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Correcting size distortion of the Dickey–Fuller test via recursive mean adjustment
Statistics & Probability Letters 60 (2002) 75 – 79
Correcting size distortion of the Dickey–Fuller test via recursive mean adjustment Steven Cook∗ Department of Economics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK Received March 2002; received in revised form June 2002
Abstract Leybourne et al. (J. Econom. 87 (1998) 191) have shown the Dickey–Fuller (J. Amer. Statist. Assoc. 74 (1979) 427) unit root test to su2er from severe oversizing in the presence of level breaks. In this paper it is shown that recursive mean adjustment can correct this distortion, even for large breaks. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Recursive mean adjustment; Unit root test; Structural breaks; Monte Carlo methods
1. Introduction Following Perron (1989), it has long been recognised that the Dickey and Fuller (1979) (DF) test can exhibit low power when applied to stationary series which are subject to a structural break. This ;nding has led to the emergence of a large literature on unit root testing in the presence of structural breaks or regime shifts. 1 In contrast to this, Leybourne et al. (1998) have recently shown the existence of a ‘converse Perron phenomenon’, with the unit root hypothesis spuriously rejected by the DF test when a break occurs under the null. In particular, it was shown that when a unit root process experiences a level break early in the sample period, the DF test can experience severe size distortion, leading to the false inference that an I(1) series with a break is I(0). In this paper it is examined whether this observed size distortion can be corrected via the use of recursive mean adjustment. Such a ;nding is of interest as the Monte Carlo results of Shin and So (2001) show that recursive mean adjustment leads to a dramatic increase in the power of unit root tests. The results obtained below show the recursively mean-adjusted DF test to be remarkably ∗
Tel.: +44-1792-295168. E-mail address: [email protected] (S. Cook). 1 See, inter alia, Bai et al. (1998), Bai and Perron (1998), Banerjee et al. (1992), Perron (1989,1990), Zivot and Andrews (1992). c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 2 8 0 - 8
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S. Cook / Statistics & Probability Letters 60 (2002) 75 – 79
robust to level breaks. The combination of high power, ease of application and robustness to structural breaks suggest that recursively mean-adjusted unit root tests should receive more widespread adoption. This paper will proceed as follows. In Section 2 the method of recursive mean adjustment for unit root tests is outlined. In Section 3 the Monte Carlo analysis is presented, with Section 4 providing some concluding remarks. 2. Recursive mean adjustment Shin and So (2001) introduce the concept of recursive mean adjustment by considering the following AR(1) model: yt − = (yt −1 − ) + t
t = 1; : : : ; T;
(1)
where t is a zero mean stationary process. Shin and So (2001) note that when || ¡ 1, the mean of yt is given as . The unknown can then be replaced by the mean of yt , calculated as n yG = n−1 yt : (2) t=1
Application of the DF test to the mean-adjusted observations (yt − y) G is achieved using the following regression: yt − yG = (yt −1 − y) G + t :
(3)
However, as Shin and So further note, replacing with yG in (1) leads to correlation between the regressor (yt −1 − y) G and t . Denoting the ordinary least-squares estimator of as ˆ0 , the resulting bias of ˆ0 has been derived by, inter alia, Tanaka (1984) and Shaman and Stine (1988) as E(ˆo − ) = −T −1 (1 + 3) + o(T −1 ):
(4)
To overcome the problem of correlation between the regressor and error, Shin and So propose the use of recursively adjusted mean yG t : t yi : (5) yG t = t −1 i=1
The recursively mean-adjusted version of (1) and (3) is then given as yt − yG t −1 = (yt −1 − yG t −1 ) + t
(6)
with the regressor now independent of t when = 1. Considering the DF test of the null = 1 in its pivotal form as (ˆ − 1)=se(), ˆ the recursively mean-adjusted DF statistic based upon the use of (6) is denoted as DFr , with the familiar DF test using (3) denoted as DF. The Monte Carlo results presented by Shin and So show DFr to display substantially higher power than DF. Indeed, the easily applied DFr test exhibits near identical power to the weighted symmetric DF test, the high power of which has been noted by Pantula et al. (1994). In the following section it will be seen whether the high power and ease of application of recursive mean adjustment are also accompanied by robustness to level breaks.
S. Cook / Statistics & Probability Letters 60 (2002) 75 – 79
77
3. Unit root processes with level breaks To analyse the behaviour of the DFr and DF tests in the presence of level breaks, the experimental design of Leybourne et al. (1998) is employed, with the data generation process (DGP) given as below: yt = st () + t ;
(7)
t = t − 1 + t ;
(8)
t ∼ i:i:d: N(0; 1); 0 for t 6 T; st () = 1 for t ¿ T;
(9) ∈ (0; 1):
(10)
The error series {t } is generated using the RNDNS procedure in GAUSS, with all experiments having performed over 20,000 replications. Following an initial condition y0 = 0, an initial 20 observations were created and discarded, resulting in an e2ective sample size of 100 observations, as employed by Leybourne et al. (1998). To further mimic the approach of Leybourne et al. (1998), the values ∈ {2:5; 5; 10} are selected for the break magnitude. Denoting the break fraction as , the break in level is therefore imposed after observation T . In contrast to Leybourne et al. (1998), where results for a subset of breakpoints are tabulated, all possible breakpoints over the range ∈ (0; 1) are considered. To calculate the empirical rejection frequencies of the unit root hypothesis for the DFr and DF tests at the 5% level of signi;cance, the appropriate critical values are drawn from Shin and So (2001) and Fuller (1996), respectively.
Fig. 1. Size distortion of DF and DFr tests ( = 2:5).
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S. Cook / Statistics & Probability Letters 60 (2002) 75 – 79
Fig. 2. Size distortion of DF and DFr tests ( = 5).
Fig. 3. Size distortion of DF and DFr tests ( = 10).
Given the large number of breakpoints considered, the arguments of Davidson and MacKinnon (1998) are followed, with the resulting empirical rejection frequencies presented graphically to ease interpretation. In Figs. 1–3 the rejection frequencies for the DFr and DF tests are presented for the three break sizes considered. The results for the DF test replicate the ;ndings of Leybourne et al. (1998), with oversizing occurring when a break is experienced early in the sample period. As the
S. Cook / Statistics & Probability Letters 60 (2002) 75 – 79
79
size of the break is increased, this size distortion becomes more severe, with sizes in excess of 50% observed for a break of = 10. In contrast, the DFr test displays remarkable robustness to level breaks, with no evidence of oversizing observed for any of the values of considered. 4. Conclusion In this paper the severe oversizing of the Dickey–Fuller unit root test in the presence of level breaks has been reconsidered. It has been shown that the relatively simple procedure of recursive mean adjustment suggested by Shin and So (2001) corrects this problem. Combined with the ease of application of the approach and its known high power, this discovery of robustness suggests that recursively mean-adjusted unit root tests should receive more widespread adoption. The performance of the recursively mean-adjusted test when applied to stationary series subject to a structural break remains the subject of future research. References Bai, J., Perron, P., 1998. Testing for and estimation of multiple structural changes. Econometrica 66, 47–79. Bai, J., Lumsdaine, R., Stock, J., 1998. Testing for and dating common breaks in multivariate time series. Rev. Econom. Stud. 65, 395–432. Banerjee, A., Lumsdaine, R., Stock, J., 1992. Recursive and sequential tests of the unit root and trend break hypotheses: theory and international evidence. J. Business Econom. Statist. 10, 271–287. Davidson, R., MacKinnon, J., 1998. Graphical methods for investigating the size and power of hypothesis tests. Manchester School 66, 1–26. Dickey, D., Fuller, W., 1979. Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74, 427–431. Fuller, W., 1996. Introduction to Statistical Time Series, 2nd Edition. Wiley, New York. Leybourne, S., Mills, T., Newbold, P., 1998. Spurious rejections by Dickey–Fuller tests in the presence of a break under the null. J. Econom. 87, 191–203. Pantula, S., Gonzalez-Farias, G., Fuller, W., 1994. A comparison of unit root test criteria. J. Business Econom. Statist. 12, 449–459. Perron, P., 1989. The Great Crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361–1401. Perron, P., 1990. Testing for a unit root in time series with a changing mean. J. Business Econom. Statist. 8, 153–162. Shaman, P., Stine, R., 1988. The bias of autoregressive coeQcient estimators. J. Amer. Statist. Assoc. 83, 842–848. Shin, D., So, B., 2001. Recursive mean adjustment for unit root tests. J. Time Ser. Anal. 5, 595–612. Tanaka, K., 1984. An asymptotic expansion associated with maximum likelihood estimators in ARMA models. J. Roy. Statist. Soc. B 46 (5), 58–67. Zivot, E., Andrews, D., 1992. Further evidence on the Great Crash, the oil price shock and the unit root hypothesis. J. Business Econom. Statist. 10, 251–270.
Correcting size distortion of the Dickey–Fuller test via recursive mean adjustment Steven Cook∗ Department of Economics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK Received March 2002; received in revised form June 2002
Abstract Leybourne et al. (J. Econom. 87 (1998) 191) have shown the Dickey–Fuller (J. Amer. Statist. Assoc. 74 (1979) 427) unit root test to su2er from severe oversizing in the presence of level breaks. In this paper it is shown that recursive mean adjustment can correct this distortion, even for large breaks. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Recursive mean adjustment; Unit root test; Structural breaks; Monte Carlo methods
1. Introduction Following Perron (1989), it has long been recognised that the Dickey and Fuller (1979) (DF) test can exhibit low power when applied to stationary series which are subject to a structural break. This ;nding has led to the emergence of a large literature on unit root testing in the presence of structural breaks or regime shifts. 1 In contrast to this, Leybourne et al. (1998) have recently shown the existence of a ‘converse Perron phenomenon’, with the unit root hypothesis spuriously rejected by the DF test when a break occurs under the null. In particular, it was shown that when a unit root process experiences a level break early in the sample period, the DF test can experience severe size distortion, leading to the false inference that an I(1) series with a break is I(0). In this paper it is examined whether this observed size distortion can be corrected via the use of recursive mean adjustment. Such a ;nding is of interest as the Monte Carlo results of Shin and So (2001) show that recursive mean adjustment leads to a dramatic increase in the power of unit root tests. The results obtained below show the recursively mean-adjusted DF test to be remarkably ∗
Tel.: +44-1792-295168. E-mail address: [email protected] (S. Cook). 1 See, inter alia, Bai et al. (1998), Bai and Perron (1998), Banerjee et al. (1992), Perron (1989,1990), Zivot and Andrews (1992). c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 2 8 0 - 8
76
S. Cook / Statistics & Probability Letters 60 (2002) 75 – 79
robust to level breaks. The combination of high power, ease of application and robustness to structural breaks suggest that recursively mean-adjusted unit root tests should receive more widespread adoption. This paper will proceed as follows. In Section 2 the method of recursive mean adjustment for unit root tests is outlined. In Section 3 the Monte Carlo analysis is presented, with Section 4 providing some concluding remarks. 2. Recursive mean adjustment Shin and So (2001) introduce the concept of recursive mean adjustment by considering the following AR(1) model: yt − = (yt −1 − ) + t
t = 1; : : : ; T;
(1)
where t is a zero mean stationary process. Shin and So (2001) note that when || ¡ 1, the mean of yt is given as . The unknown can then be replaced by the mean of yt , calculated as n yG = n−1 yt : (2) t=1
Application of the DF test to the mean-adjusted observations (yt − y) G is achieved using the following regression: yt − yG = (yt −1 − y) G + t :
(3)
However, as Shin and So further note, replacing with yG in (1) leads to correlation between the regressor (yt −1 − y) G and t . Denoting the ordinary least-squares estimator of as ˆ0 , the resulting bias of ˆ0 has been derived by, inter alia, Tanaka (1984) and Shaman and Stine (1988) as E(ˆo − ) = −T −1 (1 + 3) + o(T −1 ):
(4)
To overcome the problem of correlation between the regressor and error, Shin and So propose the use of recursively adjusted mean yG t : t yi : (5) yG t = t −1 i=1
The recursively mean-adjusted version of (1) and (3) is then given as yt − yG t −1 = (yt −1 − yG t −1 ) + t
(6)
with the regressor now independent of t when = 1. Considering the DF test of the null = 1 in its pivotal form as (ˆ − 1)=se(), ˆ the recursively mean-adjusted DF statistic based upon the use of (6) is denoted as DFr , with the familiar DF test using (3) denoted as DF. The Monte Carlo results presented by Shin and So show DFr to display substantially higher power than DF. Indeed, the easily applied DFr test exhibits near identical power to the weighted symmetric DF test, the high power of which has been noted by Pantula et al. (1994). In the following section it will be seen whether the high power and ease of application of recursive mean adjustment are also accompanied by robustness to level breaks.
S. Cook / Statistics & Probability Letters 60 (2002) 75 – 79
77
3. Unit root processes with level breaks To analyse the behaviour of the DFr and DF tests in the presence of level breaks, the experimental design of Leybourne et al. (1998) is employed, with the data generation process (DGP) given as below: yt = st () + t ;
(7)
t = t − 1 + t ;
(8)
t ∼ i:i:d: N(0; 1); 0 for t 6 T; st () = 1 for t ¿ T;
(9) ∈ (0; 1):
(10)
The error series {t } is generated using the RNDNS procedure in GAUSS, with all experiments having performed over 20,000 replications. Following an initial condition y0 = 0, an initial 20 observations were created and discarded, resulting in an e2ective sample size of 100 observations, as employed by Leybourne et al. (1998). To further mimic the approach of Leybourne et al. (1998), the values ∈ {2:5; 5; 10} are selected for the break magnitude. Denoting the break fraction as , the break in level is therefore imposed after observation T . In contrast to Leybourne et al. (1998), where results for a subset of breakpoints are tabulated, all possible breakpoints over the range ∈ (0; 1) are considered. To calculate the empirical rejection frequencies of the unit root hypothesis for the DFr and DF tests at the 5% level of signi;cance, the appropriate critical values are drawn from Shin and So (2001) and Fuller (1996), respectively.
Fig. 1. Size distortion of DF and DFr tests ( = 2:5).
78
S. Cook / Statistics & Probability Letters 60 (2002) 75 – 79
Fig. 2. Size distortion of DF and DFr tests ( = 5).
Fig. 3. Size distortion of DF and DFr tests ( = 10).
Given the large number of breakpoints considered, the arguments of Davidson and MacKinnon (1998) are followed, with the resulting empirical rejection frequencies presented graphically to ease interpretation. In Figs. 1–3 the rejection frequencies for the DFr and DF tests are presented for the three break sizes considered. The results for the DF test replicate the ;ndings of Leybourne et al. (1998), with oversizing occurring when a break is experienced early in the sample period. As the
S. Cook / Statistics & Probability Letters 60 (2002) 75 – 79
79
size of the break is increased, this size distortion becomes more severe, with sizes in excess of 50% observed for a break of = 10. In contrast, the DFr test displays remarkable robustness to level breaks, with no evidence of oversizing observed for any of the values of considered. 4. Conclusion In this paper the severe oversizing of the Dickey–Fuller unit root test in the presence of level breaks has been reconsidered. It has been shown that the relatively simple procedure of recursive mean adjustment suggested by Shin and So (2001) corrects this problem. Combined with the ease of application of the approach and its known high power, this discovery of robustness suggests that recursively mean-adjusted unit root tests should receive more widespread adoption. The performance of the recursively mean-adjusted test when applied to stationary series subject to a structural break remains the subject of future research. References Bai, J., Perron, P., 1998. Testing for and estimation of multiple structural changes. Econometrica 66, 47–79. Bai, J., Lumsdaine, R., Stock, J., 1998. Testing for and dating common breaks in multivariate time series. Rev. Econom. Stud. 65, 395–432. Banerjee, A., Lumsdaine, R., Stock, J., 1992. Recursive and sequential tests of the unit root and trend break hypotheses: theory and international evidence. J. Business Econom. Statist. 10, 271–287. Davidson, R., MacKinnon, J., 1998. Graphical methods for investigating the size and power of hypothesis tests. Manchester School 66, 1–26. Dickey, D., Fuller, W., 1979. Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74, 427–431. Fuller, W., 1996. Introduction to Statistical Time Series, 2nd Edition. Wiley, New York. Leybourne, S., Mills, T., Newbold, P., 1998. Spurious rejections by Dickey–Fuller tests in the presence of a break under the null. J. Econom. 87, 191–203. Pantula, S., Gonzalez-Farias, G., Fuller, W., 1994. A comparison of unit root test criteria. J. Business Econom. Statist. 12, 449–459. Perron, P., 1989. The Great Crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361–1401. Perron, P., 1990. Testing for a unit root in time series with a changing mean. J. Business Econom. Statist. 8, 153–162. Shaman, P., Stine, R., 1988. The bias of autoregressive coeQcient estimators. J. Amer. Statist. Assoc. 83, 842–848. Shin, D., So, B., 2001. Recursive mean adjustment for unit root tests. J. Time Ser. Anal. 5, 595–612. Tanaka, K., 1984. An asymptotic expansion associated with maximum likelihood estimators in ARMA models. J. Roy. Statist. Soc. B 46 (5), 58–67. Zivot, E., Andrews, D., 1992. Further evidence on the Great Crash, the oil price shock and the unit root hypothesis. J. Business Econom. Statist. 10, 251–270.