- Email: [email protected]
Unit-root detection allowing for measurement error
ARTICLE IN PRESS
Statistics & Probability Letters 74 (2005) 373–377 www.elsevier.com/locate/stapro
Unit-root detection allowing for measurement error Kosei Fukuda College of Economics, Nihon University, 1-3-2, Misakicho, Chiyoda-ku, Tokyo 101-8360, Japan Received 3 May 2004; received in revised form 10 November 2004; accepted 1 April 2005 Available online 24 June 2005
Abstract This paper presents a model-selection method for unit-root detection allowing for measurement error. Simulation results show that the proposed method gives better performances than the filtered least-squares method. r 2005 Elsevier B.V. All rights reserved. Keywords: Bayesian information criterion; Filtered least squares; Measurement error; Model selection; Unit root
1. Introduction Many variables used in econometric analyses are recorded with error. It is sometimes the case that observations are contaminated by such noise so that the true relationship between variables becomes somewhat indistinct. This is known as the problem of measurement error (ME). In time series econometrics literature, Blake and Camba-Mendez (1998) and Li et al. (1995) proposed the filtered least-squares (FLS) method for cointegration analysis. In their methods, the Hodrick–Prescott (Hodrick and Prescott, 1997, hereafter HP) filter is applied to the data to remove the high-frequency component before estimating the cointegrating vector. They show that the FLS method works well to correct possible ME in the cointegration analysis. To date, however, few studies have been carried out on univariate unit-root econometrics. As shown later, the autoregressive (AR) process with a unit root and MEs can be transformed into the autoregressive moving average (ARMA) process. It is well known that in the case of such E-mail address: [email protected]. 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.04.059
ARTICLE IN PRESS K. Fukuda / Statistics & Probability Letters 74 (2005) 373–377
374
a process, the Dickey–Fuller (Dickey and Fuller, 1979) tests for a unit root can be oversized (see Schwert, 1989). The simulation results suggest that the FLS method for unit root tests shows poor performances. The purpose of this paper is to show the usefulness of a model-selection method for unit-root detection allowing for ME. I examine the existence of ME by evaluating the values of the Bayesian information criterion (BIC) of a range of alternative stationary and nonstationary models with and without ME. Thus, the correct model is selected by using the minimum BIC procedure (Schwarz, 1978). Simulation results show that the proposed method selects correct models more often than the FLS method.
2. Method I consider the following process: yt ¼ xt þ ut
ðt ¼ 1; . . . ; TÞ,
bðLÞxt ¼ et ,
(1)
bðLÞ ¼ 1 b1 L bp Lp , where only fyt g is observable, fxt g is the underlying process specified as AR(p), and fut g is an ME. It is assumed that the vector ðut ; et Þ0 is independent and identically distributed with zero mean and a diagonal covariance matrix such as ! rs2e 0 , 0 s2e and that bðzÞ ¼ 0 has all roots outside the unit circle. In the proposed method, the following four alternative model classes are considered: stationary model without ME, nonstationary model without ME, stationary model with ME, and nonstationary model with ME. In the nonstationary model, Eq. (1) is modified into bðLÞDxt ¼ et , where Dxt ¼ xt xt1 . Under the assumption of Gaussian distributions, likelihoods are computed for different order models in different model classes using the state-space Kalman-filter methodologies. In the subsequent analyses the maximum lag order, bmax , is fixed as bmax ¼ 4. Thus, the best BIC model is selected among the alternative models using the minimum BIC procedure. Whether a time series contains a unit root and/or ME is determined as a result of model selection. Some comments on the identification issue are appropriate here. It is well known that it is very easy to set up the proposed model, often called unobserved component (UC) model, which is not identifiable. Under the normality assumption the identifiability of the model depends on the form of a covariance matrix (see Harvey, 1989, p. 206). The model (1) can be easily rewritten in the following reduced form: bðLÞyt ¼ et þ bðLÞut .
ARTICLE IN PRESS K. Fukuda / Statistics & Probability Letters 74 (2005) 373–377
375
Thus ARMA(p; p) model is obtained by Granger’s lemma (see Granger and Newbold, 1986, p. 29). In this case, since the number of parameters of the UC model is 2 þ p and that of the ARMA(p; p) is 2p þ 1; p 1 nonlinear restrictions on the parameters of the ARMA(p; p) model are imposed.
3. Simulation results I consider the following data-generating process (DGP): ! ut r 0
iid N . yt ¼ xt þ ut ; xt ¼ fxt1 þ et ; et 0 1
(2)
The parameter vector for f is (1, 0.95, 0.9, 0.8), and that forr is (0, 0.5, 1, 10). Sixteen DGPs in total are therefore considered. The number of replications is 1000 with T ¼ 100. Three alternative methods are considered as follows: (M1) BIC-based model selection, (M2) AIC-based model selection, and (M3) Dickey–Fuller (Dickey and Fuller, 1979) test using the general-to-specific lag-order selection procedure suggested by Ng and Perron (1995). In this simulation, the Akaike information criterion (AIC, Akaike, 1974) as well as the BIC is considered. Hannan (1980) showed, however, that while the BIC provides a strongly consistent estimator of the ARMA order, the AIC does not. In M3 all the hypothesis tests are carried out using the 5% significance level. In addition, the three FLS methods (M4–M6) are introduced. In these methods, observed series are first filtered using the HP filter, and then M3 is applied. The values of the smoothness parameter (l) for the HP filter in M4–M6 are 1, 100, and 1600, respectively. Blake and Camba-Mendez (1998) set l ¼ 100 in simulations, and many economists set l ¼ 1600 in analyzing quarterly time series. Table 1 shows frequency counts of selecting stationary models. First consider the unit-root processes. The AR (1) model with ME in (2) gives the reduced-form ARMA (1,1) model as follows: ð1 fLÞyt ¼ et þ ut fut1 ¼ et yet1 , where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð1 þ f Þr f1 þ ð1 þ f2 Þrg2 4f2 r2 2
y¼
2fr
if r40, and y ¼ 0 if r ¼ 0. In M3, therefore, the size distortion suggested by Schwert (1989) is shown in the case of f ¼ 1 and r40. For example, if r ¼ 10 (implied y ¼ 0:73), for 39% of the series stationary models are incorrectly selected. The same holds for M1 and M2, whereas frequency counts are more stable, although depending on what value of r is considered. M2
ARTICLE IN PRESS K. Fukuda / Statistics & Probability Letters 74 (2005) 373–377
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Table 1 Frequency counts of selecting stationary models f
1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.80 0.80 0.80 0.80
r
0.00 0.50 1.00 10.00 0.00 0.50 1.00 10.00 0.00 0.50 1.00 10.00 0.00 0.50 1.00 10.00
y
0.00 0.27 0.38 0.73 0.00 0.26 0.37 0.72 0.00 0.25 0.36 0.71 0.00 0.23 0.34 0.66
Note DGP: yt ¼ xt þ ut ;
M1
M2
M3
BIC
AIC
Nominal (5%)
Sizeadjusted
Nominal (5%)
0.22 0.28 0.27 0.32 0.46 0.55 0.51 0.59 0.76 0.77 0.75 0.82 0.99 0.96 0.94 0.96
0.54 0.60 0.60 0.66 0.83 0.86 0.84 0.87 0.95 0.95 0.95 0.95 0.98 0.98 0.98 0.98
0.07 0.13 0.16 0.39 0.16 0.30 0.33 0.71 0.36 0.55 0.56 0.89 0.83 0.88 0.90 0.98 !
0.22 0.28 0.27 0.32 0.43 0.54 0.54 0.64 0.72 0.78 0.79 0.85 0.97 0.97 0.97 0.97
0.07 0.08 0.07 0.12 0.14 0.12 0.12 0.28 0.27 0.24 0.21 0.52 0.47 0.47 0.48 0.80
xt ¼ fxt1 þ et ;
ut et
iid N
M4
r 0 0
1
M5
M6
Sizeadjusted
Nominal (5%)
Nominal (5%)
0.22 0.28 0.27 0.32 0.40 0.43 0.46 0.60 0.58 0.61 0.66 0.80 0.80 0.87 0.91 0.96
0.07 0.16 0.12 0.06 0.11 0.25 0.21 0.13 0.17 0.30 0.29 0.19 0.22 0.34 0.35 0.24
0.14 0.21 0.23 0.14 0.13 0.23 0.30 0.19 0.12 0.25 0.30 0.17 0.14 0.25 0.27 0.15
.
M1: BIC-based model-selection method, M2: AIC-based model selection method, M3: Dickey–Fuller test, M4–M6: Filtered least squared methods with smoothness parameter value ¼ 1, 100, and 1600, respectively. The size of M3 and that of M4 are adjusted to be identical to that of M1. y is an implied parameter value of MA(1).
incorrectly selects stationary models more often than M1. Regarding FLS methods, M4 shows good performances. Next consider stationary processes. Since the size of a model-selection method cannot be controlled, not only the nominal (size-unadjusted) power but also the size-adjusted power of M3 and that of M4 is shown. In this case, the size of M3 and that of M4 are adjusted to be identical to that of M1. Simulation results indicate that if the size-adjusted power is used, there is little difference between M1 and M3 and that M4 shows poor performances. Considering the stability acquired through changing the value of r as well, I argue that M1 performs better than the other methods, although its size is somewhat large and takes the values from 0.22 to 0.32. The other fundamental drawback of the FLS method is that it cannot avoid arbitrariness in selecting the value of the smoothness parameter and the size of hypothesis testing for a unit root and for lag-order selection. This method seems to require extensive human intervention to achieve satisfactory analyses. On the other hand, in the proposed method, the problem of statistical identification is explicitly formulated as a problem of estimation, and the semiautomatic execution is possible using the minimum BIC procedure.
ARTICLE IN PRESS K. Fukuda / Statistics & Probability Letters 74 (2005) 373–377
377
4. Conclusion This paper proposes a BIC-based model-selection method for unit-root detection allowing for ME. Simulation results show that the proposed method gives better performances than alternative methods including the FLS method. Another advantage of the BIC-based model-selection method is that it eliminates subjective judgment in time series modeling. Acknowledgements I thank an anonymous referee for valuable comments on an earlier version of this paper.
References Akaike, H., 1974. A new look at the statistical model identification. IEEE Trans. Automat. Control AC-19, 716–723. Blake, A.P., Camba-Mendez, G., 1998. Filtered least squares and measurement error. Econom. Lett. 59, 163–168. Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74, 427–431. Granger, C.W.J., Newbold, P., 1986. Forecasting Economic Time Series, second ed. Academic Press, New York. Hannan, E.J., 1980. The estimation of the order of an ARMA process. Ann. Statist. 8, 1071–1081. Harvey, A.C., 1989. Forecasting, Structural Models and the Kalman Filter. Cambridge University Press, Cambridge. Hodrick, R.J., Prescott, E.C., 1997. Post-war U.S. business cycles: an empirical investigation. J. Money Credit Banking 29, 1–16. Li, Y., Maddala, G.S., Rush, M., 1995. New small sample estimators for cointegration regression: low-pass spectral filter method. Econom. Lett. 47, 123–129. Ng, S., Perron, P., 1995. Unit root tests in ARIMA models with data-dependent methods for the selection of the truncation lag. J. Amer. Statist. Assoc. 90, 268–281. Schwarz, G., 1978. Estimating the dimension of a model. Ann. Statist. 6, 461–464. Schwert, G.W., 1989. Tests for unit roots: a Monte Carlo investigation. J. Business Econom. Statist. 7, 147–160.
Statistics & Probability Letters 74 (2005) 373–377 www.elsevier.com/locate/stapro
Unit-root detection allowing for measurement error Kosei Fukuda College of Economics, Nihon University, 1-3-2, Misakicho, Chiyoda-ku, Tokyo 101-8360, Japan Received 3 May 2004; received in revised form 10 November 2004; accepted 1 April 2005 Available online 24 June 2005
Abstract This paper presents a model-selection method for unit-root detection allowing for measurement error. Simulation results show that the proposed method gives better performances than the filtered least-squares method. r 2005 Elsevier B.V. All rights reserved. Keywords: Bayesian information criterion; Filtered least squares; Measurement error; Model selection; Unit root
1. Introduction Many variables used in econometric analyses are recorded with error. It is sometimes the case that observations are contaminated by such noise so that the true relationship between variables becomes somewhat indistinct. This is known as the problem of measurement error (ME). In time series econometrics literature, Blake and Camba-Mendez (1998) and Li et al. (1995) proposed the filtered least-squares (FLS) method for cointegration analysis. In their methods, the Hodrick–Prescott (Hodrick and Prescott, 1997, hereafter HP) filter is applied to the data to remove the high-frequency component before estimating the cointegrating vector. They show that the FLS method works well to correct possible ME in the cointegration analysis. To date, however, few studies have been carried out on univariate unit-root econometrics. As shown later, the autoregressive (AR) process with a unit root and MEs can be transformed into the autoregressive moving average (ARMA) process. It is well known that in the case of such E-mail address: [email protected]. 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.04.059
ARTICLE IN PRESS K. Fukuda / Statistics & Probability Letters 74 (2005) 373–377
374
a process, the Dickey–Fuller (Dickey and Fuller, 1979) tests for a unit root can be oversized (see Schwert, 1989). The simulation results suggest that the FLS method for unit root tests shows poor performances. The purpose of this paper is to show the usefulness of a model-selection method for unit-root detection allowing for ME. I examine the existence of ME by evaluating the values of the Bayesian information criterion (BIC) of a range of alternative stationary and nonstationary models with and without ME. Thus, the correct model is selected by using the minimum BIC procedure (Schwarz, 1978). Simulation results show that the proposed method selects correct models more often than the FLS method.
2. Method I consider the following process: yt ¼ xt þ ut
ðt ¼ 1; . . . ; TÞ,
bðLÞxt ¼ et ,
(1)
bðLÞ ¼ 1 b1 L bp Lp , where only fyt g is observable, fxt g is the underlying process specified as AR(p), and fut g is an ME. It is assumed that the vector ðut ; et Þ0 is independent and identically distributed with zero mean and a diagonal covariance matrix such as ! rs2e 0 , 0 s2e and that bðzÞ ¼ 0 has all roots outside the unit circle. In the proposed method, the following four alternative model classes are considered: stationary model without ME, nonstationary model without ME, stationary model with ME, and nonstationary model with ME. In the nonstationary model, Eq. (1) is modified into bðLÞDxt ¼ et , where Dxt ¼ xt xt1 . Under the assumption of Gaussian distributions, likelihoods are computed for different order models in different model classes using the state-space Kalman-filter methodologies. In the subsequent analyses the maximum lag order, bmax , is fixed as bmax ¼ 4. Thus, the best BIC model is selected among the alternative models using the minimum BIC procedure. Whether a time series contains a unit root and/or ME is determined as a result of model selection. Some comments on the identification issue are appropriate here. It is well known that it is very easy to set up the proposed model, often called unobserved component (UC) model, which is not identifiable. Under the normality assumption the identifiability of the model depends on the form of a covariance matrix (see Harvey, 1989, p. 206). The model (1) can be easily rewritten in the following reduced form: bðLÞyt ¼ et þ bðLÞut .
ARTICLE IN PRESS K. Fukuda / Statistics & Probability Letters 74 (2005) 373–377
375
Thus ARMA(p; p) model is obtained by Granger’s lemma (see Granger and Newbold, 1986, p. 29). In this case, since the number of parameters of the UC model is 2 þ p and that of the ARMA(p; p) is 2p þ 1; p 1 nonlinear restrictions on the parameters of the ARMA(p; p) model are imposed.
3. Simulation results I consider the following data-generating process (DGP): ! ut r 0
iid N . yt ¼ xt þ ut ; xt ¼ fxt1 þ et ; et 0 1
(2)
The parameter vector for f is (1, 0.95, 0.9, 0.8), and that forr is (0, 0.5, 1, 10). Sixteen DGPs in total are therefore considered. The number of replications is 1000 with T ¼ 100. Three alternative methods are considered as follows: (M1) BIC-based model selection, (M2) AIC-based model selection, and (M3) Dickey–Fuller (Dickey and Fuller, 1979) test using the general-to-specific lag-order selection procedure suggested by Ng and Perron (1995). In this simulation, the Akaike information criterion (AIC, Akaike, 1974) as well as the BIC is considered. Hannan (1980) showed, however, that while the BIC provides a strongly consistent estimator of the ARMA order, the AIC does not. In M3 all the hypothesis tests are carried out using the 5% significance level. In addition, the three FLS methods (M4–M6) are introduced. In these methods, observed series are first filtered using the HP filter, and then M3 is applied. The values of the smoothness parameter (l) for the HP filter in M4–M6 are 1, 100, and 1600, respectively. Blake and Camba-Mendez (1998) set l ¼ 100 in simulations, and many economists set l ¼ 1600 in analyzing quarterly time series. Table 1 shows frequency counts of selecting stationary models. First consider the unit-root processes. The AR (1) model with ME in (2) gives the reduced-form ARMA (1,1) model as follows: ð1 fLÞyt ¼ et þ ut fut1 ¼ et yet1 , where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð1 þ f Þr f1 þ ð1 þ f2 Þrg2 4f2 r2 2
y¼
2fr
if r40, and y ¼ 0 if r ¼ 0. In M3, therefore, the size distortion suggested by Schwert (1989) is shown in the case of f ¼ 1 and r40. For example, if r ¼ 10 (implied y ¼ 0:73), for 39% of the series stationary models are incorrectly selected. The same holds for M1 and M2, whereas frequency counts are more stable, although depending on what value of r is considered. M2
ARTICLE IN PRESS K. Fukuda / Statistics & Probability Letters 74 (2005) 373–377
376
Table 1 Frequency counts of selecting stationary models f
1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.80 0.80 0.80 0.80
r
0.00 0.50 1.00 10.00 0.00 0.50 1.00 10.00 0.00 0.50 1.00 10.00 0.00 0.50 1.00 10.00
y
0.00 0.27 0.38 0.73 0.00 0.26 0.37 0.72 0.00 0.25 0.36 0.71 0.00 0.23 0.34 0.66
Note DGP: yt ¼ xt þ ut ;
M1
M2
M3
BIC
AIC
Nominal (5%)
Sizeadjusted
Nominal (5%)
0.22 0.28 0.27 0.32 0.46 0.55 0.51 0.59 0.76 0.77 0.75 0.82 0.99 0.96 0.94 0.96
0.54 0.60 0.60 0.66 0.83 0.86 0.84 0.87 0.95 0.95 0.95 0.95 0.98 0.98 0.98 0.98
0.07 0.13 0.16 0.39 0.16 0.30 0.33 0.71 0.36 0.55 0.56 0.89 0.83 0.88 0.90 0.98 !
0.22 0.28 0.27 0.32 0.43 0.54 0.54 0.64 0.72 0.78 0.79 0.85 0.97 0.97 0.97 0.97
0.07 0.08 0.07 0.12 0.14 0.12 0.12 0.28 0.27 0.24 0.21 0.52 0.47 0.47 0.48 0.80
xt ¼ fxt1 þ et ;
ut et
iid N
M4
r 0 0
1
M5
M6
Sizeadjusted
Nominal (5%)
Nominal (5%)
0.22 0.28 0.27 0.32 0.40 0.43 0.46 0.60 0.58 0.61 0.66 0.80 0.80 0.87 0.91 0.96
0.07 0.16 0.12 0.06 0.11 0.25 0.21 0.13 0.17 0.30 0.29 0.19 0.22 0.34 0.35 0.24
0.14 0.21 0.23 0.14 0.13 0.23 0.30 0.19 0.12 0.25 0.30 0.17 0.14 0.25 0.27 0.15
.
M1: BIC-based model-selection method, M2: AIC-based model selection method, M3: Dickey–Fuller test, M4–M6: Filtered least squared methods with smoothness parameter value ¼ 1, 100, and 1600, respectively. The size of M3 and that of M4 are adjusted to be identical to that of M1. y is an implied parameter value of MA(1).
incorrectly selects stationary models more often than M1. Regarding FLS methods, M4 shows good performances. Next consider stationary processes. Since the size of a model-selection method cannot be controlled, not only the nominal (size-unadjusted) power but also the size-adjusted power of M3 and that of M4 is shown. In this case, the size of M3 and that of M4 are adjusted to be identical to that of M1. Simulation results indicate that if the size-adjusted power is used, there is little difference between M1 and M3 and that M4 shows poor performances. Considering the stability acquired through changing the value of r as well, I argue that M1 performs better than the other methods, although its size is somewhat large and takes the values from 0.22 to 0.32. The other fundamental drawback of the FLS method is that it cannot avoid arbitrariness in selecting the value of the smoothness parameter and the size of hypothesis testing for a unit root and for lag-order selection. This method seems to require extensive human intervention to achieve satisfactory analyses. On the other hand, in the proposed method, the problem of statistical identification is explicitly formulated as a problem of estimation, and the semiautomatic execution is possible using the minimum BIC procedure.
ARTICLE IN PRESS K. Fukuda / Statistics & Probability Letters 74 (2005) 373–377
377
4. Conclusion This paper proposes a BIC-based model-selection method for unit-root detection allowing for ME. Simulation results show that the proposed method gives better performances than alternative methods including the FLS method. Another advantage of the BIC-based model-selection method is that it eliminates subjective judgment in time series modeling. Acknowledgements I thank an anonymous referee for valuable comments on an earlier version of this paper.
References Akaike, H., 1974. A new look at the statistical model identification. IEEE Trans. Automat. Control AC-19, 716–723. Blake, A.P., Camba-Mendez, G., 1998. Filtered least squares and measurement error. Econom. Lett. 59, 163–168. Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74, 427–431. Granger, C.W.J., Newbold, P., 1986. Forecasting Economic Time Series, second ed. Academic Press, New York. Hannan, E.J., 1980. The estimation of the order of an ARMA process. Ann. Statist. 8, 1071–1081. Harvey, A.C., 1989. Forecasting, Structural Models and the Kalman Filter. Cambridge University Press, Cambridge. Hodrick, R.J., Prescott, E.C., 1997. Post-war U.S. business cycles: an empirical investigation. J. Money Credit Banking 29, 1–16. Li, Y., Maddala, G.S., Rush, M., 1995. New small sample estimators for cointegration regression: low-pass spectral filter method. Econom. Lett. 47, 123–129. Ng, S., Perron, P., 1995. Unit root tests in ARIMA models with data-dependent methods for the selection of the truncation lag. J. Amer. Statist. Assoc. 90, 268–281. Schwarz, G., 1978. Estimating the dimension of a model. Ann. Statist. 6, 461–464. Schwert, G.W., 1989. Tests for unit roots: a Monte Carlo investigation. J. Business Econom. Statist. 7, 147–160.