Estimation and Testing for the Cointegration Rank in a Threshold ...

Estimation and Testing for the Cointegration Rank in a Threshold Cointegrated System Jaya Krishnakumar and David Neto

No 2009.01

Cahiers du département d’économétrie Faculté des sciences économiques et sociales Université de Genève

Février 2009

Département d’économétrie Université de Genève, 40 Boulevard du Pont d’Arve, CH -1211 Genève 4 http://www.unige.ch/ses/metri/

Estimation and Testing for the Cointegration Rank in a Threshold Cointegrated System Jaya Krishnakumar David Neto Department of Econometrics, University of Geneva∗ January 2009

Abstract The paper generalises estimation and inference procedures for a threshold VECM with more than one cointegrating relation. We derive estimators of long-run parameters and loading factors by means of a reduced rank regression. We provide their asymptotic distributions and propose a testing procedure for the cointegrating rank. The asymptotic distributions of our test statistics are derived and tabulated. In order to improve finite sample inference, we also compute bootstrap approximation to the distribution of our test statistics. Monte-Carlo experiments are conducted to evaluate the finite-sample performance of our test and its power. We apply our methodology to the expectations hypothesis of the term structure of U.S. interest rates. Keywords: Threshold cointegration, reduced rank estimator, TUR models, cointegration rank test, term structure of interest rates.

JEL classification: C12, C13, C32, E43.

∗

Address: 40, Bd. du Pont d’Arve, CH-1211, Geneva 4, Switzerland. [email protected], [email protected]

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Email:

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Introduction

Since the pioneering work of Engle and Granger (1987), the early literature on cointegration focussed on a linear cointegrating relationship implying a continuous adjustment to the long run equilibrium. However, this may be a restrictive assumption for some market mechanisms. Thus, Caballero and Hammour (1994) discuss evidence of asymmetric responses in certain economic mechanisms according to whether the economy is undergoing recession or expansion. Government interventions may also lead to asymmetry in the adjustment towards equilibrium. For example, Central Banks may manipulate two different interest rates so that the spread does not exceed a given fluctuation band. Nonlinearities in the error correction mechanism can also be due to institutional constraints for instance when monetary authorities use a target zone device to apply an exchange rate policy. Indeed, such a target zone model was introduced by Krugman (1991) in which the long-run parity relationship remains inactive within a given range of disequilibria while it becomes active when the system crosses the boundaries of allowed fluctuations. Existence of transaction costs in financial assets markets also may prevent continuous adjustments towards the fundamentals. In such situations, it is not surprising that real data often fail to validate cointegration relations representing well known economic “laws" (that are interpreted as long run equilibria). Recent years have seen a surge of interest in the examination of the different possible ways to take into account non-linear adjustments to equilibrium. Balke and Fomby (1997) were the first authors to introduce the idea of threshold cointegration to describe asymmetric adjustment to equilibrium and derive its error correction representation, called the threshold vector error correction model (henceforth TVECM). This model typically specifies a threshold autoregressive (TAR) process for the equilibrium error. The TAR models, first investigated by Tong (1983), are particularly popular in empirical applications because of their capacity to adequately represent observed time series behaviour in terms of persistence and asymmetry. In addition, the statistical properties of TAR processes have been extensively studied in the literature outside of cointegration theory (see Tsay (1989, 1998), Chan (1993), Hansen (1996, 1997), González and Gonzalo (1998) and more recently Caner and Hansen (2001) among numerous others) and therefore many results are available for conducting inference. Following a different but less widespread line of thinking Krolzig (1999) introduces Markov-switching dynamics (developed by Hamilton, 1989) in the common trend representation of a cointegrated VAR. Recently, Camacho (2005) has proposed to directly specify Markov-switching in the long run disequilibrium process to capture different dynamics in short-term fluctuations depending on the phase of the business cycle. Subsequent to the study by Balke and Fomby (1997), several tests have been proposed in the literature on threshold cointegration. For example, Enders and Siklos (2001) use both self-excisting TAR (hereafter SETAR) and Momentum TAR (MTAR) processes to describe the disequilibrium and propose a method-

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ology to test for cointegration but without a real asymptotic test theory.1 Lo and Zivot (2001) extend the work of Enders and Siklos and test threshold adjustment in international relative prices. Kapetanios et al. (2006) have dealt with a disequilibrium which follows an exponential smooth transition process and have proposed a non-linear residual-based test for cointegration. Hansen and Seo (2002) and Gonzalo and Pitarakis (2006a) provide statistics and asymptotic theory for testing the existence of a threshold effect in the error correction model. Gonzalo and Pitarakis (2006a) also investigate the stochastic properties of the error correction model with threshold effect and in particular the stability conditions. In Seo (2006) a test for linear cointegration is developed for the equilibrium error specified as a SETAR process and the law of one price hypothesis is examined when threshold dynamics is introduced. Kapetanios and Shin (2006) investigate a test similar to that of Seo (2006) for a three-regime SETAR model where the corridor regime follows a random walk. Note that two pitfalls are combined when one wishes to test for a unit root in a threshold model where the transition variable is the lagged dependent variable itself. In addition to the well known issue of non-identification of the threshold parameter under the null hypothesis inherent in TAR models (Hansen 1996, 1997), the transition variable is also nonstationary under the null. Thus, although the model of Caner and Hansen (2001) and Gonzalo and Pitarakis (2006a) covers a large set of processes, it does not include this particular case considered in Seo (2006) and in Kapetanios and Shin (2006). However, as discussed in Caner and Hansen (2001), one could always take the lagged variation of the threshold variable in order to ensure its stationarity. Enders and Siklos (2001) note that this choice could often be suitable in economic models. Further, except for Gonzalo and Pitarakis (2006a), all the above studies assumed a single cointegrating relationship. This underlying assumption simplifies both the estimation procedure and inference. Although determining the number of cointegrating relationships for a set of integrated variables and estimating them has been an important area of research in standard cointegration theory (one can cite Johansen (1988, 1991), Phillips (1991), and Phillips and Durlauf (1986) among others), there are few developments in the threshold cointegrated context. Gonzalo and Pitarakis (2006a) propose a methodology to test for cointegrating rank in a threshold vector error correction model (TVECM) by directly estimating the unknown ranks of the coefficient matrices using a model selection approach introduced by Gonzalo and Pitarakis (1998, 2002). However, an alternative (and more traditional) rank test based on a Lagrange Multiplier (LM)-type procedure could be developed based on the works of Lütkepohl and Saikkonen (2000). This paper is therefore concerned with the estimation and inference procedures for a TVECM with more than one cointegrating relation. For this purpose, we consider the original model of Balke and Fomby (1997) in which the disequilibria are threshold processes (Enders and Siklos, 2001). We show 1 Recall that the SETAR model specifies the lagged dependent variable as the transition variable while the MTAR model uses its lagged variation.

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how a TVECM similar to the one used in Hansen and Seo (2002), Gonzalo and Pitarakis (2006a) and Seo (2006) is obtained in our case where only the loadings switch according to the regime. Estimators of the long-run parameters and loading factors are provided by adapting to our framework, the reduced rank regression method proposed by Ahn and Reinsel (1990) for the standard VECM. A test for the cointegrating rank is developed in this context. We provide the asymptotic distributions of all our estimators and test statistics, and examine the size and power of our tests statistics using Monte Carlo experiments. Both asymptotic and small-sample (Bootstrap based) critical values are derived. Finally our methodology is applied in an empirical study. The remainder of this paper is organized as follows. The model and assumptions are introduced in Section 2. Section 3 focuses on the estimation of parameters and its asymptotic properties. Inference is discussed in Section 4, in particular, a testing procedure for the threshold cointegrating rank is proposed. The asymptotic distributions of the test statistics are derived and tabulated and their approximation in finite samples is discussed. Section 5 points out the modifications to be made (in the estimation as well as in the inference procedures) when an intercept is included in the error correction model. Section 6 is devoted to Monte-Carlo experiments that evaluate the finite-sample performance of our tests and their power. For an improved small-sample inference using our rank test statistics, we perform a constrained bootstrap distribution approximation. In Section 7, we present an empirical application of our procedure to the term structure of interest rates. The last section summarizes our work and concludes.

2

Model and Assumptions

The following general notation is used. A (p × p) identity matrix is denoted by Ip . Whenever the dimension needs to be specified, we will denote a squared (p × p) matrix A as Ap and a (p × m) matrix A as A(p×m) . The projection matrix associated with a (p × m) matrix A will be denoted as PA , the orthogonal complement of A as A⊥ which is a (p × p − m) matrix of full column rank such that A0 A⊥ = 0. We will use I (1) and I (0) to represent time series that are integrated of order 1 and 0, respectively. Throughout the paper, integrals are taken over the unit interval and we use “=⇒” to indicate the convergence in distribution or the weak convergence as sample size T tends to infinity. This section presents the model and the underlying assumptions. Consider 0 0 )0 , for a p-dimensional I (1) process yt = (y1t , ..., ypt )0 partitioned as (ymt , ynt t = 1, ..., T, where ymt and ynt are (m × 1) and (n × 1) respectively (p = m+n). Assume that the generating mechanism for yt is the cointegrated system: A0 yt = umt ,

(1)

∆ynt = unt , where ut = (u0mt , u0nt )0 is an I (0) process. This triangular representation (Phillips, 1991) implies that while ynt is not cointegrated, the first set of equa4

tions represent m cointegrating regressions with cointegrating vectors A0 = (Im , − β) with β a (m × n) matrix of parameters . Let us write system (1) as a VAR process: (Ip − GL) yt = vt , (2)     Im β 0 β where vt = Dut , D = and G = . 0 In 0 In If vt has an autoregressive form B (L) vt = εt , with εt an uncorrelated I (0) process and B (L) a (p × p) polynomial of order q − 1 such that B (L) = Ip − B1 L − ... − Bq−1 Lq−1 , then the VAR representation of (2) is given by A (L) yt = εt , where εt = (ε0mt , ε0nt )0 , A (L) = B (L) (Ip − GL) = Ip − A1 L − ... − Aq Lq . The vector error correction (VEC) representation of (2) is given by ∆yt = Πyt−1 + Λ (L) ∆yt−1 + εt , (3)
Pq−1 Pq i−1 and Λ = − where Π = −A (1), Λ (L) = i i=1 Λi L k=i+1 Ak for i = 1, ..., q − 1 are (p × p) matrices of short-term parameters (Johansen, 1995). The hypothesis of cointegration
implies a matrix Π of reduced rank such that Π = ΓA0 , where Γ0 = −B (1)0 , 0 . By identification, it is straightforward to see that A1 = B1 + G, Ak = (Bk − Bk−1 G) for 2 ≤ k < q, and Aq = −Bq−1 G.

2.1

Threshold cointegration

Following the idea of Balke and Fomby (1997) who assume that the disequilibrium follows a TAR process in a bivariate cointegrated system, here we consider m disequilibria umt = A0 yt which follow a m-dimensional threshold autoregressive (TAR) process of order ` (TAR(`)). Let us define a univariate process st−d which satisfies the following assumptions. ASSUMPTION 1. st−d is a strictly stationary and ergodic sequence, whose distribution F is continuous everywhere, with d > 0, to ensure the predermination of the process.2 The multivariate TAR is introduced in Tsay (1998) and it is defined as follows in the case of two regimes: (Φ1 (L) umt ) I{st−d ≤θ} + (Φ2 (L) umt ) I{st−d >θ} = εmt , where I{·} denotes an indicator function; Φj (L) = Im − (j) φi

(j) i i=1 φi L ,

P`

(4) for j = 1, 2

and are (m × m) diagonal matrices; (εmt )k , the k-th element of εmt , is an iid sequence with zero mean, constant variance and finite 2τ moments for some τ > 2, k = 1, ..., m. The univariate process st−d , satisfying Assumption 1, is called the transition variable (with respect to the threshold θ). Thus, in this model, the magnitude of the transition variable will induce changes in regime. The threshold θ satisfies the following assumption: 2

Note that if d = 0 then the variable st has to be exogenous.

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ASSUMPTION 2. θ ∈ Θ, where Θ is a closed and bounded subset of the sample space of the variable st−d . (1)

Note that Tsay’s formulation (1998) of the multivariate TAR assumes φi (2) and φi to be diagonal matrices so that the stationary conditions of the TAR reduce to the stationary conditions of the process. For examn of each component o (j) (j) (j) ple, for a TAR(1), φ1 = diag φ11 , ..., φ1m for j = 1, 2, the conditions of (1)

(2)

(1) (2)

stationarity are given by Chan et al. (1985): φ1k < 1, φ1k < 1 and φ1k φ1k < 1, for k = 1, ..., m. Under these conditions and according to the definition of cointegration, there are m cointegrating relationships in the system described by (1) and (4), given by the rows of A0 . However, unlike the conventional cointegrated system  considered by Engle and Granger (1987) where the adjustment is (1) (2) symmetric i.e. φi = φi , here the movement towards long-run equilibrium is asymmetric in that the speed of mean reversion differs according to the value of st−d . It is important to note that, in the symmetric case, the assumption of (j) diagonal φi could appear to be restrictive for the dynamics of the disequilibria, in the sense that the process followed by umt will be a VAR without cross effects since Φ1 (L) = Φ2 (L) = Φ (L) will also be diagonal. Unfortunately, it seems difficult to relax this assumption because the stationary conditions of a (j) TAR model are not known when the matrices φi are not diagonal (see Tsay, 1998). Thus the Engle-Granger definition is a special case of the threshold cointe(1) (2) grated system (1) and (4). Note that when ` = 1, the special case φ1 = φ1 = Im implies no cointegration; umt is a pure unit root process.

2.2

Error correction representation

According to the VAR representation (1), the error vt of (2) is given by:   umt + βunt vt = Dut = . unt Assume that umt follows a stationary TAR(`) , unt is an autoregressive process of the form K (L) unt = t , where K (L) = In − K1 L − ... − Kq−1 Lq−1 and t is an uncorrelated stationary process. We can write   J (L) umt + βK (L) unt B (L) Dvt = , K (L) unt where J (L) = (Φ1 (L)) I{st−d ≤θ} + (Φ2 (L)) I{st−d >θ} ,3 and   J (L) −J (L) β + βK (L) B (L) = . 0 K (L) The threshold vector error correction representation (TVEC henceforth) associated with the DGP (1) and (4) is given by the following proposition. 3

Note that the lag operator L does not apply to the indicator function.

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PROPOSITION 1. Let umt be the TAR( `) process of (4) and unt be an AR( q − 1) process. The VEC representation of the VAR model (2), with B (L) defined as above, is given by:   ∆yt = Π(1) yt−1 I{st−d ≤θ} + Π(2) yt−1 I{st−d >θ} + Λ(1) (L) ∆yt−1 I{st−d ≤θ} (5)   + Λ(2) (L) ∆yt−1 I{st−d >θ} + εt , where (for j = 1, 2),   (j) (j) (j) (j) Π(j) = − A0 − A1 − A2 − ... − Amax(q,`) , P  Pmax(q−1,`−1) (j) i−1 (j) max(q,`) (j) Λ(j) (L) = i=1 Λi L , Λi = − A , k=i+1 k ! (j) (j) (j) (j) φ1 β (K1 + In ) − φ1 β A0 = Ip I{·} , A1 = I{·} , 0 (K1 + In ) ! (j) (j) (j) φi β (Ki − Ki−1 ) − φi β Ai = I{·} , f or 2 ≤ i ≤ min (q − 1, `) , 0 (Ki − Ki−1 ) and when min (q − 1, `) = ` :   0 β (Ki − Ki−1 ) (j) I{·} , f or ` + 1 ≤ i ≤ q − 1, Ai = 0 (Ki − Ki−1 )   0 −βKq−1 (j) Aq = I{·} , 0 −Kq−1 when min (q − 1, `) = q − 1 : A(j) q = (j) Ai

(j) φi

0 (j)

=

φi 0

(j) −βKq−1 − φi β −Kq−1 (j)

−φi β 0

! I{·} ,

! I{·} , f or q + 1 ≤ i ≤ `.

The hypothesis of cointegration implies
matrices Π(j) of reduced rank such that 0 Π(j) = Γ(j) A0 with Γ(j) = −Φj (1)0 00 . Proof. See Appendix A.1.  The threshold effect is active only if 0 < P (st−d ≤ θ) < 1, otherwise the model is reduced to the standard VECM, i.e. Π(1) = Π(2) . The above TVECM (5) is similar to the model used by Gonzalo and Pitarakis (2006a), Hansen and Seo (2002), and Seo (2006) except that here, the threshold affects all the coefficients. However, this is true only for an order of TAR greater than 1. Indeed, for ` = 1, the TVECM does not include threshold effect  on the coefficients of P (j) (j) (j) do not include lags of ∆yt since Λ1 = − i≥2 Ai and the matrices Ai i≥2

the parameters of TAR(1) in this case. 7

For simplicity of presentation and without loss of generality, we restrict our investigation to a TAR(1) on the equilibrium errors which could be written in a first difference form as follows: ∆umt = −Φ1 (1) umt−1 I{st−d ≤θ} − Φ2 (1) umt−1 I{st−d >θ} + εmt ,

(6)

for which the stationary conditions are well known. This TAR(1) could be augmented by adding lags on ∆umt as in Caner and Hansen (2001) in order to ensure the absence of correlation in the error term process. Indeed, the iid property of εmt is needed to use the weak convergence result of BT (F (θ)) = P[T r] T −1/2 t=1 I{F (st−d )≤F (θ)} εmt established in Caner and Hansen (2001). Gonzalo and Pitarakis (2006b) discuss how one can deal with serial correlation in εmt . They generalize this result taking εmt to be a finite order moving average process. In this paper, we will only deal with an iid error process εmt to derive our asymptotic properties. Given (6), the TVECM is then: ∆yt = Π(1) yt−1 I{st−d ≤θ} + Π(2) yt−1 I{st−d >θ} + Λ (L) ∆yt−1 + εt ,

(7)

 P 0 β q−1 Ki Li−1 i=1 Pq−1 . Thus, the threshold only affects the where Λ (L) = i−1 0 i=1 Ki L
loading factors Γ(j) in this error correction mechanism. Because we have ∆yt I{st−d ≤θ} + ∆yt I{st−d >θ} = ∆yt , the latter TVECM can be splitted according to both the regimes with each ∆yt I{·} following a moving-average representation: ∆yt I{·} = Cj (L) εt I{·} , for j = 1, 2, and where Cj (L) could be decomposed as Cj (L) = Cj (1) + (1 − L) Cj∗ (L) . Therefore the representation theorem (Engle and Granger, 1987, Johansen 1995) holds for each regime and the solution is given by:
Pt yt = y0 + C1 (1) I{st−d ≤θ} + C2 (1) I{st−d >θ} i=1 εi + St , 

 −1 (j)0 (j)0 Γ⊥ has rank p − m, with Ψ (L) = Ip − where Cj (1) = A⊥ Γ⊥ Ψ (1) A⊥
Pq−1 i ∗ ∗ i=1 Λi L , for j = 1, 2, and St = C1 (L) I{st−d ≤θ} + C2 (L) I{st−d >θ} εt .

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Estimation

The triangular normalization enables us to estimate the parameters of the vector correction model as in Ahn and Reinsel (1990). Rewriting (7) as:



∆yt = Γ(1)

(1)

  A0 0  (2) Γ 0 A0

yt−1 (2) yt−1

! +

Pq−1

i=1 Λi ∆yt−i

or ∆yt = ΓBYt−1 +

Pq−1

i=1 Λi ∆yt−i

8

+ εt .

+ εt ,

(8)

(1)

(2)

where yt = yt I{st−d ≤θ} and yt = yt I{st−d >θ} . Since the threshold does not affect the coefficients of the lagged variations in our TVECM, let us concentrate
out the short-term parameters Λi by regressing ∆yt and yt−1 on ∆yt−1 , ..., ∆yt−(q−1) . ∗ Let us denote as ∆yt∗ and yt−1 the residuals of the previous regressions and let ∗(1)

∗(1)

yt = yt∗ I{st−d ≤θ} and yt tion part of (7), we have:

= yt∗ I{st−d >θ} . Thus, focusing on the error correc-

∆yt∗ = ΓBY∗t−1 + ε∗t , (9)   Ωm Ωmn where V (ε∗t ) = Ω partitioned as . Vectorizing the last expresΩnm Ωn sion, we get:
∗0 ∆yt∗ = Yt−1 ⊗ Γ vec (B) + ε∗t . Let vec (B) = Hϕ + h, where H is (4mp × mn) selection matrix which selects the elements of ϕ = vec (β) consisting of long run parameters and h a 4mpdimensional vector which selects the ones in A. Under normality of εt , the log-likelihood function is given by: LT (Γ, ϕ, Ω, θ) = const. − T2 ln |Ω| −



0 −1 PT 1 ∗0 ∗0 ∗ ⊗ Γ (Hϕ + h) , ∆yt∗ − Yt−1 t ∆yt − Yt−1 ⊗ Γ (Hϕ + h) Ω 2

Hence the maximum likelihood estimator of long-run parameters, ϕ, ˆ given Γ, θ and Ω is:
0

P ∗0 ∗0 ⊗ Γ Ω−1 ∆yt∗ − Yt−1 ⊗Γ h , (10) ϕˆ (θ) = C −1 H 0 Tt Yt−1 hP
0 −1 ∗0
i T ∗0 with C = H 0 Yt−1 ⊗ Γ H a (mn × mn) matrix, and its t Yt−1 ⊗ Γ Ω variance is given by V (ϕˆ (θ)) = C −1 . A feasible estimator for ϕ is obtained by plugging in consistent estimators of Γ and Ω in (10). The
feasible estimator ϕˆ of the long-run parameters thus obtained is Op T −1+δ for 0 < δ ≤ 1. In order to derive its limiting distribution, we make some additional assumptions and get some preliminary results. Recall that the support of threshold variable, Θ, was assumed to be a   closed and bounded set : Θ = θ , θ . It is convenient to trim it such that
P (st−d ≤ θ) = τ > 0 and P st−d ≤ θ = 1 − τ. Let Ut = F (st ) which follows a uniform distribution U[0,1] . Due to the equality I{st−d ≤θ} = I{F (st−d )≤F (θ)} , we will use u = F (θ) ∈ [ τ , 1 − τ ] (See Caner and Hansen, 2001). Then, if Ut and ε∗t satisfy Assumption 3 given in Appendix A.2, Theorems 1-3, of Caner and Hansen (2001, pp. 1560-1561) could be applied to obtain the limiting distribution of the estimators of long-run parameters. THEOREM 1. Let ϕˆ (θ) be the estimator obtained from (10) for a given threshold θ, loading parameters Γ, covariance matrix Ω and short-term parameters Λ. Then, under Assumption 1 to 3 (See Appendix A.2), we have  

−1 0 T (ϕˆ (θ) − ϕ) =⇒ H 0 D (r, u) ⊗ Γ0 Γ H H vec Γ0 Ω−1/2 Q (r, u) , 9


R dB (r, u) B (r)0 , dB (r, 1 − u) B (r)0 , is a (p × 2p) ma  R u 0 trix and D (r, u) = ⊗ B (r) B (r)0 dr, a 2p-squared matrix, with 0 1−u B (r, u) and B (r) being standard p-dimensional two-parameter and one-parameter Brownian motions respectively. where Q (r, u) =

R

Proof. See Appendix A.2.



To sum up, for a given threshold, the iterative estimation procedure can be outlined as follows: For a given threshold. (1) (2) ∗ Step  1. Γ and0Γ can be consistently estimated from a regression of ∆yt ∗(1)0 ∗(2)0 ∗(1) ∗(2) on yt−1 , yt−1 , from ∆yt∗ = Π(1) yt−1 + Π(2) yt−1 + ε∗t . The triangular

normalisation implies that the first m columns of Π(1) and Π(2) are equal to ˆ (1) and Π ˆ (2) as Γ(1) and Γ(2) respectively. Thus, one can use these columns of Π estimators of the loading factors Γ(1) and Γ(2) . Step 2. Using the consistent estimator of Ω from the estimated varianceˆ (θ) = T −1 PT εˆ∗t εˆ∗0 , we obtain covariance matrix of the residuals from step 1: Ω t t a feasible estimator of ϕ from (10) . Step 3. With ϕ, ˆ one can re-estimate Γ(j) by running regression (9) . For an unknown threshold. When the threshold is unknown, it has to be searched over a grid on [ τ , 1 − τ ] . In practice, this grid is given by the values taken by the selected transition variable. However, the choice of τ is somewhat arbitrary; empirical investigations take τ to be 5, 10 or 15 percent. In general, it is chosen in order to ensure that there are enough observations in each regime and that the limits for the test statistics used in inference are nondegenerate (See Andrews 1993, Hansen, 1996). In practice, various values must be tried in empirical applications to check the robustness of the results to the selected value of τ. ˆ The concentrated likelihood function is LT (θ) = −0.5T log Ω (θ) − 0.5T p, ˆ and so an estimator of θ is given by θˆ = arg min log Ω (θ) (see Tsay, 1989, Enders and Siklos, 2001, Hansen and Seo, 2002). Statistical inference on the TVECM requires estimators of loading matrices and their asymptotic properties. As shown earlier, when the disequilibrium terms are assumed to be a m-dimensional TAR(1) process (m cointegrating relations), the loading vectors are such that the first block (of dimension m) (j) (j) is given by Γm ≡ −Φ1 (L) whereas the lower block is Γn ≡ 0, for j = 1, 2. ∗ = ε∗ . Therefore, the n last equations of the TVECM are reduced to ∆ynt nt

10

(j)

(j)

(j)

Letting umt = A0 yt , for a given A, Γm can be estimated by least squares on the first m equations of the TVECM:4 ∗ (1) (2) (2) ∗ ∆ym = u(1) m Γm + um Γm + εm ,

(11)

(j)

(j)

with V (ε∗m ) = Ωm ⊗ IT , and where um stacks the observations on umt−1 for t = 1, ..., T. We obtain:       ˆ (j) − vec Γ(j) = Im ⊗ (u(j)0 Mj 0 u(j) )−1 u(j)0 Mj 0 vec (ε∗ ) vec Γ m m m m m m  (j 0 )0 (j 0 ) −1 (j 0 )0 um um um . As εmt is of zero mean and covariance        (j) (j) ˆ (j) ˆ matrix Ωm , we have vec Γ − vec Γ = o (1) and V vec Γ = m m m p   −1  (j) (j)0 (j) (1) (2) ˆ (j) is Ω−1 . Since um and um are I (0), vec Γ m − Γm m ⊗ (um Mj 0 um )
−1/2 Op T . Therefore   −1  √ ˆ (j) − Γ(j) ) =⇒ N 0, Ωm ⊗ u(j)0 Mj 0 u(j) T vec(Γ , m m m m (j 0 )

with Mj 0 = IT − um

4



Testing

4.1

Testing no-cointegration

As the threshold is unknown, Gonzalo and Pitarakis (2006) propose to use the supremum Wald statistic to test the null hypothesis of no-cointegration H0 : 

0
0 0 Rπ = 0 where π = vec Π(1) vec Π(2) is a 2p2 -vector and R = Ip2 − Ip2 is the selection matrix:5 sup W0 =

sup

W0,T (u) ,

(12)

u∈[τ , 1−τ ]

and they provide the limiting distribution of this statistic under the null H0 : Π(1) = Π(2) = 0: sup W0 =⇒

sup u∈[ τ ,1−τ ]

1 u(1−u) tr

n R

0
0

B (r) dK (r, u)

R

B (r) B (r)

0
−1

R

0


B (r) dK (r, u)

where B (r) denotes here, a p-dimensional standard Brownian motion, K (r, u) is a Kiefer process given by K (r, u) = B (r, u) − uB (r, 1) with B (r, u) a pdimensional two-parameter standard Brownian motion. (j) ˆm Recall that according to the assumption that Φj (L) is diagonal, the sub-matrix Γ will (j) also be diagonal. This restriction on the first m columns and rows of Γ comes from the triangular specification of the system for which the last n rows of Π(j) in the TVECM are zeros. 5 See Davies (1987), Andrews and Ploberger (1994) and Hansen (1996) for a discussion on supremum statistics. 4

11

o

,

4.2 4.2.1

Rank test procedure Threshold unit root and asymmetric cointegration rank

The inference here is trickier than for a system with only one cointegrating vector considered by Enders and Siklos (2001), Hansen and Seo (2002) or more recently by Seo (2006). The simplest configuration occurs when we have the same number of cointegrating

relationships active in both regimes. Formally it means rk Π(1) = rk Π(2) ≤ p − 1 (where rk (·) stands for the rank func(1) (2) tion), with
Π 6=(2)Π
. According to the DGP given by (1) and (6), we have (1) rk Π = rk Π = m in the TVECM. This is a relatively straightforward generalization of the asymmetric adjustment for more than one cointegrating relationship. The problem becomes more complicated if we consider a possible unit root in one of the regimes of the TAR process. This possibility was initially considered in univariate TAR models by Balke and Fomby (1997) and González and Gonzalo (1998) and more recently studied by Caner and Hansen (2001) and Kapetanios and Shin (2006). For instance in a TAR(1) model, this (j) (j 0 ) local nonstationary feature occurs when φik = 1 and φik < 1 for j 6= j 0 and for some k ∈ {1, ..., m} , such that the k-th univariate TAR behaves like a unit root process in one regime. Caner and Hansen (2001) called this process a partial unit root process. González and Gonzalo (1998) investigate threshold unit root (TUR) processes and provide various useful theoretical results. In particular, they show that even when one of the two regimes has a unit root, a TUR(1) process could be covariance stationary under Assumption 3, Appendix A.2, and other additional assumptions (see González and Gonzalo, 1998). See also Gonzalo and Pitarakis (2006a) for a discussion on the covariance stationary property in presence of a unit root in one regime. Note that the presence of unit roots in the multivariate threshold model for the disequilibrium induces some consequences on ranks of Π(j) matrices in the TVECM. Following the terminology of Caner and Hansen (2001), when the disequilibrium is a TUR process, this situation can be called
one of partial cointegration.
Let us assume that rk Π(1) = p − 1 > rk Π(2) . This means that at least one column of Π(2) is equal to zero. In other words, some cointegration relationships are inactive in the second regime, i.e. one component of umt behaves like a unit root process in the second regime. In the general case of S regimes, if the TAR(`) has S unit roots (at most one unit root per regime), then S columns of the loading factor matrices Γ(j) (and hence S columns of Π(j) ) (j) will be null. This statement comes from the assumption that φi are diagonal P 0 (j)0 ` matrices Γ(j) = − I m , 00 . i=1 φi As an example, consider a 4-dimensional integrated system (1) with three cointegration relationships, i.e. yt = (y1t y2t y3t y4t )0 , A0 = (I3 − β) , where β is (3 × 1), ∆y4t = u4t with u4t is a (finite autocorrelated) stationary process, and where the vector umt = (u1t u2t u3t )0 is described by a trivariate TAR(1) process as follows:

12

(1) (1)

(2) (2)

(1) (1)

(2) (2)

(1) (1)

(2) (2)

∆u1t = ρ1 u1t−1 (u) + ρ1 u1t−1 (u) + ε1t , ∆u2t = ρ2 u2t−1 (u) + ρ2 u2t−1 (u) + ε2t , ∆u3t = ρ3 u3t−1 (u) + ρ3 u3t−1 (u) + ε3t ,   (j) (j) (1) (2) where ρi = φi − 1 , with the stationarity conditions ρi < 0, ρi < 0    (1) (2) (2) and ρi + 1 ρi + 1 < 1 for i = 1, 2, 3. Now, if we assume ρ2 = 0   (2) i.e. φ2 = 1 , then Π(2) = Γ(2) A0 will look like this: 

   (2) (2)   0 0 γ11 0 0 γ11 β11   1 0 0 β11   0 0  0   0 0 0   0 1 0 β Π(2) =  =   , 21 (2) (2) (2)     0 γ33 0 0 γ γ β 31 33 33 0 0 1 β31 0 0 0 0 0 0

such that rk Π(2) = 2 whereas rk Π(1) = 3.
Thus, for a given A, if rk (A) > rk Π(j) for some regime j = 1, 2, it means that the TAR process on the equilibrium errors zt has a unit root. Gonzalo and Pitarakis (2006a) propose to directly estimate the unknown ranks of the Π(j) matrices using a model selection approach introduced by Gonzalo and Pitarakis (1998, 2002). Here we propose an alternative methodology based on the works of Saikkonen and Lütkepohl (2000) and Lütkepohl and Saikkonen (2000). We use a Wald statistic to test the rank (m > 0) for each regime. Our procedure tests H0 : m = r0 vs. Ha : m > r0 (or m = r0 + 1). 4.2.2

(2)

γ11 0 0 0

Test Statistics

Noting that PA + PA⊥ = Ip , inserting in equation (9) and splitting the two regimes, we can write for a given matrix A (and A⊥ ): ∆yt∗ = (κ1 umt−1 + λ1 νt−1 ) I{st−d ≤θ} + (κ2 umt−1 + λ2 νt−1 ) I{st−d >θ} + ε∗t (13) where κj = Π(j) A (A0 A)−1 , λj = Π(j) A⊥ (A0⊥ A⊥ )−1 , νt = A0⊥ yt∗ . Since the ∗ both regimes are independent, we can deal with each regime separately: ∆y t =
∗ + ∆y ∗ . Then, for the regime j, under H : m = r = rk Π(j) , we ∆y1t 0 0 2t have λj = 0 and κj = Γ(j) , whereas, under the alternative, some columns of A⊥ will be cointegrating vectors so that λj 6= 0. Thus, one can base the rank test on testing H0 : λj = 0. For an easier implementation of the test, following (j)0 Lütkepohl and Saikkonen (2000), we multiply each equation by Γ⊥ to give:  ˜ 1 νt−1 + ε˜∗ if st−d ≤ θ, κ ˜ 1 umt−1 + λ ∗ t ∆˜ yt = (14) ˜ 2 νt−1 + ε˜∗ if st−d > θ, κ ˜ 2 umt−1 + λ t (j)0

where for any variable x, we denote as x ˜ = Γ⊥ x and for any parameter a, we (j)0 ˜ j = 0. The transformation denote as a ˜ = Γ⊥ a. Thus, the null becomes H0 : λ 13

(j)0

by Γ⊥ is convenient because then the limiting distribution of the resulting test statistic will only depend on p − m. On the other hand, if one uses (13) for the test, the limiting distribution of the test statistic will depend on p and m, which is cumbersome for tabulation. Piling up the observations, we have the following regression model for each regime ˜ 0 + ε˜∗ , for j = 1, 2, ∆˜ yj∗ = u(j) ˜ 0j + ν (j) λ m κ j j where ∆˜ yj∗ and ε˜∗j are (T × p − m) matrices which stack the observations on (j)

(j)0

∗0 and ε (j) stack u ∆˜ yjt ˜∗0 jt respectively. The (T × m) um and (T × p − m) ν mt−1 (j)0

and νt−1 respectively. For a given A (and hence A⊥ ), partitioned regression results yield :      
ˆ j − vec λj = Ip−m ⊗ (ν (j)0 M ν (j) )−1 ν (j)0 M vec ε˜∗ , vec λ j   (j)0 (j) −1 (j)0 ˜ 0 , M = IT − P (j) , with P (j) = u(j) u where λj = λ u um . Under the m m m j u u   null H0 : vec λj = 0, the Wald test statistic is given by:  0   −1   (j) ˆ j V vec λ ˆj ˆj . WT (u, m) = vec λ vec λ      (j)0 (j) = Γ⊥ Ωj Γ⊥ ⊗ IT with Ωj = E I{·} ε∗j ε∗0 Recalling that V vec ε˜∗j j , we get    (j)0 (j) ˆj V vec λ = Γ⊥ Ωj Γ⊥ ⊗ (ν (j)0 M ν (j) )−1 . Hence, the full expression of Wald statistic is given by: (j)

WT (u, m) = vec ε˜∗j


0 

(j)0

(j)

Γ⊥ Ωj Γ⊥

−1

  −1
⊗ M ν (j) ν (j)0 M ν (j) ν (j)0 M vec ε˜∗j .

As the threshold is unknown, we take the supremum Wald statistic over a grid set of possible values of the threshold: sup W (j) (m) =

sup u∈[τ 1−τ ]

(j)

WT (u, m) .

˜ j = 0), we will use the estimators In order to compute the test statistics (H0 : λ (j) (j) derived in Section 3, producing a feasible version with νˆt−1 = Aˆ0⊥ yt−1 in (14) (j)0 (j) ˆ (j)0 Ω ˆjΓ ˆ (j) in the Wald statistic. and replacing Γ Ωj Γ by Γ ⊥

4.2.3

⊥

⊥

⊥

Asymptotic distributions

We now derive the asymptotic distribution of the Wald statistic. For this purpose, it is convenient to write it as sup W

(j)

(m) =

sup

tr

u∈[τ , 1−τ ]



(j)0 (j) Γ ⊥ Ωj Γ ⊥

−1  0  −1   (j)0 ∗ (j)0 (j) (j)0 ∗ ν M ε˜j ν Mν ν M ε˜j .

14

THEOREM 2. Let Assumptions 1 to 3 hold. Then, under H0 , the limiting distributions of the sup-Wald statistic in regimes 1 and 2 are : sup W (j) (m) =⇒

T (j) (u) , for j = 1, 2 ,

sup u∈[τ , 1−τ ]

where T (1) (u) and T (2) (u) are defined as: o
−1 R R 0 0 0 V (r) dB (r, u) , dB (r, u) V (r) u V (r) V (r) dr o nR
−1 R R 0 0 0 V (r) dB (r, 1 − u) , T (2) (u) = tr dB (r, 1 − u) V (r) (1 − u) V (r) V (r) dr T (1) (u) = tr

nR

and B (r, u) and V (r) denote a standard (p − m)-vector two-parameter Brownian motion and a standard (p − m)-vector Brownian motion, respectively. Proof. See Appendix A.3.



The asymptotic distributions of Theorem 2 are calculated by Monte-Carlo simulations. The stochastic integrals are evaluated at 10,000 points over the argument r and 100 steps over the argument u. The critical values are computed as the empirical quantiles from 10,000 replications and reported in Table 1 for various values of p − m. Note that the critical values for sup W (1) and sup W (2) are the same due to the symmetry of both statistics. These values are reported for various ranges [ τ , 1 − τ ]. Caner and Hansen (2001) discuss the inconsistency of the tests when one chooses τ = 0 in threshold models. Indeed, because the critical value of the statistics increases as τ decreases, the rejection of the null requires a larger value of the statistics as τ tends to 0. It follows that τ should be set in the interior of (0, 1) . As it had been mentioned in the previous section, the choice of τ is somewhat tricky. In general, empirical studies try various values like 5, 10 or 15 percent. Table 1: Critical values 90% p−m [τ, 1 − τ ] [0.15, 0.85] 5.914 [0.10, 0.90] 6.210 [0.05, 0.95] 6.568 p−m [τ, 1 − τ ] [0.15, 0.85] [0.10, 0.90] [0.05, 0.95]

for the sup-Wald statistic 95% 99% 90% 1 7.446 7.743 8.043

10.601 10.854 11.194

14.209 14.614 14.909

4 39.503 40.084 40.776

42.810 43.503 43.918

95% 2

99%

90%

95% 3

99%

16.347 16.661 17.079

20.689 21.034 21.522

25.183 25.673 26.183

27.885 28.371 28.907

33.134 33.530 33.903

5 48.925 49.475 50.063

57.610 58.468 59.323

Note: Calculated from 10,000 simulations.

15

61.675 62.233 63.325

6 69.147 69.823 70.807

79.387 80.581 81.795

83.293 84.629 86.020

92.240 93.552 94.921

4.3

Bootstrap

It is known that asymptotic tests tend to be strongly biased in small samples in a threshold modelling framework (Caner and Hansen, 2001, Seo, 2006). Therefore, the distributions given in Theorem 2 should be approximated in order to investigate the finite sample properties of the test statistics. For this purpose, we follow the residual-based developed in Seo (2006) for a similar sup-Wald statistic. It consists in resampling the residuals of model (14) independently and with replacement. However, the bootstrap generation issue becomes quite thorny when some of the time-series involved are non-stationary (m < p). This is the case under our null hypothesis. Indeed, since the purpose of our bootstrap is the distribution of the sup-Wald statistic under the null, it is natural to perform the resampling from the restricted regression (14), ˜ j = 0, otherwise the bootstrap distribution will not be consistent i.e. with λ for the correct sampling distribution (Caner and  Park 2002).  Hansen, 2001, (j) ˆ εˆt , that are estiˆ Γ ˆ , θ, Then, we fit regression (9) and with the inputs A,
mates of A, Γ(j) , θ, εt , we obtain the bootstrap samples for εt , say εbt , that T 
P so that Eb εbt = 0 a.s., where Eb is are samples from εˆt − T −1 Ti=1 εˆi t=1 the expectation with respect to Pb , the probability measure which
∞provides the bootstrap probability space conditional on the realization of εbt t=1 . Using εbt b , say and starting from an initial value we obtain the bootstrap samples for yjt b . Let λ ˆ b be the estimator of λb from the regression of yjt j j b b I{·} + κbj ubmt−1 I{·} + t , ∆˜ yjt = λbj A0⊥ yjt−1



(15)



ˆ b , finally the bootstrap verand S b λˆ bj be the bootstrap covariance matrix for λ j 

0



−1





ˆb . sion of the sup-Wald statistic is defined as sup WT(j)b = vec λˆ bj S b λˆ bj vec λ j The distribution of this bootstrap statistic can be obtained by repeating the generation of bootstrap samples and collecting the test statistic values for each repetition. This distribution is regarded as an approximation of the distribution of the statistic under the null. The asymptotic convergence results of the bootstrap distribution for a similar statistic can be found in Seo (2006, Theorem 3, p.136).

5

TVECM with an intercept

So far we assumed that the DGP had zero mean in order to arrive at the results without complicating notations. However it is not a realistic assumption in practice where it is reasonable to include an intercept such that the TVECM becomes: ∆yt = µ + Π(1) yt−1 I{st−d ≤θ} + Π(2) yt−1 I{st−d >θ} + Λ (L) ∆yt−1 + εt ,

(16)

where µ is a p-vector. Then, from the representation theorem discussed in the second section for a TVECM, the constant drift will generate a linear trend 16

term in the process. Thus, yt has the representation:
Pt yt = y0 + C1 (1) I{st−d ≤θ} + C2 (1) I{st−d >θ} i=1 εi

(17)


+ C1 (1) I{st−d ≤θ} + C2 (1) I{st−d >θ} µt + St , (1)0

(2)0

Thus, if Γ⊥ µ = 0 and Γ⊥ µ = 0, the trend in yt disappears, however, this implies that µ is in the space of Γ(1) in regime 1 and Γ(2) in regime 2. Then µ could be written as Γ(1) β0 in the first regime and Γ(2) β˜0 in the second one.6 The TVECM can then be written as:   ∆yt = Γ(1) (β0 + A0 yt−1 ) I{st−d ≤θ} + Γ(2) β˜0 + A0 yt−1 I{st−d >θ} + Λ (L) ∆yt−1 + εt . (1)0

(2)0

However, if the restrictions Γ⊥ µ = 0 and Γ⊥ µ = 0 are not satisfied, the intercept will be both in the error correction term and as a autonomous growth component in ∆yt . In this more general setting, using PΓ(j) + PΓ(j) = Ip , we can decompose the ⊥

Γ(j)

(j)

parameter µ in (16) in the directions of and Γ⊥ , j = 1, 2, as follows:     (1) (2) µ = Γ(1) I{st−d ≤θ} + Γ(2) I{st−d >θ} β0 + Γ⊥ I{st−d ≤θ} + Γ⊥ I{st−d >θ} γ0 , where  −1 −1  β0 = Γ(1)0 Γ(1) Γ(1)0 µI{st−d ≤θ} + Γ(2)0 Γ(2) Γ(2)0 µI{st−d >θ} ,     (1)0 (1) −1 (1)0 (2)0 (2) −1 (2)0 γ 0 = Γ⊥ Γ⊥ Γ⊥ µI{st−d ≤θ} + Γ⊥ Γ⊥ Γ⊥ µI{st−d >θ} . Therefore the general model covering the two possibilities is:  
(1) (2) ∆y t = Γ(1) I{st−d ≤θ} + Γ(2) I{st−d >θ} β0 + Γ⊥ I{st−d ≤θ} + Γ⊥ I{st−d >θ} γ0 + Π(1) yt−1 I{st−d ≤θ} + Π(2) yt−1 I{st−d >θ} + Λ (L) ∆yt−i + εt . It is important to distinguish between the two models that are compatible with (16) because the critical values of the test are altered according to the considered scheme. While the presence of an intercept in the model does not cause any trouble for the estimation procedure, the asymptotic distributions of the test statistics need minor analytical transformations. Thus, for estimation, the intercept is merely concentrated out along with the short-term parameters (1)0 (2)0 Λi . Under the unrestricted hypothesis (Γ⊥ µ 6= 0 and Γ⊥ µ 6= 0), the intercept of the TVECM switches according to the regime as following: ∆y t =PΓ(1) µI{st−d ≤θ} + PΓ(2) µI{st−d >θ} + Γ(1) (β0 + A0 yt−1 ) I{st−d ≤θ} + ⊥ ⊥ P Γ(2) (β0 + A0 yt−1 ) I{st−d >θ} + q−1 Λi yt−i + εt . i Theorem 2 still holds except that the Brownian motion V (r) should be replaced by the corresponding demeaned Brownian motion defined as V ∗ (r) =
0
0 Since Γ(1) = −Φ1 (1)0 00 and Γ(2) = −Φ2 (1)0 00 , we can write Γ(1) = Γ(2) P, with a suitable matrix P , and hence if µ = Γ(1) β0 then µ = Γ(2) P β0 = Γ(2) β˜0 . 6

17

V1 (r) −

R

V1 (r) dr, ..., Vp−m−1 (r) −

R

Vp−m−1 (r) dr, r −

(1)0

(2)0


1 0 2

(j)0

when Γ⊥ µ 6= 0

for j = 1, 2, whereas, if Γ⊥ µ = 0 and Γ⊥ µ = 0, V (r) it should be replaced by V ∗∗ (r) = (V1 (r) , ..., Vp−m (r) , 1)0 (See Johansen, 1995). Corresponding critical values are given in Tables 2 and 3 below. (j)

Table 2: Critical values for sup-Wald statistic, for model with Γ⊥ µ 90% 95% 99% 90% 95% 99% p−m 1 2 [τ, 1 − τ ] [0.15, 0.85] 10.355 12.334 16.420 20.916 23.259 28.091 [0.10, 0.90] 10.638 12.633 16.808 21.268 23.697 28.335 [0.05, 0.95] 11.014 12.885 17.066 21.643 24.072 28.778 p−m [τ, 1 − τ ] [0.15, 0.85] [0.10, 0.90] [0.05, 0.95]

4 51.302 52.368 53.387

54.812 55.733 56.893

= 0 for j = 1, 2. 90%

95% 3

99%

34.334 35.026 35.642

37.373 38.042 38.654

43.018 43.828 44.341

5 61.879 63.021 63.543

71.810 73.191 74.613

76.181 77.391 78.747

6 84.622 86.084 87.413

96.137 98.088 99.933

100.940 102.908 105.177

111.178 112.854 114.834

(j)

Table 3: Critical values for sup-Wald statistic, for model with Γ⊥ µ 6= 0 for j = 1, 2. p−m [τ, 1 − τ ] [0.15, 0.85] [0.10, 0.90] [0.05, 0.95] p−m [τ, 1 − τ ] [0.15, 0.85] [0.10, 0.90] [0.05, 0.95]

90%

95% 1

99%

90%

95% 2

99%

90%

95% 3

99%

8.891 9.145 9.408

10.830 11.099 11.325

14.797 14.985 15.163

14.932 15.283 15.561

17.073 17.360 17.677

21.326 21.712 22.001

26.161 26.582 26.965

28.705 29.131 29.539

34.324 34.870 35.015

4 41.209 41.764 42.407

44.418 44.996 45.535

5 50.355 51.049 51.766

58.844 59.927 60.933

62.906 63.980 64.728

6 70.300 71.014 72.056

80.586 81.932 83.586

85.184 86.591 88.067

Note: Calculated from 10,000 simulations.

6 6.1

Monte-Carlo experiments Size

A Monte-carlo experiment is performed to investigate the small-sample properties of the proposed tests and to compare them with other tests for the cointegrating rank. We simulate a three-dimensional TVECM according to DGP (1) with a multivariate TAR(1) on umt , for m = 1, 2. Thus, the TVECM is the

(1) (2) (1) (2) same as (8) with K (L) = 0, Γ 6= Γ and rk Γ = rk Γ = m. This gives: (1) (2) ∆yt = Γ(1) A0 yt−1 + Γ(2) A0 yt−1 + εt , (18) 18

93.823 95.674 96.878

where yt = (y1t , y2t , y3t )0 and εt = (ε1t , ε2t , ε3t )0 . The exogenous and stationary transition variable is chosen to be an AR(1) process. We test H0 : r0 = 1 and 2 for a sample of size T = 100 generated from εt iid N (0, I3 ) . Table 4 reports rejection frequencies of the test statistic at 5% level from 1000 MonteCarlo replications using asymptotic and bootstrap critical values. The latter are obtained as described in the previous section from 5000

bootstrap replications. (1) (2) The null hypothesis H0 : rk Γ = rk Γ = r0 is rejected when
rk Γ(j) = r0 is rejected for at least one regime j = 1 or 2. The experiments are conducted for a threshold bound set with τ = 0.15. The rejection frequencies obtained with the asymptotic critical values are very large. This problem of over-rejection of the sup-Wald statistic has been already noticed in Seo (2006) and remains even when the sample increases. However, with the bootstrap critical values, small-sample properties are reasonable. Note that the rejection frequencies (based on the asymptotic critical values) for the likelihood ratio (LR) test of the null hypothesis that there are (at most) r0 cointegration relations against the alternative that only one additional cointegrating vector exists (say the λmax statistic of Johansen, 1995) are 5.1% and 6.6% for r0 = 1 and r0 = 2, respectively. Then, both tests mildly over-reject though our statistic over-rejects slightly more than the λmax statistic. Table 4: Size of Rank Tests Asymptotic Bootstrap H0 : m = r0 critical values critical values m = 1, r0 = 1 0.332 0.078 0.218 0.112 m = 2, r0 = 2 Note: T = 100. Nominal size 5%. Rejection rates from 1000 replications.

6.2

Power

We now investigate the our
power of
test under the
following three
alternatives (1) (2) (1) (2) on (18): a) rk Γ = rk Γ = 1, b) rk Γ = rk Γ = 2, and c)

rk Γ(1) = 1, rk Γ(2) = 2. Table 5 reports the power and the size-corrected power under these alternatives (i.e. using the asymptotic critical values and the

bootstrap critical values). The results show that when rk Γ(1) = rk Γ(2) , (alternatives (a) and (b)), the performance of both tests are similar. Note that the alternative associated with the test H0 : r0 = 2 is full rank of Γ(1) and Γ(2) (i.e. r0 = 3) . Under this hypothesis, the system is stationary: there are no I (1) variables among yt . The second power experiment is a more interesting one in our context. The DGP now is the partial unit root case (alternative (c)). The first disequilibrium error, u1t , follows a TUR(1) process i.e. it behaves like a unit root process in regime 1 whereas it is stationary in the second regime. On the other hand, the second disequilibrium error u2t is a stationary TAR(1) process. We also report in Table 5 the power (based on the asymptotic critical values) for the λmax statistic. While, as expected, our test does a much better job than the LR test in small samples when the cointegration ranks differ (note that LR

19

is not supposed to do a good job in such a context), the LR test is still powerful in a threshold context when the ranks are the same which is surprising. Table 5: Power of Rank Tests

DGP H0 : r0 m = 2, r0 = 1 m = 3, r0 = 2 DGP (1) H0 : r0 = 1

sup-Wald Asymptotic critical values
Bootstrap
critical values (1) (2) rk Γ = rk Γ =m

a

1.00 1.00

0.999 1.00



rk Γ(1) = 1 6= rk Γ(2) = 2

1.00

1.00

λmax

0.997 1.00

0.307

Note: T = 100. Nominal size 5%. Rejection rates from 1000 replications. When m = p = 3, Γ(j) is of full rank matrix and so the system is stationary. (1) The rejection rate of H : r = 2 is not reported in this part as this test has no 0 0 meaning here given that the DGP is the alternative (c).

a

7

Application: the term structure of interest rates

Studies using cointegration tests for an empirical verification of economic “laws” involving nonstationary variables have grown rapidly in recent years. The area of international finance is particularly rich in such relationships that have important implications for policy design and evaluation. Expectation Hypothesis of Term Structure of interest rates (EHTS) is an example of these “laws” that a policy-maker would be interested in knowing whether they can be assumed to be true or not in practice. The term structure of interest rates deals with the relationship between long-term and short-term interest rates and predicts that the spread between the two is stationary. In what follows, we apply our rank test for analyzing and testing EHTS for U.S. interest rates. (n) Let Rt be the continuously compounded n-period interest rate and let (n ) Rt 0 be a shorter-term interest rate, say n0 -period interest rate. Campbell and Shiller (1987) show that if the short-term interest rate is I (1), then EHTS implies that the long-term rate will also be I (1) and the spread will be stationary, (n) (n ) (n ,n) (n) (n ) i.e. if Rt ∼ I (1) and Rt 0 ∼ I (1), then St 0 = Rt − Rt 0 ∼ I (0) . So, there is a cointegration relationship between these two rates with the cointegration vector (1, −1). Hall et al. (1992) extend their investigations on expectations theory of the term structure to the multivariate case, considering N (n ) maturities Rt j , with n0 ≤ n1 ≤ ... ≤ nj ≤ ... ≤ nN −1 . They state that if the (n ) short-term rate Rt 0 is I (1), then all the others will also be I (1) and that the spreads with the short-term rate will all be I (0) . In other words, there will be (N − 1) cointegration relationships. The basic idea behind the introduction of threshold effect in the spreads between two interest rates is that the nonlinearity in the adjustment process stems from interventions of monetary authority. In order to test this hypothesis, we 20

use the monthly interest rate series of McCulloch and Kwon (1993) which are constructed from the prices of U.S. Treasury securities and expressed as continuously compounded zero-coupon bonds. We estimate and test 4-dimensional TVECM using a selection of bond rates with maturities ranging from 1 to 120 months for the period January 1951 - February 1991 (482 observations). It is generally agreed that interest-rates series are I(1) and do not contain time trend but one cannot ignore the possibility of a drift in the spreads due to the presence of a premium.7 Thus we remove the intercept from the error correction model and include  it only in the cointegrating equations. Defining  (n0 )

yt = Rt

(n1 )

, Rt

(n2 )

, Rt

(n3 ) 0

, Rt

, the estimated models are written as:

∆y t = µ(1) I{st−d ≤θ} + Π(1) yt−1 I{st−d ≤θ} + µ(2) I{st−d >θ} +Π(2) yt−1 I{st−d >θ} + Λ (L) ∆y t−1 +εt , where µ(1) = Γ(1) β0 , µ(2) = Γ(2) β˜0 , β0 and β˜0 are (m×1) vectors representing the intercept terms of the cointegration relations. Our transition variable is given by the smallest deviation  from the variation of the spreads: min (∆St ) = st ,where St = J0 yt and J = (1, −1, 0, ..., 0)0 ; (1, 0, −1, 0, ..., 0)0 ; ...; (1, 0, ..., 0, −1)0 . The selection of the lag d of st−d could be based on an information criterion such as the Bayesian information criterion (BIC) or the Akaike information criterion (AIC). Here we take d = 1. We now present our results for four maturities: 3-month/12-month/24month/120-month. The value of the supremum Wald test of no-cointegration (Gonzalo and Pitarakis’s statistic (12)), obtained over 336 grid points on the empirical distribution of our threshold variable, i.e. FT (st−1 ) , is sup W0,T = 317.09 which indicates a clear rejection of the null hypothesis of no-cointegration (See Table A.4 in Appendix A.4 for the critical values). Note that our grid set corresponds to 70% of the values of the transition variable which are taken as possible values of the threshold. The discarded values are the largest and smallest 15% of st−1 , i.e. τ = 0.15. Note that the conclusions of the tests do not change with τ = 5% and 10% and we only present the results for τ = 15%. Next, we go on to test the cointegration rank of Π(1) and Π(2) matrices. For this purpose, the TVECM model is estimated under the triangular normalization, over the above-mentioned grid points on the threshold parameter. For (1) (2) each estimated model, we compute the statistics sup WT and sup WT . Since the interest rates are highly heteroskedastic, standard deviations of parameter estimates are computed using the Newey-West method (Newey and West, 1987) and heteroskedasticity-consistent statistic values are obtained. The results are summarized in Table 6 below.

7

The unit root test results on each variable, which confirm the difference-stationarity, are not reported here to save space.

21

Table 6: Statistic values for rank cointegration tests H0 : r0 r0 = 1 r0 = 2 r0 = 3

sup W (1) 70.158 22.030a,b 2.426a,b,c

sup W (2) 80.135 58.844 8.661a,b,c (j)

Note: a, b, c : H 0 cannot be rejected at 1%, 5% and 10%, respectively, for Γ⊥ µ = 0 (see Table 2).

From the above results, we conclude (at the 5% level) that there are only two active cointegration relationships in the first regime and three in the second 0 ˆ (j) = −Φ ˆ j (1) 00 one. The estimates of the loadings Γ are given bellow: (1×3)



−0.364

0

0





−0.184

0

0



(0.031)

  (0.022)     0 −0.196 0    (1) (2) ˆ ˆ   , (0.030) (0.022) Γ , Γ =    0 0 −0.436  0 0 −0.211   (0.034)  (0.024)  0 0 0 0 0 0 0 and the long-run vector estimate is βˆ = (0.943, 1.004, 1.021) with standard ˆ = 0.210 (θˆ = errors (0.034, 0.024, 0.016)0 . The threshold estimate is F (θ) −0.022). The above estimates strongly suggest that the speed of adjustment is greater in the first regime, i.e. when the variation in the long-term rates gap is slightly negative.    =   

8

0

−0.394

0

Conclusion

This paper examines a threshold VECM with more than one cointegrating relation, incorporating the possibility that some cointegrating relationships are not active in some regimes, and derives estimation and inference procedures for such a model. Estimators of the long-run parameters and loading factors are obtained through reduced rank regression and their asymptotic properties are derived. A testing procedure for the cointegrating rank is proposed in the context of partial cointegration and the asymptotic distributions of the test statistics are derived and tabulated. We also obtain the bootstrap distribution approximation for our test statistics for conducting finite sample inference. Monte-Carlo experiments are conducted to evaluate the finite-sample performance of the tests and their power. Our experiment results show that in absence of partial unit root, our rank test works as well as the LR test. However, when there are unit roots or some inactive cointegrating relations in one regime, the LR test fails to identify the correct number of cointegrating relations whereas our test still does a good job. Finally we apply our methodology to test the term structure of U.S interest rates. The results reveal the presence of a unit root in one regime yielding a difference in the cointegration ranks between the two regimes. 22

9

Acknowledgements

We thank Eric Jondeau for his comments. We are grateful to two anonymous referees and an anonymous associate editor for many valuable comments. This work was supported by a grant from the Swiss National Fund for Scientific Research.

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25

APPENDIX A.1 PROOF OF PROPOSITION 1: We have A (L) = B (L) (Ip − GL) ,  where B (L) =



J (L) −J (L) β + βK (L) 0 K (L) 

and A (L) = B (L) − B (L) GL =

,

J (L) β (K (L) − K (L) L) − J (L) β 0 K (L) − K (L) L

 .

Denoting K ∗ (L) = K (L) − K (L) L = Im − K1∗ L − ... − Kq∗ Lq , with K1∗ = (K1 + Im ) , Ki∗ = (Ki − Ki−1 ) for q > i > 1 and Kq∗ = −Kq−1 , we have:  B (L) − B (L) GL =

J (L) βK ∗ (L) − J (L) β 0 K ∗ (L) 

Im 0

0 In



 −

 =

˜ ∗ (L) − (Im − J (L)) β Im − J (L) β K ˜ ∗ (L) 0 K

 ,

˜ ∗ (L) = Im − K ∗ (L) =Pq Ki∗ Li . where K i=1 Splitting A (L) according to the two regimes we have: 

˜ ∗ (L) − φ(j) (L) β φ(j) (L) β K ˜ ∗ (L) 0 K



Ip I{·} − PP (j) i i=1 φi L .

I{·} for j = 1 or 2, and where φ(j) (L) =

By identification, we obtain the terms of A (L) .  A.2 PROOF OF THEOREM 1: For proving Theorem 1, we need the following Assumption, Theorem and Lemma. ASSUMPTION 3. Consider the sequence {et , Ut }, where et is a p-dimensional vector and Ut is a univariate process whose marginal distribution is U[0,1] , and let Ft−1 be the natural filtration. We assume (i) ekt , k = 1, ..., p and Ut are strictly stationary, ergodic P∞ 1/2−1/δ and strong mixing with mixing coefficients ai satisfying i=1 ai < ∞ for some δ > 2, (ii) ekt are independent of Ft−1 for k = 1, ..., p, and (iii) E (ekt ) = 0, E |ekt |4 < ∞, for k = 1, ..., p. THEOREM A.1. Consider the p-dimensional I(1)-vector process yt such that ∆y t = et , where et satisfies Assumptions 3 with covariance matrix Ωe . Then, as T → ∞ and under Assumptions 1 to 3, we have:

26

(i) BT (u) = T −1/2

P[T r] t=1

1/2

2

I{Ut−d ≤u} et =⇒ Ωe B(r, u), on (r, u) ∈ [0, 1] ,

where T −1/2 B(r, u) ∼ N (0, ru) is a n-vector two-parameter Brownian motion. P[T r] 1/2 Symmetrically we have T −1/2 t=1 I{Ut−d >u} et =⇒ Ωe B (r, 1 − u). 0

(ii) T −1

PT

(iii) T −2

PT

=⇒ F (θ) Ωe

(iv) T −2

PT

=⇒ S (θ) Ωe

yt−1 I{Ut−d ≤u} e0t =⇒

t=1

0 t=1 I{Ut−d ≤u} yt−1 yt−1

0 t=1 I{Ut−d >u} yt−1 yt−1

R

1/2

B (r) dB (r, u) Ωe , 1/2

1/2

R


1/2 B (r) B (r)0 dr Ωe ,

R


1/2 B (r) B (r)0 dr Ωe ,

where S (θ) = 1 − F (θ) . PROOF OF THEOREM A.1: These results are direct multivariate extensions of the results of Caner and Hansen (2001, Theorems 1-3, pp. 1560-1561) for the univariate case.

 LEMMA A.1. Let assumptions 1 to 3 hold, we then have:  
0 PT 0 −1/2 0 Q (r, u) , ⊗ Γ Ω−1 (i) T1 t Yt−1 e et =⇒ vec Γ Ωe where Q (r, u) = hP

T t

(ii) T −2 H 0

R

0

dB (r, u) B (r) ,

R

0
dB (r, 1 − u) B (r) .


i 0 (Yt−1 ⊗ Γ0 ) Ωe−1 Yt−1 ⊗ Γ H =⇒ H 0 (D (r, u) ⊗ Γ0 Γ) H, 

 R u 0 0 ⊗ B (r) B (r) dr, and where B (r, u) and B (r) are 0 1−u standard p-dimensional two-parameter and one-parameter Brownian motions respectively.

with D (r, u) =

PROOF OF LEMMA A.1: For (i): Note that we can write :  
0 −1
PT PT PT 0 0 −1 0 0 −1 0 t Yt−1 ⊗ Γ Ωe et = t vec Γ Ωe et Yt−1 = vec Γ Ωe t et Yt−1 , and define W (r, u) to be the two-parameter Brownian motion associated with I{Ut−d ≤u} et , P[T r] 1/2 i.e. T −1/2 t=1 I{Ut−d ≤u} et =⇒ W (r, u) , with W (r, u) = Ωe B (r, u). Theorem A.1 can be applied to get the limit of the previous expression. We have: R R PT PT (1)0 (2)0 0 0 1 et yt−1 =⇒ dW (r, u) W (r) and T1 et yt−1 =⇒ dW (r, 1 − u) W (r) . T 0 (j) (j )0 For (ii) : By definition yt−1 yt−1 = 0 for j 6= j 0 , it is therefore easily seen that

T −2 H 0

hP

T t


i 0 (Yt−1 ⊗ Γ0 ) Ω−1 Yt−1 ⊗Γ H = e ! PT (1) (1)0 0 yt−1 yt−1 ⊗ Γ0 Ω−1 −2 0 e Γ T H H. PT (2) (2)0 0 yt−1 yt−1 ⊗ Γ0 Ω−1 e Γ

27

Once again using Theorem A.1, we get the following limit for the above expression:  
1/2 u 0 1/2 R 0 Ωe B (r) B (r) dr Ωe ⊗ Γ0 Ω−1 e Γ. 0 1−u  PROOF OF THEOREM 1:
0 PT ∗0 The estimator ϕˆ can be written as ϕ= ˆ ϕ + C −1 H 0 t Yt−1 ⊗ Γ Ω−1 ε∗t , and hence p lim (ϕ) ˆ = ϕ. Using Lemma A.1, the asymptotic distribution of T (ϕˆ − ϕ) is immediate.  A.3 PROOF OF THEOREM 2: Consider the sup-Wald 1 and let us normalize it aso follows: n statistic for
regime
−1 −1 (1)0
0 2 (1) −1 (1)0 (1)0 sup W (u, m) = tr T ν M ς T ν M ν (1) T ν Mς , −1/2

˜ with ς = ε˜∗j Ω j

˜ j = Γ(j)0 Ωj Γ(j) . Opening the first term, we get: , where Ω ⊥ ⊥ (1)

T −1 ν (1)0 M ς = T −1 ν (1)0 ς − T −1 ν˜(1)0 um

−1  (1)0 (1)0 (1) um ς um um

(1)

where um is I (0) under H0 . Since under the null, ν (1) is I (1), then we use Lemma A.1 to obtain the limit of this term as follows T −1 ν (1)0 ς = T −1

PT

I{Ut−d ≤u} νt−1 ςt0 ,

R 0 1/2 and therefore T −1 ν (1)0 ς =⇒ A0⊥ B (r) dB (r, u) Ως , since νt = A0⊥ yt∗ . For the second term, it is straightforward to see that (1)

T −1 ν (1)0 um



(1)0 (1)

um um

−1

(1)0

um ς = op (1) .

So we have: (A.3.1)

R 0 1/2 T −1 ν (1)0 M ς =⇒ A0⊥ B (r) dB (r, u) Ως .

In the same way, for the middle term of the Wald statistic, we have: (1)

T −2 ν (1)0 M ν (1) = T −2 ν (1)0 ν (1) − T −2 ν (1)0 um

 −1 (1)0 (1) (1)0 um um um ν (1) ,

= T −2 ν (1)0 ν (1) − op (1) , and from Theorem 1 we have T −2 ν (1)0 ν (1) = T −2

PT

1/2

0 =⇒ uΩς A0⊥ I{Ut−d ≤u} νt−1 νt−1

R


1/2 0 B (r) B (r) dr A⊥ Ως .

Hence: (A.3.2)

1/2

T −2 ν (1)0 M ν (1) =⇒ uΩς A0⊥

R


1/2 0 B (r) B (r) dr A⊥ Ως .

28

Finally, let us denote U (r) = A0⊥ B (r) with covariance matrix Ωu . Combining (A.3.1) 1/2 and (A.3.2) , and denoting U (r) = Ωu V (r) the limiting distribution of sup-Wald statistic is given by sup W (1) (m) =⇒

sup

tr

nR

o
−1 R R 0 0 0 V (r) dB (r, u) . dB (r, u) V (r) u V (r) V (r) dr

u∈[τ , 1−τ ]

A similar derivation can be carried out to obtain the asymptotic distribution of the sup-Wald statistic for the parameter estimators of regime 2.  A.4. Gonzalo and Pitarakis (2006) only give the sup −W0 critical values for the bidimensional case. We reproduce their values and complete their tables by providing critical values for various dimensions p and τ. Table A.4: Critical values for sup −W0 including 90% 95% 99% 90% p 1 [τ, 1 − τ ] [0.15, 0.85] 6.602 8.083 11.460 13.619 [0.10, 0.90] 7.003 8.509 11.941 14.074 8.839 12.382 14.574 [0.05, 0.95] 7.410 p [τ, 1 − τ ] [0.15, 0.85] [0.10, 0.90] [0.05, 0.95]

4 32.676 33.361 34.238

35.229 35.864 36.687

intercept. 95% 99% 2 15.549 15.918 16.391

19.701 20.298 20.558

90%

95% 3

99%

22.202 22.836 23.569

24.480 25.196 25.855

29.468 29.970 30.792

5 40.832 41.454 41.979

45.312 46.215 47.098

Note: Calculated from 10,000 simulations.

29

48.576 49.262 49.932

6 54.981 55.652 56.285

69.935 60.622 61.671

63.117 63.971 65.017

59.621 70.408 71.263

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