Modeling and forecasting aggregate stock market volatility in unstable environments using mixture innovation regressions

• DOI: http://doi.org/10.1002/for.2466
• Author: Nima Nonejad
• Date: 01 September 2017
• Published date: 01 September 2017

Received: 24 April 2014 Revised: 16 January 2017 Accepted: 31 January 2017

DOI: 10.1002/for.2466

RESEARCH ARTICLE

Modeling and forecasting aggregate stock market volatility in

unstable environments using mixture innovation regressions

Nima Nonejad

Department of Mathematical Sciences and

CREATES, Aalborg University, Aalborg,

Denmark

Correspondence

Nima Nonejad, Department of Mathematical

Sciences, Aalborg University,Fredrik Bajers

Vej 7G, 9220 Aalborg Ø, Denmark.

Email: nimanonejad@gmail.com

Abstract

We perform Bayesian model averaging across different regressions selected from a

set of predictors that includes lags of realized volatility, financial and macroeco-

nomic variables. In our model average, we entertain different channels of instability

by either incorporating breaks in the regression coefficients of each individual model

within our model average, breaks in the conditional error variance, or both. Changes

in these parameters are driven by mixture distributions for state innovations (MIA)

of linear Gaussian state-space models. This framework allowsus to compare models

that assume small and frequent as well as models that assume large but rare changes

in the conditional mean and variance parameters. Results using S&P 500 monthly

and quarterly realized volatility data from 1960 to 2014 suggest that Bayesianmodel

averaging in combination with breaks in the regression coefficients and the error

variance through MIA dynamics generates statistically significantly more accurate

forecasts than the benchmark autoregressive model. However, compared to a MIA

autoregression with breaks in the regression coefficients and the error variance, we

fail to provide any drastic improvements.

KEYWORDS

model uncertainty, mixture innovation regressions, parameter instability, stock market volatility

1INTRODUCTION

How to measure and model volatility is an important issue in

financial econometrics. The behavior of volatility has impor-

tant implications for decisions in risk management, portfolio

choice, and derivative pricing. For instance, volatility is cru-

cial to constructing optimal hedge ratios to hedge against risk

and to determine the value of commodity-based contingent

claims, which determine whether to invest in physical capital

or not. Changes in volatility can also have important conse-

quences on the economy, as evidenced bythe recent financial

crisis. Therefore, learning and understanding the economic

1A previous version of this paper was circulated under the title “A mixture

innovationheterogeneous autoregressive model for structural breaks and long

memory.”

drivers of aggregate stock market volatility is an important

topic.

However, although volatility plays a very big role in finan-

cial markets, there is a problem with using it: Volatility

is latent and cannot be directly observed and there is no

unique or universally accepted way to define it. Tradition-

ally, most studies consider volatility as an unobserved pro-

cess and seek to implement a fully specified model to esti-

mate and analyze it. Over the years, this approach has led

to well-known models such as generalized autoregressive

conditional heteroskedasticity (GARCH), stochastic volatil-

ity (SV), Markov switching (MS) and exponentially weighted

moving average (EWMA), as advocated by the Riskmet-

rics methodology. An alternative approach is to construct

an observable proxy for the unobserved volatility by using

718 Copyright © 2017 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/for Journal of Forecasting.2017;36:718–740.

NONEJAD 719

FIGURE 1 S&P 500 realized volatility and posterior estimates from AR(1)-MIA-SV.(a) Monthly S&P 500 realized volatility (black) and

earnings-to-price ratio (dotted). (b) Opposite of the quarterly S&P 500 realized volatility (black) and the real US GDP growth rate (dotted). (c)

Posterior estimates of the persistence of realized volatility for Equation 7 under Equations3 and 4 using mont hlyS&P 500 realized volatility at

h=1. (d) Posterior structural break probabilities in the persistence of realized volatility for Equation 7 under Equations 3 and 4 using monthly S&P

500 realized volatility at h=1. (e) Posterior estimates of the conditional error volatility for Equation 7 under Equations3 and 4 using mont hlyS&P

500 realized volatility at h=1. (f) Posterior estimates of the persistence of realized volatility for Equation7 under Equations 3 and 4 conditional on

two lags of volatility using monthly S&P 500 realized volatility at h=1. In panels (c), (e), and (f) blacklines indicate the poster ior mean estimates.

Dotted lines are the respective 25%and 75%posterior percentiles. In the panels, gray vertical lines indicate business cycle peaks, that is, the point at

which an economic expansion transitions to a recession, based on National Bureau of Economic Research (NBER) business cycle dating

intraperiod data. This proxy is labeled as realized volatil-

ity (RV; see Andersen, Bollerslev, Diebold, & Labys, 2001a,

2001b; Barndorff-Nielsen & Shephard, 2002a, 2002b). Real-

ized volatility is essentially a nonparametric estimate of ex

post volatility.Fur thermore, by treating volatilityas obser ved

rather than latent, we are able to model and forecast it

using simple time series techniques such as linear regression

models (see, e.g., Andersen, Bollerslev, & Diebold, 2007;

Christensen, Schmeling, & Schrimpf, 2012; Corsi, 2009; Liu

& Maheu, 2008, 2009; Paye, 2012).

In panels (a) and (b) of Figure 1 we plot the monthly and

quarterly S&P 500 realized volatility from 1960 to 2014,

see Section 2 with regards to how we define this quantity.

Clearly,realized volatility is persistent (with different degrees

of persistence) and tends to be higher during recessions than

tranquil periods. Moreover, panels (a) and (b) lead us to sus-

pect that there is a connection between realized volatility and

financial and macroeconomic variables. For example, in panel

(b), we observe that volatility and US gross domestic prod-

uct (GDP) track one another closely these figures suggest that

(i) accounting for time variation in the parameters that govern

realized volatility dynamics and (ii) discovering the link

between how future volatility responds to changes in the

underlying macroec onomic conditions can improve forecast

accuracy.