A Filtering Process to Remove the Stochastic Component from Intraday Seasonal Volatility

• Author: Robert T. Daigler,Jang Hyung Cho
• DOI: http://doi.org/10.1002/fut.21585
• Published date: 01 May 2014
• Date: 01 May 2014

AFILTERING PROCESS TO REMOVE THE

STOCHASTIC COMPONENT FROM INTRADAY

SEASONAL VOLATILITY

JANG HYUNG CHO and ROBERT T. DAIGLER*

The intraday seasonal variance pattern contains stochastic as well as deterministiccomponents.

Therefore, the estimation of information arrivals in the associated volatility process requires the

proper filtering of both of these seasonal components. However, popular current models remove

only the deterministic part of the typical U‐shape volatility. Here, we provide the first empirical

results of the importance of the stochastic component, as developed by Cho and Daigler (2012).

We show that a highly significant additional 8.5% to 12.9% of the total seasonal variance is

explained by the stochastic seasonal variance component for S&P500 futures, live cattle futures,

and the Japanese yen‐U.S. dollar spot exchange rate. Moreover, we show that the stochastic

seasonal filtering model implemented here does not create any statistical distortions of the

filtered series, as occurs with deterministic‐based seasonal adjustment processes, as well as

comparing the model examined here with the most popular current deterministic model. As part

of our analysis we examine the application of the model to macroeconomic news and out‐of‐

sample results for the model. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 34:479–495, 2014

1. INTRODUCTION

An understanding of the volatility process of a time series is important, because the volatility

process reflects the impact of information arrivals (Andersen & Bollerslev, 1997b; Gross-

man, 1976), as well as impacting the price of derivative instruments. However, it is difficult to

analyze the volatility arising from intraday information events because the informational

volatility is obscured by the inherent (daily) seasonal volatility component. In particular,

empirical findings show that a consistent daily U‐shaped seasonal volatility pattern exists for

various financial markets (Wood, McInish, & Ord, 1985; Lockwood & Linn, 1990; Daigler,

1997). Consequently,it is crucial to employ a proper seasonal adjustment procedure to identify

the size of both the seasonalized and the deseasonalized volatility components, especially

surrounding an information event.

Several methods exist to filter the seasonal variance components from the total variance

process, as discussed in section 2. We classify these methods into deterministic and stochastic

Jang Hyung Cho is an Assistant Professor of Finance, Department of Accounting and Finance, BT 854,

College of Business, San Jose State University, San Jose, California. Robert Daigler is Knight Ridder Research

Professor of Finance, Chapman Graduate School of Business, Florida International University, Miami,

Florida. We would like to thank Paul Pfleiderer for his encouragement and Zhiyao Chen and Zhiguang Wang for

suggestions on earlier versions. We also wish to thank the reviewer, whose extensive comments substantially improved

the quality of the study. An earlier draft of this study was presented at the Financial Management Meetings.

*Correspondence author, Department of Finance, RB 206, College of Business, Florida International University,

Miami, FL 33199. Tel: 305‐348‐3325, Fax: (305) 348-4245, e‐mail: daiglerr@fiu.edu

Received November 2011; Accepted September 2012

The Journal of Futures Markets, Vol. 34, No. 5, 479–495 (2014)

© 2012 Wiley Periodicals, Inc.

Published online 26 November 2012 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21585

seasonal adjustment procedures. The deterministic seasonal adjustment models, such as the

flexible Fourier form model (FFF), are widely used in empirical analysis, as explained below.

However, the intraday seasonal volatility pattern also possesses stochastically time‐varying

components, causing these stochastic seasonal components remaining in a deterministic only

seasonally adjusted series to create statistical distortions. Therefore, one must filter out the

stochastic components as well as the deterministic components from an intraday volatility

series. Given the importance of filtering both types of seasonal variance components, it is

surprising that no models or empirical studies exist that examine the stochastic seasonal

variance component. In fact, important economic and statistical reasons exist to adjust for the

significant seasonal component inherent in the U‐shape volatility curve. In particular,

Granger (1976) points out that the seasonal component obscures the movements of

economically important variables, as well as making it difficult to determine local volatility

trends. In addition, the seasonal component causes nonstationarity (Lutkepohle, 2007) and a

spurious relation between two time series if they share a common seasonal component

(Granger, 1976; Hylleberg, 1986).

This study estimates the stochastic seasonal variance components in a financial time

series, as well as separating the stochastic seasonality from the deterministic seasonality. In

addition, we compare the seasonal adjustment performance of the stochastic filtering

model presented here to the most commonly used deterministic model, namely the FFF

model. In order to estimate both the stochastic and deterministic seasonal variance series

simultaneously, we employ the Autoregressive Conditional Seasonal Variance (ARCSV)

theoretical process developed by Cho and Daigler (2012). Unlike other seasonal adjustment

procedures, the ARCSV model captures the stochastic (as well as the deterministic)

components in the seasonal pattern by assuming that the intraday seasonal variance in each

intraday time interval follows a unique autoregressive moving average (ARMA) process.

Moreover, Cho and Daigler (2012) prove that the ARCSV model does not impound statistical

distortions into the seasonally adjusted series from the filtering process, as found in other

filtering methods.

This study adds to the literature by (1) showing how to implement and use the Cho and

Daigler (2012) model for filtering seasonality, (2) empirically determining the amount of

variance that is stochastic, as well as deterministic portion, in validating the ARCSV model, (3)

illustrating by direct comparison how the ARCSV model is superior to the flexible Fourier form

(FFF) model, (4) applying the ARCSV model to macroeconomic news and providing out‐of‐

sample results, and (5) thereby showing how the AACSV model is an appropriate tool to

analyze the volatility components of high frequency data.

Our results provide the first empirical evidence for the ARCSV model, showing that the

stochastic component accounts for an average of 10.14% of the total seasonal variance

processes for the S&P500 and live cattle futures and the yen versus dollar exchange rate

(specifically 9.04%, 12.90%, and 8.48%, respectively). We also provide a unique finding that

the importance of the stochastic seasonal variance component increases when the effects of

macroeconomic news are incorporated into the filtering model. Testing for potential

distortions using spectral tests shows no statistical distortions in the resultant filtered time

series. We also provide the first empirical evidence that nonstochastic modifications to

deterministic filtering models (without including a seasonal shock term to capture the time‐

varying seasonal patterns) leads to statistical distortions in the resultant series. In particular,

the FFF model creates statistical distortions for the filtered time series if the interaction terms

between the daily volatility and the seasonal sinusoid terms are included to capture the time‐

varying seasonal pattern. Furthermore, purely deterministic filtering models that do not

include interaction terms (the FFF model in Andersen and Bollerslev (1997a) can be modeled

without these terms) and the variance mean filtering model in Engle, Sokalska, and Chanda

480 Cho and Daigler