Fixed-b Inference for Difference-in-Difference Estimation (Job Market Paper)

Transcription

1 Fixed-b Inference for Difference-in-Difference Esimaion (Job Marke Paper) Yu Sun November 7, 212 Absrac This paper provides an analysis of he sandard errors proposed by Driscoll and Kraay (1998) in linear Difference-in-Difference (DD) models wih fixed-effecs and individual-specific ime rends. The analysis is accomplished wihin he fixed-b asympoic framework developed by Kiefer and Vogelsang (25) for HAC esimaor based ess. For he fixed-n, large-t case, i is shown ha fixed-b asympoic disribuions of es saisics consruced using he DD esimaor and he Driscoll and Kraay (1998) (DK) sandard errors are differen from he resuls found by Kiefer and Vogelsang (25) and Vogelsang (212). The newly derived fixed-b asympoic disribuions depend on he choice of individual-specific rend funcions and he policy change poin λ as well as he choice of kernel and he bandwidh. Wheher ime period dummies are included does no affec he fixed-b limis. For oher regressors ha don have a srucural break, he usual fixed-b asympoic disribuions sill apply. Mone Carlo simulaions illusrae he performance of he fixed-b approximaions in pracice. 1 Inroducion This sudy focuses on fixed-b asympoic disribuions of he W ald and saisics for Difference-in- Difference (DD) esimaion in linear panel seings. Recenly, DD esimaion has become a favorable way o esimae causal relaionships. DD esimaion involves idenifying a specific inervenion or reamen (ofen a policy change or a passage of a law). Applied researchers hen compare he difference in oucomes before and afer he inervenion for groups affeced by he inervenion (reamen groups) o he same difference for unaffeced groups (conrol groups). Such panel daa ses ofen include serial correlaion and/or spaial correlaion in he cross secion. Even hough he correlaion srucure is no of ineres, he failure o accoun for he presence of serial and spaial correlaion may lead o severe disorions in he inference abou parameers of ineres. Afer Berrand e al. (24) poined ou ha sandard errors robus o serial correlaion should be considered in DD esimaion, using clusered sandard errors (see Arellano, 1987) has become a sandard mehod o deal wih serial correlaion in he DD conex. Hansen (27) exended he resuls for he radiional shor panel case, he large-n, fixed-t case, o large-n, large-t and fixed-n, large-t cases. The assumpion ha individuals in he cross-secion are uncorrelaed wih each oher is needed for he panel clusered sandard errors o be valid. In oher words, spaial 1

2 correlaion in he cross-secion is ofen ignored. Wooldridge (23) provided a useful discussion of cluser mehods. Someimes he cross-secion can be divided ino groups or clusers where i is assumed ha individuals wihin in a cluser are correlaed while individuals beween clusers are uncorrelaed. In his case, sandard errors robus o cross-secion clusering can be consruced. The number of clusers could be small, hough. In ime series economerics, he nonparameric heeroskedasiciy and auocorrelaion (HAC) covariance marix esimaor (see Newey and Wes, 1987) is widely used. To handle he spaial correlaion, robus sandard errors can be obained using he approaches of Conley (1999), Kelejian and Prucha (27), Beser e al. (28), Beser e al. (211) or Kim and Sun (211a) when a disance measure is available. Kim and Sun (211b) provides resuls on kernel HAC sandard errors in linear panel models wih individual and ime dummy variables using a disance measure. When a disance measure is eiher unavailable or unknown for he cross-secion of he panel, he Driscoll and Kraay (1998) approach can be used o obain robus sandard errors. Driscoll and Kraay (1998) esablished consisency of hese sandard errors under mixing condiions. However, he mixing condiions do no hold for he fixed-effecs esimaor. Forunaely, Gonçalves (211) esablishes consisency of he Driscoll and Kraay (1998) (DK) sandard errors for he fixed-effecs esimaor in he presence of general forms of weakly dependen crosssecion correlaion. A recen paper by Vogelsang (212) develops a fixed-b asympoic heory for es saisics based on he fixed-effecs esimaor and he DK sandard errors following Kiefer and Vogelsang (25). This paper provides an analysis of he DK sandard errors in linear DD models wih fixed-effecs and individual-specific ime rends. The analysis is accomplished wihin he fixed-b asympoic framework proposed by Kiefer and Vogelsang (25) for HAC esimaor based ess. Fixed-b asympoics are appealing because hey capure he influence of he choice of kernel and bandwidh on he sampling behavior of he sandard errors while he radiional asympoics don. Large-T framework is required in he fixed-b approach. According o he survey of DD papers in Berrand e al. (24), among 92 DD papers hey found, 1% have a leas 36 ime periods and 5% have a leas 51 ime periods. Therefore, i is feasible o use he DK sandard errors for DD esimaion o cope wih any general forms of spaial correlaion in he cross secion given ha he ime series dimension is saionary. This paper only considers he fixed-n, large-t case. The main resul of his paper is o derive fixed-b asympoic disribuions of es saisics consruced using he DD esimaor and he DK sandard errors. I is found ha he limis are differen from he fixed-b asympoic disribuions found by Kiefer and Vogelsang (25) and Vogelsang (212). The newly derived fixed-b asympoic disribuions depend on he individual-specific ime rend funcions and he policy change poin λ. When i comes o he sandard individual fixed-effecs model wih no rend, he fixed-b asympoic disribuions are he same as found in a pure ime series shif in mean model. New criical values are simulaed in his sudy and hey have a U-shape wih respec o λ. Wheher ime period dummies are included does no affec he fixed-b asympoic disribuions. For oher regressors ha don have a srucural break, he fixed-b asympoic disribuions for DK es saisics found in Vogelsang (212) sill apply. The radiional shor panel case is no included. Wih T fixed, 2

3 here is no sufficien informaion in he ime dimension for he DK approach o work. The remainder of he paper is organized as follows. The nex secion describes he DD models and es saisics. Secion 3 presens he fixed-b asympoic resuls for es saisics consruced using he DD esimaor and he DK sandard errors, and new criical values for saisics in wo special cases. Finie sample properies are examined in Secion 4. Conclusions are given in Secion 5 and proofs are given in he Appendix. Throughou he paper, x i and β denoe he full se of regressors and parameers respecively in each model. denoes he ranspose, when used in he conex of a vecor. 2 Model seup and es saisics Consider a DD model wih fixed-effecs and individual-specific deerminisic rends given by y i = f() a i + β 1 T rea i + β 2 DU + β 3 T rea i DU + u i, (1) i = 1, 2,..., N, = 1, 2,..., T, where y i and u i are scalars, f() denoes a (J 1) vecor of rend funcions, a i denoes a (J 1) vecor of individual-specific unobservable variables. 1 T rea i denoes an indicaor for individuals in he reamen group which akes one if individual i is in he reamen group. Wihou loss of generaliy, I assume ha he firs kn individuals are in he reamen group. Thus, T rea i = 1(i kn). DU denoes an indicaor for pos-policy-change ime periods which akes one afer he policy change. Tha is, DU = 1( > λt ) = 1(r > λ), where he parameer λ is he relaive posiion of he policy change poin wihin he ime sample. Boh k and λ are assumed known. Ofen ime period fixed-effecs are included which gives he model y i = λ + f() a i + β 1 T rea i + β 2 DU + β 3 T rea i DU + u i. (2) An alernaive model includes common ime rends insead of ime period fixed-effecs. The asympoic resuls for he alernaive model remain unchanged. A more general model wih addiional regressors is y i = f() a i + β 1 T rea i + β 2 DU + β 3 T rea i DU + z i γ + u i, (3) where z i is a (K 1) vecor of addiional regressors. Including ime period fixed-effecs gives he model y i = λ + f() a i + β 1 T rea i + β 2 DU + β 3 T rea i DU + z i γ + u i. (4) The focus is on esimaion and inference abou β 3, which capures he impac of a policy change on y. The ordinary leas squares (OLS) esimaor of β 3, ˆβ 3, is usually referred o as DD esimaor. Since we are primarily ineresed in he DD esimaor, we could do a de-rending ransformaion o ge rid of he unobservable variables λ and a i, similar o he fixed-effecs ransformaion. Therefore, I will call he 1 a i could be eiher random or deerminisic. Asympoic resuls will no differ because of he de-rending ransformaion. 3

4 de-rended OLS esimaor he fixed-effecs OLS esimaor in he remainder. Consider he fixed-effecs OLS esimaor of β given by ( N T ) 1 N T ˆβ = x i x i x i ỹ i, (5) where in model (1) β2 DU β =, x i = x i ˆx i = β 3 T rea i DU, ỹ i = y i ŷ i, DU = DU DU, wih ŷ i = T ( T ) 1f() y is f(s) f(s)f(s) and DU = T ( T ) 1f(). DU s f(s) f(s)f(s) T rea i drops afer he ransformaion as long as f() has an inercep. In model (2) we have Noe ha β = β 3, ỹ i = y i ŷ i 1 N x i = x i ˆx i 1 N N (y j ŷ j ), N (x j ˆx j ) = T rea i DU, wih T rea i = T rea i 1 N N T rea j = 1(i kn) k. Le DU h i = T rea i DU. Here, boh T rea i and DU drop afer he ransformaion. In model (3) we have he same ỹ i and DU as in model (1) bu differen β and x i given by β 2 β = β 3, x hi i =, z i γ where z i = z i ẑ i = z i T ( T ) 1f(). z is f(s) f(s)f(s) In model (4), ỹi, z i, DU and T rea i ake he same form as in model (2). However, β and x i now become β3 T rea i β =, x i = DU. γ Plugging (1), (2), (3) or (4) ino (9) for ỹ i yields ( N T ) 1 N T ˆβ β = x i x i x i u i. (6) z i 4

5 Le ṽ i = x i u i and define ˆv i = x i û i where û i are he OLS residuals given by û i = ỹ i x ˆβ. i As shown by Driscoll and Kraay (1998), i is possible o obain sandard errors in a panel model ha are robus o general forms of spaial correlaion in he cross-secion. These sandard errors are also robus o heeroskedasiciy and serial correlaion when he serial correlaion is weakly dependen. Define ˆ v = N ˆv i, and he parial sums of ˆ v as ˆ S = ˆ v, where r (, 1 and is he ineger par of. Le ˆ Γ j = T 1 T =j+1 ˆ v ˆ v j, and hen define T 1 ˆ Ω = ˆ Γ + k( j M )(ˆ Γj + ˆ Γ j ), which is he nonparameric kernel HAC esimaor using ˆ v, he kernel, k(x), and bandwidh M. equivalen expression of ˆ Ω is given by An where ˆ Ω = T 1 T T K sˆ v ˆ v s, s K s = k( M ). When ˆ Ω is used as he middle erm of he sandwich form of he sandard error marix, we obain he robus sandard error marix proposed by Driscoll and Kraay (1998) N ˆV = T ( T x i x i) 1 ˆ Ω( N Consider esing linear hypoheses abou β of he form H : Rβ = r, T x i x i) 1. where R is a q K marix of known consans wih full rank wih q K and r is a q 1 vecor of known consans. Define he W ald saisics as W ald = (R ˆβ r) R ˆV R 1 (R ˆβ r). 5

6 In he case where q = 1 we can define he -saisics = (R ˆβ r) R ˆV R. Noe ha q 2 in model (1) and q = 1 in model (2). In hese wo cases, he focus is on he asympoic behavior of he -saisics under null hypoheses involving resricions on he DD esimaor. For model (3) and (4), he asympoic behavior of he W ald-saisics under null hypoheses involving linear resricions on he γ vecor is also derived. 3 Asympoic heory and criical values This secion analyzes he asympoic properies of he es saisics under null hypoheses in he large-t, fixed-n case. All limis are aken as T as N held fixed. Simulaed criical values are provided. Throughou, he symbol denoes weak convergence. Boh p and p lim denoe convergence in probabiliy. The asympoic disribuions of W ald and saisics under null hypoheses are obained using large-t asympoics. This approach allows he sandard errors o be approximaed wihin he fixed-b asympoic framework developed by Kiefer and Vogelsang (25) which capures he choice of kernel and bandwidh in he asympoic approximaion. Moreover, i generaes limis ha are invarian o general forms of spaial correlaion under assumpions of covariance saionariy and weak dependence in he ime dimension. The asympoic disribuions of he saisics depend on he form of he kernel used o compue he HAC esimaors. Here only Barle kernel, k(x) = 1 x for x 1 and k(x) = for x 1, is considered. Before we proceed, some definiions are required. The random marices ha appear in he asympoic resuls are expressed in erms of he following funcions and random variables. Definiion 1. Le W (r) denoe a generic vecor of independen sandard Wiener processes. Define H F (r, λ) = 1(r > λ) N F (W ) = Q F (r, λ, W ) = r r λ H F (r, λ)dw (r), H F (s, λ)dw (s) ( F(s) 1 1F(r), ds F(s)F(s) ds) dw (s)f (s) ( ( H F (s, λ) 2 1 1N ds H F (s, λ) ds) 2 F (W ). ) 1 r F (s)f (s) ds F (s)h F (s, λ)ds The following definiion defines some random marices ha appear in he asympoic resuls. Definiion 2. Le B(r) denoe a generic vecor of Brownian bridges. If k(x) is he Barle kernel, le he random marices, P F (b, λ, Q F ), P (b, B), P 21 (b, λ, Q F, B) and P 21 (b, λ, Q F, B) be defined as follows 6

7 for b (, 1 P F (b, λ, Q F ) = 2 b P (b, B) = 2 b P 12 (b, λ, Q F, B) = 2 b P 21 (b, λ, Q F ) = 2 b 1 b Q F (r, λ, W )Q F (r, λ, W ) dr b Q F (r, λ, W )Q F (r + b, λ, W ) + Q F (r + b, λ, W )Q F (r, λ, W ) dr, B(r)B(r) dr 1 b b Q F (r, λ, W )B(r) dr 1 b B(r)Q F (r, λ, W ) dr 1 b B(r)B(r + b) + B(r + b)b(r) dr, b b Q F (r, λ, W )B(r + b) + Q F (r + b, λ, W )B(r) dr, B(r)Q F (r + b, λ, W ) + B(r + b)q F (r, λ, W ) dr. For all models, he following assumpion on he rend is sufficien o obain he main resuls of he sudy. Assumpion 1. f() includes a consan, here exiss a (J J) diagonal marix τ T and a vecor of funcions F, such ha τ T f() = F( T ) + o p(1), F i(r)dr <, i = 1,..., J, and de F(r)F(r) dr >. Assumpion 1 is fairly sandard and is he same as he assumpion used by Bunzel and Vogelsang (25). Noe ha he sandard individual fixed-effecs model is a special case wih f() = 1; he individual specific rend model is a special case wih f() = (1, ). 3.1 Models wih no addiional regressors This subsecion invesigaes he asympoic properies of he saisics in models (1) and (2). For a given sack u 1, u 2,..., u N ino an (N 1) vecor u = The following assumpion is sufficien o obain resuls for he fixed-effecs OLS esimaor based on model (1) and (2). Assumpion 2. T 1 2 rt u 1 u 2. u N u ΛW (r), where W (r) is an N 1 vecor of independen sandard Wiener processes and ΛΛ is he N N long run variance marix of u. For a given ime period, sacking he N regression errors in he same period ino a vecor akes ino accoun general forms of spaial correlaion in he cross-secion. Assumpion 2 holds under covariance saionariy and weak dependence in he ime dimension. I essenially requires ha u saisfy a funcional cenral limi heorem (FCLT). Here, ΛΛ is no resriced o be diagonal. Therefore, he assumpion allows for general forms of spaial correlaion. Saionariy is no required in he cross-secion in he large-t, 7

8 fixed-n case. This is analogous o he large-n, fixed-t case where he random sampling in he crosssecion allows general forms of serial correlaion in model, including nonsaionariy. Before we sar o derive he resuls in model (1), i is worh noing ha he -saisic on he DD esimaor is exacly he same for he following hree cases 2 : 1. one includes individual dummies only; 2. one includes ime period dummies only; 3. one includes boh individual and ime period dummies. This exac equivalence resul direcly implies ha wheher ime period dummies are included does no affec he limi of he -saisic on he DD esimaor in he sandard individual fixed-effecs model. Proofs of he exac equivalence resul are provided in he Appendix. Le 1, 1,..., 1, 1,..., 1 A = 1, 1,..., 1,,..., where A is a (2 N) marix wih all elemens in he firs row and firs kn elemens in he second row equal o one. Le G = AA. The following proposiion and lemma presen he asympoic disribuions of ( ˆβ β) and he parial sums in model (1). Proposiion 1. Suppose Assumpion 1 and 2 hold. Le W (r) denoe a (2 1) vecor of sandard Wiener processes and le Λ denoe he marix square roo of he marix AΛΛ A. In model (1), for N fixed as T he following holds: T ( ˆβ β) ( G ) 1 1 H F (r, λ) 2 dr Λ H F (r, λ)dw (r). Lemma 2. Suppose Assumpion 1 and 2 hold. Assume M = bt where b (, 1 is fixed. Le W (r) denoe a (2 1) vecor of sandard Wiener processes and le Λ denoe he marix square roo of he marix AΛΛ A. In model (1), for N fixed as T he following holds: T 1 2 ˆ Sr Λ Q F (r, λ, W ). When k(x) is he Barle kernel, from calculaions in Hashimzade and Vogelsang (28) we have ˆ Ω = 2 T b T 1 2 ˆ S ˆ S 1 T M 1 b T 2 ( ˆ S ˆ S +M + ˆ S+M ˆ S ) (7) using he fac ha ˆ ST =. The following proposiion presens he fixed-b limi of he HAC esimaor. Proposiion 3. Suppose Assumpion 1 and 2 hold. Assume M = bt where b (, 1 is fixed. Le W (r) denoe a (2 1) vecor of sandard Wiener processes and le Λ denoe he marix square roo of he marix AΛΛ A. In model (1), for N fixed as T he following holds: ˆ Ω Λ P F (b, λ, Q F )Λ. 2 The same resul can be obained by including a global inercep only. 8

9 Based on Proposiion 1 and 3, he following heorem summarizes he heoreical resuls for model (1). Theorem 1. Suppose he model does no include ime period dummies nor addiional regressors. Suppose Assumpion 1 and 2 hold. Assume M = bt where b (, 1 is fixed. Le Wq sandard Wiener processes. For N fixed as T, W ald N F (W q ) P F (b, λ, Q F q ) 1 N F (Wq ) N F (W1 ) P F (b, λ, Q F 1 ) denoe he q 1 vecor of Theorem 1 demonsraes ha pivoal es saisics are obained wihin he fixed-b framework in he presence of spaial correlaion in he cross-secion. Therefore, he saisics based on he DK sandard errors under fixed-b asympoics have broader robusness properies wih respec o correlaion in he model. The limiing disribuions differ from he pure ime series resuls obained by Kiefer and Vogelsang (25) in he following wo aspecs. Firs, he limiing disribuions of he es saisics depend on no only he choice of kernel and bandwidh, bu also he policy change poin λ and he choice of individualspecific rend funcions. Second, he asympoic resul is differen from Vogelsang (212) because DU is deerminisic. The limiing disribuions of he es saisics are idenical o he pure ime series resuls in a shif in mean model wih deerminisic rends. criical values are easily obained using simulaion mehods. The limiing disribuions are non-sandard, bu Corollary 1. Suppose model (1) is a sandard individual fixed-effecs model wih no ime rends. Tha is, f() = 1. Define λw (1) W (λ) = (λ 1) W ( λ λ 1 ). Le W q denoe he q 1 vecor of sandard Wiener processes. Then H F (r, λ) = 1(r > λ) (1 λ), N F (W ) = λw (1) W (λ) = (λ 1) W λ ( λ 1 ), r r r ( Q F (r, λ, W ) = H F (s, λ)dw (s) W (1) H F (s, λ)ds H F (s, λ) 2 1 1N ds H F (s, λ) ds) 2 F (W ). For N fixed as T, he following hold W ald N F (W q 1 T ( ˆβ β) λ (1 λ) G 1 Λ (λ 1) W λ ( λ 1 ) ) P F (b, λ, Q F q ) 1 N F (W q ), N F (W1 ) P F (b, λ, Q F 1 ) Corollary 1 provides resuls for a sandard individual fixed-effecs DD model. The limis are idenical o he pure ime series resuls in a shif in mean model. When ime period dummies are also included in he model (model (2)), he limiing disribuions of he saisics remain he same due o he exac equivalence resul. This finding is useful since empirical researchers always pu a full se of ime period dummies in heir model. 9

10 3.2 Models wih addiional regressors This subsecion analyzes he asympoic properies of he saisics in models (3) and (4). Some addiional noaions in his subsecion are needed as follows. Le I h denoe a h h ideniy marix. Le ι denoe an N 1 vecor of ones. Le e i denoe an N 1 vecor wih i h elemen equal o one and zeros oherwise, i.e. e i = (,,...,, 1,,..., ). Define a K (K + 1) marix B and a K N(K + 1) marix A i as follows B =, I K, A i = (e i B). Le ẽ 1 denoe an (K + 1) 1 vecor wih 1 s elemen equal o one and zeros oherwise, i.e. ẽ 1 = (1,,..., ). Le ē 1 denoe an (NK + 1) 1 vecor wih 1 s elemen equal o one and zeros oherwise, i.e. ē 1 = (1,,..., ). The following assumpion on addiional regressors z i is sufficien o obain resuls for he fixed-effecs OLS esimaor based on models (3) and (4). Assumpion 3. Suppose here is no srucural change for z i wihin he enire sample periods. Assume ha p lim T 1 T = µ i E(z i ) and p lim T 1 z i z i = rq i for r (, 1 where Q = N Q i and Q is nonsingular. To handle he case where addiional regressors are also included (model (3)), Assumpion 2 needs o be srenghened as follows. Sack he addiional regressors z i and rend funcions and consider he reduced form of he (T K) sacked vecor z i ha is, he linear projecion of z i ono he space spanned by he (T J) sacked vecor of rend funcions f(t ) wih an error erm as z i = f(t )b i + e i, where e i is a (T 1) vecor and b i is a (J K) vecor. I is easy o show ha z i are he OLS residuals given by z i = z i ˆb if(), where ˆb i is he OLS esimaor of b i. Define he (K + 1) 1 vecor Sack he vecors v 11,..., v NN v ii = u i (z i b i f())u i o form he N(K + 1) 1 vecor of ime series v 11 v 22 ṭ v =.. v NN. 1

11 Assumpion 4. E(u i z i ) = and T 1 2 rt v ΛW (r), where W (r) is an N(K + 1) 1 vecor of sandard Wiener processes and Λ Λ is he N(K + 1) N(K + 1) long run variance marix of v. Assumpion 3 requires ha he addiional regressors don have srucural change before and afer he policy change and ha he sample mean and sample variance-covariance marix of he addiional regressors across ime have well-defined limis. The form of Q i depends on he form of dummies included in he model and he choice of he rend funcions. Assumpion 4 allows weak exogeneiy in he crosssecion and over ime and requires a FCLT holds for v. Because Q i is no resriced o be idenical for all i and because he form of Λ Λ is no resriced o be block diagonal, he assumpions allow for heerogeneiy in he condiional heeroskedasiciy and serial correlaion as well as general forms of spaial correlaion. The following lemma shows ha h i and z i are asympoically uncorrelaed. Lemma 4. Under Assumpion 1 and 3, for N fixed and as T, he following holds In paricular, when r = 1, T 1 N T h i z i T 1 N p. h i z i Noe ha no assumpions on he correlaion beween he indicaors and he addiional regressors are required. Le R = p R11 R 12 R 21 R 22 where R 11 is a q 1 2 marix, R 12 is a q 1 K marix, R 21 is a q 2 2 marix and R 22 is a q 2 K marix. Usually we pay aenion o resricions eiher on he DD esimaor or on he addiional explanaory variables, no on boh of hem a he same ime. In oher words, we are ineresed in he cases when q 2 = and R 12 =, or when q 1 = and R 21 =. The nex heorem presens he resuls for model (3). Theorem 2. Suppose he model includes addiional regressors bu no ime period dummies. Suppose Assumpion 1, 3 and 4 hold. Assume M = bt where b (, 1 is fixed. Le W (r) denoe a (q 1 1) vecor of sandard Wiener processes. Le W q (r) denoe a (q 2 1) vecor of sandard Wiener processes. Le W (r) denoe a (2 1) vecor of sandard Wiener processes and Λ is he marix square roo of he marix (A ẽ 1 ) Λ Λ (A ẽ 1 ). For N fixed as T, he following hold: 1 (G T ( ˆβ β) HF (r, λ) 2 dr) 1 ( Λ 1 HF (r, λ)dw (r) Q 1 ( N A i) ΛW (1) If q 2 = and R 12 =, ha is, we are esing resricions on he DD esimaor, hen R = R 11, W ald N F ( W ) (P F (b, λ, Q F )) 1 N F ( W ) N F ( W ) P1 F (b, λ, Q F 1 ) 11

12 If q 1 = and R 21 =, ha is, we are esing resricions on he addiional regressors, hen R =, R 22 W ald W q (1) P q (b, B) 1 W q (1) W q(1) Pq (b, B) Theorem 2 provides some ineresing insighs ino doing inference for DD esimaor and he addiional regressors under fixed-b asympoics. If we only focus on esing resricions on DD esimaor, he limiing disribuions of he saisics urn ou o be he same as he resuls in Theorem 1. If we only wan o es resricions on he addiional regressors z i, he limiing disribuion of he saisics are idenical o he resuls in Vogelsang (212). Noe ha he limiing disribuions of saisics based on z i are invarian o he choice of rend funcions. In eiher case, he es saisics are asympoically pivoal. Neverheless, esing resricions on boh of hem a he same ime is much more complicaed. The es saisics are no longer asympoically pivoal. General forms of he limis of he es saisics are provided in he proof of Theorem 2 in he Appendix. The mos general model including boh addiional regressors and ime period dummies (model (4)) requires a sronger assumpion han Assumpion 4. To cope wih his case, Assumpion 4 needs o be srenghened in he following way. Define he K 1 vecor v ij = (z i b i f())u j. For a given j sack u j and he vecors v 1j, v2j,..., vnj ino an (NK + 1) 1 vecor v j = u j v 1j v 2j ṭ. v Nj and hen sack he vecors v 1, v 2,..., v N ino an N(NK + 1) 1 vecor v 1 v ex v 2 =. where he ex superscrip denoes an exended vecor ha includes vecors v ij for i j. Assumpion 5. E(u i z j ) = for all i, j = 1, 2,..., N and T 1 2 v N rt v ex Λ ex W ex (r), where W ex (r) is an N(NK + 1) 1 vecor of sandard Wiener processes and Λ ex Λ ex is he N(NK + 1) N(NK + 1) long run variance marix of v. Assumpion 5 requires sric exogeneiy in he cross-secion bu allows weak exogeneiy over ime. I also requires ha a FCLT hold for he exended vecor v ex. Here, Λ ex Λ ex is no resriced o be 12

13 block diagonal, which permis general spaial correlaion. Assumpion 4 and 5 indicae ha he form of exogeneiy needed depends on wheher or no ime period dummies are included in he model. Wihou ime period dummies, only weak exogeneiy is required in boh he ime and cross-secion dimensions. When ime period dummies are included, sric exogeneiy is needed in he cross-secion dimension while only weak exogeneiy is required in he ime dimension. Like resuls in model (2), including ime period dummies does no affec he limiing resuls. The following heorem summarizes he resuls for model (4). Noe ha Assumpion 4 is now replaced wih he sronger Assumpion 5. Theorem 3. Suppose he model includes boh addiional regressors and ime period dummies. Suppose Assumpion 1, 3 and 5 hold. Assume M = bt where b (, 1 is fixed. Le à = 1 k,..., 1 k, k,..., k and G = Ãà = N T rea 2 i. Le W1 ex (r) denoe a sandard Wiener processes wih long run variance Λ ex 2 1 = (à ē 1 )Λex Λ ex (à ē 1 ). For N fixed as T, he following hold: 1 ( G T ( ˆβ β) HF (r, λ) 2 dr) 1 Λ ex 1 HF (r, λ)dw1 ex (r) Q 1 ( N Aex i )Λ ex W ex (1) and he limis of he saisics are he same as given by Theorem 2. Theorem 3 demonsraes ha resuls for saisics in Theorem 2 coninue o hold when ime period dummies are included. This is consisen o he finding in model (2). 3.3 Asympoic criical values The asympoic criical values for W ald and saisics based on DD esimaor can be obained hrough Mone Carlo simulaions. To keep he analysis sraighforward, I consider he case q = 1 and focus on he sandard individual fixed-effecs model and he individual-specific rend model. The asympoic criical values are simulaed using 5, replicaions. The Wiener processes are approximaed by normalized sums of i.i.d. N(, 1) errors using 1 seps. The criical values for saisics in he sandard individual fixed-effecs model are presened in Table 1 hrough 4. The criical values for saisics in he individualspecific rend model are presened in Table 5 hrough 8. Using he Barle kernel, criical values are compued for he percenage poins 9%, 95%, 97.5%, and 99%. Righ ail criical values are given. The lef ail criical values follow from symmery around zero. The policy change poin λ goes from.1 o.9 wih sep size.1. The bandwidhs b sars from.2 o 1 wih sep size.2. The criical values are invarian o he values of k. For a given b, he criical values are symmeric around λ =.5 wih respec o λ. The minimum value occurs a λ =.5. As λ approaches zero or one, he criical values increase. This paern is he same as he pure ime series model wih a known srucural break, Cho (see 212). For a given λ, wih b =.2, criical values are close o N(, 1) regardless of he choice of rend funcions. As b grows, ails ge faer. Wih b = 1 ails are quie fa. For differen choices of rend funcions, ails ge faer in differen raes. For example, when λ =.5, in he sandard individual fixed-effecs model he criical values a 5%/2.5% ails wih b =.2 and b = 1 are 1.712/

14 and 4.781/5.958, respecively, while in he individual specific model, he criical values a 5%/2.5% ails wih b =.2 and b = 1 are 1.745/2.73 and 5.98/6.395, respecively. Therefore, ails ge faer more quickly in he individual-specific rend model. The criical values predic ha if N(, 1) criical values are used for saisics, hen for a given value of T, as bandwidh M increases, b increases and hus will over-rejec. 4 Finie sample properies In his secion he finie sample performance of he DK sandard errors is examined using a simulaion sudy. The focus here is on cases where here is spaial correlaion in he cross secion. Simulaion resuls for oher cases can be found in Sun (212). Because using radiional clusered sandard errors is he mos common mehod o conduc robus inference for DD esimaor, he asympoic approximaions given by he heorems are compared wih he radiional asympoics as well as he asympoics for radiional clusered sandard errors. clus denoes -saisics consruced using radiional clusered sandard errors and DK denoes -saisics consruced using he DK sandard errors. The daa generaing process (DGP) used for he simulaions is y i = c i + g i + β 1 T rea i + β 2 DU + β 3 T rea i DU + z i γ + u i, (8) where u i = ρu i, 1 + ε i, u i =, ε i N(, 1), cov(ε i, ε js ) = for s; z i = ρz i, 1 + e i, z i =, e i N(, 1), cov(e i, e js ) = for s. c i is he individual fixed-effecs and g i is he individual-specific simple linear rend. In all cases, all coefficiens are se o zero. Also se c i =, g i =, k =.5 and λ =.5. Only one addiional regressor z i is included and i is uncorrelaed wih u i. z i and u i are modeled as AR(1) processes wih he same auoregressive parameer. ε i and e i have spaial correlaion in he cross-secion, hough uncorrelaed over ime. In paricular, hey are consruced in he following way. For a given ime period,, N i.i.d. N(, 1) random variables are placed on a square grid. A each grid poin, ε i is consruced as he weighed sum of he normal random variable a ha grid poin, he normal random variables ha are one sep away o he lef, righ, up or down on he grid wih a weigh θ and he normal random variables ha are wo seps away in he same direcion wih a weigh θ 2. Hence, ε i is a spaial MA(2) process wih parameer θ and he disance measure is maximum coordinae-wise disance on he grid. e i is consruced in a similar way. In all cases, θ =.5. Resuls are given for sample sizes T = 1, 5, 25 and N = 9, 49, 256. The number of replicaions is 25 in all cases and he significance level is.5. Resuls are repored for he Barle kernel. Fixedeffecs OLS as discussed in secion 2 is used o esimae he model. Resuls on he DD esimaor are for DD -saisics for esing he null hypohesis H : β 3 = agains he alernaive H 1 : β 3. Resuls on z i are for z -saisics for esing he null hypohesis H : γ = agains he alernaive H 1 : γ. 14

15 Resuls wih ime period dummies included are no repored because wheher ime period dummies are included in he model generaes exacly he same resuls when z i is no included. Wih z i included, including ime period dummies shows similar paerns. Table 9 repors empirical null rejecion probabiliies for clus and DK saisics in he sandard individual fixed-effec model wih no addiional regressor z i. For DK a small selecion of bandwidhs are considered, b =.2,.6,.1,.4,.7, 1. For DK wo ses of null rejecion probabiliies are repored. The firs se is based on rejecions using he 5% N(, 1) criical value provided ha he sandard errors are consisen. The second se uses he new fixed-b criical values (adjused fixed-b criical values) obained in subsecion 3.3. For clus rejecion probabiliies are repored using he 5% N(, 1) criical value. There are several poins worh noing. In Table 9 rejecion probabiliies for clus are subsanially larger han.5. This is expeced since he radiional clusered sandard errors are no robus o he spaial correlaion in he cross-secion. For DK paerns in he rejecion probabiliies are quie differen when N(, 1) criical value is used compared o when he adjused fixed-b criical values are used. Using N(, 1) criical value, rejecion probabiliies end o be much higher han.5 and his over-rejecion problem ges worse as b increases or as ρ increases. Only when b is small, T is large, and ρ is close o zero are rejecion probabiliies close o.5. In conras, when he adjused fixed-b criical values are used, he over-rejecion problem is less severe. For a given N, T, ρ combinaion, rejecion probabiliies are above.5 wih small b and hey seadily decline as b increases. For a given value of ρ, as T increases, rejecion probabiliies approach.5 for all bandwidhs. When T = 25 and b = 1, rejecion probabiliies are no ha far from.5 even when ρ =.9, around.9. The paerns in he rejecion probabiliies of DK are similar o Vogelsang (212). As explained in Vogelsang (212), he bias in ˆ Ω consiss of wo pars. One par depends on he srengh of he serial correlaion and his bias rises as he serial correlaion becomes sronger, which explains why he overrejecion problem ges worse as ρ increases. This bias causes over-rejecion for eiher he N(, 1) criical value or he adjused fixed-b criical values. However, his bias declines as b increases. The oher par is capured by he adjused fixed-b approximaions, bu no he N(, 1) approximaions. Therefore, overrejecion becomes less severe when fixed-b criical values are used. I is shown (see Vogelsang, 28) ha as b increases, bias in ˆ Ω iniially decreases bu hen increases as b increases furher. Because of his, when b is close o one, ˆ Ω has subsanial downward bias and DK ends o over-rejec when he N(, 1) criical value is used. Overall, he N(, 1) approximaions do no reflec he influence of he bandwidh, and hus using he N(, 1) criical value may lead o severe disorions in rejecions. In conras, he fixed-b approximaions capure mos of he bias in ˆ Ω. In addiion, he par ha hey canno capure decreases as b increases. This demonsraes why he rejecion probabiliy of DK is lowes a b = 1 when adjused fixed-b criical values are used. Table 1 repors empirical null rejecion probabiliies for DD and z saisics in he individualspecific rend model wih one addiional regressor z i. For each saisic, a small selecion of bandwidhs are considered, b =.2,.6,.1,.4,.7, 1. For DD, he adjused fixed-b criical values are used. For z rejecion probabiliies are repored using he usual fixed-b criical values in Kiefer and Vogelsang (25) 15

16 and Vogelsang (212). The firs hing o noe here is ha he usual fixed-b criical values are used for z because here is no srucural break in z i. Specifically, hese criical values are invarian o he choices of rend funcions. Paerns of he rejecion probabiliies are idenical o he findings in Vogelsang (212). Now focus on he DD esimaor. The fixed-b approximaion capures he change in he choice of rend funcions when a simple linear rend is included in he model. Similar paerns of rejecion probabiliies as Table 9 are obained. 5 Conclusion This sudy derives a fixed-b asympoic heory for es saisics in Difference-in-Difference models wih fixed-effec and individual specific rends in linear panel seings. The sandard errors proposed by Driscoll and Kraay (1998) ha are robus o heeroskedasiciy, auocorrelaion and spaial correlaion are analyzed. This paper esablishes he condiions under which he DK sandard errors lead o valid ess in linear DD models wih fixed-effecs and individual-specific ime rends for he fixed-n, large-t case. I is shown ha he fixed-b asympoics for ess on he DD esimaor are differen from he resuls in Vogelsang (212), bu hey are idenical o he pure ime series resuls in a shif in mean model for he sandard individual fixed-effecs model. The ess on addiional regressors wihou a srucural break have he same fixed-b asympoic disribuions as in Vogelsang (212). The exac equivalence resul is found for he cases when only individual dummies are included, when only ime period dummies are included and when boh ses of dummies are included. As a resul, wheher ime period dummies are included in he model does no influence he asympoic disribuion. I is also shown ha he fixed-b asympoics for ess on DD esimaor depend on he individual-specific deerminisic rends included and he policy change poin λ. New criical values are simulaed for sandard fixed-effecs model and individual specific rend model. For each value of bandwidh, he adjused criical values shows a U-shaped paern in λ. Tails ge faer in differen raes for differen rend funcions. Simulaion resuls illusrae ha he use of fixed-b criical values will lead o much more reliable inference in pracice in he presence of spaial correlaion. Appendix. Proofs Proofs of he exac equivalence resul, Proposiion 1 and 3, Lemma 2 and 4, Theorem 1 2 are provided in his Appendix. Proof of he exac equivalence resul. I is sraighforward o obain T DU 2 = λ(1 λ)t, DU DU s = 16

17 DU DU s (1 λ)du (1 λ)du s + (1 λ) 2 and N T rea 2 i = k(1 k)n. Define Recall = ˆβ 3 β 3 s.e.( ˆβ 3 ). η = λ ξ = k 2 S N S N s kn T u i kn λt u i kλ N T N u i + k λt u i ks kn Ss N ks N Ss kn + S kn Ss kn = (S kn ks N )(Ss kn kss N ) N kn S N = u i, S k N = u i Consider he individual dummies case. We have ( ˆβ N T DU β = T rea i DU DU, T rea i DU ) N 1 T DU T rea i DU u i (9) Simple algebra yields ( N T DU ( N = T rea i DU DU, T rea i DU ) 1 T DU 2 1 T rea ) 1 ( i 1 k ) 1 1 = λ(1 λ)nt = T rea i T rea i k k λk(1 λ)(1 k)nt k k k 1 (1) N T DU N 1 T T rea i DU u i = DU u i = T rea i λ N T u i N λt u i = λ kn T u i kn λt u i N 1 ( λ T rea i T u i λt u i ) (11) Plugging (1) and (11) ino (9), i direcly follows In paricular, we have ˆβ β = 1 λk(1 λ)(1 k)nt ˆβ 3 β 3 = k(λ i>kn T u i Nex, consider he sandard error marix. We know N DU ˆ v = T rea i DU u i = DU S N S kn η λt i>kn u i ) η λk(1 λ)(1 k)nt. (12) 17

18 Therefore, ˆ Ω = T 1 T = T 1 T Using his formula, i follows ( 1 N T DU T T rea i DU ( 1 ) 2 k k = λk(1 λ)(1 k)n k 1 ( 1 = λk(1 λ)(1 k)n Specifically, we have T K sˆ v ˆ v s = T 1 T T S N K s DU DU Ss N s S kn T S N K s DU DU s S N S kn s Ss N S kn DU, T rea i DU 1 ( 1 ) ˆ Ω T k k ˆ Ω k 1 ) 2 s.e.( ˆβ 3 ) = T 1 T T K s DU DU s ξ 1 T (λk(1 λ)(1 k)n) 2 N S kn s T T S kn DU T rea i DU Now consider he individual and ime dummies case. Similarly we can derive S N s, S kn s DU, T rea i DU ) 1 T K s DU DU s ξ. (13) ( ˆβ N 3 β 3 = = = T T rea 2 i DU 2 ) 1 N 1 λk(1 λ)(1 k)nt η λk(1 λ)(1 k)nt For he sandard error marix, i is easy o show N T rea i (λ T T rea i DU u i T u i λt u i ) (14) and ˆ Ω = T 1 T Thus, i follows s.e.( ˆβ ( 1 3 ) = T T K sˆ v ˆ v s = T 1 N ˆ v = T N T T T rea 2 i DU 2 T rea i DU u i = DU (S kn ks N ) K s DU DU s (S kn ks N )(Ss kn kss N ) = T 1 ) 2 ˆ Ω = 1 T (λk(1 λ)(1 k)n) 2 T T T K s DU DU s ξ T K s DU DU s ξ. (15) From above, we know he op and he boom of saisics are exacly equivalen in hese wo cases. As a resul, saisics are exac equivalen in hese cases. By symmery, i is easy o show ha his exac equivalence resul holds in he case when only ime period dummies are included. 18

19 Proof of Proposiion 1. T ( ˆβ β) = (T 1 N T x i x i ) 1 (T 1 2 N T x iu i ). Using Assumpion 1 and 2, i can be shown ha T 1 2 T 1 Therefore, N N T x i u i = T 1 2 T ( x 1,..., x N ) u = T 1 2 = A T 1 2 AΛ T x i x i = T 1 T A DU u T DU T ( T 1 DU s f(s) τ T T 1 1(r > λ) = Λ H F (r, λ)dw (r) T A DU DU A = G T 1 λ T ) 1τT τ T f(s)f(s) τ T f() u ( F(s) 1 ) 1F(r) ds F(s)F(s) ds dw (r) T DU 2 T = G T DU T ( 1 T 1 DU s f(s) τ T T 1 G = G 1(r > λ) H F (r, λ) 2 dr λ T ) 1τT 2 τ T f(s)f(s) τ T f() ( F(s) 1 1F(r) 2dr ds F(s)F(s) ds) T ( ˆβ β) (G H F (r, λ) 2 dr) 1 Λ H F (r, λ)dw (r) Proof of Lemma 2. Using Assumpion 1, 2 and Proposiion 1, we obain T 1 2 ˆ Sr = T 1 2 ˆ v = T 1 2 = T 1 2 = A T 1 2 N x i u i T 1 2 (G T 1 N x i û i = T 1 2 N DU u A T 1 2 T DU 2 ) T ( ˆβ β) Λ r H F (s, λ)dw (s) r ( H F (s, λ) 2 1 ds = Λ Q F (r, λ, W ) x i ü i (T 1 ( 1 u s f(s) τ T T dw (s)f (s) ( 1N H F (s, λ) ds) 2 F (W ) N x i ũ i x i( ˆβ β) N x i x i) T ( ˆβ β) T ) 1 1 τ T f(s)f(s) τ T τ T f() DU T ) 1 r F (s)f (s) ds F (s)h F (s, λ)ds 19

20 because T 1 2 N x i û i = T 1 2 N T T x i ( u is f(s) )( f(s)f(s) ) 1 f() T N T = T 1 2 ( x i u is f(s) )( f(s)f(s) ) 1 f() T T = T 1 2 ( A DU u s f(s) )( f(s)f(s) ) 1 f() = A T 1 2 T u s f(s) τ T ( 1 T T τ T f(s)f(s) τ T ) 1 1 τ T f() DU T Proof of Proposiion 3. I direcly follows from (7), Lemma 2 and he coninuous mapping heorem ha ˆ Ω = 2 T b T 1 1 T 1 2 ˆ S T 1 2 ˆ S 1 T M 1 b T 1 (T 1 2 ˆ S T 1 2 ˆ S +M + T 1 2 ˆ S+M T 1 2 ˆ S ) 2 b 1 b b Λ Q F (r, λ, W )Q F (r, λ, W ) Λ dr = Λ P F (b, λ, Q F )Λ Λ Q F (r, λ, W )Q F (r + b, λ, W ) + Q F (r + b, λ, W )Q F (r, λ, W ) Λ dr Proof of Theorem 1. Using Proposiion 3, i direcly follows ha R (T N 1 ( R G T 1 x i x i) ˆ Ω (T N 1 T x i x i ) 1R H F (r, λ) 2 dr) 1Λ P F (b, λ, Q F )Λ ( G = P F (b, λ, Q F ( r, λ, R(G = Λ q P F (b, λ, Q F (r, λ, Wq ) H F (r, λ) 2 dr) 1 Λ W ) ))Λ q Using Proposiion 1, we have R T ( ˆβ ( ) 1 1 β) R G H F (r, λ) 2 dr Λ Wih (16) and (17), i follows ha W ald = (R ˆβ r) R ˆV R 1 (R ˆβ r) = (R T ( ˆβ β)) R (T N 1 (Λ q T = Λ q P F (b, λ, Q F q H F (r, λ)dw (r) = Λ q 1 x i x i) ˆ Ω (T N 1 H F (r, λ)dw q (r)) Λ q P F (b, λ, Q F = N F (Wq ) P F (b, λ, Q F q ) 1 N F (Wq ) When q = 1, i direcly follows ha N F (W 1 ). P F (b,λ,q F 1 ) 2 q )Λ q ) 1R H F (r, λ) 2 dr )Λ q (16) H F (r, λ)dw q (r) (17) T 1R x i x i) 1 R T ( ˆβ β) 1 Λ q H F (r, λ)dw q (r)

21 Proof of Lemma 4. T 1 T DU z i = T 1 T s=λt +1 f(s) ( T f(s)f(s) ) 1 using he fac ha T f() z i =. Hence, T 1 DU z i = o p(1). If r > λ, hen T f() z i = (18) T 1 DU z i = T 1 =λ+1 = T 1 =λ+1 z i ( p (r λ) µ i z i T 1 = (r λ)(µ i µ i) = =λ+1 f() τ T ( T 1 T τ T f(s)f(s) τ T ) 1T 1 ( F(r) 1 1(µ ) dr F(r)F(r) dr) i,,..., ) T τ T f(s)z is (19) If r λ, hen T 1 From (18), (19) and (2), i direcly follows ha DU z i = (2) and hus T 1 T 1 N DU z i = T 1 (DU DU ) z i h i z i = N 1 T rea i T 1 DU z i p (21) p. Proof of Theorem 2. The K 1 vecor z i u i can be wrien in erms of he N(K + 1) 1 vecor v as follows z i u i = (z i ˆb if())u i = ((z i b if()) (ˆb if() b if()))u i = Bv ii = A i v (τ 1 T (ˆb i b i )) τ T f()u i Using his formula i is easy o show ha T 1 2 z i u i = T 1 2 (A i v (τ 1 T (ˆb i b i )) τ T f()u i ) (ˆb i b i ) f()u i = A i T 1 2 v T 1 2 ( T τ 1 T (ˆb i b i )) T 1 2 τ T f()u i = A i T 1 2 v + T 1 2 Op (1) O p (1) = A i T 1 2 v + o p (1) A i ΛW (r) (22) 21

22 using Assumpion 1 and 4. Wih Assumpion 1, 3, 4, Lemma 4 and (22), simple algebra gives Le T ( ˆβ β) = ( N N = N T 1 T T 1 T T 1 T x i x i h i h i z i h i ) 1 ( N N T 1 2 T 1 T N T 1 T (G HF (r, λ) 2 dr) 1 Q 1 T ) x i u i h i z i z i z i 1 N N T 1 2 T 1 2 T h i u i T z i u i (A ẽ 1 ) Λ 1(r > λ) λ F(s) ds( 1 F(s)F(s) ds) 1 F(r)dW (r) ( N A i ) ΛW (1) (G = HF (r, λ) 2 dr) 1 ( Λ 1 HF (r, λ)dw (r) ( Q 1 ( N A i) ΛW (1) Λ Λ = ( N A i ) Λ which is a (K + 2) (K + 2) block diagonal marix. Using he fac ha T z i f() =, i follows ha N z i û i = = = N T z i ( N u is f(s) ( T f(s)f(s) ) 1 f() z i f() )( T f(s)f(s) ) 1 T T N ( T o p (1) f(s)f(s) ) 1 u is f(s) u is f(s) p (23) 22

23 The limis of he parial sums ˆ S are easy o obain T 1 2 ˆ S = T 1 2 = = N T 1 rt 2 T 1 rt N 2 x i ũ i (T 1 N h i (u i û i ) T z i (u i û i ) N x i x i) T ( ˆβ β) rt 1 N T 1 N h i h rt i T 1 N h i z i T ( ˆβ β) N z i z i ) 1 r F (s)hf (s, λ)ds z i h i T 1 Λ r HF (s, λ)dw (s) dw (s)f (s) ( F (s)f (s) ds r G HF (s, λ) 2 ds r Q Λ Q F (r, λ, W ) ( N A i ) ΛB(r) The limi of ˆ Ω can be wrien as ( N A i ) ΛW (r) (G HF (r, λ) 2 dr) 1 ( Λ 1 HF (r, λ)dw (r) Q 1 ( N A i ) ΛW (1) = Λ Q F (r, λ, W ) B(r) ˆ Ω = 2 T b T 1 1 T 1 2 ˆ S T 1 2 ˆ S 1 T M 1 b T 1 (T 1 2 ˆ S T 1 2 ˆ S +M + T 1 2 ˆ S+M T 1 2 ˆ S ) 2 1 Q Λ F (r, λ, W ) Q F (r, λ, W ) b Λ dr 1 B(r) B(r) b ) Q F (r + b, λ, W ) Q F (r, λ, W ) Λ dr B(r + b) B(r) = Λ P F (b, λ, Q F ) P 12 (b, λ, Q F, B) Λ P 21 (b, λ, Q F, B) P (b, B) b ( Q Λ F (r, λ, W ) Q F (r + b, λ, W ) B(r) B(r + b) R(T 1 N T x i x i) 1 ˆ Ω(T N 1 T x i x i) 1 R R11 R 12 (G HF (r, λ) 2 dr) 1 R 21 R 22 (G HF (r, λ) 2 dr) 1 R11 R 12 Q 1 R 21 R 22 R11 (G = HF (r, λ) 2 dr) 1 Λ R Q 1 12 ( N R 21 (G HF (r, λ) 2 dr) 1 Λ R Q 1 22 ( N Q 1 R11 (G HF (r, λ) 2 dr) 1 Λ R 12 Q 1 ( N A i) Λ R 21 (G HF (r, λ) 2 dr) 1 Λ R Q 1 22 ( N A i) Λ Λ P F (b, λ, Q F ) P 12 (b, λ, Q F, B) Λ P 21 (b, λ, Q F, B) P (b, B) A i) Λ P F (b, λ, Q F ) P 12 (b, λ, Q F, B) A i) Λ P 21 (b, λ, Q F, B) P (b, B) (24) 23

24 R T ( ˆβ β) R11 R 12 R 21 R 22 (G HF (r, λ) 2 dr) 1 ( Λ Q 1 ( N A i) ΛW (1) HF (r, λ)dw (r) (25) If q 2 = and R 12 =, ha is, we are esing resricions on he DD esimaor, hen R = R 11, and he limis of (24) and (25) are simplified as follows and R (T N 1 R 11 (G T R T ( ˆβ β) R 11 (G 1 x i x i) ˆ Ω (T N 1 T x i x i ) 1R H F (r, λ) 2 dr) 1 Λ P F (b, λ, Q F ) Λ (G H F (r, λ) 2 dr) 1 Λ H F (r, λ) 2 dr) 1 R 11 = Λ 1 P F (b, λ, Q F ) Λ 1 H F (r, λ)dw (r) = Λ 1 H F (r, λ)d W (r) where W(r) is a q 1 1 vecor of sandard Wiener processes and Λ 1 is he marix square roo of he marix I direcly follows ha R 11 (G H F (r, λ) 2 dr) 1 Λ Λ (G H F (r, λ) 2 dr) 1 R 11. W ald ( Λ 1 H F (r, λ)d W (r)) ( Λ 1 P F (b, λ, Q F ) Λ 1 ) 1 Λ1 H F (r, λ)d W (r) = ( H F (r, λ)d W (r)) (P F (b, λ, Q F )) 1 H F (r, λ)d W (r) If q 1 = and R 21 =, ha is, we are esing resricions on he addiional regressors, hen R =, R 22 and he limis of (24) and (25) are simplified as follows R(T 1 N T x i x i) 1 ˆ Ω(T N T 1 x i x i) 1 R N N R 22 ( Q i ) 1 ( A i ) ΛP (b, B) Λ N N ( A i ) ( Q i ) 1 R 22 = Λ 2 P (b, B) Λ 2 R T ( ˆβ N N β) R 22 ( Q i ) 1 ( A i ) ΛW (1) = Λ 2 W q (1) where W q (1) is a q 2 1 vecor of sandard Wiener processes and Λ 2 is he marix square roo of he marix N N R 22 ( Q i ) 1 ( A i ) Λ Λ N N ( A i ) ( Q i ) 1 R 22 I direcly follows ha W ald ( Λ 2 W q (1)) ( Λ 2 P (b, B) Λ 2 ) 1 Λ2 W q (1) = W q (1) P q (b, B) 1 W q (1) 24

25 Proof of Theorem 3. The key sep is o show ha he limis of T ( ˆβ β) and T 1 2 ˆ S ake he same form as in Theorem 2. Once hese resuls are obained, he res of he proof closely follows he proof in Theorem 2 and deails are omied. Wih boh rend funcions and ime period dummies in he model i follows ha z i u i = (z i ˆb if())u i N 1 N (z j ˆb jf())u i N = ((z i b if()) (ˆb if() b if()))u i N 1 ((z j b jf()) (ˆb jf() b jf()))u i N N = (z i b if())u i N 1 (z j b jf())u i (ˆb if() b if())u i + N 1 (ˆb jf() b jf())u i = v ii N 1 N v ji (ˆb i b i ) f()u i + N 1 N (ˆb j b j ) f()u i = (, e i I K 1 N, ι I K )(e i I NK+1 )v ex = A ex i v ex (τ 1 T (ˆb i b i )) τ T f()u i + N 1 Using his formula i direcly follows ha N (ˆb i b i ) f()u i + N 1 (τ 1 T (ˆb j b j )) τ T f()u i N (ˆb j b j ) f()u i T 1 2 z i u i = A ex i T 1 2 v ex T 1 2 ( T τ 1 T (ˆb i b i )) T 1 2 N + N 1 T 1 2 ( T τ 1 T (ˆb j b j )) T 1 2 τ T f()u i = A ex i T 1 2 v ex + o p (1) A ex i Λ ex W ex (r) τ T f()u i (26) using Assumpion 1 and 5. Using (21), we have T 1 N T rea i DU z i = T 1 = T 1 = T 1 = N N T rea i DU z i ẑ i 1 N N N T rea i T 1 N (z j ẑ j ) T rea i DU (z i ẑ i ) T 1 T rea i DU (z i ẑ i ) T 1 DU (z i ẑ i ) p 1 N 1 N N N (z j ẑ j ) T rea i DU N (z j ẑ j ) 25

26 Using Assumpion 1, 3 and (26) i immediaely follows ha T ( ˆβ β) ( G HF (r, λ) 2 dr) 1 = Q 1 (Ã ē 1 )Λex 1 1(r > λ) F(r) ( F(s)F(s) ds) 1 1 λ F(s)dsdW ex (r) ( N A ex i )Λ ex W ex (1) ( G HF (r, λ) 2 dr) 1 Λ ex Q 1 ( N Aex i )Λ ex W ex (1) HF (r, λ)dw ex (r) The resul for T 1 2 ˆ S is given nex. From (23) we know be shown ha Direc calculaion gives T 1 2 N (z j ẑ j )û i = N z i ũ i = T 1 2 = T 1 2 = T 1 2 = T 1 2 = N ( rt (z j ẑ j )f() )( T f(s)f(s) ) 1 N (z i ẑ i )û i = o p (1). Similarly, i can N ( T o p (1) f(s)f(s) ) 1 N z i (u i û i 1 N N z i u i T 1 2 N z i u i T 1 2 N z i u i + o p (1) T u is f(s) p N (u j û j )) = T 1 2 N (z i ẑ i 1 N T u is f(s) N (z j ẑ j ))û i N (z i ẑ i )û i + T 1 2 N z i (u i û i ) N 1 N N (z j ẑ j )û i 26

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