A Sieve Bootstrap Test for Cointegration in a Conditional Error Correction Model

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1 A Sieve Bootstrap Test for Coitegratio i a Coditioal Error Correctio Model Fraz C. Palm Stepha Smeekes Jea-Pierre Urbai Departmet of Quatitative Ecoomics Maastricht Uiversity December 5, 28 Abstract I this paper we propose a bootstrap versio of the Wald test for coitegratio i a sigle-equatio coditioal error correctio model. The multivariate sieve bootstrap is used to deal with depedece i the series. We show that the itroduced bootstrap test is asymptotically valid. We also aalyze the small sample properties of our test by simulatio ad compare it with the asymptotic test ad several alterative bootstrap tests. The bootstrap test offers sigificat improvemets i terms of size properties over the asymptotic test, while havig similar power properties. The sesitivity of the bootstrap test to the allowace for determiistic compoets is also ivestigated. Simulatio results show that the tests with sufficiet determiistic compoets icluded are isesitive to the true value of the treds i the model, ad retai correct size. JEL Classificatio: C5, C32. Keywords: sieve bootstrap, coitegratio, error correctio model. Correspodig author: Departmet of Quatitative Ecoomics, Maastricht Uiversity, P.O. Box 66, 62 MD Maastricht, The Netherlads. F.Palm@ke.uimaas.l. Previous versios of this paper have bee preseted at the Ecoometric Society Europea Meetig i Mila, August 28, at the Iteratioal Workshop o Recet Advaces i Time Series Aalysis i Cyprus, Jue 28, at the Workshop o Bootstrap ad Time Series i Kaiserslauter, Jue 28, at the coferece etitled Iferece ad Tests i Ecoometrics, A Tribute to Russell Davidso i Marseille, April 28, at a Ete Luigi Eiaudi Semiar i Ecoometrics i Rome, November 27, ad at the first workshop of the Methods i Iteratioal Fiace Network i Maastricht, September 27. We gratefully ackowledge the commets by participats at these semiars ad the commets by Aders Swese, Petti Saikkoe ad three aoymous referees. The usual disclaimer applies.

2 Itroductio I this paper we preset a bootstrap versio of the sigle-equatio error correctio model (ECM) Wald test for coitegratio origially proposed by Boswijk (994). Broadly speakig, tests proposed i the literature to test for the absece of coitegratio ca be classified i two groups. Tests that allow for more tha oe coitegratig vector uder the alterative usig for example a VAR framework, see e.g. Johase (995), ad tests that cosider sigle-equatio models ad assumig at most a sigle coitegratig vector uder the alterative. Amog the latter oes, we ca further distiguish betwee approaches based o the triagular represetatio of a coitegratio system that aturally leads to residual-based tests for coitegratio (e.g. Phillips ad Ouliaris, 99) that make use of semi-parametric correctio for edogeeity ad serial correlatio; ad those based o fully specified parametric data geeratig processes that aturally lead to sigle equatio dyamic models. The ECM test cosidered i this paper falls i this category. As already discussed i the literature, ECM tests are a attractive optio for coitegratio testig, as, cotrary to the more popular residual-based tests, ECM tests do ot suffer from imposig potetially ivalid commo factor restrictios (Kremers, Ericsso, ad Dolado, 992; Baerjee, Dolado, ad Mestre, 998; Zivot, 2). Moreover, Pesaveto (24) aalyzes several tests which have as ull hypothesis o coitegratio, icludig the residual ADF test by Egle ad Grager (987) ad the maximum eigevalue test by Johase ad Juselius (99), ad fids that amog these the ECM tests perform best i terms of power both i small ad large samples, while performig similarly as the other tests i terms of size. ECM tests thus appear to be a appealig tool of testig for coitegratio. The ECM Wald test has as mai advatage over the ECM t-test (Baerjee et al., 998) that it is more ituitive ad oe does ot have to add a redudat regressor if o particular coitegratig vector is specified. Although the Wald ECM test performs well i geeral, especially i terms of power, it still suffers from size distortios i fiite samples (see for example Boswijk ad Frases, 992). It is well kow that the bootstrap s ability to provide asymptotic refiemets ofte leads to a reductio of size distortios for hypothesis tests. Eve uder o-favorable coditios for the bootstrap, uder which it is uclear whether it provides asymptotic refiemets, such as whe dealig with ostatioary time series, the bootstrap has bee show to reduce size distortios i fiite samples (see for example the tests for uit roots cosidered i Chag ad Park, 23, Palm, Smeekes, ad Urbai, 28 or Paparoditis ad Politis, 23). Little is kow so far about the applicatio of the bootstrap to coitegratio testig i error correctio models. Swese (26) ad Trekler (26) provide theoretical ad The otable exceptio to the lack of theoretical results is Park (23), who shows that bootstrap ADF tests offer asymptotic refiemets uder the assumptio that the errors are a fiite AR process with kow order. 2

3 simulatio results o bootstrap versios of the trace test for coitegratio rak by Johase (995). Their settig differs from ours i that we a priori assume that the coitegratig rak is at most oe. Seo (26) provides aalytical ad simulatio results for a residual-based bootstrap test i a threshold vector error correctio model. Closer to our settig, Matalos ad Shukur (998) ad Ahlgre (2) cosider a bootstrap versio of the test with kow coitegratig vector by Kremers et al. (992), however they oly provide simulatio results for a simple model. I this paper we will allow for more geeral depedece over time i our model, ad we provide aalytical as well as simulatio results. Our paper relies o the sieve bootstrap itroduced by Bühlma (997), a method that ca hadle time series depedece i the form of a geeral liear process that is approximated by a autoregressive process. The sieve bootstrap method is easy to use ad performs well relative to other time series bootstrap methods, especially the block bootstrap (for a compariso betwee methods i the uit root settig, see Palm et al., 28). The coditio of liearity is fulfilled by a large class of processes, ad is eeded to validate the use of the Wald test without the bootstrap as well. The cotributio of the paper is threefold. First, we prove that the sieve bootstrap versio of the sigle-equatio Wald test of o coitegratio is asymptotically valid. The proofs are give i detail for the multivariate settig, such that proofs of other types of tests could be doe alog the same lies as preseted here. Secod, we provide simulatio results showig that the bootstrap versio of the Wald test has better properties i fiite samples tha the asymptotic test. Third, we ivestigate the sesitivity of the bootstrap to various specificatios of determiistic compoets ad alterative distributioal assumptios. The structure of the paper is as follows. Sectio 2 explais the model ad assumptios. The costructio of the bootstrap test ad the establishmet of its asymptotic validity are discussed i Sectio 3. Our simulatio study is preseted i Sectio 4. The iclusio of determiistic compoets is discussed i Sectio 5. Sectio 6 cocludes. All proofs are cotaied i Appedix A. Fially, a word o otatio. We use to deote the Euclidea orm for vectors ad matrices, i.e. v = (v v) /2 for a vector v ad M = (tr M M) /2 for a matrix M. For matrices we also use the operator orm M = max v Mv / v. W(r) = (W (r),w 2 (r) ) deotes a multivariate stadard Browia motio of dimesio ( + l). x is the largest iteger smaller tha or equal to x. Covergece i distributio (probability) is deoted by d ( ). p Bootstrap quatities (coditioal o the origial sample) are idicated by appedig a superscript to the stadard otatio. Subscripts p (or q) are used to idicate quatities depedig o approximatios of ifiite order models by models of order p (or q). For simplicity we suppress these subscripts wheever clarity allows it. 3

4 2 The model Our Data Geeratig Process (DGP) is closely related to that of Pesaveto (24). We let our ( + l)-dimesioal time series z t = (y t,x t ) be described by the process z t = µ + τt + ζ t. () The stochastic compoet ζ t is give by ζ t = (ρ )αβ ζ t + u t, (2) where u t = Ψ(L)ε t (3) with Ψ(z) = j= Ψ jz j. Furthermore we assume that ζ =. 2 The ull hypothesis is H : ρ =, there is o coitegratio. Uder the alterative H : ρ < there is coitegratio with a sigle coitegratig vector β ad the error correctio term must be preset i the equatio for y t. Also, we impose α = ad α 2 =, which follows from the triagular represetatio of the model as i Pesaveto (24) ad is eeded for idetificatio purposes. 3 These poits are formalized i Assumptio. Assumptio. We assume (i) αβ is of rak, i.e. there is a sigle ( + l)-dimesioal coitegratig vector β, (ii) β is ormalized o the coefficiet of y t, i.e. β = (, γ ), (iii) α = (, ). It is importat to remark that Assumptio is of o importace for the derivatio of the ull distributio of the tests as it oly cocers the situatio where coitegratio is preset i the system. It is however importat to derive the equivalece betwee the triagular represetatio ad the ECM form. Assumptio is also importat to eable us to focus o a sigle equatio ECM ad to rule out cases where the ECM tests would trivially have low power. This would for example occur uder the alterative if the coitegratio vector oly appears i the equatio for the coditioig variables x t. Equatio (3) shows that we take u t to be a liear process (Phillips ad Solo, 992). Assumptio 2 esures the ivertibility of u t ad the existece of momets of ε t. These assumptios are ot very striget ad ecompass may assumptios (icludig all fiite VARMA models) that are ofte used i coitegratio aalysis. 2 This assumptio is made for expositioal simplicity oly ad ca be exteded to ζ = O p(). 3 Pesaveto (24) shows that this restrictio correspods to the assumptio that x t are ot mutually coitegrated, as required uder Assumptio (i), ad are kow a priori to be I(). 4

5 Assumptio 2. We assume (i) ε t are i.i.d. with E(ε t ) =, E(ε t ε t ) = Σ ad E ε t 4 <. (ii) det(ψ(z)) for all z, ad j= j Ψ j <. By Assumptio 2 we may write Φ(L) = j= Φ jl j = Ψ(L). We may the substitute equatio () ito (2) ad apply the Beveridge-Nelso decompositio to show as i Pesaveto (24) that this model ca be rewritte i VECM form z t = (ρ )Φ()αβ (z t µ τ(t )) + τ + Φ (L) z t + ε t (4) where, Φ (L) = ( ρ) j= i=j+ Φ i αβ Φ j+ L j ad τ = Φ j + (ρ ) j= j= i=j+ Φ i αβ τ. It ca be see from the above represetatio that z t has a drift if τ, ad this drift leads to a liear tred i the coitegratig relatio if β τ. The costat µ oly appears i the coitegratig relatio; ote that the coitegratig relatio has mea zero if β µ =. Pesaveto shows that the model ca be writte i triagular form as well, which makes it a very flexible model. As we do ot eed that represetatio here, we cotiue with the VECM represetatio (4) ad coditio o x t to obtai y t = (ρ )θβ (z t µ τ(t )) + τ + π x t + π j z t j + ξ t, (5) where ξ t = ε,t Σ 2 Σ 22 ε 2,t i.i.d. (,ω 2 ) ad θ = Φ ()α Σ 2 Σ 22 Φ 2()α with Φ() = (Φ (),Φ 2 () ). 4 The advatage of this framework is that its assumptios are weaker tha what is usually assumed for tests based o a coditioal ECM, as it does ot impose that x t are weakly exogeous for β uder the alterative of coitegratio. Uder the ull however, the error correctio term does ot appear i the margial equatios, which makes a test o the error correctio term i the coditioal model a valid test for coitegratio (Boswijk, 994). 4 Note that ω 2 = σ Σ 2Σ 22 Σ2. Σ ad εt have bee partitioed coformably with yt ad xt, i.e. ε,t is a scalar ad ε 2,t is a l-dimesioal vector. 5

6 3 The bootstrap test ad asymptotics 3. Test statistic The Wald test proposed by Boswijk (994) is based o the coditioal model (5). Cosider the regressio y t = δ z t + λ D t + π x t + p π j z t j + ξ p,t, (6) where D t are the (urestricted) determiistic compoets icluded i the regressio, z t = (z t,dr t ) where Dt r are the determiistic compoets that are restricted to be equal to zero uder the ull (see Sectio 5) ad ξ p,t = j=p+ π j z t j + ξ t. If ρ =, δ = (ρ )θβ =, which leads to the test statistic T wald = ˆδ Var(ˆδ) ˆδ, (7) where ˆδ is the OLS estimator of δ i (6) ad Var(ˆδ) is its estimated covariace matrix. The ull hypothesis of o coitegratio is the rejected for large values of T wald. We let the lag legth p i regressio (6) grow to ifiity at a cotrolled rate. Assumptio 3. Let p ad p = o( /2 ) as. The limitig distributio of T wald ca be foud i Boswijk (994) for the ECM with fiite autoregressive depedece ad i Pesaveto (24) for the ifiite-order model. The asymptotic distributio of the test without the iclusio of ay determiistic compoets (ad with µ = τ = ) is give for completeess i Lemma without proof. Lemma. Uder Assumptios 2 ad 3 we have that d T wald dw (r)w(r) where T wald is defied i equatio (7). W(r)W(r) dr W(r)dW (r) 3.2 Bootstrap method The multivariate sieve bootstrap method we employ here is similar to the oe employed by Chag, Park, ad Sog (26). It is importat to ote that they study bootstrap iferece o the coitegratig regressios ad they do ot cosider bootstrap tests for o coitegratio. The full algorithm is give below. Bootstrap Algorithm. 6

7 Step : Fit a VAR(q) process to z t by OLS ad save the residuals ˆε q,t = z t ˆλ s D s t ˆΦ j z t j, (8) where D s t are the determiistic compoets icluded i this sieve estimatio (see Sectio 5 for details). Receter the residuals ˆε q,t i the case where o costat is icluded to elimiate ay drifts i the resampled series ad save the recetered residuals ε q,t = ˆε q,t ( q ) t ˆε q,t. 5 Step 2: Resample with replacemet from ε q,t to obtai bootstrap errors ε t. Step 3: Build u t recursively as u t = ˆΦ j u t j + ε t, (9) usig the estimated parameters ˆΦ j from Step, ad build z t as z t = z t + u t. () Note that it is uecessary to iclude determiistic compoets i this step, as the tests we cosider are asymptotically similar (see Remark 8 i Sectio 5). Step 4: Usig the bootstrap sample z t, obtai ˆδ from the regressio p yt = δ z t + λ Dt + π x t + πj zt j + ξp,t, () where p is the lag legth selected i the bootstrap regressio (see Remark 6) ad z t = (z t,dr t ), ad calculate the bootstrap test statistic T wald = ˆδ Var (ˆδ ) ˆδ. (2) Dt ad Dt r are the bootstrap couterparts of D t ad Dt r. I order to get the correct asymptotic bootstrap distributio, oe should always take Dt = D t ad Dt r = Dt r. Step 5: Repeat Steps 2 to 4 B times, obtaiig bootstrap test statistics Twald b,b =,...,B, ad select the bootstrap critical value c α as c α = mi{c : B b= > c) α}, or equivaletly as the ( α)-quatile of the ordered T b wald I(T b wald statistics. Reject the ull 5 I the cases where we do ot iclude a costat i this regressio the residuals may have a sample mea uequal to zero, eve though their theoretical mea is zero. As the sample mea of the residuals becomes the populatio mea of the bootstrap errors, this may lead to (uwated) drifts i the bootstrap sample. 7

8 of o coitegratio if T wald, calculated from equatios (6) ad (7), is larger tha c α, where α is the omial level of the test. We eed to allow the lag legth q i the sieve bootstrap to go to ifiity at a cotrolled rate. We will use two assumptios. Assumptio 4. Let q ad q = o((/l ) /2 ) as. Assumptio 4. Let q ad q = o((/l ) /3 ) as. We also eed a assumptio o the relative speed of the lag legths p ad q. Assumptio 5. Let p/q κ > as, where κ may be ifiite. Note that by allowig κ to be ifiite, we do ot impose the same rate o p ad q. Assumptio 5 imposes a lower boud but ot a upper boud o the rate of p (or equivaletly a upper boud but ot a lower boud o the rate of q). Remark. I Step 3 we eed to iitialize u t i (9) ad z t i (). We propose to geerate a large umber of values of u t ad delete the first geerated values. This will esure that u t is a statioary process. The iitial values i (9) will the become uimportat as the realizatio of u t will ot deped o them; hece they may be set equal to zero. A alterative is to take the first q values of u t equal to the first q values of u t ; this however does ot esure statioarity of u t. As asymptotically the effect of z disappears, we simply set z =. The logical alterative here would be to set z = z, especially i applicatios. Remark 2. Istead of estimatig the sieve uder the ull of o coitegratio (which we impose by fittig the VAR model to z t i Step ), we may also estimate it uder the alterative of coitegratio. I this case we would estimate the residuals as ˆε q,t = z t ˆλ b D s,a t ˆΦ z t ˆΦ j z t j, (3) where ˆΦ deotes the urestricted OLS estimator ad D s,a t are the determiistic compoets icluded i this alterative-based sieve estimatio. Note that eve for the same determiistic settig, D s,a t is ot ecessarily the same as Dt s i (8), as is explaied i Sectio 5 (Remark ). I the cotext of uit root testig, Paparoditis ad Politis (25) advocate the use of such a residual-based estimatio as opposed to the differece-based estimatio i (8), claimig that the residual-based tests have better power properties. We will retur to this poit i our simulatios i Sectio 4. 8

9 Remark 3. A secod alterative bootstrap strategy would be to base the sieve bootstrap o the coditioal/margial ECM model istead of the VECM/VAR model. I this case we would eed two separate equatios to estimate residuals i Step. We would estimate the residuals from the coditioal model as ˆε,q,t = y t ˆλ s, Dt, s ˆπ x t ˆπ j z t j ad the residuals from the margial model as ˆε 2,q,t = x t ˆλ s,2 Dt,2 s ˆΦ 2,j z t j for the differece-based alterative. We ca of course also costruct a residual-based versio of this test. I the simulatios i Sectio 4 we will look at these alteratives as well. Although such a approach is closer i spirit to the sigle-equatio Wald test statistic, it is basically just a reparametrizatio of the VECM approach, as the model o which the bootstrap is based is still completely specified. A alterative approach, which would be truly coditioal o x t, is to take x t as fixed ad oly resample y t. To justify such a approach we would have to assume strog exogeeity, see Va Giersberge ad Kiviet (996) for a discussio. This last approach will ot be ivestigated i this paper. Remark 4. Although estimatio uder the alterative is a optio i Step, it is ot possible to build the bootstrap sample zt i Step 3 based o the alterative hypothesis, i.e. usig z t = (I + ˆΦ )z t + u t. (4) Basawa, Mallik, McCormick, Reeves, ad Taylor (99) show that if such a alterativebased recursio is used i the uit root settig, the limitig distributio of the bootstrap test statistic is radom due to the discotiuity of the limitig distributio at the uit root. The same logic applies here, therefore the ull hypothesis of o coitegratio must be imposed i Step 3. Remark 5. To obtai the theoretical results i the ext subsectio, we set all determiistic compoets equal to zero, both i the model (µ ad τ) ad i the test (all variats of D t ). I Sectio 5 we will go ito more detail about the iclusio of determiistic compoets, ad preset some simulatio results. We cojecture that asymptotic validity still holds i the presece of determiistic compoets. Remark 6. I Step 4 we specify the lag legth i the bootstrap test regressio () as p, i order to emphasize that this lag legth does ot have to be the same as the lag legth i the origial test regressio (6). I fiite samples the performace of the bootstrap test will be better if the lag legth is allowed to be differet. Just as for the origial test regressio (ad 9

10 the sieve bootstrap), the lag legth ca be chose i practice usig iformatio criteria like AIC ad BIC. Obviously, p has to fulfil the same coditios as p. Therefore, we ca write p as p i the theoretical results, which is doe for otatioal simplicity. Remark 7. As we will see i the ext subsectio, Assumptio 4 is sufficiet to prove Theorem. However, to prove the secod result eeded for Theorem 2, we require the stroger assumptio 4. The result i the proof of Theorem 3.3 of Park (22, p. 487, lie 2), where it is stated (i Park s otatio) that p p k ˆα p,k = k α p,k + o() a.s., k= k= with ˆα p,k beig the OLS estimators of the p-th order autoregressive approximatio of the uivariate geeral liear process cosidered by Park (22) with coefficiets α p,k, does ot go through with p = o((/l ) /2 ). Oe eeds a stroger restrictio o p to make the secod part o(). 6 With our stroger Assumptio 4 oe ca show that Theorem 3.3 of Park (22) (ad cosequetly our Theorem 2) holds. 3.3 Asymptotic results I this sectio we will give the mai theoretical results eeded to show the asymptotic validity of the bootstrap test. As stated i Remark 5, we derive these results for the tests (ad DGP) without determiistic compoets. The proofs of all the results here plus additioal lemmas ca be foud i Appedix A. Most of the proofs are based o the proofs i Chag et al. (26), ad the papers they refer to. As we preset all our proofs for vector processes, the theory employed i the paper ca be used to prove validity of other multivariate bootstrap procedures as well. Note that all our bootstrap weak covergece results hold i probability as we derive all uderlyig results i probability. The first step i provig the asymptotic validity is the developmet of a ivariace priciple for the bootstrap errors ε t. Theorem. Uder Assumptios 2 ad 4, we have that r W(r) = /2 ε t d LW(r) i probability where L is a ( + l) ( + l)-dimesioal lower triagular matrix such that the Cholesky decompositio of Σ is equal to LL. 6 We thak Aders Swese for brigig this poit to our attetio i a persoal commuicatio.

11 We ca show this result by first showig that E ε t a = O p () for some a > 2, ad the referrig to Eimahl (987), who shows that a ivariace priciple holds if this coditio is met. From this result, with the help of the Beveridge-Nelso decompositio, we ca costruct a ivariace priciple for u t. Theorem 2. Uder Assumptios 2 ad 4 we have that r B(r) = /2 u t d B(r) i probability where B(r) is a ( + l)-dimesioal Browia motio such that B(r) = Ψ()LW(r). The, usig Theorem 2, we ca derive the limitig distributios of the elemets of the test statistic, ad fially show the cosistecy of the bootstrap variace estimator. With these results, we ca the preset Theorem 3 which establishes the asymptotic distributio of the bootstrap test statistic. Theorem 3. Uder Assumptios 2, 3, 4 ad 5 we have that T wald d dw (r)w(r) W(r)W(r) dr W(r)dW (r) i probability where T wald is defied i equatio (2). Note that Theorem 3 shows that the bootstrap test statistic has the same asymptotic distributio as the origial test statistic, which shows that the bootstrap test is asymptotically valid. Also ote that the test statistic is asymptotically pivotal, which meas that the bootstrap may offer asymptotic refiemets, although this does ot have to be so. 4 Simulatios We wish to study the small sample properties of our test by simulatio. We compare our test with the test based o asymptotic critical values (provided by Boswijk, 994) ad with the three alterative bootstrap tests metioed i Remarks 2 ad 3. Our bootstrap test is deoted by Tv,, where the subscript v stads for estimatio based o the VAR/VECM model, ad the for estimatio of the sieve bootstrap uder the ull. The alterative test discussed i Remark 2 is deoted by Tv,a, with the subscript a idicatig estimatio uder the alterative. Similarly, the two alteratives discussed i Remark 3 are give as Tc, ad Tc,a, where the subscript c idicates that these are based o the coditioal/margial model. Fially, the asymptotic test is deoted as T as.

12 For the simulatio study we use the same setup as Pesaveto (24). We let the bivariate series (y t,x t ) be geerated by the triagular system y t = γx t + w t, w t = ρw t + v t, x t = v 2t. (5) We take ρ = to aalyze the size of tests, ad ρ < for the power. For the local power aalysis, ρ = + c/, where, the sample size, is either 5 or. The tests are ivariat to the true value of γ as log as it is o-zero, therefore we set γ =. Furthermore we set w = x =. The errors v t = (v t,v 2t ) are geerated as ( ΦL)v t = ( + ΘL)ε t, where ε t is geerated as a i.i.d. sequece from a bivariate ormal distributio with covariace matrix r Σ =. r The exact parameter combiatios cosidered are summarized i Table. Isert Table about here We ca rewrite the above DGP i terms of the model i (2) by settig α = (,), β = (, γ) ad u t = γ vt i equatio (2). The lag legths i (6), (8) ad () are selected by BIC, with maximum lag legths of 8 for = 5 ad for =. Each geerated sample is used to perform all the tests, such that the lag legth p i (6) is always the same for all tests. Our results are based o 2 simulatios, with 999 bootstrap replicatios per simulatio. The results for the DGPs with white oise errors (Φ = Θ = ) are give i Table 2. For this case, the asymptotic test has a reasoably good size, but the bootstrap tests clearly have sizes eve closer to the omial size, especially for = 5. The rejectio frequecies of the bootstrap tests are somewhat smaller tha those of the asymptotic test uder the alteratives cosidered, but it is difficult to compare powers as sizes are ot equal. We therefore also report size-corrected powers for the asymptotic test (i the Table as T sc ). 7 The size-corrected power of the asymptotic test is close to the power of the bootstrap tests, which shows that the higher raw power of the asymptotic test is maily due to the higher size distortios. All bootstrap 7 There is o eed to correct the power of the bootstrap tests, as they have virtually o size distortios; their size-corrected powers would be almost the same as their raw powers. 2

13 tests perform similarly both i terms of size ad power, idicatig that there is o evidece of reduced power for the differece-based tests i this settig. Isert Table 2 about here Table 3 gives the results for the size of the tests for DGPs with autoregressive ad movigaverage errors. For all DGPs cosidered here, there is a clear advatage of usig the bootstrap, which virtually elimiates all size distortios except for the egative movig-average coefficiets. Agai ote that the differece betwee the bootstrap ad asymptotic test is the largest for = 5. The bootstrap tests perform fairly similarly, with a mior advatage for the differece-based tests. This is especially oticeable for the DGP with egative movig-average coefficiets. Isert Table 3 about here To illustrate the power properties for DGPs allowig for some depedece i the errors, we selected oe DGP with autoregressive ad oe with movig-average coefficiets from the set cosidered above. The results are give i Table 4. We agai have to be cautious whe comparig raw powers as the sizes vary across the tests. We see that the asymptotic test has somewhat higher rejectio frequecies tha the bootstrap tests, but as i Table 2 the differeces are due to high size distortios of the asymptotic test. This is cofirmed by the size-corrected power of the asymptotic test, which is ot better, ad i some cases cosiderably worse, tha the power of the bootstrap tests. The differece-based tests appear to have higher power tha the residual-based tests (especially for = 5 ad for alteratives close to the ull). This is quite surprisig, as it is exactly the opposite of what Paparoditis ad Politis (25) foud for uit root tests. This may possibly be a small sample pheomeo reflectig the fact that very ofte imposig ivalid restrictios may lead to improved fiite sample statistical iferece by reducig the effect of samplig errors. Isert Table 4 about here These results show that the bootstrap tests all offer sigificat size improvemets over the asymptotic test, while retaiig quite good power properties. Note that the four bootstrap tests perform similarly, with a small advatage for the differece-based tests, both i terms of size ad power. The bootstrap tests based o the coditioal-margial represetatio perform as their couterparts based o the vector represetatio, thus givig o reaso to prefer the coditioal represetatio over the more straightforward vector represetatio. As suggested by a referee, the similar performaces of the bootstrap tests based o the vector represetatio ad the coditioal-margial represetatio may be due to the ormality of the iovatios i our DGP. I order address this issue, we also performed simulatios where the ε t s are geerated from o-ormal distributios, i particular cetral χ 2 - ad 3

14 t-distributios. The simulatio results ot reported here show that the two represetatios also lead to very similar results if the variables are ot ormal. We also ivestigated the sesitivity to the form of v t. I the first aalysis we geerate the iovatios ε t as multivariate GARCH errors, which fall outside the class of processes defied by Assumptio 2. I the secod aalysis we cosider a Markov-switchig model i which the parameters of the short-ru dyamics are geerated by a Markov process. The results show that the bootstrap tests are robust agaist both types of processes. Fially, we also performed simulatios with the origial DGP usig AIC istead of BIC to select lag legths. The results show that the bootstrap tests are somewhat udersized. The oly otable improvemet of the size of the bootstrap tests with respect to lag legth selectio by BIC occurs i the case of the large egative MA parameters. The power of the bootstrap tests is adversely affected by the use of AIC. Surprisigly, the asymptotic test has larger size distortios usig AIC tha BIC. 8 5 Determiistic compoets I this sectio we will discuss how to iclude determiistic compoets i the tests. Determiistic compoets have to be icluded both i the test regressio (D t ad Dt r i equatio (6) ad their bootstrap couterparts i equatio ()) ad i Step of the bootstrap procedure (D s t i equatio (8)). We cosider the five differet optios proposed by Boswijk (994). The first optio is to simply leave out all determiistic compoets, which is the case we aalyzed before i the paper. Obviously this is oly valid if both µ ad τ i equatio () are equal to zero. The secod ad third optios (Boswijk s ξ µ ad ξ µ) arise if there is o drift i z t (τ = ). I this case we iclude a itercept i regressio (6) ad its bootstrap equivalet (). The itercept ca but eed ot be restricted to zero uder the ull of o coitegratio. I the first case D t = ad D r t =, i the secod case D t = ad D r t =. As i both cases z t does ot have a drift, there is o eed to iclude ay determiistic compoets i Step of the bootstrap procedure, hece D s t =. If the variables are geerated by a process with drift, we have to iclude a liear tred as well as a itercept i equatios (6) ad () (Boswijk s ξ τ ad ξ τ ). Agai we ca either restrict the tred to be equal to zero uder the ull, i which case D t = ad D r t = t, or we leave it urestricted, i which case D t = (,t) ad D r t =. As z t ow has a ozero mea, we iclude a costat term i equatio (8) i Step of the bootstrap procedure, i.e. we set D s t = i both cases. Remark 8. While it is possible to accout for the presece of determiistic compoets i Step 3 of the bootstrap algorithm as well, it is ot ecessary. By specifyig the tests as above, 8 The results of all the additioal simulatios discussed above ca be foud o the website 4

15 the tests are similar, i.e. their asymptotic distributios do ot deped o the true value of the determiistic compoets. Therefore, buildig the bootstrap process with or without determiistic compoets will both lead to the correct limitig distributio, as log as the determiistic specificatio i the bootstrap test regressio () is the same as the specificatio i the origial regressio (6), i.e. D t = D t ad D r t = D r t.9 Remark 9. Oe might wat to use the test with the urestricted costat term to deal with the situatio where the variables have a drift, but the drift does ot lead to a time tred i the coitegratig relatio (β τ = ). However, Boswijk (994) stresses that i this case the asymptotic distributio of the test will ot be similar ad deped o whether the drift is zero or ot. Therefore we do ot cosider this to be a viable optio. Remark. Oe ca also adapt the bootstrap procedure metioed i Remark 2 to the iclusio of determiistic compoets. As estimatio i Step is doe uder the alterative hypothesis, the iclusio of determiistic compoets is slightly differet. If we oly iclude a costat term i the regressio, the a costat term must be icluded i equatio (3) as well, hece D s,a t =. If the variables are geerated by a drift, ad a tred is added to the regressio, D s,a t = (,t). To illustrate the tests with determiistic compoets, we perform a small simulatio study. The DGP used for the simulatios correspods to the DGP used i Sectio 4, except that we ow add determiistic compoets to the triagular system as follows. y t = µ + τ t + γx t + w t, w t = ρw t + v t, (x t µ 2 τ 2 t) = v 2t. (6) Note that µ ad τ correspod to β µ ad β τ respectively i equatio (4). To reduce the size of the experimet we oly report simulatios for = 5, ad for c = ad c =. Also, we oly cosider three combiatios of Φ ad Θ: Φ = Θ = ; Φ = ad Θ = ; ad Φ = ad Θ = We restrict our attetio to the two bootstrap variats T v, ad Tv,a ad the asymptotic test T as. We cosider two models without a drift, ad two where a drift is preset. For the models without drift, a DGP with o determiistic compoets ad oe with just a costat term are chose. For the models with drift, we select oe DGP where the drift cacels out i the directio of the coitegratig vector (i.e. τ = ), ad oe where it does ot. For each model we perform the tests with every determiistic specificatio that is appropriate for that specific model. The specific values used ad the correspodig empirical rejectio frequecies ca be foud i Table 5. 9 Ureported simulatio results, which ca also be foud o the website metioed above, show that i fiite samples the tests perform the same whether or ot determiistic compoets are icluded i Step 3. 5

16 Isert Table 5 about here It ca be see from the table that the size of the bootstrap test is satisfactory for all settigs cosidered. As i Sectio 4, the ull-based test has slightly better size tha the alterative-based test i the presece of serial correlatio. The asymptotic test has agai large size distortios almost everywhere. I terms of power the coclusios are similar to those draw i Sectio 4 as well. Also, both i terms of size ad power, the rejectio frequecies for a particular determiistic specificatio of the tests (D (r) t ), are comparable across differet specificatios for the treds i the DGP (µ ad τ), cofirmig the similarity of the tests. Noticeable is that the bootstrap tests lose power if determiistic compoets are icluded uecessarily. This is very much a small sample effect, ureported simulatios for = show that this effect, although still preset, is less proouced there. The asymptotic test does ot seem to lose as much power. This ca be explaied by the fact that (cotrary to the bootstrap tests) the size distortios of the asymptotic test icrease whe determiistic compoets are added uecessarily. It also appears that the tests with urestricted determiistic compoets are slightly more powerful tha their restricted couterparts. 6 Coclusio I this paper we preset a bootstrap versio of the Wald test for coitegratio i a coditioal sigle-equatio ECM origially proposed by Boswijk (994) ad also cosidered by Pesaveto (24). A multivariate sieve bootstrap method is used to deal with depedece i the data, ad show to be asymptotically valid. We also cosider several alterative bootstrap tests, for which the asymptotic validity ca be established i a similar fashio, ad show how determiistic compoets ca be icluded i the test. The small sample properties of our bootstrap tests are studied by simulatio, ad compared to those of the asymptotic test ad several alterative bootstrap tests. All bootstrap tests clearly outperform the asymptotic test i terms of size, while retaiig good power. Our bootstrap test based o the ull hypothesis performs slightly better i terms of size ad power tha the bootstrap test based o the alterative, while the performace of the tests based o the vector represetatio is very similar to that of the tests based o the coditioal represetatio. The bootstrap tests with determiistic compoets retai excellet size properties ad are isesitive to the true value of the treds i the model as log as sufficiet determiistic compoets are icluded. The results show that our bootstrap versio of the Wald ECM test is worth beig cosidered i empirical research, as our test ca be see to improve upo the origial Wald test cosidered by Boswijk (994) ad Pesaveto (24). The Wald ECM test easily allows for other bootstrap variats as well, such as those cosidered i the simulatio study, or block bootstrap methods, which accout for somewhat more geeral DGPs. Such tests could easily 6

17 be placed i the framework preseted here. A Proofs All bootstrap weak covergece results that we preset i the followig are i probability. We do ot add this explicitly to every result i order to simplify the otatio. Also ote that we defie bootstrap stochastic order symbols Op( ) ad o p( ) i the same way as O p ( ) ad o p ( ) for the origial sample (see Chag ad Park, 23, Remark ). A paper cotaiig more detailed versios of the proofs i this appedix is available o the website I order to prove Theorem, we first eed the followig lemma. Lemma 2. Uder Assumptios 2 ad 4 we have for ay 2 < a 4 E ε t a = O p (). Proof of Lemma 2. Our proof follows Park (22, Proof of Lemma 3.2). Usig that k k a k i= x i a we have that E ε t a = ˆε q,t a ˆε q,t { 4 a ˆε q,t ε q,t a + ε q,t ε t a + ε t a + where = c(a + B + C + D ) a} ˆε q,t A = ε t a B = ε q,t ε t a C = ˆε q,t ε q,t a a D = ˆε q,t ad c = 4 a is a costat ot depedig o. Note that ε q,t is defied as ε q,t = u t Φ j u t j = ε t + j=q+ i= x i Φ j u t j. (7) We first look at A. By the weak law of large umbers, ε t a p E ε t a. As by Assumptio 2 E ε t a = O(), we have that A = O p (). a 7

18 have For B, we wish to show that E ε q,t ε t a = o(q a ). Usig Mikowski s iequality we E ε q,t ε t a j=q+ (E Φ j u t j a ) /a a E u t a j=q+ Φ j a = o(q a ). The fial step comes from Bühlma (995), where it is show i Lemma 2. that Assumptio 2 implies that j= j Φ j <. It is also show (i the proof of Theorem 3.) that j=q+ j Φ j = o() if j= j Φ j <. Cosequetly j=q+ Φ j = o(q ) as q j=q+ Φ j j=q+ j Φ j. Next we tur to C. We ca write ˆε q,t = ε q,t (ˆΦ j Φ q,j )u t j (Φ q,j Φ j )u t j (8) where Φ q,j is defied as the coefficiet of y t j i the best liear predictor of y t i terms of y t,...,y t q. The ˆε q,t ε q,t a 2 a (ˆΦ j Φ q,j )u t j We defie C = (ˆΦ j Φ q,j )u t j a a a + (Φ q,j Φ j )u t j., C 2 = ad show that C,C 2 = o p (). The we have that C q a where we use that (ˆΦ j Φ q,j ) a u t j a ( q a ) a max ˆΦ j Φ q,j j q ( u t a + t= q t= (Φ q,j Φ j )u t j u t a ) = O p ((l /) a/2 )(q a /)O p () = O p (q a (l /) a/2 ), max ˆΦ j Φ q,j = Op ((l /) /2 ) (9) j q uiformly i q < Q, where Q = o((/l ) /2 ), from Haa ad Kavalieris (986). Note that while Haa ad Kavalieris (986) show their result for the Yule-Walker estimator, (9) a 8

19 is valid for OLS as well by Theorem of Poskitt (994). To coclude this part of the proof, ote that C = o p () as q = o((/l ) /2 ). For C 2, ote that by Markov s iequality for ay ǫ > P( C 2 > ǫ) ǫ E a (Φ q,j Φ j )u t j Usig that u t is statioary, we have E (Φ q,j Φ j )u t j a E u t a Φ q,j Φ j Agai from Bühlma (995, p. 337), we have that Φ q,j Φ j c j=q+ a. Φ j = o(q ) (2) with c some costat. Hece, C 2 = o(q a ) which completes the proof for C. Fially, we look at D. We wat to show that ˆε q,t = ε q,t + o p () = ε t + o p (). Usig equatios (7) ad (8) we ca write ˆε q,t = ε t + j=q+ Φ j u t j (ˆΦ j Φ q,j )u t j Hece, what we eed to show is that j=q+ (Φ q,j Φ j )u t j. Φ j u t j p (2) p (Φ q,j Φ j )u t j (22) p (ˆΦ j Φ q,j )u t j. (23) 9

20 Note that, usig Markov s iequality, we have for ay ε > P j=q+ Φ j u t j > ǫ ǫ a E j=q+ Φ j u t j a = o(q a ) for ay 2 < a 4 which follows from the proof for B. This shows (2). To show (22), we ca use the proof of C 2 to show that P Fially, to prove (23), ote that (Φ q,j Φ j )u t j > ǫ ǫ a E ( ) (ˆΦ j Φ q,j )u t j q max ˆΦ j Φ q,j j q = O p (q(l /) /2 ) (Φ q,j Φ j )u t j ( u t + t= q t= a u t = o(q a ). which follows exactly as i the proof of C. This shows that D = o p (), ad the proof is complete. Before proceedig with the proof of Theorem, we eed oe additioal lemma to esure that the covariace matrix of the bootstrap errors correctly mimics that of the origial errors. Lemma 3. Uder Assumptios 2 ad 4 we have that Σ = E (ε t ε t ) = Σ + o p(). Proof of Lemma 3. This proof follows by first boudig Σ Σ as i the proof of Theorem 2.5 i Paparoditis (996, p. 288) ad the applyig the methods from the proof of Lemma 2 (with a = ). Proof of Theorem. Give Lemma 2, the result follows immediately from Eimahl (987), as i Chag et al. (26). Proof of Theorem 2. Usig the Beveridge-Nelso decompositio, we ca write ε t = ˆΦ()u t i= j=i ˆΦ j (u t i u t i+ ) ) 2

21 ad hece u t = ˆΨ()ε t + ˆΨ() i= j=i ˆΦ j (u t i u t i+) = ˆΨ()ε t + (ū t ū t), where ū t = ˆΨ() q i= ( q j=i ˆΦ j )u t i ad ˆΨ() = ˆΦ(). The r r r B (r) = /2 u t = /2 ˆΨ()ε t + /2 (ū t ū t ) = ˆΨ()W (r) + /2 (ū ū r ) Hece, we eed to show that ˆΦ() p Φ() (24) { } P max t /2 ū t > ǫ = o p () (25) We first show (24). Usig equatios (9) ad (2) we have that ˆΦ() Φ() + ˆΦ j Φ q,j Φ q,j Φ j + Hece ˆΦ() = Φ() + o p (). This proves (24). To prove (25), we have as i Park (22) j=q+ Φ j = O p (q(l /) /2 )+o(q ). { } { } P max t /2 ū t > ǫ P /2 ū t > ǫ (/ǫ a ) a/2 E ū t a The secod iequality follows from Markov s iequality. Hece, we have to show that a/2 E ū t a = o p (). (26) If the Yule-Walker method is used to estimate (8), the estimated autoregressio is always ivertible. Although ivertibility of the estimated autoregressio is ot guarateed for fiite samples usig OLS, the asymptotic equivalece of OLS to Yule-Walker (Poskitt, 994, Theorem ) implies that for large we ca write u t = ˆΨ j= j ε t j ad furthermore ū t = Ψ j= j ε t j, where Ψ j = ˆΨ i=j+ j. Let ū (k),t be the k-th elemet of ū t ad let Ψ (k),j be the k-th row of Ψ j. By successive applicatio of the Marcikiewicz-Zygmud iequality 2

22 (Berger, 99) ad Mikowski s iequality elemet by elemet, we have ( l+ E ū t a = E k= ū 2 (k),t l+ c a Ψ (k),j 2 k= j= ) a/2 l+ c a (l + ) a/2 a/2 c a (l + ) a/2 Ψ j 2 j= E ε t a a/2 k= E ε t a E Ψ (k),j ε t j 2 j= a/2 (27) for some costat c a ot depedig o. Phillips ad Solo (992, p. 973) show that a sufficiet coditio for Ψ j 2 = O p () is j /2 ˆΨ j = O p (). (28) This will i tur hold if (Haa ad Kavalieris, 986) q j/2 ˆΦ j = O p (). We have j /2 ˆΦ j j /2 ˆΦ j Φ q,j + j /2 Φ q,j Φ j + j /2 Φ j = O p (q 3/2 (l /) /2 ) + o(q /2 ) + O() = O p (), by (9), (2) ad Assumptio 4. Together with Lemma 2 this shows that E ū t a = O p (). (29) This cocludes the proof of this theorem. Next we eed several lemmas i order to show the limitig distributio of the bootstrap test statistic. Lemma 4. Let ξ t be the bootstrap equivalet of ξ defied i equatio (5), i.e. yt = π x t + πj z t j + ξ t. (3) The, if Assumptios 2 ad 4 hold, r /2 ξt d B ξ (r), 22

23 where B ξ (r) is a scalar Browia motio with variace ω 2, i.e B ξ (r) = ωw (r), where W (r) is the first elemet of the stadard Browia motio W(r). Proof of Lemma 4. Follows immediately from Theorem. Lemma 5. Let f deote the spectral desity ad Γ (k) the autocovariace fuctio of u t. Uder Assumptios 2 ad 4, we have ad sup f (λ) f(λ) = o p() (3) λ k= Γ (k) = k= Γ(k) + o p (). (32) Proof of Lemma 5. The spectral desity f (λ) of u t is f (λ) = 2π (I ˆΦ j e i jλ ) Σ (I Note that by Lemma 3 Σ ) (ˆΦj Φ j e i jλ ˆΦ je i jλ ). p Σ. Furthermore, q max ˆΦ j Φ q,j + Φ q,j Φ j = o p () j q by (9) ad (2). Now the result i (3) follows straightforwardly. The result i (32) follows trivially by otig that k= Γ(k) = 2πf() ad correspodigly k= Γ (k) = 2πf (). Now we ca derive the limitig distributios of the differet elemets of the test statistic. First defie w p,t = ( x t, z t,..., z t p), ad let W p = (w p,,...,w p, ), Z = (z,...,z ), Ξ p = (ξ p,,...,ξ p, ) ad Y = ( y,..., y ), ad defie their bootstrap versios accordigly. 23

24 Lemma 6. Uder Assumptios 2, 3, 4 ad 5 we have a) 2 Z Z = 2 b) Z Ξ p = z t z t z t ξ p,t d d B(r)B(r) dr (33) B(r)dB ξ (r) (34) c) ( ) Wp Wp ) ( = w p,twp,t = O p() (35) d) Z Wp = zt w p,t = O p(p /2 ) (36) e) Wp Ξ p = wp,t ξ p,t = O p (/2 p /2 ). (37) Proof of Lemma 6. The proof of part a) follows directly from the proof of Lemma 3.4 i Chag et al. (26). Next we look at b). We have zt ξ p,t zt ξ t + zt (ξ p,t ξt ). Hece, we first have to show that z t (ξ p,t ξt ) = o p (). Note that ξp,t = k=p+ π k u t k + ξ t, where πk = ˆΦ,k ˆΣ ˆΦ 2ˆΣ 22 2,k (38) ad ˆΦ k = (ˆΦ,k, ˆΦ 2,k ). As ˆΦ k = for k > q ad usig Assumptio 5, we have that j=p+ π j = k=p+ The defie ˆΨ p,j such that ξ t ξ p,t = k=p+ π j = o p (). π k u t k = j j=p+ k=p+ π k ˆΨ j k ε t j = j=p+ ˆΨ p,j ε t j, (39) 24

25 We the have that j=p+ ˆΨ p,j j k=p+ j=p+ π k j=p+ ˆΨ j k ( ) πj ˆΨ i = i= j=p+ πj O p (). (4) Defie ηt = t i= ε i, such that we ca write z t = ˆΨ()η t + (ū ū t ). The we have zt (ξ p,t ξt ) = ˆΨ()η t (ξ p,t ξt ) + = R + R 2 + R 3. ū (ξp,t ξt ) ū t (ξp,t ξt ) Give equatio (4), it ow follows very similarly as i Chag et al. (26, Proof of Lemma A.6) that R = o p(), R 2 = o p( /2 ) ad R 3 = o p(). that Furthermore, by Park ad Phillips (989, Lemma 2.), Theorem 2 ad Lemma 4, we have z t ξ t d B(r)dB ξ (r). This completes the proof of part b). For c), we wat to show that E ( w p,tw p,t ) = O p(). (4) Followig Chag ad Park (23, Proof of Lemma 3) we first wat to show that E u t i u t j Γ (i j) For this to hold it is sufficiet to show that E ( 2 = O p (). (42) u (a),t i u (b),t j Γ (ab) (i j) ) 2 = O p (), (43) 25

26 for all a,b + l, where u (a),t is the a-th elemet of u t, ad similarly Γ (ab)(i j) is the (a,b)-th elemet of Γ (i j). Aalogous to the case for uivariate time series models discussed i Berk (974, eqs (2.) ad (2.), p. 49), we have that E ( + u (a),t i u (b),t i Γ (ab) (i j) ) 2 c,d,e,f κ cdef ( k= Ψ 2 (ac),k ) /2 ( k= 2 Ψ 2 (bd),k k= Γ (ab) (k)2 ) /2 ( k= Ψ 2 (ae),k ) /2 ( k= Ψ 2 (bf),k) /2, where κ cdef = E (ε (c),t ε (d),t ε (e),t ε (f),t ) σ cdσ ef σ ce σ df σ cf σ de ad σ cd = E (ε (c),t ε (d),t ). Note that κ cdef = O p() as E ε t 4 = O p () (take a = 4 i Lemma 2). Furthermore, ( ) /2 k= Γ (ab) (k)2 = O p () through Lemma 5 ad k= Ψ 2 (ac),k = Op () as k= k/2 ˆΨ k = O p (), which we demostrated i the proof of Theorem 2, equatio (28). Now equatio (43) follows straightforwardly. Next, partitio Γ (k) coformably with y t ad x t as Γ Γ (k) = (k) Γ 2 (k) Γ 2 (k) Γ 22 (k) ad defie Γ 2 (k) = Γ 2 (k),γ 22 (k) ad Γ 2 (k) = Γ 2 (k),γ 22 (k). The defie Ω pp as Γ 22 () Γ 2 ( )... Γ 2 ( p) Ω pp = Γ 2 () Γ ()... Γ ( p) Γ 2 (p) Γ (p ) Γ () As for ay matrix M, M 2 i,j M ij 2, we ca write E wp,t w p,t Ω pp 2 E + p i= u 2,t u u t iu 2,t Γ 22 () 2 2,t Γ 2(i) p (p,p) (i=,) u 2,t u t j Γ 2 ( j) u t iu = O p ( ) + O p ( p) + O p ( p) + O p ( p 2 ) = O p ( p 2 ). We let M ij deote submatrices ito which oe ca partitio M. 2 2 t j Γ (j i) 26

27 Next we eed to show that Ω 2π pp ( ) if f (λ) = O p (). λ Let us cosider a exteded Ω pp matrix, i.e. Γ () Γ 2 () Γ ( )... Γ ( p) Γ 2 () Γ () Γ ( )... Γ ( p) Ω pp = Γ () Ω Γ pp = () Γ ()... Γ ( p) Γ (p) Γ (p ) Γ () Γ (p) Let λ = (λ,...,λ (l+)p ) be the eigevalues of Ω pp ad defie < F = if λ f (λ). The as a direct cosequece of Lemma A.2 of Chag et al. (26) we have that Ω (2πF ) = O p (). pp As Ω Ω, we kow that Ω = Op () as well. The pp ( pp w p,t w p,t which implies that ( pp ) Ω ( pp + Ω pp + Ω pp w p,t w p,t w p,t w p,t ) Ω pp Ω pp ) Ω pp ( p,t Ω pp w p,tw w p,t w p,t Ω pp w p,tw p,t ) holds for large with probability as w p,twp,t Ω pp = O p ( /2 p). As Ω pp = O p (), the result i (4) follows. Write For d) we wat to show that E zt w p,t z t u t j = = O p(p /2 ). z t u t + z t u t j 27 z t u t = z t u t + R.

28 The we ca show i much the same way as i Chag ad Park (22, Proof of Lemma 3.2) that R = O p () uiformly i j =,...,p. Note that z t u t = Op () by Phillips (988), ad that z t u 2,t = z t u t + Op (). The Write E zt w p,t E 2 zt u 2,t + p Fially, we look at e). We wat to show that wp,tξ p,t = O p( /2 p /2 ). wp,t ξ p,t = wp,t ξ t + wp,t ( ξ p,t ξt ). We ca show that u t j ( ξ p,t ξt ) = o p ( /2 ) 2 z /2 t u t j = O p (p /2 ). uiformly i j p i the same way as i Chag et al. (26, Proof of Lemma A.6). The Furthermore, E u t jξ t w p,tξ p,t 2 = p = which cocludes the proof. E u t ju t j E ξt 2 = O p (). u t jξ p,t 2 /2 = O p( /2 p /2 ), The followig lemma shows the cosistecy of the bootstrap variace estimator. Lemma 7. Let ˆω 2 be the estimator of the variace of the bootstrap errors ξ p,t i regressio (), i.e. ˆω 2 = ( y Z ˆδ ) (I W p (W p W p ) W p )( y Z ˆδ ). The ˆω 2 p ω 2 uder Assumptios 2, 3, 4 ad 5. 28

29 Proof of Lemma 7. Note that ˆω 2 = Ξ p (I W p (Wp W p ) Wp )Ξ p Ξ p (I W p (Wp W p ) Wp )Z ˆδ which we write as ˆδ Z (I W p (W p W p ) W p )Ξ p + ˆδ Z (I W p (W p W p ) W p )Z ˆδ. ˆω 2 = C 2D + E We first look at C. Write C = Ξ p Ξ p Ξ p W p (W p W p ) W p Ξ p. Usig that ˆδ = O p ( ) ad the results from Lemma 6, we have that Hece, Ξ p W p (W p W p ) W C = Ξ p Ξ p + o p(). p Ξ p Next we tur to D. We ca write D as Ξ p W p D = Ξ p Z ˆδ Ξ p W p (Wp W p ) Wp Z ˆδ. Agai usig Lemma 6 ad ˆδ = O p( ), we have D Ξ p Z ˆδ + Ξ p W p (W p Wp ) W p Ξ p = O p( /2 p /2 )O p( )O p( /2 p /2 ) = o p( /2 ). (W p Wp ) W p Z ˆδ = O p()o p( ) + O p( /2 p /2 )O p( )O p(p /2 )O p( ) = O p( ). Fially we look at E : E = ˆδ Z Z ˆδ ˆδ Z W p (W p W p ) W p Z ˆδ. As before, we use the results from Lemma 6 ad ˆδ = O p( ) to obtai E ˆδ Z Z ˆδ + ˆδ Z W p (W p Wp ) W p Z ˆδ = O p( )O p( 2 )O p( ) + O p( )O p(p /2 )O p( )O p(p /2 )O p( ) = O p( ). 29

30 Therefore, we have that ˆω 2 = ξ 2 p,t + o p(). Next we wish to show that ξ 2 p,t = ξ 2 t +o p (), for which our proof is similar as Chag ad Park (22, Proof of Lemma 3.(c)). Note that The, as ( ( ξ p,t ξ t ) 2 = ξ 2 p,t) /2 = j=p+ ( j=p+ ˆΨ p,j ε t j ˆΨ p,j 2 O p () + ξ 2 t ) /2 ( 2 = j=p+ i=p+ j=p+ i=p+ ˆΨ p,j ( ε t jε t i ) ˆΨ p,i ˆΨ p,j ˆΨp,i O p ( /2 ) = o p (). ) /2 ( ξ p,t ξt ) 2 as a cosequece from the triagle iequality it follows that ξ 2 p,t = ξ 2 t which cocludes this step. + o p() For the fial step we show that t ω 2 ξ 2 ξ 2 t ω 2 + ω 2 ω 2 = o p (), where ω 2 = E (ξt 2 ). First, we show that ξ 2 t ω 2 = op (). Note that P ( ξ 2 t ) ( ω 2 > ǫ ǫ 2 E Next, we ext show that ω 2 p ω 2. As = ǫ 2 2 ξ 2 t ω 2 ) 2 ω 2 = σ Σ 2Σ 22 Σ 2 ad ω 2 = σ Σ 2 Σ 22 Σ 2, ( E (ξ 4 t ) ( E (ξ 2 t ) ) 2 ) = O p ( ). ad Σ p Σ by Lemma 3, the result follows. This completes the proof. 3