Cointegration versus Spurious Regression and Heterogeneity in Large Panels

Transcription

1 Coitegratio versus Spurious Regressio ad Heterogeeity i Large Paels Lorezo rapai Cass Busiess School Jauary 8, 009 Abstract his paper provides a estimatio ad testig framework to idetify the source(s) of spuriousess i a large ostatioary pael. his ca be determied by two o mutually exclusive causes: poolig uits eglectig the presece of heterogeeity ad geuie presece of I () errors i some of the uits. he paper proposes two tests that complemet a test for the ull of coitegratio: oe test for the ull of homogeeity (ad thus presece of spuriousess due to some of the uits beig geuiely spurious regressios) ad oe for the ull of geuie coitegratio i all uits of the pael (ad thus spuriousess arisig oly from eglected heterogeeity). he results are derived usig a liear combiatio of two estimators (oe cosistet, oe icosistet) for the variace of the estimated pooled parameter. he paper also derives two estimators for the degree of heterogeeity ad for the fractio of spurious regressios; cosistecy is achieved as log as (; ), with o eed for special restrictio o the rate of expasio betwee ad as they pass to i ity. JEL Codes: C, C3. Keywords: Large Paels, Heterogeeity, Spurious Regressio. Cass Busiess School, Faculty of Fiace, 06 Buhill Row, Lodo ECY 8Z, el.: +44 (0) ; L.rapai@city.ac.uk

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3 Itroductio Cosider the heterogeeous pael regressio model y it = i + i x it + u it ; () where i = ; :::;, t = ; :::; ad the variables y it ad x it are both I () for each i. As far as estimatio is cocered, sometimes the pooled versio of () is employed, i.e. y it = i + x it + v it ; () either because the assumptio of homogeeity ( i = for all i) is ot rejected by the data or because the object of iterest are ot the uit-speci c slopes but the log-ru average parameter - see also the commets i emple (999). However, poolig itroduces a further compoet i the error term, ( i ) x it, which is I () ad thus makes the pael spurious, uless i = ; this was oted by Phillips ad Moo (999), who proved that uder heterogeeity, () is equivalet to a spurious regressio ad the estimate of is p -cosistet as opposed to p -cosistet which would be the case i a coitegrated pael. hus, imposig homogeeity leads to a spurious pael. O the other had, i (), for each i the error term u it ca be either I (0), ad thus the uit is a coitegratio relatioship, or I (), ad therefore uit i is geuiely a spurious regressio, irrespective of heterogeeity. his situatio could e.g. correspod to the case (ofte foud i empirical applicatios) where u it is truly statioary i accord with some theory, but it is observatioally equivalet to a I () process due to mis-speci catio. A comprehesive review of the literature o the possible causes of this is i a recet cotributio by Fuertes (008). hus, lettig [0; ] be the proportio of uits that are spurious regressios, () could be a coitegrated pael ( = 0), a pael where all uits are described by spurious regressios ( = ), or ay situatio i betwee. I light of these cosideratios, a pooled pael model ca thus be spurious due to two (ot mutually esclusive) reasos: eglected heterogeeity or geuie spuriousess of some uits. herefore, a test for the ull of pael coitegratio applied to () is i fact a test for both homo- 3

4 geeity/poolability ad = 0. he questio the arises as to how to disetagle the two possible sources of spuriousess. he purpose of this paper is to provide a step further after rejectig the ull of pael coitegratio, by providig two tests, oe for the ull of homogeeity ad oe for the ull of pael coitegratio. hese two tests make it possible to idetify the source of spuriousess i the pael. As a by-product, this paper also proposes two cosistet estimators, for the degree of heterogeeity across uits ad for the fractio of spurious regressios,. he two estimators are cosistet ad use the model i levels, (), as opposed to models i di ereces where the risk of overdi erecig is preset. Particularly, the two estimators are based o a liear combiatio of two estimators, oe cosistet ad oe icosistet, of the variace of the estimate i (). Note that i a mixed pael cotext (where some uits are coitegrated ad some are ot), estimatig the degree of heterogeeity usig data i levels is a otrivial exercise as some of the uit speci c estimates i s will be icosistet ad therefore caot be used. Estimatio of the fractio of spurious regressios has bee recetly addressed by Ng (008), where a di eret estimator is proposed to estimate (0; ]. I this paper, a cosistet estimator for is proposed ad cosistecy is show i all the iterval [0; ], icludig the boudary = 0; o special assumptios, such as uit log ru variaces i the u it s are required. Based o this estimator, a test for the ull of coitegratio H 0 : = 0 ca be costructed, which has various advatages, sice ofte coitegratio is the workig hypothesis of iterest. Results are derived joitly for (; ), ad all the asymptotics is derived allowig for cross depedece of various stregth, icludig strog cross depedece that could arise from a factor structure i the error term. he paper is orgaised as follows. Sectio discusses the model ad the mai assumptios; cosistet estimatio of the degree of heterogeeity ad of the fractio of spurious regressios is discussed i Sectio 3, ad the results cocerig testig are i Sectio 4. Sectio 5 cocludes. Notatio is fairly stadard. hroughout the paper, deotes the ordiary limit, weak covergece ad d weak covergece uder the ull d H0 4

5 hypothesis of a test, ad p covergece i probability. Stochastic processes such as W (r) o [0; ] are usually writte as W, itegrals such as R W (r) dr 0 as R W ; W, W etc. deote idepedet demeaed stadard Browia motios. We let M, M etc. such that M j < be geeric positive costats, ot depedig o or. Model ad assumptios Recall the model (), where for the sake of simplicity we cosider oly oe regressor, x it, ad its pooled versio We let y it = i + i x it + u it ; y it = i + x it + v it : u it = u () it d ( ) + u it ( d ) ; (3) with d = for i = ; :::; bc ad zero otherwise. We assume that u () o-statioary ad u ( ) it is statioary, i.e. u () it = u () it ( ) u (L) u it = u it; + u it; it is where u (L) is a lag polyomial with all roots withi the uit circle. I (), it holds that v it = u it + ( i ) x it. he regressor x it is assumed to be i.i.d. across i have the followig DGP x it = x it + e x it: Let it = [e x it; u it] 0 ad cosider the followig assumptios. Assumptio : [cross sectioal properties] there exists a ivariat - eld C such that E ( it j C) = 0 ad it j C is i.i.d. across i. Assumptio : [time series properties] (i) E k it j Ck 8+ < for some 5

6 > 0; (ii) ad a ivariace priciple holds for the partial sums of it such that for all r [0; ] b rc r a:s: p it = B i (r) + O p ; t= where B i (r) is a vector Browia motio with covariace matrix i = x;i 0 0 u;i Assumptio 3: [heterogeeous coe ciets] (i) for all i, the i s are i.i.d. with E ( i ) =, V ar ( i ) = [0; +) ad E j ij 4+ < for some > 0; (ii) f i g ad fx it ; u it g are two mutually idepedet groups. he setup cosidered here allows for a mixed pael, where bc uits are spurious regressios ad the rest of the uits are coitegratio relatioships. Note that allowig for [0; ] meas that the boudary cases whereby all uits are coitegratio/spurious relatioships ca be accommodated withi this framework. hus, we etertai the cases that (a) all uits are coitegrated, (b) all uits are spurious regressios ad (c) the pael is a mixture of coitegratig ad spurious regressios (mixed pael). Hece, the tests discussed below are robust to the boudary cases of coitegratio or spurious regressio across all uits as well as the case of a pael with mixed I (0) ad I () error terms. Likewise, allowig [0; +) meas that the results derived heceforth are valid uder both homogeeity ad heterogeeity. Note that whe > 0 model () represets a spurious regressio for all uits i sice the error term is always I () because it cotais ( i ) x it I () for all i - see also Phillips ad Moo (999a, p. 080) - ad also possibly because u it I () for some i. Assumptio cosiders a geeral speci catio for the cross sectioal properties of pael y it. Cross sectioal idepedece is allowed for, i which case it j C is i.i.d. across i for C beig the empty set. O the other extreme, strog cross sectioal depedece as could arise form a factor model is co- # : 6

7 sidered also. Lettig e.g. it = it + 0 if t with it iid across uits ad i a oradom vector of loadigs, ad cosiderig the - eld de ed by ff t g t=, cross sectioal idepedece ca be achieved by coditioig o ff t g t=. A similar argumet, to prove the pael asymptotics with commo shocks, is also used i Kao, rapai ad Urga (008). Other forms of cross sectioal depedece ca be also cosidered. Note that achievig cross sectioal idepedece by coditioig o some - eld is eeded to prove the asymptotics hereafter; essetially, the zero mea iid coditio assumed i Assumptio, together with the momet coditios i Assumptio (ii), make it possible to use a Cetral Limit heorem (CL heceforth) ad a Law of Large Numbers (LLN) for Martigale Di erece Sequeces (MDS). A similar approach was proposed, i a cross sectioal framework, by Adrews (005), ad it heavily relies o Hall ad Heyde (980). Last, it is importat to ote that whilst the presece of a commo factor structure is allowed for by Assumptio, the commo factors are required to be I (0). Although some of the asymptotics with I () commo factors has bee discussed i rapai (008), i this cotext it is ecessary to rule out the presece of commo ostatioary factors, as these would reder the error term u it I () for all uits, thereby leadig to = with o eed for estimatio. A similar argumet (with discussio) is also i Ng (008). ime depedece is assumed, as log as the Fuctioal Cetral Limit heorem holds. Assumig that the remaider is of order O p = is quite stadard ad could e.g. be proved if oe assumed a liear process with a Beveridge ad Nelso decompositio, by applyig the method of proof proposed i Phillips ad Solo (99) - see also Phillips ad Moo (999). he momet coditio i Assumptio (ii) is required to prove a Liapuov coditio i the asymptotics below. Assumptio 3 is eeded oly i order for the CL ad the LLN to hold for the i s such that e.g. P i= ( i ) p ad = P i= ( i ) d N (0; ); less strict assumptios could be cosidered as log as the CL ad the LLN hold. Let x it = x it P t x it ad y it = y it P t y it ad de e the LSDV 7

8 estimator for i () as = # # x it y it ; x it i= t= i= t= ad cosider the followig (icosistet ad cosistet respectively) estimators of the variace of d V ar d V ar = P P i= t= v it P P ; i= t= x it P h P i = i= t= x itv it h P P i i= t= x it where v it = y it xit. Heceforth, we de e x = lim P i x;i ad u = lim (bc) P bc i= u;i. he asymptotics for, d V ar ad d V ar is give i the followig heorem. Propositio Let Assumptios -3 hold, ad assume that the rst bc uits are spurious regressios. he, as (; ) with that p d with Z N (0; ). As (; ) s 5 u x d V ar p 5 d V ar 0 it holds Z; (4) u x p u x ; (5) + : (6) Propositio states that is estimated cosistetly at a rate p. his result is typical i pael spurious regressio as show by Kao (999) ad Phillips ad Moo (999a). he ovel result i (4) is the asymptotic distributio of uder the broadly geeral Assumptios -3. Note that the limitig distributio of is ormal istead of mixed ormal, cotrary to what oe 8

9 might expect i light of Adrews (005) ad Kao, rapai ad Urga (008a, 008b). his is due to the assumptio that the commo factors are statioary, ad thus disappear whe the terms that cotai them are ormalised by. Note that however, as oted by Kao (999), se is a icosistet estimator of the variace of. d Equatios (5) ad (6) provide the probabilily limits for V ar ad d V ar. As it ca be see, ad as also proved i rapai (008) buildig d o a idea i Phillips ad Moo (999), V ar estimates the asymptotic variace of d cosistetly, whilst V ar is a icosistet estimator. Note that, as it is usually the case for results that ivolve a LLN as opposed to a CL, o restrictios o the rate of expasio of ad is required here. 3 Cosistet estimatio of ad From (5) ad (6), de e for brevity the probability limits as = u x 5 ; (7) = u x + : (8) If ad were observable, the estimators of ad could be expressed as, after solvig the liear system de ed by (7) ad (8) u x = 5 + ; (9) = 5 : (0) Equatios (9) ad (0) show that (although ifeasible) there exists a direct estimator for. As far as is cocered, this caot be estimated directly, ad estimates of x ad of u are required also. From (5) ad (6), cosistet 9

10 estimates for ad are = = P i= h P P i= P i= h P t= x itv it i P i= t= x it P t= v it P ; t= x it i ; solvig the liear system etails = 5 : () As far as estimatio of is cocered, give a estimate of x, say x, after de ig d u x = 5 + ; it would be possible to estimate directly u from (7)-(8) as d u = 5 + x: () A atural estimator of x is give, also i light of (6) i Appedix, by x = 6 i= x it; (3) ad we prove that x = x + O p =, thus beig cosistet. I order to estimate, we eed to lter out u as well. Note that is is ot possible to estimate u directly, sice it is ot kow a priori which uits are spurious regressios ad which oes are coitegrated. hus let i = h P i h P t= x it t= x ity it i, ad de e u it = y it i x it. he a kerel estimator of u;i ca be costructed for each i as u;i = u;i (u i ; :::; u i ), ad we t= 0

11 ca de e HAC = HAC = h P P i= i P i= t= x it P t= x it i= i= t= t= vit u;i # v it x it ; u;i ; so that a feasible estimator of is give by he it holds that = 5 HAC + HAC x: (4) heorem Let Assumptios -3 hold, ad assume (for the sake of coveiece) that the rst bc uits are spurious regressios. Let d be de ed as d = ( if = 0 0 if > 0 ; de e d = if = zero otherwise, ad let d = 0 if = 0 ad d = otherwise. Assume that u;i = u;i + O p ( ) for i = ; :::; bc ad u;i = O p ( ) for i = bc+; :::; for some ; 0;. he as (; ) = O p p + d O p p + O d p + o p () ; (5) d u u = O p p + O p p + o p () : (6) Corollary Uder the same assumptio as heorem ad usig the same otatio, it holds that as (; ) = ( d) O p 0 + ( d ) O p; p oa + o p () : (7) mi

12 heorem states that ad d u are cosistet estimators for ad u respectively, as log as (; ). No restrictios o the expasio rate of ad is required as they pass to i ity: therefore, from a practical viewpoit, the estimators ca be applied to paels with all the possible combiatios of ad. he heorem states that = O p o p; p = mi p o for (0; ] ad > 0, ad = O p = mi ; for = 0 or = 0. hus, a discotiuity is preset i the rate of covergece (ad it ca also be expected i the limitig distributio) of whe oe of the two parameters ; is o the boudary. However, cotrary to what foud i Ng (008) usig a di eret estimatio techique, o discotiuities are foud i the rate of covergece of d u whe = 0. Cosistet estimatio of is possible accordig to Corollary,although at a slow rate that depeds essetially o ad ad also o the rate of covergece of the HAC estimators u;i. Note that (7) states that there are discotiuities i the rate of covergece (as foud i Ng, 008) at the boudaries for [0; ]. Particularly: if (0; ), the = O p 0 + O p; p oa + o p () ; mi if = 0, 0 = O p; p oa + o p () ; mi ally if =, = O p + O p p + o p () : he rate of covergece of whe (0; ) depeds o the speed of covergece of the HAC estimators, ad. Assumig,as it could be expected, that = =, the optimal rate of covergece that maximises

13 the rate of covergece of ca be foud as a solutio of mi p; p o5 ; mi which after some algebra yields = mi =4 ; =4. his provides a idicatio as to the optimal choice of the badwidth whe estimatig u;i. 3. Limitig distributio for As a acillary result, the followig theorem provides the limitig distributio of for (0; +) Corollary Let Assumptios -3 hold. As (; ), the limitig distributio of (0; +) is give by (i) whe > 0, uder 0 p d N (0; V ) ; (8) where V = 4 u x ; (9) with = E Z Z W W 5 # W Lemma (ii) whe = 0, uder p 0 d p xp () (0) 5 where () is a radom variable with a chi-squared distributio with oe 3

14 degree of freedom ad = E Z Z W 30 W # : 4 ests for coitegratio ad homogeeity his sectio provides the ext step forward after applyig to () a test for the ull of pael coitegratio. As poited out i the itroductio, () ca be a spurious pael accordig to a pael coitegratio test due to two reasos (ot mutually exclusive), amely that = 0 ad/or = 0. Whilst the former case meas that some of the uits i the pael are geuiely spurious regressio, the latter arises from eglectig heterogeeity after poolig. Formally, a test for the ull of coitegratio would be based o H 0 : = 0 ad = 0: () Whe H 0 is rejected, the three possible cases ca be cosidered:. > 0 ad = 0: the pael is homogeeous ad thus poolig is appropriate, but some of the uits are (observatioally equivalet to) spurious regressios. I this case, a atural further step is the cosistet estimatio of the fractio of spurious regressios ;. = 0 ad > 0: the pael is heterogeeous, ad thus poolig is ot appropriate. All the uits i the pael are geuie coitegratio relatioships, ad therefore the ull of pael coitegratio is rejected simply due to poolig which itroduces a otrivial I () compoet, give by ( i ) x it, i the error term. A atural step is to avoid poolig, ad possibly to estimate the degree of heterogeeity, ; 3. > 0 ad > 0: i this case, the pael is heterogeeous ad some of the uits are spurious regressios. Corollaries ad allow to estimate the fractio of spurious regressios i the pael,, ad the level of heterogeeity,. 4

15 Based o the passages i the proof of heorem, two tests are derived as a follow-up for (). 4. estig for coitegratio - H 0 : = 0 he rst test which we shall cosider is based o ( H 0 : = 0 ad (0; +) ; H A : > 0 ad [0; +) i.e. a test whose ull hypothesis is that all the uits are geuiely coitegrated, ad thus spurious regressio at a pael level arises from eglectig heterogeeity. Of course uder the alterative that some uits are spurious regressios, the possibility that the pael is homogeeous ( = 0) eeds to be etertaied also. Note that the ull hypothesis i this testig framework is that there is coitegratio, which seems atural sice coitegratio at a micro level is ormally assumed i light of some ecoomic theory. hus, the test is a test for the ull of coitegratio. he test ca be carried out by usig d u istead of, which has a slower x rate of covergece ad for which o distributioal results were derived. Sice after (6) d u x = u x + O p p + O p p + o p () ; uder H 0 : = 0 we have d u = O p = + O p =. hus, a atural x test statistic for the ull of coitegratio is (a suitably scaled trasformatio p; p o of) mi d u. Cosider also the class of local alteratives x H (; ) A : u x = c p; p o ad [0; +) ; mi where c > 0. It holds that 5

16 heorem Let Assumptios -3 hold. he, as (; ) with 0 p d u x d H0 p Z; () where Z N (0; ), = E ( i = E ) 4, ad Z Z W 5 W # : As (; ) with 0, p p; p o distributio of mi d u x that, uder H (; ) A as (; ) d u x p; p o mi d u x = O p (). Lettig the limitig as (; ) be de ed as D, we have = c + D : (3) I light of heorem, a test statistic for the ull that = 0 ca be costructed as S () = r where is a cosistet estimate of ; the S () has a stadard ormal distributio uder H 0 as (; ), as log as 0. hus, from the practical viewpoit, the test applies to paels where thetime series dimesio is larger tha the cross sectioal oe. Whe mi = p, p; p o p d u u = O x p (); however, this is ot a sharp boud ad the limitig distributio of this radom variable is ot stadard as it depeds o x the assumptios o the DGP of the x it s. Accordig to (3), the test has power versus local alteratives as (; ) for all combiatios of ad. A cosistet estimate of ca be costructed usig the micro iformatio as = d u x 4 i b ; i= 6 ;

17 where the i s are the estimates from the idividual regressios y it = i x it + u it, ad b = P i= i. 4. estig for homogeeity: H 0 : = 0 he secod test that is of iterest i this cotext is aimed at verifyig whether all the uits have the same respose to the idiosycratic covariates x it, i.e. if i = for all i. Formally, this meas ( H 0 : = 0 ad (0; ] H A : > 0 ad [0; ] : I this testig framework, H 0 states that although some uits are coitegrated ad other are spurious, the pael is homogeeous, sice the slopes i are the same across all uits. Of course, whilst uder the ull some uits must be spurious regressios ( > 0), uder the alterative we also etertai the possibility that the pael is geuiely coitegrated so that = 0. We also cosider the class of local alteratives H (; ) A : = with c > 0. It holds that c mi ; p o ad [0; ] ; heorem 3 Let Assumptios -3 hold. he, as (; ) with d H0 6 u r 0 p 0 (); (4) where = E Z Z W W 0 W # : As (; ) with p 0, p = O p (). Lettig the limitig distributio of mi ; p o as (; ) be de ed as D, we have that, 7

18 uder H (; ) A as (; ) mi ; p o = c + D : (5) heorem 3 suggests the followig test statistic for the ull that = 0 S () = r 0 6 u ; S () has a chi-squared distributio with oe degree of freedom uder H 0 as (; ), as log as p 0. he same cosideratios as for heorem for the case mi ; p o = p apply here too, ad this test too has otrivial power versus local alteratives. Estimates of u, x, ad, that are eeded to make the test feasible, are de ed i (), (3) ad () respectively. 5 Coclusios his paper provides a estimatio ad testig framework to idetify the source(s) of spuriousess i a large ostatioary pael. his ca be determied by two o mutually exclusive causes: poolig uits eglectig the presece of heterogeeity ad geuie presece of I () errors i some of the uits. he paper proposes two tests that complemet a test for the ull of coitegratio: oe test for the ull of homogeeity (ad thus presece of spuriousess due to some of the uits beig geuiely spurious regressios) ad oe for the ull of geuie coitegratio i all uits of the pael (ad thus spuriousess arisig oly from eglected heterogeeity). he results are derived usig a liear combiatio of two estimators (oe cosistet, oe icosistet) for the variace of the estimated pooled parameter. he paper also derives two estimators for the degree of heterogeeity ad for the fractio of spurious regressios; cosistecy is achieved as log as (; ), with o eed for special restrictio o the rate of expasio betwee ad as they pass to i ity. 8

19 Refereces [] Adrews, D.W.K. (005), Cross-Sectio Regressio with Commo Shocks, Ecoometrica, 73, [] Baltagi, B.H., Kao, C. ad Liu, (008), Asymptotic Properties of Estimators for the Liear Pael Regressio Model with Idividual Effects ad Serially Correlated Errors: he Case of Statioary ad No- Statioary Regressors ad Residuals, Ecoometrics Joural, 9, [3] Fuertes, A.-M. (008), Sieve Bootstrap t-test o Log-Ru Average Parameters, Computatioal Statistics ad Data Aalysis, 5, [4] Hall, P., Heyde, C. C. (980), Martigale Limit heory ad Its Applicatios, New York: Academic Press. [5] Kao, C. (999), Spurious Regressio ad Residual-Based ests for Coitegratio i Pael Data, Joural of Ecoometrics, 90, -44. [6] Kao, C., rapai, L. ad Urga, G. (008a), Asymptotics for Pael Models with Commo Shocks, WP-CEA-0-006, Cetre for Ecoometric Aalysis, Cass Busiess School. [7] Kao, C., rapai, L. ad Urga, G. (008b), Modellig ad estig for Structural Chages i Pael Coitegratio Models with Commo ad Idiosycratic Stochastic reds, mimeo, Cass Busiess School. [8] Ng, S. (008), A Simple est for No-Statioarity i Mixed Paels, Joural of Busiess ad Ecoomics Statistics, 6, 3-7. [9] Park, J.Y., Phillips, P. C. B. (999), Asymptotics for Noliear rasformatios of Itegrated ime Series, Ecoometric heory, 5, 69, 98. [0] Phillips, P. C. B., ad Moo, H. R. (999), Liear Regressio Limit heory for Nostatioary Pael Data Ecoometrica, 67,

20 [] Phillips, P. C. B., ad Solo, V. (99), Asymptotics for liear processes, Aals of Statistics, 0, [] rapai, L. (008), O the Asymptotic t-est for Large Nostatioary Pael Models, mimeo. 0

21 6 Appedix A: useful Lemmas Lemma Let Assumptios -3 hold. he, as (; ), it holds that: 4 i= i= t= x it t= bc () i= [ ( )] bc () 4 i= [ ( )] 4 t= x it a:s: = x p ; (6) 6 + O p p 4 x 45 ; (7) h i u () a:s: it = u 6 + O p p ; (8) i=bc+ t= h t= h u ( ) it i x it u () a:s: it = x u 90 + O p i=bc+ t= h x it u i a:s: = O p ; (9) i ( ) a:s: it p ; (30) = O p : (3) Proof. I order to prove the results i the Lemma, it is ot possible to use the cetral limit theory developed i Phillips ad Moo (999) due to the possible presece of cross sectioal depedece amog uits. he theoretical tools that will be employed are a Law of Large Numbers ad a Cetral Limit heorem for Martigale Di erece Sequeces (MDS LLN ad MDS CL heceforth), based o achievig cross sectioal idepedece by coditioig upo the (ivariat) - eld C - see Assumptio. Cosider (6), ad let W i = P t= x it. Coditioal o C, we have that the W i s are iid across i i light of Assumptio. For some costats M ; M <, we have E jw i j Cj M E t= x it C E j x it j Cj : t=

22 Assumptio (ii) etails that as P t= j x itj Cj = O p () - see heorem 5.3 i Park ad Phillips (999). hus, E jw i j Cj <. As, the FCL etails W i j C = W i a:s: = x R W + O p = ad a MDS LLN ca be applied whereby i= Z a:s: W i = xe W + O p ; we kow from Kao (999) that E R W = =6. As far as (7) is cocered, de e W i = P ; t= it x the, coditioal o C, the Wi s are iid across i ad for some costats M ; M < E jw i j Cj M E 4 t= x 4 it C M E 4 j x it j Cj 4 : Similar argumets as before yield P 4 t= j x itj Cj 4 = O p (). a:s: hus, E jw i j Cj < ; as, the CM yields W i j C = W i = R W + Op =. hus, the MDS LLN implies that, as (; ) 4 x i= W i a:s: = 4 xe Z W # t= + O p : Kao (999) proves that E h R W i = =45. he proof for (8) is very similar to the proof for (6), ad thus it is omitted. As far as (9) is cocered, de e W 3i = P h i ( ) ; t= u it sice coditioig o C the W3i s are idepedet, ad give that, as, W 3i j C = O p () for all i i light of the LLN for statioary variables, we have i= a:s: W 3i = E (W 3i j C) = O p :

23 h P i ; We ow tur our attetio oto (30). Let W 4i = 4 t= x itu () it the, coditioal o C W 4i is a iid sequece with E jw 4i j Cj M E M E 4 t= 4 4 x it h u () it i C = j x it j Cj 4 t= 4 = 3 u () C 4 5 ; ad Assumptio (ii) implies that as P 4 t= j x itj Cj 4 = O p () ad P 4 t= u () it C 4 = O p (). hus, E jw 4i j Cj < ; applyig the FCL a:s: R ad the CM leads to, for, W 4i j C = W 4i = x W u W + O p =, ad the MDS LLN leads to P h i= W a:s: R i 4i = x ue W W. h R i Baltagi, Kao ad Liu (008) prove that E W W = =90. Cosiderig h P i ow (3), de ig W 5i = t= x ( ), itu we have that coditioally o C the W 5i s are idepedet ad we could prove that, followig similar argumets as for (9) ad (30), E jw 5i j Cj <. implies i= it t= a:s: W 5i = E (W 5i j C) = O p : it hus, the MDS LLN 7 Appedix B: Proofs ad derivatios Proof of Propositio. See rapai (008). Proof of heorem. Assume, with o loss of geerality, that the rst bc uits are spurious regressios. I order to derive the results, recall = i= t= x it i= x it u it + t= ( i ) i= t= x it # = O p p ; 3

24 ad cosider the followig expasios: = i= i= + x itv it t= x it u it + t= ( i ) i= i= 4( i ) i= t= t= x it x it t= # x it u it t= x it i= t= 4( i ) i= x it t= x it 3 # x it u it ; (3) t= 5 ad = i= i= + t= v it u it + t= ( i ) i= ( i ) i= t= # x it u it t= ( i ) i= t= x it x it # + # i= i= t= x it u it t= x it : (33) Cosider rst (5), ad cosider the quatity P i= P t= x it. It is well kow, e.g. from Phillips ad Moo (999), that P i= P t= x it = O p ( ); Kao (999) proves that, as (; ), ( ) P i= P t= x it = x=6 + o p (). Replacig ( ) P i= P t= x it with its limit, usig (3) ad (33) 4

25 after some algebra we have = 5 = x x 4 i= t= 8 = 6 < 5 x : x 4 i= t= + ( i ) 4 5 i= x i= x ( i ) + 30 x 4 i= 30 x 4 30 x 4 x itv it x it u it t= t= x it t= 4( i ) i= + x it x it i= i= t= t= v it 3 5 u it it3 x 5 t= it3 x 5 t= # x it u it t= t= x it 3 # x it x it u it ( i ) i= t= t= i= x it u it + ( i ) i= t= 5 i= t= x it # x it u it t= #) = A + A + A 3 + A 4 + A 5 + A 6 + A 7 + A 8 + A 9 + A 0 : (34) Cosider A : A = 5 bc x i= = A + A : t= x it u it + 5 x i=bc+ x it u it t= 5

26 As (; ), (6) yields A = 5 u 90 + O p p ; where the term O p = is a samplig error - see also Phillips ad Moo (999); also, A = O p ( ). Similarly, as far as A is cocered, we have ad as (; ) A = bc i= t= = A + A ; u it + A = u 6 + O p p ; i=bc+ t= whilst A = O p ( ). hus, A + A = O p =. As far as A 3 is cocered, de e Z i = 5 x 4 t= x it Assumptio esures that Z i is idepedet of ( i ); as, we have E (Z i ) = 3 x t= x it ; u it 6 x + O p p = 6 x + O p p ; where the term O p = arises from a samplig error. hus, the sequece ( i ) Z i is a iid sequece with E ( i ) Z i = E (i ) E [Z i ] = x=6 + O p Cosiderig A 4, we have = ; the MDS LLN etails a:s: A 3 = x + O p p : 6 A 4 = Z i ; 6 i=

27 where Z i is a sequece of ozero mea, iid variables ad the MDS LLN yields P i= Z i = O p (). Sice = O p =, we have A 4 = O p ( ). As far as A 5 is cocered, we have 30 x i= ( i ) Z i = 30 bc ( i ) Z i + 30 x x i= = A 5 + A 5 ; i=bc+ ( i ) Z i where Z i = P t= x it P t= x itu it. Note that the sequece ( i ) Z i is iid zero mea, ad therefore a CL esures that A 5 = O p = ; as far as A 5 is cocered, Z i = O p ( ) ad therefore A 5 = O p p. hus, A 5 = d O p. Cosider ow A 6 ; de ig Z 3i = p d 4 P t= x it, the sequece (i ) Z 3i is iid zero mea ad thus A 6 = 30 x ( i ) Z 3i i= = O p p d O p p = d O p : As far as A 7 is cocered, de e Z 4i = so that P t= x it P t= x itu it, A 7 = 30 x = A 7 + A 7 ; bc Z 4i + i= as, E (Z 4i ) = O = ad thus 30 x i=bc+ Z 4i bc Z 4i = i= i=bc+ Z 4i = bc bc [Z 4i E (Z 4i )] + E (Z 4i ) = O p i= i=bc+ i= [Z 4i E (Z 4i )] + i=bc+ p + O p ; p E (Z 4i ) = O p + O : 3= 7

28 hus, A 7 = O p d +Op we have p d A 8 = bc ( i ) Z 5i + i= = A 8 + A 8 ;. Cosiderig A 8, de e Z 5i = P t= x itu it ; i=bc+ ( i ) Z 5i sice ( i ) Z 5i is iid zero mea, we have A 8 = O p = ad A 8 = O p p, so that A 8 = d O p. As far as A 9 is cocered, ote that p d A 9 = bc Z 5i + i= i=bc+ = A 9 + A 9 ; as, we have E (Z 5i ) = O =, ad therefore Z 5i A 9 = bc [Z 5i E (Z 5i )] + i= = O p + O p p ; bc E (Z 5i ) i= similar argumets lead to A 9 = O p +Op p. Hece, A 9 = O p + d O p p d. Last, as far as A 0 is cocered, lettig Z 6i = ( i ) P t= x it, we have E (Z 6i ) = 0, ad E jz 6i j + < ; thus, the MDS CL yields P i= Z 6i = O p ( p ) ad therefore A 0 = d O p. Puttig all together, we have = 6 x x 6 + O p p + O p + d O p p + o d p () : (35) he proof for (6) follows similar passages. We have d u = 5 + x; 8

29 ad replacig ( ) P i= P t= x it ad x with their limit, usig (3) ad (33) after some algebra we have d u = < = 6 : + 5 x 4 5 x 4 i= i= t= t= ( i ) 4 i= + 30 x 4 4 i= ( i ) i= + 30 x x 4 x itv it + x it u it + t= t= t= x it x it x it 4( i ) i= i= 4 i= t= x it x it u it t= i= i= 5 x 5 x t= t= v it 3 5 u it t= t= # x it u it t= t= x it 3 5 x it x it # x it u it + 4 ( i ) t= i= 4 ( i ) i= 5 t= x it # x it u it t= #) = B + B + B 3 + B 4 + B 5 + B 6 + B 7 + B 8 + B 9 + B 0 : (36) We have, i light of similar argumets as before, that as (; ) B = 5 u 90 + O p B = u 6 + O p p ; p ; ad therefore B + B = u 6 + O p p : 9

30 Cosider B 3, ad de e Z 6i = x it t= 5 x 4 x it ; t= as, we have E (Z 6i ) = x 5 6 x = O p p : 4 x 45 + O p p Hece B 3 = ( i ) [Z 6i E (Z 6i )] + i= = B 3 + B 3 ; ( i ) E (Z 6i ) i= ad sice ( i ) [Z 6i E (Z 6i )] is a iid zero mea sequece, the CL etails B 3 = O p =. Also, B 3 = O p =. hus, B 3 = O p p + O p p : As far as B 4 is cocered, we have B 4 = = = B 4 + B 4 ; i= Z 6i [Z 6i E (Z 6i )] + i= E (Z 6i ) similar argumets as above, together with = O p =, lead to B 4 = O p p + O p p : i= 30

31 As far as the other terms are cocered, same argumets as i the proof for (5) yield: B 5 = O p p d ; B 6 = O p ( ); B 7 = O p + d Op p d ; B 8 = O p p d ; B 9 = B 0 = O p. Puttig all together, we have d u = 6 u 6 + O p p + O p p + o p () : (37) Proof of Corollary. he proof follows a slightly di eret approach to heorem, sice HAC residuals are ivolved ad thus the estimatio errors u;i u;i eter. Recall the de itio of = = 5 + h P P i= i P i= t= x it P t= x it 5 P i= P t= x it i= i= t= i= t= vit u;i t= v it x it 6 u;i i= 6 x it i= t= v it x it + u;i i= t= t= x it vit u;i (38) : Replacig ( ) P i= P t= x it with its limit (ad omittig higher order terms for the sake of the otatio), de ig v + it = v it u;i ; ad u + it = u it= u;i, ad recallig that v it = u it + ( i ) x it x it, 3

32 (38) ca be further expressed as = = 90 x itv + 4 it + v + x it i= t= i= t= 90 x 4 it u i it x 4 i= t= x i= u;i t= x 4 i 4 it x i= u;i 4 t= x i= + 80 i # x 4 it u + it x it x i= u;i t= t= # 80 x 4 it u + it x it + x i= u;i t= t= i= + i # x it + i= u;i t= i= u;i t= + 4 i # 4 x it u + it i= u;i t= i= u;i # 4 i x it i= u;i = C + C + C 3 + C 4 + C 5 + C 6 + C 7 + C 8 + C 9 + C 0 + C + C : Cosider C. We have t= x it u;i 3 5 t= x it # u + it t= # x it u + it t= x it 3 5 C = 90 bc x i= = C ; + C ; : 4 x it u + it + 90 t= x i=bc+ 4 x it u + it t= 3

33 As far as C ; is cocered, it holds that C ; = 90 bc x i= 4 = C ;; + C ;; ; t= x it u it u;i 90 x bc i= u;i u;i u;i 4 t= x it u it u;i we kow that C ;; = + O p bc i= u;i u;i u;i 4 t= As far as C ; is cocered, ote that C ; = 90 x 90 x i=bc+ max i u;i 4 = ; as far as C ;; is cocered, we have u it x it max u;i i t= i=bc+ x it u it u;i 4 t= = O p u;i u;i u;i : bc i= 4 t= x it u it u;i x it u it = O p ( ) O p = O p : hus, C = + O p = + O p ( ) + O p ( ); clearly, the term of magitude O p ( ) is preset oly if > 0. Cosider ow C + C 8 C + C 8 = i= i u;i 4 90 x 4 t= x it + t= x it 3 5 ; ad lettig i = 90 x 4 + x it t= t= x it; 33

34 we ca write C + C 8 = bc i # i + i= u;i = C 8; + C 8; : i=bc+ i # i Sice E ( i ) = O p =, de ig i = i E ( i ) we have C 8; = bc i i= u;i = C 8;; + C 8;; ; i + u;i bc i E ( i )# i= u;i the MDS CL etails that C 8;; = O p =, ad C 8;; max je ( i )j i As far as C 8; is cocered, we have bc i i= C 8; max 4 i u;i + max 4 i u;i i=bc+ u;i i=bc+ = O p p : 3 ( i ) i 5 3 ( i ) E ( i ) 5 = O p ( ) O p p + O p ( ) O p p : hus, C +C 8 = O p p; p o = mi ; by de itio, this term is preset 34

35 oly if <. Let us ow tur our attetio to C 3 + C 9 ; we have C 3 + C 9 = = i= bc i= = C 39; + C 39; : u;i u;i i i + i=bc+ u;i i Recallig that = O p =, similar argumets as above lead to C 39; = O p 3= + O p =, ad C 39; = O p = mi p ; p o. Cosider C 4, ad de e i = ( i ) P 4 t= x it C 4 x 80 = bc i + u;i i= = C 4; + C 4; : Sice E ( i ) = E ( i ) E P 4 t= x it i=bc+ i u;i = 0 for all i ad, the MDS CL esures that P bc i= ( i = u;i ) = O p ( p ), ad thus C 4; = O p ( ). Also C 4; max i i = O p u;i i=bc+ p O p ( ) O p p = Op hus, C 4 = O p ( ). urig our attetio to C 5, ad de ig 3i = ( i ) P t= x itu + it P t= x it, we have C 5 x 80 = bc 3i + u;i i= = C 5; + C 5; : i=bc+ : 3i u;i 35

36 Sice E ( 3i ) = 0 for all i ad, the MDS CL yields C 5; = O p =. Whe u it is statioary, we have 3i = O p ( ), which implies C 5; max i u;i i=bc+ 3i = O p ( ) O p O p p : herefore, C 5 = O p = + O p =. As far as C 6 is cocered, let 4i = P t= x itu + it P t= x it C 6 x 80 = bc 4i + u;i i= = C 6; + C 6; : i=bc+ 4i u;i Note that E ( 4i ) = O = whe u it is ostatioary ad E ( 4i ) = O 3= whe u it is statioary. he we have C 6; = bc 4i + u;i i= = C 6;; + C 6;; ; bc E ( 4i ) u;i where 4i = 4i E ( 4i ). he, recallig that = O p =, applyig the MDS CL yields C 6;; = O p ( ) ad C 6;; max je ( 4i )j i = O p O p p ; i= bc i= u;i hece, C 6;; = O p = =, ad thus C 6; = O p ( )+O p = =. As far as C 6; is cocered, similar argumets as for C 5 etail C 6; = O p = + 36

37 O p 3=. Cosider ow C 7 C 7 = bc i= t= = C 7; + C 7; : u + it + i=bc+ t= uit Similar argumets as for C ; yield C 7; p + Op ( ) + O p =. As far as C 0, C ad C are cocered, these are similar to (respectively) C 5, C 6 ad C 4, ad therefore similar passages as above would prove that they have the same asymptotic magitude. Puttig all together, we have = ++( d) O p +( d ) O p p + O p p +o p () ; u;i which proves the theorem. Proof of Corollary. Cosider rst the case > 0, which correspods to (8). As (34) ad the passages thereafter show, as (; ) with 0, the terms that domiate are A 5 ad A 8 ; thus, the limitig distributio of p is give by # p 30 ( x 4 i ) x it x it u it i= t= t= = p ( 5 ( i ) x it u it i= t= x t= ( = p bc 5 ( i ) x it u it i= t= x t= ( + p 5 ( i ) x it u it x i=bc+ = p p bc Y i + O p i= : t= x it x it t= x it ( i ) i= #) #) #) # x it u it t= 37

38 he sequece fy i g bc i= is i.i.d. across i, ad, for all, it holds that E (Y i ) = E ( i ) E ( 5 x it u it x t= t= x it #) = 0: Also, ote that E jy i j + = ( i ) x it u it x t= x it t= # + ; which is ite i light of the proof of Lemma. hus, as (; ), a MDS CL holds for = P i= Y i such that bc p Y i d i= h i = E lim Y p i Z = V Z; with Z N (0; ) idepedet of V. As far as V is cocered, as Y i a:s: = ( i ) u x Z Z W W 5 W # + o p () ; ad thus E lim Y i = u x E Z Z W W 5 # W : hus, p d p q u x p Z: As far as (0) is cocered, whe = 0, (34) ad the passages thereafter etail that = O p ( ) as (; ) with p 0; the terms that domiate are, i this case, A 4 ad A 6. hus, the asymptotics of 38

39 is drive by 30 x i= x t= 4( i ) i= x it t= x it it3 x 5 t= 3 5 : Lettig Y i = ( i 4 5 x i= ) P t= x it with E (Y i ) = 0, ad recallig that t= x it t= x it 3 5 a:s: = 6 x + o p () ad that whe = 0 it holds that a:s: = 6 x Y i + o p () ; i= we have that a:s: = 36 4 x a:s: = 6 x 6 x p p i= Y i i= 30 6 Y i x x p Y i i= p Y i i= p # 30 d i + o p () ; x Y i d i + o p () i= with d i = P t= x it. Note that E (Y i d i ) = 0. Also, E jy i j + <, ad E jy i d i j + = E jy i j + E jd i j + <. hus, as (; ), the MDS CL yields d 6 x r E lim Y i v u ute lim (Y i ) # 30 d i Z ; x 39

40 with Z N (0; ); we have E lim Y i = E ( i ) E 4 x Z W # = 4 x 45 ; ad E lim Y i # 30 d i x = E ( i ) E = 4 xe Z x ( Z x Z W 30 Z W 30 W # : W ) Puttig it all together, we have v d p 6 u Z t x E x 45 Z W 30 W # (): Proof of heorem??. from (37). Uder the ull that = 0, it holds that he results i the theorem follow immediately d u = O p p + O p p + o p () ; ad thus d p; p o u = O p = mi. Cosider rst the case whereby 0, which etails d u = O p =. From (36), ad from the passages thereafter, it follows that the term that domiates i the decompositio is B 3, give by B 3 = ( i ) 4 i= t= x it 5 x t= x it 3 5 : Let Z i = P t= x it 5 ( x 4 ) P t= x it. From the proof of 40

41 Lemma we have, as Z a:s: Z i = x W 5 x Z W + O p p ; where W is a demeaed stadard Browia motio; (6) ad (7) etail that, uiformly i i E (Z i ) = x 6 5 x 45 + O p = O p : hus, lettig Z i = Z i E (Z i ), B 3 i (36) ca be rewritte as B 3 = ( i ) Zi + i= ( i ) E (Z i ) = B 3 + B 3 : i= Cosider B 3. Coditioig upo C, ( i ) Zi is a iid sequece with E ( i ) Zi = E (i ) E Z i = 0. Also, E (i ) Zi + = E (i ) + E Zi + : Assumptio 3 esures that E (i ) + <. As far as E Zi + is cocered, ote that E Zi + M x it + 4 t= x t= x 4 it + 3 ad therefore i light of Assumptio ad the passages i the proof of Lemma, E Zi + <. hus, (i ) Zi is a MDS that satis es a Liapuov coditio, ad therefore the MDS CL yields that P i= ( i ) Zi = O p ( p ) with, as (; ) p r ( i ) Zi d E i= h i ( i ) 4 lim Z i Z; 5 ; 4

42 with Z N (0; ) idepedet of Z i. As (; ), we have E h i ( i ) 4 lim Z i = E ( i ) 4 h i E lim Z i Z Z = 4 xe W 5 W # : As far as B 3 is cocered, it holds that B 3 max i je (Z i )j = O p p ; # ( i ) i= ad thus it is domiated by B 3 uder 0. Fially, recallig that d u x = d u 6 P P ; i= t= x it it holds that, uder H 0 as (; ) with 0 p d u x d x p 4 x Z: Whe 0, (37) etails p d u x = O p (). Last, cosider H (; ) A ; sice p; p o mi d u x the drift term is ozero as (; ) if a:s: p; p o = mi u + D ; x u x = p; p o: mi Proof of heorem??. he mai result for the proof is (34), which 4

43 will be costatly referred to heceforth. From (35), it emerges that = O p p + d O p p + O d p + o p () ; thus, for (0; ] ad = 0, the term that domiates whe (; ) (uder p 0) is of order O p ( ). As (34) ad the passages thereafter show, the terms that domiate are A 4, A 7 ad A 9 ; thus, the limitig distributio of is give by 4 5 i= x 4 + i= t= x it x it u it t= it3 x 5 t= 30 x Let Y i = P t= x itu it, ad ote that i light of Lemma it holds also, 4 5 x 4 i= t= x it a:s: = 6 x t= x it t= x it # 3 5 a:s: = 6 x + o p () ; # Y i + o p () : hus, after some algebra, the limitig distributio of is drive by 6 4 bc p x i= Y i i= p bc Y i ( + d i ) 5 + o p () ; i= : with d i = 30 x t= x it: 43

44 hus, the MDS CL yields d 6 E Y = i E Y i ( + d i ) = Z ; x where Z N (0; ). As E Y i = E 4 lim t= 3 x it u it 5 = u x 90 ; ad E Y i ( + d i ) = E = u xe 4 lim 3 30 x t= x it Z Z W W 3 30 t= W 3 x it u it 5 # ; ad thus v a:s: = 6 r u x x 90 u Z Z # t 9E W W 0 W Z + o p () v 6 u Z Z # = p t u E W W 0 W Z + o p () : 0 As (; ) with p = Op (). Last, cosider H (; ) A ; sice mi ; p o a:s: = mi ; p o + D ; the drift term is ozero as (; ) if = mi ; p o: 44