Journal of Econometrics

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1 Journal of Econometric 74 (23) 66 8 Content lit available at SciVere ScienceDirect Journal of Econometric journal homepage: Low-frequency robut cointegration teting Ulrich K. Müller a, Mark W. Waton a,b, a Princeton Univerity, Department of Economic, Princeton, NJ, 8544, United State b Woodrow Wilon School, Princeton, NJ, 8544, United State a r t i c l e i n f o a b t r a c t Article hitory: Received 2 December 29 Received in revied form 3 May 22 Accepted 9 September 22 Available online 9 February 23 JEL claification: C32 C2 Standard inference in cointegrating model i fragile becaue it relie on an aumption of an I() model for the common tochatic trend, which may not accurately decribe the data peritence. Thi paper conider low-frequency tet about cointegrating vector under a range of retriction on the common tochatic trend. We quantify how much power can potentially be gained by exploiting correct retriction, a well a the magnitude of ize ditortion if uch retriction are impoed erroneouly. A imple tet motivated by the analyi in Wright (2) i developed and hown to be approximately optimal for inference about a ingle cointegrating vector in the unretricted tochatic trend model. 23 Elevier B.V. All right reerved. Keyword: Stochatic trend Peritence Size ditortion Interet rate Term pread. Introduction The fundamental inight of cointegration i that while economic time erie may be individually highly peritent, ome linear combination are much le peritent. Accordingly, a uite of practical method have been developed for conducting inference about cointegrating vector, the coefficient that lead to thi reduction in peritence. In their tandard form, thee method aume that the peritence i the reult of common I() tochatic trend, and their tatitical propertie crucially depend on particular characteritic of I() procee. But in many application there i uncertainty about the correct model for the peritence which cannot be reolved by examination of the data, rendering tandard inference potentially fragile. Thi paper tudie efficient inference method for cointegrating vector that i robut to thi fragility. We do thi uing a tranformation of the data that focue on low-frequency variability and covariability. Thi tranformation We thank participant of the Cowle Econometric Conference, the NBER Summer Intitute and the Greater New York Metropolitan Area Econometric Colloquium, and of eminar at Chicago, Cornell, Northwetern, NYU, Rutger, and UCSD for helpful dicuion. Support wa provided by the National Science Foundation through grant SES-5836 and SES-678. Correponding author at: Princeton Univerity, Department of Economic, Princeton, NJ, 8544, United State. addre: mwaton@princeton.edu (M.W. Waton). See, for intance, Johanen (988), Phillip and Hanen (99), Saikkonen (99), Park (992) and Stock and Waton (993). ha two ditinct advantage. Firt, a we have argued elewhere (Müller and Waton, 28), peritence ( trending behavior ) and lack of peritence ( non-trending, I() behavior ) are lowfrequency characteritic, and attempt to utilize high-frequency variability to learn about low-frequency variability are fraught with their own fragilitie. 2 Low-frequency tranformation eliminate thee fragilitie by focuing attention on the feature of the data that are of direct interet for quetion relating to peritence. In particular, a in Müller and Waton (28), we ugget focuing on below buine cycle frequencie, o that the implied definition of cointegration i that error correction term have a flat pectrum below buine cycle frequencie. The econd advantage i an important by-product of dicarding high frequency variability. The major technical challenge when conducting robut inference about cointegrating vector i to control ize over the range of plauible procee characterizing the model tochatic common trend. Retricting attention to low frequencie greatly reduce the dimenionality of thi challenge. The potential impact of non-i() tochatic trend on tandard cointegration inference ha long been recognized. Elliott (998) provide a dramatic demontration of the fragility of tandard cointegration method by howing that they fail to control ize 2 Perhap the mot well-known example of thi fragility involve etimation of HAC tandard error, ee Newey and Wet (987), Andrew (99), den Haan and Levin (997), Kiefer et al. (2), Kiefer and Vogelang (25), Müller (27) and Sun et al. (28) /$ ee front matter 23 Elevier B.V. All right reerved. doi:.6/j.jeconom

2 U.K. Müller, M.W. Waton / Journal of Econometric 74 (23) when the common tochatic trend are not I(), but rather are local-to-unity in the ene of Bobkoki (983), Cavanagh (985), Chan and Wei (987) and Phillip (987). 3 In a bivariate model, Cavanagh et al. (995) propoe everal procedure to adjut critical value from tandard tet to control ize over a range of value of the local-to-unity parameter, and their general approach ha been ued by everal other reearcher; Campbell and Yogo (26) provide a recent example. Stock and Waton (996), Janon and Moreira (26) and Elliott et al. (22) go further and develop inference procedure with pecific optimality propertie in the local-to-unity model. An alternative generalization of the I() and I() dichotomy i baed on the fractionally integrated model I(d), where d i not retricted to take on integer value (ee, for intance, Baillie, 996 or Robinon, 23 for introduction). Fractional cointegration i then defined by the exitence of a linear combination that lead to a reduction of the fractional parameter. A well-developed literature ha tudied inference in thi framework: ee, for intance, Velaco (23), Robinon and Marinucci (2, 23); Robinon and Hualde (23) and Chen and Hurvich (23a,b, 26). A in the local-to-unity embedding, however, the low-frequency variability of the common tochatic trend i till governed by a ingle parameter, ince (uitably caled) fractionally integrated erie converge to fractional Brownian motion, which are only indexed by d. In contrat to the local-to-unity framework, thi deciive parameter can be conitently etimated, o that the uncertainty about the exact nature of the tochatic trend vanihe in thi fractional framework, at leat under the uual aymptotic. Yet, Müller and Waton (28) demontrate that relying on below buine cycle variation, it i a hopele endeavor to try to conitently dicriminate between, ay, local-to-unity and fractionally integrated tochatic procee from data panning 5 year. Similarly, Clive Granger dicue a wide range of poible data generating procee beyond the I()model in hi Frank Paih Lecture (Granger, 993) and argue, enibly in our opinion, that it i fruitle to attempt to identify the exact nature of the peritence uing the limited information in typical macro time erie. While local-to-unity and fractional procee generalize the aumption of I() trend, they do o in a very pecific way, leading to worrie about the potential fragility of thee method to alternative pecification of the tochatic trend. A demontrated by Wright (2), it i neverthele poible to conduct inference about a cointegrating vector without knowledge about the precie nature of the common tochatic trend. Wright idea i to ue the I() property of the error correction term a the identifying property of the true cointegrating vector, o that a tationarity tet of the model putative error correction term i ued to conduct inference about the value of the cointegrating vector. Becaue the common tochatic trend drop out under the null hypothei, Wright procedure i robut in the ene that it control ize under any model for the common tochatic trend. But the procedure ignore the data beyond the putative error correction term, and i thu potentially quite inefficient. Section 2 of thi paper provide a formulation of the cointegrated model in which the common tochatic trend follow a flexible limiting Gauian proce that include the I(), localto-unity, and fractional/long-memory model a pecial cae. Section 3 dicue the low-frequency tranformation of the cointegrated model. Throughout the paper, inference procedure are tudied in the context of thi general formulation of the cointegrated model. The price to pay for thi generality i that it introduce a potentially large number of nuiance parameter that characterize the propertie of the tochatic trend and the relationhip between the tochatic trend and the model I() component. In our framework, none of thee nuiance parameter can 3 Alo ee Elliott and Stock (994) and Jeganathan (997). be etimated conitently. The main challenge of thi paper i thu to tudy efficient tet in the preence of nuiance parameter under the null hypothei, and Section 4 6 addre thi iue. Uing thi framework, the paper then make ix contribution. The firt i to derive lower bound on ize ditortion aociated with trend pecification that are more general than thoe maintained under a tet null hypothei. For example, for tet contructed under a maintained hypothei that the tochatic trend follow an I() proce, we contruct lower bound on the tet ize when the tochatic trend follow a local-to-unity or more general tochatic proce. Importantly, thee bound are computed not for a pecific tet, but rather for any tet with a pre-pecified power. The paper econd contribution i an upper bound on the power for any tet that atifie a pre-pecified rejection frequency under a null that may be characterized by a vector of nuiance parameter (here the parameter that characterize the tochatic trend proce). The third contribution i implementation of a computational algorithm that allow u to compute an approximation to the lowet upper power bound and, when the number of nuiance parameter i mall, a feaible tet that approximately achieve the power bound. 4 Taken together thee reult allow u to quantify both the power gain aociated with exploiting retriction aociated with pecific tochatic trend procee (for example, the power gain aociated with the pecializing the local-to-unity proce to the I() proce), and the ize ditortion aociated with thee power gain when the tochatic trend retriction do not hold. Said differently, thee reult allow u to quantify the benefit (in term of power) and cot (in term of potential ize ditortion) aociated with retriction on the tochatic proce characterizing the tochatic trend. Section 4 derive thee ize and power bound in a general framework, and Section 5 compute them for our cointegration teting problem. The fourth contribution of the paper take up Wright inight and develop efficient tet baed only on the putative errorcorrection term. We how that thee tet have a particularly imple form when the alternative hypothei retrict the model tochatic trend to be I(). The fifth contribution of the paper i to quantify the power lo aociated with retricting tet to thoe that ue only the error-correction term rather than all of the data. Thi analyi how that, in the cae of ingle cointegration vector, a imple-to-compute tet baed only on the error-correction term eentially achieve the full-data power bound for a general tochatic trend proce, and i thu the efficient tet. Thee reult are developed in Section 6. The paper ixth contribution i empirical. We tudy the pot- WWII behavior of long-term and hort-term interet rate in the United State. While the level of the interet rate are highly peritent, a uitably choen linear combination of them i not, and we ak whether thi linear combination correpond to the term pread, the imple difference between long and hort rate. More pecifically we tet whether the cointegrating coefficient linking long rate and hort rate i equal to unity. Thi value cannot be rejected uing a tandard efficient I() tet (Wald or LR verion of Johanen (99) tet), and we how that thi reult continue to hold under a general trend proce. Of coure, other value of the cointegrating coefficient are poible both in theory and in the data, and we contruct a confidence et for the value of the cointegrating coefficient allowing for a general trend proce and compare it to the confidence et contructed uing tandard I() method. Thee reult are preented in Section Model Let p t, t =,..., T denote the n vector of variable under tudy. Thi ection outline a time domain repreentation of 4 The econd and third contribution are application of general reult from a companion paper, Elliott et al. (22), applied to the cointegration teting problem of thi paper.

3 68 U.K. Müller, M.W. Waton / Journal of Econometric 74 (23) 66 8 the cointegrated model for p t in term of canonical variable repreenting a et of common trend and I() error correction term. The common trend are allowed to follow a flexible proce that include I(), local-to-unity, and fractional model a pecial cae, but aide from thi generalization, the cointegrated model for p t i tandard. To begin, p t i tranformed into two component, where one component i I() under the null hypothei and the other component contain element that are not cointegrated. Let β denote an n r matrix whoe linearly independent column are the cointegrating vector, let β denote the value of β under the null, and y t = β p t. The element in y t are the model error correction term under the null hypothei. Let x t = δ p t where δ i n k with k = n r, and where the linearly independent column of δ are linearly independent of the column of β, o that the element of x t are not cointegrated under the null. Becaue the cointegrated model only determine the column pace of the matrix of cointegrating vector, the variable y t and x t are determined up to tranformation (y t, x t ) (A yy y t, A xx x t + A xy y t ), where A yy and A xx are non-ingular. Mot extant inference procedure are invariant (or aymptotically invariant) to thee tranformation, and, a dicued in detail below, our analyi will alo focu on invariant tet. 2.. Canonical variable repreentation of y t and x t We will repreent y t and x t in term of a common tochatic trend vector v t and an I() vector z t y t = Γ yz z t + Γ yv v t () x t = Γ xz z t + Γ xv v t, where z t i r, v t i k, and Γ yz and Γ xv have full rank. In thi repreentation, the retriction that y t i I() correpond to the retriction Γ yv =. All of the tet tatitic dicued in thi paper are invariant to adding contant to the obervation, o that contant term are uppreed in (). A a technical matter, we think of {z t, v t } T t= (and thu alo {x t, y t } T t= ) a being generated from a triangular array; we omit the additional dependence on T to eae notation. Alo, we write x for the integer part of x R, A = tr A A for any real matrix A, x y for the maximum of x, y R, for the uual Kronecker product and to indicate weak convergence. Let W( ) denote a n tandard Wiener proce. The vector z t i a canonical I() vector in the ene that it partial um converge to a r Wiener proce T /2 T z t S z W() = W z (), where S z S = z I r. (2) t= The vector v t i a common trend in the ene that caled verion of it level converge to a tochatic integral with repect to W( ). For example, in the tandard I() model, T /2 v T HdW(t), where H i a k n matrix and (H, S z ) ha full rank. More general trend procee, uch a the local-to-unity formulation, allow the matrix H to depend on and t. The general repreentation for the common trend ued in thi paper i T /2 v T H(, t)dw(t) (3) where H(, t) i ufficiently well behaved to enure that there exit a cadlag verion of the proce H(, t)dw(t).5 5 The common cale T /2 for the k vector v t in (3) i aumed for convenience; with an appropriate definition of local alternative, the invariance (4) enure that one would obtain the ame reult for any caling of v t. For example, for an I(2) tochatic trend caled by T 3/2, et H(, t) = [t ]( t)h, with the k n matrix H a in the I() cae Invariance and reparameterization A dicued above, becaue cointegration only identifie the column pace of β, attention i retricted to tet that are invariant to the group of tranformation (y t, x t ) (A yy y t, A xx x t + A xy y t ) (4) where A yy and A xx are non-ingular, but (A yy, A xx, A xy ) are otherwie unretricted real matrice. The retriction to invariant tet allow a implification of notation: becaue the tet tatitic are invariant to the tranformation in (4), there i no lo of generality etting Γ yz = I r, Γ xv = I k, and Γ xz =. With thee value, the model i y t = z t + Γ yv v t (5) x t = v t Retricted verion of the trend model We will refer to the general trend pecification in (3) a the unretricted tochatic trend model throughout the remainder of the paper. The exiting literature on efficient tet relie on retricted form of the trend proce (3) uch a I() or local-to-unity procee, and we compute the potential power gain aociated with thee and other a priori retriction on H(, t) below. Here we decribe five retricted verion of the tochatic trend. The firt model, which we will refer to a the G-model, retrict H(, t) to atify H(, t) = G(, t)s v, (6) where G(, t) i k k and S v i k n with S v S v = I k and (S, z S v ) noningular. In thi model, the common trend depend on W( ) only through the k tandard Wiener proce W v ( ) = S v W( ), and thi retrict the way that v t and z t interact. In thi model T /2 v T G(, t)dw v (t), (7) and the covariance between the Wiener proce characterizing the partial um of z t, W z, and W v i equal to the r k matrix R = S z S v. Standard I() and local-to-unity formulation of cointegration atify thi retriction and impoe additional parametric retriction on G(, t). The econd model further retrict (7) o that G(, t) i diagonal: G(, t) = diag(g (, t),..., g k (, t)). (8) An interpretation of thi model i that the k common trend evolve independently of one another (recall that W v ha identity covariance matrix), where each trend i allowed to follow a different proce characterized by the function g i (, t). The third model further retrict the diagonal-g model o that the k tochatic trend converge weakly to a tationary continuou time proce. We thu impoe g i (, t) = g S i ( t), i =,..., k. (9) The tationary local-to-unity model (with an initial condition drawn from the unconditional ditribution), for intance, atifie thi retriction. Finally, we conider two parametric retriction of G: G(, t) = [t > ]I k () which i the I() model, and G(, t) = [t > ]e C( t) () which i the multivariate local-to-unity model, where C i the k k diffuion matrix of the limiting Orntein Uhlenbeck proce (with zero initial condition). 6 6 The I() pecification in () i the ame a the I() pecification given below (2) becaue the invariance in (4) implie that the trend model are unaffected by premultiplication of H(, t) (or G(, t)) by an arbitrary non-ingular k k matrix.

4 U.K. Müller, M.W. Waton / Journal of Econometric 74 (23) Teting problem and local alternative The goal of the paper i to tudy aymptotically efficient tet for the value of the cointegrating vector with controlled rejection probability under the null hypothei for a range of tochatic trend pecification. The different order of magnitude of z t and v t in (2) and (3) ugget a local embedding of thi null hypothei againt alternative where Γ yv = T B for B a contant r k matrix, o that in model (5), T /2 T y t S z W() + B t= u H(u, t)dw(t)du. In thi parameterization, the null hypothei become H : B =, H(, t) H (2) where H(, t) i retricted to a et of function H, that, in the unretricted trend model include function ufficiently well behaved to enure that there exit a cadlag verion of the proce H(, t)dw(t), or more retricted verion of H(, t) a in (6) and (8) (). Since our goal i to conider efficient tet of the null hypothei (2), we alo need to pecify the alternative hypothei. Our reult below are general enough to allow for the derivation of efficient tet againt any particular alternative with pecified B = B and tochatic trend proce H(, t) = H (, t), H : B = B, H(, t) = H (, t) (3) or, more generally, for tet that are efficient in the ene of maximizing weighted average power againt a et of value for B and tochatic trend model H (, t). Our numerical reult, however, focu on alternative in which the tochatic trend v t i I(), o that H (, t) atifie (6) and (). Thi i partly out of practical conideration: while there i a wide range of potentially intereting trend pecification, the computation for any particular pecification are involved, and thee computational complication limit the number of alternative we can uefully conider. At the ame time, one might conider the claical I() model a an important benchmark againt which it i ueful to maximize power not necearily becaue thi i the only plauible model under the alternative, but becaue a tet that perform well againt thi alternative preumably ha reaonable power propertie for a range of empirically relevant model. We tre that depite thi focu on the I() tochatic trend model for the alternative hypothei (3), we retrict attention to tet that control ize for a range of model under the null hypothei (2). The idea i to control the frequency of rejection under the null hypothei for any tochatic trend model in H, o that the rejection of a et of cointegrating vector cannot imply be explained by the tochatic trend not being exactly I(). In thi ene, our approach i one of robut cointegration teting, with the degree of robutne governed by the ize of the et H Summary To ummarize, thi ection ha introduced the time domain repreentation of the cointegrated model with a focu on the problem of inference about the pace of cointegrating vector. In all repect except one, the repreentation i the tandard one: the data are expreed a a linear function of a canonical vector of common trend and a vector of I() component. Under the null, certain linear combination of the data do not involve the common trend. Becaue the null only retrict the column pace of the matrix of cointegrating vector, attention i retricted to invariant tet. The goal i to contruct tet with bet power for an alternative value for the matrix of cointegrating vector under a particular model for the trend (or bet weighted average power for a collection of B and H (, t)). The formulation differ from the tandard one only in that it allow the model for the trend proce under the null to be le retrictive than the tandard formulation. Said differently, becaue of potential uncertainty about the pecific form of the trend proce, the formulation retrict attention to tet that control ize for a range of different trend procee. Thi generalization complicate the problem of contructing efficient tet by introducing a potentially large number of nuiance parameter (aociated with the trend proce) under the null hypothei. 3. Low-frequency repreentation of the model Cointegration i a retriction on the low-frequency behavior of time erie, and a dicued in the introduction, we therefore focu on the low-frequency behavior of (y t, x t ). Thi low-frequency variability i ummarized by a mall number, q, of weighted average of the data. In thi ection we dicu thee weighted average and derive their limiting behavior under the null and alternative hypothee. 3.. Low-frequency weighted average We ue weight aociated with the coine tranform, where the j th weight i given by Ψ j () = 2 co(jπ). For any equence {a t } T t=, the j th weighted average will be denoted by T t /2 A T (j) = Ψ j ()a T + d = ι jt T Ψ j a t (4) T where ι jt = (2T/jπ) in(jπ/2t) for all fixed j. A demontrated by Müller and Waton (28), the weighted average A T (j), j =,..., q, eentially capture the variability in the equence correponding to frequencie below qπ/t. We ue the following notation: with a t a h vector time erie, let Ψ () = (Ψ (), Ψ 2 (),..., Ψ q ()) denote the q vector of weighting function, and A T = Ψ ()a T +d the q h matrix of weighted average of the element of a t, where Ψ () i excluded to make the reult invariant to adding contant to the data. Uing thi notation, the q r matrix Y T and the q k matrix X T ummarize the variability in the data correponding to frequencie lower than qπ/t. With q = 2, (Y T, X T ) capture variability lower than the buine cycle (periodicitie greater than 8 year) for time erie that pan 5 year (potwar data) regardle of the ampling frequency (month, quarter, week, etc.). Thi motivate u to conider the behavior of thee matrice a T, but with q held fixed. The large-ample behavior of X T and Y T follow from the behavior of Z T and V T. Uing the aumed limit (2) and (3), the continuou mapping theorem, and integration by part for the term involving Z T, one obtain T /2 Z T Z T /2 (5) V T V where vec Z N vec V with Σ VZ = Σ VV = Irq, Σ VZ Σ ZV Σ VV t= (6) [H(, t) Ψ ()]d [S z Ψ (t)] dt (7) t [H(, t) Ψ ()]d t [H(, t) Ψ ()]d dt t

5 7 U.K. Müller, M.W. Waton / Journal of Econometric 74 (23) 66 8 ince after a change of order of integration, one obtain vec V = t [H(, t) Ψ ()]d dw(t). The relative carcity of low-frequency information i thu formally captured by conidering the weak limit (2) and (3) a pertinent only for the ubpace panned by the weight function Ψ ( ), yielding (5) a a complete characterization of the relevant propertie of the error correction term z t and the common tochatic trend v t. Uing Γ yv = T B, Eq. (5) implie that Y T = Z T + T V T B and X T = V T. Thu, T /2 Y T Y Z + VB = T /2 (8) X T X V where vec Y N, Σ vec X (Y,X) (9) with Ir I Σ (Y,X) = q B I q Ir I q Σ ZV I k I q Σ VZ Σ VV Ir I q B. (2) I q I k I q 3.2. Bet low-frequency hypothei tet We conider invariant tet of H againt H given in (2) and (3) baed on the data {y t, x t } T t=. Becaue we are concerned with the model implication for the low-frequency variability of the data, we retrict attention to tet that control aymptotic ize for all model that atify (8) (2). Our goal i to find an invariant tet that maximize power ubject to thi retriction, and for brevity we will refer to uch a tet a a bet tet. Müller (2) conider the general problem of contructing aymptotically mot powerful tet ubject to aymptotic ize control over a cla of model uch a our. In our context, hi reult imply that aymptotically bet tet correpond to the mot powerful invariant tet aociated with the limiting ditribution (9). Thu, the relevant teting problem ha a imple form: vec(y, X) ha a normal ditribution with mean zero and covariance matrix that depend on B. Under the null B =, while under the alternative B. Tet are retricted to be invariant to the group of tranformation (Y, X) (YA yy, XA xx + YA xy ) (2) where A yy and A xx are noningular, and A yy, A xx, and A xy are otherwie unretricted. Thu, the hypothei teting problem become the problem of uing an invariant procedure to tet a retriction on the covariance matrix of a multivariate normal vector. 4. Bound on power and ize The general verion of the hypothei teting problem we are facing i a familiar one: Let U denote a ingle obervation of dimenion m. (In our problem, U correpond to the maximal invariant for (Y, X).) Under the null hypothei U ha probability denity f θ (u) with repect to ome meaure µ, where θ Θ i a vector of nuiance parameter. (In our problem, the vector θ decribe the tochatic trend proce under the null hypothei and determine Σ (Y,X) via (7) and (2).) Under the alternative, U ha known denity h(u). (Choice for h(u) for our problem will be dicued in Section 5.2..) Thu, the null and alternative hypothei are H : The denity of U i f θ (u), θ Θ H : The denity of U i h(u), (22) and poibly randomized tet are (meaurable) function ϕ : R m [, ], where ϕ(u) i the probability of rejecting the null hypotheiwhen oberving U = u, o that ize and power are given by up θ Θ ϕfθ dµ and ϕhdµ, repectively. Thi ection preent two probability bound, one on power and one on ize, in thi general problem. The power bound i taken from Elliott et al. (22) (EMW), and provide an upper bound on the power of any valid tet of H veru H. A dicued in EMW, thi power bound i ueful for two reaon. Firt, when the dimenion of θ i mall, numerical method developed in EMW can be ued to contruct a tet that eentially attain the power bound, o the reulting tet i the approximately efficient tet. For example, in our problem, we will ue the EMW algorithm to compute the approximately efficient tet when x t follow a localto-unity proce. The econd reaon the bound i ueful i that it can be computed even when the dimenion of θ i large, where it i infeaible to directly contruct the efficient tet uing numerical method. The reulting power bound make it poible to evaluate the potential power hortfall of exiting tet. For example, when x t follow the general tochatic trend proce (3) and there i a ingle cointegrating vector (r = ), the power of a low-frequency of tet originally propoed by Wright (2) eentially coincide with the power bound and thi allow u to conclude that the tet i approximately efficient. The econd probability bound focue on ize, and i motivated in our context by the following concern. Suppoe that H pecifie that the tochatic trend follow an I() proce, and conider a tet that exploit feature of the I() proce to increae power. Uncertainty about the trend proce mean that it i ueful to know omething about the rejection frequency of tet under null hypothee that allow for more general trend, uch a the unretricted trend model (3) or other le retricted verion decribed above. We provide a lower bound on thi rejection frequency, where a large value of thi lower bound highlight the fragility of tet that exploit a particular H to obtain more powerful inference. 4.. An upper bound on power A tandard device for problem uch a (22) i to conider a Neyman Pearon tet for a related problem in which the null hypothei i replaced with a mixture H Λ : The denity of U i f θ dλ(θ) where Λ i a probability ditribution for θ with upport in Θ. The following lemma (taken from EMW) how that the power of the Neyman Pearon tet of H Λ veru H provide an upper power bound for tet of H veru H. Lemma. Let ϕ Λ be the bet level α tet of H Λ againt H. Then for any level α tet ϕ of H againt H, ϕ Λ hdµ ϕhdµ. Proof. Since ϕ i a level α tet of H, ϕf θ dµ α for all θ Θ. Therefore, ϕf θ dµdλ(θ) = ϕf θ dλ(θ)dµ α (where the change in the order of integration i allowed by Fubini Theorem), o that ϕ i alo a level α tet of H Λ againt H. The reult follow by the definition of a bet tet. Thi reult i cloely related to Theorem 3.8. of Lehmann and Romano (25) which provide condition under which a leat upper bound on the power for tet H veru H i aociated with a leat favorable ditribution for θ, and that uing thi ditribution for Λ produce the leat upper power bound. The leat favorable ditribution Λ ha the characteritic that the reulting ϕ Λ i a level α tet for teting H veru H. Said differently, if ϕ Λ i the bet level α tet of H Λ againt H and i alo a level

6 U.K. Müller, M.W. Waton / Journal of Econometric 74 (23) α tet for teting H veru H, then ϕ = ϕ Λ, that i ϕ Λ i the mot powerful level α tet of H veru H. Unfortunately, while the tet aociated with the leat favorable ditribution olve the teting problem (22), there i no general and contructive method for finding the leat favorable ditribution Λ. With thi in mind, Lemma i tated o that Λ i not necearily the leat favorable ditribution. That i, the bound in Lemma hold for any probability ditribution Λ. The goal of the numerical analyi carried out below i to chooe Λ to approximate the leat upper bound. Importantly, even if one cannot identify the leat favorable ditribution, Lemma how that the power of ϕ Λ provide a valid bound for the power of any tet of H veru H, for any Λ A lower bound on ize under an auxiliary null Now conider the Larger auxiliary null hypothei H L : The denity of U i f θ (u), with an aociated mixture H ΛL : The denity of U i θ Θ L f θ dλ L (θ) where Λ L ha upport in Θ L. (In our problem H L will be a null hypothei that allow for a le retricted trend proce than under H. Thu if H allow only for an I() trend, H L might allow for a local-to-unity trend or one of the more general trend procee dicued in Section 2.) Conider any tet ϕ of level α under H with power of at leat β. The following lemma provide a lower bound on the rejection frequency under the auxiliary null H L. Lemma 2. (a) The problem min ϕ f θ dλ L (θ)dµ ϕ.t. ϕ f θ dλ (θ)dµ α and ϕhdµ β i olved by ϕ = [h λ fθ dλ (θ) + λ 2 fθ dλ L (θ)], where λ and λ 2 are non-negative contant aociated with the contraint. (b) Let α L = ϕ f θ dλ L (θ)dµ denote the minimized value of the objective function in (a). Let ϕ be a level α tet under H and of power of at leat β. Then up θ ΘL ϕfθ dµ α L. Proof. (a) I a variant of the generalized Neyman Pearon Lemma (Theorem 3.6. in Lehmann and Romano, 25). (b) Note that ϕ atifie the two contraint in the problem given in (a), o that ϕ fθ dλ L (θ)dµ α L, and the reult follow from up θ ΘL ϕ f θ dµ ϕ f θ dλ L (θ)dµ. Thi lemma i particularly ueful in conjunction with Lemma : Suppoe application of Lemma implie that no 5% level tet of a relatively retricted H can have power of more than, ay, 7%. Thi ugget that there could indeed exit a 5% level tet ϕ with power, ay, 67%, and one might want to learn about the ize propertie of uch tet under the more general null hypothei H L. Lemma 2 provide a way of computing a lower bound on thi ize that i valid for any tet with power of at leat 67%. So if thi ize ditortion i large, then without having to determine the cla of 5% level tet of H with power of at leat 67%, one can already conclude that all uch tet will be fragile. In the numerical ection below, we dicu how to determine uitable Λ and Λ L to obtain a large lower bound α L (λ and λ 2 are determined through the two contraint on ϕ ). 5. Computing bound In thi ection we compute the power and ize bound from the lat ection. The analyi proceed in four tep. Firt, we derive the denity of the maximal invariant of (Y, X); thi denity form the bai of the likelihood ratio. Second, ince the denity of the maximal invariant depend on the covariance matrix of (Y, X), we dicu the parameterization of Σ (Y,X) under the null and alternative hypothee. In the third tep we decribe how the mixing ditribution Λ, Λ and Λ L are choen to yield tight bound. Finally, we preent numerical value for the bound. 5.. Denity of a maximal invariant Recall that we are conidering tet that are invariant to the group of tranformation (Y, X) (YA yy, XA xx + YA xy ) where A yy and A xx are noningular, and A yy, A xx, and A xy are otherwie unretricted. Any invariant tet can be written a a function of a maximal invariant (Theorem 6.2. in Lehmann and Romano, 25), o that by the Neyman Pearon lemma, the mot powerful invariant tet reject for large value of the likelihood ratio tatitic of a maximal invariant. The remaining challenge i the computation of the denity of a maximal invariant, and thi i addreed in the following theorem. Theorem. If vec(y, X) N (, Σ (Y,X) ) with poitive definite Σ (Y,X) and q > r + k, the denity of a maximal invariant of (2) ha the form c(det Σ (Y,X) ) /2 (det V Y Σ (Y,X) V Y ) /2 (det Ω) /2 E ω [ det (ω Y ) q r det (ω X ) q r k ] where c doe not depend on Σ (Y,X), ω Y and ω X are random r r and k k matrice, repectively, with (vec ω y, vec ω x ) N (, Ω ), Ω = D YX Σ (Y,X) D YX D YX Σ (Y,X) V Y (V Y Σ (Y,X) V Y ) V Y Σ (Y,X) D YX, D YX = diag(i r Y, I k X), V Y = ( rq rk, I k Y ), and E ω denote integration with repect to ω Y and ω X, conditional on (Y, X). Theorem how that the denity of a maximal invariant can be expreed in term of abolute moment of determinant of jointly normally ditributed random matrice, whoe covariance matrix depend on (Y, X). We do not know of a ueful and general cloed-form olution for thi expectation; for r = k =, however, Nabeya (95) reult for the abolute moment of a bivariate normal yield an expreion in term of elementary function, which we omit for brevity. When r + k > 2, the moment can be computed via Monte Carlo integration. However, computing accurate approximation i difficult when r and k are large, and the numerical analyi reported below i therefore limited to mall value of r and k Parameterization of Σ (Y,X) Since the denity of the maximal invariant of Theorem depend on Σ (Y,X), the derivation of efficient invariant tet require pecification of Σ (Y,X) under the alternative and null hypothei. We dicu each of thee in turn Specification of Σ (Y,X) under the alternative hypothei A dicued above, we focu on the alternative where the tochatic trend follow an I() proce, o that H(, t) atifie (6) and (). There remain the iue of the value of B (the coefficient that determine how the trend affect Y ) and R (the correlation of the Wiener procee decribing the I() variable, z t, and the

7 72 U.K. Müller, M.W. Waton / Journal of Econometric 74 (23) 66 8 common trend, v t ). For thee parameter, we conider pointvalued alternative with B = B and R = R ; the power bound derived below then erve a bound on the aymptotic power envelope over thee value of B and R. Invariance reduce the effective dimenion of B and R omewhat, and thi will be dicued in the context of the numerical reult preented below Parameterization of Σ (Y,X) under the null hypothei From (2), under the null hypothei with B =, the covariance matrix Σ (Y,X) atifie Irq Σ Σ (Y,X) = ZV. Σ VV Σ VZ The model pecification of the tochatic trend under the null determine the rq kq matrix Σ ZV and the kq kq matrix Σ VV by the formula given in (7). Since thee matrice contain a finite number of element, it i clear that even for nonparametric pecification of H(, t), the effective parameter pace for low-frequency tet baed on (Y, X) i finite dimenional. We collect thee nuiance parameter in a vector θ Θ. Section 2 dicued everal trend procee, beginning with the general proce given in (3) with an unretricted verion of H(, t), and then five retricted model: (i) the G model in (6), (ii) the Diagonal model (8), (iii) the Stationary model (9), (iv) the localto-unity model (), and (v) the I() model (). The Appendix dicue convenient parameterization for Σ (Y,X) for thee five retricted model, and the following lemma provide the bai for parameterizing Σ (Y,X) when H(, t) i unretricted. Lemma 3. (a) For any (r + k)q (r + k)q poitive definite matrix Σ with upper left rq rq block equal to I rq, there exit an unretricted trend model with H(, t) = for t < uch that Σ = E[vec(Z, V)(vec(Z, V)) ]. (b) If r k, thi H(, t) can be choen of the form H(, t) = G(, t)s v, where (S, z S v ) ha full rank. The lemma how that when H(, t) i unretricted (or r k and H(, t) = G(, t)s v with G unretricted) the only retriction that the null hypothei impoe on Σ (Y,X) i that Σ YY = I rq. 7 In other word, ince Σ ZV and Σ VV have rkq 2 + kq(kq + )/2 ditinct element, an appropriately choen θ of that dimenion determine Σ (Y,X) under the null hypothei in the unretricted model, and in the model where H(, t) = G(, t)s v for r k Approximating the leat upper power and greatet lower ize bound We dicu two method to approximate the power bound aociated with the leat favorable ditribution from Section 4, and ue the econd method alo to determine a large lower ize bound. Firt, we dicu a generic algorithm developed in Elliott et al. (22) that imultaneouly determine a low upper bound on power, and a level α tet whoe power i cloe to that bound. The computational complexity i uch, however, that it can only be applied when θ i low-dimenional; a uch, it i ueful for our problem only in the I() and local-to-unity tochatic trend model for r = k =. Second, when the dimenion of θ i large we chooe Λ (and Λ and Λ L for Lemma 2) o the null and alternative 7 Without the invariance retriction (2), thi obervation would lead to an analytic leat favorable ditribution reult: Factor the denity of (Y, X) under the alternative into the product of the denity of Y, and the denity of X given Y. By chooing Σ VZ and Σ VV under the null hypothei appropriately, the latter term cancel, and the Neyman Pearon tet i a function of Y only. In Section 6 below we conider the efficient Y -only invariant tet and compare it power to the approximate (Y, X) power bound. ditribution are cloe in a numerically convenient metric. Two numerical reult ugget that thi econd method produce a reaonably accurate etimate of the leat upper power bound: the method produce power bound only marginally higher than the firt method (when the firt method i feaible), and when r = we find that the method produce a power bound that can be achieved by a feaible tet that we preent in the next ection. We dicu the two method in turn Low dimenional nuiance parameter Suppoe that LR Λ = h(u)/ f θ (U)dΛ(θ) i a continuou random variable for any Λ, o that by the Neyman Pearon Lemma, ϕ Λ i of the form ϕ Λ = [LR Λ > cv Λ ], where the critical value cv Λ i choen to atify the ize contraint ϕ Λ f θ dµdλ(θ) = α. Then by Lemma, the power of ϕ Λ, β Λ = ϕ Λ hdµ, i an upper bound on the power of any level-α tet under H. If Λ i not the leat favorable ditribution, then ϕ Λ i not of ize α under H, i.e. up θ Θ ϕλ f θ dµ > α. Now conider a verion of ϕ Λ with a ize-corrected critical value cv c Λ > cv Λ, that i ϕλ c = [LR Λ > cv c Λ ] with cv c Λ choen to atify the ize contraint up θ Θ ϕ c Λ f θ dµ = α. Becaue the ize adjuted tet ϕλ c i of level α under H, the leat upper power bound mut be between βλ c and β Λ. Thu, if βλ c i cloe to β Λ, then βλ c erve a a good approximation to the leat upper bound. The challenge i to find an appropriate Λ. Thi i difficult becaue, in general, no cloed form olution are available for the ize and power of tet, o that thee mut be approximated by Monte Carlo integration. Brute force earche for an appropriate Λ are not computationally feaible. Elliott et al. (22) develop an algorithm that work well (in the ene that it produce a tet with power within ε, where ε i a mall pre-pecified value) in everal problem when the dimenion of θ i mall, and we implement their algorithm here High dimenional nuiance parameter The dimenion of θ can be very large in our problem: even when r = k =, the model with unretricted tochatic trend lead to θ of dimenion q 2 + q(q + )/2 o that θ contain 222 element when q = 2. Approximating the leat upper power bound directly then become numerically intractable. Thi motivate a computationally practical method for computing a low (a oppoe to leat) upper power bound. The method retrict Λ o that it i degenerate with all ma on a ingle point, ay θ, which i choen o that the null ditribution of the maximal invariant of Theorem i cloe to it ditribution under the alternative. Intuitively, thi hould make it difficult to ditinguih the null from the alternative hypothei, and thu lead to a low power bound. Alo, thi choice of θ enure that draw from the null model look empirically reaonable, a they are nontrivial to ditinguih from draw of the alternative with an I() tochatic trend. Since the denity of the maximal invariant i quite involved, θ i uefully approximated by a choice that make the multivariate normal ditribution of vec(y, X) under the null cloe to it ditribution under the alternative, a meaured by a convenient metric. We chooe θ to minimize the Kullback Leibler divergence (KLIC) between the null and alternative ditribution. Since the bound from Lemma and 2 are valid for any mixture, numerical error in the KLIC minimization do not invalidate the reulting bound. Detail are provided in the Appendix Numerical bound Table how numerical reult for power and ize bound for 5% level tet with q = 2. Reult are hown for r = k =

8 U.K. Müller, M.W. Waton / Journal of Econometric 74 (23) Table Power and ize bound for 5% tet (q = 2). A. r = k = H H L B = 7 B = 4 R = R =.5 R =.9 R = R =.5 R =.9 RB < RB > RB < RB > RB < RB > RB < RB > unr tat unr LTU unr tat I() unr tat LTU B. r = and k 2 H H L B = 7 B = 4 R = R =.5 R =.9 R = R =.5 R =.9 ω = ω = ω = ω = ω = ω = ω = ω = ω = ω = ω = ω = unr diag unr tat unr diag LTU unr diag tat I() unr diag tat LTU C. r = 2 and k = H H L B = B = 2 R = R =.5 R =.9 R = R =.5 R =.9 ω = ω = ω = ω = ω = ω = ω = ω = ω = ω = ω = ω = unr G unr tat unr G LTU unr G tat I() unr G tat LTU Note: Non-italicized numerical entrie are upper bound on power for 5% tet for the null hypothei retricting the trend proce a hown in the firt column. The italicized entrie are lower bound on ize for auxiliary trend procee given in column 2 for 5% level tet with power greater than the power bound le 3% point. Abbreviation for the trend model in column and 2 are: (i) unr i the unretricted model given in (3), (ii) G i the G-model in (6), (iii) diag i the diagonal G-model in (8), (iv) tat i the tationary G model given in (9), (v) LTU i the local-to-unity model given in (), and (vi) I() i I() model given in (). In panel B and C, ω = tr(r B)/( R B ). Reult are baed on 2, Monte Carlo replication. (panel A), r = and k 2 (panel B), and r = 2 and k = (Panel C). 8 Numerical reult for larger value of n = r + k are not reported becaue of the large number of calculation required to evaluate the denity of Theorem 2 in large model. 8 The reult hown in panel B were computed uing the KLIC minimized value of θ for the model with r = and k = 2. The Appendix how the reulting bound are valid for k 2. Power depend on the value of B and R under the alternative, and reult are preented for variou value of thee parameter. Becaue of invariance, when r = (a in panel A and B), or k = (a in panel C), the ditribution of the maximal invariant depend on B and R only through B, R, and, if R >, on tr(r B)/( B R ). Thu, in panel A, where r = k =, reult are hown for two value of B, three value of R and for R B < and R B >, while panel B and C how reult for three value of

9 74 U.K. Müller, M.W. Waton / Journal of Econometric 74 (23) 66 8 Table 2 Comparion of KLIC minimized and approximate leat upper power bound (r = k =, q = 2). H Bound B = 7 B = 4 R = R =.5 R =.9 R = R =.5 R =.9 RB < RB > RB < RB > RB < RB > RB < RB > LTU KLIC LUB I() KLIC LUB Note: The entrie labeled KLIC are computed uing the KLIC minimization dicued in Section The entrie labeled LUB are computed uing the approximate leat upper power bound algorithm dicued in Section 5.3., and are by contruction no more than 2.5% point above the actual leat upper bound. Reult are baed on 2, Monte Carlo replication. ω = tr(r B)/( B R ) when R >. All of the reult in Table ue the KLIC minimized value of θ a decribed in the lat ubection. Table 2 compare thi KLIC-baed bound to the numerical leat upper power bound when the parameter pace i ufficiently mall to allow calculation of the numerical leat upper bound. To undertand the formatting of Table, look at panel A. The panel contain italicized and non-italicized numerical entrie. The non-italicized number are power bound, and the italicized number are ize bound. The firt column in the table how the trend pecification allowed under H. The firt entry, labeled unr correpond to the unretricted trend pecification in (3) and the other entrie correpond to the retricted trend procee dicued in Section 2. Becaue r = k =, there are no retriction impoed by the aumption that H(, t) = G(, t)s v or that G i diagonal, o thee model are not lited in panel A. Stationarity (G(, t) = G( t)) i a retriction, and thi i the econd entry in the firt column. The final two entrie correpond to the local-to-unity ( LTU ) and I() retriction. The numerical entrie hown in the row correponding to thee trend model are the power bound. For example, the non-italicized entrie in the firt numerical column how power bound for R = and B = 7, which are.36 for the unretricted null,.4 when the trend i retricted to be tationary,.5 when the trend i retricted to follow a local-to-unity proce, and.5 when the trend i further retricted to follow an I() proce. The econd column of panel A how the auxiliary null hypothee H L, correponding to the null hypothei H, hown in the firt column. The entrie under H L repreent le retrictive model than H. For example, when H retrict the null to be tationary, an unretricted trend proce ( unr ) i hown for H L, while when H retrict the trend to be I(), the le retrictive local-to-unity, tationary, and unretricted null are lited under H L. The numerical entrie for thee row (hown in italic in the table) are the lower ize bound for H L for 5% level tet under H and with power that i 3% point le than the correponding power bound hown in the table. For example, from the firt numerical column of panel A, the power bound for the I() verion of H i.5. For any tet with ize no larger than 5% under thi null and with power of at leat.47 (=.5.3), the ize under a null that allow an unretricted trend ( unr under H L ) i at leat 2%, the ize under a null that retrict the trend to be tationary i at leat 8%, and the ize under a null that retrict the trend to follow a local-to-unity proce i at leat 4%. Looking at the entrie in Panel A, two reult tand out. Firt, and not urpriingly, retricting tet o that they control ize for the unretricted trend proce lead to a non-negligible reduction in power. For example, when B = 7, and R =, the power bound i.36, for tet that control ize for unretricted trend, the bound increae to.4 for tet that control ize for tationary trend, and increae to.5 for tet that control ize for localto-unity or I() trend procee. Second, whenever there i a ubtantial increae in power aociated with retricting the trend proce, there are large ize ditortion under the null hypothei without thi retriction. For example, Elliott (998) obervation that efficient tet under the I() trend have large ize ditortion under a local-to-unity proce i evident in the table. From the table, when B = 7, R =.9, and R B >, the power bound for the null with an I() trend i.95, but any tet that control ize for thi null and ha power of at leat.92 will have ize that i greater than.5 when the trend i allowed to follow a localto-unity proce. However, addreing Elliott (998) concern by controlling for ize in the local-to-unity model, a in the analyi of Stock and Waton (996) or Janon and Moreira (26) doe not eliminate the fragility of the tet. For example, with the ame value of B and R, the power bound for the null that allow for a local-to-unity trend i.67, but any tet that control ize for thi null and ha power of at leat.64 will have a ize greater than.32 when the trend i unretricted. Panel B (r = and k = 2) and C (r = 2 and k = ) how qualitatively imilar reult. Indeed thee panel how even more fragility of tet that do not allow for general trend. For example, the lower ize bound for the unretricted trend null exceed.5 in everal cae for tet that retrict trend to be I(), local-to-unity, or tationary. When r = k =, it i feaible to approximate the leat upper power bound for the I() and local-to-unity trend retriction uing the method developed in EMW. By contruction, the approximate leat upper bound (LUB) in Table 2 are no more than 2.5% point above the actual leat upper bound, apart from Monte Carlo error. The difference with the KLIC minimized power bound are mall, uggeting that the bound in Table are reaonably tight. 6. Efficient Y -only tet The primary obtacle for contructing efficient tet of the null hypothei that B = i the large number of nuiance parameter aociated with the tochatic trend (the parameter that determine H(, t)). Thee parameter govern the value of Σ ZV and Σ VV, which in turn determine Σ YX and Σ XX. Any valid tet mut control ize over all value of thee nuiance parameter. Wright (2) note that thi obtacle can be avoided by ignoring the x t data and baing inference only on y t, ince under the null hypothei, y t = z t. Thi ection take up Wright uggetion and dicue efficient low-frequency Y -only tet. 9 We have two related goal. The firt i to tudy the power propertie of thee tet relative to the power bound computed in the lat ection. A it turn out, when r = (o there i only 9 Wright (2) implement thi idea uing a tationarity tet of the I() null propoed by Saikkonen and Luukonen (993), uing a robut covariance matrix a in Kwiatkowki et al. (992) for the tet propoed in Nyblom (989). Thi tet relie on a conitent etimator of the pectral denity matrix of z t at frequency zero. But conitent etimation require a lot of pertinent low frequency information, and lack thereof lead to well-known ize control problem (ee for example, Kwiatkowki et al., 992, Caner and Kilian, 2, and Müller, 25). Thee problem are avoided by uing the low-frequency component of y t only; ee Müller and Waton (28) for further dicuion.