1. Introduction
After
Mandelbrot (
1971) had discussed the possibility that the strength of the statistical dependence of stock prices decreases very slowly, several researchers investigated this issue empirically. For example,
Greene and Fielitz (
1977) found indications of long-range dependence when they applied a technique called range over standard deviation (R/S) analysis (
Hurst 1951;
Mandelbrot and Wallis 1969;
Mandelbrot 1972,
1975) to daily stock return series. This technique is based on the R/S statistic
, which is defined as the range of all partial sums of a time series of length
n from its mean divided by its standard deviation. For a large class of short-range dependent processes,
converges to a non-degenerate random variable if
. An analogous result with
holds for long-range dependent processes. The parameter
H is called the Hurst coefficient and is used as a measure of long-range dependence. However,
Lo (
1991) pointed out that the results obtained with this technique may be misleading because of the sensitivity of
to short-range dependence (see also
Davis and Harte 1987;
Hauser and Reschenhofer 1995) and proposed, therefore, a
Newey and West (
1987) type modification for the denominator of the R/S statistic, which is appropriate for general forms of short-range dependence. Contrary to the findings of
Greene and Fielitz (
1977) and others, he found no evidence of long-range dependence in daily and monthly index returns once the possible short-range dependence was properly taken care of. A disadvantage of
Lo’s (
1991) modified R/S analysis is its dependence on an important tuning parameter, namely the truncation lag
q, which determines the number of included autocovariances. The general conditions that ensure the consistency of the Newey and West estimator provide little guidance in selecting
q in finite samples. Additionally,
Andrews’s (
1991) data-dependent rule for choosing
q is based on asymptotic arguments.
Long-range dependence can not only be characterized by a Hurst coefficient
but also by a slowly decaying autocorrelation function
or a spectral density
f that is steep in a small neighborhood of frequency zero, i.e.,
and:
respectively. The parameter
d is called a memory parameter (or fractional differencing parameter) and is related to
H by
. It can be estimated by replacing the unknown spectral density
f in (2) by the periodogram (
Geweke and Porter-Hudak 1983) or a more sophisticated estimate of
f (
Hassler 1993;
Peiris and Court 1993;
Reisen 1994), taking the log of both sides, and regressing the log estimate on a deterministic regressor. Robustness against short-range dependence can be achieved by using only the
lowest Fourier frequencies in the regression. A popular choice for the tuning parameter
is 0.5. For the purpose of testing, the asymptotic error variance is used. Applying the log periodogram regression method of
Geweke and Porter-Hudak (
1983) to the daily returns of the 30 components of the Dow Jones Industrials index and several indices,
Barkoulas and Baum (
1996) found no convincing evidence in favor of long-range dependence, which is not surprising in light of the finding of
Mangat and Reschenhofer (
2019) that the test based on the asymptotic error variance has very low power. Unfortunately, using the standard variance formula of the least squares estimator of the slope in a simple linear regression instead of the asymptotic error variance is also problematic because it leads to overrejecting the true null hypothesis (see
Mangat and Reschenhofer 2019).
The negative results of
Lo (
1991) and
Barkoulas and Baum (
1996) are in line with the results obtained by
Cheung and Lai (
1995) with both modified R/S analysis and log periodogram regression for stock return data from eighteen countries and by
Crato (
1994) with fractionally integrated ARMA (ARFIMA) models (
Granger and Joyeux 1980;
Hosking 1981) for stock indices of the G-7 countries. Using not only the log periodogram regression with the asymptotic error variance but additionally also nonparametric techniques such as R/S analysis and modified R/S analysis as well as parametric techniques,
Grau-Carles (
2000) also found little evidence of long-range dependence in index returns but strong evidence of persistence in volatility measured as squared returns and absolute returns, respectively, which corroborates earlier findings of
Crato and de Lima (
1994) and
Lobato and Savin (
1998). In general, results obtained with ARFIMA models must be treated with caution. Firstly, the true model dimension is unknown in practice and reliable inference after automatic model selection is illusory. Secondly,
Pötscher (
2002) has shown that the problem of estimating the memory parameter
d falls into the category of ill-posed estimation problems when the class of data generating processes is too rich. For example,
Grau-Carles (
2000) considered all ARFIMA(p,q) processes with
and
, which is possibly an unnecessarily large class for return series.
While the bulk of empirical research focused on major capital markets,
Barkoulas et al. (
2000) examined an emerging capital market, namely the Greek stock market, with the log periodogram regression and obtained significant estimates of
d in the range between 0.20 and 0.30 for values of the tuning parameter
between 0.5 and 0.6. However, their sample period is relatively short and the sampling frequency is weekly rather than daily. Even less confidence-inspiring are the positive results obtained by
Henry (
2002) with monthly data from several international stock markets. Clearly, methods that have been designed for large samples should not be applied to small and medium samples. Recently, small-sample tests for testing hypotheses about the memory parameter
d have been proposed (
Mangat and Reschenhofer 2019;
Reschenhofer and Mangat 2020). When applied to asset returns, these tests produced negative results throughout.
Cajueiro and Tabak (
2004),
Carbone et al. (
2004),
Batten and Szilagyi (
2007),
Batten et al. (
2008),
Souza et al. (
2008),
Batten et al. (
2013), and
Auer (
2016a,
2016b) observed time-variability of the Hurst exponent in stock returns, currency prices, and the prices of precious metals, respectively. These apparent changes were occasionally interpreted as indications of changing market efficiency or even used for the construction of trading strategies. Although it cannot be ruled out that some erratic estimator for the memory parameter
d catches signals that are useful for trading purposes even when in fact there is no long-range dependence, there still seems to be a need for a more efficient estimator that actually allows to get some information about the true nature of the data generating process.
In general, there is always a trade-off between bias and variance. Estimators for the memory parameter
d that are based on a smooth estimate of the spectral density have typically a smaller variance and a larger bias than those based on the periodogram (
Chen et al. 1994;
Reschenhofer et al. 2020), which is advantageous in situations where the squared bias is small relative to the variance. However, in the case of high-frequency financial data, there are usually gaps between the individual trading sessions, which make it necessary to estimate
d separately for each trading session and compute the final estimate by averaging the individual estimates. Here, the variance decreases with the number of trading sessions but the bias remains fixed; hence, conventional smoothing methods, which achieve a reduction in the variance at the expense of an increase in the bias, are of no use. The goal of this paper is therefore to introduce a new method of smoothing that does not systematically have a negative impact on the bias. This method will be described in detail in the next section.
Section 3 presents the results of an extensive simulation study, which compares the performance of various estimators for the memory parameter in terms of bias, variance, and root-mean-square error (RMSE). Using limit order book data obtained from Lobster,
Section 4 searches for indications of long-range dependence both in the intraday volatility and in the intraday returns.
Section 5 provides a conclusion.
3. Simulations
In this section, we compare the new estimator
(41) for
with
Geweke and Porter-Hudak’s (
1983) estimator
, which is based on the log periodogram regression (5), and the estimators
and
, which are obtained by replacing the periodogram ordinates in (5) by simple moving averages of neighboring periodogram ordinates and lag-window estimates of the form (30) with truncation lags
,
, respectively. In the latter case, the Parzen window is used, which is given by:
With a view to the later application of the estimators to 1-min intraday returns in
Section 4, the sample size
is chosen for our simulation study because there are 390 min in a regular trading session for U.S. stocks, which starts at 9:30 a.m. and ends at 4:00 p.m. The number
K of Fourier frequencies included in the log periodogram regression is defined by setting
,
. For
, the first
Fourier frequencies of the two disjoint subseries of length
are given by
, and for
, the first
Fourier frequencies of the 10 disjoint subseries of length
are given by
. Clearly, we cannot go beyond
because at least two frequencies are required to carry out the log periodogram regression. Additionally, using frequencies outside the interval
is not an option because this would amount to an unfair advantage, particularly when there are no short-term dependencies which have to be taken into account.
With the help of the R-package ‘fracdiff’, 10,000 realizations of length
of ARFIMA(1,
d,0) processes with standard normal innovations and parameter values
and
, respectively, are generated using a burn-in period of 10,000. For each realization, the estimators
,
,
,
,
,
, are employed for the estimation of the memory parameter
d. The competing estimators are compared with respect to bias (
Table 2), variance (
Table 3), and RMSE (
Table 4).
Table 3 shows that
has indeed a smaller variance than
. The variance keeps decreasing as the number of partitions increases from two to 10.
Table 2 shows that this improvement does in general not come at the cost of a greater bias. In contrast, the reduction in the variance achieved in the case of the estimator
by increasing the degree of smoothing from
to
is for
accompanied by a dramatic increase in the bias. Overall, in terms of the RSME, the best results are obtained with
for small values of
d and with
for larger value of
d. However, this is only relevant in the standard case where only a single time series is available. When a large number of time series are examined simultaneously (as in the empirical study of
Section 4), the bias is the decisive factor and the new estimators
are therefore more appropriate than the conventional estimators
.
Since values of
such as 0.5, 0.7, or 0.9 are usually chosen to minimize the MSE for a single sample, we may suspect that the estimator
becomes more competitive in the case of multiple samples when the averaging is taken into account. This can be done by further reducing the degree of smoothing. Unfortunately, there is a limit to what can be achieved by increasing the value of
.
Table 2 shows that large biases are still obtained with the maximum possible value of
, i.e.,
. This is due to the fact that global smoothing inevitably causes local distortions and cutting off higher-order sample autocovariances is not the only source of smoothing. Downweighting the sample autocovariances with the Parzen window also has a strong smoothing effect, even when all sample autocovariances are used.
4. Empirical Results
In this section, we employ the estimators discussed in the previous sections for the search of possible long-range dependencies in intraday returns and absolute intraday returns. For this purpose, the limit order book data from 27 June 2007 to 30 April 2019 (2980 trading days) of the iShares Core S&P 500 ETF (IVV) are downloaded from Lobster (
https://lobsterdata.com). In the process of data cleaning, 27 early-closure days (the day before Independence Day, the day after Thanksgiving, and Christmas Eve) are removed as well as 9 January 2019 because of a large number of missing values. For each of the remaining days, the first mid-quotes (midpoints of the best bid and ask quotes) in each minute and the last mid-quote in the last minute are computed and subsequently used to obtain 1-min log returns. Finally, another three days are omitted because of extreme returns, namely 19 September 2008, 6 May 2010, and 24 August 2015, which leaves 2949 days for our analysis. Estimates are computed for each day, divided by the number of days, and plotted cumulatively; hence, the last values correspond to the averages of the estimates. The validity of these values is reinforced by the striking linearity of the curves. This linearity also implies that the possible long-range dependence is not changing over time; hence, there appears to be no such thing as fractal dynamics.
Figure 1a suggests that
d is close to zero in case of the 1-min log returns. The large negative values obtained with
and
as well as the comparatively inconspicuous values obtained with
can be explained with the help of the results of our simulation study. According to
Table 2, they are indicative for
. In contrast, there is strong evidence of long-range dependence in the volatility (see
Figure 1b). Most estimators suggest that the memory parameter
d is approximately in the range between 0.3 and 0.4. Only the estimator
, which is severely downward biased in case of positive
d (see
Table 2), favors a smaller value.
Visual significance of the differences between certain estimates can be ascertained just by observing the large differences between the slopes of the corresponding lines in
Figure 1 and noting the striking stability of these lines over time. However, we still might want to augment our visual analysis with a formal statistical test. A simple way to accomplish that is to calculate the difference between two estimates separately for each trading day and compare the number of positive differences to the number of negative differences (sign test). Not surprisingly, the resulting p-values are infinitesimal. For example, even in the case of the two neighboring lines corresponding to
and
in
Figure 1b, the
p-value is less than
. It is still less than
when we omit most of the trading days and use only Wednesdays in order to ensure approximate independence of the subsamples. Note that there are
1-min returns between the last 1-min return of some Wednesday and the first 1-min return of the next Wednesday plus five overnight breaks and a whole weekend. Even for a relatively large value of the memory parameter such as
, the autocorrelation of an ARFIMA(0,
d,0) process at lag
is quite small, i.e.,
Finally, in order to check the robustness of our findings against daily and weekly periodic patterns, we repeat the graphical analyses with suitably transformed data. Replacing the 1-min log returns
, by the daily differences
and the weekly differences
, respectively, ensures that any daily or weekly periodic patterns are erased while long-range dependencies remain unaffected.
Figure 1c,e are very similar to
Figure 1a, which shows that the insights gained from
Figure 1a are genuine. Analogously, comparing
Figure 1d,f with
Figure 1b, we see that the same is true for the absolute returns
5. Discussion
In this paper, we have introduced a new estimator for the memory parameter d, which is based on running a log periodogram regression repeatedly for different partitions of the data. In contrast to conventional smoothing methods, which manage to achieve a reduction in the variance at the expense of an increase in the bias, our approach does not systematically have a negative impact on the bias, which makes it particularly useful for applications where the bias is the decisive factor. For example, intraday returns are usually only available during trading hours and estimation must therefore be carried out separately for each trading day. When the individual estimates are eventually combined by averaging, the variance decreases as the sample size increases, but the bias does not change. The results of an extensive simulation study confirm the good performance of the new estimator. It outperforms all of its competitors when both bias and variance are taken into account, but the bias is weighted more heavily.
The importance of results obtained with the help of simulations is due to the fact that reliable inference on the memory parameter
is not possible under general conditions. Some asymptotic results can be obtained under very restrictive conditions though. Unfortunately, convergence is typically very slow (recall the discussion in
Section 2.2). Indeed,
Pötscher (
2002) showed that many common estimation problems in statistics and econometrics, which include the estimation of
, are ill-posed in the sense that the minimax risk is bounded from below by a positive constant independent of
and does, therefore, not converge to zero as
. In particular, he found that for any estimator
for
based on a sample of size
from a Gaussian process with spectral density
:
where
and
is the set of all ARFIMA spectral densities (
), ARFI spectral densities (
), or FIMA spectral densities (
). Furthermore, he showed that for every
, (44) holds also “locally,” when the supremum is taken over an arbitrarily small
-neighborhood of
. Finally, he established that confidence intervals for
coincide with the entire parameter space for
with high probability and are therefore uninformative. Nevertheless, it may be possible to formally derive the statistical properties of our new estimator for a rather narrow class of processes such as low order ARFI processes. However, this is left for future research. The current paper provides just a proof of concept.
In our empirical investigation of high-frequency data of an index ETF, we have applied the competing estimators to 1-min log returns and absolute 1-min log returns separately for each day. The results are quite stable over time and across estimation methods. The few deviations are due to conventional smoothing methods and can easily be explained by the size of their bias as shown in
Table 2. We may, therefore, safely conclude that significant long-range dependencies are present only in the intraday volatility but not in the intraday returns. These findings are genuine and not just due to daily or weekly periodic patterns because similar results are obtained when daily and weekly differences are investigated instead of the original intraday returns.