Threshold models: Basic concepts and new results

Transcription

1 Threshold models: Basic concepts and new results 1 1 Department of Economics National Taipei University PCCU, Taipei, 2009

2 Outline

3 1 Structural Change Model (Chow 1960; Bai 1995)

4 1 Structural Change Model (Chow 1960; Bai 1995) { β1 x y t = t + u t if t T 1 β 2 x t + u t if t > T 1

5 1 Structural Change Model (Chow 1960; Bai 1995) { β1 x y t = t + u t if t T 1 β 2 x t + u t if t > T 1 2 Markov Switching Model (Hamilton 1989)

6 1 Structural Change Model (Chow 1960; Bai 1995) { β1 x y t = t + u t if t T 1 β 2 x t + u t if t > T 1 2 Markov Switching Model (Hamilton 1989) 3 Threshold Model (Tong 1978, 1983, 1990)

7 1 Structural Change Model (Chow 1960; Bai 1995) { β1 x y t = t + u t if t T 1 β 2 x t + u t if t > T 1 2 Markov Switching Model (Hamilton 1989) 3 Threshold Model (Tong 1978, 1983, 1990) { β1 y y t = t 1 + u t if y t 1 c β 2 y t 1 + u t if y t 1 > c

8 Threshold Autoregressive models Basic Threshold Autoregressive model is as following: { β1 x y t = t + u t if z t c β 2 x t + u t if z t > c If the threshold variable z t is equal to lagged dependent variable y t 1, it is called Self-exciting autoregressive model { β1 y y t = t 1 + u t if y t 1 c β 2 y t 1 + u t if y t 1 > c

9 Basic Assumptions 1 u t is i.i.d. mean-zero sequence with a bound density function, and E u t 2γ < for some γ > 2. 2 β 1 1 and β 2 < 1 or β 1 < 1 and β 2 1 Under these assumptions satisfied, SETAR model will be stationary.

10 Smooth transition SETAR models Consider the threshold models as below: y t = β 1 y t 1 + β 2 y t 1 G(y t 1, θ) + u t We use a smooth function to make sure threshold models continuous at threshold value θ. Generally the transition functions considered in literature are 1 Logistic distribution (LSTAR model) G(y t 1, θ) = [1 + exp(κ(y t 1 c))] 1 2 Exponential distribution (ESTAR model) G(y t 1, θ) = 1 exp( κ(y t 1 c) 2 )

11 Threshold models under different types of data Cross section data { β1 x y t = t + u t if z t c β 2 x t + u t if z t > c Time series data { β1 y y t = t 1 + u t if y t 1 c β 2 y t 1 + u t if y t 1 > c Panel data { β1 y y it = it 1 + u it if y it 1 c β 2 y it 1 + u it if y it 1 > c

12 Specification tests for threshold autoregressive model Test for nonlinearity If we have accurate threshold value c, 1 Chow (1960) test 2 Ramsey (1969) RESET test 3 CUSUM test If we need to estimate threshold value c in advance, 1 Ramsey (1969) RESET test 2 CUSUM test

13 SETAR model vs. linear model (Davies 1977, 1987, Andrews and Ploberger 1991) Estimate unrestricted equation y t = β 1 y t 1 I(y t 1 c) + β 2 y t 1 I(y t 1 > c) + u t and restricted equation y t = β 1 y t 1 + u 1t. Calculate regression variance ˆσ 2 and σ 2 for two equations. The F-test can be obtain as below: ( σ(c) 2 ˆσ(c) 2 ) F n (c) = n ˆσ(c) 2

14 LSTAR model vs. ESTAR model (Luukkonen, Saikkonen and Teräsvirta 1988; Granger and Teräsvirta 1993) Auxiliary regression y t = θ 0 + θ 1 y t 1 + θ 2 y 2 t 1 + θ 3y 3 t 1 + θ 4y 4 t 1 + ɛ t (1) We use sequential method to test LSTAR and ESTAR models. 1 If We reject H LSTAR4 : θ 4 = 0, LSTAR model will be choose. 2 If We accept H LSTAR4 : θ 4 = 0 and reject H ESTAR3 : θ 3 = 0 θ 4 = 0, ESTAR model will be choose. 3 If We accept H LSTAR4 : θ 4 = 0 and H ESTAR3 : θ 3 = 0 θ 4 = 0 and reject H ESTAR3 : θ 2 = 0 θ 3 = θ 4 = 0, ESTAR model will be choose.

15 Estimation for threshold model How to obtain consistent estimator for threshold value c? Concentrated least square method (Bai 1996) Estimate the threshold value ĉ by minimizing a concentrated two stage least square criterion S n (γ) = n (y i β 1 (c)x i I(q i c) β 2 (c)x i I(q i > c)) 2 i=1

16 After c obtained, the consistent coefficients can be estimated by Cross section case 1 Nonlinear least square methods 2 Instrument variable methods (Caner and Hansen 2004) 3 LAD methods (Caner and Hansen 2002) Time series case 1 Nonlinear least square method

17 Basic Concepts Consider the SETAR model u t if c 1 y t 1 c 2 y t = ϕ 1 (y t 1 c 1 ) + u t if y t 1 < c 1 (2) ϕ 2 (y t 1 c 2 ) + u t if y t 1 > c 2 If we want to detect whether the true data generation process is I(1), we need test H 0 : ϕ 1 = ϕ 2 = 0. Pippenger and Goering (1993) find ADF test with low power for nonlinear alternative.

18 Basic Concepts Under null hypothesis satisfied, Equation (2) is equal to I(1) process. The threshold value c cannot be identified (Davies 1977, 1987). The method to solve Davies problem is to find all possible threshold values. Calculate all ADF statistic critical values under different threshold values and find a smallest one.

19 When we consider possible threshold value, there are two possible setting in threshold value space. 1 fixed space (González and Gonzalo 1997; Berben and van Dijk 1999, Caner and Hansen 2001, Seo 2004): c C, C is fixed. 2 Adaptive space (Bec, Ben Salem and Carrasco 2004, Bec, Guay and Guerre 2004, Park and Shintani 2005, de Jong, Wang and Bae 2007): c C, the upper and lower bounds of C grow at rate T 1/2.

20 From the different consideration about residual structure. 1 u t is martingale difference sequence: infimum Dickey-Fuller test (González and Gonzalo 1997; Berben and van Dijk 1999, Caner and Hansen 2001, Seo 2004) 2 u t is weakly stationary sequence: infimum ADF test (Park and Shintani 2005) or infimum Phillips-Perron test (de Jong, Wang and Bae 2007)

21 Threshold unit root tests for smooth transition models Consider the smooth transition model y t = ρy t 1 G(y t 1, θ) + u t Nonlinear least square methods (Park and Shintani 2005) to estimate coefficients and test whether H 0 : ρ = 0. Use Taylor expansion to expend G(y t 1, θ) 1 Exponential STAR model (G(y t 1, θ) = 1 exp( κy 2 t 1 )) y t = φy 3 t 1 + ɛ t Detect H 0 : φ = 0. 2 Logistic STAR model (G(y t 1, θ) = [1 + exp(κ(y t 1 c))] 1 )

22 Use Taylor expansion Test H 0 : θ 1 = θ 2 = 0 y t = θ 1 y t 1 + θ 2 y 2 t 1 + u t

23 Panel threshold model Hansen (1999) propose panel threshold model for a balanced panel. y it = β 1 x it I(q it c) + β 2 x it I(q it > c) + µ i + u it (3) He uses concentrate least square to search c. After c obtained, fixed effect transformation is used. The estimated coefficients are consistent with fixed T as n.

24 Panel smooth transition threshold unit root tests If the panel smooth threshold model with continuous transition function, Exponential distribution function 1 Chiang, Kuan and Lo 2006 y it = ρ i y it 1 [1 exp( c i y 2 it 1)] + u it Using Taylor expansion and ADF statistic to test H 0 : φ i = 0 i ki y it = φ i yit π ij y it j + ɛ it j=1 2 Cerrato, de Peretti and Sarantis 2007 y it = ρ i y it 1 [1 exp( c i y 2 it 1)] + u it u it = λ i f t + e it

25 Use Pesaran (2007) panel unit root test for cross section dependence k i y it = φ i yit hiȳ3 it 1 + g ij ȳ it j + π ij y it j + ɛ it j=0 k i j=1 Logistic distribution function He and Sandberg (2006) consider LSTAR model. They use Taylor expansion to rewrite original LSTAR model y it = α i + δ i y it 1 + ρ(y it 1 )y it 1 + ũ it Test H 0 : δ i = 0 i and ρ(y it ) = 1.

26 Estimation for panel threshold model 1 Hansen (1999) Fixed effect transformation methods 2 Shin (2006), Gørgens, Skeel and Würtz (2008) GMM approach.

27 1 Multivariate Error Correction Approach (Balke and Fomby 1997, Hansen and Seo 2002, Seo 2006)

28 1 Multivariate Error Correction Approach (Balke and Fomby 1997, Hansen and Seo 2002, Seo 2006) y 1t = βy 2t + z t y 2t = ɛ 2t z t = ρ 1 z t 1 I(q t 1 γ) + ρ 2 z t 1 I(q t 1 > γ) + ɛ 1t

29 1 Multivariate Error Correction Approach (Balke and Fomby 1997, Hansen and Seo 2002, Seo 2006) y 1t = βy 2t + z t y 2t = ɛ 2t z t = ρ 1 z t 1 I(q t 1 γ) + ρ 2 z t 1 I(q t 1 > γ) + ɛ 1t Y t = Π 1 Y t 1 I(q t 1 γ) + Π 2 Y t 1 I(q t 1 > γ) + ɛ t

30 1 Multivariate Error Correction Approach (Balke and Fomby 1997, Hansen and Seo 2002, Seo 2006) y 1t = βy 2t + z t y 2t = ɛ 2t z t = ρ 1 z t 1 I(q t 1 γ) + ρ 2 z t 1 I(q t 1 > γ) + ɛ 1t 2 Single equation Approach (Arai 2004, Gonzalo and Pitarakis , Saikkonen 2008, Saikkonen and Choi 2004)

31 1 Multivariate Error Correction Approach (Balke and Fomby 1997, Hansen and Seo 2002, Seo 2006) y 1t = βy 2t + z t y 2t = ɛ 2t z t = ρ 1 z t 1 I(q t 1 γ) + ρ 2 z t 1 I(q t 1 > γ) + ɛ 1t 2 Single equation Approach (Arai 2004, Gonzalo and Pitarakis , Saikkonen 2008, Saikkonen and Choi 2004) { y1t = βy 2t + γy 2t g(y 2t ; θ) + ɛ 1t y 2t = ɛ 2t

32 Estimation of threshold cointegration In threshold cointegration model, We use concentrate least square to estimate Π 1, Π 2 and γ 1 Given γ, the least squares estimators are ˆΠ 1 (γ) = Y t Z 1 (Z 1 Z 1 ) 1 and ˆΠ 1 (γ) = Y t Z 2 (Z 2 Z 2 ) 1, where Z 1 = Y t 1 I(q t 1 γ) and Z 2 = Y t 1 I(q t 1 > γ). 2 Calculate γ with arg min γ U U, where U = Y ˆΠ 1 (γ)z 1 ˆΠ 2 (γ)z 2. 3 Substitute ˆγ into threshold error correction model to get ˆΠ 1 and ˆΠ 2.

33 Testing for threshold cointegration models 1 Threshold cointegration against linear cointegration (Hansen and Seo 2002) Y t = Π 1 Y t 1 I(q t 1 γ) + Π 2 Y t 1 I(q t 1 > γ) + ɛ t The null hypothesis is H 0 : Π 1 = Π 2 (linear cointegration). This test is a LM-type test. 2 Threshold cointegration against no cointegration (Seo 2007) Y t = Π 1 Y t 1 I(q t 1 γ) + Π 2 Y t 1 I(q t 1 > γ) + ɛ t The null hypothesis is H 0 : Π 1 = Π 2 = 0 (no cointegration).

34 Threshold model with endogenous threshold variable Consider the threshold model { β1 x y i = i + u i if q i γ β 2 x i + u i if q i > γ (4) If we want to obtain consistent estimators of β 1 and β 2, q t cannot be correlated with u t. If q t is correlated with u t, we need new method to estimate β 1 and β 2.

35 Bias-Correction Estimator for endogenous threshold models Kourtellos, Stengos and Tan (2007) consider a threshold model with endogenous threshold variables like Equation(4). { xi β y i = 1 + u i if q i γ x i β 2 + u i if q i > γ (5) y i, x i, z i are exogenous variables. q i is an endogenous variable. The selection equation is q i = z i π + υ i. (6)

36 Define the indicator variable { 1 iff υi γ z I i = i π 0 iff υ i > γ z i π The joint distribution between u i and υ i is defined as ( ) ( ( ) ) ui σ 2 x i, z i N 0 u σ uɛ σ uɛ 1 υ i Use the relationship between u i and ɛ i. ( ) ( ) ( ɛi 1 σuυ ui = 0 1 υ i υ i )

37 Let κ 1 = σ uυ = ρ 1 σ u, and define u i = κ 1 υ i + ɛ i = κ 1 λ 1i (γ z i π) + e i where λ 1i (γ z i π) = φ(γ z i π) Φ(γ z i π) (Inverse Mills bias correction item). We may get conditional expectations for each of the regimes. E(y x 1, z 1, υ i γ z i π) = x i β 1 + κ 1 λ 1i (γ z i π) E(y x 2, z 2, υ i > γ z i π) = x i β 2 + κ 1 λ 2i (γ z i π)

38 When two regimes have the same error structure, THRET model can be estimate by y i = x i β 2 + x i (γ)ϕ + κλ i (γ z i π) + e i, (7) where x i (γ) = x i I(q i γ) and ϕ = β 2 β 1. This estimation method looks like sample-selection model. The main difference about these two model is THRET model using all data.

39 Estimation of Threshold model with endogenous threshold variable Kourtellos, Stengos and Tan (2007) use three steps to obtain consistent estimator. 1 Estimate the parameter vector π in Equation (6) by least square. 2 Estimate the threshold value ˆγ by minimizing a concentrated two stage least square criterion using ˆπ from first stage. n S n (γ) = (y i x i β 1 x i (γ)ϕ κλ i (γ z i ˆπ)) 2. i=1 3 Estimate the lease square estimates of the slop parameters based on the split samples implied by ˆγ.

40 Threshold regression with endogenous threshold and slop models When threshold model with endogenous threshold value and regressors, we need to use instrumental variables. Consider THRET model { xi β y i = 1 + u i if q i γ x i β 2 + u i if q i > γ (8) q i = z i Γ + υ i. (9) where x i = (x 1i x 2i ). x 1i is endogenous variable. We need z i = (z 1i x 2i ).

41 1 Estimate the parameter vector π in Equation (9) by least square. 2 Estimate the threshold value ˆγ by minimizing a concentrated two stage least square criterion using ˆπ from first stage. S n (γ) = n (y i ˆx i β 1 ˆx i (γ)ϕ κλ i (γ z i ˆπ)) 2. i=1 3 Estimate the lease square estimates of the slop parameters based on the split samples implied by ˆγ.

42 panel threshold model with endogenous threshold variable Consider a panel threshold model with endogenous threshold value: y it = x it I(q it γ)β 1 + x it I(q it > γ)β 2 + η i + e it (10) q it = z it π + u it. (11) Under n and T fixed, we may derive consistent estimators for β 1 and β 2.

43 Basic Assumptions 1 Assumption 1: {y it, x it, q it, e it } is strictly stationary, ergodic. 2 Assumption 2: {y it, x it, q it, e it : 1 i n, 1 t T } are from balanced panel data 3 Assumption 3: u it z it N (0, 1) 4 Assumption 4 : The joint distribution between e it and u it is defined as: [ eit u it ] ( x it, z it N 0, [ σ 2 e γ j γ j 1 ] ), where γ j is covariance between e it and u it, γ j = γ 1 when q it θ and γ j = γ 2 when q it > θ. 5 Assumption 5: n and T is fixed.

44 Estimation of panel threshold model with endogenous threshold variable Using first difference transformation to eliminate fixed effect η i in Equation (10). Estimate the parameter vector π in Equation (11) by least square. Estimate the threshold value ˆγ by minimizing a concentrated two stage least square criterion using ˆπ from first stage. n S n (γ) = ( y i (x it I(q it γ) x it 1 I(q it 1 γ))β 1 i=1 (x it I(q it > γ) x it 1 I(q it 1 > γ))β 2 κλ i (γ z i ˆπ)) 2.

45 Estimate the lease square estimates of the slop parameters based on the split samples implied by ˆγ. Why not use fixed effect transformation? Because fixed effect transformation will generate heteroskedasticity under T small. E(u it ū i )(u is ū i ) = σ2 u T. We plan to expend this panel threshold model to dynamic panel data model.

46 Future study and possible problems Panel threshold unit root case. Panel smooth transition threshold model