Long Memory of Financial Time Series and Hidden Markov Models with Time‐Varying Parameters

• Published date: 01 December 2017
• DOI: http://doi.org/10.1002/for.2447
• Author: Peter Nystrup,Henrik Madsen,Erik Lindström
• Date: 01 December 2017

Journal of Forecasting,J. Forecast. 36, 989–1002 (2017)

Published online 13 September 2016 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/for.2447

Long Memory of Financial Time Series and Hidden

Markov Models with Time-Varying Parameters

PETER NYSTRUP,1,2HENRIK MADSEN2AND ERIK LINDSTRÖM3

1

Sampension, Hellerup, Denmark

2

Department of Applied Mathematics and Computer Science, Technical University of Denmark,

Kongens Lyngby, Denmark

3

Centre for Mathematical Sciences, Lund University, Lund, Sweden

ABSTRACT

Hidden Markov models are often used to model daily returns and to infer the hidden state of financial markets. Previous

studies have found that the estimated models change over time, but the implications of the time-varyingbehavior have

not been thoroughly examined. This paper presents an adaptive estimation approach that allows for the parameters of

the estimated models to be time varying. It is shown that a two-state Gaussian hidden Markovmodel with time-varying

parameters is able to reproduce the long memory of squared daily returns that was previously believed to be the most

difficult fact to reproduce with a hidden Markov model. Capturing the time-varying behavior of the parameters also

leads to improved one-step density forecasts. Finally, it is shown that the forecasting performance of the estimated

models can be further improved using local smoothing to forecast the parameter variations. Copyright © 2016 John

Wiley & Sons, Ltd..

KEY WORDS hidden Markov models; daily returns; long memory; adaptive estimation; time-varying

parameters

INTRODUCTION

Many different stylized facts havebeen established for financial returns (see, for example, Granger and Ding, 1995a,b;

Granger et al., 2000; Cont, 2001; Malmsten and Teräsvirta, 2010). Rydén et al. (1998) showed the ability of a hidden

Markov model (HMM) to reproduce most of the stylized facts of daily return series introduced by Granger and Ding

(1995a,b). In an HMM, the distribution that generates an observation depends on the state of an unobserved Markov

chain. Rydén et al. (1998) found that the one stylized fact that could not be reproduced by an HMM was the slow

decay of the autocorrelation function (ACF) of squared and absolute daily returns, which is of great importance in

financial risk management. The daily returns do not have the long-memory property themselves—only their squared

and absolute values do. Rydén et al. (1998) considered this stylized fact to be the most difficult to reproduce with

an HMM.

According to Bulla and Bulla (2006), the lack of flexibility of an HMM to model this temporal higher-order

dependence can be explained by the implicit assumption of geometrically distributed sojourn times in the hidden

states. This led them to consider hidden semi-Markov models (HSMMs) in which the sojourn time distribution is

modeled explicitly for each hidden state so that the Markov property is transferred to the embedded first-order Markov

chain. They found that an HSMM with negative-binomially distributed sojourn times was better than the HMM at

reproducing the long-memory property of squared daily returns.

Bulla (2011) later showed that HMMs with t-distributed components reproduce most of the stylized facts as well

or better than the Gaussian HMM at the same time as increasing the persistence of the visited states and the robustness

to outliers. Bulla (2011) also found that models with three states provide a better fit than models with two states.

In Nystrup et al. (2015b), an extension to continuous time was presented and it was shown that a continuous-time

Gaussian HMM with four states provides a better fit than discrete-time models with three states with a similar number

of parameters.

The data analyzed in this paper is daily returns of the S&P 500 stock index from 1928 to 2014. It is the same

time series that was studied in the majority of the above-mentioned studies just extended through the end of 2014.

Granger and Ding (1995a) divided the full sample into 10 subsamples of 1700 observations, corresponding to a little

less than 7 years, as they believed it was likelythat with such a long time span there could have been structural shifts in

the data-generating process. Using the same approach, Rydén et al. (1998) and Bulla (2011) found that the estimated

Correspondence to: Peter Nystrup, Department of Applied Mathematics and Computer Science, Technical University of Denmark, Asmussens

Allé, Building 303B, 2800 Kongens Lyngby, Denmark. E-mail: pnys@dtu.dk

Copyright © 2016 John Wiley & Sons, Ltd.