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This paper offers a multifaceted perspective of the literature on long memory. Although the research on long memory has played an instrumental role in elevating the level of scholarly discourse on market efficiency, the authors believe that the issue of the prevalence of long memory or lack thereof remains unsettled. While long memory models should be in the econometrician’s toolbox, their use should be governed by an initial exploratory analysis of the data being studied and the context of the research questions being addressed. Mere fixation on the presence/absence of long memory without taking due cognisance of other confounding factors would pave way for confirmation bias. Consequently, this paper pinpoints the possible pitfalls and potential trade-offs in modeling long memory in asset prices. While not a comprehensive meta-analysis of the literature on long memory, this paper offers a selective bibliography of prior works on long memory that is geared to nudge researchers to exercise caution and judgement while exploring long memory in asset prices.
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Pitfalls in Long Memory Research
Kunal Sahaa,
, Vinodh Madhavanb, G. R. Chandrashekharc
aIFMR GSB, University of Madras, Chennai, India
bAmrut Mody School of Management, Ahmedabad University, Ahmedabad, India
cIFMR GSB, Krea University, Chennai, India
This paper offers a multifaceted perspective of the literature on long memory. Although the research on long
memory has played an instrumental role in elevating the level of scholarly discourse on market efficiency,
the authors believe that the issue of prevalence of long memory or lack thereof remains unsettled. While
long memory models should be in the econometrician’s toolbox, their use should be governed by an initial
exploratory analysis of the data being studied and the context of the research questions being addressed.
Mere fixation on presence/absence of long memory without taking due cognisance of other confounding
factors would pave way for confirmation bias. Consequently, this paper pinpoints the possible pitfalls and
potential trade-offs in modeling long memory in asset prices. While not a comprehensive meta-analysis of
the literature on long memory, this paper offers a selective bibliography of prior works on long memory
that is geared to nudge researchers to exercise caution and judgement while exploring long memory in asset
Keywords: Long Memory, Efficiency, Measures, Review
JEL Classification: C18, C58, G14
Revision: 1
Declarations of interest: none
1. Introduction
While studies grounded in Long Memory constitute a notable strand of literature disputing market
efficiency, such studies are not devoid of caveats. Long Memory gained traction in scholarly discourse due
to Mandelbrot’s work on asset prices using rescaled range estimation techniques (Mandelbrot & Van Ness,
1968;Mandelbrot & Wallis,1968). A lot of water has flown under the bridge since then. Meanwhile,
the literature on long memory too has become multi-dimensional in nature. While the initial works on
re-examining market efficiency using methodologies that are theoretically grounded in long memory offered
the much-needed contrast to a somewhat homogeneous literature on market efficiency; inferences from such
studies on the prevalence of long memory or lack thereof have not been unequivocal.
Corresponding Author
Email address: (Kunal Saha)
Preprint submitted to Cogent Economics and Finance Jan 30, 2020
Although there are a few review papers that discuss long memory (Baillie,1996;Gu´egan,2005;Lim &
Brooks,2011), literature seems wanting on bringing the various arguments for and against the observation
of long memory together. In this backdrop, the authors believe a snapshot of prevailing literature on long
memory in asset prices, without losing sight of the attendant contexts behind such studies, is the need of
the hour. Such a snapshot would aid researchers to take stock of the various facets of the discourse on long
memory, in a manner that would nudge them to exercise requisite caution before drawing any definitive
inference on long memory in asset prices in their future research endeavours.
While prior studies on long memory have significantly broadened the literature landscape on market
efficiency, a definitive take-away from such literature on long memory that is oblivious to other confounding
factors which can manifest as long memory would be short sighted and self-fulfilling. In short, this is as
much an attempt to sensitize researchers about the pitfalls in research on long memory, as highlighting the
prominence of long memory in the context of revisiting market efficiency.
2. Definition, Measures and Methodologies
2.1. Definition
For a second-order stationary process Xtwith an auto-covariance function γX(k), Xthas
a) Short Memory, if
b) Anti-Persistence, if
γX(k) = 0
c) Long Memory, if
γX(k)→ ∞
Long memory (or persistence) implies that a positive or negative movement is more likely to be followed
by another move in the same direction. On the other hand, for an anti-persistent process, a positive
movement is more likely to be followed by a move in the opposite direction. In other words, a persistent
process is trending whereas an anti-persistent process shows mean reversion.
Beran et al. (2016) provides a detailed review of various definitions of long memory and the conditions
in which these can be used interchangeably. The different measures of long memory are as follows.
2.2. Measures
2.2.1. Hurst Exponent
The most popular measure for long memory is the “Hurst Exponent”(denoted as H). This measure
gained traction owing to Mandelbrot and Wallis’s pioneering work on operational hydrology (Mandelbrot &
Wallis,1968). There are several methodologies that are used to calculate the Hurst Exponent. The classical
rescaled range (R/S) analysis proposed by Hurst (1951) and its subsequent variants, such as Modified R/S
analysis and Rescaled Variance (V/S) analysis, are the most prominent ones.
When 0.5< H < 1, the autocovariances are positive at all lags and the time series process is called
persistent. When 0 < H < 0.5, the autocovariances at all lags are negative and the time series process is
called anti-persistent.
2.2.2. Fractional Order of Integration
Another popular approach to ascertain long memory or lack thereof is to measure the fractional order
of integration (denoted as d) of a time series. This paved the way for ARFIMA-FIGARCH models, which
were designed to explicitly model long memory in the first and second moments (Granger & Joyeux,1980;
Hosking,1981;Baillie et al.,1996).
2.2.3. Fractal Dimension
Gneiting & Schlather (2004) describe the fractal dimension, D, of a surface as a roughness measure with
D[n, n + 1) for a surface in Rnwhere higher values can be interpreted as rougher surfaces. Technically,
Fractal dimension(D) and Hurst exponent(H) are independent of each other. Fractal dimension is a
local property, while Hurst exponent is a global property, which is used to characterize the long-memory
dependence in a time series. For self-affine processes, local properties are reflected in global ones, which
lead to the relationship D+H=n+ 1 between Dand Hfor a self-affine surface in n-dimensional space.
Table 1offers a snapshot of the above stated measures of long memory.
Table 1: Summary of measures of Long Memory
Type of Persistence H d =H0.5D= 2 H
Short Memory 0.5 0 1.5
Anti-persistence (Mean Reversion) (0,0.5) (-0.5,0) (1.5,2)
Long Memory (Persistence) (0.5,1) (0,0.5) (1,1.5)
2.3. Methodologies
Over the years, a number of methodologies have been proposed by researchers for measuring long
memory. While in many cases, long memory in conditional mean and variance are studied independently,
unified approaches to study long memory are also present (Teyssi`ere,1997).
Several popular heuristic methods to measure long memory in the first and second moments include
the Rescaled range (R/S) method, Rescaled variance (V/S) method and Detrended Fluctuation Analysis
(DFA). Prominent semi-parametric estimators of long memory include the Log Periodogram estimator
(Geweke & Porter-Hudak,1983), and Local Whittle estimator. These estimators were defined based on
linear time series models and should be used to measure long memory in the first moment. It was found
that the GPH estimator can be downward biased when used to measure long memory in volatility (Deo
& Hurvich,2001). The Whittle estimator is found to be more robust in measuring the long memory in
volatility (Hurvich & Ray,2003).
In terms of parametric modeling based approaches, Mandelbrot & Van Ness (1968) introduced the
fractional brownian motion. This was a generalisation of the standard brownian motion by incorporating
a self-similar parameter d(0.5,0.5) and provides the most basic framework for studying long memory.
Another important approach of modeling long memory is the ARFIMA - FIGARCH class of models. While
ARFIMA model is used to model long memory in the first moment (return time series), FIGARCH is used
for modeling the long memory in volatility.
The ARFIMA (p, d, q) model, which was introduced by Granger (1980); Granger & Joyeux (1980) and
Hosking (1981), is defined as
φ(L)(1 L)dyt=θ(L)t
where φ(L) and θ(L) are lagged polynomials of orders pand qrespectively, with tbeing white noise. For
the ARFIMA model, the fractional parameter dlies between -0.5 and 0.5. An ARFIMA process depicts
long memory when 0 <d<0.5, anti-persistence for 0.5<d<0, short memory for d= 0, and infinite
memory (random walk) for d= 1.
The incorporation of long memory in GARCH models was introduced by Robinson (1991) and built
upon by Baillie et al. (1996); Ding & Granger (1996) and others. Among these approaches, the more
popular FIGARCH(p, d, q) model (introduced by Baillie et al. (1996)) is defined as
φ(L)(1 L)d2
t=α0+ [1 β(L)]νt
where 0 <d<1, νt=2
thtwith htbeing the conditional variance. In ARFIMA model, the long
memory operator is applied to unconditional mean (µ) of yt, whereas in the case of FIGARCH model, it is
applied to squared errors. However, the FIGARCH model has its own nuances that need to be remembered
during application. The memory parameter of FIGARCH is actually dand increases as d0. This
happens due to the fact that memory parameter acts on the squared errors in FIGARCH. Consequently,
the Hyperbolic GARCH (HYGARCH) model was proposed by Davidson (2004).
Among other popular models on long memory include the Fractional Exponential (FEXP) model by
Beran (1993) which is an extension of the ARFIMA model, long-memory stochastic volatility models
(Breidt et al.,1998;Harvey,1998) and the Multi-Fractal model (MMAR model) by Mandelbrot et al.
The following table 2offers a snapshot of the notable methodologies pertaining to long memory along
with their original sources. These popular long memory methodologies possess theoretical antecedents in
diverse areas.
Table 2: Various Methodologies to measure Long Memory
Sl. No. Methodology Reference
1 Classical Rescaled Range Estimation (R/S Analysis) Hurst (1951); Mandelbrot & Wallis (1968)
2 Modified R/S Analysis Lo (1991)
3 V/S Analysis Giraitis et al. (2003)
4 Log Periodogram Estimator Geweke & Porter-Hudak (1983); Robinson (1994)
5 Gaussian Semi-parametric Estimator Robinson (1995)
6 Smoothed Periodogram Estimator Reisen (1994)
7 Detrended Fluctuation Analysis Peng et al. (1994); Kantelhardt et al. (2002)
8 Wavelet Analysis Mallat (1989)
9 Whittle Estimation Whittle (1951); Hou & Perron (2014)
10 Fractional Brownian Motion Mandelbrot & Van Ness (1968)
11 ARFIMA Model Granger (1980); Granger & Joyeux (1980)
12 Fractionally Integrated GARCH (FIGARCH) Baillie et al. (1996)
13 Hyperbolic GARCH (HYGARCH) Davidson (2004)
14 Fractional Exponential (FEXP) Model Beran (1993)
15 Long Memory Stochastic Volatility Breidt et al. (1998); Harvey (1998)
16 Heterogeneous Long Memory (HAR) uller et al. (1997a); Corsi (2009)
17 Time-Varying Hidden Markov Model Bulla & Bulla (2006); Nystrup et al. (2017)
18 Bayesian Long Memory Koop et al. (1997); Ravishanker & Ray (1997)
19 Multi-Fractal Long Memory Mandelbrot et al. (1997); Calvet & Fisher (2002)
20 Detrended Cross-Correlation Analysis Podobnik & Stanley (2008)
21 Fractional Cointegration Granger (1986); Johansen (2008)
3. Evidence on Presence of Long Memory
Financial Literature is divided on the presence of long memory. Long memory has been observed in most
economic variables (Hassler & Wolters,1995;Koustas & Veloce,1996), bonds (Backus & Zin,1993), stocks
(Hiemstra & Jones,1997), indices (Bollerslev & Mikkelsen,1996;Sadique & Silvapulle,2001), currencies
(Cheung,1993), commodities (Cai et al.,2001;Elder & Serletis,2008), derivatives (Helms et al.,1984;
Barkoulas et al.,1999) and other specialized instruments (Madhavan & Arrawatia,2016).
However, this evidence of long memory varies across variables and markets. While asset returns have
been shown to have weak to no evidence of long memory (especially in developed markets) (Henry,2002;
Floros et al.,2007), asset volatility has been found to show strong evidence of long memory (Bollerslev &
Mikkelsen,1996;Fleming & Kirby,2011;Mighri & Mansouri,2014). In contrast, recent studies with high
frequency data have shown the presence of anti-persistence in volatility (Gatheral et al.,2018). Literature
on trading volume seems to be consistently in support of long memory (Lillo & Farmer,2004;Lux & Kaizoji,
2007). Studies on developing markets also show conflicting results. While prior studies showed stronger
presence of long memory in developing markets (McMillan & Thupayagale,2009;Hull & McGroarty,2014),
recent studies show that some developing markets have become more efficient than a few developed ones
since the 2008 financial crisis (Sensoy & Tabak,2016;Mensi et al.,2019).
3.1. Plausible Causes of True Long Memory
While the evidence on long memory is based on statistical and heuristic tests, the discussion remains
incomplete without pointing out plausible causes of true long memory.
3.1.1. News Flow and its interpretation
Long Memory can be a manifestation of interaction between many diverse information processes and
hence is inherent to the returns process (Andersen & Bollerslev,1997). This goes against the argument of
structural breaks leading to the hyperbolic decay of autocorrelations.
The arrival of news is seen as a driver for markets. Lillo & Farmer (2004) explain that news could be
classified as external or internal. Externals news are events outside of control of market participants (e.g.,
natural calamities). Such events are known to have power law distributions. While internal news comes
under the purview of market players, the ability to understand and act on them can be complicated due to
the social dynamics such as herding behaviour. Moreover, limited attention and comprehension ability of
humans coupled with their changing preferences towards fundamental and technical analysis can generate
long memory in financial time series (Kirman & Teyssiere,2002).
These aspects can be better understood in the framework of Adaptive Market Hypothesis (AMH) (Lo,
2004,2005). Human decisions are usually made under incomplete information and are delayed due to other
factors. Such time lags in responding to news arrival can lead to autocorrelations in order inflow.
3.1.2. Market Microstructure issues and other factors
Various market microstructure based factors can lead to long memory. For example, iceberg orders
wherein large orders are split into many smaller ones before being sent to the exchange, might lead to
power law based autocorrelations in order flow mechanism (Lillo et al.,2005). Similarly, simulation studies
suggest that the observed long memory in order flow, volume and volatility can be attributed to the inherent
imitative and adaptive behaviour of various market participants (LeBaron & Yamamoto,2007,2008).
Other notable explanatory factors pertaining to long memory include bid-ask spreads, non-synchronous
trading (Campbell et al.,1997), influence of institutional investors (Gabaix et al.,2006), extent of market
openness (Lim & Brooks,2010) and speed of price adjustment (Zheng et al.,2018). Lastly, long memory
has also been attributed to the economic and institutional differences between emerging and developed
markets (Liow,2009).
4. Long Memory: Beware of False Positives
Empirical research seems divided on the debate on differentiating true and spurious long memory in
economic variables. While prevalence of long memory runs contrary to market efficiency, this section
discusses several known pitfalls that can cause false positives in long memory analysis.
4.1. Structural Breaks
Potter (1979) argued that long memory may be an artifact of non-homogeneity in the data. He referred
to several studies on precipitation and concluded that studies with homogeneous data did not support the
presence of long memory. This was also proved in other studies. For example, introducing a trend to a
stationary time series can create long memory (Bhattacharya et al.,1983). Simulations based on incorpo-
rating breaks in a data generating process (DGP) provide evidence of spurious long memory (Diebold &
Inoue,2001). Various empirical studies using financial market data have also shown the confounding effect
structural breaks can have on long memory in returns (Granger & Hyung,2004) and volatility (Liu,2000).
However, distinguishing between long memory and structural break is mathematically difficult. This is
similar to the confusion between unit root and structural breaks. For true long memory processes, several
tests for structural change may show structural break when there should be none (Kuan & Hsu,1998). On
the other hand, long memory estimators will be biased towards long memory for stationary processes with
level shifts (Perron & Qu,2010). Literature provides several avenues to check for the confounding effect of
structural breaks on long memory and also to incorporate both phenomenon in models to measure their
individual effects.
Quite a few statistical tests exist to check for this confounding effect. Most are hypothesis tests based on
LM, LR, Wald and CUSUM tests for breaks in mean (Hidalgo & Robinson,1996;Shimotsu,2006;Perron
& Qu,2010;Qu,2011;Shao,2011;Mayoral,2012;Wright,1998;Wenger et al.,2018b). Similarly, tests for
differentiating structural break from long memory in volatility are also available (Hwang & Shin,2015).
Recently, Sibbertsen et al. (2018) proposed a multivariate approach to differentiate between structural
breaks and long memory.
Model-specific studies are also available. For example, ARFIMA based models that are robust to struc-
tural breaks have been proposed (Baillie & Morana,2012;Shi & Ho,2015). Similarly, attempts to capture
and differentiate structural breaks from long memory in volatility have led to pertinent improvisations
of FIGARCH (Baillie & Morana,2009), Markov Switching GARCH (Charfeddine,2014) and HAR-RV
(Hwang & Shin,2018) class of models. Volatility specific structural break tests like the ICSS test (Inclan
& Tiao,1994) and its variants can be used with FIGARCH models to differentiate between structural
breaks and long memory (Walther et al.,2017). Lastly, Long Memory Estimators that are robust to
structural breaks have also been proposed by Hou & Perron (2014).
For a review of the literature of tests that aid in differentiating structural breaks from long memory,
the readers may refer to Sibbertsen (2004), Banerjee & Urga (2005) and Wenger et al. (2018a).
4.2. Temporal Aggregation
Temporal Aggregation refers to transformation of a time series data at a frequency lower than the
original DGP. In some cases, data is also analyzed after aggregation over a longer duration to remove
seasonal fluctuations. A typical example is data on industrial production where data is only available
quarterly. While this aids in smoothening the data points as well as filtering out the high frequency noise,
it can also manifest as spurious long memory. For example, LeBaron (2001) showed that a temporally
aggregated series created by adding up just three short memory linear time series of different time scales
can show spurious long memory.
Having said so, evidence in support of true long memory, notwithstanding temporal aggregation is
also available. For instance, Andersen & Bollerslev (1997) showed that true volatility persistence can be
attributed to temporal aggregation of heterogeneous inflow of news over time. Their work lends credence
to long memory dependence being an inherent feature of the DGP and not a spurious manifestation of
temporal aggregation (Mcmillan & Speight,2008). A notable result in this school of thought was brought
forward by Souza (2008) where it was shown that for true long memory series, temporal aggregation does
not change the estimated memory parameter.
This association between temporal aggregation and long memory has also formed the basis of a specific
class of volatility models called Heterogeneous Autoregressive models (uller et al.,1997b;Corsi,2009).
These models draw motivation from the Heterogeneous Market Hypothesis (M¨uller et al.,1993) and also
the “Mixture of Distribution Hypothesis” of Andersen & Bollerslev (1997).
To assist researchers, there are several tests to differentiate between true long memory and spurious long
memory owing to temporal aggregation (Ohanissian et al.,2008;Kuswanto,2011;Frederiksen & Nielsen,
2013;Davidson & Rambaccussing,2015).
4.3. Cross-sectional Aggregation
Just like temporal aggregation, cross-sectional aggregation can also lead to spurious observations of
long memory in time series variables. A large number of AR(1) processes can be added to produce a time
series that would show long memory under certain assumptions (Granger,1980).
Studies on inflation data have attributed the observed long memory to cross-sectional aggregation since
inflation is measured via aggregating various sectoral sub-indices that possess only short memory (Balcilar,
2004;Altissimo et al.,2009). On the other hand, prior works such as Kang et al. (2010) uncover evidence
of long memory in the stock index as well as the underlying constituent stocks.
Cross-sectional aggregation also requires the count of individual series (N) to be very large. Granger
(1980) postulated this case for N→ ∞. However, this count also varies across studies depending on other
assumptions used for the Monte Carlo simulations. While Zaffaroni (2004) used a dataset with N > 1500 to
reproduce the theoretical results, Leccadito et al. (2015) simulated long memory with N= 500 components
only. Another study by Haldrup & Vald´es (2017) showed that Nseems to depend on the extent of long
memory in the individual series. If the individual series have high long memory, an aggregate of just
250 such series can mimic that inherent long memory. However, for individual series with low levels of
persistence, even an aggregate of N= 10,000 such series did not have a similar level of long memory. In
addition, it was proved that when such a composite series is fractionally differenced, the autocorrelation
function of its residuals still exhibit hyperbolic decay. This inability of ARFIMA models to suitably capture
the long memory of the true DGP caused by cross-sectional aggregation calls for better models.
Long Memory can also be observed when a number of linear and homogeneous subsystems with short
memory are connected to create a network structure (Schennach,2018). This provides another possible
reason for long memory without non-linearity, heterogeneity, unit roots or structural breaks. Several
economic examples that can be modeled using these approach are firms in an industry and supply chain
time series.
4.4. Biases in Estimation process and related issues
Differentiating true and spurious long memory calls for researchers to exercise judgement while choosing
the estimation method. Not all estimators are equally suitable in all cases. Various studies have commented
on the properties of popular R/S statistic and its many variants in terms of size and power (Lo,1991;
Teverovsky et al.,1999). Similarly, notable prior works offer a critical review of small sample properties of
other estimators, such as, but not limited to, GPH (Agiakloglou et al.,1993), Local Whittle (Hurvich &
Ray,2003), Higuchi, Peng and Wavelet estimators (Rea et al.,2013).
Applying an AR-GARCH filter on returns would significantly reduce the spurious long memory effect,
for such a filter would, to a larger extent obviate the confounding effect of short memory (Lo,1991).
Further, use of low-frequency data can cause a downward bias on the long memory estimates (Bollerslev
& Wright,2000;Souza & Smith,2002). Also, the choice of proxies to measure volatility has an effect on
long memory estimates (Wright,2002).
Various characteristics of data also need to be reviewed before choosing methodologies. For instance,
emerging markets, having higher levels of volatility, may be more suitable for wavelet based estimation
(Ozun & Cifter,2008). In general, Local Whittle estimators are observed to be the most stable among other
long memory estimators (Taqqu et al.,1995;Hassler,2011). In addition, cyclical and seasonal patterns
in data can lead to the observation of long memory in the squared return series (Lobato,1997). Many
long memory tests assume unconditional homoscedasticity. Tests that allow for heteroscedasticity (Harris
& Kew,2017) should be used for financial time series. Similarly, absence of higher order moments also
manifest as long memory (Lobato & Savin,1998).
While actual datasets may not show these exact pathologies, issues such as the presence of heavy tails
are very much real. Consequently, it is advised to ascertain the model assumptions before employing them.
If required, more robust models whose assumptions are closer to the empirical properties of the time series
should be used.
5. Impact of Long Memory on Quality of Forecasts
The most practical benefit of using long memory models is their better forecasting ability. Literature
supports this finding across all approaches of modeling long memory, such as, fractionally integrated models
(Bhardwaj & Swanson,2006), HAR models (Corsi,2009;Wen et al.,2016), Fractal theory (Liu et al.,2019)
among others. In addition, modeling long memory along with other stylizations of the underlying data,
such as periodicity (Franses & Ooms,1997), heteroscedasicity and structural breaks (Ma et al.,2019)
have shown to improve forecasting performance. Similarly, model generalization, such as FIGARCH to
hyperbolic GARCH (Davidson,2004;Li et al.,2015) or incorporating a multivariate framework (Harris &
Nguyen,2013;Balcilar et al.,2017), also significantly increase forecasting accuracy.
Since volatility estimation and forecasting is essential in the context of risk management of financial
portfolios, Value-at-Risk (VaR) calculations also stand to benefit from long memory models (Batten et al.,
2014;Kinateder & Wagner,2014;Meng & Taylor,2018). Allowing for long memory in the cross-section of
asset returns can help create specific trading strategies that can generate significant gains (Nguyen et al.,
Notwithstanding the above stated developments on the modeling front, prior studies also nudge re-
searchers to exercise caution while modeling long memory. Notable studies have shown that similar fore-
casting results can also be approximately matched by using standard ARIMA models of very high orders
(Ray,1993). In addition, the forecasting error pertaining to over-differencing, is significantly lesser than
the forecasting error pertaining to under-differencing 1(Smith & Yadav,1994). There are more nuances
to be considered here. If the AR/MA coefficients are negative (especially for low values of d), standard
ARMA models will provide similar short term predictability. ARFIMA models would be better only for
time series with strong persistence (d/0.5) or longer term prediction (Andersson,2000;Man,2003).
Moreover, Granger & Hyung (2004) found that modeling a time series with only structural breaks and
separately with only long memory can provide similar predictive performance with long memory model
having a slight edge. Similar findings were reported for many extensions of ARFIMA models. Hence, the
model specifications would depend on the researcher’s choice between parsimonious fractional models and
the over-parametrized standard ARIMA models.
Another primary motive for employing long memory models is to study the impact of shocks to volatility
on asset prices. If the impact of such shocks are short lived and modest, it calls into question the significance
of employing long memory models for examining the DGP as has been illustrated by Christensen & Nielsen
6. Conclusion
If true long memory can be established, many modeling exercises should change. For instance, CAPM
can be modified to include fractional returns (Raei & Mohammadi,2008) and a persistent error term
(Amano et al.,2012). Similarly, modeling exercises involving endogenous variables should be geared towards
adequately capturing long memory. This would call for refinement of popular multivariate frameworks
such as Granger Causality tests (Chen,2006,2008), VAR-MGARCH (Dark,2018;Zhao et al.,2019) and
Cointegration methods (Granger,1986;Johansen,2008). In addition, implied volatility models based on
options pricing would be incomplete without incorporating long memory (Cardinali,2012).
While it cannot be denied that long memory models should be in the econometrician’s toolbox, their
use should be governed by an initial exploratory analysis of the data and the context of the research
questions. The researcher should keep in mind that to a man with a hammer, everything looks like a nail.
Multiple confounding factors, such as, but not limited to, structural breaks and aggregation, can manifest
as spurious long memory. This review hopes to nudge researchers to exercise judgement while choosing
appropriate long memory models, for inferences derived from misspecified models can lead to misleading
policy recommendations.
1Over-differencing refers to first differencing a time series whose DGP is closer to I(d). Under-differencing refers to
fractionally differencing a time series who DGP is closer to random walk (I(1)).
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... Memory characteristics (also termed persistency) in the dynamics of typical econometrics captured over time are intimately connected with cycles and thus also to the underlying processes [18]. Technically, however, these same characteristics such as long-range memory processes are challenging to analyze and interpret due to (in part) selfsimilarity and typical non-stationary properties (as they confound spurious from true signals) [19]. The Hurst constant and wavelet analyses are statistical time series tools that may be calculated in such as a way as to avoid these challenges [20]. ...
It is challenging to elucidate the effects of changes in external influences (such as economic or policy) on the rate of US drug approvals. Here, a novel approach—termed the Chronological Hurst Exponent (CHE)—is proposed, which hypothesizes that changes in the long-range memory latent within the dynamics of time series data may be temporally associated with changes in such influences. Using the monthly number FDA’s Center for Drug Evaluation and Research (CDER) approvals from 1939 to 2019 as the data source, it is demonstrated that the CHE has a distinct S-shaped structure demarcated by an 8-year (1939–1947) Stagnation Period, a 27-year (1947–1974) Emergent Period, and a 45-year (1974–2019) Saturation Period. Further, dominant periodicities (resolved via wavelet analyses) are identified during the most recent 45-year CHE Saturation Period at 17, 8 and 4 years; thus, US drug approvals have been following a Juglar/Kuznet mid-term cycle with Kitchin-like bursts. As discussed, this work suggests that (1) changes in extrinsic factors (e.g., of economic and/or policy origin) during the Emergent Period may have led to persistent growth in US drug approvals enjoyed since 1974, (2) the CHE may be a valued method to explore influences on time series data, and (3) innovation-related economic cycles exist (as viewed via the proxy metric of US drug approvals).
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It is challenging to elucidate the extrinsic effects of social, economic, policy or other substantive changes on the dynamics of US drug approvals. Here, a novel approach, termed the Chronological Hurst Exponent (CHE), is proposed, which hypothesizes that changes in intrinsic long-range memory latent within the dynamics of time series data may be temporally associated with extrinsic variables. Using the monthly number FDA Center for Drug Evaluation and Research (CDER) approvals from 1939 to 2019 as the data source, it is demonstrated that US approvals are found to have a distinct S-shaped (trichotomized) long-range CHE structure: an 8-year (1939-1947) Stagnated (random; H of 0.5), a 27-year (1947-1974) Emergent (time-varying persistent; H between 0.5 and 0.9), and the 45-year (1974-2019) Saturated (persistent; H of 1) US Approvals Hurst Cycles. Further, dominant periodicities (resolved via wavelet analyses) are identified in the Saturated Period at 17, 8 and 4 years; thus, US Approvals have been following a Juglar/Kuznet mid-term cycle with Kitchin-like bursts. This work suggests that (1) macro-factors in the Emergent Period have led to persistent growth in US approvals enjoyed since 1974, (2) the CHE may be a valued method to explore the constancy of extrinsic pressures on time series data, and (3) adds further evidence that innovation-related economic cycles exist (as supported via the proxy metric of US drug approvals).
This paper examines volatility linkages and forecasting for stock and foreign exchange markets from a novel perspective by utilizing a bivariate Markov‐switching multifractal model that accounts for possible interactions between stock and foreign exchange markets. Examining daily data from major advanced and emerging nations, we show that generalized autoregressive conditional heteroskedasticity models generally offer superior volatility forecasts for short horizons, particularly for foreign exchange returns in advanced markets. Multifractal models, on the other hand, offer significant improvements for longer horizons, consistently across most markets. Finally, the bivariate multifractal model provides superior forecasts compared to the univariate alternative in most advanced markets and more consistently for currency returns, while its benefits are limited in the case of emerging markets.
The volatility of crude oil price has a great influence on the world economy. In order to measure the crude oil price risk (VaR) and explain the dynamic relationship between investment income and risk in the oil market more clearly, this paper uses a variety of fractional GARCH models to describe typical volatility characteristics like long memory, volatility clustering, asymmetry and thick tail. The autoregressive conditional heteroscedasticity in the mean model (ARCH-M) and peaks-over-threshold model of extreme value theory (EVT-POT) are taken into account to develop a hybrid time-varying long memory GARCH-M-EVT model for calculation of static and dynamic VaR. Empirical results show that the WTI crude oil has a significantly long memory feature. All the fractional integration GARCH models can describe the long memory appropriately and the FIAPARCH model is the best in regression and out of sample one-step-ahead VaR forecasting. Back-testing results show that the FIAPARCH-M-EVT model is superior to other GARCH-type models which only consider oil price fluctuation characteristics partially and traditional methods including Variance-Covariance and Monte Carlo in price risk measurement. Our conclusions confirm that considering long memory, asymmetry and fat tails in the behavior of energy commodity return combined with effectively dynamic time-varying risk reflection such as the ARCH-M model and reliable tail extreme filter processes such as EVT can improve the accuracy of crude oil price risk measurement, provide an effective tool for analyzing the extreme risk of the tail of the oil market and facilitate the risk management for oil market investors.
In this article, we account for the first time for long memory, regime switching and the conditional time-varying volatility of volatility (heteroscedasticity) to model and forecast market volatility using the heterogeneous autoregressive model of realized volatility (HAR-RV) and its extensions. We present several interesting and notable findings. First, existing models exhibit significant nonlinearity and clustering, which provide empirical evidence on the benefit of introducing regime switching and heteroscedasticity. Second, out-of-sample results indicate that combining regime switching and heteroscedasticity can substantially improve predictive power from a statistical viewpoint. More specifically, our proposed models generally exhibit higher forecasting accuracy. Third, these results are widely consistent across a variety of robustness tests such as different forecasting windows, forecasting models, realized measures, and stock markets. Consequently, this study sheds new light on forecasting future volatility.
We examine long memory volatility in the cross-section of stock returns. We show that long memory volatility is widespread in the United States and that the degree of memory can be related to firm characteristics, such as market capitalization, book-to-market ratio, prior performance, and price jumps. Long memory volatility is negatively priced in the cross-section. Buying stocks with shorter memory and selling stocks with longer memory in volatility generates significant excess returns of 1.71% per annum. Consistent with theory, we find that the volatility of stocks with longer memory is more predictable than stocks with shorter memory. This makes the latter more uncertain, which is compensated for with higher average returns.
This study investigates the time varying efficiency of five European GIPSI stock markets, compared to global and regional U.S. markets. Using the MF-DFA approach, we show evidence of long memory in both short and long term for all markets. Furthermore, the long memory is more pronounced in the long term than in the short term. Finally, Greece is the highest inefficient market, whatever is the time horizons, while Portugal and Ireland markets are the least inefficient in the short and long term, respectively. Global and regional stock markets are less efficient than GIPSI (except Greece) markets in the short term.
Many time series exhibit “long memory”: Their autocorrelation function decays slowly with lag. This behavior has traditionally been modeled via unit roots or fractional Brownian motion and explained via aggregation of heterogeneous processes, nonlinearity, learning dynamics, regime switching, or structural breaks. This paper identifies a different and complementary mechanism for long-memory generation by showing that it can naturally arise when a large number of simple linear homogeneous economic subsystems with short memory are interconnected to form a network such that the outputs of the subsystems are fed into the inputs of others. This networking picture yields a type of aggregation that is not merely additive, resulting in a collective behavior that is richer than that of individual subsystems. Interestingly, the long-memory behavior is found to be almost entirely determined by the geometry of the network, while being relatively insensitive to the specific behavior of individual agents.
This paper examines persistence in Turkish inflation rates using data from consumer and wholesale price indices. The inflationary process in Turkey is believed to be highly inertial, which should lead to strongly persistent inflation series. Persistence of seventy-five inflation series at various aggregation levels is examined by estimating models that allow long memory through fractional differencing. The order of fractional differencing is estimated using several semiparametric and maximum likelihood methods. Persistence of each series is evaluated using the time required for a given percentage of the effect of a shock to dissipate. We find that disaggregate inflation series show no significant persistence. We found that only twelve out of seventy-five series require more than six months for 99 percent of the effect of a shock to dissipate. Thus, the paper finds evidence of spurious long memory due to aggregation.
Multivariate models with long memory (LM) in conditional correlation and volatility are proposed. The models employ a fractionally integrated version of the dynamic conditional correlation GARCH (DCC-GARCH) process (Engle, 2002), and can be used to forecast conditional covariance matrices of high dimension. The models are applied to a data set consisting of ten US stocks and out of sample forecasts over 1–80 days evaluated using statistical and economic loss functions. If intraday data is unavailable, the statistical loss function reveals that LM correlation models provide superior return covariance matrix forecasts over 20–80 days. When intraday data is available, LM correlation models provide superior forecasts of the realised covariance matrix over the same horizons, however the gains when forecasting the return covariance matrix are small. Finally, when forecasting minimum variance portfolio weights, even though the benefits from LM correlation models diminish completely, they are not consistently outperformed by any of the benchmarks.
It is well known that standard tests for a mean shift are invalid in long-range dependent time series. Therefore, several long-memory robust extensions of standard testing principles for a change-in-mean have been proposed in the literature. These can be divided into two groups: those that utilize consistent estimates of the long-run variance and self-normalized test statistics. Here, we review this literature and complement it by deriving a new long-memory robust version of the sup-Wald test. Apart from giving a systematic review, we conduct an extensive Monte Carlo study to compare the relative performance of these methods. Special attention is paid to the interaction of the test results with the estimation of the long-memory parameter. Furthermore, we show that the power of self-normalized test statistics can be improved considerably by using an estimator that is robust to mean shifts.