- Email: [email protected]
Partial unit root and linear spurious regression: A Monte Carlo simulation study
Economics Letters 118 (2013) 189–191
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Partial unit root and linear spurious regression: A Monte Carlo simulation study Lingxiang Zhang School of Management and Economics, Beijing Institute of Technology, Beijing, Haidian district, 100081, People’s Republic of China
article
info
Article history: Received 5 August 2012 Received in revised form 12 October 2012 Accepted 15 October 2012 Available online 23 October 2012
abstract In this paper, we consider both the partial unit root and the near partial unit root processes in nonlinear transition autoregression models. Our simulations show that when these time series data are used in ordinary least squares regression, spurious regression occurs. However, if we re-estimate the regression by adding an AR(1) term, spurious regression can almost be eliminated. © 2012 Elsevier B.V. All rights reserved.
JEL classification: C12 C32 C52 Keywords: Partial unit root Spurious regression Monte Carlo simulation
1. Introduction Since the studies by Granger and Newbold (1974) and Phillips (1986), much effort has been exerted in understanding linear spurious regression between independent variables. Durlauf and Phillips (1988) observe spurious correlations between random walks and linear trends. Marmol (1995, 1996) demonstrate that spurious correlations are evident in OLS regressions that involve combinations of series with integer orders of integration equal to or greater than one. Granger et al. (2001), Hassler (2003), Kim et al. (2004), and Noriega and Ventosa-Santaulària (2007) also show that spurious regressions occur in models with series generated by various combinations of different types of stationary processes (with and without linear trends and those that potentially allow for time-varying means because of structural breaks or seasonality). Marmol (1998) and Tsay and Chung (2000) indicate that spurious correlation generally occurs in regressions that involve various combinations of fractionally integrated processes (both stationary and non-stationary). Caner and Hansen (2001) and Park and Shintani (2005) consider a nonlinear transition autoregression model between two regimes, in which one regime is generated by a unit root and the other regime is a mean-reverting process. They call this model the partial unit root model. The present paper aims to examine the spurious regression between independent partial unit root processes
E-mail address: [email protected]. 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.10.018
or near partial unit root processes. Monte Carlo simulations reveal significant evidence of spurious regression between two independent partial unit root (or near partial unit root) variables. However, if we consider the auto-correlated residuals, and if we re-estimate the simulated equation using the EViews specification of AR(1) disturbance instead of presuming white noise disturbances, then the spurious regression phenomenon can almost be eliminated. The present paper is structured as follows. Section 2 provides some preliminaries. Section 3 presents the Monte Carlo simulation results. Section 4 concludes. 2. Preliminary We consider a transition autoregression model between two regimes: zt = θz zt −1 + λzt −1 π (zt −d , κ, µ) + uzt ,
(1)
where 0 ≤ θz ≤ 1 and λ < 0, such that |θ + λ| < 1; π (zt −d , κ, µ) is a real-valued transition function on R × R2 that usually takes values between zero and unity, in which zt −d is the transition variable with lag delay d ≥ 1. For simplicity, we set d = 1 in the simulations below1 ; κ is the scale parameter, µ is the location parameter, and uzt is the zero mean sequence of errors.
1 Transition variables can also be other exogenous variables, lagged difference variables, and so on. However, our simulations indicate that the spurious regression phenomenon cannot be affected by different transition variables.
190
L. Zhang / Economics Letters 118 (2013) 189–191
When θz = 1, one extreme regime of model (1) is given by a unit root process, which represents no adjustment movement. The other extreme regime is a mean-reverting process that represents linear adjustment toward a long-run equilibrium. The model can also be viewed as a nonlinear AR model of a nonlinear meanreverting process with an appropriate transition function. In such a case, we call model (1) the partial unit root model following the description by Caner and Hansen (2001). Furthermore, when θz < 1 but close to 1, we call model (1) the near partial unit root model. In this paper, we consider three commonly employed models for different transition functions, namely, the threshold autoregression (TAR), logistic smooth transition autoregression (LSTAR), and the exponential smooth transition autoregression (ESTAR). The corresponding transition functions are given respectively below:
Table 1 Rejection frequencies for tδˆ (|tδˆ | > 1.96) based on Regression (5) when yt and xt are balanced independent processes in the LSTAR models. n
θy , θx = 1.0 λ = −0.9
−0.5
100 300 500
49.2 70.7 79.4
58.1 75.7 82.1
100 300 500
61.1 77.7 83.2
63.0 79.1 83.7
(3)
LSTAR (µ, κ) = (1.0, 20.0)
ESTAR: π (zt −1 , κ, µ) = 1 − exp{−κ(zt −1 − µ) }.
(4)
100 300 500
yt = αˆ 0 + δˆ xt + εˆ t ,
(5)
yt = αˆ 1 + βˆ xt + εˆ t εˆ t = ρˆ εˆ t −1 + eˆ t ,
(6)
where both xt and yt are generated by model (1), with uxt and uyt being independent. In Eq. (5), we directly regress yt on xt without considering autocorrelation. In Eq. (6), we consider the series autocorrelation that may be exhibited in the residuals if the regression is spurious, as mentioned by McCallum (2010), by adding an AR(1) term in the regression to eliminate the first-order autocorrelation. 3. Monte Carlo simulation Based on Eqs. (1)–(4), we consider the following data generation processes (DGP) for xt and yt , where both uxt and uyt are i.i.d. N (0, 1). For the TAR model, λ ∈ {−0.9, −0.5, −0.1}, µ ∈ {−3, −1, 0, 1, 3}. For the LSTAR model, λ ∈ {−0.9, −0.5, −0.1}, (κ, µ) = (1.0, 0.0), (1.0, 1.0), (20.0, 0.0), (20.0, 1.0). For the ESTAR model, we set µ = 0 and κ = {0.1, 0.5, 1.0, 5.0}. For all DGPs, we set θx , θy equal to 1, 0.9 and to 0.7 to obtain the partial unit root and the near partial unit root, respectively. We consider the samples sizes of n = 100, 300, and 500 and compute the empirical rejection rates of the t-statistic to test the null hypotheses H0 : δ = 0 and H0 : β = 0, respectively, based on 10,000 replications and on the nominal level of 5%. For brevity, we only present the results in the LSTAR models, as shown by Tables 1–4. The results in the TAR and ESTAR models are similar to those presented in the LSTAR models and available upon request. Tables 1 and 2 detail the rejection frequencies for tδˆ based on Regression (5) when yt and xt represent the independent partial unit root and the near partial unit root processes, respectively, in the LSTAR models. When both yt and xt are partial unit root processes, the spurious regression phenomenon occurs both in the balanced and the unbalanced regression environment,2 and spurious rejection frequencies increase with the increase in sample sizes. In the particular case of λ = −0.1, the rejection frequencies
2 Here ‘‘balanced’’ means θ = θ , and ‘‘unbalanced’’ means θ ̸= θ . x y x y
−0.1
67.7 79.4 84.6
8.3 8.2 8.6
12.3 12.2 13.3
19.7 20.6 21.1
67.7 80.1 84.7
11.5 12.0 11.6
15.0 16.0 15.0
21.9 21.3 21.3
68.2 80.7 84.8
10.8 10.6 10.6
13.9 14.0 14.3
21.1 21.0 21.0
68.1 81.1 84.9
12.1 11.9 11.9
15.5 15.0 15.6
20.6 21.1 22.0
LSTAR (µ, κ) = (0.0, 20.0)
LSTAR: π (zt −1 , κ, µ) = (1 + exp{−κ(zt −1 − µ)})−1
To study the spurious regression phenomenon, we consider the following OLS regressions:
−0.5
LSTAR (µ, κ) = (1.0, 1.0)
100 300 500
2
θy , θx = 0.7 λ = −0.9
LSTAR (µ, κ) = (0.0, 1.0)
(2)
TAR: π(zt −1 , µ) = 1{zt −1 ≤ µ}
−0.1
66.8 80.5 84.7
66.8 81.3 85.1
66.7 80.7 84.8
67.3 80.4 84.5
Table 2 Rejection frequencies for tδˆ (|tδˆ | > 1.96) based on Regression (5) when yt and xt are unbalanced independent processes in the LSTAR models. n
θy = 1.0, θx = 0.7 λ = −0.9 −0.5
−0.1
θy = 0.7, θx = 0.9 λ = −0.9 −0.5
−0.1
33.6 35.1 34.8
11.1 12.4 13.3
18.6 19.1 19.2
28.9 28.4 30.2
32.8 35.3 36.8
17.4 18.6 17.8
21.3 22.6 22.1
29.8 29.9 30.2
33.5 35.5 35.1
17.1 17.3 17.2
20.4 20.6 20.7
28.3 29.1 29.6
33.3 36.1 36.4
18.2 19.7 18.8
21.7 22.2 23.3
29.4 29.9 29.5
LSTAR (µ, κ) = (0.0, 1.0) 100 300 500
15.5 17.0 17.8
23.0 25.7 25.2
LSTAR (µ, κ) = (1.0, 1.0) 100 300 500
21.6 23.7 23.6
26.8 27.5 28.6
LSTAR (µ, κ) = (0.0, 20.0) 100 300 500
21.1 23.8 23.2
24.7 27.9 27.5
LSTAR (µ, κ) = (1.0, 20.0) 100 300 500
22.6 25.3 25.7
27.3 29.2 29.4
are very high, indicating severe spurious regression. Furthermore, when both yt and xt are independent near partial unit root processes, i.e., θy , θx = 0.7 and θy = 0.7, θx = 0.9, the spurious regression phenomenon also occurs in certain situations, such as in λ = −0.1. Our simulations also show that when the parameter values of θx and θy approach unity, the spurious regression becomes more severe. However, these findings are not provided here because of space limitations. Tables 3 and 4 present the rejection frequencies for tβˆ based on Regression (6), in which a first-order autoregressive term is included to eliminate the autocorrelation of errors.3 We perform this regression based on calculations provided by the ‘‘AR(1)’’ procedure in EViews. When the simulated equation is estimated using the EViews specification of AR(1) disturbance instead of
3 We also investigate the DW statistics in Regression (5), which shows that in some cases, the fraction of times that the DW statistics are less than 1.0 are over 50%, indicating the presence of strong and serially correlated residuals.
L. Zhang / Economics Letters 118 (2013) 189–191 Table 3 Rejection frequencies for tβˆ (|tβˆ | > 1.96) based on Regression (6) when yt and xt are balanced independent processes in the LSTAR models. n
θy , θx = 1.0 λ = −0.9
−0.5
−0.1
θy , θx = 0.7 λ = −0.9
−0.5
−0.1
6.1 4.9 5.2
5.6 4.7 4.5
5.4 4.5 5.1
4.9 4.6 4.4
6.1 5.2 4.8
4.7 4.8 4.7
5.2 4.8 4.8
5.4 4.8 4.8
6.7 5.1 4.7
5.1 4.9 4.3
4.9 4.7 4.6
5.1 4.9 4.9
6.0 5.5 5.2
4.9 4.5 4.3
5.1 4.7 4.6
5.2 4.9 4.5
LSTAR (µ, κ) = (0.0, 1.0) 100 300 500
6.6 5.7 5.7
6.5 5.6 5.1
LSTAR (µ, κ) = (1.0, 1.0) 100 300 500
6.9 5.6 5.2
6.2 5.6 5.0
LSTAR (µ, κ) = (0.0, 20.0) 100 300 500
6.9 5.2 5.1
6.4 5.3 5.3
LSTAR (µ, κ) = (1.0, 20.0) 100 300 500
7.2 5.2 4.9
6.8 4.9 5.1
191
4. Conclusion This paper investigates the linear spurious regression phenomenon between two independent partial unit root or near partial unit root processes. Monte Carlo simulations reveal strong evidence of spurious regression between two independent partial (or near partial) unit root variables in the TAR, LSTAR, and ESTAR models. However, when the regressions are re-estimated by adding an AR(1) term, the re-estimation that considers potential autocorrelation tends to eliminate the appearance of non-existent relationships. This paper only examines the spurious regression phenomenon via simulations. Therefore, further theoretical research should be performed. Acknowledgments The research was funded by grants of the China Ministry of Education (No. 12YJC790268) and the Excellent Young Scholars Research Fund of Beijing Institute of Technology. References
Table 4 Rejection frequencies for tβˆ (|tβˆ | > 1.96) based on Regression (6) when yt and xt are unbalanced independent processes in the LSTAR models. n
θy = 1.0, θx = 0.7 λ = −0.9 −0.5
−0.1
θy = 0.7, θx = 0.9 λ = −0.9 −0.5
−0.1
4.5 4.2 4.5
5.8 5.2 5.2
5.5 4.8 4.8
6.3 4.9 4.9
4.6 4.1 4.4
5.8 5.2 4.9
5.8 5.0 5.0
6.4 5.1 4.7
4.5 4.4 4.8
6.5 5.4 5.3
5.4 4.8 4.7
6.2 4.9 4.6
4.6 4.4 4.1
5.6 4.9 4.9
5.9 4.9 5.0
6.1 5.0 4.8
LSTAR (µ, κ) = (0.0, 1.0) 100 300 500
5.2 4.9 5.3
5.0 4.7 4.3
LSTAR (µ, κ) = (1.0, 1.0) 100 300 500
4.9 4.9 4.5
4.9 4.5 4.6
LSTAR (µ, κ) = (0.0, 20.0) 100 300 500
4.6 4.6 4.8
4.9 4.9 4.5
LSTAR (µ, κ) = (1.0, 20.0) 100 300 500
4.7 4.9 5.2
4.5 4.7 4.8
presuming white noise disturbances, the spurious regression phenomenon is eliminated in most situations.
Caner, M., Hansen, B.E., 2001. Threshold autoregression with a unit root. Econometrica 69, 1555–1596. Durlauf, S.N., Phillips, P.C.B., 1988. Trends versus random walks in time series analysis. Econometrica 56, 1333–1354. Granger, W.J. Clive, Hyung, Namwon, Jeon, Yongil, 2001. Spurious regressions with stationary series. Applied Economics 33, 899–904. Granger, C.W.J., Newbold, P., 1974. Spurious regressions in econometrics. Journal of Econometrics 2, 111–120. Hassler, U., 2003. Nonsense regressions due to neglected time-varying means. Statistical Papers 44, 169–182. Kim, T., Lee, Y., Newbold, P., 2004. Spurious regressions with stationary processes around linear trends. Economics Letters 83, 257–262. Marmol, F., 1995. Spurious regressions between I (d) processes. Journal of Time Series Analysis 16, 313–321. Marmol, F., 1996. Nonsense regressions between integrated processes of different orders. Oxford Bulletin of Economics and Statistics 58, 525–536. Marmol, F., 1998. Spurious regression theory with nonstationary fractionally integrated processes. Journal of Econometrics 84, 233–250. McCallum, Bennett T., 2010. Is the spurious regression problem spurious? Economics Letters 107, 321–323. Noriega, A., Ventosa-Santaulària, D., 2007. Spurious regression and trending variables. Oxford Bulletin of Economics and Statistics 69, 439–444. Park, J.Y., Shintani, M., 2005. Testing for a unit root against transitional autoregressive models. Working Paper No. 05010. Department of Economics, Vanderbilt University. Phillips, P.C.B., 1986. Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311–340. Tsay, W., Chung, C., 2000. The spurious regression of fractionally integrated processes. Journal of Econometrics 96, 155–182.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Partial unit root and linear spurious regression: A Monte Carlo simulation study Lingxiang Zhang School of Management and Economics, Beijing Institute of Technology, Beijing, Haidian district, 100081, People’s Republic of China
article
info
Article history: Received 5 August 2012 Received in revised form 12 October 2012 Accepted 15 October 2012 Available online 23 October 2012
abstract In this paper, we consider both the partial unit root and the near partial unit root processes in nonlinear transition autoregression models. Our simulations show that when these time series data are used in ordinary least squares regression, spurious regression occurs. However, if we re-estimate the regression by adding an AR(1) term, spurious regression can almost be eliminated. © 2012 Elsevier B.V. All rights reserved.
JEL classification: C12 C32 C52 Keywords: Partial unit root Spurious regression Monte Carlo simulation
1. Introduction Since the studies by Granger and Newbold (1974) and Phillips (1986), much effort has been exerted in understanding linear spurious regression between independent variables. Durlauf and Phillips (1988) observe spurious correlations between random walks and linear trends. Marmol (1995, 1996) demonstrate that spurious correlations are evident in OLS regressions that involve combinations of series with integer orders of integration equal to or greater than one. Granger et al. (2001), Hassler (2003), Kim et al. (2004), and Noriega and Ventosa-Santaulària (2007) also show that spurious regressions occur in models with series generated by various combinations of different types of stationary processes (with and without linear trends and those that potentially allow for time-varying means because of structural breaks or seasonality). Marmol (1998) and Tsay and Chung (2000) indicate that spurious correlation generally occurs in regressions that involve various combinations of fractionally integrated processes (both stationary and non-stationary). Caner and Hansen (2001) and Park and Shintani (2005) consider a nonlinear transition autoregression model between two regimes, in which one regime is generated by a unit root and the other regime is a mean-reverting process. They call this model the partial unit root model. The present paper aims to examine the spurious regression between independent partial unit root processes
E-mail address: [email protected]. 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.10.018
or near partial unit root processes. Monte Carlo simulations reveal significant evidence of spurious regression between two independent partial unit root (or near partial unit root) variables. However, if we consider the auto-correlated residuals, and if we re-estimate the simulated equation using the EViews specification of AR(1) disturbance instead of presuming white noise disturbances, then the spurious regression phenomenon can almost be eliminated. The present paper is structured as follows. Section 2 provides some preliminaries. Section 3 presents the Monte Carlo simulation results. Section 4 concludes. 2. Preliminary We consider a transition autoregression model between two regimes: zt = θz zt −1 + λzt −1 π (zt −d , κ, µ) + uzt ,
(1)
where 0 ≤ θz ≤ 1 and λ < 0, such that |θ + λ| < 1; π (zt −d , κ, µ) is a real-valued transition function on R × R2 that usually takes values between zero and unity, in which zt −d is the transition variable with lag delay d ≥ 1. For simplicity, we set d = 1 in the simulations below1 ; κ is the scale parameter, µ is the location parameter, and uzt is the zero mean sequence of errors.
1 Transition variables can also be other exogenous variables, lagged difference variables, and so on. However, our simulations indicate that the spurious regression phenomenon cannot be affected by different transition variables.
190
L. Zhang / Economics Letters 118 (2013) 189–191
When θz = 1, one extreme regime of model (1) is given by a unit root process, which represents no adjustment movement. The other extreme regime is a mean-reverting process that represents linear adjustment toward a long-run equilibrium. The model can also be viewed as a nonlinear AR model of a nonlinear meanreverting process with an appropriate transition function. In such a case, we call model (1) the partial unit root model following the description by Caner and Hansen (2001). Furthermore, when θz < 1 but close to 1, we call model (1) the near partial unit root model. In this paper, we consider three commonly employed models for different transition functions, namely, the threshold autoregression (TAR), logistic smooth transition autoregression (LSTAR), and the exponential smooth transition autoregression (ESTAR). The corresponding transition functions are given respectively below:
Table 1 Rejection frequencies for tδˆ (|tδˆ | > 1.96) based on Regression (5) when yt and xt are balanced independent processes in the LSTAR models. n
θy , θx = 1.0 λ = −0.9
−0.5
100 300 500
49.2 70.7 79.4
58.1 75.7 82.1
100 300 500
61.1 77.7 83.2
63.0 79.1 83.7
(3)
LSTAR (µ, κ) = (1.0, 20.0)
ESTAR: π (zt −1 , κ, µ) = 1 − exp{−κ(zt −1 − µ) }.
(4)
100 300 500
yt = αˆ 0 + δˆ xt + εˆ t ,
(5)
yt = αˆ 1 + βˆ xt + εˆ t εˆ t = ρˆ εˆ t −1 + eˆ t ,
(6)
where both xt and yt are generated by model (1), with uxt and uyt being independent. In Eq. (5), we directly regress yt on xt without considering autocorrelation. In Eq. (6), we consider the series autocorrelation that may be exhibited in the residuals if the regression is spurious, as mentioned by McCallum (2010), by adding an AR(1) term in the regression to eliminate the first-order autocorrelation. 3. Monte Carlo simulation Based on Eqs. (1)–(4), we consider the following data generation processes (DGP) for xt and yt , where both uxt and uyt are i.i.d. N (0, 1). For the TAR model, λ ∈ {−0.9, −0.5, −0.1}, µ ∈ {−3, −1, 0, 1, 3}. For the LSTAR model, λ ∈ {−0.9, −0.5, −0.1}, (κ, µ) = (1.0, 0.0), (1.0, 1.0), (20.0, 0.0), (20.0, 1.0). For the ESTAR model, we set µ = 0 and κ = {0.1, 0.5, 1.0, 5.0}. For all DGPs, we set θx , θy equal to 1, 0.9 and to 0.7 to obtain the partial unit root and the near partial unit root, respectively. We consider the samples sizes of n = 100, 300, and 500 and compute the empirical rejection rates of the t-statistic to test the null hypotheses H0 : δ = 0 and H0 : β = 0, respectively, based on 10,000 replications and on the nominal level of 5%. For brevity, we only present the results in the LSTAR models, as shown by Tables 1–4. The results in the TAR and ESTAR models are similar to those presented in the LSTAR models and available upon request. Tables 1 and 2 detail the rejection frequencies for tδˆ based on Regression (5) when yt and xt represent the independent partial unit root and the near partial unit root processes, respectively, in the LSTAR models. When both yt and xt are partial unit root processes, the spurious regression phenomenon occurs both in the balanced and the unbalanced regression environment,2 and spurious rejection frequencies increase with the increase in sample sizes. In the particular case of λ = −0.1, the rejection frequencies
2 Here ‘‘balanced’’ means θ = θ , and ‘‘unbalanced’’ means θ ̸= θ . x y x y
−0.1
67.7 79.4 84.6
8.3 8.2 8.6
12.3 12.2 13.3
19.7 20.6 21.1
67.7 80.1 84.7
11.5 12.0 11.6
15.0 16.0 15.0
21.9 21.3 21.3
68.2 80.7 84.8
10.8 10.6 10.6
13.9 14.0 14.3
21.1 21.0 21.0
68.1 81.1 84.9
12.1 11.9 11.9
15.5 15.0 15.6
20.6 21.1 22.0
LSTAR (µ, κ) = (0.0, 20.0)
LSTAR: π (zt −1 , κ, µ) = (1 + exp{−κ(zt −1 − µ)})−1
To study the spurious regression phenomenon, we consider the following OLS regressions:
−0.5
LSTAR (µ, κ) = (1.0, 1.0)
100 300 500
2
θy , θx = 0.7 λ = −0.9
LSTAR (µ, κ) = (0.0, 1.0)
(2)
TAR: π(zt −1 , µ) = 1{zt −1 ≤ µ}
−0.1
66.8 80.5 84.7
66.8 81.3 85.1
66.7 80.7 84.8
67.3 80.4 84.5
Table 2 Rejection frequencies for tδˆ (|tδˆ | > 1.96) based on Regression (5) when yt and xt are unbalanced independent processes in the LSTAR models. n
θy = 1.0, θx = 0.7 λ = −0.9 −0.5
−0.1
θy = 0.7, θx = 0.9 λ = −0.9 −0.5
−0.1
33.6 35.1 34.8
11.1 12.4 13.3
18.6 19.1 19.2
28.9 28.4 30.2
32.8 35.3 36.8
17.4 18.6 17.8
21.3 22.6 22.1
29.8 29.9 30.2
33.5 35.5 35.1
17.1 17.3 17.2
20.4 20.6 20.7
28.3 29.1 29.6
33.3 36.1 36.4
18.2 19.7 18.8
21.7 22.2 23.3
29.4 29.9 29.5
LSTAR (µ, κ) = (0.0, 1.0) 100 300 500
15.5 17.0 17.8
23.0 25.7 25.2
LSTAR (µ, κ) = (1.0, 1.0) 100 300 500
21.6 23.7 23.6
26.8 27.5 28.6
LSTAR (µ, κ) = (0.0, 20.0) 100 300 500
21.1 23.8 23.2
24.7 27.9 27.5
LSTAR (µ, κ) = (1.0, 20.0) 100 300 500
22.6 25.3 25.7
27.3 29.2 29.4
are very high, indicating severe spurious regression. Furthermore, when both yt and xt are independent near partial unit root processes, i.e., θy , θx = 0.7 and θy = 0.7, θx = 0.9, the spurious regression phenomenon also occurs in certain situations, such as in λ = −0.1. Our simulations also show that when the parameter values of θx and θy approach unity, the spurious regression becomes more severe. However, these findings are not provided here because of space limitations. Tables 3 and 4 present the rejection frequencies for tβˆ based on Regression (6), in which a first-order autoregressive term is included to eliminate the autocorrelation of errors.3 We perform this regression based on calculations provided by the ‘‘AR(1)’’ procedure in EViews. When the simulated equation is estimated using the EViews specification of AR(1) disturbance instead of
3 We also investigate the DW statistics in Regression (5), which shows that in some cases, the fraction of times that the DW statistics are less than 1.0 are over 50%, indicating the presence of strong and serially correlated residuals.
L. Zhang / Economics Letters 118 (2013) 189–191 Table 3 Rejection frequencies for tβˆ (|tβˆ | > 1.96) based on Regression (6) when yt and xt are balanced independent processes in the LSTAR models. n
θy , θx = 1.0 λ = −0.9
−0.5
−0.1
θy , θx = 0.7 λ = −0.9
−0.5
−0.1
6.1 4.9 5.2
5.6 4.7 4.5
5.4 4.5 5.1
4.9 4.6 4.4
6.1 5.2 4.8
4.7 4.8 4.7
5.2 4.8 4.8
5.4 4.8 4.8
6.7 5.1 4.7
5.1 4.9 4.3
4.9 4.7 4.6
5.1 4.9 4.9
6.0 5.5 5.2
4.9 4.5 4.3
5.1 4.7 4.6
5.2 4.9 4.5
LSTAR (µ, κ) = (0.0, 1.0) 100 300 500
6.6 5.7 5.7
6.5 5.6 5.1
LSTAR (µ, κ) = (1.0, 1.0) 100 300 500
6.9 5.6 5.2
6.2 5.6 5.0
LSTAR (µ, κ) = (0.0, 20.0) 100 300 500
6.9 5.2 5.1
6.4 5.3 5.3
LSTAR (µ, κ) = (1.0, 20.0) 100 300 500
7.2 5.2 4.9
6.8 4.9 5.1
191
4. Conclusion This paper investigates the linear spurious regression phenomenon between two independent partial unit root or near partial unit root processes. Monte Carlo simulations reveal strong evidence of spurious regression between two independent partial (or near partial) unit root variables in the TAR, LSTAR, and ESTAR models. However, when the regressions are re-estimated by adding an AR(1) term, the re-estimation that considers potential autocorrelation tends to eliminate the appearance of non-existent relationships. This paper only examines the spurious regression phenomenon via simulations. Therefore, further theoretical research should be performed. Acknowledgments The research was funded by grants of the China Ministry of Education (No. 12YJC790268) and the Excellent Young Scholars Research Fund of Beijing Institute of Technology. References
Table 4 Rejection frequencies for tβˆ (|tβˆ | > 1.96) based on Regression (6) when yt and xt are unbalanced independent processes in the LSTAR models. n
θy = 1.0, θx = 0.7 λ = −0.9 −0.5
−0.1
θy = 0.7, θx = 0.9 λ = −0.9 −0.5
−0.1
4.5 4.2 4.5
5.8 5.2 5.2
5.5 4.8 4.8
6.3 4.9 4.9
4.6 4.1 4.4
5.8 5.2 4.9
5.8 5.0 5.0
6.4 5.1 4.7
4.5 4.4 4.8
6.5 5.4 5.3
5.4 4.8 4.7
6.2 4.9 4.6
4.6 4.4 4.1
5.6 4.9 4.9
5.9 4.9 5.0
6.1 5.0 4.8
LSTAR (µ, κ) = (0.0, 1.0) 100 300 500
5.2 4.9 5.3
5.0 4.7 4.3
LSTAR (µ, κ) = (1.0, 1.0) 100 300 500
4.9 4.9 4.5
4.9 4.5 4.6
LSTAR (µ, κ) = (0.0, 20.0) 100 300 500
4.6 4.6 4.8
4.9 4.9 4.5
LSTAR (µ, κ) = (1.0, 20.0) 100 300 500
4.7 4.9 5.2
4.5 4.7 4.8
presuming white noise disturbances, the spurious regression phenomenon is eliminated in most situations.
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