forecasting costs incurred from unit differencing fractionally integrated processes

F.ISEVIER International Journal of Forecasting 10 (1994) 507-514

Forecasting costs incurred from unit differencing fractionally integrated processes

Jeremy Smith*‘“, Sanjay Yadavb “Department of Economics, University of Warwick, Warwick, UK

hFinancial Options Research Centre, Warwick Business School, University of Warwick, Warwick, UK

Abstract

This paper investigates the cost of assuming a unit difference when the series is only fractionally integrated with an integration parameter d # 1. Studies have pointed to the low power of unit root tests against a fractionally integrated alternative, and have noted the performance of these tests is worse than against nearly integrated stationary ARMA models, due to the extra persistence associated with fractional models. We look at the gains, in terms of forecasting performance, of fitting a correctly specified ARFIMA model against a mis-specified ARIMA model and ask the question as to whether the forecasting gain offsets the computational costs of estimating the correct ARFIMA model.

Keywords: ARFIMA models; Forecasting; Fractional integration

1. Introduction

Following the paper of Granger and Joyeux (1980) there has been an increase in interest in estimating the long-term, or integration parame- ter d. Traditionally, nonstationarity focused at- tention on the distinction between stationary, d = 0.0, and nonstationary, d = 1.0, series. Frac- tional integration extended the discussion by allowing the integration parameter d to take on noninteger values. Diebold and Rudebusch (1989) estimate both the long-run and short-run parameters using fractionally integrated ARMA, denoted ARFIMA, models for a variety of different measures of U.S. aggregate output. The authors find, for most output measures, across the majority of time periods, the long-run

* Corresponding author.

integration parameter is less than unity, although imprecision, reflected in large standard error estimates, does not permit the authors to reject a value of d = 1 for many series. In their conclu- sion, Diebold and Rudebusch (1991) recommend a “more appropriate testing procedure . . . be- fore drawing conclusions about the presence of a unit root”.

The results of Diebold and Rudebusch (1989) contrast with those of Nelson and Plosser (1982), who found a unit root in almost all U.S. macro- economic aggregate series. One explanation for the different results can be found in the papers of Diebold and Rudebusch (1991) and Smith and Yadav (1994), both of whom show, for relatively small sample sizes, the low power of existing unit root tests, when the underlying process is a fractionally integrated one. Over a variety of tests that were considered, Smith and Yadav

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508 .I. Smith, S. Yadav I international Journal of Forecasting IO (1994) 507-514

(1994) find that the Augmented Dickey Fuller (ADF) test, has the lowest power. For example, for a sample size T = 128, when d = 0.8 the homoscedastic test of Lo and Ma&inlay (1989) correctly rejects the null hypothesis 59.4% of the time for a two-sided test at the 5% significance level, whereas for d = 0.9 power falls to 21.6%. In comparison the ADF test correctly rejects the false null hypothesis only 31.6% and 9.0% of the time, respectively. Similar rejection frequencies exist for d > 1.0.

Over-differencing of the data, from assuming a unit root when the series is only fractionally integrated with d < 1.0 produces a spectral den- sity which is highly peaked at the origin when looking at the level of the series, but which has no power at the origin in first differences (see Granger, 1966, for a description of the typical spectral density function of these series). Where- as under-differencing will yield a spectral density function which is highly peaked at the origin even in first differences.

This paper investigates the effects of over- or under differencing a series on forecasting per- formance over short-, medium- and long- horizons. That is, the paper investigates whether, assuming the presence of a unit root and fitting a standard ARIMA model, involves a substantial loss in terms of forecasting perform- ance, when compared with fitting an appro- priately specified ARFIMA model.

Ray (1993) undertook a forecasting compari- son between ARIFMA models, and high order AR models. The results show that high order AR models are capable of forecasting the longer term well when compared with ARFIMA models. The author disputes whether there is much gain, in a forecasting sense, from using ARFIMA as opposed to high order AR models, given the problems of generating fractionally integrated series and identifying both short-run and long-run parameters. In the analysis the author considers positive values of the integra- tion parameter d up to d = 0.45, and compares theoretical and actual forecast mean square errors from the ARFIMA model with forecast mean square errors from AR models, up to an AR(20) model, for up to 20 steps ahead.

The results of Diebold and Rudebusch (1989), suggest that the integration parameter estimates often lie between d = 0.5 and d = 1.0.’ There- fore, in looking at the forecasting performance of incorrectly specified ARIMA models applied to fractional generated series, we consider inte- gration parameters in the range 0.7<d < 1.3. For these series we assume the researcher mis- takenly applies a unit difference to the series, based on the results of preliminary unit root testing, yielding series which are fractionally integrated with parameter d = d - 1 in the range -0.3<d”<o.3.

Following Ray (1993) we compare the theoret- ical ARFIMA forecast mean square errors with the forecast mean square errors from each of the estimated ARIMA models and report the per- centage difference between the two. A variety of

integration parameters, 2 are considered and a forecast horizon of 20 periods is used, due to the fact that there are little observed differences in forecasts over longer horizons.

2. Monte Carlo simulations

We consider fractionally integrated data generating processes for the series y,, of the form (1 - L)“y, = E,, where d =0.7, 0.8, 0.9, 0.95, 1.05, 1.1, 1.2, and 1.3. This implies that (1 - L)d-‘(l - L)y, = E(, in other words (1 - L)JAy, = E,, that is, that Ay, is fractionally inte- grated of order d = d - 1, where d” = - 0.3, - 0.2,-O.l,-0.05, 0.05, 0.1, 0.2, and 0.3. The series is constructed along similar lines to that used by Diebold (1989). A vector C, of T stan- dard normal variates (T = 250) is constructed using the NAG routine GOSDDF. Given that the autocovariance for fractionally integrated series is written as

@(l - 2d)QT + d) “=f(d)r(l-d)r(7+1-d) 7.=‘-+17 *-2,

*3,...,

where r(.) is a gamma function, we calculate the

’ Values of the fractional integration parameter - 0.5 <

d < 0.5, yield stationary invertible processes.

J. Smith, S. Yadav I Innternational Journal of Forecasting 10 (1994) 507-514 509

desired T x T autocovariance matrix S. Using lower triangular. Finally, Ay, is formed as Ay = the NAG routine F07FDF a Choleski decompo- (1 - L)y = Pe. As Diebold (1990) and Granger sition of 2, 2 = PP’ is performed, where P is and Joyeux (1980) note the attraction of this

Table 1 Forecast performance of AR models applied to (1 - L)” Aq = E,, d” < 0.0

2 Step AR(O) AR(l) AR(2) horizon

AR(3) AR(4) AR(5)

-0.3 1 14.65 16.97 15.98 15.32 15.04 15.65 2 1.13 9.48 8.82 9.20 9.85 10.60

3 -0.99 -3.10 -2.37 -3.07 -2.98 -2.49 4 -1.98 -3.87 -3.70 -3.41 -2.87 -1.90 5 -0.32 5.86 5.81 5.79 6.24 7.26

6 -0.78 5.46 5.47 5.20 5.42 6.35

8 -1.07 -4.68 -4.68 -4.67 -4.80 -4.64

10 0.06 2.2Y 2.29 2.31 2.30 2.31

12 2.31 3.27 3.27 3.26 3.27 3.30

16 1.39 -1.54 -1.54 -1.54 -1.54 -1.49

20 -0.52 -2.92 -2.92 -2.92 -2.92 -2.92

-0.2

-0.1

1 8.81 6.37 5.91 5.76 6.11 6.41

2 0.47 1.00 0.83 1.44 2.17 2.45

3 -1.14 -1.19 -0.84 -0.23 0.03 0.67

4 -2.44 -2.43 -2.39 -2.05 -1.79 -1.29

5 -0.28 -0.28 -0.26 -0.26 -0.07 0.66

6 -0.67 -0.67 -0.67 -0.66 -0.50 -0.14

8 -1.10 -1.10 -1.10 -1.11 -1.12 -1.05

10 0.23 0.23 0.23 0.23 0.22 0.23

12 2.16 2.16 2.16 2.15 2.16 2.16

16 1.40 1.40 1.40 1.40 1.41 1.41 20 -0.47 -0.47 -0.47 -0.47 -0.47 -0.47

1 4.81 4.34 4.56 4.77 5.35 5.71

2 -0.29 -0.15 0.10 0.90 1.64 2.01

3 -1.38 -1.40 -1.31 -0.66 -0.20 0.43

4 -3.05 -3.04 -3.00 -2.91 -2.57 -2.04

5 -0.39 -0.39 -0.38 -0.38 -0.35 0.37

6 -0.61 -0.61 -0.61 -0.59 -0.46 -0.31

8 -1.15 -1.15 -1.15 -1.17 -1.16 -1.12 10 0.38 0.38 0.38 0.38 0.38 0.38 12 1.90 1.90 1.90 1.90 1.90 1.90 16 1.38 1.38 1.38 1.38 1.38 1.38 20 -0.45 -0.45 -0.45 -0.45 -0.45 -0.45

-0.05 1 3.62 3.71 4.16 4.48 5.12 5.49 2 -0.57 -0.53 -0.15 0.71 1.45 1.87 3 -1.47 -1.48 -1.44 -0.79 -0.24 0.37 4 -3.36 -3.36 -3.32 -3.29 -2.29 -2.37 5 -0.49 -0.49 -0.48 -0.49 -0.48 0.21 6 -0.59 -0.59 -0.59 -0.56 -0.46 -0.37 8 -1.18 -1.18 -1.18 -1.19 -1.18 -1.15

10 0.45 0.45 0.45 0.45 0.44 0.44 12 1.73 1.73 1.73 1.73 1.74 1.74 16 1.35 1.35 1.35 1.34 1.35 1.35 20 -0.45 -0.45 -0.45 -0.45 -0.45 -0.45

510 J. Smith, S. Yadav I International Journal of Forecasting 10 (1994) .507-514

method of generating the series is that the series y is formed without dependence on startup values. The number of replications undertaken is N = 10,000.

3. Results

For each replication the first 200 observations of the series Ay, are used to estimate AR models (up to an AR(5)) and these models are then used to produce forecasts over the remaining 50 observations, denoted A?,,,, k = 1, 2, . . . JO. Forecast mean square errors (FMSE) are then calculated as V*(k) = E(Ay,+, - A$,+,)2. Ray (1993) shows how to calculate the theoretical FMSE for ARFIMA models of the form x, = Ay, = (1 - L))&(, and this is denoted V(k). The percentage increase in the FMSE from the cor- rectly specified ARFIMA model over the AR(p) process is calculated as

P(k) = 100 x V*(k) - V(k)

V(k)

Table 1 reports the results of P(k) for c?- 0.3, - 0.2, - 0.1 and - 0.05, and shows that there is little to be gained in specifying the correct ARFIMA model, this is more evidently illus- trated in Fig. 1, which plots the P(k) values for k=l, 2,... ,lO, for the AR(O), AR(l), AR(3) and AR(5) models. The maximum potential gain

from fitting the correct ARFIMA model com- pared to the mis-specified ARIMA model is around 15%, which occurs for d = - 0.3 with all the AR models at the one-step-ahead forecast horizon. However, such large potential losses from fitting an incorrectly specified model dis- appears at longer forecast horizons, such that at a horizon of three steps ahead there is no apparent forecast gain from fitting the correct ARFIMA model over any of the AR processes, including the no-change AR(O) model. At all forecast horizons the order of the AR model appears to make little difference to forecasting performance. The poor performance of the high- er AR models compared to the no-change AR(O) model is surprising given the significance of the coefficient estimates in the higher order AR models across the 10,000 replications. Table 2 reports the average coefficient estimates for the AR(5) model and the theoretical coefficients obtained from approximating the ARFIMA model by an infinite order AR process for all values of 2. For 2 = - 0.3, c#+ and c$~ are highly significant, while for c?- 0.2, only r& is signifi- cant. For d” = - 0.1 and - 0.05 no AR coeffi- cients are significantly different from zero. For negative values of (i closer to zero the potential loss from fitting an incorrectly specified AR model is even smaller, with little perceptible difference between the FMSE from the AR model and the theoretical ARFIMA model, for

Table 2

Coefficient estimates for AR(5) model applied to (1 - L)d Ay, = E,

AR d” = -0.3 ci = -0.2 2 = -0.1 Li: = -0.05 d=o.os d=O.l d = 0.2 (? = 0.3

coefficients

4, -0.284 -0.194 -0.101 -0.053 0.043 0.093 0.192 0.292

(-0.300) (-0.200) (-0.100) (-0.050) (0.050) (0.100) (0.200) (0.300) 42 -0.179 -0.116 -0.060 -0.034 0.013 0.034 0.069 0.096

(-0.195) (-0.120) (-0.055) (-0.026) (0.024) (0.045) (0.080) (0.105) +, -0.122 -0.079 -0.040 -0.023 0.008 0.021 0.043 0.056

(-0.150) (-0.088) (-0.039) (-0.018) (0.015) (0.029) (0.048) (0.060) 4b, -0.092 -0.063 -0.036 -0.023 -0.001 0.009 0.024 0.034

(-0.123) (-0.070) (-0.030) (-0.014) (0.011) (0.021) (0.034) (0.040) A -0.059 -0.041 -0.024 -0.015 0.000 0.010 0.025 0.036

(-0.106) (-0.059) (-0.024) (0.011) (0.009) (0.016) (0.026) (0.030)

Standard errors for the coefficient estimates are approximately 0.073. The figures in parentheses are the theoretical coefficients

from fitting an infinite order AR model to each of the fractional series.

J. Smith, S. Yadav I bmvnational Journal of Forecasting 10 (lY94) 507-214 511

d-Is-O.2

step norizon

-w- AR(O) + AR(i) --x- AR(3) 8 AFl(5)

Fig. 1.

example, for 2 = - 0.1, the maximum potential loss is 5.7% at a one-step-ahead horizon for the highly over-parameterised AR(5) model, while for the AR(O) model the loss is only 4.8%.

The percentage difference in the FMSE, P(k),

results from forecasting d > 0.0 are reported in Table 3 and Fig. 2 graphs some of these P(k)

values for different AR models. Both Table 3 and the Fig. 2 show that there are potentially greater losses from incorrectly fitting an AR model, compared with the results of Table 1, at all forecast horizons. For example, for J= 0.3 the theoretical ARFIMA model forecasts 33% better than the AR(O) model at a one-step-ahead forecast horizon and 10% (6%) at a lo- (20)- step-ahead forecast horizon. Also evident from Table 3 is the superior performance of the higher order AR models compared to the AR(O) model, up to a forecast horizon of five steps ahead. For example, the AR(5) has a potential loss of 7.6%, 8.0% and 6.2% at the one-, lo- and_ 20-step- ahead horizons. However, only for d = 0.3 is

there any monotonic forecast gain in specifying higher order AR processes.

4. Concluding remarks

The major point to arise from this work is that over-differencing a series (due to the low power of unit root tests indicating a series is Z(l), when in fact it may be truly Z(d), 0.7<d < 1.0) and fitting an AR model, will produce a loss in forecasting performance at a horizon of one step ahead, with only a limited loss thereafter. By contrast, under-differencing a series is more costly with larger potential losses from fitting a mis-specified AR model at all forecast horizons.

However, it must be borne in mind that the results presented here exaggerate the forecast loss from AR models as we have compared estimated AR models with the true known ARFIMA model. Estimation of the ARFIMA model could make the forecasting comparison

512

Table 3

Forecast performance of AR models applied to (1 - L)” AY, = F,, 2 > 0.0

J. Smith, S. Yadav I International Journal of Forecasting 10 (1994) 507-514

Step horizon

AR(O) AR(l) AR(2) AR(3) AR(4) AR(5)

0.05 1 2 3 4

10 12 16 20

2 3 4

6 8

10 12 16 20

0.2

2 3

5 6 8

10 12 16 20

0.3

2 3 4

6 8

10 12 16 20

3.35 3.34 3.93 4.32 5.00 5.33 -0.44 -0.51 -0.19 0.66 1.36 1.88 -1.20 -1.21 -1.20 -0.66 0.00 0.54

-3.70 -3.70 -3.69 -3.66 -3.30 -2.75 -0.59 -0.59 -0.59 -0.59 -0.56 0.06 -0.40 -0.40 -0.39 -0.38 -0.30 -0.25 -1.09 - 1.09 -1.09 -1.09 -1.08 -1.09

0.63 0.63 0.63 0.63 0.62 0.60 1.41 1.41 1.41 1.41 1.42 1.42 1.29 1.29 1.29 1.29 1.29 1.29

-0.42 -0.42 -0.42 -0.42 -0.42 -0.42

4.65 3.67 4.14 4.47 5.12 5.41 0.33 0.10 0.15 0.90 1.55 2.10

-0.56 -0.61 -0.66 -0.28 0.38 0.85 -3.46 -3.46 -3.50 -3.47 -3.18 -2.67 -0.38 -0.38 -0.39 -0.40 -0.35 -0.20 -0.04 -0.04 -0.04 -0.04 0.03 0.07 -0.81 -0.81 -0.81 -0.81 -0.80 -0.83

0.86 0.86 0.86 0.85 0.84 0.82 1.38 1.38 1.38 1.38 1.38 1.39 1.35 1.35 1.35 1.35 1.35 1.35

-0.31 -0.31 -0.31 -0.31 -0.31 -0.31

12.04 5.64 5.36 5.39 5.84 5.98 5.05 3.26 2.00 2.26 2.72 3.23 3.26 2.75 1.92 1.59 2.04 2.32

-0.76 -0.86 -1.36 -1.66 -1.72 -1.43 1.78 1.75 1.53 1.31 1.22 1.48 2.32 2.31 2.21 2.03 1.98 1.91 1.33 1.33 1.29 1.20 1.11 0.99 2.51 2.51 2.50 2.44 2.35 2.24 2.43 2.43 2.43 2.42 2.40 2.36 2.42 2.42 2.42 2.41 2.40 2.38 0.78 0.78 0.78 0.78 0.77 0.75

33.18 10.36 8.29 7.63 7.67 7.57 20.08 11.29 6.80 5.87 5.90 6.11 15.87 12.38 8.45 6.43 6.25 6.10 10.09 8.81 5.90 4.10 3.11 2.81 11.08 10.59 8.78 7.30 6.45 6.01 11.47 11.26 10.09 8.84 8.07 7.43 10.07 10.03 9.51 8.65 7.96 7.30 9.86 9.85 9.62 9.10 8.54 7.96 8.97 8.97 8.88 8.64 8.30 7.92 8.49 8.49 8.47 8.37 8.20 7.95 6.48 6.48 6.48 6.43 6.34 6.16

J. Smith, S. Yadav I International Journal of Forecasting IO (1994) W7-514 513

d-1=0.1 d-1=0.05

Step Horizon

-m- AR(O) + AR(l) ++ AR(3) -E- AR@)

Fig. 2.

Diebold, F.X., 1989, Random Walks Versus Fractional Inte- gration: Power Comparisons of Scalar and Joint Tests of the Variance-Time Function, in: B. Raj (ed.), Advances in Econometrics (Kluwer Academic Publishers).

Diebold, F.X. and G.D. Rudebusch, 1989, Long memory and persistence in aggregate output, Journal of Monetary Economics, 24, 189-209.

Diebold, F.X. and G.D. Rudebusch, 1991, On the power of Dickey-Fuller tests against fractional alternatives. Econ- omics Letters, 35, 155-160.

Granger, C.W.J., 1966, The typical shape of an economic variable, Econometrica, 34, 150-161.

Granger, C.W.J. and R. Joyeux, 1980, An introduction to long-memory time series models and fractional integration, Journal of Time Series Analysis. 1, 15-29.

Lo, A.W. and A.C. MacKinlay, 1989, The size and power of the variance ratio test in finite samples: a Monte Carlo

investigation, Journal of Econometrics, 40, 203-238. Nelson, C.R. and C.I. Plosser, 1982, Trends and random

walks in macroeconomic times series: some evidence and

implications, Journal of Monetary Economics, 10, 139-

162.

Ray, B.K., 1993, Modeling long-memory processes for

optimal long-range prediction, Journal of Time Series Analysis, 14, 511-525.

Smith, J. and S. Yadav, 1994, The size and power of unit root

tests against fractional alternatives: a Monte Carlo in- vestigation, Warwick Economic Research Papers No. 419.

even less attractive in favour of the ARFIMA model due to the poorly estimated integration parameter. This problem could potentially be accentuated due to the bias in the estimation of the fractional integration parameter d in the presence of AR or MA components as demon- strated by Agiakloglou et al. (1993)’

Acknowledgements

The authors would like to thank the editor and two anonymous referees for helpful comments and suggestions.

References

Agiakloglou, C., P. Newbold and M. Wohar, 1993, Bias in an

estimator of the fractional difference parameter, Journal of Time Series Analysis, 14, 235-246.

2 However, Ray (1993) actually found that estimating the parameter d improved the forecast performance of the AR models.

514 .I. Smith, S. Yadav I International Journal of Forecasting 10 (1994) 507-514

Biographies: Jeremy SMITH: He is a lecturer in the Depart-

ment of Economics at the University of Warwick. He

received his first degree in Economics from the University of

Manchester and obtained his PhD in Econometrics from the

Australian National University. His interests lie in the areas

of unit root analysis, fractional integration, cointegration and

time series forecasting.

Sanjay YADAV: he is a research fellow in the Financial

Options Research Centre at the University of Warwick. He

received his first degree in Economics from the LSE and his

MPhil in Economics from the University of Warwick. His

research is mainly concerned with the econometric aspects of

financial modelling.