Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

Transcription

1 Nonlinear Error Correction Model and Multiple-Threshold Cointegration Man Wang Dong Hua University, China Joint work with N.H.Chan May 23, 2014 Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

2 1 Introduction 2 Least Squares Estimator (LSE) of TVECM 3 Smoothed LSE (SLSE) 4 Simulation and Empirical Studies Simulation Study Term Structure of Interest Rates Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

3 1 Introduction 2 Least Squares Estimator (LSE) of TVECM 3 Smoothed LSE (SLSE) 4 Simulation and Empirical Studies Simulation Study Term Structure of Interest Rates Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

4 Cointegration and ECM Linear Cointegration: If x 1,t and x 2,t are both I(1), and x 1,t + βx 2,t is I(0), then we say that x 1,t and x 2,t are cointegrated. Granger representation theorem: (x 1,t, x 2,t ) has a vector error correction model (VECM) representation as [ x1,t x 2,t ] = [ c1 c 2 ] [ a1 + a 2 ] [ b11 b z t b 21 b 22 ] [ x1,t 1 x 2,t 1 ] [ u1,t + u 2,t ], where z t 1 = x 1,t 1 + βx 2,t 1 is the error correction term, called the equilibrium error. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

5 Threshold Cointegration and TVECM The linear cointegration implies consistent adjustment towards long-run equilibrium. Balke and Fomby (1997) proposed threshold cointegration, assuming that the system adjustment follows a threshold model. The threshold vector ECM (TVECM): [ ] x1,t x 2,t 3 = j=1 + ( [ ] (j) [ c1 a1 + c 2 a 2 ] (j) z t 1 [ ] b11 b (j) [ ]) [ 12 x1,t 1 u1,t 1{z b 21 b 22 x t 1 in j th regime} + 2,t 1 u 2,t ] (j) Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

6 m-regime TVECM Variables: x t : p dimensional I(1) vector, cointegrated with single cointegrating vector (1, β ). z t 1 (β) = x 1,t 1 + x 2,t 1β: the error correction term, where x t = (x 1,t, x 2,t). X t 1 (β) = (1, z t 1 (β), x t 1,..., x t l ), where x t i, i = 1,..., l are the lagged difference terms. An m-regime TVECM: x t = m A jx t 1 (β)1{γ j 1 z t 1 (β) < γ j }+u t, t = l+1,..., n. (1) j=1 = γ 0 < γ 1 <... < γ m = are thresholds and A j is coefficient in the j-th regime. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

7 Multiple-threshold cointegration is widely used, but estimation theories are limited to the one threshold cointegration (two-regime TVECM). Hansen and Seo (2002) constructed the MLE whose consistency is still unknown. Seo (2011) gave the asymptotics of the LSE and SLSE. Goal: estimation of multiple threshold cointegration (m-regime TVECM with m > 2). Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

8 1 Introduction 2 Least Squares Estimator (LSE) of TVECM 3 Smoothed LSE (SLSE) 4 Simulation and Empirical Studies Simulation Study Term Structure of Interest Rates Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

9 An m regime TVECM: x t = m A jx t 1 (β)1{γ j 1 z t 1 (β) < γ j } + u t, t = l + 1,..., n. j=1 We denote: X j (β, γ) = X l (β)1{γ j 1 z l (β) < γ j }. X n 1 (β)1{γ j 1 z n 1 (β) < γ j }, y = ( x ( l+1,..., ) x n λ = vec A 1,..., A m, ), u = ( u l+1,..., u n ), Then, (( ) ) y = X1 (β, γ),..., X m (β, γ) I p λ + u. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

10 Let S n (θ) = u u, 1 LS estimator: ˆθ = argmin θ Θ S n (θ). θ = (β, γ = (γ 1, γ 2,..., γ m 1 ), λ ) Θ. 2 2-step procedure: [( Fix (β, γ), y = X1 (β, γ), X ) ] 2 (β, γ),..., X m (β, γ) I p λ + u. ˆλ(β, γ) = X 1 X1 X 1 X2 X 1 X m X1 X2... X1 X m X2 X2... X2 X m... X m X2... X m X m Step1: ( ˆβ, ˆγ) = argmin S n (β, γ, ˆλ(β, γ)), (β,γ) Θ β,γ Step2: ˆλ = ˆλ( ˆβ, ˆγ). 1 X 1 X 2. X m I p y. (2) Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

11 Assumptions The parameter space Θ is compact with max { λ z1, λ zm } and min 1 i<j m 1 { γ i γ j } bounded away from zero. 1.2 {u t } is an i.i.d. sequence of random variables with Eu t = 0, Eu t u t = Σ. 1.3 { x t, z t } is a sequence of strictly stationary strong mixing random vectors 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 X t α0 1 < and E X t 1 u t α0+ε <. Furthermore, E x t = 0, and the partial sum process, x [ns] / n, s [0, 1], converges weakly to a vector Brownian motion B with a covariance matrix Ω, which is the long-run covariance matrix of x t and has rank p 1 such that (1, β 0)Ω = 0. In particular, assume that x 2[nx] / n converges weakly to a vector of Brownian motion B with a covariance matrix Ω, which is finite and positive definite. 1.4 Let u t (ξ, γ, λ) be defined as the error u t by replacing z t (β) by z t + ξ, where ξ belongs to a compact set in R and let S(ξ, γ, λ) = E(u t(ξ, γ, λ)u t (ξ, γ, λ)). Then, assume that 1/nΣ Nonlinear u Error (ξ, γ, Correction λ)u (ξ, Model γ, and λ) Multiple-Threshold p S(ξ, γ, λ) Cointegration uniformly in May 23, / 31

12 Assumptions The probability distribution of z t has a density with respect to the Lebesgue measure that is continuous, bounded, and everywhere positive, and that the density function f(z t x 2,t ) is bounded by M > 0 for almost every x 2,t, t = 1, 2,..., n. 2.2 E[X t 1 (A0 j A0 j+1 )(A0 j A0 j+1 ) X t 1 z t 1 = γ 0 j ] > 0 j = 1, 2,..., m 1. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

13 Convergence rate of LSE Theorem 1 Under Assumptions 1 and 2, ˆθ is consistent, further n 3/2 ( ˆβ β 0 ) is O p (1) and n(ˆγ γ 0 ) = O p (1), n(ˆλ λ 0 ) is asymptotically normal. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

14 1 Introduction 2 Least Squares Estimator (LSE) of TVECM 3 Smoothed LSE (SLSE) 4 Simulation and Empirical Studies Simulation Study Term Structure of Interest Rates Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

15 Smoothed Least Squares Estimator The objective function of the LSE of a threshold model is discontinuous. Seo and Linton ( 2007) proposed to smooth it by replacing 1{s > 0} by a smooth function K(s/h), where h is bandwidth. The minimizer of the smoothed objective function is called the SLSE. Herein K(s) is smooth, bounded, with lim s K(s) = 0, lim The smoothing is reasonable because s + K(s) = 1. K(s/h) 1{s > 0}, as h 0. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

16 SLSE for an m regime TVECM: x t = Let m A jx t 1 (β)1{γ j 1 z t 1 (β) < γ j }+u t, t = l+1,..., n. (3) j=1 K t 1 (β, γ j 1, γ j ) = K( z t 1(β) γ j 1 h ) + K( γ j z t 1 (β) ) 1, h and replace 1{γ j 1 z t 1 (β) < γ j } with K t 1 (β, γ j 1, γ j ). Denote X j (β, γ) = X t 1(β)K t 1 (β, γ j 1, γ j ), then the smoothed objective function become S n = u u, where u = y [(X 1(β, γ),..., X m(β, γ)) I p ] λ. ˆθ = argminsn is defined as SLSE of m regime TVECM. θ Θ Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

17 Assumptions K is twice differentiable everywhere, K (1) is symmetric around zero, and K (1) and K (2) are uniformly bounded and uniformly continuous. Furthermore, K (1) (s) 4 ds, s 2 K (1) (s) ds, K (2) (s) 2 ds and s 2 K (2) (s) ds are finite. 3.2 K(x) K(0) has the same sign as x and for some integer ν 2 and each integer i (1 i ν), s i K (1) (s) ds <, s i 1 sgn(s)k (1) (s)ds = 0 and s ν 1 sgn(s)k (1) (s)ds Furthermore, for some ɛ > 0, lim n hs >ɛ hi ν s i K (1) (s) ds = 0 and lim n hs >ɛ h 1 s i K (2) (s) ds = {h} is a function of n and satisfies that for some sequence q 1, nq 3 0, log(nq)(n 1 6/r h 2 q 2 ) 1 0, and h 9k/2 3 n (9k/2+2)/r+ɛ α q 0, where k is the dimension of θ and r > 4 is specified in Assumptions 4. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

18 Assumptions E[ X t u t r ] <, E[ X t X t r ] <, for some r > { x t, z t } is a sequence of strictly stationary strongly mixing random vectors with mixing coefficients α m, m = 1, 2,..., that satisfy α m Cm (2r 2)/(r 2) η for positive C and η, as m. 4.3 For some integer ν 2 and each integer i such that 1 i ν 1, for all z in a neighborhood of threshold γ j, j = 1,..., m 1, for almost every x 2,t and some M <, f (i) (z x 2,t ) exists and is a continuous function of z satisfying f (i) (z x 2,t ) < M. In addition, f(z x 2,t ) < M for all z and almost every x 2,t. 4.4 The conditional joint density f(z t, z t m x 2,t, x 2,t m ) < M, for all (z t, z t m ) and almost every (x 2,t, x 2,t m ). 4.5 θ 0 is an interior point of Θ. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

19 Limiting Distribution Limiting distribution of ˆθ. Theorem 2 Under Assumptions 1-4, ( nh 1/2 ( ˆβ β 0 ) nh 1 (ˆγ γ 0 ) ) d = σq 2 10 BB σq 2 10 B σ q B... σ q B m 1 σq 2 10 B σ 2 1 q σq 2 10 B σ 2 m 1 q m 1 1 Bd m 1 j=1 σv j W j σ v1 W 1 (1) σ v2 W 2 (1). σ vm 1 W m 1 (1), and n(ˆλ λ 0 ) = N 0, E ( distribution. nh 1/2 ( ˆβ β 0 ) nh 1 (ˆγ γ 0 ) I I I m X t 1X t 1 1 Σ, ) is asymptotically independent of n(ˆλ λ 0 d ). Here = stands for convergence in Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

20 Notations in Theorem 2 are as follows: B, W 1,..., W m 1 are mutually independent Brownian motions. I j = 1{γ 0 j 1 z t 1 < γ 0 j }, here γ0 is the true value of threshold parameters. K 1 (s) = [ K (1) (1(s > 0) ] K(s)). For j = 1,..., m 1, σv 2 = E F j j z t 1 = γj 0 f Z (γj 0 ), where F j = K (1) 2 2 (X t 1 (A0 j A0 j+1 )u t) 2 + K (X t 1 (A0 j A0 j+1 )(A0 j A0 j+1 ) X t 1 ). For j = 1,..., m 1, σ 2 q j = K (1) (0)E[X t 1 (A0 j A0 j+1 )(A0 j A0 j+1 ) X t 1 ) z t 1 = γ 0 j ]f Z (γ 0 j ) and σ 2 q = m 1 j=1 σ2 q j. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

21 1 Introduction 2 Least Squares Estimator (LSE) of TVECM 3 Smoothed LSE (SLSE) 4 Simulation and Empirical Studies Simulation Study Term Structure of Interest Rates Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

22 Model specification: ( 1 x t = 0.8 ( ) z t 1 1{z t 1 < γ 10 } + ( 1 0 ) z t 1 1{z t 1 γ 20 } + u t, ) z t 1 1{γ 10 z t 1 < γ 20 } where z t 1 = x 1,t 1 + β 0 x 2,t 1, β 0 = 2. u t i.i.d N(0, I 2 ) for t = l + 1,... n, and x 0 = u 0. Case 1: γ 0 = ( 1, 1), Case 2: γ 0 = ( 3, 3). Sample sizes n=100 and 250 for case 1, and n=250 for case replications for each experiment. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

23 Notation of estimators. β: the Johansen s MLE. ˆβ, ˆγ: the LSE. ˆβ, ˆγ the SLSE. ˆγ r (ˆγ r ) the restricted LSE (SLSE) when β 0 is given. ˆβ s and ˆγ s ( ˆβ s and ˆγ s ) are the sequential LSE (SLSE). That is, the two thresholds are selected sequentially. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

24 Comparison of estimation for different n. mean sd MAE in log n=100 ˆβ β e ˆγ 1 γ ˆγ 2 γ ˆβ β e ˆγ 1 γ ˆγ 2 γ n=250 ˆβ β e ˆγ 1 γ ˆγ 2 γ ˆβ β e ˆγ 1 γ ˆγ 2 γ Conclusion: the performance of the LSE and SLSE is much improved when n. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

25 Comparison of estimation for different γs. mean sd MAE in log case 1, n=250, γ 0 = c(1, 1) ˆβ β e ˆγ 1 γ ˆγ 2 γ ˆβ β e ˆγ 1 γ ˆγ 2 γ case 2, n=250, γ 0 = c(3, 3) ˆβ β e ˆγ 1 γ ˆγ 2 γ ˆβ β e ˆγ 1 γ ˆγ 2 γ Conclusion: the performance of the LSE and SLSE declines when magnitude of γ. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

26 Result of different estimators for case I, n=250. mean sd MAE in log β β e ˆβ β e ˆβ β e βˆ s β e ˆβ s β e LSE and SLSE simultaneous estimators for γ ˆγ 1 γ e ˆγ 2 γ e ˆγ 1 γ e ˆγ 2 γ e LSE and SLSE sequential estimators for γ ˆγ s,1 γ e ˆγ s,2 γ e ˆγ s,1 γ e ˆγ s,2 γ e LSE and SLSE estimators when β 0 known ˆγ r,1 γ e ˆγ r,2 γ e ˆγ r,1 γ e ˆγ r,2 γ e Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

27 For estimators of β, The Johansen s MLE does not perform as well as the other estimators. The LSE outperforms the SLSE. For estimators of γ, The LSE ˆγ performs slightly better than the SLSE ˆγ. ˆγ and ˆγ shows superiority over the sequential estimators ˆγ s and ˆγ s respectively. ˆγ r and ˆγ r outperform the unrestricted estimators ˆγ and ˆγ. Improvement from ˆγ to ˆγ r is limited. Conclusions. Simulation results agree with the the theories developed. With our choice of K and h, the SLSE performs almost as we as the LSE and is therefore recommended. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

28 Empirical Study Campbell and Shiller (1987): the term structure of interest rates implies that long-term and short-term interest rates are cointegrated with α = (1, 1). R t and r t : interest rates of the twelve-month and 120-month bonds of the US during in the period Hansen and Seo (2002) found them to be threshold cointegrated via a two-regime TVECM estimated by the MLE: { R t = zt R t r t 1 + u 2,t, z t R t r t 1 + u 1,t, if z t , if z t 1 > 0.63, { r t = zt R t 1.04 r t 1 + u 2,t, z t R t r t 1 + u 2,t, if z t , if z t 1 > 0.63, where z t 1 = R t r t 1. The estimated cointegrating vector (1, 0.984) is very close to (1, 1) and (8%,92% ) of data fall into the two regimes. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

29 We take one step further and consider a three-regime TVECM. By the LSE, the model is estimated as: R t = r t = z t R t 1.28 r t 1 + u 1,t, if z t , z t R t r t 1 + u 1,t, if 1.58 z t 1 > 0.62, z t R t r t 1 + u 1,t, if z t 1 > 1.58, z t R t 1.28 r t 1 + u 2,t, if z t , z t R t r t 1 + u 2,t, if 1.58 z t 1 > 0.62, z t R t r t 1 + u 2,t, if z t 1 > (1, ˆβ) = (1, 0.983), data in three regimes (8%, 72%, 20%). For the SLSE, the model is estimated as: R t = r t = z t R t 1.25 r t 1 + u 1,t, if z t , z t R t r t 1 + u 1,t, if 1.58 z t 1 > 0.84, z t R t r t 1 + u 1,t, if z t 1 > 1.58, z t R t 1.22 r t 1 + u 2,t, if z t , z t R t r t 1 + u 2,t, if 1.58 z t 1 > 0.84, z t R t r t 1 + u 1,t, if z t 1 > (1, ˆβ ) = (1, 0.981), data in three regimes (5.4%, 73.15%, 21.45%). Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

30 A three-regime TVECM seems to be more reasonable: 1 Situations of R t relatively high and R t relatively low are both abnormal. 2 The three-regime models keep the data in the left regime and divide the rest to two more regimes. 3 Both the two models are different in the middle and right regimes. They both indicate: market is more sensitive to low R t than high R t. Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31

31 Thank You! Nonlinear Error Correction Model and Multiple-Threshold Cointegration May 23, / 31