论文研究-使用超高频数据评估波动率预测—来自澳大利亚股票市场的证据

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Theoretical Economics Letters, 2018, 8, 1-27

http://www.scirp.org/journal/tel

ISSN Online: 2162-2086

ISSN Print: 2162-2078

DOI:

10.4236/tel.2018.81001 Jan. 4, 2018 1 Theoretical Economics Letters

Evaluating Volatility Forecasts with

Ultra-High-Frequency Data—Evidence

from the Australian Equity Market

Kai Zhang

, Lurion De Mello, Mehdi Sadeghi

Department of Applied Finance and Actuarial Studies, Macquarie University, Sydney, Australia

Abstract

Due to the unobserved nature of the true return variation process, one of the

most challenging problems in evaluation of volatility forecasts is to find an

accurate benchmark proxy for

post

volatility. This paper uses the Austra

ian equity market ultra-high-frequency data to construct an unbiased

post

volatility estimator and then use it as a benchmark to evaluate various practi

al volatility forecasting strategies (GARCH class model based). These for

casting strategies allow for the skewed distribution of in

novations and use

various estimation windows in addition to the standard GARCH volatility

models. In out-of-sample tests, we find that forecasting errors across all model

specifications are systematically reduced if using the unbiased

post

volati

ity estimator compared with those using the realized volatility based on

sparsely sampled intra-day data. In particular, we show that the three benc

mark forecasting models outperform most of the modified strategies with di

ferent distribution of returns

and estimation windows. Comparing the three

standard GARCH class models, we find that the asymmetric power ARCH

(APARCH) model exhibits the best forecasting power in both normal and f

nancial turmoil periods, which indicates the ability of APARCH model to

capture the leptokurtic returns and stylized features of volatility in the Au

tralian stock market.

Keywords

High-Frequency Volatility, Volatility Forecasting, GARCH,

Volatility Forecast

Evaluation

1. Introduction

Volatility forecasting is a critical task in a variety of financial activities for

How to cite this paper:

Zhang, K.,

Mello

, L. and Sadeghi, M. (2018) Evaluat-

ing Volatility For

ecasts with Ultra-High-

Frequency D

ata—Evidence from the Aus-

tralian Equity Market

Theoretical Ec

nomics Letters

, 1-27.

https://doi.org/10.4236/tel.2018.81001

Received:

November 10, 2017

Accepted:

January 1, 2018

Published:

January 4, 2018

8 by authors and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

K. Zhang et al.

DOI:

10.4236/tel.2018.81001 2 Theoretical Economics Letters

different financial instruments and markets around the world. Important

financial decisions such as portfolio optimisation, derivative pricing, risk

management and financial regulation heavily depend on volatility forecasts. In

derivative pricing, such as in the estimation of the Black-Scholes option pricing

model, volatility is the only parameter that needs to be forecasted. The

prediction of volatility is also crucial in development of Value at Risk (VaR) and

a variety of systemic risk models, as well as in banking and finance regulations.

For example, according to the Basel Accord II and III, it is compulsory for all

financial institutions to predict the volatility of their financial assets to

incorporate the risk exposure of their capital requirements.

The focus of our study is on the predictive ability of the popular

AutoRegressive Conditional Heteroscedastic (ARCH) class of models that

originated from a seminal Nobel Prize-wining article by [1] [2] generalized his

framework to obtain the GARCH model. ARCH and GARCH models are

popular and standard volatility forecasting models in econometrics and finance.

Documented stylized features of variation such as the clustering and long

memory effect can be captured by GARCH class models, and the model

parameters are relatively easy to estimate. A comprehensive survey of the

GARCH family of models can be found in [3]. The current study selects three

popular GARCH class models from the literature, including the standard

GARCH, the threshold GARCH (TGARCH) of [4] and [5], and the asymmetric

power ARCH (APARCH) of [6]. In addition to the three standard models, we

consider 12 corresponding forecasting strategies associated with them, which

involves different estimation windows and errors distributions.

The predictive power of a volatility model is evaluated based on an

out-of-sample test in which the predicted volatility generated from the model is

compared with the

post

volatility measurements. Superior volatility forecasting

models are supposed to have small forecasting errors, measured as the difference

between the predicted and actual volatility. However, unlike the return, the

volatility process cannot be observed. Therefore, in out-of-sample evaluation of

volatility forecasting models, the crucial task is to find an accurate proxy for the

underlying unobserved volatility process. In the mid-1990s, a series of empirical

studies noted that while GARCH-type models are for fitting the time series

returns, they failed to explain much of the variability in

post

volatility

measured by the squared returns in out-of-sample tests. Hence, the practical

usefulness of GARCH models was challenged. [7] responds to the critique of the

model and argues that the unsatisfied empirical results are due to the noisy

volatility proxies used in these studies, that is, squared returns or absolute

returns. In out-of-sample forecast evaluation, a common approach for

evaluating the practical performance of any model is to compare the fitted

predictions derived from the model with the subsequent realizations. However,

volatility is not directly observed and dealt with as a latent variable in financial

modelling. Therefore, the squared innovation return is usually employed as a

K. Zhang et al.

DOI:

10.4236/tel.2018.81001 3 Theoretical Economics Letters

proxy for volatility realization. It is empirically documented that the correlation

between the predicted and the realized value is weak, with low

values.

However, this finding is not surprising and is considered evidence of the poor

predicting ability of a volatility model. Although in statistical terms, the squared

innovation return is an unbiased estimator of underlying variance, it is not an

accurate estimator and displays a large degree of observation-by-observation

variation. In summary, empirical findings of poor forecasting performance are

due to unsatisfied volatility proxies rather than the predictive power of GARCH

class models.

The recent availability of nearly continuous high-frequency transaction data

provides a chance to explicitly compute the market volatility by using intra-day

data, referred to as realized volatility in the literature. [7] shows that the basic

GARCH(1,1) model performs rather well when it is evaluated with a volatility

measurement constructed using high-frequency data. This stems from the fact

that high-frequency volatility is a more accurate measure of daily volatility

compared with those estimated using low-frequency data. Instead of only using

the opening or closing price of a trading day in squared daily returns,

high-frequency volatility exploits more information contained in intra-day

trading data. In the literature, five-minute intra-day data are popularly used to

construct realised volatility. However, in our study, we show that while

contamination with microstructure noise can be reduced if realised volatility is

constructed based on sparsely sampled high-frequency data, it is still a biased

estimator.

Because of the crucial role of

post

volatility measurement in evaluating

forecasting performance, in addition to using five-minutes tick data we calculate

the realized variance using ultra-high-frequency data and relying on the Two

Time Scaled Realized Volatility (TSRV) estimator proposed in the study of [11]

and [12], which is shown to be an unbiased

post

variance estimator. Unlike

an arbitrarily and subjectively selected sample frequency such as five minutes,

the TSRV employs all available high-frequency intra-day data and therefore

exploits full information about return variation contained in the

ultra-high-frequency data. Our results show that forecasting errors relying on

TSRV and ultra-high-frequency data are significantly lower than those based on

sparsely sampled intra-day data.

This paper is an attempt to mimic volatility forecasting strategies using

GARCH family models in practice, and examine natural questions arising from

employing such strategies in the stock market in Australia. These questions

include, Which model provides the smallest error? Should we allow for

heavy-tailed distributed innovations? What is the best choice for the estimation

window? Would it be a growing window, a longer rolling window or a shorter

rolling estimation window? Moreover, the performance of these volatility

forecasting strategies in periods of financial turmoil such as the Global Financial

The empirical evidence can be found in [8] [9] and [10].

K. Zhang et al.

DOI:

10.4236/tel.2018.81001 4 Theoretical Economics Letters

Crisis (GFC) of 2008 are of concern for financial practitioners. The current

paper responds to these questions by examining the empirical predictive power

of various strategies in pure out-of-sample tests, relying on a recently developed

unbiased

post

volatility proxy and using ultra-high-frequency trading data in

the Australian equity market.

Our research is also motivated by the fact that no such study has previously

been conducted on the Australian stock market. [13] model and evaluate the

monthly volatility forecasting techniques using Australian low-frequency data.

They identify several unique features that distinguishing the Australian equity

stock market from other stock markets. For instance, the top 20 firms listed on

the Australian Stock Exchange (ASX) own 80% of the total assets in the market,

whereas in the United States, the top 30 companies on the Dow Jones Industrial

Average (DJIA) index hold only 65% of the total assets. The Australian market is

dominated by energy and resources firms ([14]), which are more volatile than

industrial shares. However, regulatory guidelines developed by the Australian

Security and Investment Commission (ASIC) require disclosure of more

financial information for investors to make decisions than is required in some

other countries. This may reduce the level of information asymmetry and

expected volatility in the stock market. Overall, Australian institutional features

and settings are quite unique compared with other markets, therefore,

examining volatility forecasting issues in Australian provides particular evidence

in relation to the related literature.

In our empirical analysis using ultra-high-frequency data from the Australian

stock market, we evaluate the predictability of three commonly used

GARCH-type models and four variations of each. The three models are GARCH,

TGARCH and APARCH

. The benchmark forecasting models are each

GARCH-type model estimated with a growing estimation window and Gaussian

innovations, and the variations are the models estimated with rolling forecasting

windows (one-year and three-years rolling windows) and skewed and

heavy-tailed innovation distributions (Student-t innovations and skew Student-t

innovations). We firstly compare the predictive abilities of the variations of each

separate GARCH model to the benchmark GARCH class model, which is

estimated with normally distributed innovations in the full sample period from

January 2005 to December 2013. Then we proceed by directly comparing the

forecasting accuracy of each benchmark GARCH model to determine which one

provides the best predictive performance. The forecasting ability comparison is

based on two commonly used loss functions in the literature: the Mean Square

Error (MSE) and the Quasi-like Loss function (QL). The Diebold-Mariano test is

also implemented to examine the statistical significance of the improved forecast

accuracy. Our results show that for each individual GARCH-type model, the

benchmark usually is not significantly outperformed by the modifications. The

exception is that the skew Student t distribution improves the forecast accuracy

The Asymmetric Power ARCH model at least nests seven other GARCH-type models, see [6].

K. Zhang et al.

DOI:

10.4236/tel.2018.81001 5 Theoretical Economics Letters

of the standard GARCH model. In the direct comparison of the three benchmark

GARCH class of models, the APARCH usually gives the best forecasts in the

out-of sample period from 2005 to 2013

. We note that the volatility forecasting

procedure presented in this study is not limited to GARCH class models. In

practice, the out-of-sample predictive performance of any volatility model can

be evaluated based on our procedure.

The reminder of the paper is organized as follows: Section 2 provides a brief

discussion of the theory around measuring high-frequency volatility. Section 3

and Section 4 respectively illustrate forecasting models and detailed procedures

to compare prediction performance. Section 5 describes the daily data and tick

data used in this paper. The main empirical results are presented in Section 6

and conclusions are drawn in Section 7.

2. Measurement of Ex-Post Daily Volatility

2.1. Theoretical Set-Up

Unlike for raw return, the actual daily return volatility process usually cannot be

directly observed because there is just one daily return per trading day.

Conventionally, volatility is treated as a latent variable

in parametric models

such as GARCH-type and Stochastic volatility (SV) models that are inferred

from

post

low-frequency return data. Volatility measurement by these

models is based on specific distributional assumptions and usually involves

complex procedures in estimating model parameters. [15] introduced the

concept of realized volatility for the first time.

The realized volatility is a

non-parametric estimator that does not rely on the distribution of parameters.

In the standard form, realized volatility is the second-order sample moment, that

is, the sum of squares of the high-frequency returns over a fixed period, say, one

day.

t ti

RV r

∑

where

,ti

is the

ith

high-frequency return for day

In financial asset pricing models, the asset price is assumed to be driven by a

continuous time diffusion process,

( ) ( ) ( ) ( )

d ddXt tt tWt

µσ

= +

where

( )

is log price,

( )

is standard Brownian Motion, and

( )

and

We also examine the performance of our selected volatility forecasting strategies during the fina

cial turmoil period. The same procedure above is applied to the period of GFC in 2008. We choose

three months after the date that the Lehman Brothers Holdings filed for Chapter 11 of the United

States bankruptcy protection code as the test

period. We compare the forecast performance of

GARCH models in the crisis period with three months in early 2008 which has much lower unco

ditional daily volatility. We find that during financial turmoil, the degree of forecast losses are si

nificantly inc

reased, however, the overall ranking of the forecast does not change. The APARCH

provides the best forecast across all cases suggesting the importance of the role of negative returns in

predicting future volatility when the market is down.

Latent variables cannot be directly observed and are estimated by using other observed variables

In literature, realized volatility and realized variance are often used interchangeably.

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