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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

Abstract

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

prices.

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@ifmr.ac.in (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

0<

∞

X

k=1

γX(k)<∞

b) Anti-Persistence, if

∞

X

k=1

γX(k) = 0

c) Long Memory, if

∞

X

k=1

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 &

2

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 =H−0.5D= 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

3

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

t−htwith 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 d→0. 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.

(1997).

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.

4

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) M¨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

5

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).

6

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).

7

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 (M¨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

8

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.

9

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.,

2019).

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

(2007).

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

10

(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.

Notes

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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