2 σ 2 ) 2 ] = E[( P {ε 2 σ 2 }) 2 ]= C max P P 0. n P 2 Var(ε 4 ) Then by the Markov and Trangle nequaltes, P ˆε 2 σ 2. Also, snce (as just shown) P P 2 0tfollowsbyLemmaA3andσ2 = σ2 that ˆV P 2 n σ 4 j ˆε2 ˆε2 j P 2 = σ 4 j σ P 4 2 = o (1) + o(1) 0. Therefore, ˆT = σ2 ˆV ˆε 0 P ˆε σ 2 + P n=1 P ˆε 2 ˆV =[1+o (1)] ˆε0 P ˆε σ 2 + o (1). It follows by standard arguments that ˆε 0 P ˆε/σ 2 Slutzky Lemma. Q.E.D. d χ 2 ( G), so the concluson follows by the 5 References Ackerberg, D.A. and P. Deveraux (2003) Imroved JIVE estmators for overdentfed models wth and wthout heteroskedastcty. Workng Paer, Unversty of Calforna, Los Angeles. Anatolyev, S. and Gosodnov, N. (2009). Secfcaton testng n models wth many nstruments, Econometrc Theory, forthcomng. Andrews, D.W.. and J.H. Stock (2007) Testng wth many weak nstruments. Journal of Econometrcs 138, Angrst, J.D., G.W. Imbens, and A. rueger (1999) Jackknfe nstrumental varables estmaton. Journal of Aled Econometrcs 14, Bekker, P.A. (1994) Alternatve aroxmatons to the dstrbutons of nstrumental varable estmators. Econometrca 62,

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