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University of T oronto

School of Economics and Management

Tsinghua University

John M. Maheu

Dept. of Economics

University of Toronto

This version: March 2008

Abstract

How to measure and model volatility is an important issue in finance. Recent

research uses high frequency intraday data to construct ex post measures of daily

volatility. This paper uses a Bayesian model averaging approach to forecast realized

volatility. Candidate models include autoregressive and heterogeneous autoregres-

sive (HAR) specifications based on the logarithm of realized volatility, realized power

variation, realized bipower variation, a jump and an asymmetric term. Applied to

equity and exchange rate volatility over several forecast horizons, Bayesian model

averaging provides very competitive density forecasts and modest improvements in

point forecasts compared to benchmark models. We discuss the reasons for this,

including the importance of using realized power variation as a predictor. Bayesian

model averaging provides further improvements to density forecasts when we move

away from linear models and average over specifications that allow for GARCH

effects in the innovations to log-volatility.

∗We are grateful to Olsen Financial Technologies GmbH, Zurich, Switzerland for making the high

frequency FX data available. We thank the co-editor, Tim Bollerslev, and three anonymous referees for

many helpful comments. We thank Tom McCurdy who contributed to the preparation of the data used in

this paper, and the helpful comments from Doron Avramov, Chuan Goh, Raymond Kan, Mark Kamstra,

Gael Martin, Alex Maynard, Tom McCurdy, Angelo Melino, Neil Shephard and participants of the Far

Eastern Meetings of the Econometric Society, Beijing, and China International Conference in Finance,

Xi’an. Maheu thanks the SSHRCC for financial support.

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1Introduction

How to measure and model volatility is an important issue in finance. Volatility is latent

and not observed directly. Traditional approaches are based on parametric models such

as GARCH or stochastic volatility models. In recent years, a new approach to model-

ing volatility dynamics has become very popular which uses improved measures of ex

post volatility constructed from high frequency data. This new measure is called real-

ized volatility (RV) and is discussed formally by Andersen, Bollerslev, Diebold and Labys

(2001), Andersen, Bollerslev, Diebold and Ebens (2001) and Barndorff-Nielsen and Shep-

hard (2002a,2002b).1RV is constructed from the sum of high frequency squared returns

and is a consistent estimator of integrated volatility plus a jump component for a broad

class of continuous time models. In contrast to traditional measures of volatility, such as

squared returns, realized volatility is more efficient. Recent work has demonstrated the

usefulness of this approach in finance. For example, Bollerslev and Zhou (2002) use real-

ized volatility to simplify the estimation of stochastic volatility diffusions, while Fleming,

This paper investigates Bayesian model averaging for models of volatility and con-

tributes to a growing literature that investigates time series models of realized volatility

and their forecasting power. Recent contributions include Andersen, Bollerslev, Diebold

and Labys (2003), Andersen, Bollerslev and Meddahi (2005), Andreou and Ghysels (2002),

Koopman, Jungbacker and Hol (2005), Maheu and McCurdy (2002), and Martens, Dijk

and Pooter (2004). These papers concentrate on pure time series specifications of RV,

however, there may be benefits to model averaging and including additional volatility

proxies.

Barndorff-Nielsen and Shephard (2004) have defined several new measures of volatility,

and associated estimators. Realized power variation (RPV), is constructed from the sum

of powers of the absolute value of high frequency returns. This is a consistent estimator

of the integral of the spot volatility process raised to a positive power (integrated power

variation). Realized bipower variation, which is defined as the sum of the products of

intraday adjacent returns, is a consistent estimator of integrated volatility.

There are several reasons why RPV may improve the forecasting of volatility. Barndorff-

Nielsen and Shephard (2004) show that power variation is robust to jumps. Jumps are

generally large outliers that may have a strong effect on model estimates and forecasts.

Second, the absolute value of returns displays stronger persistence than squared returns

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Building on this work, we show empirically for data from equity and foreign exchange

markets that persistence is highest for realized power variation measures. The correlation

between realized volatility and lags of realized power variation as a function of the order

p, is maximized for 1.0 ≤ p ≤ 1.5, and not p = 2, which corresponds to realized volatility.

Compared with models using just realized volatility, daily squared returns or the intraday

range, we find that power variation and bipower variation can provide improvements.

These observations motivate a wide range of useful specifications using realized volatil-

ity, power variation of several orders, bipower variation, a jump and an asymmetric term.

We focus on the benefits of Bayesian model averaging (BMA) for forecasts of daily, weekly

and biweekly average realized volatility. BMA is constructed from autoregressive type

parameterizations and variants of the heterogeneous autoregressive (HAR-log) model of

Corsi (2004) and Andersen et al. (2007) extended to include different regressors. Choos-

ing one model ignores model uncertainty, understates the risk in forecasting and can lead

to poor predictions (Hibon and Evgeniou (2004)).3BMA combines individual model fore-

casts based on their predictive record. Therefore, models with good predictions receive

large weights in the Bayesian model average.

We compare models’ density forecasts using the predictive likelihood. The predictive

likelihood contains the out-of-sample prediction record of a model, making it the central

quantity of interest for model evaluation (Geweke and Whiteman (2005)). The empirical

results show BMA to be consistently ranked at the top among all benchmark models,

including a simple equally weighted model average. Considering all data series and fore-

cast horizons, the BMA is the dominate model. Although there are substantial gains in

BMA based on density forecasts, point forecasts using the predictive mean show smaller

improvements.

The importance of GARCH dynamics in time series models of log-realized-volatility

has been documented by Bollerslev et al. (2007). We find that Bayesian model averaging

provides further improvements to density forecasts when we move away from linear models

and average over specifications that allow for GARCH effects. For example, it provides

improvements relative to a benchmark HAR-log-GARCH model for daily density forecasts.

There are two main reasons why BMA delivers good performance. First, we show

that no single specification dominates across markets and forecast horizons. For each

market and forecast horizon there is considerable model uncertainty in all our applica-

The ranking of individual models can change dramatically over data series and forecast

horizons. Bayesian model averaging provides an optimal way to combine this informa-

tion.4The second reason, is that based on the predictive likelihood, including RPV terms

can dramatically improve forecasting power. Although specifications with RPV terms

also display considerable model uncertainty, BMA gives them larger weights when they

perform well.

3Recent examples of Bayesian model averaging in a macroeconomic context include Fern´ andez, Ley,

and Steel (2001), Jacobson and Karlsson (2004), Koop and Potter (2004), Pesaran and Zaffaroni(2005)

and Wright (2003).

4Based on a logarithmic scoring rule, averaging over all the models provides superior predictive ability

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The relative forecast performance of the specifications that enter the model average is

ordered as follows. As a group, models with RPV regressors deliver forecast improvements.

Bipower variation delivers relatively smaller improvements over models with only realized

volatility regressors. A realized jump term which is constructed from bipower variation

is important in all model formulations.

This paper is organized as follows. Section 2 discusses the econometric issues for

Bayesian estimation and forecasting. Section 3 reviews the theory behind the improved

volatility measures: realized volatility, realized power variation and realized bipower vari-

ation. Section 4 details the data and the adjustment to RV and realized bipower variation

in the presence of market microstructure noise. The selection of regressors is discussed in

Section 5. Section 6 presents the different configurations that enter the model averaging

while Section 7 discusses forecasting results as well as the role of realized power variation,

and the performance of BMA when allowing for GARCH effects. The last section con-

cludes. An appendix explains how to calculate the marginal likelihood, and describes the

algorithm to estimate volatility models with GARCH innovations.