A weighted symmetric cointegration test Steven Cook∗ and Dimitrios Vougas 20 October 2005
ABSTRACT Weighted symmetric estimation is employed to develop a new test for cointegration. Using Monte Carlo simulation, the resulting test is shown to possess greater power than alternative existing tests.
Keywords: Cointegration; Weighted symmetric estimation; Monte Carlo simulation; Test power.
∗ Dr Steven Cook, Department of Economics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP. Tel: (01792) 602106. E-mail:
[email protected].
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1
Introduction
Following the seminal research of Engle and Granger (1987), the notion of cointegration has come to occupy a central position in econometrics and time series analysis. In addition to providing a formal definition of cointegration, Engle and Granger (1987) presented a two-step OLS based approach to test for possible cointegration between time series. The testing procedure advocated involves examination of the residual series obtained from an initial static cointegrating regression between the variables of interest to the practitioner. The null of no cointegration is then examined via analysis of the order of integration of the residuals. Since its introduction, this testing procedure has received widespread application. In this paper it is examined whether the power of the frequently employed Engle-Granger test can be improved via the introduction of weighted symmetric estimation. In previous research, Park and Fuller (1995) have shown weighted symmetric estimation to result in increased power in the context of unit root testing.1 It is examined here whether the beneficial properties of weighted symmetric estimation can be extended to the analysis of cointegration. The present paper therefore develops a weighted symmetric residual-based test for cointegration before considering its power properties relative to both the seminal Engle-Granger test and the more recently proposed single equation test of Kanioura and Turner (2005). The results obtained show the development of a weighted symmetric cointegration test to result in an increase in power relative to the tests of Engle and Granger (1987) and Kanioura and Turner (2005). This paper proceeds as follows. In the following section the alternative cointegration tests considered are presented. In section [3] Monte Carlo analysis is employed to derive finite-sample critical values for the proposed weighted symmetric cointegration test for a range of sample sizes. Section [4] contains a simulation analysis of the power properties of the alternative cointegration tests considered using the Monte Carlo experimental design of Engle and Granger (1987). Section [5] concludes. 1 The appeal of weighted symmetric estimation is further supported by the simulation analysis of Leybourne et al. (2005) which considered the powers of alternative modified unit root tests, finding the weighted symmetric Dickey-Fuller test to overperform its rivals.
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2
Alternative single equation cointegration tests
2.1
The Engle-Granger test
Given two variables of interest {yt , xt }, the first step of the Engle-Granger procedure involves the estimation of the following static cointegrating regression:2
yt = dt + βxt + ²t
t = 1, .., T
(1)
where dt denotes a deterministic term which may be either an intercept (α) or an intercept and linear trend (α + βt). In the second stage, possible cointegration between the series is examined via analysis of the order of integration of the residuals {b²t } from (1) using a Dickey-Fuller test as
below:
∆b²t = (ρ − 1) b²t−1 + ν t
(2)
− 1), with the resulting The null of no cointegration (H0 : ρ − 1 = 0) is tested via the t-ratio of (ρd test statistic denoted here as τ EG .
2.2
The ECM-based F-test for cointegration
Given the above series {yt , xt } , the cointegration F-test of Kanioura and Turner (2005) is based upon the significance of the lagged level terms in the following error correction model (ECM):
∆yt = α0 + α1 ∆xt + α2 yt−1 + α3 xt−1 + ν t
(3)
Kanioura and Turner derive critical values for the joint hypothesis of no cointegration H0 : α2 = α3 = 0, noting that unlike the ECM-based t-test of Kremers et al. (1992), these values are invariant to the parameters of the specific problem examined. The resulting test statistic is denoted here as F ECM . 2
In the present paper, the properties of the alternative tests for cointegration are considered in a bivariate setting. However, all of the tests can be extended to the multivariate case.
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2.3
Weighted symmetric cointegration test
To derive a weighted symmetric cointegration test, the first stage of the Engle-Granger procedure is employed. However, the order of integration of the resulting residual series is not then assessed using a Dickey-Fuller test as in (2) above, but is instead examined using the τ ws statistic obtained as below using weighted symmetric estimation:3
τ
ws
=
bws σ −1 ws (ρ
− 1)
µX T −1
b ²2 t=2 t
+T
−1
XT
b ²2 t=1 t
¶1 2
(4)
where
b ρws
=
µX
T −1 2 b ² t=2 t
+ T −1
µ
XT
t=1
σ 2ws = (T − 2)−1 T −1 (T − 1)
b ²2t
¶−1 X
T −1 t=2
XT −1 t=2
b ²tb²t−1
b bws ²2t + ρ
XT −1 t=2
(5) b ²tb²t−1
¶
(6)
This method directly extends the weighted symmetric unit root test of Park and Fuller (1995) to the analysis cointegration.
Finite-sample critical values of the τ ws test
3
To derive finite-sample critical values for the weighted symmetric cointegration test, the following data generation process (DGP) is employed:
xt = xt−1 + ε1,t
(7)
yt = yt−1 + ε2,t
t = 1, ..., T
(8)
The two unit root processes {xt , yt } are derived using the innovation series {ε1,t , ε2,t } which are generated as pseudo i.i.d. N(0, 1) random numbers from the RNDNS procedure in GAUSS. The initial conditions of both series are set to zero (x0 = 0, y0 = 0). Using 50,000 simulations of the above DGP, critical values for the τ ws test of (4) are derived when the initial static cointegrating 3
For further information on weighted symmetric estimation in the context of unit root testing and more generally, see Park and Fuller (1995) and Fuller (1996) respectively.
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regression of (1) contains either an intercept (dt = α) or an intercept and trend (dt = α + βt). This mimics the options available for the τ EG test. The resulting critical values are reported in Table One for a range of sample sizes, T = {50, 100, 200, 400} and levels of significance (1%, 5%, 10%). TABLE ONE ABOUT HERE
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Power experiments
To explore the relative powers of the alternative τ EG , F ECM and τ ws cointegration tests, the following Monte Carlo DGP of Engle and Granger (1987) is employed:
yt + xt = u1,t
u1,t = u1,t−1 + e1,t
yt + 2xt = u2,t
u2,t = ρu2,t−1 + e2,t
t = 1, ..., T
(9) (10)
As with the earlier experiments, the innovation series {e1,t , e2,t } are generated as pseudo i.i.d. N(0, 1) random numbers using the RNDNS procedure in GAUSS, with the initial conditions of the series {u1,t , u2,t } set to zero (u1,0 = u2,0 = 0). The series {xt , yt } are cointegrated via the imposition of |ρ| < 1. The powers of the alternative tests are calculated over 50,000 simulations of the above DGP at the 10%, 5% and 1% levels of significance.4 The above sample sizes of T = {50, 100, 200, 400} are considered again, with the degree of cointegration varied across the differing sample sizes via the application of differing (near unity) values of ρ. More precisely, the values ρ = {0.80, 0.85, 0.90, 0.95} are considered. The empirical powers of the tests are reported in Tables Two and Three. In Table Two the results for the τ EG and τ ws tests correspond to the intercept model (dt = α) , while Table Three presents results for these tests in their trend model forms (dt = α + βt). The results for the F ECM tests do not vary across Tables Two and Three as this test does not consider alternative deterministic terms. Considering the power results in Table Two, it is clear that the τ ws test possesses greater power than the rival cointegration tests. In many cases the increased power exhibited by the τ ws test 4 Critical values for the τ EG and F ECM tests for the power experiments are derived using the DGP reported in section [3]. However, critical values are available for the τ EG test for any sample size using the response surface analysis of MacKinnon (1991), while Kanioura and Turner (2005) report critical values for the F ECM test for a range of sample sizes.
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is substantial. To illustrate this, consider the results for (T, ρ) = (200, 0.90) . For this design it can be seen that the τ ws test exhibits power gains of 81% and 360% respectively relative to the τ EG and F ECM tests at the 1% level of significance. At the 5% level of significance the gains in power relative to the τ EG and F ECM tests are 37% and 194%. Similar results are reported in Table Three, where the newly proposed τ ws test is again found to be the most powerful of the three tests considered. TABLES TWO AND THREE ABOUT HERE
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Conclusion
In this paper a new test for cointegration has been developed via the use of a two-step procedure employing weighted symmetric estimation. The results of Monte Carlo simulation using the experimental design of the seminal Engle and Granger (1987) show the newly proposed test to possess greater power than rival cointegration tests. Given the higher power of the test and its relative ease of application, the test should prove of interest to the practitioner.
References [1] Dickey, D. and Fuller, W. (1979) ‘Distribution of the estimators for autoregressive time series with a unit root’, Journal of the American Statistical Association, 74, 427-431. [2] Engle, R. and Granger, C. (1987) ‘Cointegration and error correction: representation, estimation and testing’, Econometrica, 55, 251-276. [3] Fuller, W. (1996) Introduction to Statistical Time Series (second edition), New York: Wiley. [4] Kanioura, A. and Turner, P. (2005) ‘Critical values for an F-test for cointegration in a multivariate model’, Applied Economics, 37, 265-270. [5] Kremers, J., Ericsson, N. and Dolado, J. (1992) ‘The power of cointegration test’, Oxford Bulletin of Economics and Statistics, 54, 325-348. [6] Leybourne, S., Kim, T. and Newbold, P. (2005) ‘Examination of some more powerful modifications of the Dickey-Fuller test’, Journal of Time Series Analysis, 26, 355-370. [7] MacKinnon, J. (1991) ‘Critical values for cointegration tests’, in Engle, R. and Granger, C. (eds) Long Run Economic Relationships, Oxford: Oxford University Press. [8] Park, H. and W. Fuller (1995), Alternative estimators and unit root tests for the autoregressive process, Journal of Time Series Analysis, 16, 415-429. 6
Table One: Finite-sample critical values for the τ ws tests
(i) Intercept model (dt = α)
T
1%
5%
10%
50
−3.942
−3.326
−3.021
100
−3.869
−3.275
−2.987
200
−3.829
−3.263
−2.977
400
−3.816
−3.251
−2.963
(ii) Trend model (dt = α + βt)
T
1%
5%
10%
50
−4.540
−3.896
−3.572
100
−4.393
−3.802
−3.505
200
−4.302
−3.758
−3.481
400
−4.285
−3.736
−3.456
Notes: The tabulated figures are finite-sample critical values for the weighted symmetric cointegration test of (4) obtained using the data generation process of (7)-(8) over 50,000 simulations.
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Table Two: A power comparison of alternative cointegration tests I
(T, ρ)
Test
1%
5%
10%
50, 0.80
τ EG
3.48
14.72
26.39
F ECM
2.17
9.64
17.75
τ ws
6.45
23.63
38.89
τ EG
8.50
29.83
47.46
F ECM
4.05
15.84
27.18
τ ws
15.86
46.17
65.29
τ EG
18.12
50.59
69.50
F ECM
7.14
23.59
36.76
τ ws
32.82
69.64
85.70
τ EG
18.58
50.75
69.76
F ECM
7.61
23.53
37.18
τ ws
30.67
69.24
85.78
100, 0.85
200, 0.90
400, 0.95
Notes:
The tabulated represent empirical rejection frequencies expressed in percentage terms
obtained using the data generation process of (9)-(10) over 50,000 simulations. The τ EG and τ ws tests include an intercept in the first stage testing equation.
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Table Three: A power comparison of alternative cointegration tests II
(T, ρ)
Test
1%
5%
10%
50, 0.80
τ EG
2.15
9.61
18.33
F ECM
2.17
9.64
17.75
τ ws
2.64
12.08
22.35
τ EG
5.03
17.87
31.25
F ECM
4.05
15.84
27.18
τ ws
6.29
23.50
38.89
τ EG
9.73
30.28
47.30
F ECM
7.14
23.59
36.76
τ ws
14.06
40.70
59.09
τ EG
8.89
29.83
47.74
F ECM
7.61
23.53
37.18
τ ws
13.23
40.79
59.63
100, 0.85
200, 0.90
400, 0.95
Notes:
The tabulated represent empirical rejection frequencies expressed in percentage terms
obtained using the data generation process of (9)-(10) over 50,000 simulations. The τ EG and τ ws tests include an intercept and linear trend in the first stage testing equation.
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