Estimation of Threshold Cointegration

Estimation of Myung Hwan London School of Economics December 2006

Outline Model Asymptotics Inference Conclusion 1 Model Estimation Methods Literature 2 Asymptotics Consistency Convergence Rates Asymptotic Distributions 3 Inference 4 Conclusion

Outline Model Asymptotics Inference Conclusion Estimation Methods Literature 1 Model Estimation Methods Literature 2 Asymptotics Consistency Convergence Rates Asymptotic Distributions 3 Inference 4 Conclusion

Model I Model Asymptotics Inference Conclusion Estimation Methods Literature Variables: x t : p-dimensional I (1) vector that is cointegrated with the cointegrating vector β. The first element of β is normalized to 1. z t (β) = x tβ : equilibrium error, or error-correction term X t 1 (β) = (1, z t 1 (β), x t 1,, x t l+1) : the regressor which is a (pl + 2)-dimensional vector. Two-regime threshold vector error correction model (Balke and Fomby 1997) { A x t = X t 1 (β) + u t, if z t 1 (β) γ (A + D) X t 1 (β) + u t, if z t 1 (β) > γ t = l + 1,..., n. That is, x t = A X t 1 (β) + D X t 1 (β) 1 {z t (β) > γ} + u t, where 1 { } is the indicator function.

Model II Model Asymptotics Inference Conclusion Estimation Methods Literature Matrix notation: Let α = vec (A) and δ = vec (D), where vec stacks rows of a matrix and y = X (β) = x l+1. x n X l (β). X n 1 (β), u = u l+1. u n, X γ (β) =, X l (β) 1 {z l (β) > γ}. X n 1 (β) 1 {z n 1 (β) > γ}. Then, u (θ, β) = y (X (β) I p ) α + ( X γ (β) I p ) δ.

Model Asymptotics Inference Conclusion Estimation Methods Literature Estimation I Classification of Parameters: Long-run parameter: the cointegrating vector β Short-run parameters: collectively we denote them as θ. Slope parameters: A and D or α and δ Threshold parameters: γ (β?) Estimation method: 1 Least squares: denoted with the superscript * 2 Smoothed Least squares ( & Linton 2005)

Model Asymptotics Inference Conclusion Estimation Methods Literature Estimation II Least Squares Estimation Objective Function: S n (θ, β) = u (θ, β) u (θ, β), where θ indicates all the short-run parameters (α, δ, γ). LS estimator: ˆθ, ˆβ = arg minsn (θ, β), θ,β where the minimum is taken over a compact parameter space. Concentrated LS estimator: for a fixed (β, γ),» ˆα (β, γ) ˆδ (β, γ)» X (β) X (β) X (β) Xγ 1 (β) X (β) = Xγ (β) X (β) Xγ (β) Xγ (β) Xγ (β) which is then plugged back into S n for optimization over (β, γ). «I p! y,

Model Asymptotics Inference Conclusion Estimation Methods Literature Smoothed LS I Smoothed Least Squares Estimation ( & Linton (2005)) Define a bounded function K ( ) satisfying that Let K t (β, γ) = K zt(β) γ K t (β, γ) : X γ (β) = lim K (s) = 0, lim K (s) = 1. s s + σ n 0 B @ for σ n 0 and replace 1 {z t (β) > γ} with X l (β) K l (β, γ). X n 1 (β) K n 1 (β, γ) 1 C A u (θ, β) = y (X (β) I p) α + (X γ (β) I p) δ. Then, the Smoothed Least Squares (SLS) estimator is ˆθ, ˆβ = arg mins n (θ, β), θ,β where S n (θ, β) = u (θ, β) u (θ, β).

Model Asymptotics Inference Conclusion Estimation Methods Literature Smoothed LS II Concentration» ˆα (β, γ) ˆδ (β, γ)» X (β) X (β) X (β) X γ (β) = X γ (β) X (β) X γ (β) X γ (β) 1 X (β) X γ (β) «I p! y.

Model Asymptotics Inference Conclusion Estimation Methods Literature Literature I Applications of : (PPP) Baum, Barkoulas, & Caglayan (2001), Taylor (2001), Enders & Falk (1998), O Connell (1998), Obstfeld & Taylor (1997), Michael, Mobay, & Peel (1997) (LOP) Lo & Zivot (2001), O Connell & Wei (1997), Parsley & Wei (1996) (Term structure) Balke & Wohar (1998), Baum & Karasulu (1998), Martens, Kofman, & Vorst (1998) Enders & Siklos (2001) (Long-run money demand) Escribano (2004)

Model Asymptotics Inference Conclusion Estimation Methods Literature Literature II Stability of threshold cointegration model - Bec & Rahbek (2004), Saikkonen (2005) Testing for threshold cointegration: Cointegration : Enders & Siklos (2001), Bec Guay & Guerre (2004), M. (2005, 2006), Park & Shintani (2005) Threshold effect (assuming the presence of cointegration) : Hansen & B. (2002) Estimation - No Distribution theory yet. Bec & Rahbek (2004) known cointegrating vector. stationary threshold models: discontinuous TAR: Chan (1993), continuous TAR: Chan & Tsay (1998), subsampling: Gonzalo & Wolf (2005) Regression: Hansen (2000), and Linton (2005) de Jong (2001, 2002) : smooth nonlinear error correction model

Outline Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions 1 Model Estimation Methods Literature 2 Asymptotics Consistency Convergence Rates Asymptotic Distributions 3 Inference 4 Conclusion

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Review of Asymptotics for stationary threshold models I Least Squares Smoothed LS Model Slope Threshold Threshold p Discontinuous n rate n rate nσ 1 n rate (Chan) Normal complex Poisson Normal Continuous n rate n rate n rate (Chan & Tsay) Normal Normal Normal p Diminishing n rate n 1 2α rate n 1 2α σn 1 rate (Hansen) Normal Function of BM Normal If the cointegrating vector is known, these results will apply.

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Consistency Assumption (1) (a) {u t } is an independent and identically distributed sequence with Eu t = 0, Eu t u t = Σ that is positive definite. Furthermore, it has a density wrt Lebesque measure that is everywhere positive. (b) { x t, z t } is a sequence of strictly stationary strong mixing random variables with mixing numbers α m, m = 1, 2,..., that satisfy α m = o ( m (α0+1)/(α0 1)) as m for some α 0 1, and for some ε > 0, E X t Xt α0+ε < and E X t 1 u t α0+ε <. Furthermore, E x t = 0 and x [ns] / n converges weakly to a vector Brownian motion B with the covariance matrix, which is the long-run covariance matrix of x t and has rank p 1 s.t. β 0 = 0. (c) E [ X t 1 D 0D 0 X ] t 1 z t 1 > 0 a.s. Condition (c) implies the discontinuity of the model

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Consistency Theorem (1) Under Assumption 1, n ( ) (ˆθ ) ˆβ β 0 and θ 0 are o p (1). In addition, assume that s 2 1 {s > 0} K (s) < M <, for any s R. Then, ( ) ) n ˆβ β 0 and (ˆθ θ 0 are o p (1). Sketch of Proof 1 Show n ˆβ β 0 = O p (1) by contradiction. 2 Show ˆθ n ˆβ β 0 and θ 0 are o p (1) based on ULLN. 3 Show the difference between S n and S n are asymptotically uniformly negligible.

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Convergence Rate Long-run vs. Short-run parameters Threshold vs. Slope parameters Smoothed vs. Unsmoothed estimator Theorem (2) Under Assumption 1, ˆβ ( = β 0 + O ) ( p n 3/2 and ˆγ = γ 0 + O ) p n 1. In consequence, ˆα and ˆδ converge at the rate of n.

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Asymptotic Distribution I Assumption (2) (a) E[ X t u t r ] <, E[ X t X t r ] <, for some r > 4, (b) { x t, z t } is a sequence of strictly stationary strong mixing random variables with mixing numbers α m, m = 1, 2,..., that satisfy α m Cm (2r 2)/(r 2) η for positive C and η,as m. (c) For some integer h 1 and each integer i such that 1 i h, all z in a neighborhood of γ, almost every, and some M <, f (i) (z ) exists and is a continuous function of z satisfying f (i) (z ) < M. In addition, f (z ) < M for all z and almost every. (d) and the conditional joint density f (z t, z t m t, t m ) < M, for all (z t, z t m ) and almost all ( t, t m ). (e) θ 0 is an interior point of Θ.

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Asymptotic Distribution II Assumption (3) (a) K is twice differentiable everywhere, K (1) ( ) and K (2) ( ) are uniformly bounded, and each of the following integrals is finite: K (1) 4, K (2) 2, v 2 K (2) (v) dv. (b) For some integer h 1 and each integer i (1 i h), v i K (1) (v) dv <, and and K (x) K (0) 0 if x 0. s i 1 sgn (s) K (1) (s) ds = 0, and s h sgn (s) K (1) (s) ds 0,

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Asymptotic Distribution III Assumption (3 cont d) (c) For each integer i (0 i h), and η > 0, and any sequence {σ n } converging to 0, lim n σi h n s i K (1) (s) ds = 0, and lim n σ 1 n (d) lim supnσn 2h < and n lim n σ 2h n σ ns >η σ ns >η σ ns >η K (2) (s) ds = 0. K (1) (s) ds = 0.

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Asymptotic Distribution IV Assumption (3 cont d) (e) For some µ (0, 1], a positive constant C, and all x, y R, K (2) (x) K (2) (y) C x y µ. (f )For some sequence m n 1, and ε > 0, ( ) 1 log (nm n ) n 1 6/r σnm 2 2 n 0 σ 3k 1 n n 3/r+ε α mn 0.

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Asymptotic Distribution V Condition (f ) serves to determine the rate for σ n. When the data are i.i.d. and the regressors possess a moment generating function, the conditions can be weakened to log (n) nσ 2 n 0, since α mn = 0 and we can set m n = 1 in this case. Although condition (e) provides permissible rates for the bandwidth selection, it may not be sharp.

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Asymptotic Distribution VI Let B be the limit Brownian motion of the partial sum process of x t, whose covariance matrix is Ω, and [ K σv 2 = E (1) 2 ( ) ] X 2 t 1 D 0 u t + K (1) 2 ( X t 1 D 0 D ) 2 0X t 1 zt 1 = γ 0 f (γ 0 ) 2 σ 2 q= K (1) (0) E ( X t 1D 0 D 0X t 1 z t 1 = γ 0 ) f (γ0 ), where K (i) is the i-th derivative of K, K (1) (s) = K (1) (s) (1 {s > 0} K (s)), and g 2 2 = ( g 2) 2

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Asymptotic Distribution VII Theorem Suppose Assumption 1-3 hold. Let W denote a standard Brownian motion that is independent of B.Then, ( ( ) ) 1/2 nσ ˆβ β0 n nσ 1 n (ˆγ γ 0 ) d σ v σ 2 q ( 1 1 0 BB 1 0 B 1 0 B ) 1 ( BdW W (1) ( ) ( [ ( ) ] 1 ˆα α0 d 1 dt 1 n N 0, E X ˆδ δ 0 d t 1 d t 1 X t 1 Σ) t 1 and they are asymptotically independent. The unsmoothed estimator ˆα and ˆδ have the same asymptotic distribution as ˆα and ˆδ. ),

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Two-step Estimation I Corollary ( ) Let ˆγ (β) be the smoothed estimator of γ when β is given. Then, ˆγ ˆβ has ( ) the same asymptotic distribution as that of ˆγ (β 0 ), which is N 0, σ2 v. σq 4 Thus, we do not need to estimate the long-run variance for the inference for γ. ( Suppose that β n = β + O ) p n 1. For example, β n can be obtained from the simple OLS of the first element of x t to the other elements of x t, as in Engle-Granger procedure. Note that, by multipying β 0 both sides of the model { A x t = X t 1 (β) + u t, if z t 1 (β) γ (A + D) X t 1 (β) + u t, if z t 1 (β) > γ,

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Two-step Estimation II we obtain a threshold autoregressive process for z t = x tβ 0 { a z t = 0 z t 1 + a 1 z t 1 + + u t, if z t 1 γ (a 0 + d 0 ) z t 1 + (a 1 + d 1 ) z t 1 + + u t, if z t 1 > γ But, the plug-in estimate ˆα (β n ) and ˆδ (β n ) do not have the same asymptotic distribution as ˆα (β 0 ) and ˆδ (β 0 ). To treat the estimate β n as the true value β 0, we first need 1 x t k K t 1 (β n, γ n ) x n t 1 (β n β 0 ) = o p (1) t

Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions Two-step Estimation III In case ( ) of Ez t = 0, we can still retain the Normality by replacing the z t 1 β with ( ( ) z t 1 β = x t 1 β = x t 1 1 n ) x s 1 β, for any n-consistent β, as in de Jong (2001). It is worth noting, however, that the asymptotic variance increases by doing this. s

Outline Model Asymptotics Inference Conclusion 1 Model Estimation Methods Literature 2 Asymptotics Consistency Convergence Rates Asymptotic Distributions 3 Inference 4 Conclusion

Inference I Model Asymptotics Inference Conclusion Asymptotic Variance Estimation: long-run covariance matrix Ω : see Andrews (1991) for example. variance of threshold estimate (σ v and σ q) : Let ˆτ t = 1 (1) X t 1 ˆβ DK t 1 ˆβ, ˆγ û t, (1) σn where û t is the regression residual, and let ˆσ 2 v = 1 n X t ˆτ t 2, and ˆσ q 2 = ˆθ σn n Qn22, where Q n22 is the diagonal element corresponding to γ of the second derivative of S n. Refer to and Linton (2005). variance of slope estimate: the same as linear regression. We do not have to estimate the density and conditional expectation by a nonparametric method. Hansen (2000) uses a nonparametric method to estimate the asymptotic variance; the introduction of a smoothing parameter is necessary.

Model Asymptotics Inference Conclusion Inference II Testing for two thresholds: To test the null of one threshold against two threshold, we can extend the SupLM test statistic of Hansen and (2002) that examines the null of no threshold. Due to the fast convergence rate of the least squares estimator of β and γ, we can treat the estimated β and γ as if known. For the p-value, the fixed regressor bootstrap and residual bootstrap can be employed as described there.

Outline Model Asymptotics Inference Conclusion 1 Model Estimation Methods Literature 2 Asymptotics Consistency Convergence Rates Asymptotic Distributions 3 Inference 4 Conclusion

Model Asymptotics Inference Conclusion Conclusion The paper establishes the consistency and convergence rates of the LSE and SLSE of the threshold vector error correction model. In particular, The LSE of the cointegrating vector estimate converges extremely fast. This validates a two-step estimation, in which the cointegrating vector is first estimated by LS and the other short-run parameters by SLS. The LSE of the slope is asymptotically not affected by the estimation of the other parameters The limit distribution of the SLSE is derived and thus providing a way of inference for the cointegrating vector. Multiple-Regime Threshold: We can estimate the thresholds simultaneously or sequentially. Hansen (1999) and Bai and Perron (1998). More than one cointegrating relation