an improvement of the gph estimator

Economics Letters 77 (2002) 137–146www.elsevier.com/ locate/econbase

A n improvement of the GPH estimator

*Jonas Andersson

Department of Information Science Division of Statistics, Uppsala University, Box 513, SE 751 20 Uppsala, Sweden

Received 10 November 2001; accepted 28 March 2002

Abstract

In this paper three semi-parametric estimators of long memory are considered. These are the commonly usedestimator of Geweke and Porter-Hudak [J. Time Series Anal. 4 (1983) 221], a modification of it and a frequency

¨domain maximum likelihood (FDML) estimator, originally proposed by Kunsch [Proceedings of the First WorldCongress 1 (1987) 67] and later thoroughly investigated by Robinson [Ann. Stat. 23(5) (1995) 1630]. Theresults show that the modified GPH estimator and the FDML estimator both have MSEs smaller than theoriginal GPH estimator, with the FDML’s estimator being the smallest. 2002 Elsevier Science B.V. All rights reserved.

Keywords: GPH estimator; Semi-parametric estimators

JEL classification: C22

1 . Introduction

The GPH estimator, see Geweke and Porter-Hudak (1983), is probably the most used estimator ofthe long memory parameter. It is based on the ordinates of the lowest frequencies of the periodogramand therefore no specification of the spectral density for high frequencies is necessary. Robustnessagainst model misspecification is, as for all other semi- or non-parametric estimators, an advantageover more parametrized models but the price for this is always a larger variance. However, it is shownin this paper that a modification of the GPH estimator leads to a decreased variance and mean squareerror.

In the next section, the definition of the long memory property and some stationary processes thatsatisfy it are presented. In Section 3, a simple improvement of the GPH estimator is suggested andanother semiparametric method, based on the spectral representation of the log likelihood, is

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138 J. Andersson / Economics Letters 77 (2002) 137–146

introduced. The estimators are then, in Section 4, investigated by means of a Monte Carlo study wherethe bias, mean squared errors and coverage probabilities of asymptotic confidence intervals arecompared. Section 5 summarizes the results.

2 . Long memory models

A stationary long memory process is defined in the following way.

Definition 1. x is a stationary long memory process if there exists constantsd [ 21/2, 1/2 ands dh jtc [ 0, ` such that its spectral densityh v satisfies:s ds d

h vs d]]lim 5 1.22d

1v→0 cv

Two processes that satisfy this condition are the general fractional noise (GFN) process (Gewekeand Porter-Hudak, 1983) and the autoregressive fractionally integrated moving average (ARFIMA)process (Granger and Joyeux, 1980; Hosking, 1981). The GFN is defined as a process with spectraldensity

h v 5 h v, d h v (1)s d s ds dGFN FN 1

where

2 22(d11)h (v, d)5s 2p G 2d sin (p(d 11)) (2)s d s dFN

`

2 22(d11)34 sin (v /2) O n 1v /2p (3)u un52`

h v, d is the spectral density of a simple fractional noise processy , defined bys d h jFN t

y 5 z 2 z (4)t t t21

t

dz 5E t 2 s dB s , (5)s d s dt

2`

whereB s is a Brownian motion andh v is a positive continuous function bounded above and aways d s d1

from zero on the interval2p, p .f gThe ARFIMA(p,d,q) model, which is a natural extension of the ARIMA(p,d,q) model has the

spectral density

h v 5 h v, d h v (6)s d s ds dARFIMA SI 2

where

2iv 2u us du e]]]h v 5 , (7)s d2 2iv 2u s duf e

J. Andersson / Economics Letters 77 (2002) 137–146 139

and

2sh

]]]]]h v, d 5 (8)s dSI 2iv 2du u2p 12 e

h v, d is the spectral density of a simple integrated processs dSI

d12 L x 5´ (9)s d t t

and ´ is a strict white noise process.h jtGeweke and Porter-Hudak (1983) showed that the function

f v 5 h v, d /h v, d (10)s d s d s dFN SI

is bounded above and below by positive numbers on2p, p which means that if a process is a GFNf gprocess it is also an ARFIMA process and vice versa. The difference between the models is thus just amatter of parametrization of the functionsh v and h v .s d s d1 2

A third type of long memory model are the ones that are specified by the direct use of the definitionof the long memory property. In the neighborhood ofv the spectral density is supposed to be

22dreasonably good approximated bycv . Robinson (1994) uses a slight generalization of theu udefinition by, instead of using the constantc, specifying a slowly varying function at infinity

h vs d]]]]lim 5 1. (11)

1 1v→0 22dS]DL vv

In order to estimate the parameters he uses the spectral distribution functionH v which satisfiess d

122d H vs d s d]]]]lim 5 1. (12)

1 1v→0 122dS]DL vv

Note that the parameterd in all these processes represents the same property, namely the slope of thespectral density in the neighborhood ofv 5 0. The estimators in the next section are all designed toestimate this parameter, regardless of which of the above processes the sample is taken from.

3 . The estimators

In this section, a review of the GPH estimator and an introduction of two new estimators are made.All three methods rely on the periodogram as an estimator of the spectral density. A sample ofNobservationsx , x , . . . , x is assumed to be available.1 2 N


3 .1. The GPH estimator

This estimator, first presented by Geweke and Porter-Hudak (1983), is based on the ARFIMAmodel but can also be formulated by the use of the definition of long memory, as in Beran (1994).Taking the logarithm of the spectral density (6) and adding the logarithm of the periodogram

N1]]I v 5 O x exp 2 iv t (13)s d s dk t k2pN t51

and ln h 0 to both sides of the equation, we gets d2

2s h 0s d2 2ivkS D]]] u uln I v 5 ln 2 2d ln 12 e (14)s dk 2p

h v I vs d s d2 k k]] ]]]]1 ln 1 ln (15)S D S Dh 0 h vs d s d2 ARFIMA k

wherev 5 2pk /n andk goes from 1 tom which is the number of ordinates to be used. The term lnk1h v /h 0 will be negligible asv → 0 . We now write the equations d s ds d2 2

y 5 e 1 dx 1j (16)k k k

where

2s h 0 I vs d s d2 k2ivkS D]]] ]]]]u uy 5 ln I v , e 5 ln , x 5 22ln 12 e andj 5 ln .s d S Dk k k k2p h vs dARFIMA k

The parameters in the regression equation are estimated by ordinary least squares estimation where thefirst two moments of the error term are (asymptotically) known, the mean is2C (the Euler number,

2which is approximately 0.577) and the variance isp /6. The mean ofj will be absorbed in thek

estimate of the intercept.The choice ofm is crucial because if too many frequencies are used, the estimator will be sensitive

to short term memory, and if just a few frequencies are used, the variance of the estimator will belarge. However, because the purpose of this paper is to compare the GPH estimator with othersemi-parametric estimators where an equivalent choice of a tuning parameter is involved, I will use

]Œthe value ofm, often used in the literature,m 5 N.Strictly speaking, the error term in the regression (16) does not fulfil the conditions usually

required, i.e. that they are independent and identically distributed, not even asymptotically (e.g.Hurvich and Beltrao, 1993) ifd ± 0. The distributional properties used in this section is therefore justapplicable whend 5 0. In Section 4 it will be studied, by means of a Monte Carlo study, how well theestimators work whend ±0.

3 .2. A modification of the GPH estimator

A fact not used in the GPH estimator is that we know more about the distribution ofj than just itsk

first two moments. It is, asymptotically, Gumbel distributed. Consequently, we can estimate the


regression by a maximum likelihood approach instead of ordinary least squares. The log likelihood tobe maximized is

m

ln L e, d; y 5O y 2 e 2 dx 2exp y 2 e 2 dx . (17)s d s s dd1 k k k kk51

The benefits of maximum likelihood estimators, such as asymptotic properties and construction oflikelihood ratio tests, can now be used.

A similar approach, which makes it easier in applied work to extend the model towards moreparametric specifications, is to match the periodogram with the spectral density rather than theirlogarithms. This is a method sometimes used for parametrically specified time series models and wasintroduced by Whittle (1953). For the case of semi-parametric long memory models the estimator was

¨suggested by Kunsch (1987) and a thorough investigation of the statistical properties was done inRobinson (1995). The function to be maximized,

m I vs dk]]ln L c, d; y 5 2O ln h v 1 , (18)s d s dS D1 k h vs dkk51

is in fact an approximation to the log likelihood of a linear Gaussian time series model ifm 5 N /2 . Af gdifference here, however, is that a small number of ordinates of the smallest frequencies are used. Thespectral density used is the one specified in the definition of long memory in Section 2. The estimatorsobtained by the maximization of (17) and (18) will henceforth be called the modified GPH estimatorand the frequency domain maximum likelihood (FDML) estimator, respectively.

Remark. Another semiparametric estimator, not investigated here, is presented by Robinson (1994)and is based on the averaged periodogram.

4 . Monte Carlo study

The reason for using a semiparametric long memory model is the aversion to specify the spectraldensity for high frequencies. Therefore it is important that a semiparametric estimator is robust toshort term memory. In this paper the short term memory is represented by an AR(1) term of theprocess. The long memory parameterd was set to20.4, 20.2, 0.2 and 0.4 and the autoregressiveparameterf was set to 0, 0.4 and 0.8. The sample sizesN were 50, 100, 1000 and 2000, the numberof replicates 1000 and the ARFIMA processes were generated by the method in Hosking (1984). The

1results are presented in Tables A.1–A.8 in Appendix A.The Monte Carlo variability can be considered in the following way. If the true coverage

probability is 95%, the number of replicates not covered by the calculated confidence intervals will bebinomial distributed Bin(1000; 0.05). Using a normal approximation we can see that the interval (36;64) will contain the number of replicates inside the confidence intervals in 95% of the times if werepeat the study many times.

1The computations were performed on the DEC Alphaserver 8200 cluster at the Department of Scientific Computing,Uppsala University.


Generally, the FDML estimator worked best followed by the modified GPH and the original GPHestimators, both with respect to bias and MSE. Regarding the coverage probabilities of the 95%asymptotic confidence intervals, the three methods gave approximately the same results. For theparameter values and sample sizes studied, the most striking observation was that the largestdifference in the performance of the estimators occurred for the largest sample size. This can beexplained by the fact that the two new estimators are based on the entireasymptotic distribution of theperiodogram while the original GPH estimator only uses the first two moments. For small samples thesuperiority of the new estimators is therefore not obvious in the same way.

5 . Conclusions

A modification of the GPH estimator has been proposed and compared with the original GPHestimator and another semiparametric estimator, here called the FDML estimator. The modified GPHestimator and the FDML estimator were shown to have smaller mean squared errors than the originalGPH estimator, the FDML estimator slightly better than the modified GPH estimator. The relativemean squared errors against the GPH estimator were 55–86% for the FDML estimator and 62–101%for the modified GPH estimator, decreasing as the sample size increased. The bias due to short termmemory was slightly smaller for the new estimators than for the original GPH estimator.

A cknowledgements

˚I would like to thank Professor Paul Newbold, Docent Anders Agren, Dr. Robert Kunst and DocentJohan Lyhagen for helpful comments.

A ppendix A

Simulation results

Table A.1. The bias and the relative (to the GPH estimator, in %) meansquared errors of the estimators

n f d GPH GPH mod. FDMLBias, MSE Bias, Rel. MSE (%) Bias, Rel. MSE (%)

50 0 20.4 0.051, 0.142 20.008, 77.2 20.009, 78.920.2 0.020, 0.142 20.030, 83.6 20.027, 76.80.2 0.008, 0.145 20.027, 85.4 20.028, 76.80.4 0.008, 0.142 20.026, 85.0 20.036, 75.9


0.4 20.4 0.187, 0.180 0.134, 71.7 0.138, 75.020.2 0.156, 0.166 0.123, 80.8 0.125, 76.30.2 0.122, 0.153 0.095, 85.1 0.088, 75.80.4 0.147, 0.171 0.116, 77.1 0.086, 73.9

0.8 20.4 0.625, 0.537 0.593, 86.0 0.587, 87.220.2 0.655, 0.573 0.623, 88.4 0.611, 85.80.2 0.595, 0.499 0.581, 91.2 0.542, 83.80.4 0.540, 0.440 0.571, 101.0 0.505, 85.0

Table A.2. The percentages of the replicates for which the true parametervalue was outside the obtained asymptotic 95% confidence interval

n f d GPH GPH FDMLmod.

50 0 20.4 4.2 5.3 4.520.2 4.8 5.5 4.80.2 4.2 7.2 6.50.4 4.6 6.7 5.4

0.4 20.4 7.4 6.6 7.020.2 4.8 8.7 8.60.2 4.6 4.9 4.80.4 7.1 6.6 5.9

0.8 20.4 35.5 44.9 46.020.2 39.9 48.3 49.00.2 33.0 42.9 39.90.4 28.0 41.7 36.9


n f d GPH GPH mod. FDMLBias, MSE Bias, Rel. MSE Bias, Rel. MSE

100 0 20.4 0.045, 0.091 20.008, 73.5 20.009, 68.220.2 0.019, 0.096 20.018, 72.9 20.023, 68.10.2 20.002, 0.092 20.027, 71.7 20.033, 66.50.4 0.019, 0.086 20.009, 75.2 20.012, 58.1

0.4 20.4 0.107, 0.086 0.076, 70.8 0.076, 67.320.2 0.080, 0.086 0.053, 73.7 0.052, 69.10.2 0.090, 0.087 0.065, 78.7 0.060, 74.90.4 0.074, 0.093 0.053, 72.9 0.028, 55.3


0.8 20.4 0.480, 0.316 0.473, 92.8 0.462, 86.420.2 0.469, 0.307 0.454, 89.0 0.428, 77.00.2 0.444, 0.284 0.440, 92.9 0.393, 75.30.4 0.435, 0.281 0.441, 94.2 0.313, 64.0



100 0 20.4 5.6 6.2 5.420.2 6.2 6.7 5.90.2 5.4 5.5 4.80.4 4.7 5.7 3.9

0.4 20.4 4.4 3.6 3.220.2 4.5 4.7 4.30.2 6.0 6.3 5.60.4 6.0 6.7 4.1

0.8 20.4 38.1 45.5 44.320.2 36.1 43.5 38.00.2 33.7 40.5 30.00.4 34.1 41.7 27.8



1000 0 20.4 0.025, 0.013 0.008, 64.9 0.004, 59.320.2 0.007, 0.013 20.005, 65.5 20.008, 62.00.2 0.000, 0.013 20.011, 65.7 20.014, 62.60.4 0.003, 0.013 20.003, 66.9 20.004, 66.6

0.4 20.4 0.034, 0.013 0.015, 64.5 0.015, 61.320.2 0.009, 0.013 20.005, 65.3 20.006, 64.30.2 0.007, 0.012 20.006, 69.2 20.007, 66.70.4 0.017, 0.013 0.006, 63.7 0.005, 63.2

0.8 20.4 0.094, 0.020 0.084, 71.6 0.084, 71.620.2 0.091, 0.019 0.082, 70.6 0.082, 70.60.2 0.086, 0.018 0.080, 73.2 0.078, 70.90.4 0.104, 0.020 0.089, 72.6 0.084, 67.6




1000 0 20.4 6.2 7.7 6.820.2 5.4 5.5 5.10.2 5.2 5.4 4.80.4 5.7 6.0 6.2

0.4 20.4 5.5 5.5 4.920.2 5.2 5.9 5.80.2 3.7 4.8 4.20.4 5.7 5.7 5.7

0.8 20.4 10.5 10.5 10.720.2 9.1 8.5 8.50.2 8.0 9.6 9.00.4 11.3 11.7 10.7



2000 0 20.4 0.023, 0.012 0.004, 65.3 0.002, 61.420.2 0.002, 0.012 20.009, 65.3 20.011, 63.20.2 20.002, 0.012 20.009, 66.6 20.010, 64.20.4 0.003, 0.013 0.004, 62.5 20.005, 62.5

0.4 20.4 0.015, 0.013 0.004, 65.3 0.004, 64.920.2 0.000, 0.014 20.007, 62.3 20.007, 61.30.2 0.004, 0.012 20.004, 63.2 20.003, 61.70.4 0.002, 0.013 20.005, 64.5 20.005, 64.2

0.8 20.4 0.058, 0.016 0.048, 66.9 0.048, 66.920.2 0.054, 0.015 0.043, 67.2 0.043, 67.20.2 0.048, 0.015 0.040, 67.1 0.040, 66.10.4 0.054, 0.016 0.046, 68.2 0.042, 64.4




2000 0 20.4 5.2 6.4 5.720.2 4.7 5.2 5.10.2 4.6 5.3 5.10.4 5.0 5.6 5.8

0.4 20.4 5.5 6.2 6.220.2 5.6 6.3 6.20.2 4.7 5.3 4.90.4 5.8 4.9 5.0

0.8 20.4 7.4 8.5 8.520.2 6.8 7.1 7.10.2 8.1 7.6 7.50.4 7.6 8.1 7.8

R eferences

Beran, J., 1994. Statistics for Long Memory Processes. Chapman and Hall, New York.Geweke, J.F., Porter-Hudak, S., 1983. The estimation and application of long memory time series models. Journal of Time

Series Analysis 4, 221–238.Granger, C.W.J., Joyeux, R., 1980. An introduction to long memory models and fractional integration. Journal of Time Series

Analysis 1, 15–29.Hosking, J.R.M., 1981. Fractional differencing. Biometrica 68, 165–176.Hosking, J.R.M., 1984. Modeling persistence in hydrological time series using fractional differencing. Water Resources

Research 20, 1898–1908.Hurvich, C.M., Beltrao, K.I., 1993. Asymptotics for the low-frequency ordinates of the periodogram of a long-memory time

series. Journal of Time Series 14, 455–472.¨Kunsch, H.R., 1987. Statistical aspects of self-similar processes. In: Prohorov, Y., Sazanov, V.V. (Eds.). Proceedings of theFirst World Congress, Vol. 1. VNU Science Press, Utrecht, pp. 67–74.

Robinson, P.M., 1994. Semiparametric analysis of long memory time series. The Annals of Statistics 22 (1), 519–539.Robinson, P.M., 1995. Gaussian semiparametric estimation of long range dependence. The Annals of Statistics 23 (5),

1630–1661.Whittle, P., 1953. Estimation and information in stationary time series. Arkiv Math 2, 423–432.

an improvement of the gph estimator

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