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This is to certify that the
dissertation entitled

FINANCIAL ECONOMETRIC MODELING OF RISK IN
COMMODITY MARKETS

presented by

Jeongseok Song a_

has been accepted towards fulfillment
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FINANCIAL ECONOMETRIC MODELING OF RISK IN COMMODITY MARKETS

By

Jeongseok Song

A DISSERTATION

Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

2004

ABSTRACT

FINANCIAL ECONOMETRIC MODELING OF RISK IN COMMODITY MARKETS
By

Jeongseok Song

This dissertation is composed of three interrelated body chapters. Its goal is to
identify underlying sources of return volatility movement and analyze important
problems in the economics of commodity markets by applying various time series
econometric models to commodity market price data.

Chapter 2 investigates stochastic properties of daily cash price changes for six
commodities: corn, soybeans, live cattle, live hogs, unleaded gasoline, and gold. We use
the FIGARCH conditional variance model and the semi-parametric local Whittle
estimation method to explore the daily cash return volatility behavior. We apply the long
memory models to the temporally aggregated daily cash returns and compare the
volatility dynamics at various sample frequencies.

Chapter 3 is concerned with commodity futures return volatility at daily and
higher sample frequencies. In particular, strong intra-day periodicity in the high
frequency return volatility is observed. We examine the high frequency futures return
volatility pattern after removing the intra-day seasonality using the Flexible Fourier Form
(FF F) filter and compare the volatility movement with the daily futures return volatility

process.

Chapter 4 introduces a newly suggested volatility measure, the realized volatility,
and applies the volatility measure to commodity futures market price data. The realized
volatility is calculated as the sum of high frequency squared returns and exhibits some
ideal statistical properties. Taking advantage of the stochastic properties of the realized
volatility measure allows us to study important economic determinants for commodity

futures market risk behavior.

Dedicated to My Parents

iv

Acknowledgements

This dissertation would not have been completed without support of many people.
However, my first and greatest thanks must go to my advisor, Professor Richard T.
Baillie for his superb guidance and encouragement. He provided insightful and

invaluable advices as well as financial supports during my graduate studies.

Also, I wish to thank Professor Robert Myers in agricultural economics
department for extending my knowledge to broader area and suggesting many helpful
comments. In addition, I am grateful to Professors Christine Amsler and Anna Maria
Herrera in economics department for their patience and generosity with respect to my

dissertation drafts.

Due to too many to list, I mention a few of colleagues in alphabetical order of
their last names to thank: Wooseok Choi, Young-Wook Han, Seung Won Kim, Jaewha
Lee, Han Sang Yi, and Kyeongwon Yoo. They delighted and assisted me for a variety of

reasons through a long journey of graduate studies.

My parents have waited for me to complete my graduate studies for a decade. It

is impossible for me to compensate their care and support by any means.

My final thanks remain for Minkyung, my beloved daughter for yielding her time
to play with me, and Minah Park, my wife, for doing many of my duties so as to let me

concentrate on my works.

[TABLE OF CONTENTS]

LIST OF TABLES ................................................................................ viii
LIST OF FIGURES ................................................................................ xii
CHAPTER 1. INTRODUCTION .................................................................. 1

CHAPTER 2. MODELING COMMODITY CASH RETURNS

2.1 . Introduction .............................................................................. 4
2.2. Long Memory, Temporal Aggregation, and Self-Similarity ..................... 6
2.3. Application to Daily Cash Returns ................................................ 16
2.4. Conclusion ............................................................................. 21

CHAPTER 3. MODELING DAILY AND HIGH FREQUENCY COMMODITY

FUTURES RETURNS
3.1 . Introduction ............................................................................ 34
3.2. Analysis of Daily Commodity Returns ............................................. 37
3.3. Analysis of High Frequency Commodity Returns ................................ 48
3.4. Local Whittle Estimation and Self Similarity ..................................... 52
3.5. Conclusion ............................................................................. 54
Appendix ........................................................................... 73

CHAPTER 4. REALIZED VOLATILITY IN COMMODITY FUTURES MARKETS

vi

 

4.1 . Introduction ............................................................................ 88

4.2. Statistical Foundations of Realized Volatility .................................... 89
4.3. Practical Issues in the Calculation of Realized Volatility ....................... 97

4.4. Stochastic Properties for Realized Volatility for Modeling and Forecast. . ...99

4.4.1. The Distributional Facts of the Realized Volatility ................... 99
4.4.2. The Long Memory of the Realized Volatility ........................ 101
4.4.3. Forecast for the Realized Volatility ................................... 102
4.5. Economic Factors for the Realized Commodity Futures Volatility .......... 105
4.5.1 Announcement effects ................................................... 106
4.5.2. Time-to-Maturity and Information Flow .............................. 109

4.5.3. Dependence between the Realized Volatilities for Different

Commodities ............................................................ 1 14

4.6. Conclusion ........................................................................... 1 17
CHAPTER 5. CONCLUSION .................................................................. 151
LIST OF REFERENCES ........................................................................ 155

vii

[LIST OF TABLES]

[CHAPTER 2]

Table 2-1: Estimated MA-FIGARCH Models for Daily Cash Returns for Corn .......... 23
Table 2-2: Estimated MA-FIGARCH Models for Daily Cash Returns for Soybean ...... 24
Table 2-3: Estimated MA-FIGARCH Models for Daily Cash Returns for Cattle ......... 25
Table 2-4: Estimated MA—FIGARCH Models for Daily Cash Returns for Hog ....... 26-27
Table 2-5: Estimated MA-FIGARCH Models for Daily Cash Returns for Gasoline ...... 28
Table 2-6: Estimated MA-FIGARCH Models for Daily Cash Returns for Gold ........... 29

Table 2-7. Semi-Parametric Long Memory Parameter Estimation:

Absolute Daily Cash Returns at Different Daily Sample Frequencies ......... 30
[CHAPTER 3]
Table 3-1: Sample Periods and Summary Statistics ............................................ 56
Table 3-2: MA-FIGARCH Estimation Results for Daily Futures Returns .................. 57

Table 3-3: Long Memory Parameter Estimation at Different Daily Sample
Frequencies ............................................................................. 58
Table 3-4: Estimated MA(1)-FIGARCH(1,d,1) model for Filtered High Frequency
Futures Returns ........................................................................ 59
Table 3-5: Long Memory Parameter Estimation at Different Intraday Sample
Frequencies ............................................................................. 60
Table A-1: Estimated MA-FIGARCH Models for Temporally Aggregated Daily Futures
Returns for Corn ....................................................................... 76

Table A-2: Estimated MA-FIGARCH Models for Temporally Aggregated Daily Futures
Returns for Soybean .................................................................. 77

viii

 

Table A-3: Estimated MA-FIGARCH Models for Temporally Aggregated Daily Futures
Returns for Live Cattle ................................................................ 78
Table A-4: Estimated MA-FIGARCH Models for Temporally Aggregated Daily Futures
Returns for Live Hogs ................................................................. 79
Table A-5: Estimated MA-FIGARCH Models for Temporally Aggregated Daily Futures
Returns for Gasoline ................................................................... 80
Table A-6: Estimated MA-FIGARCH Models for Temporally Aggregated Daily Futures
Returns for Gold ........................................................................ 81
Table A-7: Estimated MA-FIGARCH model for Temporally Aggregated Filtered High
Frequency Futures Returns for Corn ................................................ 82
Table A-8: Estimated MA-FIGARCH Model for Temporally Aggregated Filtered High
Frequency futures returns for Soybean ............................................. 83
Table A-9: Estimated MA-FIGARCH Model for Temporally Aggregated Filtered High
Frequency futures for Life Cattle futures .......................................... 84
Table A-10: Estimated MA-FIGARCH Model for Temporally Aggregated Filtered High
Frequency futures for Life Hog futures ............................................ 85
Table A-1 1: Estimated MA-FIGARCH Model for Temporally Aggregated Filtered High
Frequency futures for Gasoline futures ............................................ 86
Table A-12: Estimated MA-FIGARCH Model for Temporally Aggregated Filtered High
Frequency futures for Gold fiItures ................................................ 87

[CHAPTER 4]

ix

Table 4-1: Basic Descriptive Statistics: Unconditional Distribution of Daily Commodity
Futures Returns ....................................................................... 1 19
Table 4-2: Basic Descriptive Statistics: Distribution of Realized Volatility ............... 119

Table 4-3: Basic Descriptive Statistics: Daily Returns Standardized by Realized

Volatility ............................................................................... 119
Table 4-4: ARFIMA(0,d,0) Estimation for Realized Volatility Series ..................... 121
Table 4-5: The Local Whittle Estimation for the Long Memory Parameter ............... 122
Table 4-6: Mincer-Zarnowitz Regressions for Realized Volatilities ....................... 123

Table 4-7 (a): ARFIMA(0,d,O) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity
Markets: Corn ........................................................................ 125
Table 4-7 (b): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence
between Commodity Markets: Corn .............................................. 127
Table 4-8 (a): ARFIMA(O,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity
Markets: Soybean .................................................................... 129
Table 4-8 (b): ARFIMA(0,d,O) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence
between Commodity Markets: Soybean .......................................... 130
Table 4-9 (a): ARFIMA(O,d,O) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity

Markets: Cattle ........................................................................ 131

Table 4-9 (b): ARFIMA(0,d,0) Estimation forithe Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence
between Commodity Markets: Cattle .............................................. 132
Table 4-10 (a): ARFIMA(O,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity
Markets: Gasoline .................................................................... 133
Table 4-10 (b): ARFIMA(0,d,O) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence
between Commodity Markets: Gasoline ............... , ........................... 134
Table 4-11 (a): ARFIMA(0,d,O) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity
Markets: Gold ......................................................................... 135
Table 4-11 (b): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence
between Commodity Markets: Gold .............................................. 136

Table 4-12: Correlation among the realized volatility, squared daily return, trading

intensity, and time-to-maturity ..................................................... 137
Table 4-13: VAR Parameter Estimates (regression form) ............................ 138-139
Table 4-14: Correlation matrix for six realized volatility series ........................... 14]

xi

[LIST OF FIGURES]

[CHAPTER 2]

Figure 2-1 Correlograms for Absolute Daily Cash Returns ................................. 31
[CHAPTER 3]

Figure 3-1. Autocorrelation of Daily Live Cattle Futures .................................... 62
Figure 3-2. Autocorrelation of Daily Corn Futures ............................................ 63
Figure 3-3. Autocorrelation of Filtered Daily Corn Futures .................................. 64
Figure 3-4. Autocorrelation of Daily Soybean Futures ....................................... 65
Figure 3-5. Autocorrelation of Filtered Daily Soybean Futures .............................. 66

Figure 3-6 Correlograms for Absolute Raw and Filtered F ive-minute Returns ....... 67-69

Figure 3-7 Fitted Intraday Volatility Pattern by the F FF filtering ....................... 70-72
[CHAPTER 4]
Figure 4-1(a). Kernel Density for Realized Volatility .................................. 142-144

Figure 4-1(b). Kernel Density for Daily Returns Standardized by Realized
Volatility ........................................................................ 145-147

Figure 4-2 Realized Commodity Volatility Level ....................................... 148-150

xii

CHAPTER 1

INTRODUCTION

This dissertation is concerned with the application of some modern financial
econometric techniques to daily and high frequency commodity markets. The
econometric methods are applied to the cash and futures markets. These cash and futures
markets are an active and important financial institution in the modern economy, and the
volatility associated with commodity futures markets is an important factor for study in
risk management and in commodity trading. The market in recent years has observed
remarkable growth in trading volume, the variety of contracts, and the range of
underlying commodities. Market participants are also becoming increasingly
sophisticated about recognizing and exercising operational contingencies embedded in
delivery contracts. For all of these reasons, there is a widespread interest in models for
pricing and hedging commodity-linked contingent claims. Despite these facts, relatively
little attention has been paid to commodity markets, in comparison with the enormous
recent empirical analyses of the currency and equity markets. While commodity markets
are smaller and possibly lack the glamour of currency and equity markets, they are
nevertheless important for the agricultural sector of the economy and for maintaining

overall supply and demand conditions in the macro economy.

Chapter 2 is concerned with commodity cash market price risks. The cash
markets are characterized by the unique physical properties of commodities since cash
prices are determined by supply and demand for commodities that are subject to various
unique factors such as weather and other environmental determinants. We introduce the
long memory property with its characteristic self-similarity and study daily cash return
volatility dynamics with reflection on various characteristics of commodities such as
annual seasonality for agricultural products and distinguishing trading patterns for
livestock. Accounting for those characteristics, our empirical investigation uses the
F IGARCH and local Whittle semi-parametric estimation method to reveal that the long
memory property is evident in the daily cash return volatility.

Chapter 3 is concerned with investigating the possible existence of the long
memory feature in commodity markets. Apart from the study by Cai et a1. (2002), this
thesis appears to be the first systematic study of the phenomenon and its applicability to
and implications for commodity markets. Hence, in chapter 3 we apply the FIGARCH
and the local Whittle estimator to commodity futures market price data and also report
the estimates of various long memory volatility models. We find overwhelming evidence
for the phenomenon, which is consistent with the evidence found in the securities and ‘
currency markets. We also investigate and discuss the property of self-similarity in
commodity markets, generally finding our empirical results to be consistent with self-
similarity. It turns out to be very important to consider issues of time to maturity when
modeling volatility in these markets. Further, chapter 3 deals with high frequency
commodity futures market data. We first discuss meaningful ways of constructing high

frequency returns, and then describe the empirical properties of these. Much emphasis is

placed 6n the particularly unusual intra-day periodicity that occurs in these futures
markets and its elimination through the application of Gallant’s (1981) FF F filtering
method. Our finding is supportive of self-similarity for high frequency commodity
futures return volatility.

Chapter 4 is concerned with the relatively new measure of realized volatility,
which has recently become a competitor with the dominant GARCH model of Engle
(1982). We find some interesting features of very persistent autocorrelation, or long
memory, in the realized volatility series. The realized volatility series are also partly
determined by USDA announcement effects and the local market conditions of time to
maturity. We introduce the new concept of information flow, which is measured using
trading intensity built fiom a high frequency time dimension. We consider information
flow and time-to-maturity effects to explain realized volatility. Even allowing for these
effects, the long memory effects in the realized volatility series tend to remain. Chapter 4
also investigates the patterns of dependencies between the realized volatility series of
several different commodities. We discuss these results in the context of fractional
integration and the existence of a common structural long memory trend in the generation
of the realized volatility series.

We summarize our studies and conclude this dissertation with possible future

research in chapter 5.

CHAPTER 2
MODELING COMMODITY CASH RETURNS

2.1. Introduction

This chapter is concerned with the stochastic properties of daily commodity cash
prices for corn, soybeans, live cattle, live hogs, unleaded gasoline, and gold. This type of
analysis is an important precursor for many financial market applications, including
calculation of optimal hedge ratios, computation of Value at Risk (VaR), etc. While
previous studies have investigated the time series properties of commodity cash prices
using stable GARCH models, we are unaware of any previous investigation of the long
memory properties of daily cash series. For the successful application of financial market
analysis and policy, an investigation of the detailed properties of these asset prices seems
long overdue.

Commodities are physical products and possibly not involved in trading for
possible swift arbitrage. Commodity trading involves some transaction costs attributable
to storage and transportation that are not relevant to most financial assets. In particular.
commodity cash prices are directly determined by the supply and demand for actual
products, while commodity futures contracts are traded in order to reduce uncertainty for
the underlying commodities. Therefore, commodity cash markets are quite different

from other financial markets. Baillie and Myers (1991), and Cecchetti, Cumby, and

 

Figlewski (1988) used commodity cash and futures prices for the optimal hedge ratio
calculation. Mackey (1989), Yang and Brorsen (1992), and Burton (1993) documented
daily commodity cash prices by using nonlinear dynamic models. Yang and Brorsen
(1992) and Burton (1993) compared GARCH models with chaos models to explain
complicated commodity cash price volatility dynamics. Yang and Brorsen (1992)
considered GARCH, mixed diffusion-jump, and deterministic chaos models of cash
commodity prices and concluded that the GARCH volatility process provided the best fit.
Mackey (1989) suggested a theoretical model to argue that supply and demand for
commodities may cause nonlinear price dynamics.

The long memory property is well known to occur in squared returns, absolute
returns, and various transformations of volatility such as conditional variances and
stochastic volatility models. There are several plausible reasons for the occurrence of
long memory in absolute returns, conditional variances and other measures of volatility.
First, Granger (1980) showed how the contemporaneous aggregation of independent
AR(1) processes can lead to a long memory process as the number of cross section units
gets large. This result depends upon the autoregressive parameters having a beta
distribution in the interval (0, 1). Usually, the aggregation of N independent AR(1)
processes leads to an ARMA(N, N-l) model. However, Granger (1980) shows that this
tends to follow fractional white noise as N gets large. Extension of this aggregation
argtunent to volatility models is less than straightforward. Ding and Granger (1996)
showed that if each asset’s return is a martingale with stable GARCH( l ,1) innovations,
then the autocorrelations of the squared returns of the contemporaneously aggregated

assets will exhibit hyperbolically decaying autocorrelations, and hence the long memory

property. Also, Andersen and Bollerslev (1997a) claimed that long memory can result
from aggregated heterogenous information components in line with the Mixture-of-
Distribution hypothesis noted by Clark (1973) and Tauchen and Pitt (1983). A further
suggestion of Parke (2000) is that long memory can arise from the aggregation of shocks,
each with a different duration time. Indeed, financial markets are subject to numerous
economic factors and considerably responsive to the vast amount of information available
in the markets.

This chapter adds to the literature by investigating the long memory for
commodity market price risk and examining self-similarity to verify the long-run
temporal dependence as an original property of commodity cash price changes. We start
with the daily cash price in this chapter and continue with daily and intra-day futures
prices in chapter 3.

The remainder of this chapter proceeds as follow. Section 2 provides a brief
theoretical background for long memory, self-similarity, and temporal aggregation. In
section 3, we document empirical findings for the long memory and self-similarity by
using the FIGARCH and the local Whittle semi-parametric model for the temporally

aggregated daily cash returns motivated in section 2. Section 4 concludes the chapter.

2.2. Long Memory, Self-Similarity, and Temporal Aggregation
In this section, we discuss definitions of long memory and relate it to the concept
of self—similarity. One possible definition of long memory is as follows: if the

population autocorrelation of a time series process at lag j , denoted by p 1 , has the

following property,

lim 2 lpj|=oo, (2.1)
n

the process is said to exhibit long memory. For a sufficiently large number of lags j , a
process with autocorrelation function p j z chd’1 and a positive constant c , and where

-0.5 < d < 0.5 , can be formally defined as a stationary long memory process. We call 0'
the long memory parameter. Autocorrelation for such a type of process decays very
slowly over long time lags.

Granger and Joyeux (1980), Granger (1980), and Hosking (1981) have developed
the Autoregressive Fractionally Integrated Moving Average (henceforth, ARFIMA)
model to represent a time series process with the long memory property. Baillie (1996)
provides a comprehensive survey of the long memory theories and applications in
macroeconomics and finance. As suggested by Granger and .oneux (1980), Granger

(1980), and Hosking (1981), the ARFIMA model takes the following form,
(I
¢(L)(1—L) (y,—,u)=6(L)3,, (2.2)

where all the roots of the p’th order polynomial in the lag operator ¢(L) and the q’th

order polynomial in the lag operator 9(L) are assumed to lie outside the unit circle. The

process a, is white noise. The operator (1 - L)d is the fractional difference operator

defined as follows:

 

 

(1-L)d a{1—dL+d_(‘:_!‘IIL2_d(d—13)!(d—2)L3+...}. (2.3)

The ARFIMA process combines the stationary and invertible ARMA model which
generates 1(0) behavior with the above fractional difference operator, which adds on the

long memory behavior for the time series process. For a large lag j, there is hyperbolic
decay in the autocorrelations of the ARFIMA process and p 1- z cj2d_l where c > 0. To

describe another important property of the ARFIMA process, we consider the impulse
response weights, following Campbell and Mankiw (1987). The impulse response

weights are defined by first differencing the ARFIMA process, y, , to obtain
(1—L)y, = A(L)a, (2.4)

where A(L) = (1 — L)I—d ¢(L)_19(L). We can express the lag polynomial A (L) in

terms of the hypergeometric functions as

A(L)= F(d—1,1,1;L)¢(L)'16(L) (2.5)

where F(a,b,c;z) -=-. {F(c)/[F(a)F(b)]} {i [F(c+i)/I‘(i+1)]} and [(0) is a Gamma

i=1

function. Since F (d — 1,1,1; L) = 0 , as Gradszteyn and Ryzhnik (1980) Show, we

 

have A(1) = F(d—1,1,1;1)¢(1)_I 0(1) = 0 for d <1. The impact ofa unit innovation at

time t on the process y,+ k is then given by

1+ 21:” A j . (2.6)

Therefore, a fractionally integrated process with d <1 is mean reverting. In particular,

 

y, for 0.5 < d <1 is still mean reverting, although the process is not covariance

stationary. The long memory feature provides a flexible way of describing complicated
volatility temporal patterns, while conventional ARMA class models capture only short-
run dynamics in modeling time series and may be too strict in uncovering longer term
persistence for the series.

Another important property of the long memory process is self-similarity. The
general notion of self-similarity is that some random variables behave identically when

they are viewed at different scales on a dimension. The dimension may be space or time,

and, particularly, will be time when we analyze time series data. Consider a process y,

following the long memory property with autocorrelation pj- 2: chd'I for j lags. Given

the autocorrelation function, the corresponding spectral density for the associated process

can be expressed as follows,

2 oo .
f(/t) 427—”; p je’“ . (2.7)

Then, the spectral density is approximately of the form p 1- z chd—1 with a constant c as

xi —) 0 where 1 represents Fourier frequencies. More formally, y, is called self-similar

with a self-similar parameter H, if for any positive stretching factor c the rescaled process

0‘” ye, has the same distribution as the original process y, .

Following the formal definition and basic concept of self-similarity, we proceed

with temporal aggregation. Let R,(k) 2 21:0 (k—I) R”, _, denote temporally aggregated

returns at a k-day sample frequency. For simplicity, assume that R, = 0,2, and z, are
independent and identically distributed and 0', represents a positive and measurable

time-varying function. According to many previous empirical findings for the long

memory property for squared asset returns, we can assume that
2
FUR: I2 {Rt—1'] )z jZd—I for 0 < d < 0.5 I. The temporally aggregated returns, R,(k),

can be expanded as follows:
(’0 2 2 2 2 2
RI = Rik +le—l “I’Rtk—Z +"'+Rrk—k+l +221”, Rrk-IRrk—m- (2'8)

Since we assume that z, are independent and identically distributed, R”, —lRtk-m terms

for l at m should not matter in considering the autocorrelation below. Then, the j-th order

2 2
autocorrelation pH 121(k)] ,[ng] ] is simply the sum of autocorrelations for all the

 

' The long memory process with O < d < 0.5 shows all positive autocorrelations decaying at a hyperbolic
rate; see Baillie (1996).

10

 

2 2
possible pairs of the squared terms underlying [R,(k)] and [ng] . In other words, the

2 2
j-th order autocorrelation p[[ R510] ,[ng] ] can be obtained by summing the

(k) 2 (k) 2
autocorrelations between [R, ] and [R, 17,41] for h=—k+1,-k+2,---,k—1. After

some straightforward algebra, we have

p([RIk)]2,[RIf}]2]=k“2h=:+l(k-lhl)p([RIk)]2,[Rff}.k_,,]2]

k" 2d I
=k‘2 Z (k-Ihl)(jk+h) ‘
h=-k+l

(2.9)

k-l
where h=—k+1,—k+2,---,k—l. Note that k’2 2 (k—Ih|)=1. Further, iftime lag j

h=—k+l
is sufficiently large, we have
2 1‘4 2d—I 2d—I 2d 1
k‘ 2 (k-|h|)(jk+h) SUI) ~j ‘ (2.10)
h=—k+1

Consequently, we have

p([RI")]2,[RIfI-IZ]~12"“ (2.11)

11

According to this result, we assess that temporally aggregated squared returns
theoretically exhibit identical decaying rates for their autocorrelations regardless of the

sampling frequencies k. In other words, for sufficient lags j, the autocorrelation

2 2
between[R,(k)] and [R53] takes an identical form to the autocorrelation between

R,2 and R3,]- for different values of k. Concretely, if R,2 exhibits long memory, then

2
[RI/0] also shows the same degree of long run temporal dependence. Consistent with

the self-similarity notion discussed above, temporally aggregated squared returns Show
identical long memory behavior if their underlying squared returns follow the long
memory process. This result can carry over to a temporally aggregated absolute return

case as below.

Especially if we assume further that a, from R, = 0,2, follows log-normal

26
distribution, then R,(k)| for all 6 > 0 will exhibit identical decaying rates for

 

autocorrelation behavior. This result has been noted by Granger and Newbold (1976) and

2

2
recently confirmed by Andersen (1994). Since [RI/Y] = R,(k)

 

 

, temporally aggregated

squared returns should be one particular case of power transformed absolute

20
k 0 O I I
returns, R,( I for 6 =1. Further, Identical decaying rates for autocorrelations of

 

 

29
R,(k) l for all 6 > 0 imply that temporally aggregated absolute returns also yield the

 

12

same autocorrelation behavior as temporally aggregated squared returns, since temporally

26
aggregated absolute retum R,(k)l is just another case for 0 = 0.5.

 

The theoretical self-similarity of temporally aggregated squared returns motivates
the application of F IGARCH conditional variance model to temporally aggregated daily
cash returns. Following Baillie, Bollerslev, and Mikkelsen (1996), the FIGARCH model

is defined as follows:
0,2 = a) +fl(L)a,2 +[1—,6(L)—(1—¢(L))(1—L)d]r:,2 - (2.12)

wherefl(L) = fllL+fl2L2 +---+,6pr and ¢(L)== ¢|L+¢2L2 ~~~+¢qu. Baillie,

Bollerslev, and Mikkelsen (1996) incorporated slow hyperbolic decay into the
conditional variance modeling. The FIGARCH process considers a slowly decaying
autocorrelation for lagged squared innovations and allows for persistent impulse response
weights without involving the never-dying-out cumulative impulse response weights.

The FIGARCH model can describe conditional variance in a more flexible way by
allowing for 0 < d <1 while the IGARCH model yields unrealistically everlasting
volatility persistence and the GARCH process considers only short run dynamics for
conditional variances. To describe the features that distinguish the FIGARCH from the
GARCH and the IGARCH process, we consider the impulse response functions. Another

expression for the FIGARCH process is

13

{1—¢(L)}(1—L)" a} =w+{1—,B(L))u, (2.13)

where U, = 8,2 — 0,2 . Analogously to the impulse response function for the ARFIMA

process mentioned above, we express the first differenced 8,2 as,

(1—L)£,2=a)+7(L)v,. (2.14)

Then, we have the impulse response weights for the FIGARCH process such that

y(L) =(1—L)“" ¢(L)’l {1—fl(L)). (2.15)

By the same token, the impulse response weights for the GARCH process and the

—l
IGARCH are 7(L) = (1—L)(1—a(L)—,6(L)} {1 —,6(L)} and7(L) = {1 —,B(L)},
respectively. Since the limit of the cumulative impulse response weights is 7(1) , the

impact of past shocks on the FIGARCH volatility process from equation (2.15) is zero, as

for the GARCH process. Note that 7(1) =1— ,6 > O particularly for the IGARCH (1,1).

Further, we consider the F IGARCH cumulative impulse response weights for lag j as

I, : Zon, . (2.16)

14

According to Stirling’s approximation, the cumulative impulse response weight for lagj,

,1]- for the FIGARCH process is

I, z[(1—,B)/I‘(d)]jd'l. (2.17)

Therefore, the hyperbolic decay component is present in the cumulative impulse response
weights so that a shock to the squared residuals will decay at a very slow rate although all
the past shocks eventually will die out.

Another class of models to describe long memory volatility was suggested by
Breidt, Crato, and de Lima (1993) and Harvey (1998). They model long memory for

conditional volatility series as follows:

Y: = 0151 (2-18)

and

0,2 = 02 exp(h, ) , (2.19)

where g, is normal and independently distributed. Estimation of the stochastic volatility

model uses the state space representation and Quasi-Maximum Likelihood Estimation

(QMLE) via the Kalman filter.

15

In this chapter, by using both parametric and semi-parametric models, we
investigate whether daily commodity cash return volatility follows the long memory
process. The FIGARCH model is used to identify the long memory behavior in daily
cash return volatility parametrically while the local Whittle estimation method is chosen
for a semi-parametric counterpart to estimate the long memory parameter for the absolute

daily cash returns.

2.3. Application to Daily Cash Returns

In the previous section, we considered the theoretical relationship among the long
memory process, the self-similarity property, and temporal aggregation. We plan to
study the theoretical relationship empirically by using daily cash price data for various
types of commodities: corn, soybeans, live cattle, live hogs, gasoline, and gold.

We apply the FIGARCH model to the temporally aggregated daily cash returns to
analyze the return volatility temporal patterns across various daily sample frequencies.
We choose one-day through five-day sample frequencies because 5 trading days usually

form a week of business days. We temporally aggregate returns
R,(k) a 21:0 (,4) R,k _, at k-daily frequency by summing one-day cash returns over k- '

daily periods for k = 1, 2, 3, 4, 5. For the conditional mean, we choose MA(I) to capture
the usually small but significant autocorrelations of return levels at the first few lagsz.
The generic MA(q)-FIGARCH(p,d,q) model to estimate for daily cash returns is as

follows:

 

2 In the high frequency context, this MA(l) term is related to the market microstructure noise issue. We
will discuss this more in chapter 3 where high frequency commodity futures returns are considered.

16

y, =100Aln(P,) = p + e, + 98H (2.20)

8t = 2.th (2.21)

0,2 = a) + fl(L)0',2 +|:1—,B(L)—(1—¢(L))(1—L)d:|a,2 (2.22)

where P, is the commodity cash price, 2, is an i.i.d.(0,l) random variable,
fl(L) 2 AL + ,6sz +---+ ,6pr , ¢(L) 2 (AL +¢2L2 ---+¢qL‘7 , and L is the lag operator.

Before proceeding further, we will briefly describe some characteristics of the
commodities considered here. Daily prices for cash commodities are cash prices for the
delivery location and specifications included in the corresponding futures contracts.
These were obtained from the Futures Industry Institute data center. The agricultural
product cash markets for corn and soybeans especially seem to display different volatility
patterns due to their inherent attributes. Figure 2-1 plots the sample autocorrelations for
absolute returns of daily cash prices for all the commodities considered. The horizontal
axis represents daily lags up to 1000 days in order to consider approximately four years
of trading days. In particular, the unique patterns of corn and soybean cash return
volatility in their sample autocorrelations are worth notice. In figure 2-1, the dotted line
represents the sample autocorrelations of the absolute (raw) cash returns for corn and
soybeans. As shown in the figure, there seems to be some pronounced yearly seasonality
for the original daily cash return volatility for corn and soybeans. The peaks are observed

almost every 250-day interval, which approximately corresponds to a year of trading

17

days. Typical yearly planting and harvesting cycles for the agricultural products may be
responsible for the seasonality. Such seasonality may impede proper analysis of inherent
volatility patterns. To cope with the annual periodic patterns in daily cash return
volatility, we apply the Fourier flexible functional filtering, as introduced by Gallant
(1981). The Fourier flexible functional filtering is formally discussed in chapter 3 when
we consider high frequency commodity futures price data, since we apply the FFF
filtering to cope with strong intraday seasonality for all of the commodities. For the other
types of commodities, seasonal patterns are not observed for the sample autocorrelations
of absolute daily cash returns. Hence, the other commodities do not require filtering
before we apply the F IGARCH model to those time series data. The solid lines for the
correlograms of corn and soybeans represent the sample correlations for the filtered
returns for those commodities. As shown in Figure 2-1, seasonal patterns seem to be
markedly reduced by the F F F filtering. Another striking feature is the unusual
autocorrelation patterns of live cattle cash returns. The sample autocorrelations of live
cattle absolute cash returns appear to be very different from the others and repeatedly
deviate very much from zero.

Tables 2-1 through 2-6 present the results of applying the F IGARCH model to
cash returns for (filtered) corn, (filtered) soybeans, gasoline, cattle, hogs, and gold at
various daily frequencies. Specification tests are performed by applying the Ljung-Box
portmanteau statistic on the standardized residuals resulting from quasi-maximum

likelihood estimation for the FIGARCH model on the grounds that the test statistics

asymptotically follow 1,2,4, distribution.

18

The estimated long memory parameter in Tables 2-1 through 2-6 is strongly
statistically significant for the cash return series for corn, soybeans, gasoline, and gold.
In contrast, the long memory estimates for live cattle and hogs seem to be less significant
than those for the other commodities. In particular, daily cash prices for live cattle seem
to be constant for Wednesdays, Thursdays, and Fridays mainly, while most of the daily
cash price changes seem to occur on Mondays and sometimes on Tuesdays, according to
our preliminary data analysis. This odd data feature may be responsible for the unusual
sample autocorrelation patterns as shown in Figure 2-1. The cash prices for live hogs
also seem to involve some unusual characteristics. Although the live hog cash price
changes are found quite evenly over the week’s days, the changes seem to have strong
day-of-week effects. To capture possible day-of-week effects on daily cash price
changes, we include dummy variables for Monday, Tuesday, Thursday, and Friday in the
conditional variance specification. From our pre-estimatidn, we found that there are
considerable day-of-week effects for live hog cash return volatility. The robust t-values
for Monday, Tuesday, Thursday, and Friday3 dummies are 3.867, 4.797, 1.694, and
2.507, respectively. Also, the mean level of live hog daily cash returns exhibits a
significant level of serial correlations during the course of the MA-FIGARCH estimation.
To capture such a strong serial correlation in the mean level of live hog cash returns, we
impose MA(15)4 for the conditional mean model.

Apart from the unusual features mentioned above for the livestock, the long
memory estimates from the FIGARCH conditional variance specification from (2.20) to

(2.22) seem to be significant, and the model performs fairly in fitting the daily cash return

 

3 To avoid a dummy trap, we drop dummies for Wednesday.
4 Our informal experiment revealed that beyond 15 time lags did not seem to be statistically significant.

19

volatility. Particularly, we practiced a robust Wald test of the stationary GARCH(],l)5

null hypothesis versus a FIGARCH(1,d,1) alternative hypothesis. Under the null, the

robust Wald test statistic W will have an asymptotic [,2 distribution. Especially for a

one-day sample frequency, we reject the null hypothesis for d = 0, and thus the
GARCH(] ,1) model is rejected for most of the commodities, with the exception of the
liVestock. For the crops, gasoline, and gold, at many temporal aggregation levels the
formal statistical test supports the conclusion obtained both here and in J in and Frechette
(1994) that the FIGARCH is superior to the GARCH for modeling commodity return
volatilities6. On the other hand, the W statistics for live cattle and hogs seem to be
extremely low and less likely to reject the null hypothesis of GARCH specification at a 5-
day (i.e., weekly) sample frequency. Again, this feature can be attributed to inactive spot
market trading and the possible day-of-week effects for the livestock discussed above.

In addition, the long memory estimate levels themselves appear to be very stable
across different sample frequencies for most of the commodities, with few exceptions.
Our results imply that conditional variances of daily cash returns for each commodity
may demonstrate a similar degree of persistence at different sample frequencies. This
finding seems to be supportive of the self-similarity property discussed in section 2.

The semi-parametric local Whittle estimation methods have been suggested by
Kunch (1987) and Robinson (1995). As a robustness check for the FIGARCH estimation
results, we apply the local Whittle estimation for the long-memory parameter by using

the absolute daily cash returns. One of the motivations for the semi-parametric

 

5 For some instances, we test the null hypothesis for different GARCH specification other than
GARCH(I,1).
6 In fact, Jin and Frechette (1994) have used commodity futures price data.

20

estimation method is that, while the long memory volatility parameter estimation results
using parametric models such as ARFIMA or FIGARCH specification may be affected
by any possible short run dynamics, the semi-parametric estimation method affords
general treatment of short run temporal dependence7. We discuss the local Whittle
estimation separately in more detail in chapter 3. Table 2-7 reports the estimates of the
long memory parameter by using absolute daily cash returns.

For the absolute daily cash returns, the semi-parametric long memory estimates
seem to be qualitatively similar to the FIGARCH estimation results. For example, the
low long memory estimate levels for live cattle and hogs can be found for the local
Whittle estimation results Similarly as in the FIGARCH long memory estimates. Also,
the local Whittle estimates for the long memory parameter seem to be stable, as we found
from the FIGARCH estimation results, and supportive of self-similarity for temporally
aggregated absolute returns, as the FIGARCH estimates are stable across different sample

frequencies.

2.4. Conclusion

The long run volatility dynamics for prices of physical commodities have been
considered in this chapter. By using both parametric and semi-parametric long memory
models, we confirmed that long memory exists for daily cash return volatility and,
further, that the long memory behaviors are consistently witnessed across various daily
frequencies for most of the commodities. We observed this evidence for temporally

aggregated absolute returns and squared returns in common. This feature is consistent

 

7 In general, semiparametric estimation methods may be somewhat controversial due to their poor
performance in terms of bias and mean squared error.

21

with the theoretical self-similarity property of long memory, which implies that the
autocorrelation of the long memory process decays at the same rate regardless of the
sample frequency. Despite distinct aspects of commodity cash markets, the cash return
volatility seems to exhibit the long memory property with exceptions only for livestock,
as found in previous studies for many financial markets.

More practically, a proper understanding of cash price risks is important
information for the hedge ratio of commodity futures, since the optimal hedge ratio is the
conditional covariance between cash and futures returns divided by the conditional
variance of futures returns. Therefore, studies of conditional moments of cash price
change are very related to futures hedge modeling. Analysis of commodity futures return

volatility, using both daily and high frequency return data, follows this chapter.

22

Table 2-1: Estimated MA-FIGARCH Models for Daily Cash Returns for Corn

(The sample period: 1/02/80 - 3/30/01)

 

1 day 2 days 3 days 4 days 5 days
T 5362 2681 1787 1340 1072
It 0.0009 0.0025 0.0035 0.0042 0.0032
(0.0018) (0.0037) (0.0055) (0.0074) (0.0095)
0 0.0215 0.0006 0.0162 0.0237 0.0269
(0.0162) (0.0220) (0.0259) (0.0313) (0.0321)
d 0.2720 0.2992 0.2641 0.3215 0.2702
(0.0438) (0.0675) (0.0618) (0.0808) (0.0827)
(1) 0.0026 0.0050 0.0086 0.0102 0.0156
(0.0008) (0.0017) (0.0032) (0.0038) (0.0066)
B 0.1730 0.1607 0.1 170 0.1040 0.0864
(0.0470) (0.0702) (0.0787) (0.0917) (0.0916)
m3 -0500 -0471 -0.372 -0394 -0455
m4 6.463 5.534 4.505 5.514 5.919
Q(20) 31.662 28.498 26.122 23.336 16.723
Q2(20) 5.745 7.504 1 1.648 8.343 8.542
w 38.492 19.631 18.286 15.846 10.666

 

Key: ln(L) is the value of the maximized log likelihood; Q(20) and Q2(20) are the
Lj ung-Box statistics with 20 degree of freedom based on the autocorrelations of the
standardized residuals and autocorrelations of the squared standardized residuals. The

sample m3 and m4 are also based on the standardized residuals. T is the number of
observations.

23

Table 2-2: Estimated MA-FIGARCH Models for Daily Cash Returns for Soybean

(The sample period: 1/02/80 — 12/29/00)

 

1 day 2 days 3 days 4 days 5 days
T 5300 2650 1766 1325 1060
[1 -0.0020 -0.0017 -0.0014 -0.0003 0.0024
(0.0016) (0.0031) (0.0046) (0.0061) (0.0076)
0 -0.0336 -0.0368 -0.0368 -0.0474 -0.0677
(0.0155) (0.0210) (0.0254) (0.0309) (0.0303)
(1 0.3291 0.3397 0.3904 0.2899 0.3498
(0.0488) (0.0649) (0.1 103) (0.0671) (0.0988)
0) 0.0016 0.0029 0.0036 0.0080 0.0078
(0.0004) (0.0009) (0.0015) (0.0033) (0.003 5)
[3 0.2723 0.2753 0.3189 0.1102 0.1999
(0.0620) (0.0754) (0.131 1) (0.0910) (0.1201)
m3 -0.265 -0.256 -0.005 0.080 0.059
m4 5.152 4.538 3.772 3.509 3.761
Q(20) 22.361 26.364 15.869 19.815 21.595
Q2(20) 34.198 25.785 21.693 21.277 24.398
W 45.485 27.369 12.532 18.657 12.528

 

Key: As for table 2-1

24

Table 2-3: Estimated MA-FIGARCH Models for Daily Cash Returns for Live Cattle

(The sample period: 1/02/80 — 12/29/00)

 

 

1 day 2 days 3 days 4 days 5 days
T 4800 2400 1 600 1 200 960
[.1 0.0037 0.0019 0.0122 -0.0024 -0.0018
(0.0233) (0.0195) (0.0275) (0.1231) (0.0619)
0 0.0297 -0.0091 -0.0421 -0.0493 -0.0855
(0.0161) (0.0169) (0.0293) (0.0357) (0.0363)
(1 0.1768 0.1534 0.1385 0.0661 0.0668
(0.0930) (0.0668) (0.0696) (0.0819) (0.0792)
0) 0.1546 0.6173 1.0479 1.2694 2.6452
(0.1050) (0.3500) (0.5645) (0.9834) (1.6237)
0 0.5832 0.1557 0.1354 0.3828 0.1077
(0.0679) (0.0656) (0.0638) (0.4618) (0.6031)
4) 0.4086 0.4615 0.1891
(0.0643) (0.4907) (0.6487)
m3 -1.538 -0.873 -0.630 -0.553 0426
m4 40.253 18.439 11.856 9.119 7.564
Q(20) 28.860 18.392 25.561 27.665 41.104
Q2(20) 24.741 18.294 9.771 14.416 10.133
W 3.167 5.269 3.962 0.652 0.71 l

 

Key: As for table 2-1

25

Table 24: Estimated MA-FIGARCH Models for Daily Cash Returns for Live Hogs

(The sample period: 1/02/80 — 12/29/00)

 

1 day8 2 days 3 days 4 days 5 days
T 4551 2275 1517 1137 910
p -0.0037 -0.0152 -0.0212 -0.0174 0.0177
(0.0279) (0.0578) (0.0994) (0.1071) (0.0860)
0. -0.1524 -0.2878 -0.0692 0.0120 0.1001
(0.0170) (0.0277) (0.0284) (0.0335) (0.0369)
02 -0.2078 0.1482 0.0830 0.1 104 0.1384
(0.0153) (0.0232) (0.0275) (0.0319) (0.0343)
03 0.0463 0.0326 0.1200 0.0832 0.0086
(0.0159) (0.0228) (0.0268) (0.0312) (0.0346)
04 0.1229 0.0854 0.0862 0.0158 0.0006
(0.0159) (0.0229) (0.0274) (0.0306) (0.0341)
05 -0.0096 0.0734 0.0033 0.0147 -0.0947
(0.0165) (0.0232) (0.0269) (0.0329) (0.0341)
06 -0.0061 -0.01 15 0.0426 -0.0253 -0.0444
(0.0157) (0.0234) (0.0280) (0.0350) (0.0347) .
07 0.0562 0.0514 -0.0304 -0.0402 0.0230
(0.0158) (0.0225) (0.0284) (0.03 50) (0.0362)
03 0.0394 -0.0287 0.0223 -0.0161 -0.0255
(0.0171) (0.0224) (0.0277) (0.0368) (0.0363)
09 0.0510 0.0562 -0.0412 0.0341 0.0337
(0.0144) (0.0225) (0.0308) (0.0333) (0.0363)
910 0.0196 -0.0223 0.0327 0.0195 -0.0246

 

8 For live hogs at one-day sample frequency, we could cope with higher 02(20) statistics by including day-
of-week dummy variables. All the coefficient estimates are 0.9437 with standard error, 0.2440 for
Monday; 1.3074 with standard error, 0.2725, for Tuesday; 0.3487 with standard error, 0.2058 for Thursday:
and 0.6577 with standard error, 0.2623 for Friday. We do not consider such day—of-week effects since they
seem to collapse by temporal aggregation beyond one-day sample frequency.

26

 

 

 

(0.0174) (0.0233) (0.0321). (0.0320) (0.0342)
0.. -00151 -00014 -00301 0.0118 -0.1198
(0.0159) (0.0238) (0.0301) (0.0357) (0.0422)
0.2 -00155 -00094 -00214 -00250 -0.1273
(0.0165) (0.0222) (0.0295) (0.0340) (0.0404)
0.3 0.0496 .0.0305 0.0731 .0.0002 -0.0421
(0.0158) (0.0217) (0.0289) (0.0353) (0.0407)
0... 0.0434 0.0049 0.0326 -0.0337 -0.1069
(0.0153) (0.0209) (0.0279) (0.0339) (0.0377)
0.5 -0.0498 -00127 -00055 -0.0730 .0.0202
(0.0153) (0.0212) (0.0300) (0.0342) (0.0309)
(I 0.2083 0.1789 0.1361 0.1570 0.0981
(0.0548) (0.0395) (0.0411) (0.0574) (0.1203)
6) 0.1718 1.8094 3.0760 0.1614 1.1651
(0.1838) (0.4762) (0.8904) (0.0798) (2.2573)
[5. 0.5239 0.0914 0.0197 0.9654 0.7838
(0.2096) (0.0442) (0.0524) (0.0179) (0.3272)
82" 0.0230
(0.0266)
6. 0.4150 0.9498 0.8272
(0.1801) (0.0263) (0.2455)
m3 -0044 -0001 -0.146 0.130 0.008
m4 3.544 3.296 3.762 3.476 3.266
Q(20) 17.972 2.9234 9.510 1 1.983 6.753
Q2(20) 29.283 21.082 24.578 21.542 22.423
w 27.616 20.516 10.967 7.475 0.665

 

Key: As for Table 2-1

 

9 FIGARCH (2, d, 1) seems to fit the live hog cash daily return volatility at one-day sample frequency
fairly relative to FIGARCH (l, d, l) or FIGARCH (l, d, 0) conditional variances specifications.

27

Table 2-5: Estimated MA-FIGARCH Models for Daily Cash Returns for Gasoline

(The sample period: 1/02/91 — 12/29/00)

 

 

 

1 day 2 days 3 days 4 days 5 days
T 2509 1254 836 627 501
[.1 -0.0192 -0.0632 -0.0970 -0.1201 -0.1151

(0.0451) (0.0949) (0.1442) (0.1834) (0.2230)
0 0.0926 0.0574 -0.0107 -0.0781 -0.0829

(0.0215) (0.0329) (0.0429) (0.0458) (0.0487)
(1 0.2900 0.2968 0.3715 0.2309 0.2105

(0.0694) (0.0979) (0.3187) (0.1524) (0.0845)
00 0.8453 1.9729 1.0488 5.8006 7.9115

(0.2650) (0.7972) (1.3298) (4.9808) (4.1986)
[3 0.1726 0.1372 0.6309 0.1930 0.1898

(0.0770) (0.1211) (0.1643) (0.2527) (0.1424)
(I) 0.4135

(0.1720)

m3 -0.103 -0.389 -0.214 -0.289 0160
m4 4.739 4.320 3.734 3.931 3.371
Q(20) 28.076 24.839 32.326 23.792 26.302
Q2(20) 17.416 16.398 9.190 15.708 17.395
W 17.463 23.160 1.359 2.295 6.204

 

Key: As for table 2-1

28

Table 2-6: Estimated MA-FIGARCH Models for Daily Cash Returns for Gold

(The sample period: 1/02/80 - 12/29/00)

 

1 day 2 days 3 days 4 days 5 days
T 5283 2641 1761 1320 1056
[.1 -0.0167 -0.0316 -0.0677 -0.0796 -0.0643
(0.0096) (0.0207) (0.0298) (0.0392) (0.0559)
0 -0.0583 -0.0124 -0.0043 -0.0132 -0.0309
(0.0169) (0.0301) (0.0298) (0.0356) (0.0360)
d 0.2905 0.3438 0.2942 0.4093 0.3160
(0.0351) (0.0574) (0.0434) (0.0953) (0.1374)
00 0.0755 0.1374 0.2068 0.1316 0.4705
(0.0250) (0.0572) (0.1 155) (0.1 176) (0.5367)
0 ' 0.1512 0.1787 0.1071 0.2872 0.2477
(0.0490) (0.0694) (0.1 134) (0.1398) (0.1521)
m3 0.086 0.959 0.419 0.637 1.411
m4 9.563 15.589 8.085 10.058 18.805
Q(20) 39.257 19.918 18.735 12.869 22.962
Q2(20) 11.015 2.982 10.029 6.215 8.135
W 68.634 35.854 45.890 68.634 5.291

 

Key: AS for table 2-1

29

Table 2-7. Semi-Parametric Long Memory Parameter Estimation:
Absolute Daily Cash Returns at Different Daily Sample Frequencies.

 

1 day 2 days 3 days 4 days 5 days

Corn (Filtered)

Local Whittle 0.3397 0.3633 0.3238 0.2823 0.2536
(0.0376) (0.0471) (0.0539) (0.0592) (0.0635)

Soybean (Filtered)

Local Whittle 0.3853 0.4693 0.4629 0.4568 0.4282
(0.0378) (0.0474) (0.0541) (0.0592) (0.0638)

Live Cattle

Local Whittle 0.1534 0.1457 0.1452 0.1995 0.1992
(0.0390) (0.0489) (0.0599) (0.0612) (0.0660)

Live Hog

Local Whittle 0.2418 0.2562 0.2224 0.1947 0.1492
(0.0397) (0.0497) (0.0569) (0.0625) (0.0672)

Gasoline

Local Whittle 0.2899 0.3027 0.3590 0.4061 0.4133
(0.0481) (0.0603) (0.0689) (0.0760) (0.0818)

Gold

Local Whittle 0.4436 0.4817 0.4626 0.4810 0.5073
(0.0378) (0.0474) (0.0541) (0.0595) (0.0638)

 

Key: Asymptotic standard errors are in parentheses below corresponding parameter

estimates.

30

Fig. 2-1 Correlograms for Absolute Daily Cash Returns

Com
0.3

p
N

__-..d-"

p
_s
l

J‘ . {d
l

‘ II '.
4' ., ,' .
II It“ I I
III R -. i 1 1
l .I . _.
I H UkJ‘
fl

' IIIIIIIII'IllIlIlllIIIIIIllIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIlll

1000

Sample Autocorrelaions
p
O

 

 

 

 

 

S bear
0.4 W
0.3
LI
3 0.2-
g i
3 0.1- II, l.‘ '
'3 " ‘~ it “It
055 I1 I. II‘ I
0.0- ' ~
"III
0.1.

 

200 400 600 800 1000

Key: Dotted line and solid line indicate the sample autocorrelations for absolute daily raw
and filtered cash returns.

31

Live Cattle

0.3

 

Sample Autocorrelations

 

 

 

 

 

 

 

.02
200 400 600 1000
DaiyLag
Liv
0.4 eHogs
0.3
g 0.2.
28
<
s
E
w 0.0.
-0‘1 IIIIllllIlllllllIIIIIIIIllIIllllllllllqlllillIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

 

1000

Key: Dotted line and solid line indicate the sample autocorrelations for absolute daily raw
and filtered cash returns.

32

Unleaded Gasol'ne

 

 

 

.0
E

0.00. ' (I I

Sample Autocorrebtions

 

 

 

'0' 10 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIllll

1000

Gold

 

0.30

0.25 .

0.20.

_c>

—L

(J1
4_

Sample Autocorrelations
#75

 

18 '8

 

 

 

-0' 05 IllIIIllIIIIIIIllllllllllllllllIlllllllllllllllllllllllll"[IlllllllllIIIIIIIIIIIIIIIIIIIIIIIIlllll

Key: Dotted line and solid line indicate the sample autocorrelations for absolute daily raw
and filtered cash returns.

33

CHAPTER 3

MODELING DAILY AND HIGH FREQUENCY COMMODITY FUTURES
RETURNS

3.1. Introduction

This chapter is concerned with the stochastic properties of commodity futures
prices and applies some recent developments in volatility modeling, in particular the
F IGARCH long memory volatility model, to commodity futures returns. The volatilities
of daily futures returns are found to be well described by the F IGARCH model, with
relatively similar estimates of the long memory parameter across commodities. The
conditional means of the daily returns are close to being uncorrelated, with small
departures from martingale behavior being represented by low order moving average
models. We also estimate FIGARCH models for high frequency commodity futures
returns based on intra-day tick data. These high frequency commodity returns are
dominated by strong intra-day periodicity, hypothesized to be a result of repeated trading
day cycles resulting from the institutional features of the futures exchanges where trades
are taking place. The intra-day periodicity is removed using a deterministic Fourier
Flexible Form (F FF) filter. The filtered high frequency futures returns are also well

described by the FIGARCH process. The results of the chapter have important

34

implications for our understanding of the stochastic properties of commodity prices, and
hence for empirical applications such as optimal hedge ratio estimation, tests for futures
market efficiency, tests for the announcement effect of market news, option valuation,
farm risk portfolio management, etc.

The FIGARCH model has already been applied gainfully to exchange rates, stock
returns, inflation rates, and a range of other economic data; for examples see Baillie,
Bollerslev and Mikkelsen (1996), Bollerslev and Mikkelsen (1996), Baillie, Han and
Kwon (2002), etc. However, there have been few applications of the model to
commodities. Crato and Ray (2000) study long memory in the daily volatilities of several
agricultural commodity futures returns, along with a stock index return, currencies,
metals, and heating oil. They find strong evidence of long memory in daily commodity
futures prices, though they do not explicitly estimate FIGARCH models. I in and
Frechette (2004) estimate FIGARCH volatility models for 14 agricultural futures series
and find that FIGARCH fits the data significantly better than a traditional GARCH
volatility model. While these studies have provided valuable information on the long
memory properties of commodity futures price volatilities, much more work remains to
be done.

This chapter adds to our understanding of long memory in commodity price
volatilities in three main ways. First, while I in and Frechette (2004) argue in favor of the
FIGARCH model over the GARCH model for commodity futures volatilities, they did
not undertake a formal statistical test comparing the two models. Here we undertake a
robust Wald test, which formally compares the fit of the GARCH and FIGARCH models.

Second, in addition to the standard quasi-maximum likelihood estimator (QMLE), we

35

also apply the semi-parametric local Whittle estimator of the long memory parameter.
This provides additional information on the robustness of long-memory inferences
concerning daily commodity price volatilities. Third, in addition to daily returns we
study high frequency returns on futures contracts using intra-day tick data. This study is
the first to systematically examine volatility using high frequency commodity futures
data.lo We find that estimated models at different sampling frequencies are consistent
with the theory that commodity futures returns are “self-similar” processes, and hence
have long memory parameters that are invariant to the sampling frequency; see Beran
(1994). The “self-similarity” of the estimates of the long memory volatility parameter
across relatively short spans of high frequency data strongly suggests that the long
memory property is an intrinsic feature of the system rather than being caused by
exogenous shocks or regime shifts.

The plan of the rest of the chapter is as follows. Section 2 discusses the
application of the long memory F IGARCH volatility model to daily futures returns.
Similar to J in and Frechette (2004), we find the F IGARCH models to be econometrically
superior to regular stable GARCH models. Section 3 describes the results from the
analysis of high frequency futures returns and compares them to the daily return results.
Section 4 presents an analysis of semi-parametric local Whittle estimation of the long
memory parameter as a robustness check, and also compares estimates of the long
memory parameter across a range of different sampling frequencies. This shows that the

commodity return series display self-similarity. Section 5 offers a brief conclusion.

 

10
Cai, Cheung and Wong (2001) have analyzed high frequency gold futures. However, their approach is

somewhat informal and does not include either FIGARCH or local Whittle estimation of the long memory
parameter.

36

3.2. Analysis of Daily Commodity Returns

This section is concemed with the analysis of daily futures retums for different
commodities. We examine six commodities: corn, soybeans, cattle, hogs, gasoline, and
gold. Corn and soybeans are major annual crops that are of critical importance to US.
agriculture. These crops are related in the sense that they can be substitutes in production
and both are used heavily as animal feed. They are different, however, in that most com
is produced in the northern hemisphere, while soybeans have a significant southern
hemisphere harvest in Brazil and Argentina. This southern hemisphere harvest may
influence seasonal price and volatility patterns. Cattle and hogs are both important
livestock commodities in US. agriculture, but their different life cycles mean different
inherent price dynamics, even though we would expect a lot of similarity in the stochastic
properties of prices for these two livestock commodities. Gasoline is included to see if
results are markedly different for a natural resource-based commodity, and gold is

included as a commodity that has a central role as a store of wealth.

Data were obtained from the Futures Industry Institute data center.ll The daily
data are daily closing futures prices on major US. futures markets for the relevant
commodity, in particular, the Chicago Board of Trade for corn and soybeans, the Chicago
Mercantile Exchange for live cattle and hogs, and the New York Mercantile Exchange for
unleaded gasoline and gold. Returns are defined in the conventional manner as
continuously compounded rates of return and calculated as the first difference of the

natural logarithm of prices. To compute the futures returns, nearby contracts were used,

 

II
The Futures Industry Institute is now called the Institute for Financial Markets. For more information

and data availability, see http://www.theifm.org.

37

and then the data was switched to the next available contract nearby on the first day of

the month in which the current nearby contract expires. For consistency, returns are

always defined using the same futures contract)2 The use of nearby futures contracts to
define our firtures return series has the advantage that we are using the most actively
traded contracts to generate our return data. However, if volatility depends on time to
maturity, as might be expected in at least some instances, then switching from an expiring
futures contract to the next nearby maturity may introduce jumps into the volatility
process because of jumps in time to maturity at the switch points. We will discuss how
we allowed for the effects of these jumps in time to maturity when we outline the
econometric model further below.

The details of the sample periods used for each commodity are provided in Table
3-1, along with some summary statistics for daily returns over these periods. All the daily
data begin at the first trading day of January 1980, except for gasoline. For gasoline, we
exclude data from January 1980 through December 1990 and begin the sample period the
first trading day of January 1991. This is to avoid two periods of exceptional volatility in
gasoline prices that we argue are a result of structural shifts in the volatility process for
this commodity. The first period is 1986-87, a period in which Saudi Arabia expanded its
oil production significantly in order to discipline other OPEC countries. The second
period extends from August 1990 to December 1990 and is caused by the Iraqi invasion
of Kuwait and the subsequent Gulf War. By starting the gasoline price series in January

of 1991 we avoid having to model these structural breaks in the volatility process. All of

 

12
That is, at each point when the data switch to the next nearby maturing contract, the futures return is

defined as the difference between the natural logarithm of today’s futures price for a contract maturing at
the next nearby and yesterday’s futures price for a contract with exactly the same maturity date. In this
way, daily returns are never defined using prices from two different contracts with different maturity dates.

38

the daily data end at the last trading day of December 2000, except for corn, which ends
the last trading day in March of 2001. In all cases we used the most recent data that was
provided in the data set obtained from the Futures Industry Institute.

Previous studies by Cecchetti, Cumby and F iglewski (1988), Baillie and Myers
(1991), and Yang and Brorsen (1992) have argued that most daily cash and futures
commodity returns are well described as martingales with GARCH effects. The
possibility of mixed diffusion-jump processes has also been suggested as a way to
characterize volatility in commodity prices. Yang and Brorsen (1992) compared
GARCH, mixed diffusion-jump, and deterministic chaos models of cash commodity
prices and concluded that the GARCH volatility process provided the best fit. It is only
more recently that studies such as Crato and Ray (2000) and J in and Frechette (2004)
have begun to investigate the long memory properties of commodity volatilities.

Figures 3-1 and 3-2 plot the sample autocorrelations for the returns, squared
returns and absolute returns in daily futures prices for two representative commodities,
namely live cattle and corn. There is one noticeable difference between the crop
commodity and the livestock commodity, namely that: both squared and absolute daily
returns for corn exhibit strong yearly seasonality in their sample autocorrelations, while ,
this does not occur for live cattle. To conserve space, the corresponding graphs for the
other commodities are not shown. However, it was observed that soybeans also display
seasonality in volatility (though not as pronounced as in the case of com, perhaps because
of the influence of a southern hemisphere harvest for soybeans) while live hogs, gasoline
and gold display no seasonality in volatility (similar to live cattle). In order to analyze

the intrinsic stochastic properties of the daily corn and soybean return volatilities we filter

39

out the seasonality by using a FFF filter. '3 The sample autocorrelations for the returns,
squared returns and absolute returns for the filtered daily corn futures price series is
provided in Figure 3-3. Notice that the FFF filter has been quite effective in removing
the seasonality in the squared and absolute corn futures returns. In all subsequent analysis
of the corn and soybean return volatilities we use the filtered volatility models.

Plots of the live cattle sample autocorrelations (Figure 3-1), the FFF filtered corn
sample autocorrelations (Figure 3-3), and other commodity return sample
autocorrelations (not shown) reveal a familiar lack of autocorrelation in returns and the
marked persistence in autocorrelations of squared and absolute returns that was first
noticed by Ding, Granger and Engle (1993) for the case of stock market returns. In
particular, the autocorrelation functions for the squared and absolute returns do not
display the usual exponential decay associated with the stationary and invertible class of
ARMA models, but rather appear to be generated by a long memory process with

hyperbolic decay.

More formally, the autocorrelation at lag k, p], , tends to satisfy pk z ckZd—I as k

gets large, where c is a constant and d is the long memory parameter. This type of
persistence is consistent with the notion of hyperbolic decay and is sometimes called the
“Hurst phenomenon.” The Hurst coefficient is defined as H = d +0.5. If d = 1, so that H
= 1.5, then the autocorrelation firnction does not decay and the series has a unit root. If d
= 0, so that H = 0.5, then the autocorrelation function decays exponentially and the series
is stationary. But for 0 < d < 1, i.e. 0 < H < 1.5, the series is sufficiently flexible to allow

for slower hyperbolic rates of decay in the autocorrelations. While many stochastic

 

13
See the appendix for the details of the FFF filter.

40

processes could potentially exhibit the long memory property, the most widely used
process is the ARFIMA(p, d, q) process of Granger and Joyeux (1980), Granger (1980),

and Hosking (1981). In the ARFIMA process, a time series x. is modeled as
a(L)(1— L)dx, = b(L)£, with a(L) and b(L) being p’th and q’th order polynomials in the
lag operator L, with all their roots lying outside the unit circle, while a, is a white noise

process. The ARFIMA process is stationary and invertible in the region of

-0.5 < d < 0.5 . At high lags, the ARFIMA(p, d, q) process is known to have an

autocorrelation function that satisfies pk z ckZd'l , so that the autocorrelations may

decay at a slow hyperbolic rate, as opposed to the required exponential rate associated
with the stationary and invertible class of ARMA models. The sample autocorrelation
function of the squared and absolute daily filtered futures corn returns appears to be very
consistent with the above properties, and analogous plots for the. other commodity returns
were found to be extremely similar.

Virtually all studies of daily asset returns, including commodity assets, have
found return y. to be stationary with small autocorrelations at the first few lags, which can
be attributed to a combination of a small time-varying risk premium, bid-ask bounce,
and/or non-synchronous trading phenomena; see Goodhart and O’Hara (1997) for a
description of this issue in high frequency currency markets. On the other hand, volatility
has been found to be very persistently autocorrelated with long memory hyperbolic
decay. A model that is consistent with these stylized facts is the MA(n)-FIGARCH(p, d,

q) process,

y, =100Aln(P,)= ,u+b(L)£,, (3.1)

41

and

[1-fl(L)10.2 =w+[1—{1—4<L)1-¢(L>(1—L>"]a.2. (3.3)

where P, is the asset price, 2, is an i.i.d.(0,1) random variable, and the polynomial in the
lag operator associated with the moving average process is

b(L) = 1+ blL + b2L2 + ...+ an". The FIGARCH model in equation (3.3) can be best

motivated from noting that the standard GARCH(p, q) model of Bollerslev (1986) can be

expressed as
a." = co +0406? + M003,

where the polynomials are a(L) s alL + (2sz + + (2qu and

,B(L) 5 AL + ,62 L2 + + ,6pr . The GARCH(p, q) process can also be expressed as the

ARMA[max(p, q), p] process in squared innovations as

[1 —a(L) — ,6(L)]t:,2 = a) + [1 — ,B(L)]v,

42

where u, s 8,2 -— 0,2 and is a zero mean, serially uncorrelated process which has the

interpretation of being the innovations in the conditional variance. The F1GARCH(p, d,

q) process in equation (3.3) can also be written as
¢(L)(1— 1.)d a} = a) +[1— ,6(L)]u,, (3.4)

where ¢(L) = [1— a(L) — ,6(L)](1— L)'d is a polynomial in the lag operator. Equation

(3.4) can be easily shown to transform to equation (3.3), which is the standard
representation for the conditional variance in the F1GARCH(p, d, q) process. Further
details concerning the FIGARCH process can be found in Baillie, Bollerslev and
Mikkelsen (1996). The parameter d characterizes the long memory property of hyperbolic
decay in volatility because it allows for autocorrelation decay at a slow hyperbolic rate.
The attraction of the FIGARCH process is that for 0 < d < 1, it is sufficiently flexible to
allow for intermediate ranges of persistence, between complete integrated persistence of
volatility shocks associated with d = l and the geometric decay associated with d = 0.
The volatility model in equation (3.3) has to be slightly adjusted to accommodate
the potential jumps in volatility that can occur at contract switching points, when futures
return data are computed from a sequence of nearby futures contracts. The long spans of
daily futures returns are constructed from contracts with different maturities, and the
resulting variations (and jumps) in time to maturity may have an influence on the
volatility process. To account for possible time to maturity effects we introduce a time to
maturity variable in the formulation of the F1GARCH(I, d, 1) model in (3.3), which then

becomes

43

0'2 = a) + 50,24 + yTM, + [1 — flL — (1 — ¢L)(1— L)d 15,2 , (3.5)

where TM represents the time to maturity on the contract used to construct the futures

return for period t, and y is the associated parameter.

The above model (3.1), (3.2), and (3.5) is estimated for futures returns on our six

commodities of interest by maximizing the Gaussian log likelihood function

T
ln(L;O) = —(0.5T)1n(21t)— 0.5Z[1n(o,2) + 3.26:2 I (3.6)
t=l

where O/ =(p,61,..49,,,a),,61,..,6p,¢, ,...¢,) is the vector of unknown parameters.
However, it has long been recognized that most asset returns are not well represented by
assuming 2, in equation (3.2) is normally distributed; for examples, see McFarland, Pettit

and Sung (1982) and Booth (1987). Consequently, inference is usually based on the
quasi-maximum likelihood estimator (QMLE) of Bollerslev and Wooldridge (1992),

which is valid when z, is non-Gaussian. Denoting the vector of parameter estimates

obtained from maximizing (3.6), using a sample of T observations on equations (3.1),

A A

(3.2) and (3.5), with 2, being non-normal by OT , then the limiting distribution of OT . is
Tl/2(@r-@o)->N[0,A(@o)-lB(@o)A(@0)—ll. (3.7)

44

where A(.) and B(.) represent the Hessian and outer product gradient, respectively, and

90 denotes the vector of true parameter values. Equation (3.7) is used to calculate the

robust standard errors that are reported in the subsequent results in this chapter, with the

A

Hessian and outer product gradient matrices being evaluated at the point OT for practical
implementation.

Table 3-2 presents the results of applying the above model (3.1), (3.2), and (3.5)
to daily futures returns for the six commodities discussed earlier. The exact parametric
specification of the model which best represents the degree of autocorrelation in the
conditional mean and conditional variance of daily commodity returns, vari by
commodity. The exact model specification for each commodity is indicated by the
number of non-zero estimates provided for the polynomial in the lag operator terms in
Table 3-2. For corn and soybean futures returns, we apply F IGARCH estimation to the
FFF filtered returns (see the Appendix). Results from Box-Pierce portmanteau statistics
on the standardized residuals are at the bottom of the table. The standard portmanteau test

m
statistic, Q(m) = T(T + 2):: r} /(T — j), where r,- is the j’th order sample autocorrelation
j=1

from the residuals, is known to have an asymptotic 131— k distribution, where k is the

number of parameters estimated in the conditional mean. Similar degrees of freedom
adjustments are used for the portmanteau test statistic based on the squared standardized
residuals when testing for omitted conditional heteroscedasticity. This adjustment is in

the spirit of the suggestions by Diebold (1988) and others. The sample skewness and

45

kurtosis of the standardized residuals (m3 and m4), are also provided at the bottom of

Table 3-2.

The Ljung-Box portmanteau statistics Show that the models specified for each
commodity do a good job of capturing the autocorrelations in the mean and volatility of
the commodity return series. In each case there is no evidence of additional
autocorrelation in the standardized residuals or squared standardized residuals, indicating
that the chosen model specification provides an adequate fit. It is interesting that
autocorrelation in the mean tends to persist more for the livestock commodities of live
cattle and hogs than for the other commodities (i.e., more MA terms in the mean are
required for an adequate fit). Furthermore, these commodities also seem to require more
flexible models to capture their autocorrelation in volatility as well (i.e., more GARCH
terms required for an adequate fit). The standardized residuals from all commodities,
except perhaps live cattle and hogs, exhibit the usual features of excess kurtosis of daily
asset returns. However, this is accommodated through use of the QMLE standard errors
for inference.

The estimated MA-FIGARCH models reported in Table 3-2 seem to fit the data
well. For each commodity there is weak evidence of small moving average effects in the
mean returns. As stated earlier, this may be attributed to a combination of a small time-
varying risk premium, bid-ask bounce, and/or non-synchronous trading phenomena. The
volatility autocorrelation parameters in B(L) and ¢1(L) indicate Strong evidence of
significant serial correlation in volatilities, which is consistent with previous findings of
autocorrelated volatility in commodity returns; see Baillie and Myers (1991), J in and

Frechette (2004), and Yang and Brorsen (1992). Furthermore, the time to maturity

46

parameter is statistically significant for all commodities except gold. Gold may not
experience a time to maturity effect in volatility because its special role as a store of
wealth means that cash and futures prices move very closely together, irrespective of the
time to maturity on the futures contract. It is interesting that the time to maturity effect is
negative for corn, soybeans and gasoline, but positive for cattle and hogs. This indicates
that the upward jumps in time to maturity that occur at contract switching points reduce
the volatility of returns for corn, soybeans, and gasoline, but increase volatility in live
cattle and hogs. Apparently, live cattle and hogs are relatively more volatile further away
from the maturity date, while corn, soybeans and gasoline are relatively more stable.

In this chapter we are primarily interested in the long memory parameter d. The
estimated long memory parameters reported in Table 3-2 are strongly statistically
significant for all six futures return series, and the hypotheses that d = 0 (stationary
GARCH) and also 6! =1 (integrated GARCH) are consistently rejected for all
commodities using standard significance levels. Table 3-2 also reports robust Wald test
statistics, denoted by W, for testing the null hypothesis of GARCH versus a FIGARCH
data generating process. Under the null, Wwill have an asymptotic 1,2 distribution and,
from Table 3-2, the GARCH model is rejected for every commodity at standard
significance levels. This formal statistical test supports the conclusion obtained both here
and in J in and F rechette (2004) that FIGARCH is superior to GARCH for modeling the
conditional variances of commodity returns. Evidently, long memory is a characteristic

feature of daily commodity futures returns, and F IGARCH represents a significant

improvement over GARCH.

47

3.3. Analysis of High Frequency Commodity Returns

Considerable previous work has examined the properties of high frequency
returns in equity and currency markets, but to date very little analysis has been done on
high frequency commodity returns. The only study we are aware of is Cai, Cheung and
Wong (2001) who studied high frequency gold futures prices. Their study analyzed 5-
minute gold futures returns between 1994 and 1997, and they discovered slow hyperbolic
decay associated with the autocorrelation function of the returns. However, they used an
informal method for approximating the long memory parameter and did not estimate
formal FIGARCH models. This section of the chapter represents a first attempt at
extensive analysis of the volatility properties of high frequency commodity futures
returns using FIGARCH models.

The raw futures tick data for the analysis were obtained from the Futures Industry
Institute data center along with the daily data (see footnote 2), and correspond to the same
six commodities studied in the previous section. The prices are for real-time transaction
records, which we initially convert to 5-minute price intervals by using the last price
quoted before the end of every S-minute interval over the trading day. For 5-minute
intervals that have no price recorded we linearly interpolate between surrounding
intervals to fill in the missing data. As with all high frequency asset price analyses, there
are potential problems with data unreliability due to the sheer amount of data being used
and the fact that there is considerable noise in the series because of little trade occurring
at some of the recorded prices. However, we minimize these problems by running the
data through a filter to identify and adjust anomalous observations. This was done by

locating return observations greater than three standard deviations and evaluating these as

48

possible data errors. A carefiil check and evaluation of these observations revealed a
small number of what appeared to be data errors in the high frequency gold returns.
These were then eliminated and replaced with a linearly interpolated value using the two
contiguous observations. No errors were detected in high frequency commodity returns
other than gold. Furthermore, instead of analyzing the 5-minute interval data (which will
be the most susceptible to data errors and noise) we convert the data to lower frequencies
(IO-minute for corn and soybeans, and lS—minute for live cattle and hogs, gasoline, and
gold) to undertake the analysis. Different intervals were chosen for different
commodities because they are traded on markets that have different trading day lengths.
Hence, in order to make sure interval returns could be computed that would exhaust the
recorded daily price change but not use consecutive intervals that stretched over two
different trading days, it was convenient to use 10-minute intervals for corn and soybeans
but 15-minute intervals for live cattle, live hogs, gasoline, and gold.

An interval return during day t is defined as y“. = 100 [ln(P. n)-ln(P,,,.- 1)], where
P”, is the futures price for the n-th intra-day interval during trading day I. As with many
analyses of high frequency asset price returns, it was found that the high frequency
commodity returns display considerable intra-day periodicity, which is usually attributed
to institutional trading features. This periodicity was removed using the FFF filtering
method, which is explained in detail in the Appendix.

Figure 3-6 plots the sample autocorrelations for lags of up to 5 trading days in 5-
minute intervals displayed in the horizontal axis for the absolute returns of the unadjusted
(raw) and the filtered 5-minute intervals for all the commodity futures returns series. The

dotted line represents sample autocorrelations for the unfiltered absolute 5-minute

49

returns, while the solid line indicates the autocdrrelations for the filtered absolute 5-
minute returns. The FFF filter seems to remove much of intra-day periodicity present in
the raw absolute returns. As usual, there is a small negative but significant first-order
autocorrelation in returns, which may be due to the non-synchronous trading
phenomenon, while higher order autocorrelations are not significant at conventional
levels. The autocorrelation functions of the absolute returns also exhibit a pronounced U
shape, suggesting substantial intra-day periodicity. Similar U-shaped patterns are found
in the equity markets (Harris, 1986; Wood et al., 1985; Chang et al., 1995; and Andersen
and Bollerslev, 1997a). Figure 3-7 shows the average absolute filtered infra-day returns
within a trading day. For all the commodities, the intra-day volatility patterns display an
U-shaped pattern. Unless otherwise indicated, all remaining analyses were done on the
filtered series.

The MA-FIGARCH model (3.1) through (3.3) was estimated based on the filtered
high frequency filtered returns. AS with the daily data, the orders of the MA and GARCH
polynomials in the lag operator were chosen to be as parsimonious as possible but still
provide an adequate representation of the autocorrelation structure of the high frequency
data. For the high frequency data, MA(1)-FIGARCH(1,d,1) models proved adequate for
all commodities. Long high frequency series were constructed by splicing several nearby
futures contracts together, in the same way as described for the daily data. A time to
maturity effect in volatility was tested, similar to that found in the daily return series. For
the high frequency return data, however, the time to maturity effect was not statistically
significant and so the time to maturity effect was restricted to zero. One possible reason

for this result is that there are many fewer contract switches in the high frequency series,

50

which combines a smaller number of futures contracts than the daily futures return series.
The number of trading days and the number of intra-day periods are different across the
different commodities. This information is provided in Table 3-3.

Details of the estimated MA(1)-FIGARCH(I,d,1) high frequency models for the
six commodities are reported in Table 3-4. All the models have small but Significant
MA(l) parameter estimates, which are usually attributed to the non-synchronous trading
phenomenon. Similar features for high frequency exchange rate returns have been noted
by Andersen and Bollerslev (1997a), Goodhart and Figliuoli (1992), Goodhart and
O'Hara (1997), and Zhou (1996). The estimated long memory volatility parameter d
ranges from 0.2 to 0.3 for most of the commodities considered and is generally
statistically significant.

Similar to the daily return results, we found significant long memory volatility in
the high frequency returns data as well. In general, the long memory estimates for intra-
day return volatilities are slightly lower than those for daily returns. Furthermore, as in
the daily return models, the robust Wald statistics in Table 3-3 Show strong evidence in
favor of the FIGARCH specifications against the GARCH specifications in the high
frequency model.

Details for the FIGARCH estimation results for various daily and intradaily
sample frequencies are recorded in tables A-l through A-12 in the Appendix to this
chapter. Another remarkable observation from the detailed estimation results is that the
Robust Wald statistics W for testing the null hypothesis of GARCH specification seem to
be proportional to the sample frequency. This finding could imply that the long memory

feature becomes more pronounced as we observe price changing more frequently within a

51

particular sample period, while the long memory estimate levels themselves remain
similar across different sample frequencies. Therefore, we can conjecture that, as higher
sample frequencies are considered, the FIGARCH conditional variance specifications
become superior to a simple GARCH model that does not implement long memory

volatility.

3.4. Local Whittle Estimation and Self-Similarity

An alternative to the parametric long memory models used so far in this chapter is
the application of the semi-parametric, local Whittle estimator for estimation of long
memory parameters. The advantage of this estimator is that it allows for quite general
forms of Short run dynamics while ARFIMA and FIGARCH models are potentially
sensitive to the specifications used to represent the short-run dynamics; see Kunch (1987)
and Robinson (1995). Of course, semi-parametric estimation has its own problems, as it
is very data intensive and often exhibits poor performance in terms of bias and mean
square error. We apply local Whittle estimation as a robustness check on the FIGARCH
parametric estimates of the long memory parameter d.

A characteristic of long memory that is independent of parametric model

specification is that the spectrum of the series will be given by f ((0) ~ Griz", as

a) —-> 0 + and G is a constant. This suggests a useful objective function for estimating d

would be

j=1 j=1

Q = ln[(1/m)§w3d1(wj)]—(2d/m)§ 111(6),)

52

where [(601) is the periodogram of the series at frequency raj (see Robinson, 1995).

Solving this objective function numerically gives the local Whittle estimator of d. Note
that it is not necessary to specify the short run dynamics of the process in order to
estimate d in this framework. As shown by Robinson (1995) and others, the main

decision variable is m, the choice of the number of ordinates of the periodogram. For

consistency, it is necessary that [(1/ m) + (m/ T)] -—> 0 as T —-> 90. For asymptotic
. . . . l+2,6 2 -26
normality,1trs required that (l/m)+m [ln(m)] T —> 0 as T —> 00. In the

empirical results reported in this chapter, m is chosen as T030. Note that the asymptotic

variance of the local Whittle estimator is given by (1 / 4m).

Local Whittle estimation of the long memory volatility parameter (1' was applied to
both the daily and high frequency returns for all six commodities studied earlier.
Furthermore, both MA-FIGARCH and local Whittle estimation of d were undertaken for
a range of alternative frequencies (1 -day, 2-day, 3-day, 4-day and 5-day using the daily
data, and various return frequencies between 10 minutes and 2 hours using the high
frequency data). Estimation was undertaken over multiple frequencies to check for the
self-similarity feature. Self-similarity occurs when the magnitude of the long memory ,
parameter does not change across sampling frequencies; see Beran (1994). If the long
memory parameter is invariant across frequencies, then it suggests that the long memory
property is an intrinsic feature of the data and does not result from regime shifts or
exogenous external shocks. The self-similarity property is technically extremely difficult

to test empirically. However, one can subjectively evaluate changes in long memory

53

parameter estimates across frequencies to see whether the self-similarity feature seems to
hold in general.

Results of both FIGARCH and local Whittle estimation of the long memory
parameter d are shown for a range of daily return frequencies in Table 3-5 and for a range
of intra-day return frequencies in Table 3-6. Numbers in parentheses below the estimates
are the estimated standard errors. The first thing to notice is that FIGARCH and local
Whittle estimates of d appear quite consistent with one another, with d estimated in the
range supporting long memory in commodity return volatilities. Hence, previous
conclusions about the existence of the long memory property in commodity return
volatilities using FIGARCH appear robust to specification of alternative representations
of short-run dynamics. The second thing to notice in Tables 3-5 and 3-6 is that the long
memory parameter estimates are generally quite consistent across different return
frequencies, irrespective of whether we look at daily returns or intra-day returns. This
result is consistent with the notion of self-similarity and suggests that long memory and

hyperbolic decay are intrinsic features of commodity return data.

3.5. Conclusions

This chapter has examined the long memory volatility properties of both daily and
high frequency infra-day futures returns for six important commodities. The absolute and
squared returns all possess very significant long memory features and their volatility
processes are found to be well described as FIGARCH fractionally integrated volatility
processes. We also find small departures from the martingale in mean property. The long

memory property in absolute returns was also undertaken by semi-parametric local

54

Whittle estimation of the long memory parameter. The estimation of MA-FIGARCH
models and the application of the local Whittle estimators to absolute returns were also
computed for a range of different sample frequencies using both the daily and infra-day
high frequency returns. The long memory parameter estimates are found to be quite
robust both across estimators and across sample frequencies. This is consistent with a
finding of self-Similarity, which implies that long memory in volatility is a pervasive and
consistent feature of commodity returns, and is not just being caused by shocks or regime
shifts to the underlying price processes.

Our findings suggest that any future empirical application using daily or infra-day
commodity futures returns (for example, optimal hedge ratio estimation, tests for futures
market efficiency, tests for the announcement effect of market news, option valuation,
farm risk portfolio management, etc.) will need to account for the long memory property

in commodity return volatilities.

55

Table 3-1: Summary Statistics of Returns

Corn Soybean Cattle Hogs Gasoline Gold

 

First Day 1/02/80 1/02/80 1/02/80 1/02/80 1/02/91 1/02/80
Last Day 3/30/01 12/29/00 12/29/00 12/29/00 12/29/00 12/29/00

Sample Size 5362 5300 5306 5306 2509 5283

Mean -0.016 -0.005 0.037 0.042 0.0406 -0.0298
High 8.606 7.806 2.867 6.307 12.107 9.745
Low -10.472 -11.665 -2.812 -7.632 -30.987 -9.909
Std. Dev. 1.279 1.341 0.898 1.403 1.9594 1.227

Key: The above statistics refer to 100A ln(P,), where P, is the price of the asset in time
period t.

56

Table 3-2: Estimated MA-FIGARCH Models for Daily Futures Returns

 

Com Soybeans Cattle Hog Gasoline Gold
[.1 -0.0171 -0.0229 0.0456 0.0524 0.0097 -0.0367
(0.0152) (0.0152) (0.01 17) (0.0217) (0.0337) (0.0102)
0 0.061 8 -0.0220 * * 0.0695 -0.0247
(0.0151) (0.0144) (0.021 1) (0.0163)
(1 4 0.3154 0.3451 0.3718 0.3687 0.3179 0.2969
(0.0362) (0.0493) (0.0422) (0.0609) (0.0577) (0.0261)
00 0.2036 0.2727 0.0185 0.0621 0.7625 0.0399
(0.0473) (0.0607) (0.0141) (0.0386) (0.2151) (0.0288)
,6, 0.2542 0.3313 0.3603 0.3420 0.2852 0.1923
(0.0442) (0.0597) (0.0466) (0.0639) (0.0650) (0.0438)
,6, 0.0819 0.1206
(0.0212) (0.0202)
7 -0.1820 0.4218 0.0701 0.1933 -1.0264 0.0595
(0.0890) (0.1226) (0.0383) (0.0994) (0.2978) (0.0587)
m3 -0.003 0.016 -0.170 -0.142 -0.166 -0.097
m4 4.218 4.917 3.100 3.079 3.916 8.750
Q(20) 20.232 21.446 29.906 17.147 25.765 22.476
Q2.(20) 30.887 34.976 16.875 21.426 13.407 19.697
W 76.092 48.950 77.698 36.699 30.406 55.693

 

Key: Robust standard errors based on QMLE are in parentheses below the corresponding
parameter estimates. The diagnostic statistics Q(20) and 02(20) are the Ljung-Box
statistics based on the first 20 autocorrelations of the standardized residuals and the
autocorrelations of the squared standardized residuals respectively. The statistics m3 and
m are the sample skewness and kurtosis respectively of the standardized residuals.

The symbol * indicates that MA(S) and MA(10) models respectively were estimated for
live cattle and live hogs respectively. The parameter estimates are not reported to
conserve space.

57

Table 3-3: Summary Statistics for Five Minute Futures Returns

 

Number of Number of First Last time

trading days intraday intervals time period period
Corn 471 44 9:40 13:15
Soybeans 409 44 9:40 13: 1 5
Gasoline 401 63 10:00 15:00
Live Cattle 405 45 9:20 13:00
Live Hogs 400 45 9:20 13:00
Gold 401 72 8:30 14:25

Corn Soybean Cattle Hogs Gasoline Gold

First Day 5/03/99 5/03/99 5/03/99 5/03/99 5/03/99 5/03/99
Last Day 3/30/01 12/28/00 12/28/00 12/28/00 12/28/00 12/28/00
Sample Size 20724 17996 18225 18000 25263 25842

Mean -0.003 -0.001 0.013 0.027 0.033 -0.009
High 14.706 14.721 7.231 22.422 33.416 25.168
Low -15.783 -14.846 -7. l 60 -23.530 -32.308 -28.664

Standard Dev. 1.658 1.571 0.819 1.982 2.463 1.052

58

 

Table 3-4: Estimated MA-FIGARCH model for Filtered High Frequency Futures

 

Returns
Corn Soybean Cattle Hog Gasoline Gold
Sample 10 min. 10 min. 15 min. 15 min. 15 min. 15 min.
frequency
p -0.0030 -0.0023 0.0031 0.0091 0.0145 -0.0037
(0.0017) (0.0021) (0.0015) (0.0032) (0.0039) (0.001 1)
0 -0.1560 -0.0659 -0.0525 -0.0490 -0.0274 -0.0750
(0.01 12) (0.0120) (0.0144) (0.0158) (0.0127) (0.0134)
(1 0.2429 0.2213 0.2097 0.3503 0.1843 0.2047
(0.0368) (0.0329) (0.0367) (0.0620) (0.0218) (0.0421)
0) 0.0014 0.0018 0.0024 0.0030 0.0449 0.0026
(0.0004) (0.0005) (0.0019) (0.0012) (0.0062) (0.0006)
0 0.8866 0.8736 0.4234 0.7242 0.0572 0.2534
(0.0339) (0.0309) (0.3885) (0.0816) (0.0291) (0.1666)
0 0.8314 0.8279 0.3450 0.5485 0.3573
(0.0462) (0.0417) (0.3810) (0.0964) (0.1726)
m3 0.366 0.106 -0.1 1 1 -0.199 -0.134 0.048
m4 6.882 7.335 4.728 6.138 5.151 8.508
Q(20) 28.673 19.973 17.685 23.944 34.336 25.893
Q2(20) 16.592 15.556 7.917 24.340 17.650 11.566
W 43.548 45.173 32.716 31.944 71.753 23.619

 

Key: As for Table 3-2

59

Table 3-5: Long Memory Parameter Estimation at Different Daily Sample

 

Frequencies.
1 days 2 days 3 days 4 days 5 days

Corn

F IGARCH 0.3154 0.2734 0.3096 0.3312 0.2510
(0.0362) (0.0460) (0.0670) (0.0833) (0.0671)

Local Whittle 0.4072 0.3446 0.3052 0.3122 0.2359
(0.03 76) (0.0471 ) (0.0539) (0.0592) (0.0635)

Soybeans

FIGARCH 0.3451 0.3403 0.4052 0.3096 0.3294
(0.0493) (0.0780) (0.1385) (0.0783) (0.0921)

Local Whittle 0.3902 0.3688 0.3918 0.3356 0.3260
(0.0378) (0.0474) (0.0541) (0.0592) (0.0638)

Live Cattle

FIGARCH 0.3718 0.4399 0.4208 0.4335 0.4747
(0.0422) (0.0863) (0.0821) (0.0984) (0.1395)

Local Whittle 0.3866 0.3383 0.3361 0.3234 0.3226
(0.0378) (0.0472) (0.0539) (0.0592) (0.0603)

Live Hogs

FIGARCH 0.3687 0.3041 0.3085 0.2900 0.241 1
(0.0609) (0.0578) (0.0659) (0.0935) (0.0805)

Local Whittle 0.4061 0.3416 0.3609 0.2987 0.2455
(0.0378) (0.0472) (0.0539) (0.0592) (0.0603)

Gasoline

FIGARCH 0.3179 0.3140 0.2851 0.2999 0.2052
(0.0577) (0.0707) (0.1041) (0.1430) (0.0874)

Local Whittle 0.2935 0.2548 0.2967 0.2722 0.2400
(0.0481) (0.0603) (0.0689) (0.0760) (0.0818)

Gold

FIGARCH 0.2969 0.3435 0.2754 0.3357 0.3108
(0.0261) (0.0520) (0.0400) (0.0470) (0.0857)

Local Whittle 0.4323 0.3766 0.3565 0.3659 0.3432
(0.0378) (0.0474) (0.0541) (0.0595) (0.0638)

60

Table 3—6: Long Memory Parameter Estimation at Different Intraday Sample

 

 

 

 

 

 

Frequencies.
Corn 10 min- 20 min- 55 min- 1 hr. 50 min.
FIGARCH 0.2429 0.1919 0.2196 0.0814
(0.0368) (0.0430) (0.0951) (0.1556)
Local Whittle 0.1941 0.2037 0.1622 0.1652
(0.0255) (0.0324) (0.0462) (0.0595)
Soybeans 10 min. 20 min- 55 min. 1 hr. 50 min.
FIGARCH 0.2213 0.2689 0.2431 0.31 1 1
(0.0329) (0.0560) (0.0758) (0.1378)
Local Whittle 0.2533 0.2365 0.1706 0.1448
(0.0268) (0.0340) (0.0486) (0.0625)
Live Cattle 15 min. 25 min- 45 min- 1 hr. 15 min.
FIGARCH 0.2097 0.2580 0.2519 0.2483
(0.0367) (0.0492) (0.0806) (0.0986)
Local Whittle 0.2128 0.1833 0.1796 0.1421
(0.0307) (0.0366) (0.0451) (0.0540)
Live Hogs 15 min. 25 min. 45 min. 1 hr. 15 min.
FIGARCH 0.3503 0.3987 0.3936 0.4045
(0.0620) (0.0835) (0.1 127) (0.1405)
Local Whittle 0.2993 0.3414 0.2988 0.2802
(0.0308) (0.0368) (0.0453) (0.0543)
Gasoline 15 min. 35 min- 45 min. 1 hr. 45 min.
F IGARCH 0.1843 0.2672 0.2215 0.2191
(0.0218) (0.0556) (0.0590) (0.0828)
Local Whittle 0.2243 0.1876 0.1870 0.2436
(0.0274) (0.03 76) (0.0401) (0.0543)
Gold 15 min- 45 min- 1 hr. 30 min. 2 hr. min.
F IGARCH 0.2047 0.2870 0.3167 0.4403
(0.0421) (0.3087) (0.1 180) (0.2742)
Local Whittle 0.3832 0.3818 0.2704 0.2486
(0.0261 ) (0.0381) (0.0486) (0.0540)

 

-0.

-0.

.25

.20

.15.]

. 102
.054

.00...

-20
.15-
.10..
.051

.00-

Figure 3-1. Autocorrelation of Daily Live Cattle Futures

8

 

9

.02

9

8

200 -00 :0. :00 1000

 

[— Autocorr. of Returns]

 

 

 

05

 

 

V'YUI'VVVIT'YUI'VV'EVVT'fiTUYl'U'U'Y‘II'I'VUIIIV'I‘UYIIIIYIEI'VIIVY'YIFTIYIVY'YEIII"VWII'YIVII'YU

[— Autocorr. of Squared Returns]

 

 

 

 

 

O5...
200 400 300 800 1000

 

I — Autocorr. of Absolute Returngl

 

62

Figure 3-2. Autocorrelation of Daily Corn Futures

0.08

 

0.06

0.04

 

0.00 . [.‘H
-0.02

-0. 04

 

20. 4o. :00 200 1000

 

L— Autocorr. of Returnj

 

 

 

 

 

IU‘IVVUYIT'V1"'VI'VYYT'T‘T'I'V'V'TV'T!VIVVTVY'TTVYVUIU ''' E III I'VVVVIUIUI' '''' E ''' I'UITVUII'V'frV'

[— Autocorr. of Sqaured Returng

 

 

 

 

-0. 2

 

IVIIII!IIII'UV‘IIUI'VVUYIYT'VIU'IU[UV'F‘IVI'IUVYTI—VTTW‘IIIWVIVU,VIVVIVI"lVUVUIIYVIIIIIIIYYTYIYYYY

200 400 600 800 1 000

 

L— Autocorr. of Absolute Returns]

63

 

- -0.

-0.

-—0.

-0.

—0.

 

.25
.20-
.15-
.104
.05-

.OO_1

9.9.9.999

Figure 3-3. Autocorrelation of F iltered Daily Corn Futures

 

04—

 

 

 

YTTTI‘VTT‘VY'1‘VIVI'Y'YTIY‘Y'I'U'I'V'V'EVYFVIUI'VIIV'YIYY‘V!'T'UI"'Y'UVVIIV'IU ‘VlI'YY'YIUVVVIU'VV

I:- Autocorr. of Returns]

 

 

 

 

 

05..
200 400 600 800 1000

 

[— Autocorr. of Squared Returns]

25

 

20..

15-

10—

05-

00‘

05.4

 

10

 

 

711'[YYVYIIVYTIVYYYIVY'IIVYYTrYTjY'YYIIEI'UVIVIIVIYVYU[I'IIIV'Vvi‘UtT'YY'V]"VV!YTYVIUIUYIYYTVIVVTY

1000

 

]— Autocorr. of Absolute Returns]

 

-0.
-0.

9.9.9999

.9

10

Figure 3-4. Autocorrelation of Daily Soybean Futures

 

 

 

 

 

VVVI'FYVVIYVIY'VY'YETTYHI'VVY'II'IIVVVVE'VY'I'V'T'fifi'VYYY VTWF'V'IT'VIIVVIV!'VI"'UVVITVVT'VTV'

[— Autocorr. of Returns]

 

 

 

 

YV'I'I'YVII‘VV'V'IIEYYYI'VVIVr'V'YIYYV'IVYV'IY‘VYITY'YIWYY"V'V'V‘VYIVVYY' '''' E '''' 'V'VVIVVYVIYYYV

]— Autocorr. of Squared Returns]

 

 

 

 

 

YV'I'V‘UY'V'VU'Imh'l‘l'VVYIY'VI‘IIVI vvrvlvvrv'vv—v—v'vrr TY‘rY—VTI'IIVVFTTYEIVVIIII'VIIVVTI—VTV‘

200 4 1000

 

[— Autocorr. of Absolute Returns]

65

 

Figure 3-5. Autocorrelation of Filtered Daily Soybean Futures

0.06

 

0.04

0. 02

 

0.00

 

-0. 02

 

-0. 04

 

-—0.06
20. 4o. :00 :00 1000

 

] — Autocorr. of Returns]

 

 

 

 

 

'YYYI'VVVI'TTY'VV‘U!YVVY'Y'TIVIVIV'l'VY'ITTYVIIYTYIYYYIIV'1' VI'VIVVIYIVIVVII"VIfiY'l'VI'l'UU'

6 800 '"1000

 

] — Autocorr. of Squared Returns]

 

 

 

-0. 1

 

V'IY‘V'UVIYYVI'II'Vl'Yfi1IY'YI'VY'I'IVY'V'UV'I'I‘I'VVYII'YIE'V'V‘YV'I[7"1"'Y'IY'Y'I‘Y'VIYYYIFYYYY

1000

 

[— Autocorr. of Absolute Returns]

66

Figure 3-6 Correlograms for Absolute Raw and Filtered Five-minute Returns

0. 25

 

0.201

p
_L
(.11
1

I
1‘1
r
'i ”I I‘ a"
II II 1. I1
\ l I [I I! I. l
I I! I I I ‘ I \ I
. -« . ' I \ 1 I 1 I
I r
I I | I |
t
‘ 1

Sample Autocorrelations
p
5‘5

p
1

,1
- 3

"
C -
a."
4

. . ‘
U \A I‘ ‘ |. 'U

i "I. MN!"

 

g“:
8

 

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIllllllllllIIIIIIIIIIIIIIIIIIIIIIIIIllllllllllllllllllllllIIIIIIIIIIIIIIIIIIIIIII

Five-Minute Lags: Com

0.25

 

0.20 _

0.15-

I.
r I
' I IN 1‘ I
I H I
. -r I : I I I \ I“
I 1 i l I [I I I I
I t I I. r ‘ I l ,I
t I
h I I : fl h I .1 |‘ I
b \ I a 0 |
I . 1 \
‘ Ir I I I
\ I
8 0 05 . . . ' "
, - ' 1
I 1 I
s ‘ I ~
< I ' 4
i‘ I. . \ I “
I \
£- 1 I r 1 | \\ r
- 1 1 I :1 I I ‘ 1
E ' ' , I. I \ f' I ‘ I
w II I I I I ’ I J \ I
I 4! \ F I I ‘ n r l
(D I’ o l I -. I ‘ . \ 1
I111" , t I: I! | . ,
t r ’\’.’ I. I \ ‘01
005 .. ~t
. d

 

-0.10

 

 

lllIlIllIIIIIIIIllllIlllIIIIIIIIII|IIIlIIIIIIIIIlllllllllllllIIIIIIIIIIIII'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

Five-M'nute Lags: Soybean

Key: Dotted line and solid line indicate the sample autocorrelations for absolute raw and
filtered 5-minute futures return returns, respectively.

67

 

0. 20

 

0.15-

Sample Autocorrelations
. .o
5‘5

 

 

 

I‘I 1551 ‘ ’ ”I
o 05- l 541 13. ‘1' '1 ,5
“I. " I ' , '1 I ‘ 1"
I If: l 1‘} . .1 ‘ I ‘
II“ ,“JIII' ‘I . 1 1} I 1 1 I
I “III. "‘I "new 11.". II 1 '1‘? \I‘III
0.00. I ‘1” ' '1' '
50 1 150 200
Five-M'nute Lags: Live Cattle
0.30
0.25.
g 0.20-
g 0.15.
E
E . I‘: I.
£ 0.10. 1*. I' ‘ '11, m l
E In 1’ '1 u ‘ ”l I I
8 11' "\f ’z'I' '1‘ 1" \,“ a 1. . fl ‘1 I I"
o 05 I“: If ‘ VIEW.” In," I 111,31 '
. .1 . I, I ‘1 ‘ IA": ‘1 4 '
0.00-

 

 

50 1 150
Five-M'nute Lags: Live Hogs

Key: Dotted line and solid line indicate the sample autocorrelations for absolute raw and
filtered S-minute futures return returns, respectively.

68

 

0.25

 

0.20 -

p
A
(1'!
1

 

g .
. 1
‘
§ ‘. g
S 0.10 _ g ..
I 'l 'g ‘
' II 1 ‘ r
8 n v :: v'. .
S i 1' ‘u I' V} l’ ‘| : ‘, u'
< ' I | I, 't J \ I
g. 0.05 — 'v 3,; a 4 V. 'I' l,‘ I ‘. I
' "u f | ' ' 5‘ ‘l 1.1
E I I ‘. f ‘ ’ t t
l g ‘ ' | 5
C?) A. ' ‘ v ' ' '
o m a I. ‘\ , 1 ll
. - l. I"! \. I’ | 'g l . L '
WIV’ '3 ~ I. “ 1‘ h a. ‘\ ‘:
l“ ‘ 1.: "*5 I “t ' l '
JV U! "1"" ”J"

 

 

ITTTIWTIIlllllllllllllllllll'lllllllllIllllllTIlFTllllllll

50 100 150 200 250 300
Five-M'nute Lags: Gasoline

0.5

 

p
.5
1

Sample Autocorrelations
.0
OJ

 

 

 

 

IIWIITTI[IHIIll”ll”VIII[TillllllllllllllllHIITITIIIIUIIIITIIHIIH

50 100 150 200 250 300 350
Five-MhuteLags: Gold

Key: Dotted line and solid line indicate the sample autocorrelations for absolute raw and
filtered 5-minute futures return returns, respectively.

69

 

Figure 3-7 Fitted Intraday Volatility Pattern by the F F F filtering
0.30

 

0.25 _

.o .o
a“ 8
l 1

Average Absolute Five Min. Return
9:
8

 

 

_o
8

 

lllllilllllllrlllllTTTTIlllllllllllerTfil

5 10 15 20 25 30 35 40
Intervals Within the Day: Corn

p
()1

 

Avrg. Absolute Returns
0 .o .o
N f” f‘

p
_s

 

 

F.)
o

 

ill]lllTIllllITITTIFTTTIITIIITTTTII—TTTIrTl

510152025303540
Intervals With'n the Day: Soybean

7O

 

.099
8??

Avrg. Absohte Five-Minute Returns
§

 

p
B

 

 

D
N

TIIIITjTIl—rll ITTTTIIIIITTIIIIITTFIIFI[IIll

510123202530354045
Intervals With'n the Day: Live Cattle

 

p
O)
1

_O
(’1
1

Avrg. Absolute F lye-Minute Returns
9 o
‘3’ f‘

 

F3
N

 

 

lllfllllllllllllTllllllITlll'lllllllllllllr

51015202530354045
Intervals Within the Day: Live Hogs

71

 

 

p p p
l l l

Avrg. Absolute Five-minute Returns
.o
00

 

p
N

 

 

0.40

lllllllllllllllllllllllllllllllllllllllllIIIIIIIIIIIIIIIIIITT

51015202530354045505560
Intervals With'n the Day: Gasolne

 

0.35-

s:
.8

0. 25 _

0.20-

0.15_

Avrg. Absolute FiveMinute Returns

0.10

 

 

lllllllllllllllllllllllllllIlllllllllllll[Illlllllllllllllllllllllll

510152025303540455055606570
Intervals Within the Day: Gold

72

 

 

Appendix

The regular opening and closing of commodity markets and the institutionalized
features of lunch hours and so forth give rise to strong intra-day periodicity that is readily
observable from the recurrent U-shaped patterns in the correlograms of the squared and
absolute returns data. This is similar to but difi‘erent from the currency markets where
world-wide trading occurs. Following Andersen and Bollerslev (1998), we first remove
these deterministic intra-day periodicities by applying Gallant’s Flexible Fourier Form

(FFF) filter; see Gallant (1981) and (1982). The estimated model becomes
J’m = E(y,,,,)+(a, sl,n 21.nN-l/2 ) (Al)

where E ( y,, n) is the unconditional mean of returns, 0', is the conditional variance of
daily returns, SM is a deterministic function to represent intra-day seasonality, 2,,n is an

i.i.d(0,1) process, which is independent of the daily volatility process 0', , and N is the

number of return intervals per day. From equation (Al),
x”, = Zln I y”, — E(y,,,,) | —ln(0',2)+ ln(N) = ln(s,,,,2)+ ln(zzm).

The observable variable xm is regressed on a nonlinear function of the time interval n,

and daily volatility 0', is pre-estimated from the MA-FIGARCH model using the daily

futures return, equivalently,

73

xr,=n f(9§9’ n)+u,,,,,

where “Ln = ln(zzm) - E [ln(22,,,,)] is an i.i.d.(0,1) process and the functional form for f

is

2

n

f(9;,’n)= 20!}. {#Oj'i'lulji N] +#2j— N

F —0 2
+Zp—1,6€,k[ pc.os(p27rn/N)+6Sp sin(p27m/N)] (A2)

N N
where N,=(l/N)Zi=(N+1)/2, and N2 =(1/N)Zi2 =(N+1)(2N+1)/6. On taking
i=1 i=1

the variable x”, as the dependent variable, the parameters in equation (A2) were

estimated by OLS. The intra-day periodicity for interval n, on day t is then estimated as

gm, = T.[exp(f,,n nil/[2121,” / N)Zn=l,N exp(f,‘n /2)]. (A3) '

The 10- or 15-minute high frequency returns are then filtered by the estimated intra-day

periodicity series s”, to generate the filtered returns, which are defined as

ytm = yt,n /§t,n ' (A4)

74

 

The same filtering approach is also used to remove yearly seasonality existing in
daily absolute returns. We use the sum of squared daily returns for each year as a
substitute for the conditional volatility factor of the corresponding year since the number
of sample years is less than 30 for all the commodities and is too short to model
conditional variances properly. Alternatively, since we have a sufficient number of daily
return observations within each year, the sum of squared daily returns yield desirable ex-
post volatility measures for the associated year. The volatility measure is called “realized
volatility” in the literature, including Andersen, Bollerslev, Diebold, and Labys (2001 ,
2003). They provided theoretical support for the use of the realized volatility measure
and empirically showed the forecasting and modeling performance of the volatility
measures in comparison to parametrically estimated conditional variances. The realized
volatility series for the commodity futures market is analyzed in chapter 4. The details
for the Fourier flexible functional regressions for the filtering are not reported here, but

are available upon request.

75

Table A-1: Estimated MA-FIGARCH Models for Temporally Aggregated Daily
Futures Returns for Corn

(The sample period: 1/02/80 — 3/30/01)

 

1 days 2 days 3 days 4 days 5 days
T 5362 2681 1787 1340 1072
p -0.0171 -0.0271 -0.0305 -0.0226 -0.0684
(0.0152) (0.0321) (0.0481) (0.0674) (0.0853)
0 0.0618 0.0206 0.0122 0.0225 0.0172
(0.0151) (0.0212) (0.0249) (0.0291) (0.0309)
d 0.3154 0.2734 0.3096 0.3312 0.2510
(0.0362) (0.0460) (0.0670) (0.0833) (0.0671)
0) 0.2036 0.5326 0.7719 0.6386 0.9331
(0.0473) (0.1660) (0.2935) (0.3850) (0.7374)
13 0.2542 0.1788 0.2451 0.2848 0.1426
(0.0442) (0.0583) (0.0855) (0.1083) (0.0969)
y -O.1820 -0.3458 -07737 02777 0.5884
(0.0890) (0.2751) (0.4779) (0.7623) (1.2967)
m3 -0003 -0020 0.060 0.1 16 0.139
m4 4.218 3.955 3.932 3.836 4.002
Q(20) 20.232 23.658 19.151 17.220 13.021
Q2(20) 30.887 21.600 21.215 15.462 15.388
w 76.092 35.281 21.362 15.793 13.399

 

Key: As for Table 3-2

76

Table A-2: Estimated MA-FIGARCH Models for Temporally Aggregated Daily

Futures Returns for Soybean

(The sample period: 1/02/80 — 12/29/00)

 

1 days 2 days 3 days 4 days 5 days
T 5300 2650 1766 1325 1060
p. 0.0229 -0.036O -0.0614 -0.0623 -0.0218
(0.0152) (0.0302) (0.0464) (0.0598) (0.0750)
0 -00220 -0.0234 .0.0090 -0.0250 0.0457
(0.0144) (0.0206) (0.0252) (0.0297) (0.0304)
d 0.3451 0.3403 0.4052 0.3096 0.3294
(0.0493) (0.0780) (0.1385) (0.0783) (0.0921)
(0 0.2727 0.5874 0.9308 1.5733 1.2153
(0.0607) (0.1934) (0.3575) (0.5494) (0.6612)
13 0.3313 0.2894 0.3571 0.1907 0.1695
(0.0597) (0.0961) (0.1625) (0.0981) (0.1164)
y -O.4218 -O.866O -1.9046 -2.3028 -1.0082
(0.1226) (0.3349) (0.4880) (0.8459) (1.1923)
m3 0.016 0.033 0.104 0.238 0.215
m4 4.917 4.347 3.700 3.426 3.862
Q(20) 21.446 18.607 14.427 17.265 15.823
Q2(20) 34.976 27.228 16.050 18.887 17.339
w 48.950 19.027 8.561 15.644 12.791

 

Key: As for Table 3-2

77

Table A-3: Estimated MA-FIGARCH Models for Temporally Aggregated Daily

Futures Returns for Live Cattle
(The sample period: 1/02/80 — 12/29/00)

 

1 days 2 days 3 days 4 days 5 days
T 5306 2653 1768 1326 1061
u . 0.0456 0.0948 0.1482 0.1882 0.2244
(0.01 17) (0.0213) (0.0308) (0.0446) (0.0525)
9 It
(1 0.3718 0.4399 0.4208 0.4335 0.4747
(0.0422) (0.0863) (0.0821) (0.0984) (0.1395)
0) 0.0185 0.0485 0.1 161 0.3017 0.3993
(0.0141) (0.0360) (0.0801) (0.1507) (0.1989)
[31 0.3603 0.3867 0.3401 0.4168 0.4051
(0.0466) (0.0901) (0.0878) (0.1037) (0.1380)
[32 0.0819 0.1174 0.0959 0.0617 0.0164
(0.0212) (0.0270) (0.0310) (0.0324) (0.0513)
y 0.0701 0.0304 -0.0674 -0.6927 -0.7675
(0.0383) (0.1186) (0.2362) (0.3897) (0.5539)
m3 -0.170 -0.186 -0.195 -0.225 -0.310
7724 3.100 3.275 3.175 3.274 3.290
Q(20) 29.906 13.801 15.224 15.729 9.886
Q2(20) 16.875 14.327 12.698 15.723 17.826
W 77.698 25.975 26.259 19.404 1 1.583

 

Key: As for table 3-2. (*) indicates that we omitted MA(5) coefficient estimates here

since they are not important to our argument in the current chapter.

78

Table A-4: Estimated MA-FIGARCH Models for Temporally Aggregated Daily

Futures Returns for Live Hogs

(The sample period: 1/02/80 - 12/29/00)

 

1 days 2 days 3 days 4 days 5 days
T 5306 2653 1768 1326 1061
p 0.0524 0.0987 0.1526 0.2184 0.2408
(0.0217) (0.0467) (0.0625) (0.0867) (0.1 164)
9 11:
d 0.3687 0.3041 0.3085 0.2900 0.241 1
(0.0609) (0.0578) (0.0659) (0.0935) (0.0805)
(0 0.0621 0.3920 0.5817 1.5508 2.9756
(0.03 86) (0.1905) (0.31 16) (0.5974) (1.0538)
Br 0.3420 0.2895 0.2526 0.2225 0.1020
(0.063 9) (0.0634) (0.073 1) (0.0964) (0.0797)
132 0.1206 0.0335 0.0168 -0.0349 -0.0680
(0.0202) (0.0291 ) (0.0500) (0.0471) (0.0645)
y 0.1933 0.0117 0.0930 -1.3551 -2.5735
(0.0994) (0.4768) (0.5817) (0.9273) (1.2072)
m3 -0.142 -0.278 -0.248 -0.349 -0.226
m4 3.079 3.636 3.622 3.734 3.648
Q(20) 17.747 16.775 1 1.670 8.097 1 1.780
Q2(20) 21.426 19.681 16.352 21.553 6.912
W 36.699 27.644 21.951 9.615 8.974

 

Key: As for table 3-2. (*) indicates that we omitted MA(lO) coefficient estimates
here since they are not important to our argument in the current chapter.

79

Table A-5: Estimated MA-FIGARCH Models for Temporally Aggregated Daily

Futures Returns for Gasoline

(The sample period: 1/02/91 — 12/29/00)

 

1 days 2 days 3 days 4 days 5 days
T 2508 1254 836 627 501
p. 0.0097 0.0148 0.0252 0.0806 0.1 140
(0.0337) (0.0692) (0.1063) (0.1366) (0.1820)
0 0.0695 0.0303 -0.0176 0.0017 0.0050
(0.0211) (0.0286) (0.0386) (0.0373) (0.0472)
d 0.3179 0.3140 0.2851 0.2999 0.2052
(0.0577) (0.0707) (0.1041) (0.1430) (0.0874)
00 0.7625 1.6346 3.4137 3.8882 8.2542
(0.2151) (0.5864) (1.6571) (2.6813) (3.4729)
(3 ' 0.2852 0.2949 0.2026 0.2829 0.0715
(0.0650) (0.0835) (0.1189) (0.1727) (0.1221)
y 4.0264 -2.2562 4.1021 -5.1878 -7.9280
(0.2978) (0.8571) (2.0028) (3.2309) (4.8035)
m3 -O.166 -0132 -0270 0.037 0.086
7714 3.916 3.444 3.716 3.467 3.375
Q(20) 25.765 20.823 17.493 22.675 24.154
02(20) 13.407 20.628 23.396 12.190 14.627
w 30.406 19.745 7.495 4.400 5.509

 

Key: As for Table 3-2

80

Table A-6: Estimated MA—FIGARCH Models for Temporally Aggregated Daily
Futures Returns for Gold
(The sample period: 1/02/80 — 12/29/00)

 

1 days 2 days 3 days 4 days 5 days
T 5283 2641 1761 1320 1056
p -0.0367 -0.0702 -0. 1050 -0.1295 -0. 1 351
(0.0102) (0.0200) (0.0301) (0.0393) (0.0550)
0 -0.0247 -0.0166 -0.0342 -0.0216' -0.0557
(0.0163) (0.0262) (0.0384) (0.0390) (0.0380)
d 0.2969 0.3435 0.2754 0.3357 0.3108
(0.0261) (0.0520) (0.0400) (0.0470) (0.0857)
(0 0.0399 -0.0368 -0.0526 -0.3467 -0.6776
(0.0288) (0.0465) (0.1867) (0.1012) (0.3057)
[3 0.1923 0.2221 0.0805 0.1890 0.2219
(0.0438) (0.0465) (0.1426) (0.0837) . (0.1269)
y 0.0595 0.3543 0.5537 1.1286 2.9869
(0.0587) (0.1567) (0.3100) (0.3815) (1.6484)
m3 -0.097 0.1 18 0.031 0.248 0.726
m4 8.750 7.674 6.833 6.734 12.135
Q(20) 22.476 23.539 26.282 16.538 23.584
Q2(20) 19.697 26.551 14.605 10.474 8.722
W 55.693 61.185 22.961 42.882 5.570

 

Key: As for Table 3-2

81

TableiA-7: Estimated MA(l)-FIGARCH(1,d,l) model for Temporally Aggregated
Filtered High Frequency Futures Returns for Corn
(The sample period: 5/03/99 - 3/30/01)

 

10 minute 20 minute 55 minute 1 10 minute
T 10362 5181 1884 942
p -0.0030 -0.0072 -0.0126 -0.0220
(0.0017) (0.003 8) (0.01 10) (0.0222)
0 -0.1560 -0.0512 -0.0082 -0.0142
(0.01 12) (0.0158) (0.0276) (0.0370)
(1 0.2429 0.1919 0.2196 0.0814
(0.0368) (0.0430) (0.0951) (0.1556)
(0 0.0014 0.0147 0.0135 0.0934
(0.0004) (0.0107) (0.0134) (0.1416)
0 0.8866 0.4406 0.7952 0.6742
(0.0339) (0.3437) (0.1824) (0.3633)
¢ 0.8314 0.3647 0.7172 0.6372
(0.0462) (0.3341) (0.2076) (0.3489)
m3 0.366 0.068 0.026 -0.265
7724 6.882 9.105 6.102 6.463
Q(20) 28.673 16.375 12.699 17.424
Q2(20) 16.592 13.106 6.181 19.453
W 43.548 19.845 5.340 0.391

 

Key: As for Table 3-2

82

 

Table A-8: Estimated MA(l)-FlGARCH(l,d,l) Model for Temporally Aggregated
Filtered High Frequency futures returns for Soybean
(The sample period: 5/03/99 — 12/28/00)

 

10 minute 20 minute 55 minute 110 minute
T 8998 4499 1636 818
p -0.0023 -0.006 -0.0136 -0.0231
(0.0021) (0.0043) (0.0122) (0.0247)
0 -0.0659 0.0027 -0.0042 -0.03 14
(0.0120) (0.0158) (0.0257) (0.0385)
(1 0.2213 0.2689 0.2431 0.31 1 1
(0.0329) (0.0560) (0.0758) (0.1378)
03 0.0018 0.0036 0.0184 0.0274
(0.0005) (0.0014) (0.0123) (0.0232)
[3 0.8736 0.8377 0.6989 0.7143
(0.0309) (0.0414) (0.1262) (0.0783)
(b 0.8279 0.7182 0.5075 0.4165
(0.0417) (0.0537) (0.1210) (0.1446)
m3 0.106 0.049 0.033 0.107
7724 7.335 6.480 4.802 4.409
Q(20) 19.973 14.269 18.634 17.421
Q2(20) 15.556 20.063 26.673 25.046
W 45.173 23.065 10.302 5.399

 

Key: As for Table 3-2

Table A-9: Estimated MA(l)—FIGARCH(p,6,q) Model for Temporally Aggregated
Filtered.5-minute returns for Live Cattle futures
(The sample period: 5/03/99 — 12/28/00)

 

15 minute 25 minute 45 minute 75 minute
T 6075 3645 2025 1215
p 0.0031 0.0056 0.0080 0.0151
(0.0015) (0.0026) (0.0049) (0.0086)
0 -0.0525 .0.0253 -0.0431 0.0696
(0.0144) (0.0180) (0.0230) (0.0351)
(1 0.2097 0.2580 0.2519 0.2483
(0.0367) (0.0492) (0.0806) (0.0986)
0) 0.0024 0.0020 0.0029 0.0046
(0.0019) (0.0012) (0.0016) (0.0032)
13 0.4234 0.6557 0.7716 0.7722
(0.3885) (0.1583) (0.0882) (0.0887)
0 0.3450 0.5147 0.6269 0.6064
(0.3810) (0.1613) (0.1211) (0.1105)
m3 -0.1 1 1 -0029 -0042 0.133
m4 4.728 4.862 4.894 3.927
Q(20) 17.685 18.159 25.909 23.868
Q2(20) 7.917 13.639 13.347 10.007
w 32.716 27.537 9.775 6.108

 

Key: As for Table 3-2

Table A-10: Estimated MA(l)-FlGARCH(p,6,q) Model for Temporally Aggregated
Filtered.5-minute returns for Live Hog futures
(The sample period: 5/03/99 — 12/28/00)

 

15 minute 25 minute 45 minute 75 minute
T 6000 3600 2000 1 200
p 0.0091 0.0155 0.0270 0.0402
(0.0032) (0.0058) (0.0105) (0.0171)
0 -0.0490 -0.0124 -0.0389 0.0063
(0.0158) (0.0214) (0.0252) (0.0349)
(1 0.3503 0.3987 0.3936 0.4045
(0.0620) (0.0835) (0.1 127) . (0.1405)
0) 0.0030 0.0033 0.0088 0.0083
(0.0012) (0.0014) (0.0050) (0.0076)
0 0.7242 0.7698 0.6719 0.7349
(0.0816) (0.0474) (0.1079) (0.0956)
0 0.5485 0.5081 0.4046 0.4788
(0.0964) (0.0658) (0.0921) (0.1297)
m3 -0.199 -0.232 -0.283 -0.100
m4 6.138 6.064 5.465 5.058
Q(20) 23.944 20.766 17.932 15.106
Q2(20) 24.340 9.11 1 23.959 25.850
W 31.944 22.779 12.205 8.287

 

Key: As for Table 3-2

Table A-ll: Estimated MA(l)-FIGARCH(p,6,q) Model for Temporally Aggregated
Filtered.5-minute returns for Gasoline futures
(The sample period: 5/03/99 — 12/28/00)

 

15 minute 35 minute 45 minute 105 minute
T 8421 3609 2807 1203
p. 0.0144 0.0295 0.0337 0.0683
(0.0039) (0.0092) (0.0121) (0.0294)
0 -0.0270 -0.0127 -0.0320 0.0209
(0.0127) (0.0184) (0.0191) (0.0258)
d 0.1725 0.2672 0.2215 0.2191
(0.0180) (0.0556) (0.0590) (0.0828)
0) -0.0745 0.0220 0.0522 0.1721
(0.0121) (0.0077) (0.0242) (0.1 181)
0 -0.6661 0.6872 0.5713 0.4257
(0.1391) (0.0675) (0.1 1 14) (0.1646)
11) -0.7085 0.5473 0.3915 0.1805
(0.1249) (0.0832) (0.1067) (0.1596)
m3 -0.129 -0.214 -0.152 -0.333
7724 5.078 4.804 4.498 4.213
Q(20) 34.733 22.744 30.990 20.415
Q2(20) 15.11 1 10.238 12.435 25.223
W 71.753 36.224 17.067 8.795

 

Key: As for Table 3-2

Table A-l2: Estimated MA(l)—FIGARCH(p,6,q) Model for Temporally Aggregated
F iltered.5-minute returns for Gold futures
(The sample period: 5/03/99 — 12/28/00)

 

15 minute 45 minute 1 hr. 30 min. 2 hr.
T 9624 3208 1604 1203
1.1 -0.0037 -0.0134 -0.0195 -0.0284
(0.001 1) (0.0037) (0.0107) (0.01 14)
0 -0.0750 —0.0368 -0.0130 -0.0130
(0.0134) (0.0365) (0.243 7) (0.0481)
d 0.2047 0.2870 0.3167 0.4403
(0.0421) (0.3087) (0.1291) (0.2742)
0) 0.0026 0.0032 0.0024 0.0223
(0.0006) (0.0471 ) (0.0029) (0.0152)
[3 _ 0.2534 0.6250 0.7714 0.1646
(0.1666) (5.3126) (0.2648) (0.1522)
(11 0.3573 0.6230 0.6747
(0.1726) (5.0333) (0.3695)
m3 0.048 0.552 1.354 0.792
7714 8.508 13.644 18.805 9.259
Q(20) 25.893 27.094 29.200 21.872
Q2(20) 11.566 33.598 12.597 31.362
W 23.619 0.8643 6.0146 2.5788

 

Key: As for Table 3-2

87

CHAPTER 4

REALIZED VOLATILITY IN COMMODITY FUTURES MARKETS

4.1. Introduction

This chapter considers the new concept of realized volatility (RV), which is
constructed from high frequency returns. We initially describe the new measurement and
its properties and then apply the idea to commodity futures markets for the six important
commodities considered in chapters 2 and 3. This study appears to be the first analysis
using these concepts for commodity futures markets.

One interesting finding in this study is that the pure volatility measure known
as realized volatility has almost ideal long memory features, which is consistent with
previous work of Anderson, Bollerslev, Diebold and Labys (2002), who examined
currency markets. We find that the commodity realized volatility is very well described
as a fractionally integrated process and, furthermore, appears to follow a Gaussian
distribution. At this level, our results are quite similar to the preceding literature, which-
has applied the concept of realized volatility to currency markets. However, unlike
previous studies, we suggest particular factors that may possibly generate and interact
with realized volatility series. In the context of commodity markets, these factors include
the time to maturity of the futures contract and also the arrival of important economic
news. Also, and particularly importantly, we consider a new concept: information flow,

which depends on the total number of transactions at each high frequency interval of the

88

market being active. This information flow variable turns out to be simple to compute
and highly correlated with the measurement of realized volatility. In addition, this
chapter examines the dependency structures between the realized volatilities for the
different commodity futures data. This gives a clear indication of the mutual
dependencies between the factors driving agricultural-type commodities such as corn
and soybeans, while there is unsurprisingly little relationship between less related
commodities. There is also some evidence of fractional cointegration between the
realized volatility of com and that of soybeans.

In this chapter, section 2 provides formal and theoretical background information
for the concept of realized volatility. In section 3, we introduce and briefly discuss all of
the possible issues relevant to realized volatility before the empirical investigation
below. Section 4 investigates the stochastic properties of realized volatility to model and
forecast the volatility measurement. Various important economic factors relevant to

commodity futures markets are considered in section 5. Section 6 concludes the chapter.

4.2. Statistical Foundations of Realized Volatility:

Before defining the concept of realized volatility to be used in this chapter, it is
important to recognize that, historically, there has been an awareness of the desirability
of measuring the volatility associated with a continuous time diffusion process. In
particular, Merton (1980) and Nelson (1992) argue that, under the theoretical
assumption of a continuous diffusion process, the inherent volatility can be best
measured by integrating high frequency returns data. Indeed, the finance literature has

long focused on issues like instantaneous variance in the context of option pricing. The

89

relevant previous studies include Hull and White (19887), Melino (1994), Scott (1987),
and Wiggins (1987). The basic idea of realized volatility is that it can approximate the
theoretical quadratic variation when the sample frequency within the time interval
considered is sufficiently high, and in turn, it can provide a consistent estimator for the
true latent volatility factor. These ideas are descendants of the approach of Porterba and
Summers (1986), French, Schwert, and Stambaugh (1987), and Schwert (1989), who
used daily returns to construct a measure of monthly volatility. Hence, the high
frequency returns data can be used to construct a measure of integrated volatility by

summing squared intra-day returns.

However, the measurement of realized volatility is difficult due to the fact that
high frequency returns have a number of contaminating or complicating factors. In
particular, as seen in chapter 3 of this dissertation, the high frequency commodity returns
data are intimately involved with market microstructure factors, including a bid-ask
spread and pronounced intra-day periodicity. Hence, the high frequency returns are first
filtered by Gallant’s (1981) Flexible Fourier Form (FFF) method before subsequent
analysis, as was done in chapter 3. There is also the issue of spreads and jumps occurring
at certain times. Before discussing the issues and practical problems with the
implementation of the concept of realized volatility in commodity markets, we will first

define the mathematical foundations of this concept.

The quadratic variation theory provides the theoretical foundation of realized
volatility as a model-free unbiased estimator of conditional variance. Quadratic variation
is a measure of the sample path oscillation for a special class of stochastic processes,

known as semi-martingale processes, which have finite variations along their paths. For

90

semi-martingale process X (t) along the sample path (6 [0, T] with a positive integer T,

we define the quadratic variation process as follows:

[[X(r),X(t)]] a X(r)2 —2L’IX(8_)dX(s), 0 St 3 T, (4.1)

where the notation X _ indicates the process whose value at s is X _ = limu _, s,“ S (X H ).

We assume that these processes have a finite variation on [0, T]. We also assume that the

stochastic integral IHdX = {fiH (s) dX (s)} is well defined for semi-martingale
te[0,T]

processes X. Further, we define X(t,h) E X(t)—X(t—h) for OS hsr s T. We

proceed to the following important properties in interpreting the quadratic variation as a

volatility measure.
Property (i):

If we define an increasing sequence of {0,z'm,0,rm,l,....} so that
0 S Tm,o S rmJ S ..., over the fixed time interval [0,T] with supjzl (Tm,j+1 -— rm,j)—> 0

and sup jzl rm, j —) T for m —> 00 with probability one, then we have

{X(0)Y(0)+z,-21[X(’ A 71".} )—X(t A Tm,j—l )][X(t A z""’J')—X(t A Tm’j’l )J}

lim
m—no

—> [[X(t),X(t)]] , (4-2)

91

where the convergence is uniform on [0, T] in probability and (z A x) denotes the

minimum of the quantities z and x. . The lefi-hand side of equation (4.2) above can be
approximated by the sum of squared returns at a sufficiently fine sample frequency, and
the right-hand side in (4.2) represents the quadratic variation measure according to the

definition in (4.1).

Property (ii):

If X (t) is a locally square integrable local martingale,
2
E[X(t,h) —([[X(t),X(z)]]—[[X(t—h),X(z—h)]])| F,_,,] = 0, 0 < h _<_: s T, (4.3)

where F,_h is the information set available at time (t-h).

The formal description of quadratic theory can be utilized for a deep appreciation
of the properties of a continuous time return process. A continuous arbitrage-free price
process for general financial assets is well known to be a special type of semi-martingale,
and thus, quadratic theory can be applied to this process under the assumptions
mentioned above. The price process can be decomposed into a local martingale and a

predictable finite variation process. The local martingale process is an “unpredictable”

innovation. Then we can express the arbitrage-free logarithmic price process p(t) over

the interval [0,T] as follows:

p(’)-P(0)=M(’)+A(1)~ (4.4)

92

where M (t) is a local martingale and A (r) is a locally integrable and predictable
process of finite variation which is deterministic drifi for the price process. Since A (t) is
fully predictable and deterministic, [[A (t), A (t)]| = 0 can be implied. This argument is

intuitively equivalent to the fact that the variance of deterministic components should be

zero and conditional mean is of no import in considering conditional variance. Then, we

are allowed to focus only on martingale terms M (t) in considering the quadratic

variation of p(t) as follows:

(p(,),p(t)y=(M(.),M(.)). (45>

According to the quadratic variation theory and the assumption of a semi-martingale
price process, we can define the h-period quadratic variation for the continuous price

process as follows:

Qvaao)a[primal-172041121:-h)1
a flM(r),M(r)]|-|[M(t—h),M(r —h)]]

(4.6)

From equation (4.2), it is implied that the quadratic variation can be approximated by the

sum of squared high frequency returns for a given interval [1 — h,t]. Based on this

notion, we define the h-period Realized Volatility measure at time t,

93

RI/(h)(r)-=-Zi=,,mhrf(m)(t—h+(i/m)) fori=1,2,...,m(h—1),mh, (4.7)

where rk,(m)(t—h+(i/m)) is equal to p(t—h+(i/m))—p(t—h+(i-1)/m). In fact,

the realized volatility in equation (4.7) is an empirical approximation for the left-hand
side of equation (4.2) over the interval [t - h,t] and, in turn, converges to the h-period
quadratic variation defined in equation (4.6) by the property in (4.2). Consequently, the
realized volatility is a consistent estimator of the theoretical volatility measure measured
by the quadratic variation.

Another important notion of the realized volatility is that the volatility measure

provides a model-free unbiased estimator of the conditional variance. This fact can be

clarified from the property stated in equation (4.3). If we assume that M (t) is a' locally

square integrable martingale and make use of the property shown in equation (4.3), then

the conditional variance of p(t) is reduced to the conditional expectation of the squared

martingale term as follows:

ar(p(t,h)1 n4.) 2 E(M(t)2 14.).)—E(M(t)1m)2
E(M(l)2|1‘;-h) (4.8)

E[[[M1)M(t)1|[[M(t—h).M(t—h)]l)lf}_h].

The first identity in equation (4.8) is simply a definition of the conditional variance for

the arbitrage price process, and the second equality is due to the assumption of a

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martingale M (t). The last equality in equation (4.8) results from equation (4.3). The

term inside the expectation on the right hand side of the last equality in equation (4.8) is
the same as the volatility measure defined in (4.6). Consequently, the conditional
variance of the compounded return process over [t-h, t] interval is equivalent to the
conditional expectation of the quadratic variation over the interval. As discussed above,
the quadratic variation can be approximated by the realized volatility, as the number of
sub-sample periods within a given interval is sufficiently large. Thus, we state that the
realized volatility is an unbiased estimator for the conditional variance of the
compounded returns. We have not specified any functional form in our claim, (4.8) and
have instead utilized only the properties of the quadratic variation measure and the
assumption of a martingale price process based on the arbitrage-free price. On the other
hand, conventional GARCH models assume a parametric form to model conditional
variances. Thus, the realized volatility can be said to be a model-free unbiased estimator
for conditional variances.

In the theoretical asset and derivative pricing studies, it is frequently assumed that

logarithmic prices follow an univariate diffusion.

dp(t) =,u(t)dt+cr(t)W(t) (4.9)

where W (t) is a standard Brownian motion. Equivalently, we can rewrite the equation

as follows:

95

p(t)—p(t —h) = I_h,tp(s)ds + I_h,ta(s)dW(s). (4.10)

By using the standard stochastic differential equation algebra and Ito’s Lemma, it follows

that

[dp(t):|2 = I_h,ta(s)2 ds. (4.11)

a(t)2 can be termed an instantaneous volatility under the diffusion set-up, and, by using
the volatility defined in (4.6), Q var}, (t)/h is close to a(t)2. Therefore, the integral of

0'(t)2 over the interval [t-h, t] is approximately equal to Qvarh (t) as follows:
Qvarh (t) E flM(t),M (01] —|IM (t — h).M (t — 17)]! = £41,102 (s)ds . (4.12)
Taking a conditional expectation for equation (4.12), then we have
E(Qvarh(’)|Fz-h)=€(I_h,,02(5)dS|Fz—h)- (4-13)

The expected value of the integral metric on the right-hand side of equation (4.13), called
“integrated volatility” in the literature, is of especially central interest in option pricing

studies, as in Hull and White (1987), Melino (1994), Scott, and Wiggins (1987). As

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shown above, the realized volatility over the interval [t-h, t] is an unbiased estimator of

the conditional expectation of Qvarh (t). Accordingly, the realized volatility provides an

unbiased estimator of the integrated volatility measure for pricing derivatives securities
and options.

In particular, for our applications to commodity markets, we consider the case of
a one-day horizon being indicated by h, since the daily time horizon is the sample
frequency that is of central interest for risk management, asset pricing, and portfolio

allocation. In particular, the realized volatility in our study is defined as follows:
RV, = 0.5111(2 ,=I,,,,,,rk2(m) (z — h +(i/m))). (4.14)

By using the properties of the quadratic variation under the assumption of a continuous
arbitrage-free price process, we found that the realized volatility is a consistent estimator

of true latent volatility and is a simple, unbiased estimator of conditional variances.

4.3. Practical Issues in the Calculation of Realized Volatility

High frequency commodity return data have some unique features and also some
features which may be shared with other asset markets, such as the possibility of jumps
and discontinuities in the volatility process arising from major economic announcements.
One of the most important issues in realized volatility calculation is determining how to
model the persistence of the realized volatility. We pursue this issue by applying the long

memory model to the volatility measures. We also investigate the distributional

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properties of a realized volatility that is constructed from five-minute commodity futures
returns. The realized volatility measure is also used to derive the distribution of daily
commodity futures returns standardized by the new volatility measure. In addition, we
analyze various important economic factors that may affect realized volatility dynamics
by considering commodity-specific announcements, the time-to-maturity for the
commodity futures contracts, and an information flow variable. Our approach is
influenced by several previous studies. In particular, Andersen and Bollerslev (1998a)
and Andersen, Bollerslev, Diebold, and Vega (2002) have documented news
announcement effects on the five-minute return volatility for the US Dollar-Deutsch
Mark exchange rate. Also, Bauwens, Omrane and Giot (2003) directly analyzed the news
announcement effects on the realized volatility of Euro-Dollar foreign exchange returns.
In this chapter, we consider commodity-specific announcements as we analyze the news
effects on realized volatility using various time series methods. It appears from this study
that some commodities seem to depend on specific announcements, while major foreign

currencies rely on common macro news.

Unlike conventional financial assets, the volatility of commodity futures contracts
with different delivery dates appears to have its own discernible characteristics due to
possible seasonal patterns of the underlying physical products. Samuelson (1965) argues
that futures price volatility is likely to increase as the contract approaches maturity, which
has become known as the “Samuelson effect.” In that sense, we consider the time-to-
maturity effect on commodity futures return volatility. This time-to-maturity effect has
been documented in a variety of commodity futures market studies, such as Anderson

(1985), Milonas (1986), and Serletis (1992). Further, we consider information flow

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together with time to maturity in order to study the relationship between the realized
volatility and the time to maturity.

Another economic issue for the realized volatility analysis in this chapter is
mutual interdependence between the volatility measures for different commodities.
Intuitively, it seems reasonable that various physical aspects of the commodities
underlying the futures market may be related to their futures return volatilities. For
example, it is possible that the commodity futures return volatilities belonging to similar
commodity products have some dependence on one another. Later, we consider a time
series model to describe the contemporaneous interdependence between different
commodities’ futures return realized volatilities.

As noted before, realized volatility from commodity futures is one representation
of Volatility that does not require a parametric model and can be easily forecasted by a

simple time series econometric model.

4.4. Stochastic Properties for Realized Volatility for Modeling and Forecasting
4.4.1. The Distributional Facts of the Realized Volatility

The distributions of asset returns have been an important issue since
unconditional distributions of most asset returns are usually fat-tailed, and such a feature
has motivated conditional distributions relevant to various GARCH conditional variance
modelings. However, the conditional distributions still remain leptokurtic, although they
are less leptokurtic relative to the unconditional distributions. Turning to the distribution
issue, we describe distributional characteristics of the realized volatility process for the

commodities in Tables 4-2 and 4-3, while Table 4-1 shows that the unconditional

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distribution of daily commodity futures returns is leptokurtic. In contrast to the raw daily
commodity returns, (i) realized volatilities appear to follow a normal distribution, and (ii)
daily commodity returns standardized by the realized volatility are also close to normal
random variables. To be precise, we assume that return process can be modeled as

follows:
’1 = 0'1 81. (4.15)

where r, represents return at time t, 0, denotes the time-varying conditional standard
deviation, and s, is independently and identically distributed with a zero mean and unit
variances for simplicity. The traditional GARCH model estimates conditional variance,
0.2, by assuming parametric form. As found in many previous studies, the distributions
of returns standardized by the GARCH estimated conditional variances seem to still have
higher kurtosis than a normal distribution, although the kurtosis seems to be lower than
an unconditional return distribution. As shown in Table 4-3, the kurtosis of daily
commodity returns noticeably decreases after standardization by the realized volatility.
In particular, the excess kurtosis for gold futures returns is remarkably reduced from 28.9
to 3.39 by standardization using the realized volatility. For live hogs, the associated
kurtosis is decreased from 7.04 to 2.92. The fact that standardized daily commodity
futures returns are normally distributed is supportive of the theoretical assumption of an
underlying continuous-time diffusion, which is found in many mathematical finance
studies. As shown from Figure 4-1 (a), the kernel density graphs for corn, soybean,

cattle, and gasoline realized volatilities are supportive of a normal distribution, while the

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density graphs for live hog and gold realized volatilities are still quite leptokurtic. The
kernel densities for the distributions of daily returns standardized by the realized
volatilities are presented in Figure 4-1 (b). For the standardized futures returns, the
kernel density functions look similar to those for the normal distributions for all the
commodities considered. From the realized volatility levels in Figures 4-2 (a) through
(1‘), we can observe some peaks in the realized live hog volatility and pronounced jumps
for the realized gold volatilities. This feature is consistent with the exceptional
leptokurtic distribution of the realized volatility for live hogs and gold. Other than this

abnormal data feature, the realized volatility seems to follow a normal distribution.

4.4.2. The Long Memory of Realized Volatility

One of the well-known facts of asset return volatility is that it is very persistent,
while returns underlying the volatility are serially uncorrelated. As discussed in Baillie
(1996), the ARFIMA model is a conventional parametric form used to describe slowly
decaying time series processes. The theoretical background for long memory has been
discussed in more detail in chapters 2 and 3. In the current chapter, we estimate a simple
ARFIMA(0,d,0) model to estimate the long memory parameter for the realized volatility.
For completeness of the long memory estimation, we also use the local Whittle semi-
parameter approach that is explained in chapter 3. The estimation results for the
ARFIMA and the local Whittle method are shown in Tables 4-4 and 4-5, respectively.
The long memory parameter estimates for the commodities are in the range of 0.2 to 0.3
in most cases. Some exceptions can be found for estimates for hogs and gold greater than

0.3. The long memory estimate values for all the commodities seem to be very similar

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for both parametric and semi-parametric methods. As shown in Figure 4-2, the realized
volatilities for hogs and gasoline seem to include some significant changes. Their higher
long memory estimates may be due to some possible structural breaks. This issue would
benefit from independent research but is not pursued firrther here. For the other
commodities without unusual data features, the long memory estimates are within a

stable range.

4.4.3. Forecast for Realized Volatility

Based on the theoretical background discussed in section 2, the realized volatility
generated from a sufficient number of high frequency sample returns is a consistent
estimator of true latent volatility factor under the assumption of an arbitrage-free price
process. In that sense, we are allowed to treat the realized volatility as an observable
proxy for the true underlying volatility and assess that the future realized volatility
measure is the “volatility” to be forecasted. We can evaluate various volatility forecasts
by considering which model provides the closest forecast to the realized volatility
measure. Following Andersen, Bollerslev, Diebold, and Labys (2003) and Andersen and
Bollerslev (1998b), we evaluate the forecasting performance by using a simple least
square regression. The regression approach was originally employed in the literature to
evaluate forecasting of the conditional mean in Mincer and Zamowitz (1969). The

generic evaluation regression can be set up as follows:

V, =a0+alCV,_] +61, (4-16)

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where V. is the future volatility at time t, C V... is the one-period ahead forecast generated
under alternative conditional variance models, and e, is an error term. In principle, it can
be implied that do and a. should be equal to zero and unity, respectively, if we correctly

specify a forecasting model for the future volatility factor, for which

E (V, |Q,_l ) = a0 + alC V,_1 . As we assess how the future realized volatility is to be

forecasted, a natural candidate for a good volatility forecast is the one generated from the
past realized volatility time series, since they are very persistent, as illustrated by the long
memory estimation results for the ARFIMA and the semi-parametric estimation shown in
Tables 4-4 and 4-5. The other alternative volatility forecast is generated from the
GARCH-estimated conditional variances that have been elaborated by many previous
studies since Engle (1982) and Bollerslev (1986).

We empirically compare the forecasting performance of the ARFIMA forecast
with the GARCH forecast by running the following three OLS regression set-ups as

shown in Table 46

RV: = 170 +b1RVARFIMA.z—1 + 8: (4-17)

and

RV: = b0 + bZGGARCH,t—I + 51 ~ (4'18)

where R V A RFIMA‘ ,_l denotes one period ahead forecasts from the ARFIMA(0,d,0) model

using the past realized volatility series and O'GARCHJ4 forecasts from the GARCH( 1 ,l)

103

 

 

model using the compounded daily futures returns. Based on the robust standard errors
in Table 4-6, all the b1 estimates for the regression set-up (4.17) are not significantly
different from one another, although the b0 estimates are significantly different from zero
only for live cattle and gasoline. Our finding implies that the ARFIMA forecasting
model is correctly specified in most cases. The forecast ability of the GARCH estimated
conditional variance model'4 can be evaluated by using the set-up (4.18). According to
Table 4-6, all the b2 estimates for the GARCH forecasts are more different fiom one
another than the corresponding estimates for the ARFIMA forecasts. In particular, the b2
estimates for the regression (4.18) seem to be significantly different from one another for
live cattle, live hogs, gasoline, and gold. The GARCH conditional variance model for
those commodities seems to be mis-specified. Thus, it can be implied that using
historical realized volatility series with the ARFIMA model seems to provide more
correctly specified conditional variances than the GARCH model. We have some mixed
evidence for R squares, since the R squares for (4.17) are higher than for (4.18) for com,
soybeans, and live cattle, while we found the opposite to be true of the other
commodities.

For a more fair comparison of the forecasting performance of the ARFIMA model
and the GARCH model, we regress the realized volatility of the ARFIMA forecast and

the GARCH estimated conditional variances jointly as follows:

R V: = 170 + b1R VARFIMA,r—1 + 1’20 GARCH,t—1 + 5}- (4J9)

 

'4 We also performed comparisons of the ARFIMA forecasts with the FIGARCH estimated conditional
variances. The results made no meaningful difference and, thus, are not reported separately in the current

paper.

104

Including both types of forecasts seem to improve forecasting performance quite
significantly relative to the individual regressions of (4.17) and (4.18), since the adjusted
R squares are higher than those for (4.17) and (4.18)”. All of the b. estimates, except for
those for corn, seem to be significantly different from one. The b2 coefficient estimates
for the GARCH forecast are also significantly different from one. However, the sum of
the b. and b2 estimates for the regression equation (4.19) seems to be close to one for the
commodities, with the exception of gold futures. This result implies that a linear
combination of the ARFIMA forecast and the GARCH forecast may jointly serve to
specify the correct forecasting model and may yield improved forecasting ability with
higher R squares. Our findings suggest that the ARFIMA forecasts can provide a correct
forecasting model when we forecast the firture realized volatility for commodity futures,

and their forecasting performance is not inferior to the GARCH model.

4.5. Economic Factors for the Commodity Futures Realized Volatility

An important possibility is that economic variables are relevant factors with
which to describe commodity futures return volatility. To consider various types of
economic factors under an integrated framework, we estimate a simple ARFIMA(0,d,0)
model for each realized volatility with announcement dummies, time-to—maturity
variables, and another commodity’s realized volatility. This is a joint estimation for all of
the considered coefficient estimates, including the long memory parameter. The

estimation model takes the following form:

 

'5 We report the adjusted R squares for (4-8) for apprOpriate interpretation, since R squares generally tend
to increase with the number of regressors.

105

61
(1-1.) (y, —’U_Zi=—1,26ili —y -TM, —)ex,)= 5,, (4.20)

where TM, is the time-to-maturity variable, 1,- indicates i-days after the relevant
announcements, 6, denotes the coefficient for 1,- , and x, is the realized volatility of a

counterpart commodity that is considered. The variable TM. is calculated as the ratio of
the number of remaining trading days (as of day t) before the futures contract’s expiration
to the total number of trading days within the "nearby" contract, so that the time-to-

maturity variable is scaled between zero and one.

4.5.1. Announcement Effects

Intuitively, it is reasonable that commodity-specific announcements may have a
meaningful relation with the relevant commodity volatility, while more general economic
announcements can indirectly affect commodity markets. Recent empirical studies,
including Andersen, Bollerslev, Diebold, and Vega (2003) and Andersen and Bollerslev
(1998a), document the effect of macroeconomic announcements and news on the five-
minute DM-US dollar return volatility. Cai, Cheung, and Wong (2001) studied how
various relevant economic announcements influence five-minute gold futures return
volatility. In this section, we focus on more specific announcements for the commodities
in order to investigate the announcement effect together with other economic factors. An
important extension from the previous studies is that we use the realized volatility

measure.

106

We use the monthly announcement for our analysis since, for example, quarterly
announcements are too sparse over the current sample period to extract sufficient
information for daily volatility, and also, weekly announcements are not available for the
commodities considered here. Since including more general announcement dummies
may reduce parameter estimation efficiency and the degrees of freedom without making a
difference for the results of the estimation, we therefore only include the most relevant
announcements for each commodity. We select the following announcements for each
commodity:

(l) the monthly crop production report for corn and soybeans;

(2) the monthly cattle report for live cattle;

(3) the utility capacity report for unleaded gasoline; and

(4) the production price index announcement for gold.

The hog announcements are quarterly rather than monthly-based for the sample periods
of our data set, so we do not consider the announcement effects for live hogs in the
current section. Since we are only considering other economic effects jointly with the
announcements, we do not consider live hogs any further in the rest of this chapter. To
analyze dynamic patterns of the announcement timing effect on the volatility, we classify
announcement timing effects further as (1) pre-announcement effects, (2)
contemporaneous effects, and (3) post-announcement effects. The pre-announcement
argument is to capture any possible news-leakage effect prior to the announcements. To
consider these three types of announcement timing effects, we assign dummy variables
for one day before, the day of, one day after, and two days after the relevant

announcements. The estimation results are presented in Tables 4-7 through 4-1 1, in

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panels (a) and (b). We mainly discuss empirical findings recorded in panel (a) in Tables
4-7 through 4-11, since those announcement results are qualitatively similar to those in
panel (b) in Tables 4-7 through 4-11. We practiced the likelihood ratio test to evaluate if
inclusion of the announcement dummies could yield meaningful different estimation
results. The hypothesis to be tested here is that all the coefficients for the announcement
dummies are equal to zero. For the corn futures realized volatility, the coefficient
estimates seem to be significant and negative for one day before the announcement.
According to these results, the realized volatility of corn seems to decrease one day
before the monthly crop reports. Particularly for the realized soybean volatility, the
likelihood ratio test statistics for the hypothesis 8.] = 80 = 6) = 52 = 0 are significant, and
therefore the hypothesis is rejected. Therefore, including the announcement dummies in
the ARFIMA estimation of the realized soybean volatility seems to make a significant
difference in terms of the maximized log likelihood values. For soybeans, the
contemporaneous announcement and one day after announcement dummy coefficient
estimates are significant and positive, while the estimates for the pre-announcement
dummy variable are significant but negative. Such a negative pre-announcement effect is
a commonality with the corn realized volatility mentioned above. For the live cattle
realized volatility, the hypothesis 6-. = 80 = 5. = 82 = 0 is for some instances rejected at a
10 percent significance level. Thus, including the announcement dummies can contribute
to some degree to an explanation of the realized volatility. However, our findings show
that none of the individual announcement coefficient estimates for live cattle are
significant. For the gasoline realized volatility, the individual coefficient estimates for

one day before and two days after announcements appear to be significant and negative,

108

while the hypothesis 5-. = 60 = 6. = 62 = 0 cannot be rejected by the likelihood ratio test.
We found marginally significant one day after announcement estimates for the gold
futures realized volatility but insignificant likelihood ratio test statistics for the hypothesis
5-. =80=8.=82=0.

We have mixed evidence for announcement effects for the realized volatilities in
the presence of the time-to-maturity effect and possible relationships between different
commodities’ realized volatilities. Especially for soybeans, we found that the monthly

crop production reports seem to significantly affect this crop’s realized volatility.

1

4.5.2. Time to Maturity and Information Flow

We spliced multiple futures contracts to construct a long series of futures return
series. Switching from an expiring futures contract to the next nearby maturity may
introduce jumps into the volatility process because of jumps in time to maturity at the
switch points. In the current section, we consider the effect on the realized volatility
series of varying times to maturity from different contracts. One well-known hypothesis
is that futures return volatility rises as contracts approach their maturity, as Samuelson
(1965) assessed. The intuition behind the Samuelson hypothesis is that there is little
information flow that resolves uncertainty about futures prices in the far distant future.
On the other hand, as we come closer to the maturity date. we become more sensitive to
information that influences the final level of the futures price. Some previous studies
support the Samuelson hypothesis, while other research failed to find evidence for the
hypothesis. Anderson and Danthine (1983) and Anderson (1985) argue that the

Samuelson hypothesis is generally not true unless we have information flows

109

incorporated into the model. Anderson and Danthine (1983) introduce information flows
into their theoretical model and demonstrate that the resolution of uncertainty is the
source of increased volatility in futures prices. Milonas (1986) and Galloway and Kolb
(1996) found some mixed evidence for the Samuelson hypothesis by using futures price
series of financial assets and commodities. Chen, Duan, and Hung (1999) tested and
considered the Samuelson effect to model optimal hedging under the GARCH framework
by using daily spot and futures stock index data. They found the Samuelson hypothesis
unsupported by their empirical findings.

According to the results in panel (a) in Tables 4-7 through 4-11, the time-to-
maturity coefficient estimates for corn, soybeans, and cattle realized volatilities are
significant and positive. It can be implied that, for example, the realized soybean
volatility is reduced as the contracts approach their expiration dates. Seemingly, our
findings may be inconsistent with the Samuelson hypothesis, since the numerical sign of
the estimates should be negative according to that hypothesis. However, we use only
nearby parts of the commodity futures contracts and thus do not observe the early time
periods of the contracts. Therefore, we should be careful in interpreting the resulting
significantly positive coefficient estimates for time-to-maturity variables. More relevant
discussion of this topic will follow when we consider information flow. Particularly,
lower firtures return volatility immediately prior to expiration may be due to the liquidity
effect. On the other hand, the time-to-maturity coefficient estimates are not significant
and are negative-signed for the realized volatilities associated with gasoline and gold

futures returns.

110

To consider an underlying factor in the relation between time to maturity and
realized volatility, we introduce information flow following Anderson & Danthine (1983)
and Anderson (1985), who argued the importance of information flow in explaining the
Samuelson hypothesis. In a different line of previous studies including Andersen (1998),
Tauchen and Pitt (1983), and Clark (1973), daily trading volumes were used as an
information flow measure on the grounds of the mixture-of-distribution hypothesis.
Andersen and Bollerslev (1997a) supported the long memory of high frequency US-DM
return volatility based on the mixture-of-distribution hypothesis. In our study, we use the
trading intensity within each day, which can be informative about how often trading
occurs due to new information arriving at the relevant market. To measure trading
intensity, we simply calculate the percentage of five-minute intervals associated with
actual transactions out of the total sub-period intervals for each day. Not all of the intra-
day intervals involve real-time trading since transactions occur unevenly. This is so-
called “non-synchronous trading” in market microstructure literature.

To diagnose any possible relations among realized volatility, trading intensity,
and time to maturity, we display correlations between those factors in Table 4-12. We
focus on corn, soybeans, and cattle to reflect on their features which are seemingly
inconsistent with the Samuelson hypothesis, as we mentioned above. We chose contracts
with relatively long lifetimes for those commodities. From Table 4-12, we observe a
negative correlation between time to maturity and realized volatility when we consider
the whole contract periods. All the signs for the correlation data are negative, excepting
only the November 2000 soybean contract. This feature is consistent with the Samuelson

hypothesis. In contrast, the correlations are positive for most of the commodities if we

111

use nearby contract,s as shown from the ARFIMA model estimation (4.20). An
apparently positive correlation: between time to maturity and realized volatility may be
caused by the use of nearby contracts rather than a real inconsistency with the Samuelson
hypothesis.

Another noteworthy thing is that the correlation between trading intensity
(information flow) and realized volatility is very strong and positive, if we consider the
whole contract periods. Motivated by this fact, we add the information flow variable into

the ARFIMA model (4.20) and specify another time series model as follows:
d .
(1 — L) ( y, — p -Zi=_1’26,-I,~—y-trmat, - ,Bx, -—¢9« flow, ) = 5,, (4.21)

where flow. is the number of five-minute intervals with actual transactions divided by the
number of total subperiods within a trading day. The coefficient estimates for the trading
intensity variable from the ARFIMA estimation for (4.21) are presented in panel (b) in
Tables 4-7 through 4-1 1, and they are statistically significant and positive for all of the

commodities. It is worth notice that the time-to-maturity coefficient ,6 estimates for

corn, soybeans, and cattle are no longer significant using the set-up that includes the
information flow variable, although those commodities showed significant time-to-
maturity estimates for the original model (4.20). On the other hand, the time-to-maturity
coefficient estimates for the gasoline and gold realized volatilities are insignificant for the
estimation model (4.20) without considering the information flow variable. However,
those estimates become significant and negative after including the information flow

variable in the ARFIMA model as in (4.21).

112

 

According to our findings, trading intensity as information flow proxy seems to
explain a sizeable portion of realized volatility, while time to maturity has become less
relevant to the volatility for those commodities. This result is consistent with the
theoretical claim by Anderson and Danthine (1983) that time to maturity could matter to
futures return volatility since information flow is linked to the volatility. According to
the theory, information flow is a driving factor channeling between time to maturity and
realized volatility, and therefore, once information flow is taken into account in
explaining the volatility measure, we may be able to observe some genuine time-to-
maturity effects on the realized volatility, if any.

Another theory relevant to our findings is the mixture-of-distribution hypothesis.
In particular, Clark (1973) theoretically asserted that daily returns are generated from
many intra-day returns within a day, and variance in the daily price change is
proportional to the number of daily transactions, although he used the daily trading
volume to embody the number of daily transactions. Our finding of a significantly
positive relation between the realized volatility and the trading intensity within a day can
be theoretically justified by the mixture—of-distribution hypothesis since trading intensity
reflects effective intra-day price changes, and those intra-day price changes underlie the
realized volatility.

In addition, the comparison between the realized volatility and the squared daily
return, one of the usual daily volatility measures, can be made in terms of time to
maturity and information flow issues. The last two columns in Table 4-12 show (i) a
correlation between trading intensity and squared daily returns, and (ii) a correlation

between the time-to-maturity variable and squared daily returns. Table 4-12 shows that

113

 

 

correlations between trade intensity and squared daily returns are very low and even

negative in some instances. This is markedly in contrast with the strong and high

correlations between trading intensity and the realized volatility. The average value of

the correlations between trading intensity and the realized volatility is 0.5902. From the

correlation results, we can imply that squared daily returns may not fully reflect the

relationship between information flow and daily futures volatility. On the other hand,

time to maturity and squared daily returns are negatively correlated to a less significant

degree than time to maturity and realized volatilities if we consider the whole contract ”A
periods. The data for the correlations between time to maturity and squared returns are i
even positive for five out of 13 instances, while only one correlation between time to

maturity and the realized volatility is positive. From the correlation check, we can imply

that the realized volatility measure is more consistent with the Samuelson hypothesis, as

well as the theoretical linkage between volatility and information flow, than the squared

daily return.

4.5.3. Interdependence Between the Realized Volatilities for Different Commodities
Another economic dimension in analyzing the realized volatility is possible
interdependence between different commodity volatilities. The volatility linkage
between different markets is one of the active issues in empirical finance, since it is
informative for portfolio management, derivative pricing, and risk management. Brunetti

and Gilbert (2000) have found two similar gasoline price volatility processes correlated

114

by using a bivariate FIGARCH model in a fractional cointegration context”. Fleming,
Kirby, and Ostdiek (1998), Kodres and Pritsker (2002), and Fleischer (1998) examine
volatility linkages by considering the relation between volatility and information flow.
Fleming, Kirby, and Ostdiek (1998) argue that common information can cause a strong
volatility linkage for stock, bond, and foreign currency markets. Andersen, Bollerslev,
Diebold, and Labys (2003) applied the VAR model to fractionally differenced realized
volatilities constructed from DM-US Dollar, Yen-Dollar, and DM-Yen exchange rate
returns for the purpose of forecast modeling, and they found that many of the VAR
coefficient estimates are significant. However, we use the fi'actional VAR estimation
approach to diagnose any lead and lag relations across different commodities. We
fractionally difference the realized volatilities by using the long memory parameter
estimates from the ARFIMA (0,d,0) model shown in Table 4-4 and apply a VAR model
to those fractionally differenced realized volatility series. The estimation results are
presented in Table 4-13. According to our findings, most of the estimated VAR
coefficients do not seem to be significant, therefore implying that, after we control the
long memory feature for the realized volatility process, there may not be significant lead
and lag interdependence between the commodities considered here. Based on this
finding, we allow ourselves to concentrate on contemporaneous relations and include a
counterpart commodity in the ARFIMA model for each realized commodity volatility.
Table 4-14 shows a correlation matrix for the realized volatilities for corn, soybeans, live
cattle, live hogs, gasoline, and gold. Higher correlations between corn and soybeans and

between live cattle and hogs seem to be sensible since they belong to the same category

 

'6 Since we can have an observable volatility measurement, it is possible to test fractional cointegration
relations directly by using several semi-parametric approaches suggested by Robinson and Marinucci
(1999). This is an area of possible future research, but is not pursued further here.

115

of products. As shown in Tables 4-7 through 4-11, we simultaneously estimate the long

memory parameter and the coefficient for linear relations between two realized volatility

 

series in the presence of the announcement dummy variables, time-to-maturity variables,
and information flow variables discussed above.

In this chapter, we directly employ realized volatility measures to consider any
possible volatility linkage across different or similar types of commodities, since the

realized volatility is an observable volatility proxy, unlike the stochastic volatility model

"7"“""'“I§ a.“

Fleming, Kirby, and Ostdiek (1998) used to estimate latent volatility factors. We find

mixed evidence for contemporaneous relations across different commodity futures

 

markets. Our results show that the realized volatilities of corn and soybeans exhibit
mutually significant interdependence. The long memory estimates for the realized
volatility of corn are lower when they are evaluated together with soybeans than they are
when evaluated with other commodities. Hence, it can be implied that the realized
volatilities of corn and soybeans share long memory time trends and that there is a
possible fractional cointegration relation between com and soybean realized volatilities”.
This finding does not seem to change with respect to different specification choices,
either in (4.20) or in (4.21). On the other hand, the long memory estimates for the cattle
realized volatility are slightly lower when considered in conjunction with the hog realized

volatility than when other commodities are considered as its counterparts.

 

'7 Recent literature on the fractional cointegration tests includes Marinucci and Robinson (2001) under a
semi-parametric framework and Dueker and Starz (1998) using a vector ARFIMA model.

116

4.6. Conclusion

The cumulative sum of squared intra-day returns can be a model-free and
consistent estimator of the true volatility factor on the grounds of quadratic variation
theory and the assumption of a continuous arbitrage-free price process under conditions
of regularity, as we discussed in this chapter. More importantly, the realized volatility
measure provides an observable volatility factor. In this chapter, we identified stochastic
properties of the realized volatility and used the volatility measure to consider economic
factors in analyzing daily futures return volatilities in major commodity futures markets.
The main statistical finding is that the commodity realized volatility process is normally
distributed and exhibits slowly decaying temporal dependences. This is consistent with
the long memory volatility findings from many previous studies of exchange and stock
markets, for example, those which produce FIGARCH estimation results using exchange
rates and stocks. Based on these stochastic properties of realized volatility, we used an
ARFIMA type model to analyze the presence of the announcement effect, the time-to-
maturity effect (accounting for information flow), and contemporaneous interdependence
between different commodity markets. Our findings are that there is significant
contemporaneous interdependence between the realized volatilities of corn and soybeans,
and that information flow is a key factor in commodity realized volatility, consistent with
the futures return volatility model suggested by Anderson and Danthine (1983) and
Clark’s (1973) mixture-of-distribution hypothesis. After all the factors have been
elaborated, the long memory dynamic patterns remain, and thus, slowly decaying
volatility seems to be intrinsic for the commodity futures markets considered in this

chapter.

117

 

With access to longer sample periods of high frequency commodity price data,

there would be more opportunities to study various aspects of realized volatility

 

dynamics. First, the out-of-sample forecasting performance of a simple ARFIMA model
could be evaluated using realized volatilities. Second, a data set encompassing a longer
time span may contain firrther nonlinearities, such as structural breaks; it would be
possible to test the realized volatility series being considered for these nonlinearities. If
structural change in realized volatility series were to be detected, then it would be
possible to adjust to account for the breaks and to reconsider the temporal dynamic

patterns of realized volatility. One last but important research possibility would be to

 

relate realized volatility to market micro-structure issues. For example, transaction
occurrences at tick time intervals and the corresponding durations may provide more
insights into the relationship between information flow and volatility. We leave these

issues for future studies.

118

Table 4-1: Basic Descriptive Statistics: Unconditional Distribution of Daily
Commodity Futures Returns

 

 

 

 

 

 

 

 

 

 

Corn Soybean Live Cattle Live Hggs Gasoline Gold

No. of Obs. 471 409 405 400 401 401
Mean -0.0414 -0.0179 0.0538 0.1205 0.1899 -0.0615
Median -0.0856 -0.0907 0.0649 0.1494 0.2843 -0.0878
Maximum 3 .2232 3.9491 1.4069 6.1608 6.3237 8.6236
Minimum -4.1582 -3.6473 -l.3549 -6.3229 -5.0272 -5.6716
Std. Dev. 0.9658 1.0958 0.5138 1.265 1.7899 0.9209
Skewness 0.1579 0.2837 0.1 132 -0. 1692 -0.2364 1.7325
Kurtosis 4.0318 3.9841 3 .0026 7.0436 3.2606 28.9425

 

Table 4-2: Basic Descriptive Statistics: Distribution of Realized Volatility

 

 

 

 

 

 

 

 

 

Corn Soybean Live Cattle Live Hog Gasoline Gold

No. of Obs. 471 409 405 400 401 401
Mean 0.0818 0.0337 -O.6834 0.1 138 0.5745 -0.4305
Median 0.0882 0.0155 -0.681 0.0829 0.5536 -0.5078
Maximum 1.1441 0.898 0.3411 1.5969 1.4124 1.7832
Minimum -1.0067 -0.8194 -1.5099 -2.526 -0.1352 -l.1943
Std. Dev. 0.2925 0.295 0.2895 0.4084 0.2574 0.4136
Skewness -0.0536 0.2203 0.0051 -0.3 881 0.2253 1.51 14
Kurtosis 3 .7882 2.9894 3.316 7.0984 3.1453 6.6992

 

 

Table 4-3: Basic Descriptive Statistics: Daily Returns Standardized by Realized

 

 

 

 

 

 

 

 

 

 

Volatility

Corn Soybean Live Cattle Live Hog Gasoline Gold

No. of Obs. 471 409 405 400 401 401
Mean -0.0786 -0.0451 0.1033 0.1128 0.1362 -0.1159
Median -0.0905 -0.1202 0.1311 0.1496 0.1789 -0.1550
Maximum 1.9920 2.5650 2.4143 2.9544 2.5809 3 .2496
Minimum -2.5425 -2.3841 -2.1058 -2.3745 -2.3058 -2.0908
Std. Dev. 0.7988 0.9280 0.9299 0.9128 0.9285 0.8119
Skewness 0.0353 0.1795 0.0708 -0.0467 -0.0754 0.3923
Kurtosis 2.8075 2.6010 2.4013 2.9226 2.5930 3.3921

 

 

 

 

 

Key: For the basic description of each realized volatility series, we use the whole
sample period of high frequency data. In contrast, we should use the realized

119

volatility series with common trading days for the joint estimation for the
ARFIMA(0,d,0) below. Thus, the numbers of sample observations for the ARFIMA
estimation in Tables 4-4 through 4-11 are different from the sample numbers in
Tables 4-1 through 4-3 above.

 

 

120

Table 4-4: ARFIMA(0,d,0) Estimation for Realized Volatility Series

(1-1m, - .0 = 8.
Corn Soybean Cattle Hog Gasoline Gold

 

Sample (5/03/99 (5/03/99 (5/03/99 (5/03/99 (5/03/99 (5/03/99
Period 42/28/00) 42/28/00) 42/28/00) -12/28/00) -12/28/00) -12/28/00)
1.1 0.0699 0.0049 -0.6548 0.1204 0.5609 .0.5395
(0.0545) (0.0554) (0.0540) (0.1046) (0.0402) (0.1456)
d 0.2460 0.2591 0.2702 0.3409 0.2325 0.3961
(0.0403) (0.0385) (0.0356) (0.0607) (0.0390) (0.0550)
0’2 0.0703 0.0725 0.0660 0.1093 0.0565 0.1 148
(0.0053) (0.0051) (0.0050) (0.0211) (0.0043) (0.0133)
ln(L) -35.582 41.472 .23.255 421.157 6.744 430.604
m3 0.075 0.094 0.025 4.205 0.289 1.131
m4 3.221 2.916 3.206 15.494 3.272 6.203
Q(20) 29.729 20.658 17.784 15.867 9.112 21.984
Q2(20) 31.441 29.764 19.998 27.349 12.924 28.223

 

Key: Robust standard errors based on QMLE are in parentheses below the corresponding
parameter estimates. The diagnostic statistics Q(20) and Q2(20) are the Ljung-Box
statistics based on the first 20 autocorrelations of the standardized residuals and the
autocorrelations of the squared standardized residuals respectively. The statistics 7713 and
m4 are the sample skewness and kurtosis respectively of the standardized residuals.

121

Table 4-5: The Local Whittle Estimation for the Long Memory Parameter

 

Local Whittle Estimates for

 

Realized Volatility
Corn 0.2644
Soybean 0.2419
Cattle 0.2805
Hog 0.3404
Gasoline 0.2374
Gold 0.3756

 

Key: We use the sample size powered to 0.8 for the bandwidth.

122

Table 4-6: Mincer-Zamowitz Regressions for Realized Volatilities
(One-day-ahead Forecast)

 

18

 

b0 b1 132 R square
Corn 0.0171 1.0013 0.1572
(0.0153) (0.1220)
-0.7726 0.8885 0.0745
(0.1566) (0.1655)

-0.2847 0.8758 0.3221 0.1599
(0.1494) (0.1452) (0.1640)

Soybeans 0.00146 1.03492 0.1625
(0.0125) (0.1314)
-0.8189 0.7875 0.1600
(0.1194) (0.1 104)

-0.4804 0.6219 0.4569 0.1833
(0.1260) (0.1427) (0.1196)

Live Cattle -0.20762 0.88849 0.2501
(0.0392) (0.0742)
-2.5065 3.6166 0.2141
(0.2120) (0.4191)

.0.9971 0.6473 1.31 0.2536
(0.3587) (0.1285) (0.5994)

Live Hogs 0.0021 1.0413 0.3494
(0.0175) (0.0987)
-0.8197 0.7660 0.3532
(0.0847) (0.0740)

-0.4676 0.5511 0.4286 0.3678
(0.1313) (0.1612) (0.1244)

Gasoline 0.2239 0.8345 0.1409
(0.0481) (0.1 107)
-0.4683 0.5857 0.1485
(0.1253) (0.0686)

-0.415 0.5758 0.4199 0.1898
(0.1067) (0.1086) (0.0631)

Gold -0.0561 0.9758 0.3255
(0.0498) (0.1019)
-0.7893 0.3845 0.3373
(0.0428) (0.0495)

-0.4508 0.5005 0.2284 0.3665
(0.0998) (0.1488) (0.0445)

Key: The table reports OLS parameter estimates for Mincer-Zamowitz regressions of
realized volatility on a constant and forecasts from different models. The OLS regression

 

'8 We used the adjusted R squares for the regression including both the ARFIMA forecast and the GARC H
conditional variance for more accurate interpretation since R squares generally tends to increase with the
number of regressors.

123

is RV. = be + b. RVARFIMA. + b2 C VGA RC”. + u.. The robust standard errors are reported
in the parenthesis. RV. is 0.5*ln(E,-=. J. r,,,-) where A is the number of intraday returns
within each trading day. RVARFIMA,t is the forecasted value of RV. using ARFIMA(0,d,0)
model and CVGARCH. is the GARCH estimated conditional variances. To evaluate the
ARFIMA forecast alone, we restrict b2 =0. To evaluate the GARCH forecast alone, we
restrict b. =0.

124

 

 

Corn

Table 4-7 (a): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity Markets:

 

Dependent variable: Corn realized volatility

 

Soybeans Cattle Hogs Gasoline Gold
11 -0.0286 -0.0333 -0.0592 -0.0516 -0.01 31
(0.0450) (0.0690) (0.0620) (0.0680) (0.0607)
(1 0.1658 0.2170 0.2227 0.2163 0.2220
(0.0369) (0.0387) (0.0394) (0.0390) (0.0390)
02 0.0568 0.0662 0.0656 0.0662 0.0658
(0.0043) (0.0049) (0.0048) (0.0049) (0.0049)
6-. -0.0699 -0.1 177 -0.1 166 -0.1 182 -0.1094
(0.0431) (0.0497) (0.0489) (0.0497) . (0.0499)
60 -0.03 80 0.0197 0.0300 0.0208 0.0191
(0.0503) (0.0523) (0.0518) (0.0524) (0.0521)
6. -0.0291 0.0142 0.0186 0.0142 0.0204
(0.0649) (0.0631) (0.0599) (0.0639) (0.0632)
62 0.1046 0.0850 0.0892 0.0863 0.0889
(0.0561) (0.0627) (0.0633) (0.0625) (0.0622)
y 0.2104 0.2305 0.2421 0.2310 0.2231
(0.0549) (0.0664) (0.0668) (0.0664) (0.0656)
[3 0.3720 0.0138 0.0757 0.0161 0.0598
(0.0465) (0.0477) (0.0431) (0.0575) (0.0343)
ln(L) 5.731 -23.866 -22.037 -23.860 -22.651
m3 -0.13 5 0.075 0.082 0.078 0.084
m4 3.170 3.137 3.112 3.153 3.139
Q(20) 30.454 29.537 26.063 29.358 29.183
Q2(20) 26.895 27.232 30.1 19 26.618 26.398
LR Test 6.57 7.104 7.602 7.242 6.762

 

Key: As for Table 4-4. The estimation is based on

 

(1 — L)d (y, - p — 21:4 2 6.1,- —7 TM, — fix, ) = a, where TM. is time-to-maturity

variable, and 1,- indicates for i-days after the relevant announcements, and
6,- denotes for the coefficient for 1,. x, is the realized volatility for a counterpart

commodity considered. The variable TM. is calculated as the ratio of the number
of remaining trading days as of day t before the futures contract expiration to the

125

 

total number of trading days within the "nearby" contract so that the time-to-
maturity variable is scaled between zero and one.

126

Table 4-7 (h): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence between

Commodity Markets: Corn

 

Dependent variable: Corn realized volatility

 

Soybeans Cattle Hogs Gasoline Gold
11 -0.6857 -0.7836 -0.8174 -0.83 80 -0.7799
(0.1502) (0.1541) (0.1572) (0.1639) (0.1578)

d 0.1796 0.2296 0.2331 0.2292 0.2385
(0.0444) (0.0418) (0.0420) (0.0423) (0.0434)

02 0.0510 0.0586 0.0582 0.0586 0.0585
(0.0036) (0.0042) (0.0042) (0.0042) (0.0042)

6-. -0.0377 -0.0797 -0.0793 -0.0812 -0.0745
(0.0437) (0.0497) (0.0493) (0.0494) (0.0498)

60 -0.0049 0.0452 0.0588 0.0496 0.0490
(0.0502) (0.0518) (0.0516) (0.0521) (0.0522)

6. 0.0085 0.0514 0.0549 0.0514 0.0551
(0.0514) (0.0509) (0.0495) (0.0519) (0.0519

62 0.1073 0.0861 0.0918 0.0906 0.091 1
(0.0540) (0.0596) (0.0599) (0.0593) (0.0589)

y -0.0545 -0.0937 -0.0784 -0.0908 -0.0956
(0.0672) (0.0773) (0.0768) (0.0772) (0.0777)

[3 0.3416 0.0496 0.0709 0.0449 0.0425
(0.0444) (0.0453) (0.0374) (0.0525) (0.0339)

0 0.8646 1.0322 1.0143 1.0266 1.0135
(0.1733) (0.1791) (0.1770) (0.1799) (0.1808)

ln(L) 26.884 -0. l 34 1.187 -0.287 0.034

m3 0.240 0.387 0.397 0.407 0.400

m4 2.985 2.988 3.036 3.010 3.005
Q(20) 26.735 28.544 23.647 28.310 27.842
Q2(20) 15.995 28.977 31.075 28.753 29.019

 

Key: As for Table 4-4. The estimation is based
on(l — L)d(y,-p—Zi=_126,-1,--y-TM,—,6x,—0-flow,)= e, where TM. is

time-to-maturity variable, and I ,- indicates for i-days after the relevant
announcements, and 5,- denotes for the coefficient for 1,. x, is the realized

127

volatility for a counterpart commodity considered. The variable TM. is calculated
as the ratio of the number of remaining trading days as of day t before the futures
contract expiration to the total number of trading days within the "nearby"
contract so that the time-to-maturity variable is scaled between zero and one.
flow. is the number of five-minute intervals with actual transactions divided by the
number of total subperiods within a trading day.

128

Table.4-8 (a): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity Markets:

Soybean

 

Dependent variable: Soybean realized volatility

Corn

11 -0.0808
(0.0526)

(1 0.2188
(0.0374)

0'2 0.0590
(0.0044)

6-. -0.071 1
(0.0585)

60 0.1656
(0.0471)

6. 0.1248
(0.0536)

62 -0.0832
(0.0458)

7 0.1 127
(0.0535)

0 0.3684
(0.0504)

ln(L) -1.327

m3 0.246

m4 3.127
Q(20) 24.099
Q2(20) 19.150

00’

LR Test 20.124

Cattle

-0.0281
(0.0767)

0.2525
(0.0377)

0.0677
(0.0051)

-0.1 193
(0.0655)

0.1610
(0.0491)

0.1231
(0.0495)

-0.0486
(0.0527)

0.1436
(0.0599)

0.0701
(0.051 1)

-28.238
0.186
3.160

28.412

25.281

0..

19.230

Hogs
-0.0772
(0.0645)

0.2528
(0.0385)

0.0679
(0.0051)

-0.1 1 89
(0.0644)

0.1730
(0.0499)

0.1247
(0.0496)

-0.0433
(0.0526)

0.1413
(0.0599)

0.0312
(0.0429)

-28.841
0.201
3.161

28.774

27.717

‘0‘

20.212

Gasoline

-0.141 1
(0.0747)

0.2562
(0.0364)

0.0672
(0.0051)

-0.1247
(0.0653)

0.1640
(0.0479)

0.1221
(0.0489)

-0.0401
(0.0508)

0.1380
(0.0588)

0.1242
(0.0561)

-26.651
0.245
3.205

31.350

25.461

OD.

19.856

Gold

.0042
(0.0684)

0.2547
(0.0371)

0.0676
(0.0051)

-0.11 l 1
(0.0656)

0.1672
(0.0486)

0.1293
(0.0506)

-0.0428
(0.0510)

0.1317
(0.0588)

0.0582
(0.0369)

-28.015
0.193
3.224

28.741

26.377

0..

19.242

Key: As for table 4-7 (a). (m) represents significant the Likelihood Ratio test
statistic at one percent level.

129

Table 4-8 (b): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence between .
Commodity Markets: Soybean

 

Dependent variable: Soybean realized volatility

 

Corn Cattle Hogs Gasoline Gold
.1 -0.4625 -0.4930 -0.5338 -0.5603 -0.4870
(0.1756) (0.1831) (0.1817) (0.1800) (0.1841)

(1 0.261 1 0.2902 0.2901 0.2922 0.2927
(0.0488) (0.0435) (0.0442) (0.0426) (0.0431)

02 0.0574 0.0653 0.0658 0.0653 0.0657
(0.0044) (0.0050) (0.0051) (0.0051) (0.0051)

6-. -0.0494 -0.0907 -0.0915 -0.0978 -0.0874
(0.0560) (0.0622) (0.0614) (0.0628) (0.0628)

60 0.1795 0.1749 0.1894 0.1799 0.1833
(0.0468) (0.0489) (0.0498) (0.0485) (0.0492)

6. 0.1391 0.1403 0.1414 0.1378 0.1431
(0.0568) (0.0542) (0.0544) (0.0536) (0.0549)

62 -0.0662 -0.0309 -0.0250 -0.0238 -0.0263
(0.0436) (0.0488) (0.0487) (0.0473) (0.0475)

y -0.0522 -0.0644 -0.0585 -0.0505 -0.0576
(0.0919) (0.0943) (0.0941) (0.0929) (0.0940)

[3 0.3554 0.0932 0.0363 0.1060 0.0390
(0.0492) (0.051 1) (0.0414) (0.0550) (0.0363)

9 0.5073 0.6369 0.6065 0.5716 0.5795
(0.2224) (0.2202) (0.2206) (0.2191) (0.2218)

ln(L) 3.837 -21.309 -22.523 -21.076 -22.438

m3 0.266 0.220 0.238 0.263 0.227

m4 3.303 3.280 3.312 3.359 3.370
Q(20) 24.637 28.745 28.692 30.750 28.825
Q2(20) 21 .446 26.225 29.634 26.361 28.436

 

Key: As for table 4-7 (b)

130

Table 4-9 (a): ARFIMA(0,d,0) Estimation for the Announcement effect,
Time-to-maturity effect, and Contemporaneous Dependence between Commodity
Markets:

Cattle

 

Dependent variable: Cattle realized volatility

Corn Soybeans Hogs Gasoline Gold

0 -0.8165 -0.8182 -0.8207 -0.8399 -0.7899

(0.0595) (0.0588) (0.0573) (0.0644) (0.0634)

d 0.2537 0.2521 0.2452 0.2555 0.2561

(0.0324) (0.0327) (0.0343) (0.0326) (0.0328)

62 0.0597 0.0595 0.0593 0.0596 0.0592

(0.0042) (0.0042) (0.0042) (0.0042) (0.0041)

6-. 0.1139 0.1169 0.1074 0.1104 0.1197

(0.0755) (0.0748) (0.0752) (0.0752) (0.0750)

60 -00394 -0.0310 -0.0408 -0.0374 .0.0379

(0.0597) (0.0591) (0.0599) (0.0607) (0.0611)

6. -0.0008 0.0068 -0.0058 0.0026 0.0154

(0.0530) (0.0550) (0.0544) (0.0540) (0.0545)

62 -0.1170 -0.1075 -0.1165 -0.1 109 -0.1 126

(0.0607) (0.0623) (0.0602) (0.0610) (0.0600)

y 0.3241 0.3233 0.3156 0.3198 0.3315

(0.0626) (0.0627) (0.0615) (0.0633) (0.0625)

13 -00070 0.0550 0.0644 0.0448 0.0683

(0.0440) (0.0446) (0.0369) (0.0572) (0.0390)

ln(L) -3.856 -3173 -2.406 -3.510 -2099
m3 0.018 0.035 -0003 0.015 -0013

m4 2.940 2.926 2.924 2.918 2.880

Q(20) 27.495 26.498 24.246 28.064 27.914
Q2(20) 17.899 16.619 18.749 18.605 15.898
LR Test 8064‘ 7.620 7.644 8.438‘ 8.538‘

 

Key: As for table 4-7 (a)

131

Table 4-9 (h): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence between
Commodity Markets: Cattle

 

Dependent variable: Cattle realized volatility

 

Corn Soybeans Hogs Gasoline Gold

.1 -2.2860 -2.2778 -23152 -2.2885 -2.2474

(0.1473) (0.1464) (0.1474) (0.1512) (0.1445)

d 0.2248 0.2221 0.1924 0.2265 0.2299

(0.0331) (0.0332) (0.0389) (0.0337) (0.0339)

0'2 0.0480 0.0479 0.0472 0.0480 0.0476

(0.0035) (0.0035) (0.0034) (0.0035) (0.0034)

6-. 0.1039 0.1099 0.0979 0.1042 0.1 122

(0.0612) (0.0611) (0.0621) (0.0614) (0.0608)

60 -0.0614 0.0553 -0.0662 -0.0624 -0.0623

(0.0511) (0.0500) (0.0488) (0.0511) (0.0519)

6. 0.0701 0.0781 0.0651 0.0722 0.0852

(0.0553) (0.0556) (0.0564) (0.0556) (0.0558)

62 -0.0975 -0.0893 .0.0973 -0.0950 .0.0949

(0.0564) (0.0580) (0.0556) (0.0568) (0.0557)

y 0.0293 0.0317 0.0128 0.0301 0.0404

(0.0591) (0.0592) (0.0578) (0.0597) (0.0590)

[3 0.0283 0.0566 0.0954 0.0295 0.0634

(0.0393) (0.0405) (0.0412) (0.0517) (0.0367)

0 1.7114 1.7015 1.7387 1.6983 1.6970

(0.1629) (0.1625) (0.1648) (0.1628) (0.1598)

ln(L) 38.403 39.097 41.906 38.370 40.093
m3 0.088 0.095 0.040 0.078 0.028
m4 3.044 3.025 3.033 3.019 3.006
Q(20) 33.359 32.712 29.739 34.787 32.848
Q2(20) 25.928 25.355 24.515 26.001 27.595

 

Key: As for table 4-7 (b)

132

 

 

Table 4-10 (a): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity Markets:
Gasoline

 

Dependent variable: Gasoline realized volatility

Corn Soybeans Cattle Hogs Gold
11 0.5866 0.5838 0.6478 0.5826 0.6108
(0.0507) (0.0504) (0.0607) (0.0519) (0.0536)
d 0.2373 0.2384 0.2468 0.2435 0.2425
(0.0418) (0.0412) (0.0399) (0.0412) (0.0415)
(32 0.0555 0.0549 0.0550 0.0553 0.0552
(0.0042) (0.0041) (0.0041) (0.0042) (0.0041)
6-. -0. 1097 -0.0961 -0.1 170 -0.1 137 -0.1056
(0.0489) (0.0477) (0.0470) (0.0500) (0.0493)
60 -0.0252 -0.0175 -0.0186 -0.0324 -0.0221
(0.0468) (0.0471 ) (0.0477) (0.0472) (0.045 8)
6. -0.0015 0.0169 0.0024 -0.0050 -0.001
(0.0539) (0.0554) (0.0559) (0.0532) (0.0527)
62 -0.1133 -0. 1048 -0.1105 -0.1141 -0. 1093
(0.0618) (0.0594) (0.0632) (0.0609) (0.0616)
y -0.0308 -0.0309 -0.0428 -0.0315 -0.0318
(0.0514) (0.0517) (0.0506) (0.0513) (0.051 1)
[3 0.0007 0.0920 0.0862 0.0385 0.0532
(0.0471 ) (0.0452) (0.0510) (0.0356) (0.0355)
ln(L) 10.348 12.438 12.000 10.907 11.531
m3 0.287 0.303 0.297 0.309 0.285
m4 3.213 3.189 3.126 3.240 3.173
Q(20) 9.099 9.671 9.034 8.865 9.719
Q2(20) 10.941 12.602 13.329 1 1.193 11.080
LR test
statistic 6.866 6.1 12 7.362 7.108 7.246

 

Key: As for table 4-7 (a)

133

Table 4-10 (h): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence between
Commodity Markets: Gasoline

 

Dependent variable: Gasoline realized volatility

Corn Soybeans Cattle Hogs Gold

.1 4.0029 0.9843 0.9394 0.9969 0.9728

(0.2086) (0.2082) (0.2106) (0.2105) (0.2098)

d 0.2024 0.2037 0.2132 0.2075 0.2071

(0.0401) (0.0397) (0.0399) (0.0425) (0.0399)

0'2 0.0471 0.0467 0.0467 0.0470 0.0469

(0.0038) (0.0037) (0.0037) (0.0038) (0.0038)

6-. 0.1489 0.1389 0.1557 0.1513 -0.1461

(0.0454) (0.0444) (0.0438) (0.0457) (0.0457)

60 0.0439 0.0383 0.0379 0.0480 0.0417

(0.0470) (0.0477) (0.0475) (0.0473) (0.0462)

6. 0.0422 0.0283 -0.0387 0.0443 0.0418

(0.0454) (0.0471) (0.0473) (0.0451) (0.0446)

62 0.1250 0.1187 0.1224 0.1255 0.1219

(0.0502) (0.0493) (0.0517) (0.0500) (0.0507)

y 0.1864 0.1841 0.1969 0.1861 -0.1861

(0.0469) (0.0475) (0.0464) (0.0469) (0.0469)

B 0.0079 0.0697 0.0784 0.0212 0.0393

(0.0423) (0.0424) (0.0484) (0.0398) (0.0326)

0 1.8066 1.7820 1.7981 1.7969 1.7920

(0.2280) (0.2281) (0.2257) (0.2308) (0.2286)

ln(L) 42.291 43.687 43.878 42.471 43.049
m3 0.406 0.425 0.402 0.427 0.392
m4 3.510 3.468 3.430 3.523 3.493
Q(20) 7.158 7.873 7.378 7.391 7.658
Q2(20) 13.014 13.033 15.750 12.680 14.289

 

Key: As for table 4-7 (b)

134

Gold

Table 4-11 (a): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, and Contemporaneous Dependence between Commodity Markets:

 

Dependent variable: Gold realized volatility

 

54

50

51

ln(L)
m3
m4
Q(20)
02(20)

LR test
statistic

Corn

0.5267
(0.1471)

0.3997
(0.0583)

0.1126
(0.0130)

0.0167
(0.0713)

0.0079
(0.0700)

0.1239
(0.0651)

-0.0363
(0.0622)

-0.0173
(0.1 163)

0.1340
(0.0571)

-l26.814
1.123
6.177

20.654
28.617

2.774

Soybeans

0.5075
(0.1461)

0.3967
(0.0583)

0.1129
(0.0131)

-0.0329
(0.0724)

0.0000
(0.0672)

0.1274
(0.0646)

0.0284
(0.0615)

-0.0365
(0.1 159)

0.1130
(0.0592)

-l27.382
1.138
6.213

21.082
28.985

3.172

Key: As for table 4-7 (a)

Cattle

0.4562
(0.1499)

0.4006
(0.0577)

0.1131
(0.0132)

-0.0331
(0.0715)

-0.0018
(0.0660)

0.1256
(0.0651)

0.0400
(0.0631)

0.0267
(0.1 162

0.1012
(0.0661)

-127.814
1.168
6.269

21.109
28.738

3.036

135

Hogs
-0.5239
(0.151 1)

0.4029
(0.063 8)

0.1137
(0.0129)

0.0325
(0.0724)

0.0018
(0.0674)

0.1248
(0.0659)

-0.0376
(0.0625)

0.0292
(0.1 160)

0.0382
(0.1030)

-128.734
1.109
6.006

19.187
28.937

3.052

Gasoline

0.5954
(0.1548)

0.4006
(0.0581)

0.1128
(0.0133)

-0.0427
(0.0728)

-0.0044
(0.0686)

0.1229
(0.0649)

0.0309
(0.0613)

0.0182
(0.1130)

0.1322
(0.0686)

-127.269
1.131
6.367

21.165
26.186

2.980

Table 4-11 (h): ARFIMA(0,d,0) Estimation for the Announcement effect, Time-to-
maturity effect, Information Flow, and Contemporaneous Dependence between
Commodity Markets: Gold

 

Dependent variable: Gold realized volatility

 

Corn Soybeans Cattle Hogs Gasoline

.1 -2.2811 -2.2563 -2.2270 -2.3143 -2.3558

(0.4323) (0.4396) (0.4409) (0.4484) (0.4259)

d 0.4030 0.4000 0.4035 0.4092 0.4021

(0.0641) (0.0637) (0.0631) (0.0714) (0.0633)

0'2 0.1004 0.1010 0.1009 0.1011 0.1005

(0.0119) (0.0121) (0.0121) (0.0117) (0.0122)

6-. 0.0486 0.0631 0.0632 0.0622 0.0723
(0.0709) (0.0714) (0.0707) (0.0718) (0.0719)

60 0.0216 0.0147 0.0162 0.0116 0.0188

(0.0695) (0.0692) (0.0668) (0.0711) (0.0689)

6. 0.1194 0.1225 0.1208 0.1187 0.1183

(0.0603) (0.0608) (0.0608) (0.0622) (0.0605)

62 0.0350 0.0300 0.0383 0.0357 0.0299

(0.0603) (0.0596) (0.0605) (0.0601) (0.0595)

y 0.2022 0.2156 0.21 14 0.2174 0.2021

(0.1255) (0.1249) (0.1253) (0.1279) (0.1242)

13 0.1 185 0.0776 0.0893 0.0557 0.1237

(0.0552) (0.0563) (0.0640) (0.1002) (0.0658)

0 1.9566 1.9455 1.9659 1.9921 1.9681

(0.4324) (0.4400) (0.4437) (0.4473) (0.4299)

ln(L) 404.679 405.746 ' 405.559 405.964 404.892
m3 1.299 1.320 1.344 1.276 1.308
m4 6.473 6.557 6.596 6.178 6.685
Q(20) 26.239 26.326 26.951 24.996 27.286
02(20) 28.877 28.831 28.529 32.131 26.776

 

Key: As for table 4-7 (b)

136

Table 4-12: Correlation among the realized. volatility, squared daily return, trading
intensity, and time-to-maturity

 

 

Corr.
Corr. Corr. Corr. Corr. Corr. Between
Between Between Between Between Between Squared
Item Contract Realized Realized Realized Time-to- Squared Returns
Volatility Volatility Volatility maturity Returns and
and and and and and Time-to-
Time-to- Time-to- Trading Trading Trading maturity
maturity maturity Intensity Intensity Intensity (Whole)
(Whole) (Nearby) (Whole) (Whole) (Whole)
Corn 2000.05 0313 0.220 0.501 -0.833 0.053 0.048
Corn 2000.07 -0.521 0.348 0.608 -0.684 0.1 12 -0.105
Corn 2000.09 -0.650 0.403 0.857 -0.794 0.240 -0.132
Soybean 2000.05 ' -0.067 0.202 0.221 -0.861 -0.030 0.109
Soybean 2000.07 -0.015 0.062 0.178 -0.630 0.008 0.035
Soybean 2000.08 -0.399 0.092 0.552 -0.856 -0.01 1 0.027
Soybean 2000.09 0477 -0.380 0.668 -0.852 0.079 -0.043
Soybean 2000.1 1 0.049 0.339 0.561 -0.128 0.056 0.038
Cattle 2000.04 0522 0.292 0.737 -0.856 0.179 -0.092
Cattle 2000.06 0400 0.635 0.665 -0.814 0.056 0.072
Cattle 2000.08 0429 0.426 0.662 -0.732 0.014 -0.038
Cattle 2000.10 -0.510 0.289 0.637 -0.836 0.169 -0.160
Cattle 2000-12 0755 0.056 0.820 -0.902 0.324 -0.321

 

137

Table 4-13: VAR Parameter Estimates (reggsion form)
X Corn Soybean Cattle Hogs Gasoline Gold

Const. 0.0099 -0.0129 -0.1596 0.0252 0.1597 0.0121
(0.0337) (0.0346) (0.0326) (0.0421) (0.0306) (0.0417)

Lagl
Corn -0.0710 0.0233 0.0181 -0.1596 -0.0531 -0.0058
(0.0580) (0.0595) (0.0562) (0.0725) (0.0526) (0.0718)

Soybean 0.0487 -0.0544 0.027 -0.019 0.0671 0.1000
(0.0567) (0.0582) (0.0550) (0.0709) (0.0515) (0.0702)

Cattle -0.021 -0.0471 -0.0658 -0.0487 0.0576 0.0784
(0.0550) (0.0565) (0.0534) (0.0689) (0.0500) (0.0681)

Hogs 0.0224 0.0509 0.022 0.0397 0.0351 0.1881
(0.0427) (0.0439) (0.0415) (0.0535) (0.0388) (0.0529)

Gasoline 0.0879 -0.0458 0.0247 0.0360 -0.0281 -0.057
(0.0598) (0.0614) (0.0580) (0.0748) (0.0543) (0.0740)

Gold -0.059 0.0338 -0.0056 -0.0352 0.0131 -0.0752
(0.0430) (0.0442) (0.0417) (0.0538) (0.0391) (0.0533)

Lag2
Corn 0.1144 0.0548 0.0532 -0.0201 0.0504 0.0964
(0.0580) (0.0596) (0.0563) (0.0726) (0.0527) (0.0718)

Soybean -0.0119 -0.1047 -0.0046 0.0256 0.022 -0.0917
(0.0568) (0.0583) (0.0551) (0.0711) (0.0516) (0.0703)

Cattle 0.003 0.0955 0.0036 0.0366 0.0336 0.0783
(0.0548) (0.0563) (0.0531) (0.0686) (0.0497) (0.0678)

Hogs 0.0097 0.0749 0.015 0.0208 0.0314 0.0591
(0.0429) (0.0440) (0.0416) (0.0536) (0.0389) (0.0531)

Gasoline -0.0969 0.0664 -0.0231 0.0561 0.0049 -0.1081

138

 

(0.0594) (0.0610) (0.0576) (0.0743) (0.0539) (0.0735)

Gold 0.0089 0.0144 0.062 0.0598 0.0428 0.0031
(0.0429) (0.0441) (0.0416) (0.0537) (0.0390) (0.0531)

Lag3
Corn -0.0337 0.0402 -0.0132 0.0978 -0.0335 -0.0194
(0.0582) (0.0597) (0.0564) (0.0728) (0.0528) (0.0720)

Soybean 0.0884 0.0573 0.0434 0.02 0.0291 0.0752
(0.0568) (0.0583) (0.0551) (0.0710) (0.0515) (0.0703)

Cattle 0.0968 0.0124 0.0347 0.0669 0.0076 0.0346
(0.0551) (0.0566) (0.0534) (0.0689) (0.0500) (0.0682)

Hogs 0.0289 0.0554 0.0412 0.0725 0.0199 0.0729
(0.0431) (0.0442) (0.0418) (0.0539) (0.0391) (0.0533)

Gasoline 0.0285 0.0098 0.0193 0.0977 0.0485 0.0571
(0.0598) (0.0614) (0.0580) (0.0748) (0.0543) (0.0740)

Gold -0.011 0.0505 -0.0027 0.0036 -0.0287 0.1055
(0.0426) (0.0437) (0.0413) (0.0533) (0.0386) (0.0527)

Lag4
Corn 0.0326 0.0601 -0.0082 -0.036 0.1221 0.0235
(0.0573) (0.0589) (0.0556) (0.0718) (0.0521) (0.0710)

Soybean 0.0113 0.0174 0.106 0.0171 0.045 0.1459
(0.0563) (0.0578) (0.0546) (0.0705) (0.0511) (0.0697)

Cattle -0.0283 0.0001 0.0256 0.1546 -0.0285 -0.0275
(0.0555) (0.0570) (0.0539) (0.0695) (0.0504) (0.0688)

Hogs 0.0216 0.0911 0.0142 0.0529 0.011 0.0583
(0.0433) (0.0444) (0.0420) (0.0541) (0.0393) (0.0536)

Gasoline 0.0906 0.0104 0.0552 -0.029 0.0497 -0.0481
(0.0597) (0.0613) (0.0579) (0.0747) (0.0542) (0.0739)

139

 

Gold 0.0215 0.0145 -0.0651 0.0427 0.018 0.005
(0.0422) (0.0434) (0.0410) (0.0529) (0.0384) (0.0523)

Lag5
Corn -0.017 -0.0045 0.017 01346 -0.0034 -0.0511
(0.0571) (0.0586) (0.0554) (0.0714) (0.0518) (0.0707)

Soybean 0.0195 0.0327 0.0682 0.0587 0.0264 0.0355
(0.0568) (0.0583) (0.0551) (0.0711) (0.0516) (0.0703)

Cattle 0.1075 0.0706 -0.0084 -0.0303 0.0077 -0.0035
(0.0554) (0.0569) (0.0537) (0.0693) (0.0503) (0.0686)

Hogs 0.0252 0.0535 0.0552 0.0421 -0.0083 0.0251
(0.0431) (0.0442) (0.0418) (0.0539) (0.0391) (0.0533)

Gasoline 0.0539 0.0167 0.0976 0.0662 0.0305 0.0963
(0.0594) (0.0610) (0.0577) (0.0744) (0.0540) (0.0736)

Gold 0.0279 -0.0458 0.0172 0.028 0.0301. 0.1167

(0.0411) (0.0422) (0.0399) (0.0515) (0.0373) (0.0509)
Key: The standard errors are in parentheses.

140

Table 4-14: Correlation matrix for six realized volatility series

 

Corn Soybean Cattle Hog Gasoline Gold
Corn 1.00000 0.43224 -0.00395 0.02822 0.04000 0.07547
Soybean 0.43224 1.00000 0.12513 0.12368 0.08430 0.10974
Cattle -0.00395 0.12513 1.00000 0.28375 -0.04924 0.03887
Hogs 0.02822 0.12368 0.28375 1.00000 -0.10063 -0.02247
Gasoline 0.04000 0.08430 -0.04924 -0.10063 1 .00000 0.042 74
Gold 0.07547 0.10974 0.03 887 -0.02247 0.04274 1.00000

141

Figure 4-1(a). Kernel Density for Realized Volatility

Kernel Density (Epanechnikov, h = 0.1529)
1 .6

 

1.2-

0.8J

0.4-

 

 

 

0'0 I I I I 7
-1 .0 0.5 0.0 0.5 1.0

Realized Volatility: Corn Futures

Kernel Density (Epanechnikov, h = 0.1744)
1 .6

 

1.2-

0.8-

0.4-

 

0.0

 

 

I I 1 V

-0.5 0.0 0.5 1.0

Realized Volatility: Soybean

142

Kernel Density (Epanechnikov, h = 0.1735)
1.4

 

1.2-

1.0-

0.8—

0.6-

0.4-

0.24

 

 

0.0 . . .

 

Realized Volatility: Live Cattle

Kernel Density (Epanechnikov, h = 0.2170)
1.2 '

 

1.0-

0.8-

0.6-

0.4J

0.2-

 

 

0.0 A . . l A

 

_.

Realized Volatility: Live Hog

143

2.0

Kernel Density (Epanechnikov, h = 0.1533)

 

1.5-

1.0-

- 0.5—

 

0.0

 

I I I I

0.0 0.5 1.0 1.5

Realized Volatility: Unleaded Gasoline

Kernel Density (Epanechnikov, h = 0.1976)

 

1.4

1.2-

1.0-

0.8—

0.6-

0.4-

0.24

 

0.0

 

Realized Volatility: Gold

144

 

 

Figure 4-1 (b). Kernel Density for Daily Returns Standardized by Realized Volatility

Kernel Density (Epanechnikov, h = 0.4643)

 

0.6

0.54

0.4 _

0.3..

0.2 —

0.1—

 

0.0

 

 

-3

0.5

I I

-2 -1

0-1
.8
N4

Standardized Daily Corn Futures Return
by the Realized Volatility

Kernel Density (Epanechnikov, h = 0.5548)

 

0.4 3

0.3-

0.2 —

0.1-

 

0.0

 

 

Standardized Daily Soybean Futures Return
by the Realized Volatility

145

Kernel Density (Epanechnikov, h = 0.5570)

 

 

 

 

 

 

0.4
0.3 -
0.2 -
0.1 -
0'0 I I T 1 F
-2 -1 0 1 2
Standardized Daily Cattle Futures Return
by the Realized Volatility
Kernel Density (Epanechnikov, h = 0.5212)
0.5
0.4 4
0.3 -
0.2 4
0.1 -
0’0 I f I I I T
-2 -1 0 1 2 3

Standardized Daily Hog Futures Return
by the Realized Volatility

146

 

 

Kernel Density (Epanechnikov, h = 0.5573)

 

0.5

0.4 -

0.3 4

0.2 -

0.1-

 

0.0

 

0.5

I I I I

-2 -1 0 1 2 3

.4

Standardized Gasoline Daily Futures Return
by the Realized Volatility

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0.4 4

0.3-

0.2 -

0.1-

 

0.0

 

I I I I

-2 -1 0 1 2 3

Standardized Gold Daily Futures Return
by the Realized Volatility

147

 

 

Figure 4-2 Realized Commodity Volatility Level

Corn Futures

 

 

 

 

-1°5 IIIIIIIIIIIIIIIIIIIIIIIIIIIII]II[IIIIIIIIIIIIIIIIIIIIITIIWITIIIIIWIIIWIFIIIIIIITIIIIIIIII

1 00 200 300 400

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1 .0

 

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-0.5 —

 

 

 

 

 

-1'0 ”'Tlfi'rl“”IIII'IWI'I”"I'"'I‘”'I””I””1””I'”'I‘"W”“I””I'”II
50 100 150 200 250 300 350 400

148

 

 

Live Cattle Futures

 

 

 

 

 

-2.0

 

IIIIIIIIIIIIITIIIIIIIIIIIIIIII[IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
50 100 150 200 250 300 350 400

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IIIIYTIIIIIIIIIIIIII'IIIIITWI[IIII[IITIIIIIIIIIIIIIIII[ITTIIIITIIIIIIIIIII
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149

Gasoline Futures

 

1.6
1.2—

0.8-

 

 

0.4-

 

 

0.0 4

 

 

0.4 ............................ .- ..
I50 100 150 200 250 300 350 400

 

Gold Futures

 

 

 

 

5'01 100 150 200 250 300 350 400

150

 

 

CHAPTER 5

CONCLUSION

This dissertation has studied commodity market price risks by using various time
series econometric models. The empirical investigation in this dissertation is focused on
commodity markets but is extensive in that we analyze the volatility dynamics for 1)
daily cash and futures price changes, and 2) higher frequency futures returns. Further, we
employed parametric, semi-parametric, and new volatility modeling in our investigations
of the commodity return volatility movements. In this final chapter, we list and discuss
important factors considered in this dissertation. Possible fixture research is discussed at
the end of this chapter.

First, we found evidence for slowly decaying autocorrelations for daily
commodity cash and futures as well as for intra-daily futures return volatility. This
observation is consistent with much previous evidence from conventional financial asset
return volatility studies. Our findings imply that commodity price risk patterns seem to
be similar to financial asset risk behavior, despite some unique characteristics of the
commodity markets based on the physical properties of the various types of products. In
particular, we observed the long memory phenomena for cash and futures returns at
various sample frequencies within the same sample period. This result is consistent with

one of the theoretical properties of the long memory process, “self-similarity.” We

151

 

 

utilized both the parametric FIGARCH model and the semi-parametric local Whittle
estimation to identify the long memory return volatility feature. Our findings in chapter 2
and 3 are supportive of long-run temporal dependence in commodity price risks at daily
and high frequency sample frequencies.

We used high frequency price data in particular in this dissertation. Tick sample
frequency data have become more available recently due to developments in computer
technology. Motivated by the mixture-of-distribution hypothesis and empirically well-
known volatility persistence, recent and active studies have highlighted high frequency
return data to pursue deeper understandings of return volatility patterns. However, the
use of high frequency data necessarily involves market microstructure issues to be
resolved in order to analyze the intrinsic volatility dynamics. In chapter 3, we applied the
Flexible Fourier Form filtering to remove strong intra-day volatility periodicity, one of
the market microstructure biases.

Secondly, we applied a newly suggested volatility measure to increasingly
available high frequency return data. The so-called “realized volatility” is easy to
calculate since the measure is the sum of the squared high frequency returns. This
volatility measure can provide important implications of data which are consistent with
financial theory for option pricing and derivative modeling. According to formal
quadratic variation theory, the realized volatility measure is a consistent estimator for true
latent volatility factors. Taking advantage of this observable volatility measure, we can
enrich volatility dynamics without relying on the complicated parametric form of
traditional GARCH models. In chapter 4, we confirmed that the realized volatility

measure can provide some enhanced frameworks for commodity market risk

152

management. We considered important economic determinants such as commodity-
specific announcements, time to maturity, information flow, and volatility linkage across
markets.

After applying various classes of econometric models at different sample
frequencies, we consistently witnessed long memory in the return volatility. The slowly
decaying autocorrelations seem to be an intrinsic property of the commodity return
volatility.

Since we have uncovered the commodity return volatility for both cash and
futures markets, a stage for further risk management modeling is ready. Baillie and
Myers (1991) noted that the optimal futures hedge ratio (OHR) is time-varying, and they
calculated the OHR using a bivariate GARCH model. Since the optimal hedge ratio is
defined to be a ratio of the conditional covariance between cash and futures to the
conditional variance of futures, proper modeling of conditional moments is important in
calculating the optimal hedge ratio. One possibility for further study is to build a
bivariate futures hedge model implementing the long memory property. This idea faces
some modeling issues since conditional variance matrices in multivariate contexts may
involve additional time—varying components. In previous studies on regular GARCH
models, constant conditional correlation matrices were assumed. We could allow for
more flexible forms of conditional correlation structures that may yield more implications
for the optimal hedge ratio modeling, as Tse and Tsui (2002) assume time-varying
correlations for a multivariate GARCH model.

Although it is true that cash and futures prices are not necessarin cointegrated,

there may be a possible cointegration relation between their squares. If cointegration

153

 

 

.“J

exists between cash and futures price volatilities, it will be necessary to include a lagged
error correction term in the bivariate FIGARCH model. In particular, Brunetti and
Gilbert (2000) considered fractional cointegration in a bivariate FIGARCH set up to
study the relationship among volatilities in closely-related oil markets. Cointegration
analysis of commodity cash and futures return volatilities seems to create room for
improvement of the optimal hedge ratio.

Another possible extension is to use the realized volatility (RV) for daily futures
variance and any of the previous methods for computing the covariance between cash and
futures price changes. AS studied in chapter 4, the realized volatility constructed from
high frequency commodity futures price data could provide relatively accurate
conditional variances without relying on the parametric form of the GARCH model.
Therefore, incorporating realized volatility into the hedge model may afford a simple
framework for the optimal hedge ratio calculation. The authors are in the process of

producing further work on the issues above.

154

References:

Admati, AR. and P. Pfleiderer (1988), “A Theory of Intraday Patterns: Volume and Price
Volatility”, Review of Financial Studies, 1, 3-40.

Admati, AR. and P. Pfleiderer (1989), “Divide and Conquer: A Theory of Intraday and
Day-of-Week Mean Effects”, Review of Financial Studies, 1, 189-224.

Akgiray, V. and G.G. Booth (1987), “Compound Distribution Models of Stock Returns:
An Empirical Comparison”, Journal of Financial Research, 10, 269-80

Akin, RM. (2003), “Matru'ity Effects in Futures Markets: Evidence from Eleven
Financial Futures Markets”, UC Santa Cruz Working Paper.

Andersen, T.G. (1994), “Stochastic Autoregressive Volatility: A Framework for
Volatility Modeling”, Mathematical Finance, 4, 75-102.

Andersen, T.G. and T. Bollerslev (1997a), “Heterogeneous information arrivals and
return volatility dynamics: uncovering the long-run in high frequency returns”,
Journal of Finance 52, 975-1005.

Andersen, T.G. and T. Bollerslev, (1997b), “Intraday Periodicity and Volatility
Persistence in Financial Markets”, Journal of Empirical Finance, 4, 1 15-158.

Andersen, T.G. and T. Bollerslev (1998a), “ Deutsche Mark-Dollar Volatility: Intraday
Activity Patterns, Macroeconomic Announcements and Longer Run
Dependencies”, Journal of Finance, 53, 219-265.

Andersen, T.G. and T. Bollerslev (1998b), “Answering the Skeptics: Yes, Standard
Volatility Models Do Provide Accurate Forecasts”, International Economic
Review, 39, 885-905.

Andersen, T.G., T. Bollerslev, and F.X. Diebold (2004), “Parametric and Nonparametric
Volatility Measurement”, In L.P. Hansen and Y. Ait-Sahalia (eds.), Handbook of
Financial Econometrics, Amsterdam: Northholand, forthcoming.

Andersen, T.G., T. Bollerslev, F.X. Diebold and H. Ebens (2001a), “The Distribution of
Realized Stock Return Volatility”, Journal of Financial Economics, 61, 43-76.

Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2001b), “The Distribution of
Realized Exchange Rate Volatility”, Journal of American Statistical Association ,
96, 42-55.

Andersen, T G, T Bollerslev, F.X. Diebold and P. Labys (2001c), “Exchange Rate

Returns Standardized by Realized Volatility are (Nearly) Gaussian”,
Multinational Finance Journal, 4, 159-179.

155

 

Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2003), “Modeling and
Forecasting Realized Volatility”, Econometrica, 71, 529-626.

Andersen, T.G., T. Bollerslev, F.X. Diebold and C. Vega (2002), “Micro Effects of
Macro Announcements: Real-Time Price Discovery in Foreign Exchange”,
American Economic Review, 93, 38-62.

Anderson, R. (1985), “Some Determinants of the Volatility of Futures Prices”, Journal of
Futures Markets, 5, 331-348.

Anderson, R.W., and J. Danthine (1983), “The Time Pattern of Hedging and the
Volatility of Futures Prices”, Review of Economic Studies 50, 249-266.

Baillie, R.T. and T. Bollerslev (1989), “The Message in Daily Exchange Rates: A
Conditional Variance Tale”, Journal of Business and Economic Statistics, 7, 297-
306.

Baillie, R.T. and T. Bollerslev (1991), “Intra-Day and Inter Market Volatility in Foreign
Exchange Rates”, Review of Economic Studies, 58, 565-585.

Baillie, R.T., Y. Han, and T. Kwon ( 2002), “Further Long Memory Properties of
Inflationary Shocks”, Southern Economic Journal, 68, 496-510.

Baillie, R.T. and RJ. Myers (1991), “Bivariate GARCH estimation of the optimal
commodity futures hedge”, Journal of Applied Econometrics, 6, 109-124.

Baillie, R T, Bollerslev, T and H-0 Mikkelsen (1996), “F ractionally integrated
generalized autoregressive conditional heteroskedasticity”, Journal of
Econometrics, 74, 3-30.

Bauwens, L., W.B. Omrane, and P. Giot (2003), “News announcements, market activity
and volatility in the Euro-Dollar foreign exchange market “, Journal of
International Money and Finance, forthcoming

Bhattacharya, R. N., Gupta, V. K., Waynrire, E. (1983), “The Hurst effect under trends”.
Journal of Applied Probability 20, 649-662.

Beran, J. (1994), Statistics for Long Memory Processes, Chapman and Hall, London.

Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity”,
Journal of Econometrics, 31, 307-327.

Bollerslev, T and J.M. Wooldridge (1992), “Quasi-maximum likelihood estimation and
inference in dynamic models with time varying covariances”, Econometric
Reviews, 11, 143-172.

156

Booth, G.G. and V. Akgiray (1987), “Compound Distribution Models of Stock Returns:
An Empirical Comparison”, Journal of Financial Research, (Fall 1987), 269-80.

Breidt, F .J ., N. Crato, and P.J.F. de Lima (1993), “Modeling Long Memory Stochastic
Volatility”, Journal of Econometrics, 73, 325-334.

Brunetti, C. and Gilbert, C. (2000), “Bivariate FIGARCH and Fractional Cointegration”,
Journal of Empirical Finance, 7, 509-530.

Burton, M. (1993), “Some Illustrations of Chaos in Commodity Markets”, Journal of
Agricultural Economics, 44, 38-49.

Cai, J, Cheung Y-L and M.C.S. Wong (2001), “What moves the gold market?”, Journal
of Futures Markets, 21, 257-278.

Campbell, J.Y. and N.G. Mankiw (1987), “Are Output Fluctuation Transitory?”,
Quarterly Journal of Economics, 102, 857-880.

Cecchetti, S.G, Cumby, RE. and S. Figlewski (1988), “Estimation of optimal futures
hedge”, Review of Economics Statistics, 70, 623-630.

Change, E.C., Jain, RC, and PR. Locke (1995), “S&P 5001ndex Futures Volatility and
Price Around the NYSE Close”, Journal of Business, 68, 61-84.

Christensen, BJ. and NR. Prabhala (1998), “The Relation Between Implied Volatility
and Realized Volatility”, Journal of Financial Economics, 50, 125-150.

Chen, Y..I., J.C. Duan, and M.W. Hung, 1999, “Volatility and Maturity Effects in the
Nikkei Index Futures”, Journal of Futures Markets, 19(8), 895-909.

Clark, PK. (1973), “A Subordinate Stochastic Process Model with Finite Variance for
Speculative Prices”, Econometrica, 41, 135-155.

Dacorogna, M.M., U.A. Muller, R.J. Nagler, R.B. Olsen and ON. Pictet (1993), “A
geographical model for daily and weekly seasonal volatility in the foreign
exchange markets”, Journal of International Money and F inance, 12, 413-438.

Diebold, F.X. (1988), Empirical Modeling of Exchange Rate Dynamics, New York,
Springer Verlag-

Ding, Z. and C.W.J. Granger. (1996), “Modeling volatility persistence of speculative
returns”, Journal of Econometrics, 73, 182-215.

Ding, Z., C.W.J. Granger, and RF. Engle. (1993), “A Long memory property of stock
market returns and a new model”, Journal of Empirical Finance, 83-106.

157

Drost, F .C. and T.E.Nijman (1993), “Temporal aggregation of GARCH processes”,
Econometrica, 61, 909-927.

Dueker, M. and R. Startz (1998), “Maximum-Likelihood Estimation of Fractional
Cointegration With An Application to US. and Canadian Bond Rates”, Review of
Economics and Statistics, 83, 420-426.

Ederington, L., Lee, J.H. (1993), “How markets process information: news releases and
volatility”, Journal of Finance, 1 160 — 1 191.

Engle, RF. (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the
Variance of UK. Inflation”, Econometrica, 50, 987-1008.

Engle, RR, and T. Bollerslev (1986), “Modeling the Persistence of Conditional
Variances”, Econometric Reviews, 5, 1-50.

Fama, E.F. (1965), “The Behavior of Stock Market Prices”, Journal of Business, 14, 319-
338.

Fleming, 1., C. Kirby, and B. Ostdiek (1998) “Volatility and Common Information in
Stock, Bond, and Money Markets”, Journal of Financial Economics, 49, 111-
137.

Fleischer, P. (1998),“Volatility and Information Linkages Across Markets and
Countries”, Australian Journal of Management, 28, 251-272.

Flemming, J., C. Kirby, and B. Ostdiek (1998), “Information and volatility linkages in the
stock, bond, and money markets”, Journal of Financial Economics, 49, 111-137.

French, K.R., G.W- Schwert, and RF. Stambaugh (1987), “Expected Stock Returns and
Volatility”, Journal of Financial Economics, 19, 3-29.

Gallant, AR. (1981), “On the Bias in Flexible Function Forms and an Essentially

Unbiased Form: the Fourier Flexible Form”, Journal of Econometrics, 15, 21 l-
245.

Gallant, A.R. (1982), “ Unbiased Determination of Production Technologies”, Journal of
Econometrics, 20, 285-323.

Galloway, T. M., and R.W. Kolb, 1996, “Futures Prices and the Maturity Effect”, Journal
of Futures Markets, 16(7), 809-28.

Goodhart, C. and L. Figliuoli (1992), “The geographical location of the foreign exchange
market: a test of the island hypothesis”, Journal of International and Comparative
Economics, 1, 13-28.

158

 

Goodhart, C .A.E. and M. O'Hara (1997), “High frequency data in financial markets:
issues and applications”, Journal of Empirical Finance, 4, 73-114.

Gradszteyn, LS. and I.M. Ryzhnik (1980), “Tables of Integrals, Series, and Products”, 4th
ed. Academic Press, New York, NY.

Granger, C.W.J. (1980), “Long Memory Relationships and the Aggregation of Dynamic
Models”, Journal of Econometrics, 14, 227-238.

Granger, C.W.J. (1981), “Some Properties of Time Series Data and Their Use in
Econometric Model Specification”, Journal of Econometrics, 16, 121-130.

Granger, C.W.J. and Z. Ding (1996), “Variety of Long Memory Models”, Journal of
Econometrics, 73, 61-77.

Granger, C .W.J . and RF. Engle (1993), “A Long Memory Property of Stock Market
Returns and A New Model”, Journal of Econometrics, 73, 182-215.

Granger, C.W.J. and P. Newbold (1976), “Forecasting Transformed Series”, Journal of
the Royal Statistical Society, Series B 38, 189-203.

Han, L., J. Kling, and C. Sell (1999), “Foreign Exchange Futures Volatility: Day-of-the-
Week, Intraday, and Maturity Patterns in the Presence of Macroeconomic
Announcements”, Journal of Futures Markets, 19(6), 665-693.

Harris, L. (1986), “A Transaction Data Study of Weekly and Intradaily Patterns in Stock
Returns”, Journal of Financial Economics, 16, 99-117.

Havey, AC. (1998), “Long Memory in Stochastic Volatility”, in J. Knight and J. Satchel]
(eds.), Forecasting Volatility in Financial Markets, London, UK, Butterworth-
Heinemann.

Hong, H. (2000), “A Model of Returns and Trading in Futures Markets”, Journal of
Finance, 55(2), 959-988.

Hosking, J .R.M. (1981), “Fractional Differencing”, Biometrika, 68, 165-176.

Hsieh, D.A. (1989a), "Testing for Nonlinear Dependence in Daily Foreign Exchange
Rates", Journal of Business, 62, 339-368.

Hsieh, D.A. (1989b), “ Modeling Heteroscedasticity in the Daily Foreign Exchange
Rates”, Journal of Business and Economic Statistics, 3, 307-317.

Hurvich, CM, and W.W. Chen (2000), “An Efficient Taper for Potentially

Overdifferenced Long Memory Time Series”, Journal of Time Series Analysis,
21, 155-180.

159

Hull, J. and A. White (1987), “The Pricing of Options on Assets with Stochastic
Volatilities”, Journal of Finance, 42, 381-400.

J in, H]. and D.L. Frechette (20002), “Forecasting Agricultural Cash Price Movements in

the Presence of Long-Term Memory”, American Journal of Agricultural
Economics, 86, 432-443

Kodres, LE. and M. Pritsker (2002), “A rational expectations model of financial
contagion”, Journal of Finance, 57, 769—799.

Kunsch, HR. (1987), “Statistical Aspects of Self-similar Process”, In Proceedings of the
First World Congress of the Bernoulli Society, vol-1, 67-74.

Lamourex, CG. and W. D. Lastrapes (1990), “Persistence in Variance, Structural

Change, and the GARCH model”, Journal of Business and Economic Statistics, 8,
225-234.

LeBaron, B. (1999), “Technical Trading Rule Profitability and Foreign Exchange
Intervention”, Journal of International Economics, 49, 125-143.

Lo,’ A.W. (1991), “Long Terms Memory in Stock Market Prices”, Econometica, 59,
1279-1313. '

Mackey, MC. (1989), “Commodity Price Fluctuations: Price Dependent Delays and

Nonlinearities as Explanatory Factors”, Journal of Economic Theory, 48, 497-
509.

Marinucci, D. and PM. Robinson (2001), “Semiparametric Fractional Conitegraion
Analysis”, Journal of Econometrics, 73, 303-324.

Martens, M., and J. Luu (2002), “Testing the mixture of distributions hypothesis using
“realized” volatility”, Erasmus University Working Paper.

Mandelbrot, BB. (1963), “The Variation of Certain Speculative PriCes”, Journal of
Business, 36, 394-419.

McFarland, J W, Pettit, R. Richardson, Sam K Sung (1987), “The Distribution of Foreign
Exchange Price Changes: Trading Day Effects and Risk Measurement”, Journal
of Finance, 42, 189-194.

Maddala, GS. and H. Li (1996), “Bootstrap Based Tests in Financial Models”, in G. S.
Maddala and C R Rao (eds), Handbook of Statistics, 14, 463-488, Elsevier.

160

Melino, A. (1994), “The Estimation of Continuous-Time Models in Finance”, in CA.
Sims, ed., Advanced in Econometrics, Sixth World Congress Volume 11,
Cambridge University Press.

Merton, RC. (1980), “On Estimating the Expected Returns on the Market: An
Exploratory Investigation”, Journal of Financial Economics, 8, 323-361.

Milhoj, A. (1987), "A Conditional Variance Model for Daily Observations of an
Exchange Rate", Journal of Business and Economic Statistics, 5, 99-103.

Milonas, NT. (1986), “Price Variability and The Maturity Effect in Futures Markets”,
Journal of Futures Markets, 3, 443-460.

Mincer, J. and V. Zarnowiz (1969), “The Evaluation of Economic Forecasts”, in J
Mincer, ed., Economic Forecasts and Expectations (New York: NBER).

Muller, U.A., M.M. Dacorogna, R.B. Olsen, O.V. Pictet, M. Schworz and C. Morgenegg
(1990), "Statistical study of foreign exchange rates, empirical evidence of price

change law and intraday analysis”, Journal of Banking and Finance, 14, 1 189-
1208.

Nelson, DB. (1992), “Filtering and Forecasting with Misspecified ARCH Models 1:
Getting the Right Variance with the Wrong Model”, Journal of Econometrics, 52,
61-90.

Parke, W.R. (1999) “What Is Fractional Integration?”, Review of Economics and
Statistics, 81, 628-632

Porterba, J. and L. Summers (1986), “The Persistence of Volatility and Stock Market
Fluctuations”, American Economic Review, 76, 1124-1141.

Robinson, RM. (1995), “Gaussian Semiparametric Estimation of Long Range
Dependence”, Annals of Statistics, 23, 1630-1631.

Scott, L. (1987), “Option Pricing when the Variance Changes Randomly: Theory,
Estimation, and An Application”, Journal of Financial and Quantative Analysis,
22, 419-436.

Samuelson, RA. (1965), “Proof that Properly Anticipated Price Fluctuate Randomly”,
Industrial Management Review, 6, 41-49.

Serletis, A. (1992), “Maturity Effects in Energy Futures”, Energy Economics, April, 150-
157.

Schwert, G.W. (1989), “Why Does Stock Market Volatility Change Over Time?”,
Journal of Finance, 44, 1115-1153.

161

Tauchen, GE. and M. Pitts (1983), “The Price Variability-Volume Relationship on
Speculative Markets”, Econometrica, 51, 485-505.

Taqqu, MS. and V. Teverovsky (1997), “Robustness of Whittle Estimators for Time
Series with Long Range Dependence”, Stochastic Models, 13, 723-757.

Taqqu, MS. and V. Teverovsky (1998), “On Estimating the Intensity of Long Range
Dependence in Finite and Infinite Variance Time Series”, 177-217, A Practical
Guide to Heavy Tails”, (edited by R Adler, R. Friedman and MS. Taqqu),
Birkhauser, Boston, MA.

Tse,Y.K. and Tsui, A.K.C. (2002) "A multivariate GARCH model with time-varying
correlations", Journal of Business and Economic Statistics, 20, 351 - 362.

Wiggins, M. (1987), “Option Values under Stochastic Volatility: Theory and Empirical
Estimates”, Journal of Financial Economics, 35, 23-45..

Wood, RA, McInish, TH, and J .K. Ord (1985), “An Investigation of Transaction Data
for NYSE stocks”, Journal of Finance, 40, 723-741.

Yang, S., and B.W. Brorsen (1992), “Nonlinear Dynamics of Daily Cash Prices”,
American Journal of Agricultural Economics, 74, 706-715.

Zhou, B (1996), “High Frequency Data and Volatility in Foreign Exchange Rates”,
Journal of Business and Economic Statistics, 14, 45-52.

162

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