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Testing the unit root hypothesis against TAR nonlinearity using STAR-based tests
Economics Letters 112 (2011) 19–22
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Testing the unit root hypothesis against TAR nonlinearity using STAR-based tests Robert Sollis ∗ Newcastle University Business School, Newcastle University, Ridley Building, Queen Victoria Road, Newcastle upon Tyne, NE1 7RU, United Kingdom
article
info
Article history: Received 16 March 2010 Received in revised form 16 February 2011 Accepted 18 March 2011 Available online 29 March 2011
abstract This paper investigates the finite-sample power of STAR-based unit root tests when the data generation process is a globally stationary three-regime TAR model. Unit root tests are proposed derived from STAR models that nest TAR models. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C22 C12 Keywords: Unit roots Nonlinearity Power
1. Introduction Threshold autoregressive (TAR) and smooth transition autoregressive (STAR) models have proved to be popular for modeling a wide variety of time series data. From the TAR family of models the three-regime model with a unit root central regime and stationary outer-regimes has been found to be particularly useful for applications involving real exchange rates, since the presence of transaction costs suggest that small deviations from the law of one price are not corrected, but that large deviations are corrected; see for example Obstfeld and Taylor (1997). Similarly, from the STAR family of models the globally stationary exponential-STAR (ESTAR) model with a unit root at equilibrium has proved to be popular in empirical applications due to its ability to model processes in which small deviations from a long-run attractor are not corrected but large deviations are; see for example Taylor et al. (2001). Tests of the unit root hypothesis against globally stationary ESTAR nonlinearity with a unit root at equilibrium have been proposed by Kapetanios et al. (2003). Tests of the unit root hypothesis against globally stationary three-regime TAR nonlinearity with a unit root central regime have been proposed by Kapetanios and Shin (2006). Both types of test are popular in practice. While the ESTAR model approximates the three-regime TAR model it is important to recognize that the two models are not actually nested within each other; the ESTAR model is incapable of generating three clearly defined regimes, while the three-regime
∗
Tel.: +44 (0)1912228584; fax: +44 (0)1912226838. E-mail address: [email protected].
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.03.008
TAR model is incapable of generating smooth adjustment between regimes. Since practitioners might not have any prior information as to which model is the most appropriate for their data set, in this paper we investigate the finite-sample power of the ESTARbased unit root tests proposed by Kapetanios et al. (2003) when the data generation process (DGP) under the alternative hypothesis is a globally stationary three-regime TAR model with a unit root central regime. Unit root tests derived from STAR models that do nest three-regime TAR models with a unit root central regime are also proposed. 2. STAR unit root tests 2.1. t-tests of Kapetanios et al. (2003) Kapetanios et al. (2003) employ the following ESTAR specification,
1yt = Et (γ , yt −d )ρ yt −1 + εt Et (γ , yt −d ) = 1 − exp(−γ
y2t −d
(1)
) γ ≥0
(2)
where εt is an independently and identically distributed (i.i.d.) sequence with zero mean and constant variance, and where |1 + ρ| < 1 is required to rule out explosive behavior. Kapetanios et al. (2003) focus on the case of d = 1. Under this restriction and assuming γ > 0, for yt −1 → 0, it is effectively a unit root process, but for yt −1 → ±∞, yt it is a stationary AR (1) process with AR parameter (1 + ρ). Globally, it is stationary but locally it can behave like a unit root process. This model can be extended to deal with a non-zero mean and a deterministic trend by replacing yt with y˜t = yt − µ ˆ or y˜ t = yt − αˆ − βˆ t, where E (yt ) = µ, and
20
R. Sollis / Economics Letters 112 (2011) 19–22
µ, ˆ α, ˆ βˆ are consistent sample estimates of the relevant population parameters. It can also be extended to deal with higher-order autocorrelations in the same way as the Dickey–Fuller test (Dickey and Fuller, 1979), by augmenting with lags of 1yt ,
1yt = Et (γ , yt −d )ρ yt −1 +
k −
βi 1yt −i + εt .
(3)
i =1
Note that if γ = 0 then it contains a unit root. Hence Kapetanios et al. (2003) propose testing the unit root hypothesis by testing H0 : γ = 0 in (3) against the alternative hypothesis H0 : γ > 0 (where ρ < 0 is assumed to hold). Under this null hypothesis the parameter ρ is unidentified, which prevents the asymptotic distribution of a test statistic being derived using conventional analytical methods. However, following Luukonen et al. (1988) this problem can be circumvented by replacing Et (γ , yt −1 ) in (3) with a first-order Taylor series expansion around γ = 0. On re-arranging, this gives the auxiliary model,
1yt = δ y3t −1 +
k −
βi 1yt −i + εt
Fig. 1(a). Second-order logistic function: fast transition.
(4)
i=1
and the relevant null hypothesis is H0 : δ = 0. Let tNL , tNL,c and tNL,ct denote the tests for the zero-mean, non-zero mean and deterministic trend cases respectively. 2.2. Unit root tests derived from STAR models that nest three-regime TAR models Consider the following STAR model for yt that, instead of the exponential function utilizes a second-order logistic function,
1yt = St (γ , θ , yt −d )ρ yt −1 + εt St (γ , θ, yt −d ) = [1 + exp(−γ (
(5)
y2t −d
− θ ))]
−1
γ ≥0
(6)
Fig. 1(b). Second-order logistic function: moderately fast transition.
where |1 + ρ| < 1, εt ∼ i.i.d.(0, σ ). As in Kapetanios et al. (2003) assume d = 1. (5) is an LSTAR model, although the logistic function is second-order rather than first-order as in conventional √ LSTAR models. In this model, for γ → ∞, if yt −1 < − θ or √ yt −1 > θ , yt is√a stationary AR √ (1) process with AR parameter (1 + ρ). But if − θ < yt −1 < θ , yt is a unit root process. Note therefore that (5) with d = 1 nests the following three-regime TAR model with symmetric outer regimes, if 2
1yt = ρ yt −1 + εt if |yt −1 | > r 1yt = εt if |yt −1 | ≤ r
(7) (8)
where r is a positive threshold. (5) does not nest an ESTAR model, but for small and moderate values of γ , (5) is capable of generating exponential-type nonlinearity that closely approximates ESTAR nonlinearity. A unit root test derived from (5) should therefore have power against globally stationary three-regime TAR nonlinearity with a unit root central regime, and globally stationary ESTAR nonlinearity with a unit root at equilibrium. To clarify, in Figs. 1(a), 1(b) and 1(c) some example transitions from the second-order logistic function (6) are plotted, along with a transition from the exponential function (2) for comparison. The ability of (6) to emulate both a discrete indicator function and an exponential function is clear in these figures. Following Kapetanios et al. (2003), to allow for a non-zero mean or deterministic trend replace yt with the de-meaned or de-trended series, and to allow for additional dynamics (5) can be augmented with the lags of 1yt ,
1yt = St (γ , θ , yt −d )ρ yt −1 +
k −
Fig. 1(c). Exponential function: moderately fast transition.
root hypothesis, but in this case θ and γ are unidentified. Therefore in both of these cases conventional asymptotic analysis cannot be used to derive the asymptotic distributions for appropriate test statistics. If we replace St (γ , θ , yt −d ) in (9) with a first-order Taylor expansion around γ = 0, then on re-arrangement this gives a form that can be used for testing,
1yt = φ1 yt −1 + φ2 y3t −1 +
k −
βi 1yt −i + ηt .
(10)
i=1
βi 1yt −i + εt .
(9)
i=1
The unit root hypothesis is H0 : γ = ρ = 0. Clearly, under this null hypothesis θ is unidentified. Note that H0 : ρ = 0 is also a unit
The auxiliary model (10) has only one additional term compared to the auxiliary models of Kapetanios et al. (2003) (see Eq. (4)). The relevant null hypothesis is in this case H0 : φ1 = φ2 = 0
(11)
R. Sollis / Economics Letters 112 (2011) 19–22
21
Table 1 Critical values for Fs , Fs,c , and Fs,ct . T
10%
5%
1%
10%
Fs 100 200 Asymptotic
3.686 3.641 3.564
4.587 4.448 4.378
6.505 6.450 6.185
Fs = (Rβˆ − r)′ σˆ 2 R
−
xt x′t
R′
(Rβˆ − r)/m
(12)
− (1yt − φˆ 1 yt −1 − φˆ 2 y3t −1 )2 /(T − 2).
(13)
t
The asymptotic distribution of Fs can be established using wellknown convergence results. For brevity a full proof is omitted, but is available on request. The distribution can be written as the following functionals of Brownian motion (⇒ denotes weak convergence, W (r ) is the standard Brownian motion on r ∈ [0, 1]), Fs ⇒ h′ Q−1 h/2σ 2
(14)
where, h ≡
[
1 2
σ {W (1) − 1} 2
2
∫ 1 2 W (r )2 dr σ 0 Q≡ 4∫ 1 σ W (r )4 dr 0
10%
σ σ4 σ6
4
1 4
W (1) − 4
1
∫
∫0 1
3 2
1
∫
W (r ) dr 2
] (15)
0
W (r )4 dr W (r )6 dr
.
(16)
0
Let Fs,c and Fs,ct denote the test statistics allowing for a non-zero mean and deterministic trend. Asymptotic and finitesample critical values for all three tests obtained by Monte Carlo simulation with 10,000 replications are given in Table 1. 3. Monte Carlo simulation results 3.1. Finite-sample power experiments To investigate the finite-sample power of the Kapetanios et al. (2003) tests (tNL , tNL,c and), and the Fs , Fs,c , and Fs,ct tests proposed here against the alternative hypothesis of globally stationary threeregime TAR nonlinearity with a unit root central regime the following DGP is used,
φ1 yt −1 + ut if yt −1 < r1 yt = yt −1 + ut if r1 ≤ yt −1 ≤ r2 φ2 yt −1 + ut if yt −1 > r2
5%
1%
6.980 7.273 7.131
5.775 5.787 5.727
6.818 6.799 6.717
9.185 9.082 8.617
Table 2 The power of alternative STAR-based tests against three-regime TAR nonlinearity with a unit root central regime at the 5% nominal size. r2
tNL
tNL,c
tNL,ct
Fs
Fs,c
Fs,ct
−0.15 −0.90 −1.65 −2.40 −3.15 −3.90 T = 200
0.15 0.90 1.65 2.40 3.15 3.90
0.602 0.608 0.633 0.661 0.669 0.641
0.249 0.250 0.242 0.228 0.200 0.200
0.154 0.154 0.146 0.132 0.119 0.112
0.366 0.236 0.220 0.207 0.206 0.210
0.274 0.267 0.245 0.212 0.189 0.178
0.180 0.174 0.158 0.139 0.125 0.118
−0.15 −0.90 −1.65 −2.40 −3.15 −3.90
0.15 0.90 1.65 2.40 3.15 3.90
0.905 0.910 0.923 0.941 0.957 0.965
0.612 0.618 0.632 0.638 0.609 0.548
0.402 0.407 0.413 0.403 0.357 0.295
0.804 0.819 0.786 0.737 0.701 0.674
0.772 0.763 0.728 0.660 0.572 0.485
0.576 0.564 0.514 0.442 0.355 0.281
T = 100
where xt = (yt −1 , y3t −1 )′ , m = 2, R is a 2 × 2 identity matrix, βˆ = (φˆ 1 , φˆ 2 )′ where φˆ 1 and φˆ 2 are the OLS estimates of φ1 and φ2 , r = (0, 0)′ and,
′
5.141 5.181 5.046
r1
−1
−1
t
σˆ 2 =
1%
Fs,ct
4.223 4.255 4.169
which can be tested using an F -test. The alternative stationary regions are φ1 < 0 and φ1 + φ2 < 0. The test statistic (assuming k = 0) can be written,
5%
F s,c
(17)
where ut ∼ nid(0, 1). Results are computed for and at the 5% nominal size for different combinations of threshold values ranging from r1 , r2 = {−0.15, 0.15} to r1 , r2 = {−3.90, 3.90} in steps of 0.75. Table 2 gives the results assuming φ1 = φ2 = 0.9, and using 10,000 replications. 3.2. Results from finite-sample power experiments It can be seen in Table 2 that for T = 100 the power of tNL is in all cases much greater than the power of Fs , despite not nesting the
relevant TAR model. Note that the alternative hypothesis is onesided but the F -tests have two-sided rejection regions. Although the t-tests are computed from misspecified models the rejection region is one-sided and this is seen to have a positive impact on the power of the t-tests relative to the F -tests. Note also that the auxiliary models used to compute tNL are more parsimonious than the auxiliary models used to compute Fs ; so there are more degrees of freedom available. Overall, these effects seem in this case to be sufficient to counteract the detrimental impact of the t-tests being computed from misspecified models. When constants and deterministic trends are entertained the power of the t-tests and F -tests are similar. For small threshold values and therefore a small central regime, Fs,c and Fs,ct perform better than ts,c and tNL,ct , but for large threshold values the opposite is true. For the larger sample size T = 200 the power of all of the tests increases. For example, the power of tNL is over 90% for all combinations of threshold values considered, while the power of Fs ranges from approximately 67% to 81%. For the zero mean cases, the t-tests of Kapetanios et al. (2003) have superior power against three-regime TAR nonlinearity than the F -tests proposed here. Note however that for the case of a non-zero mean and small threshold values, Fs,c has greater power than the corresponding ttest tNL,c . For the larger sample size and allowing for a deterministic trend, Fs,ct has greater power than tNL,ct , apart from for the largest threshold values. The power experiments discussed above employ the same DGP that Kapetanios and Shin (2006) use to investigate the finitesample power of their Wsup , Wavg and Wexp tests. It is interesting to compare the results here with theirs (see Kapetanios and Shin, 2006, Table 3). Such a comparison reveals that for the DGPs considered, the STAR-based tests of Kapetanios et al. (2003) and those proposed here are very competitive. 4. Conclusions This paper investigates testing the unit root hypothesis against the alternative of globally stationary three-regime TAR nonlinearity with a unit root central regime, using STAR-based tests. It is found that for moderate sample sizes (e.g. T = 200)
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R. Sollis / Economics Letters 112 (2011) 19–22
the STAR-based tests have good finite-sample power against this alternative. Furthermore they compare well with tests of the unit root hypothesis developed by Kapetanios and Shin (2006), which are directly computed from three-regime TAR models rather than STAR approximations. Despite being computed from misspecified models, for moderate and large sample sizes the STAR-based ttests of Kapetanios et al. (2003) are sufficiently powerful to be used for identifying TAR nonlinearity, and in some cases are more powerful than F -tests derived from a generalized STAR model that does nest TAR nonlinearity. While generalizing nonlinear models can help accommodate different forms of nonlinearity, it is important for practitioners to recognize that when testing for simple forms of nonlinearity it can be beneficial to use a more restrictive and possibly misspecified model.
References Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431. Kapetanios, G., Shin, Y., 2006. Unit root tests in three-regime SETAR models. Econometrics Journal 9, 252–278. Kapetanios, G., Shin, Y., Snell, A., 2003. Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics 12, 359–379. Luukonen, R., Saikkonen, P., Teräsvirta, T., 1988. Testing linearity against smooth transition autoregressive models. Biometrika 75, 491–499. Obstfeld, M., Taylor, A.M., 1997. Nonlinear aspects of goods market arbitrage and adjustment: Hecksher’s commodity points revisited. Journal of the Japanese and International Economies 11, 441–479. Taylor, M.P., Peel, D.A., Sarno, L., 2001. Nonlinear mean reversion in real exchange rates: towards a solution to the purchasing power parity puzzles. International Economic Review 42, 1015–1042.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Testing the unit root hypothesis against TAR nonlinearity using STAR-based tests Robert Sollis ∗ Newcastle University Business School, Newcastle University, Ridley Building, Queen Victoria Road, Newcastle upon Tyne, NE1 7RU, United Kingdom
article
info
Article history: Received 16 March 2010 Received in revised form 16 February 2011 Accepted 18 March 2011 Available online 29 March 2011
abstract This paper investigates the finite-sample power of STAR-based unit root tests when the data generation process is a globally stationary three-regime TAR model. Unit root tests are proposed derived from STAR models that nest TAR models. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C22 C12 Keywords: Unit roots Nonlinearity Power
1. Introduction Threshold autoregressive (TAR) and smooth transition autoregressive (STAR) models have proved to be popular for modeling a wide variety of time series data. From the TAR family of models the three-regime model with a unit root central regime and stationary outer-regimes has been found to be particularly useful for applications involving real exchange rates, since the presence of transaction costs suggest that small deviations from the law of one price are not corrected, but that large deviations are corrected; see for example Obstfeld and Taylor (1997). Similarly, from the STAR family of models the globally stationary exponential-STAR (ESTAR) model with a unit root at equilibrium has proved to be popular in empirical applications due to its ability to model processes in which small deviations from a long-run attractor are not corrected but large deviations are; see for example Taylor et al. (2001). Tests of the unit root hypothesis against globally stationary ESTAR nonlinearity with a unit root at equilibrium have been proposed by Kapetanios et al. (2003). Tests of the unit root hypothesis against globally stationary three-regime TAR nonlinearity with a unit root central regime have been proposed by Kapetanios and Shin (2006). Both types of test are popular in practice. While the ESTAR model approximates the three-regime TAR model it is important to recognize that the two models are not actually nested within each other; the ESTAR model is incapable of generating three clearly defined regimes, while the three-regime
∗
Tel.: +44 (0)1912228584; fax: +44 (0)1912226838. E-mail address: [email protected].
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.03.008
TAR model is incapable of generating smooth adjustment between regimes. Since practitioners might not have any prior information as to which model is the most appropriate for their data set, in this paper we investigate the finite-sample power of the ESTARbased unit root tests proposed by Kapetanios et al. (2003) when the data generation process (DGP) under the alternative hypothesis is a globally stationary three-regime TAR model with a unit root central regime. Unit root tests derived from STAR models that do nest three-regime TAR models with a unit root central regime are also proposed. 2. STAR unit root tests 2.1. t-tests of Kapetanios et al. (2003) Kapetanios et al. (2003) employ the following ESTAR specification,
1yt = Et (γ , yt −d )ρ yt −1 + εt Et (γ , yt −d ) = 1 − exp(−γ
y2t −d
(1)
) γ ≥0
(2)
where εt is an independently and identically distributed (i.i.d.) sequence with zero mean and constant variance, and where |1 + ρ| < 1 is required to rule out explosive behavior. Kapetanios et al. (2003) focus on the case of d = 1. Under this restriction and assuming γ > 0, for yt −1 → 0, it is effectively a unit root process, but for yt −1 → ±∞, yt it is a stationary AR (1) process with AR parameter (1 + ρ). Globally, it is stationary but locally it can behave like a unit root process. This model can be extended to deal with a non-zero mean and a deterministic trend by replacing yt with y˜t = yt − µ ˆ or y˜ t = yt − αˆ − βˆ t, where E (yt ) = µ, and
20
R. Sollis / Economics Letters 112 (2011) 19–22
µ, ˆ α, ˆ βˆ are consistent sample estimates of the relevant population parameters. It can also be extended to deal with higher-order autocorrelations in the same way as the Dickey–Fuller test (Dickey and Fuller, 1979), by augmenting with lags of 1yt ,
1yt = Et (γ , yt −d )ρ yt −1 +
k −
βi 1yt −i + εt .
(3)
i =1
Note that if γ = 0 then it contains a unit root. Hence Kapetanios et al. (2003) propose testing the unit root hypothesis by testing H0 : γ = 0 in (3) against the alternative hypothesis H0 : γ > 0 (where ρ < 0 is assumed to hold). Under this null hypothesis the parameter ρ is unidentified, which prevents the asymptotic distribution of a test statistic being derived using conventional analytical methods. However, following Luukonen et al. (1988) this problem can be circumvented by replacing Et (γ , yt −1 ) in (3) with a first-order Taylor series expansion around γ = 0. On re-arranging, this gives the auxiliary model,
1yt = δ y3t −1 +
k −
βi 1yt −i + εt
Fig. 1(a). Second-order logistic function: fast transition.
(4)
i=1
and the relevant null hypothesis is H0 : δ = 0. Let tNL , tNL,c and tNL,ct denote the tests for the zero-mean, non-zero mean and deterministic trend cases respectively. 2.2. Unit root tests derived from STAR models that nest three-regime TAR models Consider the following STAR model for yt that, instead of the exponential function utilizes a second-order logistic function,
1yt = St (γ , θ , yt −d )ρ yt −1 + εt St (γ , θ, yt −d ) = [1 + exp(−γ (
(5)
y2t −d
− θ ))]
−1
γ ≥0
(6)
Fig. 1(b). Second-order logistic function: moderately fast transition.
where |1 + ρ| < 1, εt ∼ i.i.d.(0, σ ). As in Kapetanios et al. (2003) assume d = 1. (5) is an LSTAR model, although the logistic function is second-order rather than first-order as in conventional √ LSTAR models. In this model, for γ → ∞, if yt −1 < − θ or √ yt −1 > θ , yt is√a stationary AR √ (1) process with AR parameter (1 + ρ). But if − θ < yt −1 < θ , yt is a unit root process. Note therefore that (5) with d = 1 nests the following three-regime TAR model with symmetric outer regimes, if 2
1yt = ρ yt −1 + εt if |yt −1 | > r 1yt = εt if |yt −1 | ≤ r
(7) (8)
where r is a positive threshold. (5) does not nest an ESTAR model, but for small and moderate values of γ , (5) is capable of generating exponential-type nonlinearity that closely approximates ESTAR nonlinearity. A unit root test derived from (5) should therefore have power against globally stationary three-regime TAR nonlinearity with a unit root central regime, and globally stationary ESTAR nonlinearity with a unit root at equilibrium. To clarify, in Figs. 1(a), 1(b) and 1(c) some example transitions from the second-order logistic function (6) are plotted, along with a transition from the exponential function (2) for comparison. The ability of (6) to emulate both a discrete indicator function and an exponential function is clear in these figures. Following Kapetanios et al. (2003), to allow for a non-zero mean or deterministic trend replace yt with the de-meaned or de-trended series, and to allow for additional dynamics (5) can be augmented with the lags of 1yt ,
1yt = St (γ , θ , yt −d )ρ yt −1 +
k −
Fig. 1(c). Exponential function: moderately fast transition.
root hypothesis, but in this case θ and γ are unidentified. Therefore in both of these cases conventional asymptotic analysis cannot be used to derive the asymptotic distributions for appropriate test statistics. If we replace St (γ , θ , yt −d ) in (9) with a first-order Taylor expansion around γ = 0, then on re-arrangement this gives a form that can be used for testing,
1yt = φ1 yt −1 + φ2 y3t −1 +
k −
βi 1yt −i + ηt .
(10)
i=1
βi 1yt −i + εt .
(9)
i=1
The unit root hypothesis is H0 : γ = ρ = 0. Clearly, under this null hypothesis θ is unidentified. Note that H0 : ρ = 0 is also a unit
The auxiliary model (10) has only one additional term compared to the auxiliary models of Kapetanios et al. (2003) (see Eq. (4)). The relevant null hypothesis is in this case H0 : φ1 = φ2 = 0
(11)
R. Sollis / Economics Letters 112 (2011) 19–22
21
Table 1 Critical values for Fs , Fs,c , and Fs,ct . T
10%
5%
1%
10%
Fs 100 200 Asymptotic
3.686 3.641 3.564
4.587 4.448 4.378
6.505 6.450 6.185
Fs = (Rβˆ − r)′ σˆ 2 R
−
xt x′t
R′
(Rβˆ − r)/m
(12)
− (1yt − φˆ 1 yt −1 − φˆ 2 y3t −1 )2 /(T − 2).
(13)
t
The asymptotic distribution of Fs can be established using wellknown convergence results. For brevity a full proof is omitted, but is available on request. The distribution can be written as the following functionals of Brownian motion (⇒ denotes weak convergence, W (r ) is the standard Brownian motion on r ∈ [0, 1]), Fs ⇒ h′ Q−1 h/2σ 2
(14)
where, h ≡
[
1 2
σ {W (1) − 1} 2
2
∫ 1 2 W (r )2 dr σ 0 Q≡ 4∫ 1 σ W (r )4 dr 0
10%
σ σ4 σ6
4
1 4
W (1) − 4
1
∫
∫0 1
3 2
1
∫
W (r ) dr 2
] (15)
0
W (r )4 dr W (r )6 dr
.
(16)
0
Let Fs,c and Fs,ct denote the test statistics allowing for a non-zero mean and deterministic trend. Asymptotic and finitesample critical values for all three tests obtained by Monte Carlo simulation with 10,000 replications are given in Table 1. 3. Monte Carlo simulation results 3.1. Finite-sample power experiments To investigate the finite-sample power of the Kapetanios et al. (2003) tests (tNL , tNL,c and), and the Fs , Fs,c , and Fs,ct tests proposed here against the alternative hypothesis of globally stationary threeregime TAR nonlinearity with a unit root central regime the following DGP is used,
φ1 yt −1 + ut if yt −1 < r1 yt = yt −1 + ut if r1 ≤ yt −1 ≤ r2 φ2 yt −1 + ut if yt −1 > r2
5%
1%
6.980 7.273 7.131
5.775 5.787 5.727
6.818 6.799 6.717
9.185 9.082 8.617
Table 2 The power of alternative STAR-based tests against three-regime TAR nonlinearity with a unit root central regime at the 5% nominal size. r2
tNL
tNL,c
tNL,ct
Fs
Fs,c
Fs,ct
−0.15 −0.90 −1.65 −2.40 −3.15 −3.90 T = 200
0.15 0.90 1.65 2.40 3.15 3.90
0.602 0.608 0.633 0.661 0.669 0.641
0.249 0.250 0.242 0.228 0.200 0.200
0.154 0.154 0.146 0.132 0.119 0.112
0.366 0.236 0.220 0.207 0.206 0.210
0.274 0.267 0.245 0.212 0.189 0.178
0.180 0.174 0.158 0.139 0.125 0.118
−0.15 −0.90 −1.65 −2.40 −3.15 −3.90
0.15 0.90 1.65 2.40 3.15 3.90
0.905 0.910 0.923 0.941 0.957 0.965
0.612 0.618 0.632 0.638 0.609 0.548
0.402 0.407 0.413 0.403 0.357 0.295
0.804 0.819 0.786 0.737 0.701 0.674
0.772 0.763 0.728 0.660 0.572 0.485
0.576 0.564 0.514 0.442 0.355 0.281
T = 100
where xt = (yt −1 , y3t −1 )′ , m = 2, R is a 2 × 2 identity matrix, βˆ = (φˆ 1 , φˆ 2 )′ where φˆ 1 and φˆ 2 are the OLS estimates of φ1 and φ2 , r = (0, 0)′ and,
′
5.141 5.181 5.046
r1
−1
−1
t
σˆ 2 =
1%
Fs,ct
4.223 4.255 4.169
which can be tested using an F -test. The alternative stationary regions are φ1 < 0 and φ1 + φ2 < 0. The test statistic (assuming k = 0) can be written,
5%
F s,c
(17)
where ut ∼ nid(0, 1). Results are computed for and at the 5% nominal size for different combinations of threshold values ranging from r1 , r2 = {−0.15, 0.15} to r1 , r2 = {−3.90, 3.90} in steps of 0.75. Table 2 gives the results assuming φ1 = φ2 = 0.9, and using 10,000 replications. 3.2. Results from finite-sample power experiments It can be seen in Table 2 that for T = 100 the power of tNL is in all cases much greater than the power of Fs , despite not nesting the
relevant TAR model. Note that the alternative hypothesis is onesided but the F -tests have two-sided rejection regions. Although the t-tests are computed from misspecified models the rejection region is one-sided and this is seen to have a positive impact on the power of the t-tests relative to the F -tests. Note also that the auxiliary models used to compute tNL are more parsimonious than the auxiliary models used to compute Fs ; so there are more degrees of freedom available. Overall, these effects seem in this case to be sufficient to counteract the detrimental impact of the t-tests being computed from misspecified models. When constants and deterministic trends are entertained the power of the t-tests and F -tests are similar. For small threshold values and therefore a small central regime, Fs,c and Fs,ct perform better than ts,c and tNL,ct , but for large threshold values the opposite is true. For the larger sample size T = 200 the power of all of the tests increases. For example, the power of tNL is over 90% for all combinations of threshold values considered, while the power of Fs ranges from approximately 67% to 81%. For the zero mean cases, the t-tests of Kapetanios et al. (2003) have superior power against three-regime TAR nonlinearity than the F -tests proposed here. Note however that for the case of a non-zero mean and small threshold values, Fs,c has greater power than the corresponding ttest tNL,c . For the larger sample size and allowing for a deterministic trend, Fs,ct has greater power than tNL,ct , apart from for the largest threshold values. The power experiments discussed above employ the same DGP that Kapetanios and Shin (2006) use to investigate the finitesample power of their Wsup , Wavg and Wexp tests. It is interesting to compare the results here with theirs (see Kapetanios and Shin, 2006, Table 3). Such a comparison reveals that for the DGPs considered, the STAR-based tests of Kapetanios et al. (2003) and those proposed here are very competitive. 4. Conclusions This paper investigates testing the unit root hypothesis against the alternative of globally stationary three-regime TAR nonlinearity with a unit root central regime, using STAR-based tests. It is found that for moderate sample sizes (e.g. T = 200)
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the STAR-based tests have good finite-sample power against this alternative. Furthermore they compare well with tests of the unit root hypothesis developed by Kapetanios and Shin (2006), which are directly computed from three-regime TAR models rather than STAR approximations. Despite being computed from misspecified models, for moderate and large sample sizes the STAR-based ttests of Kapetanios et al. (2003) are sufficiently powerful to be used for identifying TAR nonlinearity, and in some cases are more powerful than F -tests derived from a generalized STAR model that does nest TAR nonlinearity. While generalizing nonlinear models can help accommodate different forms of nonlinearity, it is important for practitioners to recognize that when testing for simple forms of nonlinearity it can be beneficial to use a more restrictive and possibly misspecified model.
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