Heeroskedasiciy Auocorrelaion Robus Inference in ime Series Regressions wih Missing Daa Seung-Hwa Rho imohy J. Vogelsang Louisiana Sae Universiy Michigan Sae Universiy Absrac In his paper we invesigae he properies of heeroskedasiciy auocorrelaion HA robus es saisics in saionary weakly dependen ime series regression seings when here is missing daa. We focus on saisics consruced using nonparameric kernel HA esimaors and we obain fixedb asympoic resuls. We characerize he ime series wih missing observaions as ampliude modulaed series following Parzen 963. For esimaion and inference his amouns o plugging in zeros for missing observaions. We also invesigae an alernaive approach where he missing observaions are simply ignored. here are hree main heoreical findings. Firs, when he missing process is random and saisfies srong mixing condiions, HA robus and Wald saisics compued from he ampliude modulaed series follow he usual fixed-b limis as in Kiefer and Vogelsang 2005. Second, when he missing process is non-random, he fixed-b limis depend on he locaions of missing observaions bu are oherwise pivoal. hird, when missing observaions are ignored we obain he surprising resul ha fixed-b limis of he robus and Wald saisics have he sandard fixed-b limis wheher he missing process is random or non-random. We discuss mehods for obaining fixed-b criical values wih a focus on boosrap mehods. We find ha he naive i.i.d. boosrap is he mos effecive and pracical way o obain he fixed-b criical values when daa is missing especially when he boosrap condiions on he locaions of he missing daa. Keywords: Missing daa; Unequally spaced; Irregularly observed; Serial correlaion robus Deparmen of Economics, srho@lsu.edu Deparmen of Economics, vj@msu.edu
INRODUION I is no unusual o encouner a ime series daa se wih missing observaions. Mos of he imes series lieraure in dealing wih missing daa focuses on he esimaion of dynamic models where he goal is o forecas missing observaions. However, in he relaively simple conex of ime series regression, here appears o be a sparsiy of work relaed o missing daa issues. In paricular, lile appears o be known abou he impac of missing daa on heeroskedasiciy auocorrelaion HA robus ess in regression seings. his paper aemps o fill his void by analyzing he impac of missing daa on robus ess based on nonparameric kernel esimaors of long run variances. Following Kiefer and Vogelsang 2005 we focus on obaining fixed-b resuls for he robus ess. In addiion o capuring he impac of he long run variance esimaor s kernel and bandwidh on he robus es saisics, he fixed-b limis also capure he impac of he locaions of he missing daa on he robus es saisics when eiher he missing process is non-random or one condiions on he missing locaions. In siuaions where he more radiional approach ha seeks o obain consisency resuls for variance esimaors would be problemaic, fixed-b heory delivers useful approximaions. Following Parzen 963 we characerize missing observaions as being driven by a missing process ha is a 0- binary variable. In erms of a regression model, he Parzen 963 approach amouns o plugging in zeros for missing observaions. ime series wih zeros in place of missing daa have been labeled ampliude modulaed series by Parzen 963 which we adop hroughou he paper. Because of he zeros, ampliude modulaed series are inuiively sensible because he ime disances beween he observaions remain preserved. his would seem paricularly relevan for HA robus esing based on nonparameric kernel esimaor Newey and Wes 987 and Andrews 99 given ha hose esimaors employ quadraic forms wih weighs ha depend on he ime disances of pairs of observaions. Soon afer Parzen 963 inroduced he noion of modeling missing daa wih he ampliude modulaed series approach, many auhors invesigaed he impac of missing daa on he consisen esimaion of specral densiy funcions. For example, Scheinok 965 and Bloomfield 970 consider esimaing a specral densiy funcion of he observed process wih missing daa wih independen Bernoulli and dependen Bernoulli missing processes respecively. Neave 970 esimaes a specral densiy funcion wih iniially scarce daa. Laer work by Dunsmuir and Robinson 98 invesigaed he consisen esimaion of he specral densiy of he underlying laen process. While HA robus inference makes use of specral esimaion mehod, wih he excepion of a recen working paper by Daa and Du 202, here 2
appears o be no aemp in he lieraure o link his earlier lieraure on specral densiy esimaion wih regression inference in he case of missing daa. Daa and Du 202 used he ampliude modulaed series approach o invesigae robus inference in ime series regression seings. heir approach is based on radiional asympoic heory for HA robus ess which appeals o he consisency of he HA esimaors. In he case of non-random missing locaions, he radiional approach becomes complicaed because of he need o consisenly esimae he long run variance of he laen process. While his is possible using resuls in Dunsmuir and Robinson 98, i is no clear how o obain a posiive definie variance esimaor. In any case, given ha i is now well esablished ha fixed-b heory provides beer approximaions han he radiional approach see Jansson 2002, Sun e al. 2008, and Gonçalves and Vogelsang 20, obaining fixed-b resuls for he missing daa case is pruden. here are hree main heoreical findings in he paper. Firs, when he missing process is random and saisfies srong mixing condiions, HA robus and Wald saisics compued from he ampliude modulaed series follow he usual fixed-b limis as in Kiefer and Vogelsang 2005. Second, when he missing process is non-random, he fixed-b limis depend on he locaions of missing observaions bu are oherwise pivoal. herefore, he fixed-b criical values ha one would use in he ampliude modulaed series approach depends on wheher he missing process is bes viewed as random or non-random. hird, a seemingly naive alernaive o he ampliude modulaed series approach is o simply ignore he missing daa. One migh reasonably conjecure ha ignoring he missing daa would be problemaic for robus inference. Surprisingly we find ha he fixed-b limis of he robus and Wald saisics have he sandard fixed-b random variable wheher he missing process is random or non-random. Here, ignoring he problem missing daa has no negaive consequences and generaes he advanage of robusness o wheher he missing process is random or non-random. he res of he paper is organized as follows. Secion 2 defines he model and he ampliude modulaed series es saisics in he presence of missing daa. Secion 3 develops fixed-b asympoic resuls for he ampliude modulaed series es saisics for boh random and non-random missing processes. Because he random and non-random missing processes require differen regulariy condiions hey are reaed separaely. Simulaion of he asympoic criical values is discussed wih a focus on boosrap mehods. Following Gonçalves and Vogelsang 20, we find ha he naive i.i.d. boosrap is a paricularly good opion for obaining valid fixed-b criical values. Finie sample performance of he ampliude modulaed series ess for boh random and non-random missing processes are examined in Secion 4 by Mone 3
arlo simulaions. Aenion is focused on he relaive performance of simulaed asympoic criical values wih boosrap criical values. Secion 5 analyzes he approach of ignoring missing observaions and makes some comparisons wih he ampliude modulaed series approach. Secion 6 concludes and formal proofs are given in he Appendices A-. 2 MODEL AND ES SAISIS onsider a regression model wihou missing observaions, y = x β + u =, 2,...,, where β is a k vecor of regression parameers, x is a k vecor of regressors, and u is a mean zero random process. When here are missing observaions, is he underlying laen process. In he presence of missing observaions, we characerize he missing process as a binary variable. Le a be a missing process where daa is observed a ime a = 0 daa is missing a ime. Wheher we rea he missing process as non-random or random depends on he srucure of he daa and he reason why he observaions are missing. We consider boh sochasic and non-sochasic missing processes. Wih he missing process, a, we define he regression model wih missing observaions as y = x β + u, y = a y, x = a x, u = a u, =, 2,...,. 2 haracerizing he missing process as a 0- binary variable and consrucing regression model as 2 is one of he sandard approaches of reaing missing observaions in panel daa regression models. In ime series, Parzen 963 firs characerized ime series wih missing daa using a dummy variable and modeled observed process as 2. However his approach has no become sandard in ime series which is For simpliciy, we assume ha he dependen variable and he independen variables are missing a he same ime poins. 4
surprising because 2 can be hough of as a naural way of formulaing a regression model wih missing observaions when here is no paricular ineres in forecasing he missing daa. Model 2 is inuiively sensible. Because he zeros are plugged in for missing observaions, he rue ime disances beween observaions are preserved. A a concepual level his would appear imporan when we are using nonparameric kernel covariance marix esimaors. Parzen 963 labelled he ime series in model 2 as ampliude modulaed AM series because he original ime series are ampliude modulaed by he missing process a. We adop he same language here. hroughou our analysis we assume ha he laen regression model saisfies exogeneiy, Ex u = 0, and we assume ha he mechanism generaing he missing daa does no generae an endogeneiy problem, i.e. we assume ha Ex u = 0 or equivalenly Ea x u = 0. his allows us o focus on he impac of missing daa on robus inference assuming ha β is idenified by he observed daa. he focus of his paper is on inference regarding β based on he ordinary leas squares OLS esimaor of β. Inference is carried ou o be robus o he form of he heeroskedasiciy and serial auo correlaion. he OLS esimaor of β is given by ˆβ = x x x y. Plugging in for y gives he well known expression ˆβ β = x x x u = x x v where v = x u. he impac of serial correlaion on ˆβ comes hrough v and robus sandard errors can be obained using a nonparameric kernel esimaor of he asympoic variance of /2 v of he form ˆΩ = ˆΓ 0 + j k M ˆΓ j + ˆΓ j, where ˆΓ j = =j+ ˆv ˆv j are he sample auocovariances of ˆv = x û wih û = y x ˆβ he OLS residuals of he AM series, and ˆΓ j = ˆΓ j for j < 0. Here, kx is a kernel funcion such ha kx = k x, k0 =, kx, kx is coninuous a x = 0, k2 x <, and M is he bandwidh parameer. Noice ha ˆΩ is he usual long run variance esimaor ha is obained afer simply seing ˆv = 0 for any 5
daes for which daa is missing. his can be seen mechanically by noing ha ˆv = x û = a x û. Using well known algebra, we can rewrie ˆΩ as ˆΩ = s= k s ˆv ˆv M s. 3 Because ˆΩ is compued using he AM series, he ime disances, s, beween observed daa poins are preserved which is concepually sensible. In addiion, ˆΩ will be posiive definie wih appropriae choices of he kernel funcion, e.g. he Barle, Parzen or quadraic specral QS kernels. Suppose we are ineresed in esing he null hypohesis, H 0 : r β o = 0 agains H A : r β o = 0, where r β is a q vecor q k of coninuously differeniable funcions wih a firs derivaive marix R β = r β / β. We analyze he following Wald saisic, W = r ˆβ [ R ˆβ ˆQ ˆΩ ˆQ R ˆβ ] r ˆβ, where ˆQ = x x. he case where one resricion is being esed, q =, we can also use a - saisic of he form = r ˆβ R ˆβ ˆQ ˆΩ ˆQ R ˆβ. 3 ASSUMPIONS AND ASYMPOI HEORY In his secion we derive he asympoic behavior of he OLS esimaor, ˆβ, he HA esimaor, ˆΩ, and he HA robus wald es, W, defined in Secion 2 for he case of weakly dependen covariance saionary ime series. We presen resuls for boh random and non-random missing processes. Resuls for random and non-random missing processes are reaed separaely as hey require differen regulariy condiions. We firs sae resuls for he random missing process followed by resuls for he non-random missing process. Alhough we briefly discuss radiional asympoic heory for HA robus ess based on consisency of he HA esimaors, we are mainly ineresed in obaining fixed-b asympoic approximaions as proposed by Kiefer and Vogelsang 2005. In he fixed-b asympoic framework he bandwidh of he covariance marix esimaor is modeled as a fixed proporion, b, of he sample size. his is in conras o he radiional approach where he bandwidh is modeled as increasing slower han he sample size. he advanage of he fixed-b approach is ha he resuling asympoic approximaions for he es saisics depend on he choice of kernel and bandwidh. In he radiional approach he kernel and bandwidh 6
choice do no appear in he asympoic approximaion. he fixed-b approach is herefore more accurae han he radiional approach because he fixed-b approach capures much of he impac of he sampling disribuion of he HA esimaor on he es saisic. For heoreical evidence on he superior performance of he fixed-b approach see Jansson 2002, Sun e al. 2008, and Gonçalves and Vogelsang 20. Because he fixed-b asympoic disribuions depend on he kernels used o compue he HA esimaors, some random marices ha appear in he asympoic resuls need o be defined. Here we follow he noaion of Kiefer and Vogelsang 2005. Definiion. Le h > 0 be an ineger and B h r denoe a generic h vecor of sochasic processes. Le he random marix, Pb, B h, be defined as follows for b 0, ]. ase i : if k x is wice coninuously differeniable everywhere, P b, B h k r s 0 0 b 2 b Bh r B h s drds, ase ii : if k x is coninuous, k x = 0 for x and k x is wice coninuously differeniable everywhere excep for x =, P b, B h k r s r s <b b 2 b Bh r B h s drds + k b where k = lim h 0 [k k h /h], b 0 Bh r + b B h r + B h r B h r + b dr, aseiii : if k x is he Barle kernel, P b, B h 2 b 0 B h r B h r dr b b Bh 0 r + b B h r + B h r B h r + b dr. hroughou, he symbol denoes weak convergence of a sequence of sochasic processes o a limiing sochasic process and p denoes convergence in probabiliy. We also use he following noaion. Le Q = Ex x and Q = Ex x. Le v = x u and define Ω = Λ Λ = Γ 0 + Γ j + Γ j, where Γ j = Ev v j and Λ is he lower riangular marix based on he holesky decomposiion of Ω. Similarly, le and v = a v and define Ω = ΛΛ = Γ 0 + Γ j + Γ j, where Γ j = Ev v j and Λ is he lower riangular marix based on he holesky decomposiion of Ω. he marix Ω is he long run variance marix of he laen vecor v whereas Ω is he long run variance marix of he AM series vecor v. We derive resuls under he assumpions ha he laen processes are near epoch dependen NED on some underlying mixing process and ha he missing process is srong mixing. We follow he definiions in Davidson 2002. Le he L p norm of x be defined as x p = E x p p. Also, le denoe he Euclidean 7
norm of he corresponding vecor or marix. For a sochasic sequence ε, on a probabiliy space Ω, F, P, le F +m m = σε m,..., ε +m, such ha F m +m m=0 is an increasing sequence of σ-fields. We say ha a sequence of inegrable random variables w is L p -NED on ε if, for p > 0, w Ew F +m m p < d ν m, where ν m 0 and d is a sequence of posiive consans. For a sequence a, le F = σ..., a, a, and similarly define F +m = σa +m, a +m+,.... he sequence is said o be α-mixing if lim m α m = 0, where α m = sup sup G F,H F+m PG H PGPH. A sequence is α-mixing of size ψ 0 if α m = Om ψ for some ψ > ψ 0. Similarly, a sequence is L p -NED of size φ 0 if ν m = Om φ for some φ > φ 0. 3. Random Missing Process When he missing process is random, he asympoic heory is driven by he observed AM series. If he AM series saisfies condiions required for fixed-b asympoic heory, hen he HA esimaor and he robus Wald es of he null hypohesis follow he usual fixed-b asympoic limis as obained by Kiefer and Vogelsang 2005. he following high-level assumpions are sufficien for his purpose. Assumpion R.. 2. [r] [r] /2 x x rq, r [0, ]. v ΛW k r, r [0, ]. Assumpion R saes ha a uniform in r law of large numbers LLN holds for x x. As long as x is covariance saionary and weakly dependen, his assumpion is a fairly general condiion. Assumpion R2 saes ha a funcional cenral limi heorem FL holds for he normalized parial sum of he AM series v. Below Assumpion R, which is in erms of he laen process and he missing process raher han he AM series iself, is sufficien for Assumpion R. Assumpion R.. For some r > 2, x 2r < for all =,.... 2. x is a weakly saionary sequence L 2 NED on ε wih NED coefficien of size 2r r 2. 3. v r <, and Ev = 0 for all =, 2,.... 8
4. v is a mean zero weakly saionary sequence L 2 -NED on ε wih NED coefficien of size 2. 5. a, ε is a α-mixing sequence wih α mixing coefficien of size 2r r 2. 6. a is a weakly saionary process ha is independen of x, u. 7. Ω = lim Var /2 a v is posiive definie. Under Assumpion R, he laen process saisfies condiions sufficien for he fixed-b asympoic heory o go hrough. In paricular, under Assumpion R, for all r 0, ], [r] x x rq and /2 [r] v Λ W k r. In erms of he laen process, Assumpions R are relaively weak. For example, Phillips and Durlauf 986 saes ha if v is L 2+δ bounded saionary process δ > 0 and srong mixing hen he parial sums of v saisfies he FL. he L 2-NED condiion in Assumpion R 4 is acually weaker han his condiion. Hence, he presence of he missing observaions generally does no require addiional assumpions on he laen process. Assumpion R 6 is relaively srong and saes ha he missing locaions are no relaed o he laen process. his assumpion is sufficien for ˆβ o be consisen for β because i implies if Ex u = 0, hen Ex u = Ea x u = Ea Ex u = 0. In addiion, Assumpions R 5 and R 6 ensure ha he LLN and FL ha hold for he laen processes exend o he observed AM series, i.e. Assumpions R 5 and R 6 ensure ha Assumpions R hold. Wih hese assumpions we can sae our main resul for he esimaor and saisics based on he AM series when he missing process is random. heorem. Under Assumpion R he following hold as. a. Asympoic Behavior of OLS ˆβ β Q ΛW k = N0, Q ΩQ. b. Fixed-b approximaion of HA esimaor Le B k r denoe a k vecor of sochasic processes defined as B k r W k r rw k, for all r 0, ]. Assume M = b where b 0, ] is fixed. hen, ˆΩ ΛP b, B k Λ, where he form of Pb, B k depends on he ype of kernel via Definiion. 9
c. Fixed-b asympoic disribuion of ess Under H 0, W W q P b, B q Wq and when q =, W P b, B. Because Assumpions R imply Assumpions R, heorem direcly follows from Kiefer and Vogelsang 2005 alhough direcly esablishing heorem a is easy. If we plug in y = x β + u o he OLS esimaor ˆβ, hen ˆβ = β + x x v. herefore we can wrie ˆβ β = x x /2 v. and he limi is obained by using Assumpions R wih r =. While he fixed-b approximaion is more useful han he radiional resul ha relies on a consisency resul for ˆΩ, one could easily obain radiional resuls for he Wald and -saisics under similar regulariy condiions. 2 heorem shows ha when he missing process is random, one can simply plug in zeros for he missing observaions and conduc sandard fixed-b inference reaing he zeros as hough hey were observed daa. Given a paricular sample wih ime periods of daa including he zeros, rejecions would be compued relaive o fixed-b criical values obained by Kiefer and Vogelsang 2005. he criical values are funcions of he kernel and he value of b = M/ where M is he bandwidh used o compue ˆΩ. he fixed-b asympoic disribuions in Kiefer and Vogelsang 2005 are non-sandard. While i is relaively easy o simulae from he asympoic disribuions, more user-friendly mehods are available for he compuaion of criical values and p-values. For he case of he Barle kernel, Vogelsang 202 has developed a numerical mehod for he easy compuaion of sandard fixed-b criical values and p-values for any significance level. For oher kernels comprehensive numerical approaches have no been developed. Kiefer and Vogelsang 2005 do provide criical value funcions for popular significance levels bu heir funcions do no allow he compuaion of p-values. A good alernaive for he compuaion of fixed-b criical values and p-values is he boosrap. Gonçalves and Vogelsang 20 showed ha he 2 onsisency of ˆΩ requires a slighly sronger assumpion han Assumpion R. For example, Andrews 99 requires v o be a fourh order saionary process, and Hansen 992 requires v o be mixing wih size 2 + δr + δ/2r 2. Assumpion R in his secion is sufficien for Hansen 992. See Appendix A for he proof. 0
naive moving block boosrap has he same limiing disribuion as he fixed-b asympoic disribuion under regulariy condiions similar o hose used here. he boosrap works wih boh a fixed block lengh l or a block lengh ha increases wih he sample size bu a a slower rae l 2 / 0. In paricular, for he case of l = he block boosrap becomes an i.i.d. boosrap. herefore, he resuls of Gonçalves and Vogelsang 20 indicae ha valid fixed-b criical values can be obained via a simple i.i.d. boosrap mehod. As shown in he nex subsecion, he fixed-b limi of he robus saisics becomes more complicaed under he assumpion ha he missing locaions are non-random. In his case he boosrap becomes he ideal ool for obaining fixed-b criical values on a case by case basis in pracice. herefore, i is useful o provide some deails on he implemenaion of he boosrap. Define he vecor ω = y, x ha collecs dependen and explanaory variables. Le l N l be a block lengh and le B,l = ω, ω +,..., ω +l be he block of l consecuive observaions saring a ω. Draw k 0 = /l blocks randomly wih replacemen from he se of overlapping blocks B,l,..., B l+,l o obain a boosrap resample denoed as ω = y, x, =,...,. Noice ha we are resampling from he AM series he zeros are included. he boosrap es saisics, W and, are defined as W = r ˆβ r ˆβ [R ˆβ ˆQ ˆΩ ˆQ R ˆβ ] r ˆβ r ˆβ and = r ˆβ r ˆβ R ˆβ ˆQ ˆΩ ˆQ R ˆβ, where ˆQ = ˆΩ = x x, k s/m ˆv ˆv s, and ˆβ is he OLS esimae from he regression of y on x, and ˆv = x y x ˆβ. Noice ha boosrap saisics use he same formula as W and and his is wha makes his boosrap approach naive. Le p denoe he probabiliy measure induced by he boosrap resampling condiional on a realizaion of he original ime series. If [r] x x p rq for some Q and /2 [r] v p Λ W k r for some Λ, hen because he fixed-b asympoic disribuion of he Wald es saisics is pivoal, he limiing disribuion of W coincides wih he limiing disribuion of W, independenly of Λ and Q. We show
ha srenghening Assumpion R 3-5 o R 3-5 is sufficien for his purpose. Assumpion R. 3. v r+δ <, r > 2. 4. v is a weakly saionary L 2+δ NED on ε wih ν m of size. 5. a, ε is a α-mixing sequence wih α m of size 2+rr+δ r 2. his srenghening is necessary for boosrap resamples o saisfy condiions required for he FL. Also noe ha excep for he assumpions relaed o he missing process, he oher assumpions are idenical o he hose in Gonçalves and Vogelsang 20, which implies ha he exisence of missing observaions does no change he assumpions required for he laen process for he boosrap o provide valid criical values. See Gonçalves and Vogelsang 20, p764-766 for deails. Hence, in general, as long as he missing process saisfies he srong mixing condiion, he naive moving block boosrap provides valid criical values. We formally sae he resul below. Proofs are provided in he Appendix A. heorem 2. Le W and be naive boosrap es saisics obained from he moving block boosrap resamples. Suppose ha he block size l is eiher fixed as or l as such ha l 2 / 0. Le b 0, ] be fixed and suppose M = b. hen, under Assumpion R wih Assumpion R 3-5 srenghened o Assumpion R 3-5, as, W p W qpb, B q r W q and p W. Pb, B r 3.2 Non-random Missing Process When he missing process is non-random, missing locaions are fixed and hence he asympoic behavior of he esimaors and saisics depend on he locaions of missing observaions. We firs define he srucure of he iming of he missing observaions. 2
Definiion 2. We characerize an arbirary daa se wih missing observaions as follows. From = o = we observe daa, from = + o = 2 daa are missing, from = 2 + o = 3 we observe daa and so forh. Le he number of missing clusers be <. For simpliciy, we assume ha daa are observed a = and =. 3 hus, in general for n =,...,, from = + o = daa are missing whereas from = + o = +2 daa are observed see Figure 2. For noaional purposes, le 0 = 0 and 2+ =. daa missing daa = = = 2 = 3... daa = 2 = Figure : Daa wih missing observaions When he missing process is non-random, he asympoic heory is driven by he laen process. his is because he laen process is he only random process and wha maers is wheher he laen process saisfies condiions required for fixed-b asympoic heory. he following assumpions are sufficien for us o obain a fixed-b resul. Assumpion NR.. he missing/observed cuoffs saisfy lim n is non-random and finie, i.e., <. = λ n, n = 0,,..., 2 +, where he number of cuoffs [r] 2. x x rq, r [0, ]. 3. [r] /2 v Λ W k r, r [0, ]. Assumpion NR saes ha he number of observaions in a missing or observed block is a fixed proporion of he sample size wih he number of missing blocks also fixed. his is no mean o be a descripion of he way daa is gahered bu is simply a naural mahemaical ool for obaining approximaions ha depend on he locaions of he missing and observed daa. Once we have he daa, we know he number of missing blocks and he locaions of missing observaions as a proporion of he sample size and we are aking hem as given informaion. 3 his assumpion is only for noaional simpliciy. he resuls of his paper go hrough wihou his assumpion. 3
he oal number of observed ime periods is a and using Assumpion NR we can quanify he proporion of he ime periods ha have observed daa as λ = lim a = 2+ i+ λ i. 4 Assumpion NR2 saes ha a uniform in r LLN holds for x x. Assumpion NR3 saes ha he FL holds for he scaled parial sums of v. We now sae more primiive condiions ha are sufficien for Assumpions NR o hold: Assumpion NR.. he missing/observed cuoffs saisfy lim n is non-random and finie, i.e., <. = λ n, n = 0,,..., 2 +, where he number of cuoffs 2. For some r > 2, x 2r < for all =,.... 3. x is a weakly saionary sequence L 2 NED on ε wih NED coefficien of size 2r r 2. 4. v r <, and Ev = 0 for all =, 2,.... 5. v is a mean zero weakly saionary sequence L 2 -NED on ε wih NED coefficien of size 2. 6. ε is a α-mixing sequence wih α mixing coefficien of size 2r 7. Ω = lim Var /2 v is posiive definie. r 2. Assumpion NR is he same as Assumpion R excep for he properies relaed o he missing process a. Recalling ha in erms of he laen process, all ha Assumpion R required was he condiions sufficien for he laen process o saisfy fixed-b asympoic heory, his is naural. We now sae our main resuls when he missing process is non-random. Noe ha for wo numbers r and s, r s denoes he minimum of r and s. he proof of heorem 3 is given in Appendix B. heorem 3. Le W k 2+ j+ W k λj and le B k r, λ i be a k vecor of sochasic processes defined as B k r, λ i λ < r λ + + j+ W k r λj r λj λ W k, for r 0, ]. Under Assumpion NR, as, a. Asympoic Behavior of ˆβ ˆβ β λq Λ W k = N 0, λ Q Ω Q 4
b. Fixed-b asympoic approximaion of ˆΩ Assume M = b where b 0, ] is fixed; hen ˆΩ Λ P b, B k λ i Λ. c. Fixed-b asympoic disribuion of W Under H 0, W W qp b, B q λ i W q and when q =, W P b, B λ i. Using he asympoic normaliy resul in heorem 3a, one could pursue a radiional inference approach which would require a consisen esimaor of he asympoic variance. he challenge would be consrucing a consisen esimaor of he laen process long run variance marix, Ω. Using resuls from Dunsmuir and Robinson 98 a consisen esimaor of Ω can be consruced as ˆΩ = ˆΓ 0 + j k ˆΓ j + ˆΓ j M where ˆΓ j = =j+ ˆv ˆv j/ =j+ a a j. Because ˆΓ j is consruced using he effecive sample size of he sequence v v j here is no guaranee ha ˆΩ will be posiive definie even if kernels like he Barle, Parzen and QS are used. Besides only providing a relaively crude approximaion for es saisics, he difficuly in consrucing a posiive definie esimaor of ˆΩ makes he radiional approach even less appealing in pracice. In conras he fixed-b approach shows ha one can simply use ˆΩ o consruc valid es saisics because under fixed-b heory, ˆΩ is asympoically proporional o Ω when he locaions of missing daa are non-random. Even hough ˆΩ is no an esimaor of Ω, i can sill be used o consruc es saisics because fixed-b heory shows ha ˆΩ scales ou Ω. Looking closely a he resul given by heorem 3b we see ha he fixed-b limi of ˆΩ is similar bu is noiceably differen from he limi obained for he case of missing a random. he sochasic process B k r, λ i is differen han he Brownian bridge Br and depends on he locaions of he missing/observed daa. herefore, criical values for he limiing random variables given by heorem 3c are differen from he criical values given by he sandard fixed-b limis given by heorem c. 5
Given he locaions of he missing daa, he non-sandard disribuion in heorem 3c can be compued by simulaion mehods because he limiing disribuions are sill funcions of Brownian moions. Alhough his mehod is feasible, i can be pracically inconvenien because asympoic criical values would have o be simulaed on a case by case basis depending on he locaions of he missing daa. In his siuaion he boosrap is a more convenien mehod for obaining fixed-b criical values. Because he locaions of missing daa are reaed as non-random, we need he boosrap resampling scheme o preserve he missing locaions. his means ha blocking is no pracical because blocks will shuffle he locaions of he missing daa upon resampling. Insead he i.i.d. boosrap is more appropriae where boosrap samples are creaed by firs sampling wih replacemen from he observed daa and creaing a boosrap sample wih he same missing locaions as he original daa. Specific deails are as follows. Define ω = y, x, =,...,, ha collecs he dependen and independen variables of he AM series. Among hose observaions collec only he observed daa, which we denoe ω, =,...,, = a. Resample observaions wih replacemen from ω and ge boosrap resample which we denoe ω, =,...,. Fill in he observed locaions wih resampled daa ω and leave he missing locaions as zeros. his way we consruc an i.i.d. resample wih missing locaions fixed. Denoe his i.i.d. resample as ω = y, x, =,...,. he naive boosrap es saisics W and are compued as W = r ˆβ r ˆβ [R ˆβ ˆQ ˆΩ ˆQ R ˆβ ] r ˆβ r ˆβ and = r ˆβ r ˆβ R ˆβ ˆQ ˆΩ ˆQ R ˆβ, where ˆβ is he OLS esimae from he regression of y on x, ˆQ = / x x, and ˆΩ = / k s/m ˆv ˆv s, where ˆv = x y x ˆβ. Because we resample from observed ime periods only, his resampling can be hough of as resampling from he laen process ω y, x. We do no know he value of ω when a = 0 and hus we are resampling from observaions no he full number of ime periods. However, because he resampling is based on i.i.d. draws, his boosrap resample has essenially he same properies as an i.i.d. resample of he laen process. We could ake anoher independen draws from ω and fill in he missing locaions of ω. all his filled-in resample ω. hen by consrucion ω = a ω where ω can be 6
viewed as a sample from he laen process given he i.i.d. resampling. If he boosrap process, ω, saisfies a x x using heorem 3c i follows ha rq for some Q and b /2 v Λ W k r for some Λ, hen W p W q Pb, B q λ i W q, wih W q = 2 + j+ W q λ j where λ m 2 + m=0 are he missing locaions in he boosrap resample, is he number of missing clusers in he boosrap resample, and p denoes he probabiliy measure induced by he boosrap resampling condiional on a realizaion of he original ime series. Because he missing locaions of he boosrap resamples are configured o be idenical o he missing locaions of he daa, i follows ha λ j = λ j and =. herefore, W p W qpb, B q λ i W q, which is he same fixed-b limi of W as in heorem 3c. his asympoic equivalence is mainly due o he fac ha he limiing disribuion in heorem 3c is pivoal wih respec o Λ and Q so ha W has an asympoic disribuion equivalen o W even hough Λ and Q are poenially differen from Λ and Q. Obviously, and have equivalen asympoic approximaions as well. Srenghening Assumpion NR 3-5 o NR 3-5 is sufficien for ω Assumpion NR. o saisfy condiions a and b above. 3. v r+δ <, r > 2. 4. v is a weakly saionary L 2+δ NED on ε wih ν m of size. 5. ε is a α-mixing sequence wih α m of size 2+rr+δ r 2. See Gonçalves and Vogelsang 20 for he proofs. Here he resul of Gonçalves and Vogelsang 20 direcly applies because hese assumpions are made abou he laen process which has nohing o do wih he missing process when he missing locaions are non-random. A formal saemen of his boosrap resul is in he following heorem. heorem 4. Le W and be naive boosrap es saisics compued from he i.i.d. boosrap resample wih he locaions of missing observaions fixed and idenical o he missing locaions of he real daa. Le 7
b 0, ] be fixed and suppose M = b. hen, under Assumpion NR wih Assumpion NR 3-5 srenghened o Assumpion NR 3-5, as, W p W qpb, B q r, λ i W q and p W Pb, B r, λ i. 4 FINIE SAMPLE PERFORMANE In his secion we use Moel arlo simulaions o evaluae he finie sample performance of he fixed-b asympoic approximaion of he HA robus Wald es defined in Secion 3. 4. Daa Generaing Process We consider a simple locaion model for he laen process given by, y = β + u, u = ρu + ε i.i.d.n 0,, u = 0, ρ 2 ε, wih =, 2,..., so ha is he ime span. We se β = 0 and ρ 0, 0.3, 0.6, 0.9. For he random missing process we model a as a Bernoullip process, i.e. Pa = = p, wih p 0.3, 0.7. We provide resuls for he ime span 50, 00. For non-random missing process, we consider where daa are missing in wo clusers = 2 due o World War I from 94 o 98 and World War II from 939 o 945. We generae daa boh yearly and quarerly where ime spans from 9 o 946. For yearly daa, his means ha 2 observaions are missing ou of = 36 observaions. For quarerly daa, his implies ha 48 observaions are missing ou of = 44 ime periods. herefore for boh cases he missing process is a [r] = 0 when r λ, λ 2 ] λ 3, λ 4 ] and a [r] = oherwise, wih λ = /2, λ 2 = 2/9, λ 3 = 7/9, and λ 4 = 35/36. See Figure 2 for yearly daa. 8
Missing 9 94 98 99 =3 2 =8... 938 3 =28 Missing 939 945 4 =35 946 Figure 2: Missing due o World War I and World War II : Yearly daa Wih he laen process and he missing process generaed above we obain he AM series y = x β + u, where y = a y, x = a, u = a u. 4.2 es saisics and riical Values Wih he daa generaing process defined in Secion 4., we consider esing he null hypohesis ha β = 0 agains he alernaive β = 0 a a nominal level of 5%. he HA robus -saisic for β is = ˆβ = x2 ˆΩ x2 ˆβ = a2 ˆΩ a2 ˆβ, 2 a ˆΩ where ˆβ = x 2 x y = a a y = a y, and ˆΩ = k i j /[b] ˆv i ˆv j wih ˆv = a y ˆβ. For World War missing process we use bandwidh M, 3, 6, 9,..., 36 for yearly daa and M 4, 2, 24, 36,..., 44 for he quarerly daa. Hence we have he same b = M/ for he wo cases. For he Bernoulli missing process we use M, 5, 0,..., 45, 50 for = 50 and M 2, 0, 20,..., 90, 00 for = 00 and b for = 50 and = 00 are he same. Using 0, 000 replicaions, we compue empirical rejecion probabiliies. We rejec he null hypohesis whenever > c or rejec he null whenever < l c or > r c if l c = r c where c is a criical value. As shown from heorems c and 3c, he es saisics have differen asympoic disribuions depending on wheher he missing process is random or non-random. Hence criical values are calculaed differenly for he wo cases. When he missing process is random, c is he 97.5% percenile of he sandard fixed-b asympoic 9
disribuion derived by Kiefer and Vogelsang 2005 See heorem c. From Secion 3. we know ha we can compue he asympoic criical values by eiher simulaing he disribuion iself R c or by he naive moving block boosrap R boo,l c, c R boo,r. o evaluae he finie sample performance we use boh of he mehods o ge he criical values. For he naive moving block boosrap, we use block lengh l = he i.i.d. boosrap. From he original random sample of observaions, y,..., y, we ge 999 boosrap resamples, y B, a B..., y B he boosrap -saisic as where ˆβ B = ˆβ B. hen R boo,l c a B y B, a B,B =,..., 999. For each boosrap resample we compue B ˆβ = B ˆβ a B 2 ˆΩ, B, and ˆΩ B = s= k s /[b] ˆv B ˆv B s, where ˆv B is he 0.025 quanile and c R boo,r is he 0.975 quanile of B, B =,..., 999. = a B y B When he missing process is non-random, c is he 97.5% percenile of he disribuion derived in heorem 3c. From Secion 3.2 we know ha criical values can be compued eiher by simulaing he limiing disribuion NR c or by naive i.i.d. boosrap NR boo,l c, c NR boo,r by heorem 4. Because he limiing disribuion depends on missing locaions, he naive i.i.d. boosrap is more convenien in pracice. However, o illusrae he relaive finie performance, we compue he criical values using boh mehods. From he original sample of observaions, y,..., y, we pull ou he daa from he observed ime periods, ỹ,..., ỹ, = a. From hese observaions, we resample observaions wih replacemen. Repeaing his procedure 999 imes we obain resamples which we denoe ỹ B,..., ỹ B, B =,..., 999. By filling observed locaions wih resampled daa, ỹ, and filling missing locaions wih zeros, we obain he i.i.d. boosrap resamples, which we denoe y B,..., y B, for B =,..., 999. We compue he naive boosrap -saisic as where ˆβ B = hen NR boo,l c ab y B is he 0.025 and NR boo,r c B ˆβ = B ˆβ 2 a ˆΩ B and ˆΩ B = s= k s /[b] ˆv B ˆv B s, where ˆv B = a y B ˆβ B. is he 0.975 quanile of B, B =,..., 999. Noe ha we are using a raher han a in his case because we are condiioning on he locaions of he missing daa when resampling. 20
4.3 Finie Sample Performance We illusrae he non-random missing process case firs. Figures 4-7 show empirical rejecion probabiliies compued from 0, 000 replicaions using AM series for he World War missing processes defined in Secion 4.. Since he missing process is non-random, by heorem 3c he HA robus es saisics have a fixed-b asympoic disribuion ha depends on he missing locaions. riical values are obained by he naive i.i.d. boosrap wih locaions of missing observaions fixed labeled AM: condiional-boosrap or by direcly simulaing he limiing fixed-b disribuions labeled AM: condiional-fixed-b. In addiion o hese wo criical values we consider criical values obained by he naive i.i.d. boosrap ha does no condiion on missing locaions labeled AM: boosrap and by simulaing he sandard fixed-b limi in Kiefer and Vogelsang 2005 labeled AM: fixed-b for comparison alhough hese wo criical values are no heoreically valid. he firs hing we can noice is ha he fixed-b criical values ha rea missing locaions fixed, eiher simulaed or compued by boosrap ha condiion on he missing locaions, have less size disorions han he sandard fixed-b criical values for boh Barle and QS kernels. I appears ha when he sample size is smaller using he criical values ha rea missing locaions fixed maers more. For example, comparing ρ = 0.9-yearly daa- = 36 case Figure 4 o ρ = 0.6-quarerly daa- = 44 case Figure 5, we can find ha he former case shows bigger difference beween he criical values ha condiion and do no condiion on missing locaions. 4 Also, i appears ha he advanage of using criical values ha condiion on missing locaions increases when he serial correlaion is high. For example, ρ = 0.9 case in Figure 4 shows bigger difference beween he rejecion raes ha are compued from criical values ha condiion and do no condiion on he missing locaions han he oher values of ρ. 5 Similar endencies can be found for QS kernel as well. he simulaion resuls also sugges ha one may gain by boosrapping raher han simulaing he fixed-b disribuion especially when he serial correlaion is high and/or he sample size is small. he empirical rejecion probabiliies from naive i.i.d. boosrap wih missing locaions fixed have less size disorion han he empirical rejecion probabiliies obained by simulaing he fixed-b disribuion in heorem 3c. 4 We are comparing ρ = 0.9 of yearly daa o ρ = 0.6 of quarerly daa because for yearly daa o be comparable o he quarerly daa ρ = 0.9 should be convered o ρ = 0.9 4 0.65. 5 Noe ha for he figure we have differen scale for ρ = 0.9 and he oher values of ρ. 2
he empirical rejecion raes from Bernoulli missing process obained from 0, 000 replicaions using he AM series are given in Figures 9-. When he missing process is random, he usual fixed-b limi in Kiefer and Vogelsang 2005 is valid. Hence criical values can be obained by simulaing his disribuion or by he naive i.i.d. boosrap. As wih he non-random missing process, for comparison, we also consider criical values obained from he naive i.i.d. boosrap condiional on he missing locaions. Resuls for he random missing process show ha he condiional fixed-b limi in general performs no worse han he usual fixed-b limi when he criical values are compued by boosrap. here exiss negligible difference beween he condiional and non-condiional fixed-b limi compued from boosrap when = 50 and 70% of he daa are missing. However, if we eiher increase he sample size or decrease he proporion of missing observaions he he difference disappears. ompared o he non-random missing process, he simulaion resuls for random-missing process show ha i is more imporan o use boosrap criical values raher han he simulaed ones. When = 50 and p = 0.3 Figure 8 he condiional fixed-b limis obained from he simulaion and he boosrap show large difference. When = 00 and p = 0.7 Figure he difference decreases bu sill exiss. In sum, when he missing process is non-random using he condiional fixed-b limi maers. hoosing beween simulaed criical values and boosrapped criical values maers less bu here may exis some gain when he daa is highly correlaed by using condiional boosrap. When he missing process is random, i appears condiioning on locaions causes no harm when he condiional fixed-b limi is obained by condiional boosrap. Given hese findings, we sugges he use of condiional on locaions criical values in pracice, especially he boosrap version.. 5 WHEN MISSING OBSERVAIONS ARE IGNORED In pracice an empirical researcher migh be emped o simply ignore any missing daa problems and esimae he ime series regression wih he daa ha is observed. From he perspecive of esimaing β his has no consequences because one obains he same esimaor of β as is obained when missing observaions are replaced wih zeros. For he compuaion of long run variance esimaors, ignoring missing daa maers because he ime disances beween observaions is skewed for many pairs of ime periods. hus, robus es saisics are compuaionally differen when ignoring missing daa verses replacing missing daa wih zeros. A reasonable conjecure is ha ignoring missing daa invalidaes inference using HA robus es saisics. Surprisingly, fixed-b asympoic heory suggess oherwise. As we show in 22
his secion, ignoring he missing daa leads o HA robus ess ha have sandard fixed-b limis. his is rue wheher he missing process is random or non-random. In conras o he AM series approach, he empirical researcher does no have o worry abou robusness o wheher missing daes are bes viewed as random or non-random. 5. Models and es saisics In erms of he regression model, ignoring missing observaions amouns o sacking only he observed observaions as if hey are equally spaced in ime. See Figure 3. aking ou he missing observaions from he laen process and relabeling observed observaions, he regression model becomes y ES = x ES β + u ES =, 2,..., ES, 5 where ES = a is he number of non-missing observaions. Following Daa and Du 202 we call his model he equal space ES regression model. daa missing daa y ES y ES 5 y6 ES y8 ES... daa y ES ES 3 y ES ES Figure 3: Equal Space Regression Model As wih he AM series, he ES regression model uses only he observed daa. No aemp is made o forecas or proxy missing observaions. However unlike he AM series, he original ime disances beween observaions are no preserved in he ES regression model. he disance beween he h and s h observaions in erms of he laen process is no necessarily s bu is insead equal o a i s a i which is he number of observed daa poins beween ime periods and s. Only when here are no missing observaions beween ime periods and s will he ime disance remain s in 5. he OLS esimaor of he ES regression model is defined as ˆβ ES = ES x ES x ES ES x ES y ES. 23
Recall ha missing observaions are replaced wih zeros in he AM series and missing observaions are deleed in he ES regression model. Because he only difference beween x x and ES comes from he missing observaions which are se o zeros, i follows ha x x By he same reasoning, x y = ES xes y ES. herefore, i follows ha = ES xes x ES xes x ES. ˆβ = ˆβ ES. Hence, in erms of he OLS esimaor, he ES regression model provides he same esimae of β as he AM series. Le Ω ES = lim Var /2 ES ves, where v ES esimaor for Ω ES is defined as = x ES u ES. hen, he usual kernel based HA ˆΩ ES = ˆΓ 0 ES ES + j k ˆΓ ES j M ES + ˆΓ ES j, where ˆΓ ES j = ES ES =j+ ˆvES ˆv j ES are he sample auocovariances of ˆvES = x ES û ES and û ES = y ES x ES ˆβ ES are OLS residuals from he ES regression model. As before, kx is a kernel funcion such ha kx = k x, k0 =, kx, coninuous a x = 0, and k2 x < and M ES is he bandwidh. 6 By well known algebra we can rewrie ˆΩ ES as ˆΩ ES = ES ES n= ES m= n m k M ES ˆv n ES ˆvES m. Recall ha ˆv = x û where û are he OLS residuals from he AM series. herefore, by consrucion ˆv = x û = a x y x ˆβ which implies ha ˆv = 0 a missing daes. Because he ES regression model and he AM series share he same ˆβ, ˆv ES is obained by dropping missing observaions from ˆv. 7 Using hese facs, we can recas ˆΩ ES, which is a weighed sum of ˆv ES n ˆvES m, insead as a weighed sum of ˆv ˆv s because all he elemens of ˆv ES ˆv s ES are found among hose of ˆv ˆv s wih he remaining elemens of ˆv ˆv s being zeros. he only complicaion ha arises when rewriing ˆΩ ES in erms of ˆv ˆv s lies in maching he kernel weighs used on ˆv ˆv s o hose ha are used by ˆΩ ES which are differen from he weighs used by ˆΩ. 6 We denoe he bandwidh of he ES regression model as M wih subscrip ES because if we fix b = M ES/ ES, hen M ES depends on he ime span of he ES regression model ES. 7 o be more specific, ˆv = ˆv ES g wih g = a i whenever a =. ˆv ES g for all g =,..., ES can be defined his way. When a i = 0, here is no erm in he ES regression model ha maches ˆv i which is zero because missing observaions are dropped in he ES regression model. 24
he ime disance beween ˆv and ˆv s in he ES regression model is a i s a i and we can rewrie ˆΩ ES as ˆΩ ES = ES s= k a i s a i ˆv ˆv s. M ES Recall ha he HA esimaor of he AM series is given by 3. Boh ˆΩ ES and ˆΩ are weighed sums of ˆv ˆv s,, s =,...,, bu wih differen weighs. For he ES regression model, by aking ou he missing observaions, he ime disances beween observaions become shorer han he rue ime disances, s, unless here are no missing observaions beween and s. herefore, ˆΩ ES gives ˆv ˆv s weighs a leas as big as ˆΩ if he same bandwidh is used: k a i s a i/m ES k s/m if M = M ES. We now revisi esing he null hypohesis H 0 : r β 0 = 0 agains H A : r β 0 = 0. We define he equal spaced HA robus Wald saisic as W ES = ESr ˆβ [ R ˆβ ˆQ ES ˆΩ ES ˆQ ES R ˆβ ] r ˆβ, and when q =, -saisics of he form ES = ES r ˆβ R ˆβ ˆQ ES ˆΩ ES ˆQ ES R ˆβ, where ˆQ ES = ES ES xes can wrie W ES as W ES = r ˆβ x ES. Noe ha x x = ES [ ES xes x ES R ˆβ ˆQ ˆΩ ES ˆQ R ˆβ ] r ˆβ. Oher han he scaling facor ES/ and ˆΩ ES, he oher erms in W ES implies ˆQ ES = / ES ˆQ. We herefore are idenical o W. herefore, in erms of es saisics, choosing beween he AM series saisics and he ES saisic boils down o choosing he kernel weighs when compuing he HA esimaor. 5.2 Asympoic heory As wih he AM series, we are mainly ineresed in he fixed-b asympoic limis of W ES null hypohesis H 0 defined in Secion 5.. and ES under he 25
5.2. Non-random missing process We firs consider he non-random missing process case. Because he fixed-b asympoic disribuions depend on he kernels used o compue he HA esimaors, we need o define some random marices ha appear in he asympoic resuls. he random marices in Definiion no longer work here because he kernel weighs in ˆΩ ES are differen from hose of ˆΩ. In fac because he kernel weigh for ˆv ˆv s depends on he number of missing observaions beween and s, unlike he random marices defined in Definiion, he random marices ha appear in he asympoic approximaion of ˆΩ ES depend on he missing locaions λ i 2+ i=0. Noe ha for wo numbers r and s, r s denoes he minimum of r and s and r s denoe he maximum of r and s. Definiion 3. Le h > 0 be an ineger. Le λ i 2+ i=0 be given by Assumpion NR. and le λ be given by 4. Le B h r, λ i denoe a generic h vecor of sochasic process ha depends on λ i. Le he random marix, P ES b, B h λ i, be defined as follows for b 0, ]: ase i : if k x is wice coninuously differeniable everywhere, P ES b, B h λ i b 2 λ 3 l=0 B h r, λ i B h u, λ i ] dudr λ+ λ [ [ ] λ2l+ k λ 2l λb + j+ r λ j 2l+ j+ u λ j ase ii : if k x is coninuous, k x = 0 for x and k x is wice coninuously differeniable everywhere excep for x =, P ES b, B h λ i b 2 λ 3 λ+ λ2l+ l=0 λ [ λ 2l r u < bλ + l j= l+ j λ j [ k λb + j+ r λ j 2l+ ] j+ u λ j B h r, λ i B h u, λ i ] drdu + k bλ 2 n λ2l+ l=0 [ λ λ 2l bλ j=2l+ j λ j < u λ + bλ j=2l+ j λ j ] B h u + bλ + j=2l+ j λ j, λ i B h u, λ i + B h u, λ i B h u + bλ + j=2l+ j λ j, λ i du, where k = lim h 0 [k k h /h], ase iii : if k x is he Barle kernel, P ES b, B h λ i 2 bλ λ+ 2 λ λ2l+ λ bλ B h r, λ i B h r, λ i dr bλ 2 n l=0 [ λ 2l B h u, λ i B h u + bλ + λ k k, λ i k λ k u λ + bλ k λ k ] + Bh u + bλ + λ k k, λ i B h u, λ i du. When he missing process is non-random, he asympoic heory for he ES regression model is based on Assumpion NR which is he same assumpion ha he AM series resuls are based on. heorem 5 26
below provides he asympoic limis of ˆΩ ES and W ES ES when q = when he missing process is nonrandom. Because he ES regression model and he AM series model have he same OLS esimaor, we do no resae he asympoic resul of he OLS esimaor given by heorem 3a. he proof for heorem 5 is provided in Appendix. heorem 5. Le W k be defined as in heorem 3. Le B k r, λ i be a k vecor of sochasic processes defined as B k r, λ i + λ < r λ + j+ W k r λj r λj λ W k, for r 0, ]. Assume M ES = b ES where b 0, ] is fixed. hen under Assumpion NR, as, a. Fixed-b asympoic approximaion of ˆΩ ES ˆΩ ES Λ P ES b, B k λ i Λ, b. Fixed-b asympoic disribuion of W ES under H 0, W ES W q [ λp ES b, B q λ i ] W q and when q =, ES W λp ES b, B λ i. Alhough he difference of he limis of ˆΩ and ˆΩ ES is in he form of he funcions P and P ES because of he differen relaive disances beween observaions, i is proporional o Ω and is a funcion of B q r, λ i like ˆΩ. Similar o W, W ES has a limiing disribuion ha is non-sandard and depends on he locaions of he missing daa bu remains pivoal wih respec o Ω and Q. Surprisingly, i urns ou ha he asympoic disribuion in heorem 5b is equivalen o he sandard fixed-b asympoic disribuion in Kiefer and Vogelsang 2005 wih b = M ES/ ES. o esablish his resul we firs consider he special case where he laen process is i.i.d. When he laen process is i.i.d., he ES regression model is a ime series regression wih ES observaions and here is no serial correlaion in he daa. herefore, W ES is he usual HA saisic compued wih ES observaions. For M ES = b ES, W ES has he usual fixed-b limi because he resuls of Kiefer and Vogelsang 2005 direcly apply. Inuiively 27