On long memory origins and forecast horizons

• Published date: 01 August 2020
• Date: 01 August 2020
• Author: J. Eduardo Vera‐Valdés
• DOI: http://doi.org/10.1002/for.2651

Received: 8 November 2017 Revised: 22 July 2019 Accepted: 3 January 2020

DOI: 10.1002/for.2651

RESEARCH ARTICLE

On long memory origins and forecast horizons

J. Eduardo Vera-Valdés1,2

1Department of Mathematical Sciences,

Aalborg University, Aalborg, Denmark

2CREATES, Aarhus, Denmark

Correspondence

J. Eduardo Vera-Valdés, Department of

Mathematical Sciences, Aalborg

University, Skjernvej4A, 9220 Aalborg

Øst, Denmark.

Email: eduardo@math.aau.dk

Abstract

Most long memory forecasting studies assume that long memory is generated by

the fractional difference operator.We argue that the most cited theoretical argu-

ments for the presence of long memory do not imply the fractional difference

operator and assess the performance of the autoregressivefractionally integrated

moving average (ARFIMA) model when forecasting series with long memory

generated by nonfractional models. We find that ARFIMA models dominate in

forecast performance regardless of the long memory generating mechanism and

forecast horizon. Nonetheless, forecasting uncertainty at the shortest forecast

horizon could make short memory models provide suitable forecast perfor-

mance, particularly for smaller degrees of memory. Additionally, we analyze

the forecasting performance of the heterogeneous autoregressive (HAR) model,

which imposes restrictions on high-order AR models. We find that the structure

imposed by the HAR model produces better short and medium horizon forecasts

than unconstrained AR models of the same order.Our results have implications

for, among others, climate econometrics and financial econometrics models

dealing with long memory series at different forecast horizons.

KEYWORDS

ARFIMA, cross-sectional aggregation, forecasting, HAR model, long memory, nonfractional

memory

1INTRODUCTION

Long memory analysis deals with the notion of series with

long-lasting correlations—that is, series with autocorrela-

tions that decay at a hyperbolic rate instead of the standard

geometric one. One of the first works on long memory is

due to Hurst (1956). He studied the long-term capacity of

reservoirs for the Nile and recommended to increase the

height of a dam to be built given his observations on cycles

of highs at the river. As found by Hurst, failing to account

for the presence of long memory can lead to inaccurate

forecasts. If the data are best modeled by a long memory

process, then forecasts computed with standard models

would be too optimistic, in the sense that they would pre-

dict a return to normal events faster than what we would

observe in reality. A dam built based on a short memory

forecast would be more prone to overflow that one built

based on a long memory forecast, hence increasing the risk

of a catastrophic event. Hurst's work highlights the impor-

tance of developing appropriate forecasting tools to deal

with the presence of long memory.

In the time series literature, the autoregressive fraction-

ally integrated moving average (ARFIMA) class of models

remains to be the most popular, given its appeal of bridg-

ing the gap between the stationary ARMA models and

the nonstationary ARIMA model by the use of the frac-

tional difference operator. Moreover, some effort has been

directed to assess the performance of the ARFIMA type

of models when forecasting fractional long memory pro-

cesses. Nonetheless, no consensus has been formed.

Ray (1993) calculates the percentage increase in mean

squared error (MSE) from forecasting fractionally inte-

grated, FI(d), series with AR models. She argues that the

MSE may not increase significantly, particularly when we

Journal of Forecasting. 2020;39:811–826. wileyonlinelibrary.com/journal/for © 2020 John Wiley & Sons, Ltd. 811

VERA-VALDÉS

do not know the true long memory parameter, d.Crato

and Ray (1996) compare the forecasting performance of

ARFIMA models against ARMA alternatives and find that

ARFIMA models are in general outperformed by ARMA

alternatives for short forecast horizons. On real data,

Martens et al. (2009) show that for daily realized volatility

for forecast horizons of up to 20 days it seems to be ben-

eficial to use a flexible high-order AR model instead of a

parsimonious but stringent fractionally integrated model.

On the other hand, Barkoulas and Baum (1997)

find improvements in forecasting accuracy when fitting

ARFIMA models to Eurocurrency returns series, particu-

larly for longer horizons. By allowing for larger data sets

of both financial and macro variables, and considering

larger forecast horizons, Bhardwaj and Swanson (2006)

find that ARFIMA processes generally outperform ARMA

alternatives in terms of forecasting performance.

One thing that most forecasting comparison studies

have in common is the underlying assumption that long

memory is generated by the fractional difference opera-

tor. There are two predominant theoretical explanations

for the presence of long memory in the time series liter-

ature: cross-sectional aggregation of dynamic, persistent

micro-units (Granger, 1980); and shocks of random dura-

tion (Parke, 1999). As argued in Section 2, neither of these

sources of long memory implies an ARFIMA specifica-

tion and thus do not follow from the fractional difference

operator. The question addressed in this paper is whether

an ARFIMA model serves as a good approximation for

forecasting purposes when the long memory generating

mechanism is different from the fractional difference oper-

ator.

Moreover, as argued by Baillie et al. (2012), a prac-

titioner's goals will generally include making forecasts

over both short and long horizons. As an example, the

surge of climate econometrics as a way to address climate

change relies on the construction of long horizon forecasts.

Thus we analyze the forecasting performance of short

and long memory models at several forecast horizons.

In this sense, we extend previous studies to larger fore-

cast horizons relevant to climate change analysis. We find

that ARFIMA models achieve better forecasting perfor-

mance than short memory alternatives for all long mem-

ory generating mechanisms and forecast horizons. More-

over, the superior performance of the ARFIMA model

gets exacerbated the higher the degree of memory of the

processes. Nonetheless, forecast uncertainty at short hori-

zons is such that short memory models achieve compara-

ble forecast performance to ARFIMA models, particularly

for smaller degrees of memory. Finally, we find that the

restrictions imposed by the heterogeneous autoregressive

(HAR) model produce better short and medium forecast-

ing performance than unconstrained AR alternatives.

This paper proceeds as follows. In Section 2, we present

the long memory generating processes considered, and

show that the most cited theoretical explanations for the

presence of long memory do not imply an ARFIMA speci-

fication. Section 3 describes the design of the Monte Carlo

analysis used for the forecasting study, while Section 4

presents the main results. Furthermore, Section 5 dis-

cusses the results from the forecasting exercise in a

bias–variance tradeoff context. Sections 6 and 7 show that

the insights gained from the Monte Carlo simulations hold

on real data and address some practical considerations.

Finally, Section 8 presents the conclusions.

2LONG MEMORY GENERATING

PROCESSES

In this section, we present the long memory generating

processes considered in this work. All processes studied

are long memory in the covariance sense formalized below.

Definition 1. Let xtbe a second-order stationary pro-

cess with autocovariance function 𝛾x(k).Thenxtis said

to exhibit long memory in the covariance sense if

𝛾x(k)≈C1k2d−1as k→∞,(1)

with d∈(0,1∕2),andC1a constant.

Above, for h(x)≠0, g(x)≈h(x)as x→∞denotes

that g(x)∕h(x)converges to 1 as xtends to ∞.

Note from the definition that long memory in the covari-

ance sense relates to the rate of decay of the autocorrela-

tions; see Haldrup and Vera-Valdés (2017) for a discussion

on other definitions. For applied purposes, the fitted mod-

els try to mimic the rate of decay of the autocorrelations

to assess the importance that past observations have on

future realizations. In this context, the models use this

information to produce better forecasts. Thus we deem

the covariance sense to the appropriate definition of long

memory for this work.

2.1 Fractional difference operator

Weinclude fractionally integrated processes in the analysis

as a benchmark. Granger (1980) and Hosking (1981) pro-

posed to use the standard binomial expansion to decom-

pose the fractional difference operator,(1−L)d,inaseries

with coefficients 𝜋𝑗=Γ(𝑗+d)∕[Γ(d)Γ(𝑗+1)] for 𝑗∈N.

That is, they propose to study a series given by

(1−L)dxt=𝜖t,(2)

where 𝜖tis a white noise process, d∈(−1∕2,1∕2),andLis

the lag operator.

For d∈(0,1∕2), it can be shown that these coeffi-

cients decay at a hyperbolicrate, which in turn translates to

812