Nonlinear error correction based cointegration test in panel data

Economics Letters 157 (2017) 1–4

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Economics Letters journal homepage: www.elsevier.com/locate/ecolet

Nonlinear error correction based cointegration test in panel data Tolga Omay a , Furkan Emirmahmutoglu b, *, Zulal S. Denaux c a b c

Department of Economics, Atilim University, Ankara, Turkey Department of Econometrics, Gazi University, Ankara, Turkey Department of Economics and Finance, Valdosta State University, GA, USA

highlights • • • • •

A nonlinear ECM based cointegration test is proposed. The proposed test is the first nonlinear error correction based test in the panel cointegration literature. The nonlinear ECM with logistic transition functions implies asymmetric adjustment. This study utilizes a sieve bootstrap method for cross-section dependency problem. Simulation results confirm the superiority of the nonlinear ECM-based test in finite samples performance.

article

info

Article history: Received 21 November 2016 Received in revised form 15 May 2017 Accepted 17 May 2017 Available online 22 May 2017

a b s t r a c t We propose a nonlinear error correction-based cointegration test in a panel data setting and provide their small sample properties. © 2017 Elsevier B.V. All rights reserved.

JEL classification: C2 C4 C12 C15 Keywords: Nonlinear error correction model Sieve bootstrap Modified Wald test Cross section dependency

1. Introduction Over the last two decades, Error Correction Model (ECM) based tests have attracted considerable attention in the cointegration literature (e.g. Kremers et al., 1992; Banerjee et al., 1998) where the leading approach in cointegration literature is still residualbased tests. However, residual-based tests assume that a commonfactor restriction must be satisfied and ignore potential dynamic knowledge contained in the ECM. Additionally, Kremers et al. (1992) and Banerjee et al. (1998) argue that a failure in satisfying common-factor restriction may cause a significant power loss for residual-based tests. To this end, ECM-based tests are more powerful than the residual-based tests if the restriction is invalid. author. * Corresponding E-mail addresses: [email protected] (T. Omay), [email protected] (F. Emirmahmutoglu), [email protected] (Z.S. Denaux). http://dx.doi.org/10.1016/j.econlet.2017.05.017 0165-1765/© 2017 Elsevier B.V. All rights reserved.

In this study, we propose a new asymmetry test for nonlinear cointegration in LSTAR type ECM framework in a panel data setting. In contrast, Omay et al. (2014) suggest a residual-based nonlinear panel cointegration test by employing the Ucar and Omay (2009) methodology. Their study uses the ESTAR type of nonlinearity (symmetric and size nonlinearity with well-defined stationarity conditions). However, it is not possible to implement LSTAR nonlinearity (asymmetric and sign nonlinearity with not well-defined stationarity conditions) in a residual-based test without providing the well-defined stationarity properties. Therefore, by using the flexible structure of the ECM-based methodology our contribution to the literature is twofold; introducing asymmetric sign nonlinearity (LSTAR model) and first nonlinear ECM-based test in a panel setting. This study also makes use of approaches by Abadir and Distaso (2007) and Palm et al. (2010) to tackle mixed alternative hypotheses and cross-section dependency, respectively.

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T. Omay et al. / Economics Letters 157 (2017) 1–4

2. Model and assumptions Let zit

′

= (yit , x′it ) be I (1) process where yit and xit

=

(x1it , . . . , xkit ) is a scalar variable and k-dimensional vector, respectively. This paper aims to analyze at most one conditional cointegration relationship between yit and xit (e.g., Kapetanios et al., 2006, KSS). Hence, we consider the following conditional smooth transition ECM (ST-ECM) for 1yit and the marginal model for 1xit in the panel data.1 ′

1yit = φi1 ui,t −1 + φi1 ui,t −1 F (sit ; γi , ci ) + ∗

pi ∑

δij 1zi,t −j ′

j=1

+ ωi′ 1xit + eit (i = 1, . . . , N ; t = 1, . . . , T )

(2.1)

1xit = vit

(2.2)

where ui,t −1 = yi,t −1 − βi′ xi,t −1 with βi being a k-dimensional ∗ vector of individual cointegration parameters; φi1 and φi1 represent linear and nonlinear adjustment parameters for the conditional model, respectively. The nonlinear parameters ci and γi in F (sit ; γi , ci ) represent the threshold value and the speed of transition between regimes, respectively. Our model assumes that the transition between regimes is induced by a logistic function F (sit ; γi , ci ) = [1 + exp {−γi (sit − ci )}]−1 − 0.5;

γi > 0.

(2.3)

Here, the state variable sit = ui·t −1 denotes deviation from longrun equilibrium. Thus, the conditional model (2.1) can be called a panel logistic ST-ECM (PLSTR-ECM), which allows for asymmetric effects of positive and negative deviations from the equilibrium relative to the threshold ci . We present the following assumptions for the development of our new test. Following Westerlund (2007), the error processes ′ wit = (eit , v′it ) satisfy the following assumption: Assumption 1. (a) wit is independent and identically distributed (i.i.d.) across both i and t, with E (wit ) = 0 and Var (eit ) = σei2 < ∞ (b) Var (vit ) = Σixx > 0 (c) E(eit vjs ) = 0 for all i, j, t and s. −1/2

∑[Tr]

Under Assumption 1, the partial sum processes T t =1 wit satisfy the following multivariate invariance principle individually for each cross-section: [Tr] 1 ∑

√

T

1/2

wit ⇒ Σi

Bi (r )

as T → ∞

(2.4)

t =1 ′

where ⇒ denotes weak convergence and Bi = (Bie , B′iv ) is a (k + 1) vector standardized Brownian motion defined on r ∈ [0, 1]. Moreover, Σi represents a block diagonal covariance matrix with elements σe2i and Σi22 . Assumption 1(a) states that the individual errors are independent across cross-sections. Assumption 1(b) implies that there is no cointegration among xit . Assumption 1(c) requires that the vector of regressors, xit , is independent of the regression error, eit , which is satisfied if xit is strictly exogenous. Then, the covariance matrix of Bi is equal to the long-run covariance matrix of wit . Therefore, Bie and Biv will be independent since the long-run covariance between eit and vit is zero. However, if xit is weakly exogenous for the parameters of the conditional model, but not strictly exogenous, then the long-run correlation between the partial sums of eit and vit is nonzero since eit is correlated with the future of vit . The result 1 See Appendix A for the motivation of our model.

is that Bie and Biv are no longer independent under the null of no cointegration. To solve this problem, Saikkonen (1991) proposes to correct for serial correlation by augmenting the conditional model (2.1) with leads of 1xit .2 Additionally, we make the following assumption for nonlinearity: Assumption 2. (a) Transition function F (·) is asymptotically no greater than a linear function of xit . ∗ (b) φi1 < 0. According to KSS, Assumption 2(a) deals with nonlinearity of the underlying conditional model (2.1). Assumption 2(b) guarantees that the underlying nonlinear ECM in (2.1) is globally station∗ < 0 for each i. ary provided that φi1 + φi1 2.1. Testing procedure and test statistic We test the null hypothesis of no error correction (leads to no∗ = 0 for all i, against cointegration), which is H0 = φi1 = φi1 the alternative hypothesis of nonlinear error correction (leads to nonlinear cointegration) in (2.1). However, the parameter γi is not identified under the null. Traditionally, we used an auxiliary regression which is( the first-order Taylor series approximation ) around γi = 0 of F ui,t −1 ; γi , ci to solve the nuisance parameter problem:

1yit = ρi1 uˆ i,t −1 + ρi2 uˆ 2i,t −1 +

pi ∑

δij′ 1zi,t −j + ωi′ 1xit + e˜ it

(2.5)

j=1

∗ where ρi1 = φi1 − 41 φi1 γi ci and ρi2 = 41 φi1∗ γi . The null hypothesis of no error correction becomes H0 : ρi1 = ρi2 = 0 for all i and is ∗ equivalent to H0 = φi1 = φi1 = 0 for all i in (2.1). The alternative hypothesis

)

(

∗ H1 : φi1 < 0, φi1 <0

for some i

(2.6)

for some i.

(2.7)

is equivalent to H1 : ρi1 ̸ = 0, ρi2 < 0

Notice that one parameter is one-sided under the alternative while the other is two-sided. Abadir and Distaso (2007) show that the standard Wald statistic for testing of mixed alternatives would be inappropriate. Then they propose a modified version of Wald (MWALD) test statistic, which states that the one-sided parameter is orthogonalized with respect to transformed two-sided parameter. ′ Let the parameter vector of interest be ρi = [ρi1 ρi2 ] . Using the notation of Abadir and Distaso (2007), H0 : ρi1 = ρi2 = 0 can be represented by H0 : hi (ρi ) = hi1 (ρi )

[

hi2 (ρi )

]′

[ = ρi1

for ∀i = 1, . . . , N

ρi2

]′

[ = 0

]′

0

(2.8)

against H1 : hi1 (ρi ) ̸ = 0 or hi2 (ρi ) < 0.

(2.9)

The standard individual Wald statistic based on the Hessian matrix (H) is Wi = hi (ρˆ i )′ Vˆ i−1 hi (ρˆ i ) 2 See also Banerjee et al. (1998) and KSS.

(2.10)

T. Omay et al. / Economics Letters 157 (2017) 1–4

where (2 × 2) matrix Vˆ i =

[

⏐ ∂ hi (ρi ) ⏐ ˆ −1 (−H) ′ ∂ρi ⏐ρ =ρˆ i i ] [ vˆ

0

1

( ) = hi1 ρˆ i [

vˆ i12 vˆ i22

0

[ ( )] Vˆ i1.2 hi1.2 ρˆ i 0

with

][

( )] h1i ρˆ i ( ) h2i ρˆ i 0 1

( )2 ( )2 −1 hi1 ρˆ i = hi1.2 ρˆ i Vˆ i1.2 + vˆ i22

hi1 ρˆ i

( )] ( ) hi2 ρˆ i

( )2 −1 ( )2 τic = hi1.2 ρˆ i Vˆ i1−.12 + 1hi2 (ρˆ i )<0 hi2 ρˆ i vˆ i22 where 1A denotes indicator function, where condition A is satisfied and zero otherwise. Let hi1 (ρi ) be ρi1 and hi2 (ρi ) be ρi2 , we can obtain individual modified test statistic as follows:

vˆ i21 τic = ρˆ i1 − ρˆ i2 vˆ i22

)2 ( )−1 ρˆ 2 vˆ 2 + 1ρˆ i2 <0 i2 . vˆ i11 − i21 vˆ i22 vˆ i22

(2.12)

Then we propose the group mean statistic as

τ¯ c =

N 1 ∑

N

τic .

(2.13)

i=1

The independency assumption between units given in the previous section is restrictive. Therefore, it is likely to be violated in practice. Our aim is to obtain the empirical distribution of the (τ¯ c ) statistic using a sieve bootstrap approach which allows more general dependence structure such as contemporaneous dependency between cross-sectional units (e.g., Westerlund, 2007, Chang and Nguyen, 2012). This approach permits us to simultaneously deal with time series dependency (serial correlation) in the error processes as well as cross-sectional dependency, and is valid for a relatively fixed N which tolerates the sample size traditionally used in the empirical studies. This study used sieve bootstrap algorithm which is a nonlinear panel extension of the ECM-based cointegration proposed by Palm et al. (2010) in multivariate time series setting. (i)3 Estimate the following panel VAR(qi ) process and obtain the ′ ˆ t = (w ˆ 1t , . . . , w ˆ Nt ) residuals w

ˆ it = 1zit − w

πˆ ij 1zi,t −j . ′

(2.14)

j=1

Then, obtain the centered fitted residuals

˜t =w ˆ t − (T − q − 1)−1 w

z∗it = z∗i,t −1 + υ ∗it

(2.17) ∗′ ′

and partition z∗it = (y∗it , xit ) . (v) Calculate the bootstrap version (τ¯ c ∗ ) of τ¯ c using the bootstrap samples z∗it following regression

1yit = ∗

ρi1∗ uˆ ∗i,t −1

+

ρi2∗ uˆ 2i,∗t −1

+

pi ∑

′

ψij∗ 1z∗i,t −j

j=1

∗′

+ ωi 1xit + error ∗

(2.18)

ˆ ∗′ ∗

where uit = yit − βi xit and βi is the estimate of the cointegration vector in the bootstrap sample. (vi) Repeat steps (ii)–(v) B times. Select the bootstrap critical value as the (1 − α )-quantile of the ordered τ¯bc ∗ statistics. Reject the null of no cointegration if τ¯ c > cα∗ .

ˆ∗

∗

ˆ∗

3. Monte-Carlo simulations This section conducts Monte-Carlo simulations to evaluate the finite-sample performance of our test τ¯ c and compare it with the test of Omay et al. (2014),4 say t¯NL . 3.1. Design of data generating process The data are generated from the following bivariate PLSTRECM:

2.2. Sieve bootstrap algorithm

qi ∑

(2.16)

∗

(2.11)

−1 and Vˆ i1.2 = vˆ i11 where hi1.2 (ρˆ i ) = hi1 (ρˆ i ) − hi2 (ρˆ i )vˆ i21 vˆ i22 ( − ) 2 vˆ i21 vˆ i22 , which is an estimate of the asymptotic variance of hi1.2 ρˆ i . Generally, MWALD statistic of Abadir and Distaso (2007) is given by

(

πˆ ′ij υ∗i,t −j + w∗it .

(iv) Construct the bootstrap samples z∗it recursively using

vˆ i21 vˆ i22 ( )] ]−1 [ hi1 ρˆ i 0 ( ) vˆ i22 hi1.2 ρˆ i −1

qi ∑ j=1

])−1 [

1

(ii) Generate the N-dimensional vector w∗t = (w∗1t , . . . , w∗Nt ) by ˜ t. resampling from the centered residual vector w (iii) Generate υ ∗it recursively from w∗it as

υ∗it =

]−1 [

vˆ i22

0

ρi =ρˆ i

3 ′

vˆ i12 vˆ i22 , (2.10) can be rewritten

elements vijk . If we define Vˆ i as vˆ i11 i21 quadratic forms as

[ [ ( ) ( )] vˆ i11 Wi = hi1 ρˆ i hi2 ρˆ i vˆ i21 [ ( ) ( )] = hi1 ρˆ i hi2 ρˆ i ([ ][ −1 1 vˆ i21 vˆ i22 Vˆ i1.2 ×

]

⏐ ∂ hi (ρi )′ ⏐ ∂ρi ⏐

∑ t

ˆt w

(2.15)

where q = max(qi ). 3 For the alternative bootstrap algorithm, we replace Step1 by calculating the residuals from conditional/marginal models, see Palm et al. (2010).

) ( 1yit = φi ui,t −1 + φi∗ ui,t −1 F ui,t −1 ; γi , ci + ωi 1xit + eit

(3.1)

1xit = vit

(3.2)

where uit = yit − βi xit , γi ∼ i.i.d. U(0.01, 1) and ci = 0. Under the null, H0 : φi = φi∗ = 0, whereas φi = −0.1 and φi∗ ∼ i.i.d. U(−1, −0.01) under the alternative. We report only the simulation results for the detrended data.5 The number of replications is 20000. For all the experiments, the data with T + 50 are generated, where the first 50 observations are discarded to reduce the effect of the initial conditions. We consider three scenarios: The first examines the common-factor restriction (COMFAC) problem; the others look into the weak and strong forms of the cross-section dependency (CSD) problem. S16 (COMFAC) The innovations eit and vit are generated by

( ) eit

vit

∼ i.i.d. N

([ ] [ 2 σ 0 , ei 0

0

0

σvi 2

])

.

(3.3)

In (3.1), we choose βi = 1 and σe2i = 1. We consider two different values of ωi . If the restriction is satisfied, then ωi = 1, otherwise ωi = 0.5. To control the degree of violation by signalto-noise ratio, we use σv2i ∈ {1, 4}. 4 See Appendix A for details. 5 The results for demeaned data are available upon request. 6 Under the no CSD, the critical values of τ¯ c and t¯ are obtained via simulations NL and are available upon request.

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T. Omay et al. / Economics Letters 157 (2017) 1–4

S2 (Weak CSD) eit and vit are generated as AR(1) process eit = θi ei,t −1 + εit

Table 1 Empirical sizes of τ¯ c and t¯NL under COMFAC.

(3.4)

vit = ρi vi,t −1 + ηit

(3.5) ′

′

where εt = (ε1t , . . . , εNt ) and ηt = (η1t , . . . , ηNt ) are independent N-dimensional multivariate normal random vectors. To ensure weak CSD, the error terms εt ’s in (3.4) are generated through the relation εt = Lζt , where ζt ∼ i.i.d. N(0, IN ), LL′ = Σ and Σ is a Toeplitz matrix. We use the following different Σ matrices: N = 5;

Toeplitz (1.00; 0.70; 0.50; 0.30; 0.10)

) 1.00; 0.70; 0.60; 0.50; 0.40; 0.30; Toeplitz 0.20; 0.10; 0.05; 0.01 ⎛ ⎞ 1.00; 0.70; 0.65; 0.60; 0.55; 0.50; ⎜ 0.45; 0.45; 0.40; 0.40; 0.35; 0.35; ⎟ Toeplitz ⎝ . 0.30; 0.30; 0.25; 0.25; 0.20; 0.20; ⎠ 0.15; 0.15; 0.15; 0.10; 0.10; 0.05; 0.05 (

N = 10;

N = 25;

Similarly, we generate ηt ’s in (3.5) by using the same Σ matrices as above. The AR coefficients θi ’s and ρi ’s are selected as i.i.d. U(−0.8, 0.8) and i.i.d. U(0, 0.8), respectively. Unlike S1, ωi and βi in (3.1) are i.i.d. U(0, 1) and they are mutually independent of each other. Therefore, the COMFAC restriction is invalid. The other parameters are generated as S1. S3 (Strong CSD) eit and vit are generated as follows: eit = ci ft + ξit

(3.6)

( ) ξit = εit + θi εi,t −1 εit ∼ i.i.d. N 0, σε2i with σε2i = 1 ( ) vit = ηit + ρi ηi,t −1 ηit ∼ i.i.d. N 0, ση2i with ση2i = 1

(3.7)

T

50

N

τ¯ c

t¯NL

100

τ¯ c

t¯NL

τ¯ c

200 t¯NL

5 10 25

5.60 5.72 5.91

4.76 5.05 5.09

5.46 5.50 5.46

5.34 5.35 5.11

5.01 5.28 5.66

4.81 4.98 5.21

Table 2 Empirical sizes of τ¯ c and t¯NL under weak CSD. T

50

N

τ¯ c

t¯NL

100

τ¯ c

t¯NL

τ¯ c

200 t¯NL

5 10 25

5.64 5.21 5.66

5.16 5.22 5.44

5.39 5.08 5.48

5.16 5.20 5.42

5.34 5.20 5.65

5.16 5.07 5.78

Table 3 Empirical sizes of τ¯ c and t¯NL under strong CSD. T

50

N

τ¯ c

t¯NL

100

τ¯ c

t¯NL

τ¯ c

200 t¯NL

5 10 25

5.70 5.25 6.28

7.38 9.57 10.70

5.83 5.72 5.93

7.00 8.94 9.96

5.17 5.09 5.32

6.80 8.33 9.54

4. Conclusions This paper contributes to the literature by proposing a new ECM-based nonlinear cointegration test in panels. Monte-Carlo simulations confirm the superiority of using the nonlinear ECMbased test in finite samples performance.

(3.8)

where ft ∼ i.i.d. N (0, 1) is a common factor that generates CSD. The factor loading ci are drawn from i.i.d. U (1, 4) to ensure the strong CSD. Also, θi ∼ i.i.d. U(−0.5, 0.5) and ρi ∼ i.i.d. U(0, 0.5). The error processes (εit , ηit )′ and ft are mutually independent of each other. The other parameters are identical with S1. In both CSD forms, we apply the Sieve method to obtain bootstrap critical values of the τ¯ c statistic.7

Acknowledgements We acknowledge the financial support provided by the Turkish Scientific and Technological Research Council which enabled Furkan Emirmahutoglu to visit Valdosta State University to complete his research project (program code: 2219). We also thank an anonymous referee for the valuable comments. Appendix A. Supplementary data

3.2. Results Tables 1–3 tabulate the empirical size results of τ¯ c and t¯NL tests. Additional empirical power results are presented in Appendix A (see Tables A1–A3). The sizes of the tests for S1 are reported in Table 1. We find that both tests behave quite similarly in terms of size. Table A1 presents power performances of both tests under the COMFAC. As expected, our test is superior to the t¯NL test for all combinations of N and T regardless of whether COMFAC is valid or not. The empirical sizes of the tests for S2 and S3 are presented in Tables 2–3, respectively. For the weak CSD case, Table 2 shows that both nonlinear tests have correct size like S1. However, we clearly see that t¯NL becomes considerably oversized for large values of N and T in the presence of the strong CSD. In contrast, the empirical size of our test, τ¯ c , is close to the nominal size for almost all T and N. Finally, we investigate the power performances of the nonlinear tests under two CSD forms. The power results are provided in Tables A2–A3. Overall, our nonlinear ECM-based test continues to perform well and is more powerful than t¯NL test for all values of N and T for two CSD forms.

7 For each of simulation experiments we use Warp speed bootstrap method.

Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2017.05.017. References Abadir, K.M., Distaso, W., 2007. Testing joint hypotheses when one of the alternatives is one-sided. J. Econometrics 140, 695–718. Banerjee, A., Dolado, J.J., Mestre, R., 1998. ECM tests for cointegration in a single equation framework. J. Time Series Anal. 19, 267–283. Chang, Y., Nguyen, C.M., 2012. Residual based tests for cointegration in dependent panels. J. Econometrics 167, 504–520. Kapetanios, G., Shin, Y., Snell, A., 2006. Testing for cointegration in nonlinear smooth transtion error correction models. Econometric Theory 22, 279–303. Kremers, J.J., Ericson, N.R., Dolado, J.J., 1992. The power of cointegration tests. Oxford Bull. Econ. Stat. 54, 325–347. Omay, T., Hasanov, M., Ucar, N., 2014. Energy consumption and economic growth: Evidence from nonlinear panel cointegration and causality tests. Appl. Econometrics 34, 36–55. Palm, F.C., Smeekes, S., Urbain, J.P., 2010. A Sieve bootstrap test for cointegration in a conditional error correction model. Econometric Theory 26, 647–681. Saikkonen, P., 1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21. Ucar, N., Omay, T., 2009. Testing for unit root in nonlinear heterogeneous panels. Econom. Lett. 104, 5–7. Westerlund, J., 2007. Testing for error correction in panel data. Oxford Bull. Econ. Stat. 69, 709–748.