Improved Estimation of the Memory Parameter

In this paper, it is proposed to estimate the memory 
parameter of a potentially long-range dependent time series by applying 
goodness-of-fit tests to the cumulative normalized periodogram in the 
neighborhood of frequency zero. The results of an extensive simulation study 
show that this new estimator performs well compared to conventional 
frequency-domain estimators which are based on the Whittle likelihood or are 
obtained from the popular log periodogram estimator by trimming, smoothing, and 
utilizing non-Fourier frequencies, respectively. In an empirical investigation 
of log absolute daily index returns, we find evidence of long-range dependence 
with values of the memory parameter in the range between 0.2 and 0.3 both in 
developed and developing stock markets. There are no indications of long-range 
dependence in the case of the original index returns.


Introduction
In various fields, there are numerous reports of varying reliability claiming to have found empirical evidence of long memory, including hydrology [1] [2] [3] [4] [5], meteorology [6] [7] [8] [9] [10], geophysics [11] [12] [13], psychology [14] [15], economics [16] [17] [18], and finance [19]- [28]. For example, Hurst [1] estimated an exponent, which measures the degree of long-range dependence, for river levels and many other geophysical time series and obtained in most cases values that are much greater than the value 0.5 characteristic for short-range dependence. Applying a new frequency-domain test, which is more robust against transitory effects than conventional tests, to an annual time series of global surface air temperature, Reschenhofer [10] found further evidence of nonstationarity (global warming). Kiss et al. [13] detected long-range correla-Theoretical Economics Letters tions of extrapolar total ozone. A Fourier analysis performed by Aks, and Sprott, [14] on the time series of reversals in an psychological experiment on the human visual system showed evidence of pink noise, which is characterized by a spectral density that is inversely proportional to the frequency. Examining the persistence of various annual and quarterly measures of aggregate economic activity with fractionally integrated ARMA (ARFIMA) models, Diebold and Rudebusch [16] obtained estimates of the memory parameter d, which is related to Hurst's exponent H by , mostly in the range between 0.5 and 0.9, which indicates that macroeconomic shocks are persistent. Finally, Cajueiro and Tabak [21] observed decreasing estimates of H in emerging stock markets and interpreted their findings as a tendency towards market efficiency over time.
Long memory of a discrete-time stationary process can be characterized by a slowly decaying autocorrelation function satisfying where 0 0.5 d < < and 0 C > . The memory parameter d, which is also called fractional differencing parameter, measures the degree of long-range dependence. A simple example of a process, for which property (1) holds, is fractionally integrated white noise. It satisfies the difference equation where L is the lag operator and t u is white noise with mean zero and variance 2 σ [29] and its autocorrelation function is given by ( ) ( ) ( ) ( [30], pp. 466-467), where Γ denotes the gamma function. Granger and Joyeux [31] and Hosking [32] introduced the more general class of autoregressive fractionally integrated moving average (ARFIMA) processes by extending (2) to ( ) ( ) ( ) where the infimum is taken over all estimators ˆn d based on a sample of size n and the supremum is taken over all possible data generating processes. Furthermore, he showed that for every 0 y Y ∈ , (5) holds also "locally", when the supremum is taken over an arbitrarily small L1-neighborhood of 0 y . Finally, he established that confidence intervals for d coincide with the entire parameter space for d with high probability and are therefore uninformative. Thus, drawing inferences about d appears to be a futile exercise unless restrictive assumptions on Y are imposed. Fortunately, there are applications, which do not require very rich classes of data generating processes. For example, in the case of daily return series, the class of ARFIMA processes with small p and 0 q = seems to be adequate. Fittingly, simulation studies carried out to compare different tests and estimators for d focused on ARFIMA processes with Smith et al. [34] carried out Monte Carlo simulations, which also included the problematic case where both p and q are greater than zero, in order to compare the performance of the maximum likelihood (ML) procedure used by Dahlhaus [42] and Sowell [43], which estimates the memory parameter d simultaneously with the AR parameters j φ and the MA parameters j θ , with that of two semiparametric estimation methods, which yield only estimates of d. Both of the latter methods are frequency-domain methods. The first frequency-domain estimator is obtained from Geweke and Porter-Hudak's [44] log periodogram regression by trimming out the contributions from the very lowest frequencies [45] and the second is a variant [46] of Robinson's [47] average periodogram method. The results of their simulations suggest that the ML estimator is superior provided that the order (p,q) of the ARFIMA model is correctly specified. However, the ML estimator will in general be inconsistent if the model is misspecified. In contrast, consistency of the semiparametric estimators was established by Robinson [47] in case of the average periodogram method and by Hurvich et al. [48] in case of the log periodogram regression. Another advantage of the semiparametric estimators is that they are available in closed form and therefore do not require numerical methods.
In their simulation study, Reisen et al. [39] additionally investigated frequency-domain estimators that are based on the smoothed periodogram [49]. The results indicate that smoothing is indeed advantageous but trimming is not. Reschenhofer [50] explored another way to improve the performance of the log periodogram estimator. Including also log periodogram ordinates at non-Fourier frequencies, he achieved a significant decrease in the root mean square error. Again, the omission of the very lowest frequencies had a negative effect. This paper is concerned with the estimation of the memory parameter d. A new frequency-domain estimator is proposed, which is inspired by a test for long-range dependence recently introduced by Mangat and Reschenhofer [40].
In contrast to earlier studies, which evaluated the performance of different estimators mainly in terms of the mean squared error (MSE) or the root-mean-square

Frequency-Domain Estimation of the Memory Parameter
In Subsections 2.1 -2.5, we briefly review various frequency-domain estimators for the memory parameter d, which will serve as benchmarks in the simulations presented in Section 3, before we introduce our new estimator in Subsection 2.6.

Log Periodogram Regression
The spectral density of the ARFIMA process (4) is given by Taking logarithms and adding ( ) Since the ARMA component

Trimming
In the simple case where the observations 1  only the indices are fixed whereas the frequencies move closer to frequency zero as the sample size n increases, which poses a problem particularly for ARFIMA spectral densities because they have either a zero (when d < 0) or a pole (when d > 0) at frequency zero. Indeed, Künsch [51] showed for d > 0 that the asymptotic distribution of ( ) j J ω depends on j. Furthermore, Hurvich and Beltrao [52] and Robinson [47] showed that for both d < 0 and d > 0, the normalized periodogram ordinates ( ) j J ω , 1 j K ≤ ≤ , are asymptotically neither independent nor identically distributed when n → ∞ but the indices j stay fixed (for bounds on the asymptotic bias of the normalized periodogram and the covariance between normalized periodogram ordinates at different frequencies see [53] [54] [55]). However, Künsch [51] showed that the standard asymptotic results still hold for the Fourier frequencies 1 , , → . An obvious modification of the log periodogram regression is therefore to trim the first H Fourier frequencies [45]. We denote the resulting estimator by ˆt r GPH d .

Smoothing
Hassler [56], Peiris and Court [57], and Reisen [49] proposed to replace the periodogram ordinates ( ) of ( ) 6 , , The truncation point m determines the smoothness of the estimate. For consis- tency it is required that m → ∞ and 0 m n → as n → ∞ . When the resulting estimator, which is based on the smoothed periodogram, is compared with the previous estimators, it is important to bear in mind that its performance depends not only on K but in addition also on m. In our simulation study, we will use the tuning parameters α, which determines the number K n α   =   of included periodogram ordinates, and β, which determines the truncation point m n β   =   .

Non-Fourier Frequencies
Reschenhofer [50] modified the log periodogram regression by including non-Fourier frequencies. In this case, the amplitude R of the sinusoid can no longer be estimated by where Â and B are the least squares (LS) estimates obtained by regressing which is only identical to (7)

Whittle Likelihood
The task of carrying out a fair comparison between competing estimators with different numbers of tuning parameters becomes even more difficult when dif-  [25] used different submodels of the ARMA(2,1) model.
While it is quite understandable when we do not pursue rather special approaches like nonparametric estimation based on the adjusted-rescaled-range [1] [59], which involve nonstandard asymptotics [60] [61], it is essential that we include ML estimation in one way or another. The fairest way to do so is to use the frequency-domain likelihood (Whittle likelihood) and focus on the narrow fre- [62]. Assuming that the periodogram ordinates ( ) ( ) 1 , ,  are approximately independent exponential with means ( ) ( ) 1 , , in the neighborhood of frequency zero because of the constancy of ( ) [53]. The estimator obtained by minimization of (17) over a set D of possible values of d is denoted by ˆW d .

Goodness-of-Fit Testing
Mangat and Reschenhofer [40] reduced the problem of testing hypotheses about the memory parameter d to a problem of goodness-of-fit testing. Observing that the random variables

Simulation Study
In this section, the performance of the different estimators for the memory parameter d is evaluated with a simulation study.   Tables 1-6. Table 1 and Table 2 show the mean bias and the RMSE, respectively, for the case n = 100. Analogously, Table  3 & Table 4 and Table 5 & Table 6 show the results for the cases n = 300 and n = 3000, respectively.
The conventional log periodogram estimator serves as the main benchmark.                  narrow-band Whittle estimator ˆW d also perform quite well in terms of the bias. Although the mean squared error is just the sum of the squared bias and the variance and therefore strikes a fair balance between the bias and the variance, it sometimes makes sense to focus largely on only one of the two aspects. While the variance is in our simulation study typically large compared to the squared bias, the relationship is reversed in our empirical study of stock returns (see Section 4), where we perform a rolling analysis (in order to assess the stability of the estimates over time) and put the individual estimates together afterwards. In such a case, it is clearly the bias which matters more because the variance decreases steadily as the sample size increases whereas the bias remains fixed. As far as the bias is concerned, the results of our simulation study show that smoothing does not help. We may therefore expect that particularly the empiri-

Empirical Results
Studying emerging stock markets, Cajueiro and Tabak [21], Hull and McGroarty [65] and Auer [27] observed time-varying estimates of the Hurst exponent H. Batten et al. [25] and Auer [28] took things a step further. Assuming that fractal dynamics does in fact exist in precious metal returns, they explored possible trading strategies that are based on local estimates of H. In contrast, Reschenhofer et al. [66] found no evidence of long-range dependence, neither in stock index returns nor in gold returns. Mangat and Reschenhofer [40] and Reschenhofer and Mangat [41] developed formal statistical tests of hypotheses about d or H and applied them to stock index returns and gold returns. Again, they found no evidence of long-range dependence let alone fractal dynamics. In contrast to conventional tests, which are based on the assumption that both the length of the time series and the number of used periodogram ordinates are large and are therefore unsuitable in case of a rolling analysis, their tests require only a small number of periodogram ordinates. While we may therefore not expect to obtain estimates of the memory parameter that differ significantly from zero in the case of daily stock returns, there is a priori a much better chance of finding evidence of the presence of long memory in volatility. Accordingly, we will analyze not only the (log) returns, which are obtained as the differences of successive log prices, but also at the log absolute returns. Using log absolute returns instead of absolute returns or squared absolute returns for the investigation of volatility has the advantage that we do not have to work with extremely skewed distributions.
In our empirical study, we look for indications of long-range dependence both in developed and developing stock markets. For this purpose, six major world indices, two from America, Europe, and Asia, respectively, were downloaded  Figure 1 shows that the estimates obtained from subseries of length 300 are consistently in a very small range around zero.
The discrepancies between the estimates obtained with different estimators on the one hand or with the same estimator for different stock market indices on the other hand are therefore of no significance.

Discussion
In this paper, we have converted the test of Mangat and Reschenhofer [40] into an estimator for the memory parameter which is easy to use and highly competitive. The results of our extensive simulation study show that this new estimator performs well both in terms of the RMSE and the bias. Overall, it shows the best performance together with the Whittle estimator. The estimators based on the smoothed periodogram cannot compete when the second tuning parameter β is fixed. Clearly, the possibility to fiddle about with the second tuning parameter β gives these estimators an unfair advantage over their competitors. Choosing an unsuitable value for this parameter can lead to a severe bias, which is confirmed in our empirical investigation of the long-range properties of international daily index returns. Interpreting the empirical findings properly with the help of the results of our simulation study, we conclude that the log absolute returns are long-range dependent with the memory parameter in the range between 0.2 and 0.3 in contrast to the original returns which show no indications of long-range dependence.
In conclusion, the main points of this paper are as follows. We have introduced a simple frequency-domain estimator for the memory parameter, provided evidence of its good performance relative to conventional estimators in terms of bias and RMSE, pointed out some shortcomings of the popular lag window estimators, and used the new estimator successfully to confirm the absence of long memory in stock returns and to corroborate the presence of long memory in volatility.