COINTEGRATING SMOOTH TRANSITION

COINTEGRATING SMOOTH TRANSITION REGRESSIONS WITH A STATIONARY TRANSITION VARIABLE MARCELO C. MEDEIROS, EDUARDO MENDES, AND LES OXLEY A BSTRACT. Nonlinear regression models have been widely used in practice for a variety of time series applications. For purposes of analyzing univariate and multivariate time series data, in particular, the Smooth Transition Regression (STR) models have been shown to be very useful for representing and capturing asymmetric behavior. However, in most cases STR models have been applied stationary processes where all regressors are weakly exogenous. On the other hand, STR models with nonstationary variables are not yet fully understood as previous studies in the literature considered only cointegrated STR models where the transition variable is also nonstationary. In this paper we study smooth transition cointegrating regressions where the transition variable is stationary. This is important, for instance, to model changing equilibrium over the business cycle. We derive asymptotic results for both static nonlinear least squares (NLS) and dynamic nonlinear least squares (DNLS) estimators.

1. I NTRODUCTION Nonlinear regression models have been widely used in practice for a variety of time series applications (see Granger and Ter¨asvirta (1993) for some examples in Economics). For purposes of analyzing univariate and multivariate time series data, in particular, the Smooth Transition Regression (STR) models, initially proposed, in its univariate form, by Chan and Tong (1986) and further developed in the papers by Luukkonen, Saikkonen, and Ter¨asvirta (1988) and Ter¨asvirta (1994,1998), have been shown to be very useful for representing and capturing asymmetric behavior1. van Dijk, Ter¨asvirta, and Franses (2002) provides a useful recent review of time series STR models. Date: May 21, 2008. The term “smooth transition” in its present meaning first appeared in a paper by Bacon and Watts (1971). They presented their smooth transition model as a generalization to models of two intersecting lines with an abrupt change from one linear regression to another at some unknown change-point. Goldfeld and Quandt (1972, p. 263–264) generalized the so-called two-regime switching regression model using the same idea. 1

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M. C. MEDEIROS, E. MENDES, AND L. OXLEY

Most STR models have been applied to univariate process and stationarity of the variables is usually imposed. Under stationarity, the independent variables have been assumed to be weak exogeneous with respect to the parameters of interest in all cases. Under the assumption of exogeneity, the standard method of estimation is nonlinear least squares and the asymptotic properties of the estimators have been discussed in Mira and Escribano (2000), Suarez-Fari˜nas, Pedreira, and Medeiros (2004), and Medeiros and Veiga (2005). Nonlinear least squares is equivalent to quasimaximum likelihood or, when the errors are Gaussian, to conditional maximum likelihood. Although stationarity is imposed in the vast majority of applications, Choi and Saikkonnen (2004a,b) considered the case of STR models with cointegrated variables. However, the authors considered the case where the transition variable is also nonstationary. The purpose of this paper is to extend the results in Choi and Saikkonnen (2004a,b) to the case where the transition variable is integrated of order zero, I(0). This is of particular interested when, for example, equilibria among macroeconomic variables change depending on the state of the economy as represented, for instance, by the output gap or output growth. The paper is organized as follows. In Section 2 we present the model and state the main assumptions. Estimation of parameters and the the derivation of the asymptotic results are considered in Section 3. Section 4 presents a Monte Carlo simulation in order to evaluate the finite sample properties of the estimators. An empirical application is presented in Section 5. Finally, Section 6 concludes the paper. All technical proofs are relegated to the Appendix.

2. T HE M ODEL AND M AIN A SSUMPTIONS Consider the following model

(1)

yt = F˜ (zt , xt ; θ) + ut = F (zt ; θ)′ x ˜ t + ut ,

t = 1, 2, . . . ,

where F (·; ·) is a continuous bounded function, x ˜′t = (1, x′t ), whith xt a p×1, I(1) random vector, ut is a zero mean stationary error term, zt is a stationary m × 1 random vector and θ ∈ Θ is a k × 1 parameter vector.

COINTEGRATING SMOOTH TRANSITION REGRESSIONS WITH A STATIONARY TRANSITION VARIABLE

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The model in (1) can be used to specify nonlinear cointegration relations with a stationary variable controlling the threshold. We can rewrite model 1 as yt = µ +

(2)

p X i=1

αi xit +

p X

βj gj (zt ; νi )xjt ,

t = 1, 2, . . . ,

j=1

where x ˜t = (1, x1t , . . . , xpt )′ , θ = (µ, α1 , . . . , αp , β1 , . . . , βp , ν1′ , . . . , νp′ )′ . A straightforward generalization of model (2) is allowing j vary only in a subset of 1, . . . , p. E XAMPLE 1. Assume (3)

g1 (zt ; ν) = g1 (zt ; γ, c) = (1 + e−γ(zt −c) )−1 ,

and

g2 (·; ·) = 0.

Let p = 2, m = 1, and θ = (µ, α1 , α2 , β1 , γ, c)′ . Here, the model is equivalent to the STR model, with nonstationary regressors. The cointegration coefficient for x1t varies smoothly between α1 and α1 + β1 , depending on how positive or negative is (zt − c). We follow up by presenting some of the assumptions required for this model. Note that these assumptions are not the weakest possible, but are valid for a myriad of models of interest. A SSUMPTION 1. (4)

xt = xt−1 + vt ,

t = 1, 2, . . . ,

where vt is a zero mean stationary process, and x0 = 0 or is randomly drawn from a density independent of T. Define the following stationary (k+1)×1 vector process wt′ = [ut , ∂θ F˜ (zt , vt ; θ)−E(∂θ F˜ (zt , vt ; θ))]2. Now we make the following assumptions about this vector, following Phillips and Durlauf (1986). A SSUMPTION 2. Each element of the process {wt }∞ 1 , obeys the following assumptions: (a) E|wi |β < ∞, i = 1, . . . , (k + 1), for 2 ≤ β < ∞ (b) {ωi }∞ 1 , i = 1, . . . , (p + m + 1), is either uniform mixing of size −β/(2β − 2) or strong mixing of size −β/(β − 2), for β > 2. 2f (θ) = ∂ F˜ (z , v ; θ) − E(∂ F˜ (z , v ; θ)) = (x′ , g (z , ν ˆ1 )v1t ), . . . , gp (zt , νˆp vpt − t t t t t 1 t ˆ1 )v1t − E(g1 (zt , ν θ θ t E(gp (zt , νˆp vpt ), βˆ1 ∂ν1 g1 (zt νˆ1 )v1t − E(βˆ1 ∂ν1 g1 (zt νˆ1 )v1t ), . . . , βˆp ∂νp gp (zt νˆp )vpt − E(βˆp ∂νp gp (zt νˆp )vpt ))

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M. C. MEDEIROS, E. MENDES, AND L. OXLEY

Assumption 2 is the same employed by Phillips and Durlauf (1986) and holds for a variety of Pt ∞ processes. They also show that the partial sum process St = 1 wi constructed from {wt }1

satisfies the multivariate invariance principle (under assumption 3). Specifically, for r ∈ [0, 1], define WT (r) = T −1/2 S[T r] then (5)

WT (r) ⇒ B(r),

as T → ∞,

where B(r)′ = [Bu (r), Bf (r)′ ] is a multivariate Brownian Motion with covariance matrix Ω = limT →∞ T −1 E(ST ST′ ), and (6)



Ω=

ωu

′ ωuf

ωuf

Ωf



,

accordingly to the partitions of the process wt . The convergence of E(ST ST′ ) = E(wt wt′ )+

PT

′ ′ i=2 [E(w1 wi )+E(wi w1 )] as T

→ ∞ implies the

process wt has a continuous spectral density function fww (λ), also Ω = 2πfww (0). We assume fww (λ) satisfies the following assumption. A SSUMPTION 3. The spectral density matrix fww (λ) is bounded away from zero. We also make the following assumptions about the function A SSUMPTION 4. (i) The parameter space Θ ⊂ Rk is compact. (ii) F (zt ; θ) : Rm × Θ 7→ Rp is three times continuously differentiable on Rm × Θ∗ where Θ∗ is an open set cointaining Θ (iii) F (zt ; θ), is bounded on Rp Although these assumptions are not the weakest possible, it is satisfied by a myriad of functions of interest. Assumption 4.a is standard. We also make an assumption about the identifiability of this model. A SSUMPTION 5. F (·; ·) is identifiable with respecto to θ in a sense that, for θ1 6= θ2 ∈ Θ, F (·; θ1 ) 6= F (·; θ2 ) a.e.

COINTEGRATING SMOOTH TRANSITION REGRESSIONS WITH A STATIONARY TRANSITION VARIABLE

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3. E STIMATION AND A SYMPTOTIC P ROPERTIES In this paper we consider the Nonlinear Least Squares (NLS) estimation procedure use in other transition models with integrated variables (Choi and Saikkonen 2004a, Chang and Park 2003). Choi and Saikkonen (2004a) use the triangular array asymptotic because some of the parameters are not identifiable when the usual asymptotics is used. In our case, the variable controling the threshold is stationary; hence, we do not have this problem and the usual asymptotics is used. The nonlinear least squares estimator θˆ of θ0 , the true parameter vector, is obtained by minimizing the function QT , defined above, with respect to θ ∈ Θ T X

QT (θ) =

(7)

(yt − F˜ (zt ; θ))2 .

t=1

Define the following derivatives with respect to the parameter vector θ: (8)

∂ QT (θ)|θ∗ ∂θ T X = − (yt − F˜t,θ∗ )F˜˙t,θ∗

Q˙ T (θ∗ ) =

t=1

(9)

∂2 QT (θ)|θ∗ ∂θ∂θ′ T T X X ˙ ˙ ˜t,θ∗ ˜ ˜ ∗ ∗ (yt − F˜t,θ∗ )F¨ Ft,θ Ft,θ − =

¨ T (θ∗ ) = Q

t=1

t=1

˜t,θ = where F˜t,θ = F˜ (zt ; θ), F¨

∂ F˜ (zt ,xt ;θ) ∂θ

˜t,θ = and F¨

∂ 2 F˜ (zt ,xt ;θ) . ∂θ∂θ′

With F˜ in (1). Moreover, we

have the first order Taylor expansion (10)

ˆ = Q˙ T (θ0 ) + Q ¨ T (θT )(θˆ − θ0 ), Q˙ T (θ)

where θT is a point between θˆ and θ0 We can derive the asymptotic distribution of θˆ from (10). First we show that (11)

−1 ′¨ Γ−1 T H QT (θ0 )HΓT →d A > 0 and

′ ˙ − Γ−1 T H QT (θ0 ) →d L0 ,

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M. C. MEDEIROS, E. MENDES, AND L. OXLEY

for an appropriate normalizing sequence ΓT and an orthonormal matrix H. Under suitable conditions (theorem 10.1 in Wooldridge (1994)),we can show that −1 −1 −1 ′ ˙ ′¨ ΓT H ′ (θˆ − θ0 ) = −(Γ−1 T H QT (θ0 )HΓT ) (ΓT H QT (θ0 )) →d A0 L0 .

(12)

Assumption (iii) of theorem 10.1 in Wooldridge (1994) requires that (a) ΓT δ = T −δ ΓT → 0, for some δ > 0; (b) for θ ∈ ΘT , −1 ′ ¨ ¨ max kΓ−1 T δ H (Qt (θ) − Qt (θ0 ))HΓT δ k →p 0

(13)

Θ0

where Θ0 ≡ {θ ∈ Θ : kΓT δ (θ − θ0 )k ≤ 1}. ˆ In the following theorem we state the asymptotic distribution of the NLS estimate θ.

T HEOREM 1. Under assumptions (1)–(5), 

(14) ΓT 

µ ˆ−µ





  →d H    θˆ − θ0

R1

′ 0 B2 (r)dr R1 kδk/2 kδk2 /3 kδk 0 rdB2′ (r) R1 R1 R1 ′ 0 B2 (r)dr kδk 0 rdB2 (r) 0 B2 (r)B2 (r)dr   Bu (1)   R1   × , kδk 0 rdBu (r)   R1 0 B2 (r)dBu (r) + κ

1

kδk/2

−1    

where δ = E(vt′ , F˙tθ ), B ′ = [Bu , B2′ ], with B2 = H2∗ ′ Bf , is a brownian motion with covariance P ∗ ′ ˙ matrix H ′ ΩH, and the nuisance parameter κ = ∞ j=1 (E(v1 F1θ H2 uj ). The k − 1 × k − 2 matrix H2∗ is such that H ∗ = [δ/kδkH2∗ ] is a orthogonal matrix, 

H=

1

0

0 H∗

 

and



  ΓT =  

T 1/2

0

0

0

T 3/2

0

0

0

Ik−2 T



  . 

Note the asymptotic distribution depends on a nuisance parameter κ in a complicated way, which renderd the estimator inefficient, and oftentimes inadequate for hypothesis testing. If ut

COINTEGRATING SMOOTH TRANSITION REGRESSIONS WITH A STATIONARY TRANSITION VARIABLE

7

and vt′ F˙ (zt , xt ; θ0 ) are uncorrelated, the nuisance paramters vanishes and the distribution is mixed normal. 3.1. Efficient Estimation. In this section we will introduce the efficient 2-stage estimator based on adding leads-and-lags of vt′ F˙ (zt ; θ0 ) to the regression. We adopt the same method used by Choi and Saikkonen (2004a) and others, including leads and lags of vt′ F˙ (zt ; θ0 ) in the NLS regression. Following Saikkonen (1991) we can write the error term as ut =

(15)

∞ X

π ′ fˆt−j + et ,

j=∞

ˆ − E(∂θ F (zt , vt , θ))), ˆ where et is a zero-mean stationary process, fˆt′ = (∂θ F (zt , vt , θ) such that P ∞ ′ ) = 0 for all t = 0, ±1, ±2, . . ., and E(et v˜t−j j=−∞ (1 + |j|)kπj k < ∞. The expectral density function of et if given by the formula fee = fuu − f uf ff−1 f ff u and the long run variance is given

by ωe = ωu − ωuf Ω−1 f ωf u. The model in (1) can be rewritten as (16)

yt = F˜ (zt , xt ; θ) +

X

πj′ fˆt−j + eKt ,

X

πj′ fˆt−j .

|j|≤K

with eKt = et +

(17)

|j|>K

We use K = o(T 3 ) as in Saikkonen (1991) and Choi and Saikkonen (2004a) to avoid truncating erros in the asymptotics. Note this model can be rewritten as (18)

′ ′ ˆ fˆt−K + . . . + πK ft+K + eKt . yt = F˜ (zt , xt ; θ) + π−K

ˆ t′ = (F˜˙ ˆ, fˆ′ , . . . , fˆ′ ), and π ′ = (π−K , . . . , πK ). The Gauss-Newton equation Write D t−K t+K t,θ for the NLS estimator can be written as     θˆ(1) θˆ  = + (19) π ˆ 0 ˆ where u ˆt = yt − F˜ (zt , xt ; θ).

TX −K

t=K+1

ˆ tD ˆ t′ D

!−1

TX −K

t=K+1

ˆ tu D ˆt ,

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M. C. MEDEIROS, E. MENDES, AND L. OXLEY

T HEOREM 2. Suppose assumptions of theorem 1 hold, Ekwt ⊗ wt ⊗ wt ⊗ wt k < ∞, and K → ∞ P such that K 3 /T → 0 and T 1/2 |j|>K kπj k → 0. Then 

(20)ΓT 

µ ˆ(1) − µ





  →d H    θˆ(1) − θ0

R1

′ 0 B2 (r)dr R1 kδk/2 kδk2 /3 kδk 0 rdB2′ (r) R1 R1 R1 ′ 0 B2 (r)dr kδk 0 rdB2 (r) 0 B2 (r)B2 (r)dr   Be (1)   R1   ×  kδk 0 rdBe (r)  ,   R1 0 B2 (r)dBe (r)

1

kδk/2

−1    

where Be is a Brownian motion independent of B2 = H2∗ Bf , with variance ωe .

At this point, a heuristic proof is presented. Note that (19) is the least squares estimator of θ ˆ t on yt . By multiplying both sides of the equation(19) by a suitable and π for the regression of D orthnormal matrix, we can apply the results of Saikkonen (1991) and the proof of theorem 1 to conclude the claim. A modification of this theorem where the auxiliary regression depends only on xt and vec(zt x′t )− E(vec(zt x′t )) is being developed.

4. S IMULATION In this section we consider a simulation study to check the finite sample properties of the nonlinear least squares (NLS) and dynamic nonlinear least squares (DNLS) estimators. The baseline model for simulation is given as yt = (0.5 + 2xt ) + (1.5 + 4.5xt ) g (zt ; 10, 0) + u1t , (21)

xt = xt−1 + u2t , zt = 0.4zt−1 + u3t ,

COINTEGRATING SMOOTH TRANSITION REGRESSIONS WITH A STATIONARY TRANSITION VARIABLE

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TABLE 1. Monte Carlo Simulations: Bias and Mean Squared Error. The table shows the the average bias and the mean squared error for the parameter estimates for the NLS and DNLS estimator. Case 1, T = 100 1, T = 200 2, T = 100 2, T = 200

Case 1, T = 100 1, T = 200 2, T = 100 2, T = 200

Bias 0.01 0.00 -0.49 -0.48

α0 MSE Bias -0.00 0.02 0.00 0.01 -0.02 0.25 -0.01 0.24

Nolinear Least Squares α1 MSE Bias MSE Bias 0.00 -0.01 0.00 0.05 0.00 -0.00 -0.00 0.02 0.00 0.99 0.02 0.99 0.00 0.97 0.01 0.95

MSE 0.01 0.00 0.01 0.00

Bias 0.05 0.01 -0.20 -0.05

Bias 0.01 0.00 -0.32 -0.18

Dynamic Nonlinear Least Squares α0 α1 MSE Bias MSE Bias MSE Bias -0.00 0.02 0.00 -0.01 0.00 0.05 0.00 0.01 0.00 -0.00 -0.00 0.02 -0.02 0.18 0.00 0.69 0.02 0.79 -0.01 0.14 0.00 0.47 0.01 0.45

MSE 0.01 0.00 0.01 0.00

Bias 0.05 0.01 -0.19 -0.04

γ

c MSE 0.32 0.08 0.91 0.44

Bias -0.00 0.00 0.00 0.00

MSE 0.32 0.08 0.81 0.24

Bias -0.00 0.00 0.00 0.00

γ

MSE 0.00 0.00 0.00 0.00

c MSE 0.00 0.00 0.00 0.00

where

(22)

     1 ρ12 ρ13   0     ut = (u1t , u2t , u3t )′ ∼ NID   , 14 ρ12 1 ρ23  .   0  ρ13 ρ23 1

We consider the following cases: (1) Exogenous regressors: ρ12 = ρ13 = ρ23 = 0; (2) Endogenous regressors: ρ12 = ρ13 = ρ23 = 1; The results are shown in Table 1 5. E MPIRICAL A PPLICATION 6. C ONCLUSIONS R EFERENCES BACON , D. W., AND D. G. WATTS (1971): “Estimating the Transition Between Two Intersecting Lines,” Biometrika, 58, 525–534. C HAN , K. S., AND H. T ONG (1986): “On Estimating Thresholds in Autoregressive Models,” Journal of Time Series Analysis, 7, 179–190. C HANG , Y., AND Y. PARK (2003): “Index model with integrated time series,” Journal of Econometrics, 114, 73–106. C HOI , I., AND P. S AIKKONEN (2004a): “Cointegrating Smooth Transition Regressions,” Econometric Theory, 20, 301–340.

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M. C. MEDEIROS, E. MENDES, AND L. OXLEY

(2004b): “Testing Linearity in Cointegrating Smooth Transition Regressions,” Econometrics Journal, 7, 341– 365. G OLDFELD , S. M., AND R. Q UANDT (1972): Nonlinear Methods in Econometrics. North Holland, Amsterdam. ¨ G RANGER , C. W. J., AND T. T ER ASVIRTA (1993): Modelling Nonlinear Economic Relationships. Oxford University Press, Oxford. ¨ L UUKKONEN , R., P. S AIKKONEN , AND T. T ER ASVIRTA (1988): “Testing Linearity Against Smooth Transition Autoregressive Models,” Biometrika, 75, 491–499. M EDEIROS , M., AND A. V EIGA (2005): “A Flexible Coefficient Smooth Transition Time Series Model,” IEEE Transactions on Neural Networks, 16, 97–113. M IRA , S., AND A. E SCRIBANO (2000): “Nonlinear Time Series Models: Consistency and Asymptotic Normality of NLS Under New Conditions,” in Nonlinear Econometric Modeling in Time Series Analysis, ed. by W. A. Barnett, D. Hendry, S. Hylleberg, T. Ter¨asvirta, D. Tjøsthein, and A. W¨urtz, pp. 119–164. Cambridge University Press. PARK , Y., AND P. P HILLIPS (1988): “Statistical inference in regressions with integrated processes: Part I,” Econometric Theory, 4, 468–497. P HILLIPS , P., AND S. D URLAUF (1986): “Multiple time-series regression with integrated processes,” Review of Economic Studies, 53, 473–496. S AIKKONEN , P. (1991): “Asymptotically efficient estimation of cointegration regressions,” Econometric Theory, 7, 155–188. ˜ , M., C. E. P EDREIRA , AND M. C. M EDEIROS (2004): “Local Global Neural Networks: A New S UAREZ -FARI NAS Approach for Nonlinear Time Series Modeling,” Journal of the American Statistical Association, 99, 1092–1107. ¨ T ER ASVIRTA , T. (1994): “Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models,” Journal of the American Statistical Association, 89, 208–218. (1998): “Modelling Economic Relationships with Smooth Transition Regressions,” in Handbook of Applied Economic Statistics, ed. by A. Ullah, and D. E. A. Giles, pp. 507–552. Dekker. VAN

¨ D IJK , D., T. T ER ASVIRTA , AND P. H. F RANSES (2002): “Smooth Transition Autoregressive Models - A Survey

of Recent Developments,” Econometric Reviews, 21, 1–47. W OOLDRIDGE , J. M. (1994): “Estimation and Inference for Dependent Process,” in Handbook of Econometrics, ed. by R. F. Engle, and D. L. McFadden, vol. 4, pp. 2639–2738. Elsevier Science.

A PPENDIX A. L EMMAS ∞ L EMMA 1. Let the {ut }∞ 1 be a zero-mean stationary process and let {xt }1 be a m-vector process

satisfying ∆xt = δ + vt ,

COINTEGRATING SMOOTH TRANSITION REGRESSIONS WITH A STATIONARY TRANSITION VARIABLE

11

with x0 = 0 w.l.o.g.. Define wt′ = [ut , vt′ ], and assume {wt }∞ 1 satisfies assumption 2. Define the covariance matrix   ∞ ′ X ω1 Ω21 ′   Ω= = E[w1 w1 ] + (E(w1 wj′ ) + E(wj w1′ )) = Σ + Λ + Λ′ , Ω21 Ω2 j=2 and partition the matrices Σ and Λ accordingly. Let H = (h1 H2 ) be a orthogonal matrix, with h1 = δ/kδk, where kδk2 = δ ′ δ. Write x ˜t = H ′ xt = (˜ x1t , x ˜′2t )′ , and define the m-vector Brownian motion B = [B1 B1′ ] with covariance matrix 

Ω∗ = 

H2′ Ω′21

ω1

Ω21 H2 H2′ Ω2 H2



.

Then, −1 PT

(a) T −1/2 ΓT

(c) Γ−1 T

→d  R 1 0

kδk/2 B2 (r)dr

 

 R1 ′ (r)dr 2 /3 rB kδk kδk 2 0   ˜t x ˜′t Γ−1 R1 R1 t=1 x T →d kδk 0 rB2 (r)dr 0 B2 (r)B2′ (r)dr   R1 kδk 0 rdB1 (r)  =x ˜t ut →d  R 1 ′ 0 B2 (r)dB1 (r) + H2 (Σ21 + Λ21 )

−1 PT

(b) ΓT

˜t t=1 x





with ΓT = diag(T 3/2 , T, . . . , T ).

Proof. This result is an straightforward application of lemma 2.1 in Park and Phillips (1988) and the details of the proof will be omitted. First note that the stochastic trend of xt is accumulated in x ˜1t and x ˜2t is a driftless nonstationary process. Then we can express the sums in two parts. (a) T −1/2 Γ−1 T

T X t=1



x ˜t = 

T −2 T

PT

˜1t t=1 x

PT −3/2

˜2t t=1 x

 

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M. C. MEDEIROS, E. MENDES, AND L. OXLEY

Then, T −2

T X

x ˜1t = T −2 /kδk

t=1

and T (b)

PT −3/2

˜2t t=1 x

δi2

R1 0

T X

T X

1 + op (1) →d δ ′ δ/2kδk = kδk/2

B2 (r)dr, follows from lemma 2.1 (a). T X m X ( δi2 t2 )2 + op (1) →d δ ′ δ/3

x ˜21t = T −3 /kδk2

t=1

T −5/2

T X t=1

i=1

→d

T −3

m X

t=1 i=1

x ˜1t x ˜2t = T −5/2 /kδk

t=1

m X

δi2

i=1

T

−2

T X

x ˜2t x ˜′2t

T X

Z

1

0

1

Z

rB2 (r)dr

0

t=1

→d

t=1

t˜ x2t + op (1) →d kδk B2 (r)B2′ (r)dr

where the results follow from lemma 2.1 (b), (c) and (d). (c) T

−3/2

T X

x ˜1t ut = T

−3/2

/kδk

t=1

T −1

m X i=1

T X t=1

x ˜2t ut →d

Z

0

1

δi2

T X t=1

tut + op (1) →d

Z

1

rdB1 (r)

0

B2 (r)dB(r) + H2′ (Σ21 + Λ21 )

where the result follows from lemma 2.1 (e).



C OROLLARY 1. Assume assumptions 1, 2 and 3 hold. Let At = A(zt ) : Rm 7→ Rp×k be a continuous function of zt . Then  kδk/2 P ′  (a) T −1/2 Γ−1 At xt →d  R T 1 B2 (r)dr 0  R1 ′ (r)dr 2 /3 rB kδk kδk P 2 0 T −1 ′ ′   (b) Γ−1 R1 R1 t=1 At xt xt At ΓT →d T kδk 0 rB2 (r)dr 0 B2 (r)B2′ (r)dr   R1 kδk rdB (r) P 1 0 T ′  R  (c) Γ−1 I i = 1 At xt ut →d T = 1 ′ (Σ + Λ ) B (r)dB (r) + H 2 1 21 21 2 0

COINTEGRATING SMOOTH TRANSITION REGRESSIONS WITH A STATIONARY TRANSITION VARIABLE

13

where H and ΓT are defined in lemma 1. The k × 1 vector Brownian motion B = [B1 B2′ ] has covariance matrix Ω, defined for ξt′ = [ut , x′t At H2 ]. 

Ω=

ω1

Ω′21

Ω21

Ω2



 = E[ξ1 ξ1′ ] +

∞ X (E(ξ1 ξj′ ) + E(ξj ξ1′ )) = Σ + Λ + Λ′ . j=2

Proof. This result is a straightforward application of lemma 1. Write the process ∆˜ x′t = ∆x′t At = vt′ At −E(vt′ At )+E(vt′ At ) = v˜t +δ. The result follows from items (a), (b) and (c) respectively. 

A PPENDIX B. T HEOREMS B.1. Proof of theorem 1. We show this result by satisfying the conditions (i) – (iv) of theorem 10.1 in Wooldridge (1994). Conditions (i) and (ii) are satisfied by the assumptions and continuity of QT (θ). ¨ t (θ) defined in section 3. We will show that (1) Γ−1 H ′ PT F¨ ˜ Consider Q˙ T (θ) and Q t t=1 tθ0 →p 0.

Hence, it follows that (2)

−1 −1 ′ ′¨ Γ−1 T H Qt (θ0 )HΓT = ΓT H

(23)

T X ′ F˜˙tθ0 F˜˙tθ HΓ−1 T , 0 t=1

which converges in distribution to

(24)

    

R1

′ 0 B2 (r)dr R1 kδk/2 kδk2 /3 kδk 0 rdB2′ (r) R1 R1 R1 ′ rdB (r) B (r)dr kδk 2 2 0 B2 (r)B2 (r)dr 0 0

1

kδk/2



  , 

′ where δ and B2 (r) are as stated in the theorem. Then, we have to show that (3) Γ−1 T H

converges in distribution to

(25)

where κ is defined in the theorem.

    

R1 0

B1 (1) R1 kδk 0 rdB1 (r)

B2 (r)dB1 (r) + κ



  , 

˜˙ ′ t=1 Ftθ0 xt ut

PT

14

M. C. MEDEIROS, E. MENDES, AND L. OXLEY

It remains to show condition (iii). For that, write ¨ T (θ) − Q ¨ T (θ0 ) = A(θ) − B(θ) + C(θ), Q where (26)

A(θ) =

T X ′ ′ ) − F˜˙tθ F˜˙tθ (F˜˙tθ F˜˙tθ t=1

(27)

B(θ) =

T X ˜tθ − F¨ ˜tθ )ut (F¨ 0 t=1

(28)

C(θ) =

T X ˜tθ (F˜tθ − F˜tθ ). F¨ 0 t=1

We show that (4),for θ ∈ ΘT , −1 −1 ′ −1 −1 ′ −1 ′ Γ−1 T γ H A(θ)HΓT γ , ΓT γ H B(θ)HΓT γ , ΓT γ H C(θ)HΓT γ →p 0

uniformly. We start by writing the derivatives of F˜ (zt , xt ; θ) with respect to θ. (29)

(30)

(31)

′ F˜˙tθ = (1, x′t F˙tθ )



˜tθ =  F¨ 

′ = F˜˙tθ F˜˙tθ

1

0

0

0 F¨tθ

 

x′t F˙tθ

′ x ′ ˙ ˙′ F˙tθ t Ftθ xt xt Ftθ

 

and (32)

′ xt = (x′t , G′tθ · x′t , β1 g˙ 1 x1 , . . . , βp g˙ p xp )′ F˙tθ

COINTEGRATING SMOOTH TRANSITION REGRESSIONS WITH A STATIONARY TRANSITION VARIABLE



0



0

  ¨ Ftθ =  0 0 diag(g˙ i xi )  0 diag(g˙ i′ xi ) diag(βi g¨i′ xi )

(33)

(34)

0

′ F˙tθ xt x′t F˙tθ



xt x′t

  = −  −

xt (Gtθ · xt

)′

(Gtθ · xt )(Gtθ · xt

   

xt {[βi gi′ xi ]} )′

(Gtθ ·

xt ){[βi gi′ xi ]}

{[βi gi xi ]}{[βj gj′ xj ]}

−

15

    

where G′tθ = (g1 (zt , ν1 ), . . . , gp (zt , νp ))

(35) (36)

g˙ i =

(37)

g¨i =

∂ gi (zt ; νi ) ∂ν ∂2 gi (zt ; νi ) ∂ν∂ν ′

The dependence on νi and zt are ommited for ease of notation. PT ˜˙ ˜˙ ′ ′ Ftθ Ftθ0 HΓ−1 (1) Show Γ−1 T →d A0 . Write T H 

H=

1

0

0

H∗

 



and ΓT = 

T 1/2

0

0

Γ∗

 

Then, we can rewrite the previous sum as  P T −1/2 Γ∗ −1 H Tt=1 x′t F˙tθ0   −1 −1 ∗ ′ PT ∗ ∗ ′ ′ ∗ ˙ ˙ − Γ H t=1 Ftθ0 xt xt Ftθ0 H Γ 

1

which converges weakly to (24). This result follows from corollary 1, with At = F˙tθ0 . P (2) Show Γ−1 H ′ T F˜˙ u → L . Given Γ and H, rewrite this sum as T

T =1

tθ0 t

d

0

t

 P T −1/2 Tt=1 ut   −1 ∗ ′ PT ′ ∗ ˙ Γ H t=1 Ftθ0 xt ut 

which converges weakly to (25). This result also follows from 1, with At = F˙tθ0 and κ = P ˙′ H2∗ ′ ∞ j=1 E(F1θ0 v1 uj ).

16

M. C. MEDEIROS, E. MENDES, AND L. OXLEY

−1 −γ ′ ′ x can be represented as (3)(a) Show T −γ Γ−1 →p 0. Note that x ˆt = F˙tθ t T H A(θ)HΓT T PT PT P T a integrated process with drift, then x ˆt , x ˆt x ˆ′t and x ˆt ut converges at a higher rate PT PT P than xt , xt x′t and T xt ut respectively. We can see from equation (34) that A(θ) has

only elements of the form aij (θ) = 0, aij (θ) = p(θ)q(zt , θ)xit and aij (θ) = p(θ)q(zt , θ)xit xjt ,

where p(θ) is a polinomial in θ, and q(zt , θ) is a continuous function of zt and θ. Then, for any θ ∈ ΘT , kθ − θ0 k < ǫ implies kaij (θ) − aij (θ0 )k < aij ∗ < ∞. Consequently, the convergence P ′ x x′ F˙ , which converges. Then, rates of A(θ) are smaller than the convergence rates of T F˙tθ t t tθ −1 −γ ′ T −γ Γ−1 →p 0 for some γ > 0. T H A(θ)HΓT T

For (b) and (c), the same rational applies. C.Q.D (M. C. Medeiros) D EPARTMENT

OF

E CONOMICS , P ONTIFICAL C ATHOLIC U NIVERSITY

OF

R IO

DE JANEIRO ,

R IO DE JANEIRO , RJ, B RAZIL . E-mail address: [email protected] (E. F. Mendes) D EPARTMENT OF S TATISTICS , N ORTWESTERN U NIVERSITY, E VANSTON , IL, U.S.A. (L. Oxley) D EPARTMENT OF E CONOMICS , C ANTERBURY U NIVERSITY, C HRISTCHURCH , NZ