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Do long-memory models have long memory?
International Journal of Forecasting 16 (2000) 121–124 www.elsevier.com / locate / ijforecast
Do long-memory models have long memory? Michael K. Andersson* National Institute of Economic Research, P.O. Box 3116, SE-103 62 Stockholm, Sweden
Abstract This paper examines the predictability memory of fractionally integrated ARMA processes. Very long memory is found for positively fractionally integrated processes with large positive AR parameters. However, negative AR parameters absorb, to a great extent, the memory generated by a positive fractional difference. An MA parameter may also reduce the predictability memory substantially, even if the parameter alone provides hardly any memory. 2000 International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved. Keywords: ARMA; Fractional integration; Prediction horizon
1. Introduction This paper investigates the predictability memory of fractionally integrated ARMA (ARFIMA) time series processes. The quantity is useful when deciding upon (maximum) prediction horizon and may be defined by how long (in time) a shock influences the process. We compute the predictability memory through the Granger and Newbold (1986, p. 310) measure. Fractionally integrated processes are said to be long-memory, due to their slow hyperbolic autocorrelation and impulse response decay, compared to the faster geometrical decay of ARMA processes. The calculations suggest that long-memory processes often have quite long *Fax: 146-8-453-5980. E-mail address: [email protected] (M.K. Andersson)
predictabilty memory and that fractional integration extends the prediction memory of an ARMA process. However, this is not true for all parameter combinations examined. Moreover, two examples demonstrate that model misspecification strongly affects the estimated memory.
2. A measure of predictability memory A time series hx t j follows an ARFIMA s p,d,qd process if: p
s1 2 f1 B 2 ? ? ? 2 fp B ds1 2 Bd dxt 5s1 1 u1 B 1 ? ? ? 1 uq B qda t
(1)
where B is the backshift operator defined by B j x t 5 x t2j and a t is iid with mean zero and variance s 2a , `. If the lag polynomials have all roots outside the unit circle and d lies in the
0169-2070 / 00 / $ – see front matter 2000 International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved. PII: S0169-2070( 99 )00040-0
M.K. Andersson / International Journal of Forecasting 16 (2000) 121 – 124
122
interval (21 / 2, 1 / 2), the process is both invertible and stationary. See Granger and Joyeux (1980) and Hosking (1981) for an introduction to fractional integration. Under stationarity, the process variance s 2x and l-step forecast error variance j (l) of (1) can conveniently be written:
Oc , Oc , `
s 2x
2
5 sa
2 j
c0 ; 1
2 j
c0 ; 1
j50 l21
j (l)
5 s 2a
(2)
j50
The coefficients of (2) are given by the Wold representation of (1), that is:
s1 1 c1 B 1 c2 B 2 1 ? ? ?d q s1 1 u1 B 1 ? ? ? 1 uq B d 5 ]]]]]]]]]] p s1 2 f1 B 2 ? ? ? 2 fp B ds1 2 Bd d Granger and Newbold (1986, p. 310) suggest a measure of predictability of a stationary sfinite varianced series x t under squared error loss as the quantity R l2ux 5 1 2 jsld /s x2 . When jsld and s x2 are sufficiently close, that is R 2L 1ux # 1 2 c, the predictability memory of a particular process is L if:
jsLd , cs 2x # jsL 1 1d
(3)
The intuition of the decision rule (3) is that the model forecasts are not more accurate than the sample mean after L steps. Alternative measures, such as the mean lag or the half-life of a shock (the median lag) are not feasible for fractionally integrated processes since the quantities goes to infinity when d . 0. Furthermore, the memory may be set according to the test of ¨ Oller (1985), which is similar to our rule. 3. The memory of ARFIMA (1,d,1) processes The predictability memory is illustrated for the ARFIMA s1,d,1d specification: s1 2 f Bds1 2 Bd d x t 5s1 1 u Bda t
a t | Ns0,1d
The differencing parameter is chosen as d 5 h0, 0.1, 0.2, 0.3j, and the parameters f and u are set between 2 0.9 and 10.9. The constant c of the decision rule (3) is set to 0.95 and all computations are summarized in Fig. 1. ARMA models sd 5 0d have short predictability memory unless the AR parameter is close to unity, an AR(1) with parameter 0.9 has memory 14. An MA parameter may alter the memory only slightly, e.g. the memory decreases when an MA parameter of opposite sign is added to the process. A fractional difference may imply a long predictability memory. However, ARFIMA processes with 0 , d # 0.2 (represented Fig. 1b and c) behave similarly as ARMA processes, but a large positive / negative AR parameter prolongs / restricts the memory. When d 5 0.3, the memory amounts 37 for fractional noise, to 86 if a positive MA parameter is added, and a maximum of 1051 is attained for ARFIMA s1, 0.3, 0d processes. The combination of positive AR parameters and a fractional difference generally implies a long predictability memory. Negative AR and MA parameters are rare in practice, but they restrict the memory. As a matter of fact, if the process contains large negative AR and MA parameters the predictability memory is similar to the corresponding ARMA (1, 1) process. When d . 0.3, the predictability memory grows dramatically. 4. Examples The examples demonstrate the effects of (wrongly) using / not using fractional integration. For the ARMA estimation we use maximum likelihood and the Geweke and Porter-Hudak (GPH, 1983) estimator is utilized for the fractional specification.
4.1. Example 1: model misspecification Consider (DGPs):
the
data
generating
processes
M.K. Andersson / International Journal of Forecasting 16 (2000) 121 – 124
123
Fig. 1. The surfaces report the memory (left axis) as a function of the autoregressive parameter (bottom axis) and the moving average parameter (right axis).
s1 2 0.7Bdx t 5 a t ,
t 5 1, . . . ,100,
(4)
and s1 2 Bd 0.3 x t 5 e t ,
t 5 1, . . . ,100
(5)
For the AR(1) process (4), a fractional model is estimated to s1 2 0.4Bds1 2 Bd 0.3 x t 5 aˆ t (see Agiakloglou, Newbold & Wohar, 1993). This model has a predictability memory of 112 compared to 4 for the non-fractional AR model. Hence, a forecaster would be mislead to accept a spurious predictability memory. For the fractional noise DGP (5) an estimated non-fractional ARs pd process will on average (based on 10 000 generated and estimated series) look like s1 2 0.364Bdx t 5 e˜ t , when p is selected by the BIC, and s1 2 0.261B 2 0.118B 2dx t 5 eˇ t using the AIC. In both cases the predictability memory equals unity, whereas the memory using the fractional estimation procedure amounts to 37.
4.2. Example 2: an empirical example We estimate the predictability memory of the storage capacity, that is inflow minus outflow, of the Columbia River reservoir, the Albeni Falls Forebay. The storage capacity ss td data, courtesy of US Army Corps of Engineers, NWD, cover January 1 to November 21, 1998 and consist of daily observations (see Fig. 2). The autocorrelation function exhibits a slow decay, however, not as slow as that of a unit root process, and a prominent seasonal pattern at a periodicity of 7 days. The two models:
S1 20.52B 20.15B 20.18B Ds 5 v S1 20.16B 20.25B Ds1 2 Bd0.46s 5 w 7
(0.05 )
7
(0.06 )
14
(0.05 )
(0.05 )
t
(6)
t
14
0.06
(0.19 )
t
t
(7)
appear to fit the data well and leave no structure in the residual series vt and w t . According to the
M.K. Andersson / International Journal of Forecasting 16 (2000) 121 – 124
124
Fig. 2. Daily inflow minus outflow of the Albeni Falls Forebay reservoir, Columbia River.
estimated predictability memory of model (6) the series may be predicted 4 days ahead and 359 days due to model (7). Acknowledgements I thank two anonymous referees, Sune ¨ ¨ Karlsson, Lars-Erik Oller and Timo Terasvirta. The usual disclaimer applies. References Agiakloglou, C., Newbold, P., & Wohar, M. (1993). Bias in an estimator of the fractional difference parameter. Journal of Time Series Analysis 14, 235–246.
Granger, C. W. J., & Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–29. Granger, C. W. J., & Newbold, P. (1986). Forecasting economic time series, 2nd ed., Academic Press, San Diego, CA. Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68, 165–176. ¨ Oller, L. -E. (1985). How far can changes in business activity be forecasted? International Journal of Forecasting 1, 135–141.
Do long-memory models have long memory? Michael K. Andersson* National Institute of Economic Research, P.O. Box 3116, SE-103 62 Stockholm, Sweden
Abstract This paper examines the predictability memory of fractionally integrated ARMA processes. Very long memory is found for positively fractionally integrated processes with large positive AR parameters. However, negative AR parameters absorb, to a great extent, the memory generated by a positive fractional difference. An MA parameter may also reduce the predictability memory substantially, even if the parameter alone provides hardly any memory. 2000 International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved. Keywords: ARMA; Fractional integration; Prediction horizon
1. Introduction This paper investigates the predictability memory of fractionally integrated ARMA (ARFIMA) time series processes. The quantity is useful when deciding upon (maximum) prediction horizon and may be defined by how long (in time) a shock influences the process. We compute the predictability memory through the Granger and Newbold (1986, p. 310) measure. Fractionally integrated processes are said to be long-memory, due to their slow hyperbolic autocorrelation and impulse response decay, compared to the faster geometrical decay of ARMA processes. The calculations suggest that long-memory processes often have quite long *Fax: 146-8-453-5980. E-mail address: [email protected] (M.K. Andersson)
predictabilty memory and that fractional integration extends the prediction memory of an ARMA process. However, this is not true for all parameter combinations examined. Moreover, two examples demonstrate that model misspecification strongly affects the estimated memory.
2. A measure of predictability memory A time series hx t j follows an ARFIMA s p,d,qd process if: p
s1 2 f1 B 2 ? ? ? 2 fp B ds1 2 Bd dxt 5s1 1 u1 B 1 ? ? ? 1 uq B qda t
(1)
where B is the backshift operator defined by B j x t 5 x t2j and a t is iid with mean zero and variance s 2a , `. If the lag polynomials have all roots outside the unit circle and d lies in the
0169-2070 / 00 / $ – see front matter 2000 International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved. PII: S0169-2070( 99 )00040-0
M.K. Andersson / International Journal of Forecasting 16 (2000) 121 – 124
122
interval (21 / 2, 1 / 2), the process is both invertible and stationary. See Granger and Joyeux (1980) and Hosking (1981) for an introduction to fractional integration. Under stationarity, the process variance s 2x and l-step forecast error variance j (l) of (1) can conveniently be written:
Oc , Oc , `
s 2x
2
5 sa
2 j
c0 ; 1
2 j
c0 ; 1
j50 l21
j (l)
5 s 2a
(2)
j50
The coefficients of (2) are given by the Wold representation of (1), that is:
s1 1 c1 B 1 c2 B 2 1 ? ? ?d q s1 1 u1 B 1 ? ? ? 1 uq B d 5 ]]]]]]]]]] p s1 2 f1 B 2 ? ? ? 2 fp B ds1 2 Bd d Granger and Newbold (1986, p. 310) suggest a measure of predictability of a stationary sfinite varianced series x t under squared error loss as the quantity R l2ux 5 1 2 jsld /s x2 . When jsld and s x2 are sufficiently close, that is R 2L 1ux # 1 2 c, the predictability memory of a particular process is L if:
jsLd , cs 2x # jsL 1 1d
(3)
The intuition of the decision rule (3) is that the model forecasts are not more accurate than the sample mean after L steps. Alternative measures, such as the mean lag or the half-life of a shock (the median lag) are not feasible for fractionally integrated processes since the quantities goes to infinity when d . 0. Furthermore, the memory may be set according to the test of ¨ Oller (1985), which is similar to our rule. 3. The memory of ARFIMA (1,d,1) processes The predictability memory is illustrated for the ARFIMA s1,d,1d specification: s1 2 f Bds1 2 Bd d x t 5s1 1 u Bda t
a t | Ns0,1d
The differencing parameter is chosen as d 5 h0, 0.1, 0.2, 0.3j, and the parameters f and u are set between 2 0.9 and 10.9. The constant c of the decision rule (3) is set to 0.95 and all computations are summarized in Fig. 1. ARMA models sd 5 0d have short predictability memory unless the AR parameter is close to unity, an AR(1) with parameter 0.9 has memory 14. An MA parameter may alter the memory only slightly, e.g. the memory decreases when an MA parameter of opposite sign is added to the process. A fractional difference may imply a long predictability memory. However, ARFIMA processes with 0 , d # 0.2 (represented Fig. 1b and c) behave similarly as ARMA processes, but a large positive / negative AR parameter prolongs / restricts the memory. When d 5 0.3, the memory amounts 37 for fractional noise, to 86 if a positive MA parameter is added, and a maximum of 1051 is attained for ARFIMA s1, 0.3, 0d processes. The combination of positive AR parameters and a fractional difference generally implies a long predictability memory. Negative AR and MA parameters are rare in practice, but they restrict the memory. As a matter of fact, if the process contains large negative AR and MA parameters the predictability memory is similar to the corresponding ARMA (1, 1) process. When d . 0.3, the predictability memory grows dramatically. 4. Examples The examples demonstrate the effects of (wrongly) using / not using fractional integration. For the ARMA estimation we use maximum likelihood and the Geweke and Porter-Hudak (GPH, 1983) estimator is utilized for the fractional specification.
4.1. Example 1: model misspecification Consider (DGPs):
the
data
generating
processes
M.K. Andersson / International Journal of Forecasting 16 (2000) 121 – 124
123
Fig. 1. The surfaces report the memory (left axis) as a function of the autoregressive parameter (bottom axis) and the moving average parameter (right axis).
s1 2 0.7Bdx t 5 a t ,
t 5 1, . . . ,100,
(4)
and s1 2 Bd 0.3 x t 5 e t ,
t 5 1, . . . ,100
(5)
For the AR(1) process (4), a fractional model is estimated to s1 2 0.4Bds1 2 Bd 0.3 x t 5 aˆ t (see Agiakloglou, Newbold & Wohar, 1993). This model has a predictability memory of 112 compared to 4 for the non-fractional AR model. Hence, a forecaster would be mislead to accept a spurious predictability memory. For the fractional noise DGP (5) an estimated non-fractional ARs pd process will on average (based on 10 000 generated and estimated series) look like s1 2 0.364Bdx t 5 e˜ t , when p is selected by the BIC, and s1 2 0.261B 2 0.118B 2dx t 5 eˇ t using the AIC. In both cases the predictability memory equals unity, whereas the memory using the fractional estimation procedure amounts to 37.
4.2. Example 2: an empirical example We estimate the predictability memory of the storage capacity, that is inflow minus outflow, of the Columbia River reservoir, the Albeni Falls Forebay. The storage capacity ss td data, courtesy of US Army Corps of Engineers, NWD, cover January 1 to November 21, 1998 and consist of daily observations (see Fig. 2). The autocorrelation function exhibits a slow decay, however, not as slow as that of a unit root process, and a prominent seasonal pattern at a periodicity of 7 days. The two models:
S1 20.52B 20.15B 20.18B Ds 5 v S1 20.16B 20.25B Ds1 2 Bd0.46s 5 w 7
(0.05 )
7
(0.06 )
14
(0.05 )
(0.05 )
t
(6)
t
14
0.06
(0.19 )
t
t
(7)
appear to fit the data well and leave no structure in the residual series vt and w t . According to the
M.K. Andersson / International Journal of Forecasting 16 (2000) 121 – 124
124
Fig. 2. Daily inflow minus outflow of the Albeni Falls Forebay reservoir, Columbia River.
estimated predictability memory of model (6) the series may be predicted 4 days ahead and 359 days due to model (7). Acknowledgements I thank two anonymous referees, Sune ¨ ¨ Karlsson, Lars-Erik Oller and Timo Terasvirta. The usual disclaimer applies. References Agiakloglou, C., Newbold, P., & Wohar, M. (1993). Bias in an estimator of the fractional difference parameter. Journal of Time Series Analysis 14, 235–246.
Granger, C. W. J., & Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–29. Granger, C. W. J., & Newbold, P. (1986). Forecasting economic time series, 2nd ed., Academic Press, San Diego, CA. Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68, 165–176. ¨ Oller, L. -E. (1985). How far can changes in business activity be forecasted? International Journal of Forecasting 1, 135–141.