- Email: [email protected]
On unit root testing with smooth transitions
Computational Statistics & Data Analysis 51 (2006) 797 – 800 www.elsevier.com/locate/csda
Short communication
On unit root testing with smooth transitions Dimitrios V. Vougas∗ Department of Economics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Received 5 April 2005; received in revised form 9 July 2006; accepted 10 July 2006 Available online 1 August 2006
Abstract Improved critical values are calculated for Dickey–Fuller-type t ratio unit root tests against trend stationarity about non-linear trend, which is based on one deterministic smooth transition (STR) function. Simulation employs fine grid-search over both STR parameters to find accurate staring values, as well as constrained optimization. In addition, two new parsimonious models are introduced. Finally, an application of the test to the log of Real per capita GNP of USA is provided. © 2006 Elsevier B.V. All rights reserved. Keywords: Nonlinear trend; Deterministic smooth transition; Unit root; Structural change
1. Introduction In unit root testing, the need for alternative trend specification has been recognized by several authors. Indicative references on models with segmented line trends and a single or multiple breaks include Perron (1989, 1990), Rappoport and Rechlin (1989), Perron and Vogelsang (1992), Zivot and Andrews (1992), Lumsdaine and Papell (1997), and Bai and Perron (1998). On the other hand, Leybourne et al. (1998) (LNV, hereafter) employ generic deterministic trend via models based on a deterministic logistic smooth transition (STR). This type of trend modeling can incorporate broken or unbroken trend lines, allowing for STR as well as abrupt break. Note that the trend models employed by LNV are deterministic and must be distinguished from models with stochastic STR, that is smooth transition autoregressive (STAR) models (see Terasvirta, 1994), that are nonlinear in the autoregressive parameters. Saikkonen and Lutkepohl (2001, 2002) also employ non-linear trend modeling for level shifts and provide some asymptotic theory. Greenaway et al. (1997, 2000) employ methods similar to the ones used by LNV to model growth of various European countries. However, neglected numerical issues necessitate re-calculation of critical values for the LNV tests. For this, detailed preliminary grid search over both STR parameters, as well as constrained optimization, is required. This paper provides such accurate critical values, based on constrained optimization via sequential quadratic programming (SQP). Starting values are calculated via preliminary grid-search over both STR parameters. Furthermore, two new parsimonious (but (highly) non-linear) models (called Model D and E, respectively) are introduced. Model D captures discovered/emerging trend, possibly after a level shift, while Model E allows for highly non-linear trend. Finally, the log of Real per capita GNP of USA (see Nelson and Plosser, 1982) is employed in an empirical example. ∗ Tel.: +44 179 2602102; fax: +44 179 2295872.
E-mail address: [email protected]. 0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2006.07.015
798
D. V. Vougas / Computational Statistics & Data Analysis 51 (2006) 797 – 800
The paper is organized as follows: Section 2 presents LNV-type trend modeling. Section 3 describes the re-furbished estimation approach, derivation of new critical values, and an example using Real per capita GNP of USA, the series of Nelson and Plosser (1982). Finally, Section 4 concludes. 2. STR econometric models LNV introduce general deterministic non-linear trend models that are based on the deterministic logistic STR trend function. The STR is defined as St (, ) = [1 + exp{−(t − n)}]−1 ,
> 0, t = 1, . . . , n,
(1)
for a sample of n observations. Parameter ∈ (0, 1) indicates the fraction of the sample at which the transition occurs and ( 0) determines the speed of transition. Note that is an important parameter, since when = 0 parameter is not identified. Sollis et al. (1999) propose an alternative STR. The models employed by LNV, referred to as Models A, B, and C, respectively, are yt = fAt A + uAt = 1 + 2 St (, ) + uAt = A xAt + uAt , (2A) yt = fBt B + uBt = 1 + 1 t + 2 St (, ) + uBt = B xBt + uBt , (2B) yt = fCt C + uCt = 1 + 1 t + 2 + 2 t St (, ) + uCt = C xCt + uCt , (2C) with A = , , 1 , 2 = , , A , B = , , 1 , 1 , 2 = , , B , and C = , , 1 , 1 , 2 , 2 = , , C . Note the implicit definitions A = [1 , 2 ], B = 1 , 1 , 2 , and C = 1 , 1 , 2 , 2 . Also note the definitions xCt = xCt (, ) = 1, St (, ) , xBt = xBt (, ) = 1, t, St (, ) , and xCt = xCt (, ) = 1, t, St (, ), tS t (, ) . Both Models A and B are restricted/parsimonious versions of Model C and may be used for efficient estimation and accurate unit root inference. If 1 = 2 = 0, Model A is useful, while if 2 = 0, Model B is useful. If 1 = 2 = 0, then an additional interesting (and relatively non-linear) model (named Model D) arises, namely yt = fDt D + uDt = 1 + 2 tS t (, ) + uDt = D xDt + uDt , (2D) with D = , , 1 , 2 = , , D , D = 1 , 2 , and xEt = xEt (, ) = 1, tS t (, ) . Model D is suitable to capture discovered/emerging trend, possibly after a level shift. Cook and Vougas (2004) use Model D in a simulation study. A final restricted/parsimonious (but highly non-linear) version of Model C arises when 1 = 0 only, namely yt = fEt E + uEt = 1 + 2 + 2 t St (, ) + uEt = E xEt + uEt (2E) with parameters E = , , 1 , 2 , 2 = , , E , E = 1 , 2 , 2 , and xEt = xEt (, ) = 1, St (, ), tS t (, ) . Model choice in practice is an open issue and far beyond the scope of this paper. Nevertheless, critical values for all possible models are calculated. The original LNV approach focuses on unit root testing and estimates Eq. (2i) (i =A, B, C, D, E) first. In the sequel, an (augmented) Dickey–Fuller-type unit root test is applied to residual uˆ it (any i) with no deterministic component. Assume that uit follows an AR (pi ) with uit = i1 uit−1 + · · · + ipi uit−pi + εit
any i.
(3)
Then, with pi > 1 (li = pi − 1 > 0), it follows that uit = i uit−1 + with i =
pi
k=1 ik
li j =1
ij uit−j + εit pi
− 1 and ij = −
uˆ it = i uˆ it−1 +
li
any i,
k=j +1 ik
ij uˆ it−j + εit
(4)
for j = 1, . . . , li . The associated auxiliary autoregression is
any i,
(5)
j =1
and i = 0 is the unit root hypothesis to be tested via a t-ratio denoted i . When pi = 1, no augmentation is required.
D. V. Vougas / Computational Statistics & Data Analysis 51 (2006) 797 – 800
799
3. New critical values and empirical example Without loss of generality, yt is generated as follows: yt = ut ,
ut = ut−1 + εt , u0 = 0, εt ∼ NID(0, 1),
(6)
and all models are fitted, using constrained optimization via SQP and the SQPSolve procedure of GAUSS䉸 . The implied restrictions 0 and 0 1 are always imposed. First, a fine grid search over both and is undertaken to roughly minimize n t=1
uˆ 2it (, ),
any i.
(7)
For simplicity, the search region for is set to [0, k] for some positive integer k. The (0, 1) interval for is also split into k points, evaluating at h/(k + 1) (h = 1, . . . , k), with step 1/(k + 1). The same step is also employed in the search over . Thus, the number of required function evaluations is k 2 (k + 1). Any 4 to 19 may be selected to k between maintain a manageable number of function evaluations. The selected pair ˆ min , ˆ min , say, is then used as a pair of starting values in accurate minimization of nt=1 uˆ 2it (, ) over both and . New null critical values are derived by simulating pseudo-normal random walks, with zero starting values, fitting the desired model, and applying a simple Dickey–Fuller test to the resulting residual series. The choice of k is 9 in this simulation study to provide relatively accurate starting values for all non-linear constrained minimizations. The number of replications is 200 000, although a smaller number of replications provides similar critical values (accurate to two decimal digits). (Critical values based on 50 000 replications are available on request.) Considered sample sizes are n ∈ {25, 50, 100, 250, 500}. Table 1 presents the new accurate critical values for all t ratio tests. Table 1 Null critical values n
0.200
0.100
0.050
0.025
0.010
A
25 50 100 250 500
−3.83 −3.64 −3.56 −3.44 −3.36
−4.32 −4.05 −3.93 −3.81 −3.73
−4.77 −4.40 −4.25 −4.12 −4.04
−5.18 −4.73 −4.53 −4.39 −4.30
−5.70 −5.12 −4.87 −4.70 −4.61
B
25 50 100 250 500
−4.71 −4.36 −4.20 −4.04 −3.93
−5.23 −4.77 −4.55 −4.38 −4.28
−5.72 −5.12 −4.86 −4.68 −4.57
−6.16 −5.44 −5.14 −4.94 −4.83
−6.74 −5.86 −5.48 −5.25 −5.13
C
25 50 100 250 500
−5.09 −4.68 −4.49 −4.33 −4.22
−5.65 −5.10 −4.86 −4.68 −4.57
−6.17 −5.47 −5.18 −4.97 −4.86
−6.66 −5.81 −5.45 −5.23 −5.11
−7.30 −6.22 −5.78 −5.53 −5.42
D
25 50 100 250 500
−3.57 −3.43 −3.36 −3.31 −3.27
−4.06 −3.84 −3.75 −3.68 −3.64
−4.51 −4.20 −4.08 −4.00 −3.94
−4.93 −4.53 −4.37 −4.27 −4.21
−5.46 −4.94 −4.72 −4.60 −4.52
E
25 50 100 250 500
−4.49 −4.19 −4.05 −3.90 −3.80
−5.02 −4.61 −4.42 −4.25 −4.16
−5.50 −4.97 −4.73 −4.55 −4.46
−5.95 −5.30 −5.01 −4.82 −4.72
−6.51 −5.71 −5.36 −5.12 −5.03
800
D. V. Vougas / Computational Statistics & Data Analysis 51 (2006) 797 – 800
Finally, the log of Real per capita GNP of USA (the series of Nelson and Plosser, 1982) is employed to clarify the usefulness of the critical values as well as the two new models. All Models A, B, C, D, and E are fitted to this series with one lagged first-order difference added, in line with previous studies. Note that A = −3.65 which rejects the unit root null at 20% significance level only, B = −3.45, C = −4.54, D = −4.43 which rejects the unit root null at 5% significance level, and E = −4.58 which rejects the unit root null at 20% significance level only. It appears that for this series, Model D provides a better restricted version of Model C than all other models, since it rejects the unit root more strongly. In fact, if one estimates Model C jointly with AR(2) error for this series (not reported), one finds insignificant 1 and 2 , explaining why Model D is an efficient and parsimonious re-parameterization of Model C for this series. Note that LNV do not discuss model selection; they only consider lag length selection for an assumed model. This issue needs further research. However, it becomes apparent that parsimonious versions of the main model (C) are required to fully assess the presence of a unit root in an empirical time series. 4. Conclusions This paper calculates accurate critical values for LNV-type unit root tests. They must be used for correct inference in practice, along with the proposed estimation/test derivation method. In addition, two parsimonious and (highly) non-linear models are introduced to complete the set of possible STR models. The Real per capita GNP of USA (see Nelson and Plosser, 1982) is employed to demonstrate the usefulness of the new critical values and one of the new models. Finally, it is stressed that the proposed approach, with the detailed preliminary grid search, provides a data dependent approach to endogenize a break or a transition, making it superior to existing methods in the literature. References Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78. Cook, S., Vougas, D.V., 2004. On the finite-sample size distortion of smooth transition unit root tests. Statist. Probab. Lett. 70, 175–182. Greenaway, D., Leybourne, S., Sapsford, D., 1997. Modelling growth (and liberalisation) using smooth transition analysis. Econom. Inquiry 35, 798–814. Greenaway, D., Leybourne, S., Sapsford, D., 2000. Smooth Transitions and GDP Growth in the European Union. The Manchester School, vol. 68, pp. 145–165. Leybourne, S., Newbold, P., Vougas, D., 1998. Unit roots and smooth transitions. J. Time Ser. Anal. 19, 83–97. Lumsdaine, R., Papell, D., 1997. Multiple trend breaks and the unit root hypothesis. Rev. Econom. Statist. 79, 212–218. Nelson, C.R., Plosser, C.I., 1982. Trends and random walks in macroeconomic time series. J. Monetary Econom. 10, 139–162. Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401. Perron, P., 1990. Testing for a unit root in a time series with a changing mean. J. Busi. Econom. Statist. 8, 153–162. Perron, P., Vogelsang, T.J., 1992. Nonstationarity and level shifts with an application to purchasing power parity. J. Busi. Econom. Statist. 10, 301–320. Rappoport, P., Rechlin, L., 1989. Segmented trends and non-stationary time series. Econom. J. 99, 168–177. Saikkonen, P., Lutkepohl, H., 2001. Testing for unit roots in time series with level shifts. Allgemeines Statistisches Archiv 85, 1–25. Saikkonen, P., Lutkephol, H., 2002. Testing for a unit root in a time series with a level shift at unknown time. Econometric Theory 18, 313–348. Sollis, R., Leybourne, S., Newbold, P., 1999. Unit roots and asymmetric smooth transitions. J. Time Ser. Anal. 20, 671–677. Terasvirta, T., 1994. Specification, estimation, and evaluation of smooth transition autoregressive models. J. Amer. Statist. Assoc. 89, 208–218. Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil-price shock, and the unit root hypothesis. J. Busi. Econom. Statist. 10, 251–270.
Short communication
On unit root testing with smooth transitions Dimitrios V. Vougas∗ Department of Economics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Received 5 April 2005; received in revised form 9 July 2006; accepted 10 July 2006 Available online 1 August 2006
Abstract Improved critical values are calculated for Dickey–Fuller-type t ratio unit root tests against trend stationarity about non-linear trend, which is based on one deterministic smooth transition (STR) function. Simulation employs fine grid-search over both STR parameters to find accurate staring values, as well as constrained optimization. In addition, two new parsimonious models are introduced. Finally, an application of the test to the log of Real per capita GNP of USA is provided. © 2006 Elsevier B.V. All rights reserved. Keywords: Nonlinear trend; Deterministic smooth transition; Unit root; Structural change
1. Introduction In unit root testing, the need for alternative trend specification has been recognized by several authors. Indicative references on models with segmented line trends and a single or multiple breaks include Perron (1989, 1990), Rappoport and Rechlin (1989), Perron and Vogelsang (1992), Zivot and Andrews (1992), Lumsdaine and Papell (1997), and Bai and Perron (1998). On the other hand, Leybourne et al. (1998) (LNV, hereafter) employ generic deterministic trend via models based on a deterministic logistic smooth transition (STR). This type of trend modeling can incorporate broken or unbroken trend lines, allowing for STR as well as abrupt break. Note that the trend models employed by LNV are deterministic and must be distinguished from models with stochastic STR, that is smooth transition autoregressive (STAR) models (see Terasvirta, 1994), that are nonlinear in the autoregressive parameters. Saikkonen and Lutkepohl (2001, 2002) also employ non-linear trend modeling for level shifts and provide some asymptotic theory. Greenaway et al. (1997, 2000) employ methods similar to the ones used by LNV to model growth of various European countries. However, neglected numerical issues necessitate re-calculation of critical values for the LNV tests. For this, detailed preliminary grid search over both STR parameters, as well as constrained optimization, is required. This paper provides such accurate critical values, based on constrained optimization via sequential quadratic programming (SQP). Starting values are calculated via preliminary grid-search over both STR parameters. Furthermore, two new parsimonious (but (highly) non-linear) models (called Model D and E, respectively) are introduced. Model D captures discovered/emerging trend, possibly after a level shift, while Model E allows for highly non-linear trend. Finally, the log of Real per capita GNP of USA (see Nelson and Plosser, 1982) is employed in an empirical example. ∗ Tel.: +44 179 2602102; fax: +44 179 2295872.
E-mail address: [email protected]. 0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2006.07.015
798
D. V. Vougas / Computational Statistics & Data Analysis 51 (2006) 797 – 800
The paper is organized as follows: Section 2 presents LNV-type trend modeling. Section 3 describes the re-furbished estimation approach, derivation of new critical values, and an example using Real per capita GNP of USA, the series of Nelson and Plosser (1982). Finally, Section 4 concludes. 2. STR econometric models LNV introduce general deterministic non-linear trend models that are based on the deterministic logistic STR trend function. The STR is defined as St (, ) = [1 + exp{−(t − n)}]−1 ,
> 0, t = 1, . . . , n,
(1)
for a sample of n observations. Parameter ∈ (0, 1) indicates the fraction of the sample at which the transition occurs and ( 0) determines the speed of transition. Note that is an important parameter, since when = 0 parameter is not identified. Sollis et al. (1999) propose an alternative STR. The models employed by LNV, referred to as Models A, B, and C, respectively, are yt = fAt A + uAt = 1 + 2 St (, ) + uAt = A xAt + uAt , (2A) yt = fBt B + uBt = 1 + 1 t + 2 St (, ) + uBt = B xBt + uBt , (2B) yt = fCt C + uCt = 1 + 1 t + 2 + 2 t St (, ) + uCt = C xCt + uCt , (2C) with A = , , 1 , 2 = , , A , B = , , 1 , 1 , 2 = , , B , and C = , , 1 , 1 , 2 , 2 = , , C . Note the implicit definitions A = [1 , 2 ], B = 1 , 1 , 2 , and C = 1 , 1 , 2 , 2 . Also note the definitions xCt = xCt (, ) = 1, St (, ) , xBt = xBt (, ) = 1, t, St (, ) , and xCt = xCt (, ) = 1, t, St (, ), tS t (, ) . Both Models A and B are restricted/parsimonious versions of Model C and may be used for efficient estimation and accurate unit root inference. If 1 = 2 = 0, Model A is useful, while if 2 = 0, Model B is useful. If 1 = 2 = 0, then an additional interesting (and relatively non-linear) model (named Model D) arises, namely yt = fDt D + uDt = 1 + 2 tS t (, ) + uDt = D xDt + uDt , (2D) with D = , , 1 , 2 = , , D , D = 1 , 2 , and xEt = xEt (, ) = 1, tS t (, ) . Model D is suitable to capture discovered/emerging trend, possibly after a level shift. Cook and Vougas (2004) use Model D in a simulation study. A final restricted/parsimonious (but highly non-linear) version of Model C arises when 1 = 0 only, namely yt = fEt E + uEt = 1 + 2 + 2 t St (, ) + uEt = E xEt + uEt (2E) with parameters E = , , 1 , 2 , 2 = , , E , E = 1 , 2 , 2 , and xEt = xEt (, ) = 1, St (, ), tS t (, ) . Model choice in practice is an open issue and far beyond the scope of this paper. Nevertheless, critical values for all possible models are calculated. The original LNV approach focuses on unit root testing and estimates Eq. (2i) (i =A, B, C, D, E) first. In the sequel, an (augmented) Dickey–Fuller-type unit root test is applied to residual uˆ it (any i) with no deterministic component. Assume that uit follows an AR (pi ) with uit = i1 uit−1 + · · · + ipi uit−pi + εit
any i.
(3)
Then, with pi > 1 (li = pi − 1 > 0), it follows that uit = i uit−1 + with i =
pi
k=1 ik
li j =1
ij uit−j + εit pi
− 1 and ij = −
uˆ it = i uˆ it−1 +
li
any i,
k=j +1 ik
ij uˆ it−j + εit
(4)
for j = 1, . . . , li . The associated auxiliary autoregression is
any i,
(5)
j =1
and i = 0 is the unit root hypothesis to be tested via a t-ratio denoted i . When pi = 1, no augmentation is required.
D. V. Vougas / Computational Statistics & Data Analysis 51 (2006) 797 – 800
799
3. New critical values and empirical example Without loss of generality, yt is generated as follows: yt = ut ,
ut = ut−1 + εt , u0 = 0, εt ∼ NID(0, 1),
(6)
and all models are fitted, using constrained optimization via SQP and the SQPSolve procedure of GAUSS䉸 . The implied restrictions 0 and 0 1 are always imposed. First, a fine grid search over both and is undertaken to roughly minimize n t=1
uˆ 2it (, ),
any i.
(7)
For simplicity, the search region for is set to [0, k] for some positive integer k. The (0, 1) interval for is also split into k points, evaluating at h/(k + 1) (h = 1, . . . , k), with step 1/(k + 1). The same step is also employed in the search over . Thus, the number of required function evaluations is k 2 (k + 1). Any 4 to 19 may be selected to k between maintain a manageable number of function evaluations. The selected pair ˆ min , ˆ min , say, is then used as a pair of starting values in accurate minimization of nt=1 uˆ 2it (, ) over both and . New null critical values are derived by simulating pseudo-normal random walks, with zero starting values, fitting the desired model, and applying a simple Dickey–Fuller test to the resulting residual series. The choice of k is 9 in this simulation study to provide relatively accurate starting values for all non-linear constrained minimizations. The number of replications is 200 000, although a smaller number of replications provides similar critical values (accurate to two decimal digits). (Critical values based on 50 000 replications are available on request.) Considered sample sizes are n ∈ {25, 50, 100, 250, 500}. Table 1 presents the new accurate critical values for all t ratio tests. Table 1 Null critical values n
0.200
0.100
0.050
0.025
0.010
A
25 50 100 250 500
−3.83 −3.64 −3.56 −3.44 −3.36
−4.32 −4.05 −3.93 −3.81 −3.73
−4.77 −4.40 −4.25 −4.12 −4.04
−5.18 −4.73 −4.53 −4.39 −4.30
−5.70 −5.12 −4.87 −4.70 −4.61
B
25 50 100 250 500
−4.71 −4.36 −4.20 −4.04 −3.93
−5.23 −4.77 −4.55 −4.38 −4.28
−5.72 −5.12 −4.86 −4.68 −4.57
−6.16 −5.44 −5.14 −4.94 −4.83
−6.74 −5.86 −5.48 −5.25 −5.13
C
25 50 100 250 500
−5.09 −4.68 −4.49 −4.33 −4.22
−5.65 −5.10 −4.86 −4.68 −4.57
−6.17 −5.47 −5.18 −4.97 −4.86
−6.66 −5.81 −5.45 −5.23 −5.11
−7.30 −6.22 −5.78 −5.53 −5.42
D
25 50 100 250 500
−3.57 −3.43 −3.36 −3.31 −3.27
−4.06 −3.84 −3.75 −3.68 −3.64
−4.51 −4.20 −4.08 −4.00 −3.94
−4.93 −4.53 −4.37 −4.27 −4.21
−5.46 −4.94 −4.72 −4.60 −4.52
E
25 50 100 250 500
−4.49 −4.19 −4.05 −3.90 −3.80
−5.02 −4.61 −4.42 −4.25 −4.16
−5.50 −4.97 −4.73 −4.55 −4.46
−5.95 −5.30 −5.01 −4.82 −4.72
−6.51 −5.71 −5.36 −5.12 −5.03
800
D. V. Vougas / Computational Statistics & Data Analysis 51 (2006) 797 – 800
Finally, the log of Real per capita GNP of USA (the series of Nelson and Plosser, 1982) is employed to clarify the usefulness of the critical values as well as the two new models. All Models A, B, C, D, and E are fitted to this series with one lagged first-order difference added, in line with previous studies. Note that A = −3.65 which rejects the unit root null at 20% significance level only, B = −3.45, C = −4.54, D = −4.43 which rejects the unit root null at 5% significance level, and E = −4.58 which rejects the unit root null at 20% significance level only. It appears that for this series, Model D provides a better restricted version of Model C than all other models, since it rejects the unit root more strongly. In fact, if one estimates Model C jointly with AR(2) error for this series (not reported), one finds insignificant 1 and 2 , explaining why Model D is an efficient and parsimonious re-parameterization of Model C for this series. Note that LNV do not discuss model selection; they only consider lag length selection for an assumed model. This issue needs further research. However, it becomes apparent that parsimonious versions of the main model (C) are required to fully assess the presence of a unit root in an empirical time series. 4. Conclusions This paper calculates accurate critical values for LNV-type unit root tests. They must be used for correct inference in practice, along with the proposed estimation/test derivation method. In addition, two parsimonious and (highly) non-linear models are introduced to complete the set of possible STR models. The Real per capita GNP of USA (see Nelson and Plosser, 1982) is employed to demonstrate the usefulness of the new critical values and one of the new models. Finally, it is stressed that the proposed approach, with the detailed preliminary grid search, provides a data dependent approach to endogenize a break or a transition, making it superior to existing methods in the literature. References Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78. Cook, S., Vougas, D.V., 2004. On the finite-sample size distortion of smooth transition unit root tests. Statist. Probab. Lett. 70, 175–182. Greenaway, D., Leybourne, S., Sapsford, D., 1997. Modelling growth (and liberalisation) using smooth transition analysis. Econom. Inquiry 35, 798–814. Greenaway, D., Leybourne, S., Sapsford, D., 2000. Smooth Transitions and GDP Growth in the European Union. The Manchester School, vol. 68, pp. 145–165. Leybourne, S., Newbold, P., Vougas, D., 1998. Unit roots and smooth transitions. J. Time Ser. Anal. 19, 83–97. Lumsdaine, R., Papell, D., 1997. Multiple trend breaks and the unit root hypothesis. Rev. Econom. Statist. 79, 212–218. Nelson, C.R., Plosser, C.I., 1982. Trends and random walks in macroeconomic time series. J. Monetary Econom. 10, 139–162. Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401. Perron, P., 1990. Testing for a unit root in a time series with a changing mean. J. Busi. Econom. Statist. 8, 153–162. Perron, P., Vogelsang, T.J., 1992. Nonstationarity and level shifts with an application to purchasing power parity. J. Busi. Econom. Statist. 10, 301–320. Rappoport, P., Rechlin, L., 1989. Segmented trends and non-stationary time series. Econom. J. 99, 168–177. Saikkonen, P., Lutkepohl, H., 2001. Testing for unit roots in time series with level shifts. Allgemeines Statistisches Archiv 85, 1–25. Saikkonen, P., Lutkephol, H., 2002. Testing for a unit root in a time series with a level shift at unknown time. Econometric Theory 18, 313–348. Sollis, R., Leybourne, S., Newbold, P., 1999. Unit roots and asymmetric smooth transitions. J. Time Ser. Anal. 20, 671–677. Terasvirta, T., 1994. Specification, estimation, and evaluation of smooth transition autoregressive models. J. Amer. Statist. Assoc. 89, 208–218. Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil-price shock, and the unit root hypothesis. J. Busi. Econom. Statist. 10, 251–270.