A Note on Unit Root Tests and GARCH Errors: A Simulation Experiment1
Amélie CHARLES1 and Olivier DARNÉ2 1
PRISM, University of Paris 1–Sorbonne, France
[email protected] 2
EconomiX, University of Paris X–Nanterre, France
[email protected] Keywords: unit root test; GARCH model. Mathematics Subject Classification: 62F03; 62M10. ABSTRACT This paper re-examines the Monte Carlo experiments in Seo (1999) [Journal of Econometrics, 91, 113-144] for unit root tests with GARCH errors. We report a Monte Carlo study with data generated from various GARCH(1,1) processes where 0.8 ≤ α + β < 1 and β > α. In this case, the Dickey-Fuller test works better than the Seo test.
1. INTRODUCTION Recently, much attention has been devoted to testing the unit root hypothesis in the presence of conditionally heteroscedastic errors of the GARCH form as proposed in Bollerslev (1986) (Pantula, 1988; Ling and Li, 1998, 2003; Li, Ling and McAleer, 2002; Ling, Li and McAleer, 2003; Rodrigues and Rubia, 2005; Wang, 2006)2 . There is general agreement in the literature that the Dickey-Fuller (1979) [DF] tests are asymptotically robust to heteroscedasticity, so that the effects of GARCH errors are at most a finite-sample problem. Kim and Schmidt (1993) and Haldrup (1994) also have shown that conditional heteroscedas1
Correspondance to Amélie Charles, PRISM, University of Paris 1–Sorbonne, 17 rue de la Sorbonne,
75005 Paris, France; Email:
[email protected] 2 Barassi (2005) studied the behavior of the stationary test developed by Kwiatkowski et al. (1992) in the presence of GARCH errors.
1
ticity does not affect the small-sample property of the DF unit root test seriously unless the process is nearly degenerate (i.e. α + β is close to one). However, these tests ignore the information coming from conditional heteroscedasticity. Therefore, Seo (1999) proposed unit root tests with GARCH errors based on maximum likelihood estimation, which estimates the autoregressive unit root and the GARCH parameters jointly. He showed that his test works relatively better than the DF test when the data contains conditional heteroscedasticity only from Monte Carlo experiments. The Monte Carlo experiments for unit root tests performed by Seo (1999) are based on various GARCH(1,1) processes where the parameter α is more often superior to the parameter β. Of his ten GARCH(1,1) processes only two had β > α. However, as shown by van Dijk, Franses and Lucas (1999), the GARCH parameters for which β > α are considered as typical values for estimated GARCH(1,1) models. Poon and Granger (2003) reviewed 93 papers on the forecasting volatility in which this condition is satisfied. Therefore, we propose to re-examine the Monte Carlo experiments performed by Seo (1999) by comparing the property of size and power of the maximum likelihood estimation t-statistic developed by Seo (1999) and the DF (1979) t-statistic. For that, we simulate some autoregressive models with GARCH(1,1) errors where 0.8 ≤ α + β < 1 and β > α. Indeed, we call attention to models where the roots of the GARCH polynomial are close to unity, which seems to characterize many empirical studies of high frequency data (see, e.g., Bollerslev and Engle, 1993; Engle and Bollerslev, 1986). The outline of this paper is as follows. Section 2 gives the background to the Seo (1999) test. In Section 3 we examine the finite sample performance of the Seo and Dickey-Fuller test when the error term follows a GARCH(1,1) process with 0.8 ≤ α + β < 1 and β > α. A brief summary concludes.
2. THE SEO TEST
2
Consider a time series yt generated from an AR(k) model as follows: ∆yt = βyt−1 +
k−1 X
γi ∆yt−i + ut
(1)
i=1
and the errors follow a GARCH(p,q) process: φ(L)σt2 = ω + ζ(L)u2t−1
(2)
where φ(L) = 1 − φ1 L − . . . − φq Lq , and ζ(L) = ζ1 + ζ2 L + . . . + ζp Lp−1 . Seo (1999) proposed unit root tests with GARCH errors based on maximum likelihood estimation [MLE], which estimate the autoregressive unit root and the GARCH parameters jointly. The asymptotic distribution of the MLE t-statistic is defined as: √ R R ρ 01 B1 dB1 + 1 − ρ2 01 B1 dB2 qR MLE ⇒ 1 2 0 B1 ds
(3)
where B1 (s) and B2 (s) are independent standard Brownian motions (Seo, 1999). In fact, this distribution is a combination of the Dickey-Fuller t-distribution and the standard normal, depending on the relative weight ρ. Indeed, as the relative weight ρ becomes close to zero, the distribution approaches the standard normal. As ρ becomes close to one, the distribution approaches the Dickey-Fuller t-distribution3 . The relative weight ρ can be estimated as follows: ρˆ = with κ ˆ=
1 ˆ ξ) + σ (ˆ κ − 1)A(α, ˆ 2 /ˆ ρ2
¶ T µ 1X uˆt 4 , T t=1 σˆt à 2
ρˆ =
with σ ˆ2 = T 1X 1 T t=1 σ ˆt2
!−1
T µ 2 2 ¶ σ ˆ uˆt−j 1X ξˆ2 = , T t=1 σ ˆt4
ˆ ξ) = lim A(α,
m→∞
3
T 1X σ ˆ2 T t=1 t
m X
for j ≥ 1 α ˆ k2 ξˆk2 ,
k=1
If ρ = 1, the distribution is the same as Dickey and Fuller (1979).
3
(4)
where T is the number of observations, and α(l) =
ζ(L) φ(L)
=
P∞
k=1
αk Lk . The relative weight
depends on the magnitude of the GARCH effect, the fourth moment κ, and σ ˆ 2 /ˆ ρ2 .
3. SIMULATION RESULTS In order to investigate the empirical sizes and powers of the test statistics in Dickey and Fuller (1979) [DF] and Seo (1999) [MLE], we generate data sets from the AR(1) model with zero mean and GARCH(1,1) errors: yt = µ + φ1 yt−1 + εt where
q
εt = η t ht
εt ∼ N (0, ht )
ηt ∼ i.i.d. N (0, 1) ht = ω + αε2t−1 + βht−1 where ω > 0, α ≥ 0, β ≥ 0. For the GARCH(1,1) process, the condition for strict stationarity is E[ln(αηt2 + β)] < 0, the condition for a finite variance is α + β < 1, and the condition for a finite fourth moment under normality is 3α2 + 2αβ + β 2 < 1.
We study the finite sample properties of MLE and DF for φ1 = 0.8, 0.9, 0.95, 0.99, 1.0, ω = 1 − α − β, and (α, β) = (0.2, 0.6), (0.15, 0.7), (0.1, 0.8), (0.05, 0.9), (0.09, 0.9) such that 0.8 ≤ α + β < 1 and β > α. All experiments are based on 10,000 replications, and the sample sizes to be examined are T = {250, 500, 1000}4 . The larger values of T are chosen for their empirical relevance, especially with high-frequency financial data. The empirical sizes and powers of the test statistics, DF and MLE, are summarized in Tables 1-3 at the 5% significance level. For T = 250, the power of the MLE test is better than power of the DF when φ1 is close to one. This result confirms the finding of Seo (1999). However, when φ1 moves to one the 4
The first 100 observations of each series were discarded in order to avoid possible dependence of the
results on the initial conditions.
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DF test is (slightly) more powerful. For example, the DF test rejects 96.48% of the null hypothesis while the MLE test rejects 93.90% for φ1 = 0.90 and (α, β) = (0.10, 0.80). For T = 500 and 1000, the power of DF test is more often greater than that of the MLE test, except when φ1 = 0.99. The empirical sizes of the DF test is better than that of the MLE test whatever the GARCH effect and the sample size. For example, the DF test and the MLE test reject 5.64% and 7.70% of the null hypothesis, respectively, for (α, β) = (0.15, 0.70) for T = 500. Seo (Table 1, p. 127, 1999) obtained the same results when 0.8 ≤ α + β < 1 and β > α by simulating only two GARCH(1,1) processes. Seo showed that the MLE test is better than the DF test when β < α from eight simulated GARCH(1,1) processes. However, the GARCH parameters for which β > α are considered as typical values for estimated GARCH(1,1) models (van Dijk et al., 1999; Poon and Granger, 2003). This finding moderates the conclusions obtained by Seo (1999) who argues that the MLE test performs better than the DF test5 .
Therefore, the Dickey-Fuller test works better than the Seo test when the data contains conditional heteroscedasticity where 0.8 ≤ α + β < 1 and β > α, i.e. typical values for estimated GARCH(1,1) models in economic and financial data.
ACKNOWLEDGMENT We would like to thank the anonymous referee for very helpful comments and suggestions.
BIBLIOGRAPHY Barassi, M. (2005). On KPSS with GARCH errors. Economics Bulletin, 3, 1–12. 5
This finding also moderates the result obtained by Ling, Li and McAleer (2003). These authors proposed
a MLE-based procedure (Method C) which follows similarly the Seo (1999) test and showed that their test is more powerful than the DF test from a Monte Carlo simulation of the test performance, including several cases where α < β. However, Ling et al. (2003) only used 1000 replications.
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Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327. Bollerslev, T., Engle, R.F. (1993). Common persistence in conditional variances: Definition and representation. Econometrica, 61, 167–186. 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. Engle, R.F., Bollerslev, T. (1986). Modelling the persistence of conditional variances. Econometric Reviews, 5, 1–50. Haldrup, N. (1994). Heteroscedasticity in non-stationary time series, some Monte Carlo evidence. Statistical Papers, 35, 287–307. Kim, K., Schmidt, P. (1993). Unit root tests with conditional heteroscedasticity. Journal of Econometrics, 59, 287–300. Kwiatkowski, D., Phillips, P.C.B., Schmidt P., Shin, Y. (1992). Testing the null of stationarity against the alternative of a unit root. Journal of Econometrics, 54, 159–178. Li, W.K., Ling, S., McAleer, M. (2002). Recent theoretical results for time series models with GARCH errors. Journal of Economic Survey, 16, 245–269. Ling, S., Li, W.K. (1998). Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroscedastic errors. Annals of Statistics, 26, 84–125. Ling, S., Li, W.K. (2003). Asymptotic inference for unit root processes with GARCH(1,1) errors. Econometric Theory, 19, 541–564. Ling, S., Li, W.K., McAleer, M. (2003). Estimation and testing for unit root processes with GARCH(1,1) errors: Thoery and Monte Carlo evidence. Econometric Review, 22, 179–202. Pantula, S.G. (1988). Estimation of autoregressivemodels with ARCH errors. Sankhya B, 6
50, 119–138. Poon, S-H., Granger, C.W.J. (2003). Forecasting volatility in financial markets: A review. Journal of Economic Literature, 41, 478–539. Rodrigues, P.M.M., Rubia, A. (2005). The performance of unit root tests under leveldependent heteroskedasticity. Economics Letters, 89, 262–268. Seo, B. (1999). Distribution theory for unit root tests with conditional heteroskedasticity. Journal of Econometric, 91, 113–144. van Dijk, D., Franses, P.H., Lucas, A. (1999). Testing for ARCH in the presence of additive outliers. Journal of Applied Econometrics, 14, 539–562. Wang G. (2006). A note on unit root tests with heavy-tailed GARCH errors. Statistics and Probability Letters, 76, 1075–1079.
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Table 1: Powers and sizes for unit root processes with GARCH(1,1) errors. T = 250. φ1 (α, β)
Test
0.80
0.90
0.95
0.99
1.0
(0.20, 0.60)
DF
99.98
95.65
48.42
8.15
5.45
MLE
99.70
96.22
64.56
14.10
7.81
DF
99.98
96.04
48.65
7.89
5.52
MLE
99.63
95.25
61.77
12.61
8.06
DF
100.00
96.48
48.21
7.85
5.29
MLE
99.46
93.90
56.27
12.23
7.55
DF
100.00
96.54
47.65
8.31
5.47
MLE
99.34
92.43
51.20
11.06
7.11
DF
99.97
95.24
49.43
8.92
7.34
MLE
99.27
90.69
52.10
12.17
8.76
(0.15, 0.70)
(0.10, 0.80)
(0.05, 0.90)
(0.09, 0.90)
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Table 2: Powers and sizes for unit root processes with GARCH(1,1) errors. T = 500. φ1 (α, β)
Test
0.80
0.90
0.95
0.99
1.0
(0.20, 0.60)
DF
100.00
100.00
96.19
13.79
5.26
MLE
99.98
99.92
98.21
23.64
7.12
DF
100.00
100.00
96.27
13.53
5.64
MLE
99.99
99.91
97.73
22.67
7.70
DF
100.00
100.00
96.43
13.41
5.83
MLE
99.97
99.88
96.98
20.22
7.85
DF
100.00
100.00
96.36
12.90
4.94
MLE
99.85
99.70
95.22
17.12
6.85
DF
100.00
99.97
95.19
15.27
7.92
MLE
99.91
99.63
93.55
20.97
8.85
(0.15, 0.70)
(0.10, 0.80)
(0.05, 0.90)
(0.09, 0.90)
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Table 3: Powers and sizes for unit root processes with GARCH(1,1) errors. T = 1000. φ1 (α, β)
Test
0.80
0.90
0.95
0.99
1.0
(0.20, 0.60)
DF
100.00
100.00
99.99
33.84
5.00
MLE
100.00
99.99
100.00
54.87
7.00
DF
100.00
100.00
100.00
33.69
5.45
MLE
100.00
100.00
100.00
53.12
7.53
DF
100.00
100.00
100.00
33.24
4.94
MLE
100.00
99.96
99.99
49.93
7.26
DF
100.00
100.00
100.00
33.75
5.42
MLE
100.00
99.96
99.94
43.05
7.42
DF
100.00
100.00
99.99
37.01
5.17
MLE
100.00
99.97
99.69
50.59
6.60
(0.15, 0.70)
(0.10, 0.80)
(0.05, 0.90)
(0.09, 0.90)
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