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MODELS FOR LONG MEMORY AND HIGH FREQUENCY FINANCIAL
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YOUNG WOOK HAN

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MODELS FOR LONG MEMORY AND HIGH FREQUENCY FINANCIAL TIME
SERIES

By

Young Wook Han

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

2001

 

 

 

ABSTRACT

MODELS FOR LONG MEMORY AND HIGH FREQUENCY FINANCIAL TIME
SERIES

By

Young Wook Han

This dissertation is composed of five distinct chapters, all of which model and
examine the long memory properties in different financial time series data.

Chapter 2 considers the use of a long memory volatility process, FIGARCH,
in representing Deutsche Mark - US $ spot exchange rate returns for both high and
low frequency returns data. The Flexible Fourier Form (FFF) filter and a FIGARCH
type model is used to represent the volatility process.

Chapter 3 is concerned with the econometric modeling and appropriate
specification of models to describe exchange rates in a target zone. A long memory
GARCH model with a jump process generated by either a Bernoulli or Poisson
process is used for the daily FF-DM returns data.

Chapter 4 examines one of the earliest recorded periods of central bank
intervention in the 19203 foreign exchange market. A relatively new set of daily data
for four currencies is examined and returns are found to be close to martingales with
unusually persistent volatility processes, which are represented by FIGARCH models.
The effect of intervention by the French government to support the French franc is
represented by dynamic dummy variables. There is strong evidence that the
intervention
was successful for about six months before the FF again depreciated. In contrast, the

corresponding effects of the intervention on the conditional variance seem to be

insignificant. The intervention was not associated with increasing the volatility of the
currencies. Hence, the overall conclusion appears to be that the intervention was
successful in the short run, but was unable to prevent substantial depreciation over a
six month horizon.

Chapter 5 is concerned with modeling inflation for nine countries. The
inflation series appear to have dual long memory features in both their first and
second conditional moments. This chapter implements a combined ARFIMA-
FIGARCH model to represent the inflation series. For nearly all of the countries,
there is strong evidence of long memory features in both the conditional mean and
variance.

Chapter 6 is concerned with the bivariate relationship between high frequency
time series for the DM-$ spot exchange rate and the corresponding US and German
30 day Eurobond interest rate differential. The well known forward premium
anomaly critically depends on the order of integration of the interest rate differential,
i.e. forward premium. The use of parametric ARFIMA-FIGARCH models and also
semi parametric local Whittle estimation methods is to estimate the order of fractional
integration. The high frequency forward premium is found to have finite cumulative
impulse response weights, but to be non stationary and to have very persistent
autocorrelations. This chapter concludes with a brief discussion of the implications of

the results for the resolution of the forward premium anomaly.

This dissertation is dedicated to
My wife, Hee Soo, my daughter, Seung Yeon and my parents
For their love, encouragement and trust

iv

ACKNOWLEDGMENTS

Without the assistance and encouragement of many people, I could not finish
this work and present it in this form.

First of all, my sincerest thanks goes to Professor Richard T. Baillie, my
mentor. His incredible expenditure of time spent on my behalf, his technical advice,
his generous financial support and his general guidance made my dissertation
possible. He gave me valuable comments and suggestions on my dissertation, and
also he helped me to navigate the rough water of the whole dissertation process. His
help and comments were invaluable.

I owe a special thanks to Professor G. Geoffrey Booth in the Finance
Department and Professor Peter J. Schmidt in the Economics Department. They gave
me very helpful comments for my dissertation and helped me to complete it.

I also like to thank Dr. Tae-Go Kwon and my fellow graduate students, Chirok
Han, Hoon Kim, Jong Byung Jun and Rehim Kilic, for their advise and friendship.
Especially, Dr. Kwon gave me great help in studying Econometrics and the GAUSS
program language.

The greatest thanks, however, goes to my lovely family. My wife, Hee-Soc,
has been a constant source of inspiration and encouragement. Her dedication and
support have been so great and they helped move my dissertation forward. My
daughter, Seung-Yeon, was very patient and sympathetic even when I was neglectful
in taking care of her.

And, my parents have always supported and encouraged me in everything. Their
love, patience and trust are things that I would never be able to repay. Thanks to all

of them

 

 

for their love and encouragement.
Finally, I gratefully acknowledge support from the National Science

Foundation under grant #DMS 0071619.

vi

[TABLE OF CONTENTS]

LIST OF TABLES ....................................................................................................... ix

LIST OF FIGURES ..................................................................................................... xi

CHAPTER 1. INTRODUCTION: REVIEW OF LONG MEMORY MODELS

AND HIGH FREQUENCY DATA

1.1. Review of Long Memory Models ............................................................... 2
1.2. A New Type of High Frequency Data ........................................................ 3
1.3. The Plan of the Dissertation ....................................................................... 5

CHAPTER 2. HIGH FREQUENCY DATA ANALYSIS: F IGARCH

REPRESENTATIONS AND NON LIN EARITIES

2.1. Introduction ................................................................................................. 9
2.2. Analysis of Low Frequency Daily Returns ............................ 11
2.3. Analysis of High Frequency Returns ........................................................ 16
2.4. Nonlinear Dependencies in High Frequency Returns ............................... 23
2.5. Conclusion ................................................................................................ 26

CHAPTER 3. APPLICATION OF LONG MEMORY GARCH MODELS

WITH JUMP PROCESSES TO DESCRIBE TARGET ZONE

EXCHANGE RATES
3.1. Some Stylized Facts of Nominal Exchanges in Target Zones ................. 37
3.2. Analysis of FF-DM Exchange Rate .......................................................... 38
3.3. Discussion for Future Research ............................................................... 43

CHAPTER 4. INTERVENTION IN THE 19205 CURRENCY MARKETS

4.1. Introduction ............................................................................................... 51

4.2. The 19203 Foreign Exchange Market ....................................................... 52

vii

4.3. Analysis of Daily Currency Returns ............................................... . ......... 55
4.4. Effects of Intervention ............................................................................. 58

4.5. Conclusion .................................................................... ..................... 61

CHAPTER 5. MODELING INFLATION WITH DUAL LONG MEMORY

MODELS
5.1. Introduction .............................................................................................. 76
5.2. Conditional Mean of Inflation ........................................... . ...... ..... '77
5.3. The ARFIMA-FIGARCH Model ........................................................... 82
54. Estimated Models of Inflation .. ............................................................... 85
5.5. Conclusion ......... . ....................................................................... . ................ 86

CHAPTER 6. HIGH FREQUENCY PERSPECTIVE ON THE FORWARD

PREMIUM ANOMALY
6.1. Introduction ..................................... . ........................................... . . ...... l 13
6.2. International Finance Relationships ..................................... . ................ 114
6.3 Analysis of the High Frequency Forward Premium ........ . ............. ,. ........ 119
64. Semi Parametric Estimation .......... . ...... .................. ...... . ............... 122
6.5. Conclusion ..... . ...................................................................................... 124
LIST OF REFERENCES ................ . ......................................................................... 138

viii

[LIST OF TABLES]

[CHAPTER 2]
Table 2.1: Estimated MA(1)-FIGARCH(p,6,q) Models for Daily DM-$ Returns ...... 27

Table 2.2: Estimated MA(1)-FIGARCH(p,8,q) Models for Filtered 30 Minute
DM-$ Returns ......................................................................................... 28-29

Table 2.3: Correlation Dimension Estimates on the Residuals from the MA(1)-
FIGARCH(1,5,0) Model for Filtered 30 Minute DM—$ Returns

........................................................................................................................ 30
Table 2.4: BDS Test on the Residuals from the Estimated MA(1) - FIGARCH(1,8,0)
Model for Filtered 30 minute DM - $ Returns ............................................. 30
[CHAPTER 3]
Table 3.1: Estimation of Models for Daily F F -DM Spot Returns ......................... 45-46
Table 3.2: Estimation of Models for Weekly FF-DM Spot Returns ........................... 47
[CHAPTER 4]

Table 4.1: Estimated MA(1)—FIGARCH(p,5,q) Models for Daily Spot Returns
in the 19203 ................................................................................................... 63

Table 4.2: Estimated Martingale—FIGARCH(p,5,q) Models for Weekly Spot Returns
in the 19203 ................................................................................................... 64

Table 4.3: Estimated MA(1)-FIGARCH(p,5,q) Models for Daily Spot Returns

in the 19203 with Dummy Variable in the Mean for France's Intervention

on March 11, 1924 ........................................................................................ 65
Table 4.4: Estimated MA(1)-FIGARCH(p,5,q) Models for Daily Spot Returns

in the 19203 with Dummy Variable in the Mean and Variance
for France's Intervention on March 11, 1924............... ............................... 66

[CHAPTER 5]
Table 5.1: Estimated ARFIMA Models for Countries' Monthly Inflation Rates ........ 87

Table 5.2: Estimated Higher Order ARFIMA Models for France and UK Monthly
Inflation Rates ....................... . ...................................... . ......................... 88-89

ix

 

Table 5.3: Estimated ARFIMA-FIGARCH Models for Countries' Monthly Inflation
Rates ....................................................................................................... 90-91
[CHAPTER 6]

Table 6.1: Estimated ARFIMA-FIGARCH Models
for Filtered 30 minute Differenced Interest Rate Differential ............. 126-127

Table 6.2: Local Whittle Estimators of Long Memory Parameters
for High Frequency Forward Premium ..................................................... 128

[LIST OF FIGURES]

[CHAPTER 2]

Figure 2.1: Correlograms of Daily DM-$ Returns ....................................................... 31
Figure 2.2: Correlograms of 30 Minute DM-$ Returns ............................................... 32
Figure 2.3: Intraday Averages of Absolute 30 Minute Returns ................................... 33
Figure 2.4: Correlograms of Filtered 30 Minute Returns ............................................ 34

Figure 2.5: Correlograms of Standardized Residuals for Filtered 30 Minute Returns

........................................................................................................................ 35
[CHAPTER 3]
Figure 3.1: The FF-DM Spot Exchange Rate for March 14, 1979 through
October 10, 1995 ......................................................................................... 48
Figure 3.2: Interest Rate Differential of French 30-day Eurorate minus Germany
equivalent from March 14, 1979 through October 10, 1995 ....................... 49
[CHAPTER 4]
Figure 4.1a: Daily FF-BP Spot Exchange Rates from May 1, 1922
through May 30, 1925 ............................................................................. 67
Figure 4.1b: Daily BF-BP Spot Exchange Rates from May 1, 1922
through May 30, 1925 ............................................................................. 68
Figure 4.10: Daily IL-BP Spot Exchange Rates from May 1, 1922
through May 30, 1925 ................................................................................ 69
Figure 4.1d: Daily US-BP Spot Exchange Rates from May 1, 1922
through May 30, 1925 ................................................................................ 70
Figure 4.2: Correlograms of Daily F F -BP Spot Returns ............................................ 71
Figure 4.3: Correlograms of Daily BF-BP Spot Returns ............................................ 72
Figure 4.4: Correlograms of Daily IL-BP Spot Returns ............................................. 73
Figure 4.5: Correlograms of Daily US—BP Spot Returns ............................................. 74

xi

[CHAPTER 5]

Figure 5.1a: Graph of US CPI Inflation ....................................................................... 92
Figure 5.1b: Autocorrelations of US CPI Inflation ...................................................... 93
Figure 5.1c: Autocorrelations of Differenced US CPI Inflation .................................. 94
Figure 5.1d: Autocorrelations of Differenced Belgium CPI Inflation ......................... 95
Figure 5.1e: Autocorrelations of Differenced France CPI Inflation ............................ 96
Figure 5.1f: Autocorrelations of Differenced Germany CPI Inflation ........................ 97
Figure 5.1 g: Autocorrelations of Differenced Italy CPI Inflation ............................... 98
Figure 5.1h: Autocorrelations of Differenced Japan CPI Inflation .............................. 99
Figure 5.1i: Autocorrelations of Differenced Korea CPI Inflation ............................ 100
Figure 5.1j: Autocorrelations of Differenced UK Inflation ....................................... 101

Figure 5.2: Autocorrelations of Transformations of the Residuals
of US CPI Inflation .................................................................................. 102

Figure 5.3: Cumulative Impulse Response Weights for Conditional Mean and
Conditional Variance of US CPI Inflation .............................................. 103

Figure 5.4: Cumulative Impulse Response Weights for Conditional Mean and
Conditional Variance of Belgium CPI Inflation ....................................... 104

Figure 5.5: Cumulative Impulse Response Weights for Conditional Mean and
Conditional Variance of France CPI Inflation .......................................... 105

Figure 5.6: Cumulative Impulse Response Weights for Conditional Mean and
Conditional Variance of Germany CPI Inflation ..................................... 106

Figure 5.7: Cumulative Impulse Response Weights for Conditional Mean and
Conditional Variance of Italy CPI Inflation ............................................. 107

Figure 5.8: Cumulative Impulse Response Weights for Conditional Mean and
Conditional Variance of Japan CPI Inflation ........................................... 108

Figure 5.9: Cumulative Impulse Response Weights for Conditional Mean and
Conditional Variance of Korea CPI Inflation ........................................... 109

Figure 5.10: Cumulative Impulse Response Weights for Conditional Mean and
Conditional Variance of UK CPI Inflation ............................................. 110

xii

Figure 5.11: Cumulative Impulse Response Weights for Conditional Mean and

Conditional Variance of US Median CPI Inflation ................................ 111
[CHAPTER 6]
Figure 6.1a: Daily DM-$ Spot Returns from January 4, 1984 through

December 31, 1998 ....................................................................................... 129
Figure 6.1b: Daily Forward Premium from January 3, 1984 through
December 31, 1998 ....................................................................................... 129

Figure 6.2: Rolling Five-Year DM-$ Unbiasedness Regression ............................... 130
Figure 6.3a: Intraday 30 Minute DM-$ Spot Returns for 1996 ................................. 131
Figure 6.3b: Intraday 30 minute Forward Premium for 1996 .................................... 131
Figure 6.4a: Averaged 30 Minute Absolute DM-$ Spot Returns .............................. 132
Figure 6.4b: Averaged 30 Minute Absolute Differenced DM-$ Forward Premium
.................................................................................................................................... 132
Figure 6.4c: Averaged 30 Minute Absolute DM-$ Spot Returns

from 08:00 GMT to 20:00 GMT ............................................................. 133
Figure 6.4d: Averaged 30 Minute Absolute Differenced DM-$ Forward Premium

from 08:00 GMT to 20:00 GMT ............................................................. 133
Figure 6.5: Correlograms of Intraday Forward Premium .......................................... 134
Figure 6.6: Correlograms of Intraday Differenced Forward Premium ...................... 135
Figure 6.7: Correlograms of Filtered Intraday Differenced Forward Premium ......... 136
Figure 6.8: Correlograms of Filtered Intraday Differenced Forward Premium

Residuals ................................................................................................. 137

xiii

[Chapter 1] Introduction:

Review of Long Memory Models and High Frequency Data

1.1. Review of Long Memory Models

There has been considerable literature on econometric studies of long memory,
fractionally integrated processes that are associated with hyperbolically decaying
autocorrelations and impulse response weights. See Robinson (1994) and Baillie (1996)
for surveys of the property of long memory models. The initial interest in long memory
models seems to have come from the physical sciences, e.g. hydrology and climatology in
19503. Originally Hurst (1951,1952) analyzed geophysical time series autocorrelations
such as riverflow data and tried to understand the persistence of the autocorrelations. The
presence of the long memory can be defined from an empirical approach in terms of the
persistence of observed autocorrelations. Even though the extent of the persistence is
consistent with a stationary process, the autocorrelations take much longer to decay than
the exponential rate associated with the ARMA class of models. So, even when the
autocorrelation function is viewed as a stochastic process, it shows persistence that is
neither an 1(0) process nor 1(1) process. The greatest advantage of long memory process
is that they avoid the knife-edge distinction between 1(0) and 1(1) process. Hence the
model implies different long run predictions and effects of shocks.

Since 1980, many econometricians have used the long memory model to model
the macroeconomic data and financial data, and found substantial evidence that long
memory processes can describe quite well time series such as Real GDP, inflation rates,
forward premium and interest rate differentials. Particularly, in the discrete time long
memory fractionally integrated I(d) class of processes proposed by Adenstedt (1974),
Granger (1980,1981), Granger and Joyeux (1980) and Hosking (1981), the effects of

shocks to the mean occur at a slow hyperbolic rate of decay when the fractional

 

differencing parameter (1, is 0 < d < 1, as opposed to extremes of 1(0) exponential decay
associated with the stationary and invertible ARMA class of process, or the complete
persistence resulting from an 1(1) process.

Since the ARCH model and the GARCH model were introduced by Engle (1982)
and Bollerslev (1986), many empirical studies have found that the extreme degree of
persistence of shocks to the conditional variance process. Similarly to the issues
concerning the proper modeling of long run dependencies in the conditional mean of
economic time series data, the same questions have occurred in the modeling of
conditional variances. Subsequent empirical studies showed that long memory processes
are very successfiil in modeling the volatility of the asset price and power transformation
of returns. They detected the presence of the long memory in the autocorrelations of
squared or absolute returns of various financial asset prices with different methods. For
example, Breidt et al.(1998), Crato and de Lima (1994), and Harvey (1993) used the
LMSV (long memory stochastic volatility) model for the analysis of the stock returns and
exchange rates, Baillie et al.(1996) applied the FIGARCH (Fractionally Integrated
Generalized Autoregressive Conditional Heteroskedasticity) model to exchange rates, and
Bollerslev and Mikkelsen (1996) applied the FIEGARCH process to stock prices. This
feature of long memory in the conditional variance appears to be related to the presence
of long memory in the mean of interest rate differentials or forward premia and can

potentially provide important insights into the forward anomaly and the pricing of risk.

1.2. New Type of High Frequency Data

Financial data have been the subject of many studies, and most of the work has

analyzed daily or lower frequency data. In the last few years, however, the empirical
studies have dealt with higher frequency intraday prices since gathering financial data has
become easier due to the fast developing computer technology. In particular, foreign
exchange markets, which have no one geographical location and no business hour
limitation provide a set of complete intraday time series data covering a worldwide 24
hour market. The exchange rates used for most studies are the quotes from large data
suppliers such as Reuters. Since the actual transaction prices and trading volume are not
known to the public, quotes are intended to be used by market participants as a general
indication of the markets and these indicative prices appear to closely match the true
prices in the markets. This new type of high frequency intraday prices is important for
the empirical analysis of the foreign exchange markets. First, the large number of
observations enhances the significance of the statistical study. Second, it cam increase
the ability to analyze finer details of the behavior of different market participants.

As presented in the papers by Muller et a1. (1990), and Dacorogna et a1. (1993),
the properties of this new type of high frequency data differ from those of the daily or the
lower frequency data. The new data shows daily and weekly seasonal heteroskedasticity,
so that there is a seasonal behavior of volatility rather than the prices themselves. The g
daily seasonality appears to be particularly significant. The pattern is clearly correlated to
volume of trading in the main financial markets around the world. Chapter 2 and 6
found similar seasonality in high frequency 30minute data and used the Flexible Fourier
Form (FFF) adjustment method as suggested by Andersen and Bollerslev (1997).

When the new type of high frequency data is used, there can arise "database

holes" due to human and technical errors in the communication. So, in order to obtain the

 

 

 

prices at a time 1‘ within a hole, it appears to be more appropriate to use the linear
interpolation method for interpolating in a series with independent random increments. I
followed this method for the analysis of the 30 minute data in this dissertation as
suggested by Muller et al. (1990) and Andersen and Bollerslev (1997). For more details
about this new type of high frequency data, see Muller et al. (1990), Baillie and
Bollerslev (1991), Goodhart and F igliuoli (1991), Dacorogna et a1. (1993), Bollerslev and

Domowitz (1993) and Andersen and Bollerslev (1997).

1.3. The Plan of the Dissertation

The plan of the rest of this dissertation is as follows; chapter 2 considers the use
of a long memory volatility process, FIGARCH, in representing Deutsche Mark - US 8
spot exchange rate returns for both high and low frequency returns data. The Flexible
Fourier Form (FFF) filter is applied to eliminate intra day periodicity in the spot returns
series and a F IGARCH type model is used to represent the volatility process. The
FIGARCH model is found to be the preferred specification for both high frequency and
daily returns data, with similar values of the long memory volatility parameter across
frequencies, which is indicative of returns being generated by a self similar process.
Various tests for non-linearity are applied to the residuals of the model for the high
frequency returns. No evidence is found to suggest that the procedure for filtering the
high frequency returns to remove the intraday periodicity has induced any nonlinearities
in the residuals; hence the FIGARCH specification is found to be adequate.

Chapter 3 is concerned with the econometric modeling and appropriate

specification of models to describe exchange rates in a target zone. The EMM

 

approach of estimating continuous time models is compared with a model in discrete
time, provided by taking the relatively simple formulation of a long memory GARCH
model with a jump process generated by either a Bernoulli or Poisson process. Many of
the intrinsic features of the FF -DM exchange rate behavior are captured quite well by this
model.

Chapter 4 examines one of the earliest recorded periods of central bank
intervention in the 19203 foreign exchange market. A relatively new set of daily data for
four currencies is examined and returns are found to be close to martingales with
unusually persistent volatility processes, which are represented by FIGARCH models.
The effect of intervention by the French government to support the French franc is
represented by dynamic dummy variables. The overall conclusion appears to be that the
intervention was successful in the short run, but was unable to prevent substantial
depreciation over a six month horizon.

Chapter 5 is concerned with modeling inflation for nine countries. Several
previous studies have found fractionally integrated, or long memory behavior in the
conditional mean of inflation. This chapter notes that extremely similar phenomena are
also apparent in the squared and absolute values of the residuals from fractionally filtered
inflation series. Hence the inflation series appears to have dual long memory features in
both their first and second conditional moments. This chapter suggests a combined
ARFIMA-FIGARCH model as originally proposed by Kwon (1997) to represent the
inflation series. The model is then estimated for CPI inflation series for several different
industrialized countries, including the US. For nearly all of the countries, there is strong

evidence of long memory features in both the conditional mean and variance. 1 note some

 

of the implications for modeling inflation.

Chapter 6 is concerned with the bivariate relationship between high frequency
time series for the DM—$ spot exchange rate and the corresponding US and German 30
day Eurobond interest rate differential. The Flexible Fourier Form (FFF) filter is applied
to eliminate intra day periodicity in the spot returns series and a FIGARCH type model is
used to represent the volatility process. Corresponding analysis for the high frequency
interest rate differential, i.e. forward premium, is harder due to the very persistent
autocorrelation in both the first two conditional moments. The issue of the order of
integration of the forward premium is at the heart of the controversy concerning the
forward premium anomaly. The use of parametric ARFIMA and FIGARCH models and
also semi parametric local Whittle estimators are implemented to estimate the order of
fractional integration. A fractional filter is applied to the conditional mean of the forward
premium and the intra day periodicity is extracted by another F FF filter. The high
frequency forward premium is found to have finite cumulative impulse response weights,
but to be non stationary and to have very persistent autocorrelations. The chapter
concludes with a brief discussion of the implications of the results for the resolution of

the forward premium anomaly.

[CHAPTER 2]

High Frequency Data Analysis: FIGARCH Representations and Non Linearities

2.1. Introduction

This chapter is concerned with some of the intriguing features of high frequency
foreign exchange rates. In particular, I explore some aspects of the property of long
memory, persistent volatility that has become a well documented feature of these
markets; e.g. see Andersen and Bollerslev (1997b, 1998) and Dacorogna et a1 (1993). I
focus on the long memory volatility parameter obtained by estimating various F IGARCH
models from both high and low frequency returns data. While such models have been
found to provide good descriptions of daily return volatility, little is known of their
adequacy in dealing with higher frequency data. Hence this chapter investigates the
general appropriateness of the FIGARCH specification for many different frequencies of
DM-$ returns. Second, I also wish to see if the FIGARCH model is consistent with the
theory that returns are a self similar process, which implies the long memory parameter is
invariant to the sampling frequencies; see Beran (1994). Self-similar processes were first

introduced to statistics by Mandelbrot and van Ness (1968) and Mandelbrot and Wallis

(1968,1969). A stochastic process {Yt}teR+ is called self-similar with self-similarity
parameter I-I, if for any c > 0 the stochastic process {thhem is equal in distribution to
the process {CH Yt}teR+, If Yt has stationary increments Xi = Yi - Y H {ieN}, then
covariances Rk = cov{Xi , Xi+k} = In-“ exp(ikx)f(x)dx where f(x) is the power spectrum

are,

Rk=oz<|k+1I”‘-2Ik|2”+Ik-1I2“>/2,

where 02 = var(Xi). For extensive surveys on self-similar processes, see Taqqu (1988).

Apart from the standard diagnostic tests, the appropriateness of the FIGARCH
model for 30minute data is also investigated for the presence of further non linearities.
This seems especially important given the possibility that the filtering method for
removing intra day periodicity may have potentially induced spurious relationships. The
estimation of the correlation dimension and the results from the BDS test fail to find any
evidence of significant non-linearity in the residuals. The overall conclusion is that the
FIGARCH model appears to be a good specification for the filtered returns series.

One theory advanced by Mikosch and Starica (1999) and others is that frequently
occurring regime shifts can give rise to the long memory property. The finding of long
memory in high frequency data is important; since it is very unlikely that regime shifts
continuously occur every few weeks or months, it seems that the long memory property is
an intrinsic feature of the system rather than being due to exogenous shocks which lead to
regime shifts.

The plan of the rest of this chapter is as follows; section 2 discusses the
application of the long memory volatility, FIGARCH model to daily and still lower
frequency data. This model is found to be econometrically superior to regular stable
GARCH models. The FIGARCH model for the daily data is also important in
constructing the filtering procedure on the 30minute data. Section 3 discusses the basic
properties of the high frequency data and the presence of long memory and intra day
periodicity in the autocorrelation functions of the squared and absolute returns. The
Flexible Fourier Form (F FF ) filter is applied in an attempt to remove deterministic intra-

day periodicity. The estimates of MA(1)-FIGARCH(1,5,0) models for 30 minute returns,

10

one hour and up to eight hour returns are presented. The estimates of the long memory
volatility parameters are broadly consistent with returns being generated by a self similar
process. Section 4 then describes tests for non linearity, which tend to confirm the
appropriateness of the filtering procedure and the use of the FIGARCH approach. Section

5 provides a brief conclusion.

2.2. Analysis of Low Frequency Daily Returns

This section is concerned with the analysis of daily returns from 1979 through
1998 and the estimation of a FIGARCH model to describe daily volatility. The model for
daily returns provides an interesting comparison with the models for high frequency
results and throws some light on the possible self similarity of DM—$ returns. Also, the
model for the daily volatility process is required to filter the raw 30minute returns to
remove the strong intra day periodicity. The set of daily DM-$ spot returns used in this
study were provided by the Federal Reserve Bank of Cleveland for the sample period of
March 14, 1979 through December 31, 1998, which correspond to the origin of the EMS
(European Monetary System), and the relaxation of capital controls. Excluding weekends
and holidays, this realizes a sample of 4,989 daily observations. The autocorrelation
function of the daily returns, squared returns and absolute returns are plotted in figure 2.1.
Analogously to the 30 minute data, the autocorrelations of the squared returns and
absolute returns again exhibit the familiar slow, persistent decay; albeit without the strong
intra day periodicity.

The model that is postulated to describe the returns process is then,

11

 

(1) Rt = 100.Aln(St) = at + 98...,

(2) 8t = Zth,

(3) at = co + 13021.] + [1 - BL - (1 - «purl - D5182.

where zt is i.i.d.(0,1) and returns are specified to follow an MA(1) process, while the
conditional variance process czt, in equation (3), is represented by a F IGARCH

(Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic)
process, as developed by Baillie, Bollerslev and Mikkelsen (1996). The above
FIGARCH(1,8,1) process is sufficiently general that it can generate very slow hyperbolic
rate of decay in the autocorrelations of squared returns.

The F1GARCH(p,6,q) process for {at }can be defined by

[1 - B<L>iozi = w + [1 - B(L)- <p<L><1 - L)5]821,

where 0 < 5 < 1, and all the roots of [l - B(L)] and q)(L) lie outside the unit circle. Thus,

the conditional variance of at is simply given by

at = coil - 13(1) 1" +{1-[1-B(L)1"<p(L)(1-L)5]}82t

= coil -B(1)1" + Most.

12

where ML) = ML + XZLZ + . For the F1GARCH(p,5,q) process to be well defined and

the conditional variance to be positive almost surely for all t, all the coefficients in the

infinite ARCH representation must be nonnegative; i.e., Wk 55 0 for k = l, 2, . The

second moment of the unconditional distribution of at is infinite, and the F IGARCH

process is clearly not weakly stationary; a feature it shares with the IGARCH class of
processes. However, the FIGARCH(p,5,q) process is strictly stationary and ergodic for

0 S 5 .<_ 1. This can be proved by the direct extension of the proofs for the IGARCH case
which are presented in Nelson (1990) and Bougerol and Picard (1992). When 5 = 0, p =
q = 1, then equation (3) reduces to the standard GARCH(1,1) model; and when 5 = p = q
= 1, then equation (3) becomes the Integrated GARCH, or IGARCH(1,1) model, and
implies complete persistence of the conditional variance to a shock in squared returns.

The FIGARCH process has impulse response weights,

0% = m/(1 - B) + Moat.

where M . kd'], which is essentially the long memory property, or "Hurst effect" of

hyperbolic decay. This FIGARCH process implies a slow hyperbolic rate of decay for
lagged squared innovations and persistent impulse response weights. The attraction of
the FIGARCH process is that for 0 < 5 < 1, it is sufficiently flexible to allow for
intermediate ranges of persistence. For the covariance stationary GARCH model and the
FIGARCH model with 0 < 5 < 1, shocks to the conditional variance will ultimately die

out in a forecasting sense. However, there are important differences in the shock

13

dissipation for d = 0 and 0 < 5 < 1. Whereas shocks to the GARCH process die out at a

fast exponential rate, for FIGARCH model, M will eventually be dominated by a

hyperbolic rate of decay. Thus, even though the cumulative impulse response function
converges to zero for 0 at 5 < 1, the fractional differencing parameter provides important
information regarding the pattern and speed with which shocks to the volatility process

are propagated. The simpler F1GARCH(1,5,0) process is of the form,

oz.= w +BGZ,-I+[1-BL -<1- Lfiszt,

and has corresponding impulse response weights, ozt = (1)/(1 - B) + k(L)szt; and for large
lag k, M . [(1-B)/F (d)]kd'1. The equations (1) through (3) are estimated by using non-

linear optimization procedures in GAUSS to maximize the Gaussian log likelihood

function,

(4) ln(€) = -<T/2)1n(2n) - (1/2)2F1,Ti1n(ozt)+ 82162.].

Since most return series are not well described by the conditional normal density in (4),
subsequent inference is consequently based on the Quasi Maximum Likelihood

Estimation (QMLE) technique of Bollerslev and Wooldridge (1992), where

(5) T“2<'é~r-eo) =>N{0, A<eo>"B(eo>A(eo>"}.

l4

 

and A(.) and B(.) represent the Hessian and outer product gradient respectively; and 00
denotes the true parameter values.

Results of the estimated models for the DM-$ returns one day through seven days
of temporal aggregation are presented in table 2.1. Hence the returns are computed every
k days, where k = 1, 2,...., 7. The estimate of the long memory parameter, 5, for daily data
is 0.38. This estimate is very close to a semi parametric estimate of the long memory
parameter obtained for the absolute values of daily DM-$ returns by Andersen and
Bollerslev (1997b). Various tests for specification of the daily model were performed.
Especially, the sample period for the daily returns model includes some periods of
financial market crisis, such as the equity market meltdown of October 19, 1987 and the
EMS crisis of September, 1992. Consistent with other studies, I regard these episodes as
being part of the same generating process, rather than signaling a shift to a new regime.
For this reason, I resist including dummy variables or any other mechanism of inducing a
"better fit" to the sample period. In particular, a robust Wald test of a stationary
GARCH(1,1) model under the null hypothesis versus a FIGARCH(1,5,1) model under the
alternative hypothesis has a numerical value of 25.37, which shows a clear rejection of
the null when compared with the critical values of a chi squared distribution with one
degree of freedom. Hence there is strong support for the hyperbolic decay and persistence
as opposed to the conventional exponential decay associated with the stable GARCH(1,1)
model. Tests of model diagnostics are performed by the application of the Box-Pierce

portmanteau statistic on the standardized residuals. The standard portmanteau test statistic
Qm = sz=1,m rJ-z, where rj is the j-th order sample autocorrelation from the residuals is

known to have an asymptotic chi squared distribution with m-k degrees of freedom,

15

where k is the number of parameters estimated in the conditional mean. Similar degrees
of freedom adjustment are used for the portmanteau test statistic based on the squared
standardized residuals when testing for omitted ARCH effects. This adjustment is in the
spirit of the suggestions by Diebold (1988) and others. A sequence of diagnostic
portmanteau tests on the standardized residuals and squared standardized residuals failed
to detect any need to further complicate the model.

Table 2.1 also shows that the estimates of 5 are statistically significant at the .05
percentile for one through to seven days. In a recent study of ten years of high frequency
DM—$ and Yen-S returns, Andersen, Bollerslev, Diebold and Labys (2000) have
constructed model free measures of volatility for different temporal aggregations and
conclude in favor of significant volatility clustering, i.e. ARCH effects, for monthly data.
Their finding contrasts with previous studies by Diebold ( 1988), Baillie and Bollerslev
(1989) and Christoffersen and Diebold (1998), who tended to find that monthly exchange
rate returns were close to being Gaussian and independently distributed. However, as
noted by Andersen, Bollerslev, Diebold and Labys (2000), their measure of integrated
volatility should remain highly serially correlated even at a monthly level. The results
reported in table 2.1 are consistent with the notion of self similar returns process with the
same long memory volatility parameter, 5, up to seven days. Although the estimated 5
parameter varies from .38 to .23 the range of values is well within the two robust standard
error bands. Drost and Nijman (1993) have provided a theoretical treatment of the effect
of temporal aggregation of an underlying high frequency GARCH(1,1) process. As yet no
corresponding results exist for the FIGARCH process. Although this evidence is not

conclusive, it does seem that ARCH effects may be present in low frequency return data.

16

2.3. Analysis of High Frequency Returns

This section is concerned with the set of 30 minute DM-$ spot exchange rate data
provided by Olsen & Associates of Zurich, in which Reuter FXF X quotes are taken every
30 minutes for the complete calendar year of 1996. The sample period is 00:30 GMT,
January 1, 1996 through 00:00 GMT, January 1, 1997. Each quotation consists of a bid
and an ask price and is recorded in time to the nearest second. Following the procedures
of Muller et a1 (1990) and Dacorogna et al (1993), the spot exchange rate for each
30minute interval is determined as the linearly interpolated average between the
preceding and the following quotes. Hence the 30minute return series is defined as the
difference between the midpoint of the logarithmic bid and ask rates. For example, if at
time 0:30:00, the preceding bid-ask price pair is 1.4334-1.4341, and the following quote

is 1.4330-1.4335, then the interpolated exchange rate (Sm) at 0:30:00 would be

(6) 3,,“ = exp{(1/2)*[ln(l.4334)+ln(1.434l)] + (1/2)*[ln(1.4330)+1n(1 .4335)]}.

Then the n-th 30 minute spot return for day t is, R1,“ = ln(St’n)-ln(St,n_1). It has become

fairly standard in this literature to remove atypical data associated with slower trading
patterns during weekends. Hence returns from Friday 21 :00 GMT through Sunday 20:30
GMT are excluded. However, returns for holidays occurring during the sample are
retained in order to preserve the number of returns associated with one week. In
particular, the eventual sample used in subsequent analysis contains 262 trading days,
each with 48 intervals of 30 minute duration; which realizes a total of 12,576

observations for the DM-$ returns for the 262 days.

17

Figure 2.2 plots the first 240 autocorrelations for the returns, squared returns and
absolute returns of the unadjusted (raw) 30 minute DM-$ exchange rates for 1996. The
usual Tm asymptotic standard errors for the sample autocorrelations are not strictly valid
for a process with ARCH effects and are no more than useful guidelines. As presented in
many previous studies with high frequency data such as Andersen and Bollerslev (1997a),
Goodhart and F igliuoli (1992), Goodhart and O'Hara (1997) and Zhou (1996), there is a
small, negative but very 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. However, the autocorrelation functions of the
squared and absolute returns exhibit a pronounced U shape pattern, associated with
substantial intra day periodicity. Similar U-shaped patterns can be also found in the equity
markets, see Harris (1986), Wood et al.(1985), Chang et al.(1995) and Andersen and
Bollerslev (1997a). Similarly to the findings of Granger and Ding (1996), this pattern is
particularly strong in absolute returns; and the general pattern is consistent with the
studies of Wasserfallen (1989), Muller et a1 (1990), Baillie and Bollerslev (1991),
Dacorogna et al (1993) and Andersen and Bollerslev (1998). The pattern is generally
attributed to being due to the effects of the opening of the European, North American and
Asian markets superimposed on each other. This U-shaped pattern of volatility has been
previously noted by Foster and Viswanthan (1990). A further representation of this
phenomenon is provided by figure 2.3, which shows the absolute 30minute returns for
each of the 48 intervals, averaged over all the days in the year. The highest average
absolute returns occur between periods 26 and 34, which correspond to 1:00pm and

5:00pm GMT. It shows a great difference in the volatility over the day, and the pattern is

18

closely related to the market activities of the various financial centers around the world.
In order to remove the strong intra day periodicity, this study follows a similar

approach as Andersen and Bollerslev (1997b), and uses a two step estimation method,

whereby the intra day periodicity is first removed by applying Gallant's (1981, 1982) F F F

approach. In particular,

(7) Rt,n = B(Rt,n) + (or St,n Zt,n NW2),

where E(Rt,n ) is the unconditional mean of returns, 01 is the conditional variance of

daily returns, st,“ is a deterministic function to represent intra day seasonality, 2”, is an

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

number of return intervals per day. From equation (7), we define,

xtn = ZlnllRm - Edit,“ )I] - 1n<oz t) + 1n<N> = ln(sz in.) + WZ tn ).

Thus, the variable, xm can be generated by replacing B(RLn ) with the sample mean of
the returns, Rt," and 62¢ with the estimates, ozt from a daily volatility model. Then the
generated variable, 32,,” is regressed on a non linear function of the time interval n,

on day t; i.e.

321,11 : f(e; ta n) + ut,n9

19

where um = ln(zzt,n ) - E[ln(zzt,n )] is an i.i.d.(0,1) process and the functional form of

n .
Xt’n IS,

(8) f(9; E II) = No + Min/Ni + Hznz/Nz + AkaOJI) + 23k=i.3 GkaG, 11-10

+ Zp=1,k[5c,p.cos(p2rtn/N) + 5S,p.sin(p21mfN)],

where N1 = N"zi:.,N i = (N+1)/2, N2 = N“2,=,,N i2 = (N+1)(2N+1)/6, and Ik(t,n) is an
indicator variable which represents the occurrence of an event k on day t at interval n.
These events include US. economic announcements of retail sales, trade balances,
unemployment, and Producer Price Index (PPI) and Consumer Price Index (CPI). The
indicator function is equal to unity when an announcement of the above occurs and is

zero otherwise. After some experimentation it was also decided to include a lagged
indicator variable, Dk, which is unity for the two hours immediately following the event

and is zero otherwise. The dates and times for the US. economic announcements were

obtained from the section "Week Ahead" of the weekly magazine, Business Week. On
treating the variable xm as the dependent variable, the parameters in the equation (8) were

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

(9) 32.,“ = Truman/2)] / mum new exp(ftn/zn,

where ft,“ denotes the OLS predicted values in Eq (8). The 30 minute returns are then

20

 

filtered by the estimated intra day seasonality series, ASL”, to generate the filtered returns,

which are defined by

(10) Rt,“ = Rt,“ / ’sm .

Note that Andersen and Bollerslev (1998) used a little different filtered returns data,

(Rm - R) / am where R is the sample mean and st," is estimated intraday periodicity

component from the similar Flexible Fourier Form method. But, they found the

 

correlogram of the filtered absolute returns, | Rt’n - R l / 3t,“ , is very similar to that of the

filtered absolute returns, | Rt,“ | / '3‘”, , presented in Andersen and Bollerslev (1997b).

Now, the returns have been filtered to remove their intra day periodicity and it is possible
to analyze their properties. Figure 2.4 presents the autocorrelations of the raw, squared
and absolute filtered 30 minute DM-$ returns. It is clear that the intra day periodicity is
dramatically reduced in the autocorrelations of the squared and absolute filtered retum.
However, the autocorrelations also exhibit extreme persistence associated with the feature
of long memory.

A generalization of the daily model, which is also based on the continuously

compounded returns for sampling frequency k, includes an MA(1)-FIGARCH(1,5,1)

formulation,
(11) Rm = 11 + 8t,n + 98t,n-l a
(12) 8t,n : zt,n OIt,n 9

21

(13) can = w + Boztni + [1 - BL - <1 - <pL)(1 - L)618t,n2,

where zm is an i.i.d.(0,1) process, and where the two time indices are t = 1,.. 262 days,

and n = 1,.. 48/k intra day periods, where k = 1, 2, 3, 4, 6, 8, 12 and 16. Results for
estimating the above model for the six different frequencies over k, within the day are
presented in table 2.2. The estimated long memory volatility parameter, 5, is to be found
within the range of .14 to .24 for the 30 minute, one hour, 90 minutes, two hour, three
hour, four hour, six hour and eight hour sampling frequencies. The estimate of 5 is highly
significant for all the return series.

In many previous studies the GARCH(1,1) model has been used to represent the
volatility process. The results in table 2.2 indicate that the F IGARCH model generally
appears to be the more appropriate specification. For the 30 minute return series, a robust
Wald test of the stationary GARCH(1,1) null hypothesis versus a FIGARCH(1,5,0)
alternative hypothesis has a numerical value of 63.00, which provides an overwhelming
rejection of the GARCH(1,1) formulation. Hence, as with the daily data, the tests
generally imply strong support for the long memory F IGARCH model. Tests for mis-
specification do not reveal any obvious deficiencies with the model, and in particular
correlograms of the standardized residuals from the MA(1)-F1GARCH(1,5,0) models for
the filtered 30 minute returns are plotted in figure 2.5 and are generally indicative of
uncorrelated residuals. The estimates of the long memory parameter are relatively stable
across the high frequency data from 30 minutes to 8 hours; and in this sense are

comparable with the semi parametric estimates of long memory from 5 minute absolute

22

 

returns obtained by Andersen and Bollerslev (1998).

Again it is worth noting that an interesting implication of the robustness of the
F IGARCH model and importance of the long memory volatility parameter on relatively
short spans of high frequency data, strongly suggests that the long memory property is an
intrinsic feature of the system rather than being due to exogenous shocks which lead to

regime shifts.

2.4. Nonlinear Dependencies in High Frequency Returns

The above analysis has used the FFF adjustment to remove intra day periodicity
and has used the MA(1)-FIGARCH(1,5,0) process to represent the high frequency
returns. However, it is still possible that higher order non linearities are present in the
high frequency returns and are not captured by the model. Even though the F FF method
appears to be very successful in removing the intra day periodicity which is closely
related to the cycle of the market activity in figure 2.2, no one procedure is clearly
superior. Dacorogna et a1. (1993) used the sum of three polynomials corresponding to the
distinct geographical locations of the foreign exchange market. As Andersen and
Bollerslev (1997b) suggested, there may well be some cyclical correlated components left
in the [Rnnl series so that they may induce spurious short run dynamics in the return
volatility. Note that in figure 2.4, there still remains very small periodicity at every 48
intervals, the daily frequency and also a slightly higher peak at intervals 240 which is
about 5 days or 1 week, implying the minor day of the weekly effect which is not
considered in this study. Hence it is of interest to apply further tests for omitted non linear

effects. One way of searching for temporal dependence in time series data is the

23

 

estimation of the correlation dimension, which is based on the notion of correlation
integral. For a given time series {x(t): t = 1, ...., T} of D dimensional vectors, the so-

called correlation integral C(e) is defined as

C(S) = LimT—>oo[2/T(T'1)] 21.90% xj),

where 18(x,y) is an indicator function which is equal to one if ”X — y|| < s, and is zero
otherwise, and where H.” denotes the sup-norm. Intuitively, the correlation integral C(e)
measures the fractions of the pairs of points {x(t)} that are within an a distance from
each other. If the series is really random, then the correlation integral will diverge
without band as the dimension increases. This method considers the possibility that the
nonlinearities in the high frequency returns are produced by a low dimensional chaotic
attractor. This estimation of correlation dimensions is a method widely used to test for
chaos in time series data. It is based on the Grassberger-Procaccia correlation integral.
Grassberger and Procaccia (1983) define the correlation dimension of the time series

{x(t)} of an embedding dimension M as,

DM = Limg_,0 [ln{C(s)}/ln(e)].

The correlation dimension technique produces estimates and uses graphical analysis,
which typically requires very large data sets, that are more common in physics, as

opposed to economics and finance. The technique has been implemented on economic

24

data by Ramsey et a1 (1990) and others. As noted by Hsieh (1989) and Ramsey et al
(1990), the correlation dimension estimated from small sample sizes can be very
misleading, and has a downwards bias, thus increasing the probability of erroneously
concluding in favor of finding low-dimensional chaos. Table 2.3 reports the correlation
dimension estimates for the residuals from the model given by equations (11) through
(13) for the 30 minute returns. Liu, Granger and Heller (1992) and Cecen and Erkal
(1996) have discussed the interpretation of these estimates. It is clear in table 2.3 that
there is little convergence in dimension estimates, and they do not appear to indicate any
strong low dimensional deterministic dependence in the residuals.

A further measure of nonlinear dependence is the BDS test, of Brock, Dechert,
Scheinkman and LeBaron (1996). This test attempts to distinguish between an i.i.d. series
and a series with deterministic or stochastic dependence. Given a time series xt, for t = 1,

...., T which is an i.i.d. sequence, it can be shown that

Cn(8) = Ci(8)",

where Cn(s) is the correlation integral. By estimating C1(s) and Cn(s) by the sample

values C 13(8) and Cn,T(e), the BDS test statistic can be written as,

(14) Ema) = Twictire) - Cristi/omits).

where on,—r(e) is an estimate of the asymptotic standard error of the numerator in equation

25

(14). Then under the i.i.d. null hypothesis, Brock et a1 (1996) prove that B“; I N(0,1). The
embedding dimension, M, was chosen to be in the range of 2 through 10, while a was
fixed in the range of .25 through 1.25. Table 2.4 reports the BDS test results for the
residuals fiom the estimated model in equations (11) through (1 3) from the 30 minute
filtered returns. Except for the value of s = .25 and M $ 6, the test statistics consistently
fail to reject the i.i.d. null for the residuals. Overall there is no evidence from these tests
for non linearity to indicate model mis-specification. Hence there is no reason to doubt
the validity of the FFF filtering procedure and appropriateness of the estimated MA(1)-

FIGARCH(1,5,0) model.

2.5. Conclusion

This chapter has considered one year of high frequency DM-$ returns, and also
twenty years of daily and lower frequency data. The 30 minute returns series were filtered
by the Flexible Fourier Form (FFF) method to remove intra day periodicity. The long
memory volatility processes, FIGARCH, is found to provide a good representation of
both the high frequency and the daily DM—$ returns data. Two tests for non linearity are
presented to fiirther test the appropriateness of the FF F filtering procedure and also the
imposition of the MA-FIGARCH model. The residuals from the model are found to be
free of any significant non linear effects, and the F IGARCH model appears to
successfully account for the dynamics of the return series. Interestingly, the estimates of
the long memory volatility parameter in the FIGARCH models are very close across time

aggregations, suggesting that long memory volatility is an intrinsic feature of the system.

26

Table 2.1 : Estimated MA(1)-FIGARCH(p,5,q) Models for Daily DM-S Returns
R = 100*2i=(t-1)k,tk[ln(Si) - ln(Si-1)] = u + at +98“
ct = fit (it where Q is i.i.d.(0,1) process
6.2 = (o + not} + [1 -BL - (1 — (pL)(l — L)5]et2

where t = 4989/k, k = 1,....,7

 

 

l-day 2-day 3-day 4-day 5-day 6-day 7-day
no of obs. 4989 2494 1663 1247 997 831 712
p. 0.0028 -0.0026 -0.0006 0.0116 0.0104 0.0671 -0.0040
(0.0089) (0.0188) (0.0303) (0.0411) (0.0484) (0.0628) (0.0775)
0 0.0126 0.0074 0.0475 -0.0005 0.0791 0.0993 0.1256
(0.0150) (0.0206) (0.0264) (0.0270) (0.0313) (0.0367) (0.0403)
5 0.3826 0.3475 0.2665 0.2907 0.2395 0.2370 0.2323
(0.0760) (0.0916) (0.0753) (0.0839) (0.0740) (0.0750) (0.0841)
(1) 0.0207 0.0865 0.1944 0.2309 0.3962 0.4831 0.5865
(0.0088) (0.0311) (0.0811) (0.1010) (0.1826) (0.2302) (0.2789)
B 0.5803 0.2672 0.2106 0.2352 0.1593 0.1544 0.1341
(0.0833) (0.0981) (0.0792) (0.0863) (0.0794) (0.0776) (0.0808)
(p 0.2943 - - - - - -
(0.0580) - - - - - -
ln(L) -4968.454 -3388.357 -2606.957 -2162.524 -1829.959 -1598.865 -1415.321
Skewness -0.142 -0.082 -0.021 0.030 -0.051 0.027 -0.039
Kurtosis 4.496 4.441 4.102 4.429 3 .766 3 .376 3 .668
Q(20) 35.209 33.710 24.705 23.157 15.653 10.432 23.672
Q2(20) 14.380 14.792 14.642 17.145 20.664 17.621 15.917

 

Key : ln(L) is the value of the maximized log likelihood. The numbers in parenthesis
indicate the asymptotic robust QMLE standard errors of the corresponding parameter
estimates. The Q(20) and Q2(20) are the Ljung-Box test statistics at 20 degrees of
freedom based on the standardized residuals and squared standardized residuals. The
sample skewness and kurtosis are based on the standardized residuals.

27

 

 

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29

Table 2.3: Correlation Dimension Estimates on the Residuals from the MA(1) -
FIGARCH(1,5,0) Model for Filtered 30 minute DM - $ Returns

 

 

M C(M)
1 0.69
2 2.07
3 2.99
4 3.75
5 4.30
6 4.82
7 5.19
8 5.48
9 5.67
10 5.80

 

Key : M is the embedding dimension and C(M) is the correlation dimension
statistic estimated for dimension M.

Table 2.4: BDS Test on the Residuals from the Estimated MA(1)-
FIGARCH(1,5,0) Model for Filtered 30 minute DM - $ Returns

 

 

8 0.25 0.50 0.75 1.00 1.25
M = 2 -0.46 -0.12 -0.67 -0.07 0.12
M = 3 -0.51 -0.04 -0.07 0.03 0.30
M = 4 -0.12 -0.13 -0.25 -0.10 0.19
M = 5 1.39 -0.13 -0.36 -0.25 -0.03
M = 6 3.79 -0.07 -0.31 -0.30 -0.15
M = 7 7.72 —0.27 -0.35 -0.31 -0.17
M = 8 23.30 -0.27 -0.44 -0.35 -0.20
M = 9 63.81 -0.08 -0.44 -0.39 -0.24
M = 10 198.27 0.54 -0.47 -0.43 -0.34

 

Key : M is the embedding dimension and a is the distance.

30

 

 

 

 

 

 

 

 

 

 

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33

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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35

[Chapter 3]

Application of Long Memory GARCH Models with Jump Processes to Describe
Target Zone Exchange Rates

36

 

This chapter is concerned with the econometric modeling and the specification of
models to describe exchange rates in target zone and some international finance issues
concerning the interpretation of the model. Many of the issues and models in this chapter
are related to those in Chapter 2. Since many of the world's exchange rates are not freely
floating, but are in managed regimes or target zones, it is of interest to see how the

methods described in Chapter 2 work in this context.

3.1. Some Stylized Facts of Nominal Exchange Rates in Target Zones

Since March 1979, most of the nations of the European Union have participated in
a "Target Zone" system of exchange rate management known as the Exchange Rate
Mechanism (ERM) of the European Monetary System (EMS). So, each currency has a
central rate expressed in terms of the European Currency Unit. These central rates
determine a grid of bilateral central rates, around which fluctuation margins are
established. To keep the exchange rate within these margins, participating countries are
obliged to intervene in the foreign exchange market when the bilateral exchange rate
reaches the boundary of this band. Instead of intervening, it is also possible to realign the
parities, if all of the members of the EMS agree. See Vlaar and Palm (1993) and Neely
(1998) for the details of target zone model.

While the most appropriate model specification for a target zone exchange rate is
a debatable issue, there is some reduced form or stylized facts that are worth
consideration. This is worth tabulating for future work. In particular, there are some
striking and important differences between the stylized facts of exchange rates within a

target zone and freely floating exchange rates. Form the studies by Baillie and Bollerslev

37

(1989), Hsieh (1989) and others, the following stylized facts appear to hold for freely
floating nominal exchange rates:

(i) are close to martingale difference sequences with daily returns being almost
uncorrelated,

(ii) have persistent volatility processes for relatively high frequency returns, with
close to symmetric volatility processes,

(iii) possess excess kurtosis in an almost symmetric unconditional returns density,

(iv) have approximately independent and Gaussian distributed unconditional
returns for monthly and still lower frequencies.

From the studies of Vlaar and Palm (1993) and Neely (1998), the following
contrasting facts hold for nominal exchange rates in a Target Zone:

(a) have negatively autocorrelated daily and weekly returns,

(b) have very persistent volatility process for relatively high frequency returns,

(c) have excess kurtosis in a skewed and non-symmetric returns density,

(d) and have returns that are subject to jumps and realignments.

At the minimum, one challenge of any model of an exchange rate in a target zone

is to be consistent with the above stylized facts.

3.2. Analysis of FF - DM Exchange Rates

In a recent study, Chung and Tauchen (2000) analyzed the F F -DM spot exchange
rate. By using the EMM (Efficient Method of Moment) proposed by Gallant and
Tauchen (1996), they estimated an implicit band of the exchange rate. They treated the

upper and lower bands as unobservable parameters and estimated them from the observed

38

data. Then they tested the Krugman's target zone model (1991) and the Delgado and
Dumans' intra-marginal intervention model (1991) based on the implicit band. Also, they
examined whether the F F-DM exchange rate under the ERM can be characterized in
terms of a stochastic differential equation system. They found some evidence that a
model with intra-marginal intervention and an implicit band can describe the dynamics of
the F F -DM exchange rate process but the stochastic volatility model which is a
continuous time model without a band, is not appropriate for the description of the
observed data. They used the data recorded from July, 1987 through July, 1993; and use
342 observations at the weekly frequency, to "avoid weekend effects". It is somewhat
unclear as to the nature of the "weekend effects" in this context. The full sample of the
FF -DM after capital controls had been relinquished, until the advent of the Euro, was
from March 14, 1979 through December 31, 1998. As can be seen in figure 3.1, which
extends to October 10, 1995, the FF-DM rate started at around 2.3 when capital controls
were released, and the French franc consistently depreciated until 1987 when it plateaud
at approximately 3.4 until the dawn of the Euro. It is probably important to also include
the period around August and September 1993, when there was so much interesting
activity in the market, with the FF almost being forced out of the ERM system, although
clearly the change in the nature of the bands might well affect the level at which
intramarginal intervention would occur.

Similarly to Chung and Tauchen (2000), Ball and Roma (1993) and others have
also favored continuous time models and in particular have used jump diffusion processes
and the Ornstein-Uhlenbeck (O-U) process. Ball and Roma (1993) proposed a general

continuous time bivariate jump-diffusion representation for the exchange rates of

39

 

 

 

European currencies. They modeled the fluctuation within bilateral limits by appropriate
diffusion dynamics and posited discontinuous variation in the level of the fluctuation
band to have a jump structure. Comparing the fit of alternative models, they found some
evidence of mean reversion inside the bands for the exchange rates. I

Efficient estimation of the parameters in continuous time processes is generally
challenging and to give an alternative perspective on the continuous time formulation, it

is also interesting to consider a model in discrete time, provided by the relatively simple

formulation,

(1) y. = 100*A1n<s.)=s. +92 .-1 +u+1~vt,

(2) 8t = Zt 0t,

(3) oz. = w +1361] + [1 -BL - (1 -<pL)<1 -L)"]ezt,

where zt is i.i.d.(0,1), and St is the FF-DM spot exchange rate. Hence the FF-DM returns

are specified to follow an MA(1) process, with a jump intensity parameter of 9», where

0 < 9» < 1, and is specified to be generated by either a Bernoulli or Poisson distribution. In
Bernoulli distribution, the success probability (a) of jumps is constant from one trial to
the next and the successive trials are independent. So, the distribution for x successes of
jumps in n trials is the Binomial distribution. Then, the mean and variance of x are no

and na(1 - a), respectively. But, in Poisson distribution, the number of trials becomes

40

 

large at the same time and the success probability of jumps becomes small so that the
mean no. is stable. Then, the distribution has the limiting form of the Binomial
distribution. Vlaar and Palm (1993) presented that the results from both Bernoulli
distribution and Poisson Distribution are very similar but the Bernoulli distribution is
preferred because it does not contain the infinite sum and it is easier to estimate. The

jump size is given by the random variable V, which is assumed to be NID(v, 52). The

above model is an amalgam of the features of previous target zone exchange rate models
and floating rate models. In particular, the model is in part motivated by Vlaar and Palm
(1993) in using a jump diffusion GARCH process to model combined volatility dynamics
and realignments. A subsequent study by Nieuwland, Verschoor and Wolff (1994)
considered the mean reversion properties of ERM rates with excess kurtosis; and used a
jump GARCH model with a conditional t density. Neely (1999) focused on information
about the credibility of the target zones to allow a time varying probability for the jump
diffusion GARCH models. The volatility process is a FIGARCH(1,5,1) conditional

variance model 02k as given by Baillie, Bollerslev and Mikkelsen (1996) and as

described in Chapter 2. The equations (1) through (3) are estimated by using non-linear
optimization procedures to maximize the Gaussian log likelihood function. As presented
in Vlaar and Palm (1993), the log likelihood function for Bernoulli distribution has the

following form,

ln(cb) = -(T/2)1n(2n) + >3 t=1,T1n{[(1 - Wot 1*expt-(e. + tvf/zozt 1 +

[Molt + 52 omrexpi-(et - (1 Joel/moi + 5202]}.

41

And, the log likelihood function for Poisson distribution can be written as,

ln(c.» = -Tx - (T/2)1n(2n) + 2. =1,T1n{zj=o,4 MU «oz. + 82. pm]

*exp[-(8t - (i -x)v>2/2(02. + 62.91}.

Virtually all previous studies have restricted attention to weekly data, before the
September, 1992 crisis. In order to illustrate the magnitude of the non linearities involved
in this area, the above model was estimated for daily covering the period March 14, 1979
through October 10, 1995, and also for the same weekly data used by Chung and Tauchen
(2000) . The form of the likelihood under the Normal-binomial and Normal-Poisson jump
processes is basically the same as given by Vlaar and Palm (1993), with some
straightforward extensions to include the conditional variance process in our formulation.
Asymptotic standard errors are calculated from QMLE as before. The results of the
estimated models for the daily data are reported in table 3.1. Many of the intrinsic
features of the F F-DM exchange rate behavior are captured quite well by the model.
There is evidence of significant negative autocorrelation and hence predictability of
returns. The estimated model also exhibits very persistent volatility dynamics, and
extreme excess kurtosis. The models in the second and third columns of table 3.1 also
include a jump intensity parameter, 9», is allowed to follow either a Bernoulli or a Poisson
distribution. For the Bernoulli distribution, )M is forced in the (0,1) interval by estimating a
parameter 7t = [l + exp(k)]". The interpretation of the estimated 7t parameter of .036
indicates the probability of a jump occurring; so that within the sample of 17 years of

4,151 daily observations, the implied number of days with a jump is 149, which is

42

approximately 9 per year. Table 3.2 presents some corresponding results for precisely the
same period of weekly data as analyzed by Chung and Tauchen (2000) . As expected, the
use of weekly data results in a less persistent volatility process with less departures from
Gaussianity. With the weekly models the estimated 1» parameter is .18, which implies a
total number of 62 jumps from the 7 years of 342 weekly observations; which is again
about 9 per year. One interesting issue concerns the interpretation of the jumps and
whether or not they correspond to realignments, or intramarginal interventions. Without
more detailed information on central bank behavior, it is difficult to distinguish these

effects.

3.3. Discussion for Future Research

The above univariate analyses on daily discrete time models and Chung and
Tauchen (2000) '5 continuous time version can be criticized on the grounds of being
"curve fitting". What do we really learn from these econometric estimations which is of
interest to international finance? Also, from a modeling perspective what is the

appropriate model for comparison here? The K-TZ, Krugman Target Zone model of,

St : kt + TEt(dSt/dt),

for r > O and with kt being Brownian Motion is something of a straw man for comparison
purposes, since there is considerable empirical evidence that the fundamentals, kt, are not
a random walk. So it may not mean too much to compare the Chung and Tauchen (2000)

model with the K-TZ model.

43

While finance theory is often traditionally based on continuous time processes;
there is frequently little direct motivation for this choice of modeling paradigm. In fact,
the interpretation of the type of model employed by Chung and Tauchen (2000) is not
completely straightforward. In particular, it is unclear how predictable returns are in the
Chung and Tauchen (2000) models. From the discrete time univariate analysis above, the
target FF-DM rate has considerable departures from martingale behavior. Second, the
type of persistence in the Chung and Tauchen (2000)-model is also unclear.

One interesting discovery of Chung and Tauchen (2000) is evidence for an
implicit band operating, which can be interpreted as intramarginal intervention. A
fascinating issue concerns how the intramarginal intervention is imposed, with the word
"intervention" having its broadest usage here. However, the route of sterilized
intervention of central bank's reserves, has generally been found to be seriously deficient
in terms of supporting econometric evidence; see Baillie and Osterberg (1997). Also, the
interest rate differential on 30 day Eurobonds as shown in figure 3.2, when combined
with the spot rate in studies of uncovered interest rate parity, is generally found to reveal
the classic forward premium anomaly, e.g. Mark and Wu ( 1997), so that specific interest
rate changes do not seem able to explain the FF-DM rate being successfully kept within
the bands. So the type of monetary policy information being used is unclear, as is the
most appropriate model specification for the fundamentals. Presumably, the types of
intervention being used by the French and German monetary authorities cover the broad
range of monetary policies. However, the Chung and Tauchen (2000) study is an

interesting step in modeling the continuous time dynamics of intramarginal intervention.

44

Table 3.1 : Estimation of Models for Daily FF-DM Spot Returns

Yt = 100*Aln(St) = 5t + 95M +14 + XVt,
at = ztot, where zt is i.i.d.(0,1).

cl. = 6 + 002.-. + [1 - BL - (1 -<pL)<1 -L)“1826

 

 

 

MA( 1) MA(1 )-Bemoulli MA( 1 )-Poisson
FIGARCH(1,5,1) GARCH(1,1) GARCH(1,1)
p -0.0016 -0.0007 .0.0014
(0.0024) (0.0018) (0.0019)
k - 3.3007 -
- (0.2993) —
x - [0.036] 0.0485
- - (0.0162)
1) - 0.1354 0.1263
— (0.0817) (0.0677)
52 - 0.9344 0.5683
- (0.4637) (0.2852)
0 -03243 03529 0.3535
(0.0379) (0.0187) (0.0188)
d 0.9386 - -
(0.1060) - _
(0 0.0027 0.0038 0.0038
(0.0008) (0.0008) (0.0007)
(1 - 0.2529 0.2553
- (0.0323) (0.0313)

 

45

Table 3.1 (continued)

 

 

 

MA(1) MA(1)-Bemoulli MA(1)-Poisson
FIGARCH(1,6,1) GARCH(1,1) GARCH(1 , 1)
[3 0.7167 0.5517 0.5313
(0.0631) (0.0638) (0.0652)
(0 0.1532 - -
(0.11 17) - -
ln(L) 650.430 1269.719 1276.777
Skewness 1.820 0.501 1.969
Kurtosis 28.162 36.394 28.230
Q(20) 34.977 38.294 40.743
Q2(20) 20.880 36.233 31.024

 

Key: Daily F F-DM spot returns are covering the period March 14, 1979 through
October 10, 1995 with a total of 4,151 observations. For the jump process, a jump
intensity of 2», where 0 < X < l, is specified to be generated by either a Bernoulli or
Poisson process. The jump size is given by the random variable Vt which is assumed
to be NID(v, oz). The rest of table is the same as table 2.1.

46

 

 

Table 3.2 : Estimation of Models for Weekly FF-DM Spot Returns

yt = 100*Aln(St) = at + Get.) +11 + kvt,

St = ztot, where z, is i.i.d.(0,1),

2
C51: 03 + (1321-1 + BGZt-l

 

 

 

MA(1) MA(1)—Bemoulli MA(l)-Poisson
GARCH(1,1) GARCH(1,1) GARCH(1,1)
p. 0.0039 .0.0253 -0.036O
(0.0120) (0.0231) (0.0202)
k - 1.5037 -
- (1.2288) -
7. - [0.182] 0.3659
- - (0.2782)
1) - 0.1630 0.1121
- (0.0969) (0.0661)
62 - 0.1750 0.0989
— (0.1305) (0.0601)
0 -O.1563 -O.1823 -O.1854
(0.0645) (0.0567) (0.0540)
00 0.0050 0.0103 0.0077
(0.0048) (0.0089) (0.0058)
6 0.1116 0.0951 0.0881
(0.0597) (0.0719) (0.0622)
B 0.8334 0.5719 0.5750
(0.0919) (0.2219) (0.2151)
ln(L) -37.245 -17.276 -16.414
Skewness 0.814 0.615 0.908
Kurtosis 6.104 3.063 7.185
Q(20) 18.019 18.069 17.705
Q2(20) 30.276 44.492 47.661

 

Key: Weekly FF-DM spot returns are covering the period July, 1987 through
July, 1993 with a total of 342 observations. The diagnostic statistics are as for

table 2.1.

47

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49

[Chapter 4]

Intervention in the 1920s Currency Markets

50

4.1. Introduction

The literature in international finance typically defines intervention as the actions
of central banks buying and selling currencies in the foreign exchange market in an
attempt to influence future currency movements. Since 1980 there have been many
occasions when there has been heavy intervention in freely floating markets such as the
DM - US. dollar and Yen - US. dollar, and also in many of the European ERM markets
where currencies are frequently in a dirty floating system between assigned bands. Much
of the empirical work in this area is for the DM - US. dollar rates where intervention data
are readily available. This has spawned considerable literature on the transmission
mechanism between sterilized intervention and the currency markets; e. g. signaling
effects, portfolio balance models and effects on risk premium have been analyzed by
Ghosh (1992), Dominguez and Frankel (1993), Kaminsky and Lewis (1996), Baillie and
Osterberg (l997a,b) and many others.

The historical origins of intervention as a policy tool is not entirely clear, although
it is known that during WWI, the bank J .P. Morgan intervened in the British pound
market at the request of the British Treasury. Also, in 1917 there were several incidents of
when the US. Treasury attempted to influence exchange rates. Prior to 1914 when the
Gold Standard was abandoned, there were no intervention in terms of currencies, since
central banks were principally concerned with the value of their currencies in terms of
gold.

This foreign exchange market in the early 19208 is one of the most interesting
periods in the history of international finance. First, it is one of the earliest recorded

episodes of deliberate central bank intervention in support of a currency. Second, it

51

represents one of the most turbulent periods in the history of foreign exchange rates,
which was significantly influenced by the hyperinflation in Germany and the apparent
spillovers to neighboring countries.

The plan of the rest of this chapter is as follows: section 2 describes some of the
relevant history of the 19203 foreign exchange market and the details of the French
intervention, or "bear squeeze" as it is sometimes known. Section 3 analyzes a relatively
new set of daily exchange rates for four countries during this period. This set of daily data
has some interesting parallels and differences with post Bretton Woods era data. Long
memory volatility processes, known as FIGARCH are found to be the most appropriate
specification for these currencies. Section 4 generalizes these models to include the
effects of intervention by the French government. There is strong evidence that the
intervention was successful for about six months before the FF again depreciated and lost
approximately 50% of its value. The overall conclusion in section 5 appears to be that
intervention while successful in the short run, was unable to change prevent substantial
depreciation over a six month horizon. Hence the episode of the 19205 can be viewed as
the first of many occasions when intervention is little more than a waste of the monetary
authorities reserves in an apparently futile attempt to only postpone an inevitable

depreciation.

4.2. The 1920s Foreign Exchange Market
Einzig (1937, 1962) provided an invaluable documentation of the main economic
and political events of the 19205, and also constructed weekly data for exchange rates.

The currency markets for this period were clearly less technologically sophisticated than

52

today's markets, and also lacked the highly developed derivative markets. Furthermore,
the total daily trading volume would have been minuscule in comparison with the
estimated daily turnover of $1,200bn for the late 19903. Although little is known about
the 19203 markets, it is almost certain that capital movements were a relatively small
percentage of transactions. Also, in the early 19203 the world economy was still
recovering from the effects of World War I, and there was the additional uncertainty of
whether the war reparations would be paid to France following the extreme hyperinflation
in Germany. The events in Germany, with its implications for the budget deficit and
future economic growth in France triggered substantial depreciation of the French franc.
The period beginning in early 1923 witnessed speculative attacks on the French franc and
several other European currencies. This led to the French government organizing a
number of "bear squeezes" in the hope of deterring future speculation, where secret loans
where negotiated from commercial banks to support the FF. The situation in 1924 is best
described by Einzig (1937, pp. 280-281), "The speculative campaign attained its climax
on March 11, when the franc touched 117 [per pound]. Then followed one of the most
memorable recoveries in the history of foreign exchange. It began with rumors of the
conclusion of credits abroad. These rumors were subsequently confirmed by the
announcement that a British banking group, headed by Lazard Brothers & Co., had
granted the French government a credit of ,4,000,000 and a few hours later an American
Banking Group headed by J .P. Morgan & Co. had granted a credit of $100m. The banks
acting as agents for the French government began to buy francs heavily in an over sold
market. Before very long francs become practically unobtainable. When speculators

realized that the game was up, many of them tried frantically to cover their short positions

53

 

at all costs.

The process of bear covering continued through April, and by the end of the
month the spot rate was under 68 and the forward discount has declined to about 600. for
three months. Even at that rate, however, it was undervalued compared with its discount
rate parity, which shows that many bears still refused to cut their losses and were carrying
their positions. Their views of the temporary nature of the recovery were justified by
subsequent developments. Following the defeat of M. Poincare at the General Election,
the franc became distinctly weaker, and by the end of May it was once more over 84 to
the British Pound."

From the information provided by Einzig and other observers at the time, it is not
completely clear if the French government intended or succeeded in sterilizing the
interventions. Since the French government negotiated loans from British and US.
commercial banks and then proceeded to buy FF, it seems clear the US. and British
money stocks were unchanged. The response from the French money supply is less clear.
If the French government increased its holding of FF it would contract the French money
stock and the intervention would not appear to be sterilized. However, the French interest
rates do not appear to have changed at the time of the intervention suggesting that the
interventions were probably sterilized.

Another feature of the early 19203 was the rapid depreciation of the German mark
following the severe hyperinflation and explosion of the money supply process in
Germany. In January 1922 there were approximately 800 marks to the pound; by May
1922 there were 1500 and by September 1923 the mark had depreciated to 45 million

marks to the pound. At this point the market ceased to be quoted. Following World War I

54

...-

.— —

“Hull-

 

the French government substantially increased its expenditures to repair the regions of the
country destroyed in the war. Subsequent domestic French inflation was compounded by
the difficulty in collecting war reparations from Germany and finally, in the early 19203,
international confidence in the French franc began to deteriorate and by November 1923
heavy sales of the franc occurred in the Amsterdam market, which quickly led to similar
activity in the London market. By March 1924 the French franc had depreciated almost
50 percent and on March 11, French Premier Raymond Poincaré launched a "bear
squeeze" by negotiating secret loans from US. and British banks, who then purchased
large quantities of francs. From a level of 117.00 francs to the pound on March 11, 1924,
the franc then appreciated to 89.81 francs to the pound the following week. Similar
events, leading to another bear squeeze, occurred in July 1926. The events surrounding
the FF were quickly transmitted to other the Belgian franc which rapidly depreciated in
February and March 1924. Einzig (1962) suggested in terms of the Belgian and French
francs sharing co-movements is due to their similar economic and political situations
while Einzig (1937) has suggested a linkage due to psychological factors. Also, Aliber
(1962) argued that investors over this period were influenced by Purchasing Power Parity

considerations.

4.3. Analysis of Daily Currency Returns

This study uses daily exchange rate data, the collection of which was organized by
the late Patrick McMahon from German newspapers. The series are from the London
market from May 1, 1922 through May 30, 1925 on the spot and 30 day forward

exchange rates of Belgium (BL), France (FR), Italy (IT), and the US. (US) vis a vis the

55

British pound. The observations include Saturdays and hence there are six observations
per week leading to a total of 966 observations for each currency. Phillips, McFarland
and McMahon (1996) used McMahon's daily data and considered tests of forward rate
unbiasedness. The technical innovation in their work was to implement F M-LAD (Fully
Modified Least Absolute Deviations) estimation in the forward rate unbiasedness
regressions which allowed for the extreme outliers in the data. The FM-LAD (Fully
Modified Least Absolute Deviations) estimation is a semiparametric procedure that treat
nuisance parameters like serial dependence effects in a non-parametric way but regression
coefficients parametrically. In this respect, this approach is quite different from the
quasi-maximum likelihood technique used by Baillie, Bollerslev and Redfearn (1993) to
cope with non-Gaussian data in a univariate GARCH analysis of weekly foreign
exchange returns. This study focuses on a different econometric aspect; namely the most
appropriate specification for the volatility process of the daily returns data. Within the
univariate models, the effects of intervention are also examined both in the mean and in
the volatility process.

Previously the only data was a series of 162 weekly observations from February
25, 1922 through March 25 , 1925 and recorded in Einzig (193 7). In particular, Britain,
Holland and Switzerland returned to a gold standard in 1925 so all the data series are
truncated at March 25, 1925 to facilitate comparability. This data was used by Baillie,
Bollerslev and Redfearn (1993) who found that the weekly returns were well described as
martingales with GARCH volatility processes, where the standard GARCH processes
appeared to be explosive in many cases. The use of the F IGARCH specification on the

daily data was found to alleviate this particular difficulty. Previous authors, such as

56

F renkel (1979) have used the monthly equivalents of the 1920s data to test for market
efficiency.

The graphs of the spot exchange rates for the daily FF, BF, IL and US are plotted
in figures 4.1(a) through 4.1(d). The autocorrelation function of the daily returns, squared
returns and absolute returns of the FF are plotted in figure 4.2. Corresponding
autocorrelations for the other three currencies are extremely similar and they are
represented in figure 4.3 through figure 4.5. While the returns of the FF are close to being
uncorrelated after the first lag, the autocorrelations of the squared returns and absolute
returns exhibit the slow, persistent decay that is typical of asset prices determined in
speculative auction markets, see Ding, Granger and Engle (1993). The model that is

postulated to describe the return process in this study is,

(1) Rt =100.A1n(s,)=e,+0e,_,,
(2) 81: Z106
<3) ozt=m+Bozn +11 -13L-<1 -<pL)(1 -L)5]ezt,

where zt is i.i.d.(0,1). As for in Chapter 2 and 3, we adopt a specification with an MA(1)

process, With condltlonal vanance process 0 t, in equation (3) g1ven by a

FIGARCH(1,5,1) process. Again QMLE technique is used for the subsequent inference.

Results of the estimated models for the daily 19203 spot exchange rate returns are

57

presented in table 4.1. The estimate of the long memory parameter, 8, for daily data is in
the range of .74 to .92 for the four currencies. Table 4.1 also shows that the estimates of 5
are statistically significant at the .01 percentile, with a robust Wald test of the stationary
GARCH(1,1) null hypothesis versus a FIGARCH(1,0,1) alternative being
overwhelmingly rejected. Hence there is strong support for the hyperbolic decay and
persistence as opposed to the conventional exponential decay associated with the stable
GARCH(1,1) model. The models reported in table 4.1 for the four currencies appear
satisfactory and do not suffer from any obvious misspecifications. Interestingly, despite
the primitive market conditions in existence in the 19205 and the extreme turbulence, the
estimated models in table 4.1 reveal temporal dependencies that are very similar in nature
to today's markets. A further comparison is provided by estimating corresponding models
for Einzig's (193 7) set of 162 weekly observations. The results are reported in table 4.2
and again robust Wald tests indicate the superiority of the F IGARCH specification over

the GARCH estimated by Baillie, Bollerslev and Redfearn (1993).

4.4. Effects of Intervention

There has been considerable recent interest in the use of central bank intervention,
as a policy tool by central banks in an attempt to affect exchange tares. In particular,
there has been very little evidence to suggest that intervention is directly effective in
relation to spot exchange rates, through either a monetary, or a portfolio balance
transmission channel. Instead, the prevailing evidence indicates that intervention can
sometimes affect spot exchange rates. In the case of the 19205 data, quite precise

information exists on the dates and occurrence of the intervention, but little is known

58

about the exact magnitudes. Hence the model used is of the form,

R1 = 100.Aln(St) = [(1/(1 ' )LL)]D{ + 81+ 08 {-1 ,

81 = Zt Gt,

oz. = no + 002... + [1 -BL - <1 -ngz

where 2, is i.i.d.(0,1), and D is the dummy variable for intervention and is one when

intervention occurs and is zero otherwise. In this specification the impact multiplier is 01,
the total multiplier is 01/(1 - 2»), and the mean lag is N(l - 7»). The estimation of
intervention models for the conditional mean are reported in table 4.3 and the estimated 01
parameter is found to be extremely significant for all four currencies. For France the
impact multiplier is -9.11 implying an appreciation of FF vis a vis the pound. The
estimated discount parameter, k, is .75, which implies the total multiplier of -36.44 for the
intervention on March 11, 1923. While the intention of the French intervention appears to
clearly have been to curtail intervention against the FF, it can be seen from table 4.3 to
have had corresponding effects on neighboring currencies. In particular, the intervention
also was successful in the short run of leading to appreciations of the BF and IL.
However, the intervention also lead to a depreciation of the US $ as speculative funds
were apparently withdrawn from the US $ to purchase the rapidly appreciating FF, BF

and IL.

59

Table 4.4 shows corresponding effects of the dynamic intervention variable in the

conditional variance. The model used is of the form,

R = 100.Aln(St ) = [010 /(l - 10L)]Dt + at + 98 (-1 ,

8t =Zt0ta

oz. = w +1362.-. + [Oh/(1 8.1310. + [1 -13L - (1 413182.,

where zt is i.i.d.(0,1), and D is the dummy variable for French intervention. The impact

multipliers on the conditional mean and the conditional variance process are 010 and 011
respectively. The estimation of intervention models for the conditional mean and the
conditional variance are reported in table 4.4. While the estimated 010 parameter for the
conditional mean process is found to be extremely significant for all three currencies, the
estimated 011 parameter for the conditional variance process is not significant at all. Thus,
the French intervention seems to have no effects on the conditional variance process. In
contrast to the results reported by Chang and Taylor (1998), Baillie and Osterberg
(l997a,b) and Goodhart and Hesse (1995) for the post Bretton Woods era, intervention in
the 19205 was not associated with either increasing or decreasing market volatility as

measured by the conditional variance.

60

4.5. Conclusion

This chapter examines some of the characteristics of the foreign exchange market
in the 19205 floating period and the effects of French intervention on the spot exchange
rates. The spot exchange returns for the four currencies (Belgium, France, Italy and US)
against the British Pound in this period appear to exhibit properties similar to those on
modern foreign exchange markets. They are well described by martingales and possess
very persistent time dependent heteroskedasticity. Again the long memory volatility
process, FIGARCH model is found to be more appropriate than the regular GARCH
model.

The effect of intervention by the French government to support the French franc is
represented by dynamic dummy variables. To examine the effects of French intervention
on March 11, 1924 the distributed lag operator is included in the model. The estimated
parameter for the intervention in the conditional mean process is found to be very
significant for all four currencies so that the intervention was successful for about six
months before the FF again depreciated. Thus, the French intervention appears to be very
successful in short run of appreciating FF and have corresponding effects on the BF and
IL, while it caused to depreciate US 55. In contrast, the corresponding effects of the
intervention on the conditional variance seems to be insignificant, implying the
intervention was not associated with the volatility of the currencies. Thus, these results
show that the intervention was successful in the short run, but was unable to prevent
substantial depreciation over a six month horizon so that they appear to cast some doubt
on the efficacy of intervention and the acceptance of intervention as a policy tool. This

episode of the 19205 can be viewed as the first of many occasions when intervention is

61

little more than a waste of the monetary authorities reserves in an apparently futile

attempt to only postpone an inevitable depreciation.

 

62

Table 4.1 : Estimated MA(1)-FIGARCH(p,6,q) Model for Daily Spot

Returns in the 19205
R = 100.Aln(St) = u + 81+ 98“,
at = zt ot where z”, is i.i.d.(0,1) process
6.2 = 0) + 1301.12 + [1 - BL (1 - <pL)(l — L)8]8(2

where t = l, , 966.

 

 

Belgium. France Italy USA
11 0.0462 0.0878 0.0351 -0.0004

(0.0223) (0.0228) (0.0148) (0.0058)
0 0.0885 0.0845 0.1262 0.1386

(0.0570) (0.0487) (0.0494) (0.0483)
5 0.9196 0.7354 0.6541 0.8644

(0.1756) (0.1833) (0.1338) (0.1728)
(0 0.0131 0.0183 0.0280 0.0043

(0.0074) (0.0130) (0.0117) (0.0014)
[3 0.7331 0.6478 0.3548 0.4932

(0.1567) (0.2192) (0.1319) (0.1307)
(0 - 0.2251 - -

- (0.1 122) - -

ln(L) -1404.226 -1301.236 -773.736 229.808
Skewness -0.331 0.252 0.347 1.088
Kurtosis 7.540 5.224 6.207 8.923
Q(20) 19.534 23.428 26.247 24.806
Q2(20) 16.390 22.901 9.072 7.110
W5=0 27.423 16.091 23.888 25.026

 

Key : As for table 2.1.

63

Table 4.2 : Estimated Martingale - FIGARCH(p,6,q) Model for Weekly Spot
Returns in the 19205

yt = 100*AIn(St) = H. + St

at = zt ot where 2!,“ is i.i.d.(0,1) process

o.2= co + 136.12 +11 -BL (1 -<pL)(1 - Lflef

wheret= 1, , 161

 

 

Belgium. France USA
1.1 0.2655 0.3937 0.0510
(0.1470) (0.1450) (0.0151)
5 0.5334 0.6179 0.6755
(0.0807) (0.2005) (0.1384)
0) 0.3047 0.3489 0.0594
(0.1099) (0.0185) (0.0296)
[3 0.9254 0.9310 0.2415
(0.0098) (0.0125) (0.3063)
(1) 1.1130 1.0998 -
(0.0445) (0.0341) -
ln(L) -386.855 -392.298 .142.193
Skewness 0.794 0.665 0.636
Kurtosis 4.546 4.430 5.620
Q(20) 21.214 21.519 22.938
Q2(20) 11.709 8.857 17.333
W5.—.0 57.068 60.246 4.051

 

Key : As for table 2.1.

64

Table 4.3 : Estimated MA(1)-FIGARCH(1,5,0) Model for Daily Spot Returns

in the 19205 with Dummy Variable in the Mean for France's
Intervention on March 11, 1924.

yt = 100*Aln(St) = [J + [a/(l ' )v L)] Dt + at + 98p]
at = zt ot where z”, is i.i.d.(0,1) process
of= w + 1301-12 +11 -131. -(1- Lfief

where t=1,...966 and D is one on March.11, 1924 and zero otherwise.

 

 

Belgium. France Italy USA
11 0.0466 0.0952 0.0372 -0.0011
(0.0221) (0.0239) (0.0030) (0.0058)
01 -6.8186 -9.1129 -1.1892 0.5405
(1.4279) (0.5716) (0.0315) (0.0184)
)1 0.8215 0.7580 0.7459 0.0548
(0.1225) (0.0381) (0.3840) (0.0257)
0 0.0874 0.0872 0.1285 0.1360
(0.0587) (0.0501) (0.0494) (0.0480)
6 0.9183 0.5581 0.6508 0.8684
(0.2013) (0.0926) (0.1330) (0.1687)
0) 0.0134 0.0245 0.0286 0.0042
(0.0074) (0.0274) (0.0121) (0.0013)
[3 0.7320 0.2563 0.3463 0.4998
(0.1791) (0.1181) (0.1303) (0.1317)
ln(L) -1401.358 —1292.666 -771.547 234.820
Skewness -0.339 0.270 0.367 1.077
Kurtosis 7.562 5.461 6.194 8.945
Q(20) 18.920 22.070 25.190 24.320
02(20) 16.872 35.703 9.371 6.793

 

Key : As for table 2.1.

65

Table 4.4 : Estimated MA(1)-FIGARCH(1,6,0) Model for Daily Spot Returns
in the 19205 with the Dummy Variable in the Mean and the Variance
for France's Intervention on March 11, 1924.
Yt = 11 + [010 /(1-7\0L)l D1+ 81+ 9881
at = zt at where zm is i.i.d.(0,l) process

6.2: (0 + [011 /(1—711L)] Dt + [36.3- + [1 - BL - (1-L)5]e.2

where t=1,...966 and D[ is one on March 11, 1924 and zero otherwise.

 

 

Belgium. France Italy
[1 0.0459 0.0872 0.0391
(0.0226) (0.0225) (0.0151)
(10 -4.7496 -8.5079 -0.9929
(1.4972) (0.5650) (0.4117)
M 0.9028 0.7944 0.8545
(0.0661) (0.0381) (0.1498)
0 0.0766 0.1063 0.1266
(0.0579) (0.0468) (0.0494)
8 0.9225 0.7741 0.6416
(0.1826) (0.2005) (0.1384)
00 0.0154 0.0486 0.0284
(0.0080) (0.0185) (0.0124)
011 1.6575 1.4064 0.2238
(1.6384) (0.8404) (0.2705)
1, 0.9605 0.9898 0.8855
(0.0135) (0.0075) (0.0376)
[3 0.7135 0.4335 0.3327
(0.1668) (0.2413) (0.1334)
ln(L) -1398.274 -1276.440 -769.771
Skewness -0.320 0.268 0.387
Kurtosis 7.847 4.514 6.239
Q(20) 19.199 22.898 25.802
Q2(20) 18.559 21.192 8.967

 

Key : As for table 2.1.

66

 

 

 

 

 

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74

[CHAPTER 5]

Modeling Inflation with Dual Long Memory Models

75

5.1. Introduction

Many previous studies have considered the properties of the univariate time series
representation of monthly inflation. A central issue in much of this research has been the
degree of persistence of the shocks, and is related to the controversy concerning the
possible existence of a unit root in inflation. In particular, Ball and Cecchetti (1990),
Brunner and Hess (1993), Barsky (1987) and Nelson and Schwert (1977), have argued
that US inflation contains a unit root so that shocks to inflation are completely persistent.

Alternatively, Hassler and Wolters (1995), Baillie, Chung and Tieslau (1996) and Baum,
Barkoulas and Caglayan (1999) have found evidence that inflation is fractionally
integrated. The fractionally integrated model implies that the autocorrelations and
impulse response weights of inflation exhibit very slow hyperbolic decay. The above
articles provide quite consistent evidence across countries and time periods that inflation
is fractionally integrated with a differencing parameter which is significantly different
from zero and unity. These studies have either estimated the long memory parameter by
semi-parametric procedures, or alternatively from estimating ARFIMA models.

The contribution of this chapter is to note that very similar long memory
properties are also present in the second moment of inflation. In particular, the squared
and absolute values of inflation residuals from applying a fractional filter to the
conditional mean, also possess long memory. An implication of this finding is that the
conditional variance of inflation can be modeled as a long memory ARCH process.
Hence inflation has the rather curious, and hitherto undetected property, of persistence in
both its first and second conditional moments.

The plan of the rest of this chapter is as follows; section 2 briefly summarizes the

76

standard ARFIMA model which has the long memory property in the mean. The model
is estimated for the CPI inflation series of eight different countries, including the US and
also for a new median weighted CPI series. These results support previous findings of
long memory in the conditional mean and investigation of the residuals of the model
provides evidence suggestive of similar long memory behavior in the squared and
absolute standardized residuals. Section 3 introduces a model that is sufficiently flexible
to handle the type of long memory behavior encountered in inflation; namely a hybrid
ARFIMA-FIGARCH model, which generates the long memory property in both the first
and second conditional moments of the inflation process. Section 4 then reports
estimates of ARFIMA-FIGARCH models for the eight separate countries CPI inflation
series and also for an alternative measure of inflation that has recently been proposed that
is based on the US median weighted CPI inflation. The hybrid long memory model is
generally found to be the most appropriate representation for the inflation series. The
estimated model implies the eventual mean reversion of both the conditional mean and

conditional variance following the impact of shocks.
5.2. Conditional Mean of Inflation
The conditional mean of inflation is assumed to be generated by ARFIMA(p,d,q)

model. Granger and J oyeux (1980), Granger (1980) and Hosking (1981), the

ARFIMA(p,d,q) model is defined as,

(I) <D(L)(1 -L)“(y- - u) = «Us.

77

where B(et) = o, 13(82t ) = 02 and B(e, at) = 0 for 5 ¢ t; and (D(L) = (1 - o-L opLP),
9(L) = (1 + 91L + ...+ OqL“) and have all their roots outside the unit circle. The Wold

decomposition, or infinite order moving average representation of this process is given by

yt = 2j=0,00\i,j 8t-j’

and the infinite order autoregressive representation is given by

Yt = 2120,00 79' Yt-j + 81-

F or high lag j, these coefficients decay at a very slow hyperbolic rate, i.e. Wj . cljd'1 and rt]-

. Oz] d". Similarly, the autocorrelation coefficient at large lagj is pj . c3] 1, where c], 02

and C3 are constants. For -O.5 < d < 0.5 the process is stationary and invertible and yt is

said to be fractionally integrated of order (1, or I(d). Hence the parameter (1 represents the
degree of "long memory" behavior for the series. For 0.5 S d < 1, the process does not
have a finite variance, but for d < 1 the impulse response weights are finite, which implies
that shocks to the level of the series are mean reverting.

Previous studies by Hassler and Wolters (1995), Baillie, Chung and Tieslau

(1996) and Baum, Barkoulas and Caglayan (1999) have considered inflation to be

fractionally integrated. The series of monthly inflation is defined as yt = 100.Alog(CPIt),

where CPI, is monthly Consumer Price Index. The Baillie, Chung and Tieslau (1996)

study used CPI series from January 1947 through September 1990. In this chapter, the

78

US series extends from January, 1947 through September, 1998, and the other countries
from February, 1957 through either September, 1998, or October, 1998; while the series
for Korea was from February, 1970 through October, 1998.

While it is traditional to measure inflation as the differenced logarithm of the CPI
series, some other definitions have also been proposed. In particular, Bryan and Cecchetti
(1999) and Bryan, Cecchetti and Wiggins (1997) have argued that since disaggregate
price data generally possess high kurtosis, the standard method of taking arithmetic
averages of prices may not be the most appropriate procedure for measuring inflation.
They suggest several alternative measures, including the trimmed mean and also the
median of the first differenced logarithm of the CPI series. In order to widen this study to
include a potentially interesting new measure of inflation, this chapter also considers US
median weighted inflation, which extends from January, 1967 through September, 1998.

The US CPI inflation series and the first 100 autocorrelations of the levels of US
inflation are plotted in figures 5.1a and 5.1b respectively; while the autocorrelations of
first differenced inflation are graphed in figure 5.1c. The autocorrelations of the inflation
series possess the very slow decay associated with fractionally integrated processes.
Furthermore, the autocorrelations of the differenced US inflation series in figure 5.10
display some negative values at low lags, which is strongly suggestive of over-
differencing. Very similar plots for the other countries are presented in figures 5.1d
through 5. 1 j.

The results from estimating ARFIMA models for the different countries are given
in table 5.1. For the US an ARFIMA(1,d,O) model was found to provide an adequate

representation of the inflation series, with the estimate of (1 being .39 and a robust

79

asymptotic standard error of .06. The other countries inflation series all exhibited

considerable seasonality. Accordingly, the seasonal ARFIMA model,

(2) <D(L)(1 - L>“(y- - u) = e(L>(1 - @132)...

was also estimated. Again (D(L) and 9(L) have all their roots outside the unit circle and
also -1 < 9 < 1. Interestingly, for two countries, France and the UK, the seasonality was
quite strong and the most parsimonious model was found to also require seasonally
differencing the inflation series before estimating the model in equation (2). This
transformation seemed necessary in order to produce a more parsimonious ARFIMA
model; while the application of tests for the presence of a seasonal unit root was
inconclusive, probably due to the power of the test statistics. Without the use of seasonal
differencing, a higher order seasonal ARMA structure was required. These models
contained more parameters than the seasonally differenced ones and the results are
reported in table 5.2. The use of the seasonal differencing operator did not significantly
change the estimated value of the long memory parameter for any of the countries. This
suggests that the estimates of the long memory parameter appear robust to specification
of the short run dynamics.

The point estimates of the long memory parameter in the conditional mean for the
regular inflation series were found to be in the range of .21 through .44. The model for
the new US median weighted inflation series was quite similar to the regular US CPI
inflation series, albeit with a slightly higher estimated long memory parameter of .59 and

rather less kurtosis in its standardized residuals. Inference in the estimated models is

80

based on QMLE, so that robust asymptotic standard errors for the estimated parameters
appear beneath corresponding MLEs. Robust Wald tests indicated overwhelming
rejection of both the d = O and d = 1 hypotheses. The application of QMLE in this study
differs to that of Baillie, Chung and Tieslau (1996) who estimated ARFIMA-
GARCH(1,1) models with conditional student t densities. The QMLE procedure is
described for the more general ARFIMA-FIGARCH model in section 5.3. While the
estimated models in table 5.1 appear to adequately describe the dynamics in the
conditional mean, the residuals clearly possess ARCH effects.

An indication of the rather unusual properties of inflation can be heuristically
observed from the autocorrelations of the residuals from the previously estimated
ARFIMA models. One interpretation of the residuals is that they are formed from a filter
containing a fractional component plus short memory components. The first 100
autocorrelation coefficients of the US inflation residuals are plotted in figure 5.2 and are
consistent with the hypothesis of being generated by an uncorrelated process. However,
figure 5.2 also plots the autocorrelations of the squared and absolute values of the
residuals, or fractionally filtered series. Interestingly, the autocorrelations of the squared
and absolute residuals display extremely persistent autocorrelation that is also suggestive
of a form of long memory behavior. The nature of the autocorrelations of the squared and
absolute residuals are extremely similar to those of many observed financial market return
series. This fact was originally noted by Ding, Granger and Engle (1993) for equity
returns, and by Dacorogna et a1 (1993) for exchange rates. Some of these stylized facts

are consistent with the FIGARCH model of Baillie, Bollerslev and Mikkelsen (1996).

81

 

 

5.3. The ARFIMA-FIGARCH Model
A model that is capable of representing the dual long memory features in both the
conditional mean and in the conditional variance is the ARFIMA(p,d,q)-

FIGARCH(P,6,Q) model, given by

(3) ¢(L)(1 - L)d(yt - M) = 9(L)(1 - ®L'2)81-

(4) 8: = 2M-

 

(5) we, = co + [1 - B(L) - <p<L><1 - max.

where (D(L) and 0(L) are as defined earlier in equation (1), while B(L) = (1 - [31L -...-
BQLQ), (p(L) = (1 + (plL + ...+ (ppLP) and have all their roots outside the unit circle. Also,
Emzt = O, Var--125 1, while E11 is the expectations operator conditioned on a sigma field
set of information at time t-1, and 02, is the conditional variance and is a positive, time-
varying and measurable function with respect to the information set which is available at
time t-l. If 02- = o), a constant, the process reduces to the ARFIMA(p,d,q) model of
Granger and J oyeux (1980) and Hosking (1981). Then yt will be covariance stationary
and invertible for -O.5 < d < 0.5 and will be mean reverting for d < 1.

When 9(L) = (p(L) = l, the process reduces to the FIGARCH(P,5,Q) conditional
variance process of Baillie, Bollerslev and Mikkelsen (1996), and the conditional

variance, 02-, has a slow hyperbolic rate of decay in terms of lagged squared innovations.

82

However, ARFIMA-FIGARCH process will have an infinite unconditional variance for
all (1 given a 5 O. This fact is discussed in the context of the pure FIGARCH model by
Baillie, Bollerslev and Mikkelsen (1996); the presence of the F IGARCH volatility
process imposes an undefined unconditional variance independent of the dynamics in the
conditional mean.

Assuming conditional normality, the logarithm of the likelihood can be expressed

in the time domain as

(6) ln(C) = - (T/2)1n(27t) -(1/2)2t=1,r[1n(021)+ eztc'ztl-

 

The QMLE of the parameters are obtained by an analogous methodology to that described
by Baillie, Bollerslev and Mikkelsen (1996) and in chapter 2, where the likelihood
function is maximized with respect to the vector of parameters denoted by )1 conditional
on initial conditions and the pre-sample values of 82-, t = O, -1, -2,.... are fixed at the
sample unconditional variance. The initial observations yo, y-1, y-2, are also assumed
fixed, in which case minimizing the conditional sum of squares function will be
asymptotically equivalent to MLE. This procedure is known as minimizing the
Conditional Sum of Squares (CSS) function and is widely used in similar models; e. g. see
Baillie, Chung and Tieslau (1996) among others. The consistency and asymptotic
normality of the QMLE has only been established for specific special cases of the
ARFIMA and/or FIGARCH model. Li and McLeod (1986) consider the
ARFIMA(p,d,q)-homoskedastic model with u either zero or known, and show the MLE

are T”2 consistent and asymptotically normal. Dahlhaus (1988, 1989) and Moehring

83

(1990) have extended the proof to the case of the ARFIMA(p,d,q)-homoskedastic model
with u unknown. They show that the parameter estimates are asymptotically normal, with
the ARMA parameter estimates again being T"2 consistent, while the MLE of u is T” 211
consistent. For the conditional variance process, asymptotic normality and T“2
consistency has only been derived for the IGARCH(1,1) model by Lee and Hansen (1994)
and Lumsdaine (1996). Their proofs require zt in equation (4) to be stationary and
ergodic, together with three other relatively mild conditions on 2:. While simulation
evidence for F IGARCH and other complicated parametric ARCH models suggests

QMLE to be consistent and asymptotically normal, a fully general theoretical treatment is

 

as yet unavailable. Consequently, I conjecture that with u unknown, the limiting

distribution of the QMLE is,

(7) D-(‘M - 10) => NIO. {DT"A(IO>"B<AO>A<>10>DT" }"L

where A(.) and B(.) are the Hessian and outer product gradient respectively, when
evaluated at the true parameter values 710, and diag(DT) = [Tl/Z'd, Tm, ...... , Tm].

Even though Kwon (1997) initially performed a detailed Monte Carlo study for
three different designs of ARFIMA-FIGARCH models, I retested similar simulation
studies with different sample sizes and found the adequacy of this estimation method. In
particular, the application of the QMLE appears satisfactory with relatively small
parameter estimation biases and the use of the asymptotic t-test also appears to be very
good. For more details of the theoretical description and Monte Carlo study of the model,

see Kwon (1997).

84

5.4. Estimated Models of Inflation

Given the above, some hybrid ARFIMA-FIGARCH models were estimated for
the monthly US inflation series. Details of the most appropriate models are given in table
5.3. The estimated value of the long memory parameter in the conditional mean is
generally similar to that of the simple ARFIMA-homoskedastic model, and is
significantly different from zero or one. As for table 5.1, the estimated long memory
conditional mean parameter, (1, lies in the range of .23 to .42; while the US median

weighted inflation series has an estimated d of .61, but is less than two robust standard

 

errors away from .50. There is evidence that a model with 0.5 < d < 1 can still be
efficiently estimated by QMLE; or alternatively estimated on the differenced series; see
Smith, Sowell and Zin (1993), Baillie, Chung and Tieslau (1996) and part of section 4 of
Baillie (1996) for a discussion of related issues. For Belgium, France, Italy, Japan, UK
and US the robust Wald statistics are 6.85, 4.70, 26.51, 9.03, 5.05, 4.15, and 4.16
respectively so that the tests can overwhelmingly reject the hypothesis that 6 = 0,
indicating strong evidence of long memory in the conditional variance as well as the
conditional mean. For Germany, the robust Wald statistic is 3.02 and the hypothesis of
stable GARCH(1,1) cannot be rejected at the .05 level. For the other countries, the
FIGARCH model is the preferable parameterization.

The implied impulse responses for both the conditional mean and the conditional
variance of the US are given in figures 5.3a and 5.3b respectively. Again, extremely
similar results for the other countries and US median weighted inflation are presented in
figures 5.4 through 5.11. In general the various diagnostic statistics all indicate the

appropriateness of modeling long memory in both the first two conditional moments for

85

the eight inflation series.

5.5. Conclusion

This chapter has noted that monthly CPI inflation for eight different industrialized
countries appear to have long memory behavior in both its first and second conditional
moments. This is the only economic variable which appears to have this property. I
suggest a parametric ARFIMA-FIGARCH model to represent the dual long memory
phenomenon; and a detailed simulation study reveals the QMLE procedure to work well

for inferential purposes in this new model.

 

An interesting issue for future research concerns the reasons for the finding of
long memory in data series and whether extensions of the aggregation arguments in
Granger (1980) and Ding and Granger (1996) can account for this phenomena in
inflation. In particular, since the CPI series are aggregates of two digit industry
classifications, an interesting area for future research concerns the behavior of different

levels of aggregation of the contemporaneous price series.

86

Table 5.1: Estimated ARFIMA Models for Countries' Monthly Inflation Rates

(1 -<I>L)(1 - L)“(y- - u) = (1 + e-L + 9sz )(1 + 6L”)...

at are i.i.d. (0, co).

 

 

Belgium France Germany Italy Japan Korea UK US USm
p 0.277 0.007 0.237 0.296 0.257 0.932 0.004 0.368 0.288
(0.081) (0.170) (0.063) (0.194) (0.184) (0.446) (0.033) (0.224) (0.105)
d 0.290 0.409 0.211 0.407 0.440 0.418 0.312 0.391 0.599
(0.036) (0.103) (0.042) (0.044) (0.129) (0.094) (0.049) (0.063) (0.120)
(I) - - - - - - - -0.134 -0.100
- - - - - - - (0.077) (0.125)
01 - - - - -0.349 - - - -
- - - - (0.121) - - - -
02 - - - - -0.235 - - - -
- - - - (0.046) - - - -
O 0.169 -0.533 0.280 0.145 0.291 0.189 -0.766 0.048 -
(0.036) (0.065) (0.034) (0.045) (0.038) (0.053) (0.042) (0.058) -
a) 0.100 0.108 0.089 0.152 0.449 0.582 0.225 0.102 0.024
(0.009) (0.033) (0.009) (0.014) (0.037) (0.0633) (0.033) (0.010) (0.003)
ln(L) -125.39-150.47 -104.27 -237.83 -509.24 -396.14 -328.86 -173.40 72.23
Q(20) 81.08 27.23 38.55 23.87 28.39 32.52 21.74 30.82 55.83
Q2(20) 69.17 128.35 21.22 133.33 142.83 133.91 22.12 647.02 312.07
Skewness 0.10 -0.82 0.58 0.91 0.72 0.25 1.18 0.18 -0.09
Kurtosis 4.67 11.35 5.54 5.41 4.45 5.00 11.32 7.45 5.20
Wd=1410.88 112.73 336.40 314.44 25.87 45.46 125.36 58.39 36.85

 

Key : The first eight columns refer to monthly CPI inflation series, while Usm
indicates the median weighted inflation series described in the text. The rest of table

is the same as table 2.1.

87

 

Table 5.2: Estimated Higher Order ARFIMA Models for France and UK

(1 - (DL - (1)sz -....(D12L‘2)(1 - L)"(y.. - p) = (1 + oL‘2)a.,

at are i.i.d. (0, (0).

Monthly Inflation Rates

 

 

France UK
p 0.6229 0.2264
(0.4695) (0.2767)
d 0.3991 0.3168
(0.1372) (0.0456)
(D1 0.0267 -0.0053
(0.0891) (0.0150)
(D2 0.0199 -0.0087
(0.0706) (0.0111)
(D3 0.0252 0.0066
(0.0567) (0.0194)
(D4 -0.0282 0.0027
(0.0382) (0.0152)
(1)5 -0.0285 -0.0033
(0.0350) (0.0104)
(D6 0.0427 0.0103
(0.0532) (0.0172)
(D7 0.0092 -0.0003
(0.0344) (0.0130)
(D3 -0.0351 -0.0130
(0.0414) (0.0102)

 

88

 

Table 5.2 (continued)

 

 

France UK
(D9 0.0099 -0.0212
(0.0333) (0.0153)
(1)10 -0.0342 0.0092
(0.0478) (0.0107)
(1)11 -0.0138 -0.0038
(0.0414) (0.0109)
(D12 0.7573 0.9658
(0.1751) (0.0233)
6) -0.5362 -0.7728
(0.2751) (0.0531)
co 0.0928 0.2134
(0.0126) (0.0307)
ln(L) -115.300 —323.271
Q(20) 17.351 26.325
Q2(20) 184.246 27.170
Skewness 0.238 1.378
Kurtosis 10.223 11.368

 

 

Key: yt is the monthly CPI inflation series for France and UK. The rest of table is
the same as table 2.1.

89

Table 5.3 : Estimated ARFIMA-FIGARCH Models for Countries' Monthly
Inflation Rates

(1 - 9Lx1 - L)“(y- - u) = <1 + M + 9sz )(1 + 61.12)...
Stth-l N N(096t2)9

[1 - BLIofi = w +11 -BL - «pm - 19512-2

 

Belgium France Germany Italy Japan Korea UK US USm

 

1.1 0.246 —0.002 0.248 0.095 0.146 0.591 0.031 0.518 0.269
(0.077) (0.034) (0.072) (0.141) (0.179) (0.390) (0.058) (0.266) (0.072)

d 0.281 0.347 0.226 0.362 0.412 0.352 0.364 0.414 0.617
(0.035) (0.062) (0.042) (0.036) (0.116) (0.096) (0.057) (0.081) (0.063)

 

(1) - - - - - - - —0.205 -0.300
- - - - - - - (0.083) (0.069)
01 - - - - -0357 - — — -
- - - - (0.113) - - - -
02 — - - - -O.285 - - — —
- - - - (0.042) - - - -

0 0.201 -0.681 0.278 0.214 0.350 0.248 -0.728 0.141 —
(0.034) (0.049) (0.031) (0.038) (0.034) (0.040) (0.035) (0.035)

8 0.324 0.331 0.256 0.678 0.317 0.949 0.633 0.644 0.871
(0.135) (0.153) (0.147) (0.132) (0.101) (0.422) (0.314) (0.316) (0.164)

03 0.011 0.001 0.004 0.001 0.033 0.025 0.001 0.001 0.001
(0.008) (0.002) (0.004) (0.002) (0.030) (0.029) (0.002) (0.001) (0.001)

(3 0.256 0.899 0.848 0.773 0.253 0.735 0.893 0.811 0.670
(0.138) (0.280) (0.060) (0.070) (0.122) (0.336) (0.078) (0.124) (0.152)

(p - 0.859 0.710 0.275 - - 0.635 0.467 -
- (0.383) (0.113) (0.144) - - (0.153) (0.123) -

 

90

Table 5.3 (continued)

 

Belgium France

Germany Italy Japan Korea

UK US

USm

 

ln(L) -106.70 -63.63

Q(20) 74.69 19.49 34.46

02(20) 26.37 31.09 20.94

Skewness 0.40 -0.02 0.81

Kurtosis 4.29 6.69 5.78

W5=0 6.85 4.70 3.02

16.91

17.24

0.63

4.63

26.51

-98.32 -177.68 -471.60 -356.86

33.76 24.36
3.76 7.23
0.69 0.82
4.35 5.84
9.03 5.05

21.93 34.83

10.80 16.63

0.21 0.09

4.74 4.50

4.15 4.16

-284.74 -65.62 230.39

24.61

17.37

0.29

3.52

28.18

 

Key : As for table 5.1.

91

 

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111

[Chapter 6]

High Frequency Perspective On the Forward Premium Anomaly

112

6.1. Introduction

This chapter is concerned with the analysis of high frequency data from the
foreign exchange market and 30 day US. and German Eurobond rates. Our interest is
primarily in the relationship between these two markets and the use of high frequency
data to throw more light on the notorious forward premium anomaly, which has remained
a continuing thorn in the understanding of international financial markets. Our analysis is
based on one year (1996) of 30minute data on these two series which provides
complementary information to the high frequency series and is important in deriving a
series of daily volatility for the extraction of the intra day periodicity.

In contrast to previous studies, our emphasis is on bivariate analysis of high
frequency data and I consider analogous properties of the interest rate differential. The
analysis of the high frequency forward premium series is considerably complicated by the
presence of extremely persistent autocorrelation in both its conditional mean and in its
conditional variance. Following Maynard and Phillips (1998) and Baillie and Bollerslev
(2000), it seems that a central aspect of the possible resolution of the forward premium
anomaly lies in the determination of the order of integration of the forward premium.
Hence I devote considerable attention to this issue and estimate a fully parametric dual
long memory ARFIMA-FIGARCH process, and also the semi parametric local Whittle
estimators. The general conclusion is that there is evidence that the forward premium is
stationary and fractionally integrated. The final section describes the forward premium
anomaly on the high frequency perspective.

The plan of the chapter is as follows; section 2 outlines the basic financial

economic relationships theorized to exist between spot exchange rates and the interest

113

 

rate differentials. Particular attention is given to the forward premium anomaly. Section 3
considers the issue of modeling the high frequency forward premium series, which is far
more challenging due to the extreme persistence, or borderline non-stationarity in its
mean. The FF F method is applied to the fractionally filtered forward premium series as in
chapter 2 for the analysis of the high frequency spot returns. A parametric ARFIMA-
FIGARCH model with long memory features in both mean and variance is estimated. The
resulting estimates of the long memory parameter in the conditional mean is found to be

broadly consistent with estimates obtained by the local Whittle estimator in Section 4.

 

The applications of these different methods to the high frequency series implies

stationarity of the forward premium. Section 5 concludes briefly.

6.2. International Finance Relationships

One motivation for the high frequency analysis of spot and forward exchange
rates is an attempt to throw more light on the forward premium, or forward discount
anomaly. This paradox refers to the widespread result that the returns on freely floating
exchange rates are invariably negatively correlated with the lagged forward premium. The
implication that an appreciating currency results for the country with the higher rate of
interest, is generally interpreted as being due to (i) the existence of a time varying risk
premium, (ii) peso problem effects, and or (iii) the irrational behavior of market

participants. See Engel (1996) for a recent survey of this literature.

For the rest of this article, St,” denotes the spot exchange rate at the end of the n'th

hour on day t, while Ft,n,l refers to the forward exchange rate also at the end of the n'th

hour on t, which is for delivery at any hour on day t+l. Corresponding logarithmic values

114

are denoted by lower case variables letters and all the rates have the US. dollar as the
numeraire currency. Further, iml denotes the dollar return on an l-period risk free dollar
denominated bond at the end of hour n on day t, whereas i‘W is the foreign currency

return on a risk free bond denominated in terms of DM. The Uncovered Interest rate

Parity (U IP) condition states,

(1) Et(ASt+l,j) = (ft,j,l - St,j) = (it,j,l - i*t,j,l)a

for j = 1,.., n and where B(.) denotes an expectation conditioned on the sigma field y. =
{st,n_1, imp”, i‘t,n-1”[, .....} Hence the quantity (it’jJ - 1*th ) is the expected rate of
appreciation (depreciation) of the currency 1 business days in the future. The contract on
going long, or short in the forward market can be legally closed during any hour j on day
t+ l for j = 1,...n. See Levine (1989) and Riehl and Rodriguez (1977) for institutional
details of these contract calculations. The UIP hypothesis requires the joint assumptions
of rational expectations, risk neutrality, free capital mobility and the absence of taxes on

capital transfers. Hence expected real returns in the forward market must be zero, i.e.

(2) Et[(Ft,n,I ' St+l,n)/Pt+l,n] = 0.

where Pu, denotes the domestic dollar price level. By Taylor series expansion of equation

(2) to second order terms (Hansen and Hodrick (1983)),

115

 

(3) Et(5t+,n) — ft,n.l : '(1/ 2)*Vart(s t+l,n) + C0Vt(st+[,n Pt+l,n),
where Pt,n refers to the logarithmic price level. Note that, even under rational expectations

and risk neutrality the right hand side of equation (3) contains the two conditional second
moment terms. Generally, the discrete time, consumption based asset pricing model,
provides a risk adjusted equivalent to equation (2), which emphasizes real returns over

the current and future consumption streams of the representative investor,

(4) Et{[(Ft,n,l — St+l,n)/Pt+l,n 1* U(Ct+[,n)/U (Ct,n)} = 0.

where U(Ct+l,n)/U(Ct,n) and is the intertemporal marginal rate of substitution. The

analogue to equation (3) is,

(5) Et(5t+l,n) — ft,n,l : '(1/2)*Vart(st+l,n) + C0Vt(St+I,n Pt+l,n) + C0Vt(5t+R,n Qt+l,n).

where qwm = ln(Qw,n ) and is the logarithm of the intertemporal marginal rate of

substitution. The last term, pm] = Covt(st+/,n qt+1,n), is a time dependent risk premium.

Many empirical tests are based on the daily, weekly or monthly level of

aggregation which implies the relationship,

(6) St+l — St = 011 + Bl(ft,l — St) + ut+l,l-

The UIP hypothesis implies that a. = 0 and [31 = 1. If the sampling frequency is equal to

116

 

the maturity time of the forward contract, so that l = 1, then u...” will be serially

uncorrelated. Regressions of the form of (6) invariably find the estimated slope
coefficients to be negative. Previous explanations of the forward premium anomaly have
included "peso problem" effects, e.g. Evans and Lewis (1995), and the importance of
learning and heterogenous beliefs, e. g. Frankel and Froot (1987). Another possibility is
that estimation of equation (6) may involve a mis-specification since the conditional
variance and covariance terms featured in the above equation have been neglected.

Suppose a risk premium is present, then

 

(7) Et(St+l,n " St,n) = (anJ _ St,n) + pt,n,la

and from Fama (1984), a population value of B1< 0 implies that Cov[Et(st+1 — st)*pt3/] < 0

and also that Var(pt,,) > Var[Et(st+; — st)]. Hence a negative [3R coefficient implies a

negative sample covariance between the risk premium and the expected rate of
appreciation.

The daily DM-$ spot rates and German and US. 30 day Eurobond interest rate
differentials between January, 3 1984 through December 31, 1998 are graphed in figures
6.1a and 6.1b. The data provided by the Federal Reserve Bank of Cleveland. Excluding
weekends and holidays, this realizes a sample of 3,282 observations. The interest rate
differential has three main phases with the DM being at a discount until 1990., then at a
premium until 1993 and then at a discount for the remainder of the sample. Hence the

interest rate differential exhibits very long swings around an apparent equilibrium of

117

parity, and not surprisingly generates very persistent autocorrelation. An illustration of the
forward premium anomaly can be illustrated from analyzing 215 monthly DM-$ rates
from January 1973 through November 1995. The estimated slope coefficient in equation
(6) is -2.23, with an OLS standard error of 0.99. This anomalous finding is entirely
consistent with the evidence reported in the existing literature for other currencies.

Figure 6.2 depicts estimates of B from 208 five-year rolling regressions, with the first
estimate obtained by beginning at March, 1973 and using a total of sixty observations
through February, 1978. The next estimate was obtained by using data from April, 1973

through March, 1978, until the final estimate was based on data from December, 1990

 

through November, 1995. There are extensive periods when the rolling OLS estimator of
B is significantly negative and the largest negative coefficient in this period is -13.00.
Monthly observations on the DM-$ spot and one-month forward rates were
previously been analyzed by Baillie and Bollerslev (1994) and Crowder (1994). Of
course, the estimated [3 coefficients tend to be highly cross correlated over time and
across countries. The extreme persistence of the autocorrelations of the forward premium
has even persuaded some authors that it may contain non-stationary components; e. g.
Evans and Lewis (1995) and Crowder (1994), while Baillie and Bollerslev (1994) provide
evidence for the fractional integration of the forward premium, so that all shocks
eventually dissipate but at a very slow hyperbolic rate of decay. Baillie and Bollerslev

(2000) and Maynard and Phillips (1998) have attempted to explain the regression results

 

in terms of (7) being unbalanced, in the sense that the orders of integration of the
dependent and explanatory variables are not the same. In this situation the test of UIP

with R = 1 could be based on the regression,

118

(8) As... = a + B [1 -<1 -L>'*1 1 <1 - L)" <p(L)(fi - s.) + um.

and test that a = O, [3 = 1 and that u. is serially uncorrelated. Formal theoretical results
along these lines based on the notion of fractional cointegration have recently been
pursued by Maynard and Phillips (1998). Baillie and Bollerslev (2000) view the anomaly
as mainly being a statistical phenomenon that occurs because of the very persistent
autocorrelation in the forward premium. They calibrate an economic model of UIP which

generates long memory in the forward premium and show that the OLS estimates of the

 

slope parameter estimates in the anomalous regression are centered around unity. The
OLS estimates have a very slow rate of convergence to its true value of unity. An
implication of this is that empirical work with sample sizes are likely to find very
dispersed slope coefficient estimates. Interestingly, the coefficient estimates from the
rolling regression appears to be close to zero particularly around the year 1996, which is

the same period as that of the high frequency data.

6.3. Analysis of the High Frequency Forward Premium

There has recently been considerable literature on univariate analysis of high
frequency financial market data, especially in foreign currency markets, e. g. Dacorogna et
a1 (1993) and Andersen and Bollerslev (1997b, 1998). Before reporting the details of the
bivariate analysis of the high frequency series, I first focus on the more familiar univariate
analysis of spot returns and interest rate differentials. In Chapter 2, the detail description
of high frequency data and the analysis of the high frequency 30 minute DM-$ spot

returns are reported. Hence, this chapter presents only the analysis of the corresponding

119

 

30 minute interest rate differentials and their results.

The 30 minute DM-$ spot returns for 1996 are plotted in Figure 6.3a, and the
corresponding 30 minute interest rate differential, i.e. the forward premium is graphed in
figure 6.3b. While the daily DM-$ spot returns for 1996 appears to be quite similar to the
long span of sixteen years of daily returns, there is a distinct difference to the forward
premium which remained at a US. discount throughout 1996. The implied expected
German depreciation of the exchange rate did not occur within this period. However, the
sample autocorrelations of the high frequency forward premium are extremely persistent,
which is consistent with long spans of daily and monthly series.

In order to highlight the difference between these markets, figure 6.4a gives the
averaged 30 minute absolute DM-$ spot returns, where the averaging occurs over each of
the 262 days in the sample for each of the 48 intra day periods. Figure 6.4b presents the
corresponding averaged 30 minute absolute differenced forward premium. Unlike the
foreign exchange market the Eurobond market has long periods of virtual inactivity and
hence it is only economically meaningful to analyze the forward premium between the
times of 8:30 through 20:00 each working day, which realizes a total of 6,288
observations. Figures 640 and 6.4d present the corresponding absolute spot returns and
differenced forward premium for these periods. Hence the focus on the relationship
between these markets will be conducted on the sample of 6,288 observations when both
markets are simultaneously open.

While the high frequency spot returns have many of the classic features of asset
prices determined in speculative auction markets as presented in chapter 2, the properties

of the interest rate differential are quite different. The autocorrelations for regular,

120

squared and absolute forward premium are exhibited in the three panels of figure 6.5, and
analogous displays for the differenced forward premium and the filtered forward
premium are in figures 6.6 and 6.7 respectively. In comparison with the correlograms for
spot returns in figure 22, clearly the extreme persistence of the autocorrelations for the
forward premium in levels renders it quite different to that of the spot returns. The
correlogram for the differenced forward premium in figure 6.6 possesses a large negative
autocorrelation at lag one, which is synonymous with either overdifferencing or anti-
persistence in the sense of long memory. However, the squared and absolute differenced

forward premium possesses similar intra day seasonal characteristics to the spot returns

 

series. As did for the analysis of 30 minute spot returns in chapter 2, the intra day
seasonality is filtered by the Flexible Fourier Form (F FF) method. Then, an appropriate

model for the filtered forward premium is ARFIMA(p,d,q) - FIGARCH(P,5,Q) model.

The model can be given by

(9) ’91,. =y.,n/a.,n,

(10> (1-L>“'§.,n = u+e.,n+ee.,n-.,

(11) 8t,n =Zt,n 0t,n .

(12) oz... = co + B02... +[1-BL-(1-wLX1-L)5]et,n2,

 

121

where a In is the intraday periodicity components estimated from FFF method, 2 Ln is

another i.i.d.(0,1) process, and where the two time indices are t = 1,.. 262 days, and n =
1,.. K, intra day periods, so that K = 24/k, for k = 1, 2, 3, 4, 6, 12. Results for estimating
the above model for the six different frequencies over k, within the day are presented in
table 6.1. The autocorrelations of the filtered 30 day forward premium innovations,
squared innovations and absolute innovations are given in the panels of figure 6.8. The
results indicate that the parametric version of the model for the interest rate differential is

strongly supportive of fractional integration.

 

6.4. Semi Parametric Estimation

There are various issues concerning the application of the semi parametric
estimators of the long memory parameter. The GPH estimator is known to perform poorly
in terms of bias and RMSE in the presence of misspecification of short run 1(0) dynamics.
Also, little is known about the behavior of these estimators in the presence of time
dependent heteroskedasticity and/or unconditional return densities with infinite variance.
The only other relevant study is that of Taqqu and Teverovsky (1997), who find the local
Whittle estimator to perform better than its competing semi parametric estimators with
i.i.d. stable Paretian density. In particular, Robinson (1995), Taqqu and Teverovsky
(1997,1998) show that the estimator performs well especially in cases of infinite
variances.

The local Whittle estimator only specifies the parametric form of the spectral

density when X is close to zero. Hence f(v) —> g(d)|v|'2d, as v —) 0, and g(d) is some

 

function of d. The estimator is equivalent to Whittle except that it only assumes the

122

behavior of the spectral density up to some frequency 21tm/n. The long memory parameter

d is estimated by minimizing

(13) R(d) = ln[m'l 2H,, 1(v,)v,-2d] - 2dm" 2H... ln(vj).

The semi parametric local Whittle estimation method is applied to estimate the
long memory parameters of the filtered 30 minute differenced forward premium and the
absolute filterde 30 minute differenced forward premium. Results are reported in table
6.2 and the estimated parameters of d across various frequencies are generally consistent
with those from a parametric ARFIMA-FIGARCH estimation method in table 6.1. Thus,
there is a supporting evidence that the high frequency forward premium is indeed a
stationary I(d) process.

An interesting issue for future research concerns the Monte Carlo evidence on the
performance of these estimators in the presence of homoskedastic and heteroskedastic
innovations. In one experiment the dgp was an ARFIMA(O,d,O)—FIGARCH(1,6,0) so that
the unconditional variance not defined even in the presence of conditional
heteroskedasticity. These experiments are different, but related to those of Taqqu and
Teverovsky (1997) who sometimes used stable Paretian densities but always considered
i.i.d. densities. This case can explore the more realistic setting of non i.i.d.innovations
with time dependent heteroskedasticity. In this context, Kwon (1997) presented some
simulation results for parametric ARFIMA(0,d,0)-FIGARCH(1,8,1) model and
ARFIMA(0,d,1)-FIGARCH(1,5,1) model. He used three different designs with different

values of 5, and then found that MLE method works extremely well in terms of

123

estimating both the true d and 5 parameters and the Monte Carlo study is very supportive

of the asymptotic theory.

6.5. Conclusion

The forward premium anomaly implies that in regression of the change in the spot
exchange rates on the forward premium, negative slop coefficient have invariably been
reported throughout the literature. In chapter 2, I have analyzed the high frequency data
especially in the year 1996, 30 minute DM-$ spot returns and in the previous sections, I
have analyzed the corresponding interest rate differentials based on 30 day US and
Germany Eurobond rates. The results from the analysis of the high frequency data
verified the empirical facts that spot rates are approximate martingale differences, i.e. I(O)
process, with very persistent volatility and forward premium, or interest rate differential,
are fractional differences, i.e. I(d) process where O.5< (1 <1.0, so that it has long memory
property. In terms of order of integration, they are "unbalanced". Hence on the high
frequency perspective the anomalous regression estimates reported throughout the
literature do not appear to provide convincing statistical evidence to accept the forward
premium anomaly. The so called forward premium anomaly may be viewed as a
statistical artifact form the long memory property in the forward premium. This is in
accordance with the results of Baillie and Bollerslev (2000) from the stylized UIP model
at the lower frequency, the monthly level. In particular, a rolling regression in Baillie
and Bollerslev (2000) found the estimated [3 to be close to zero around the year of 1996
and it appears that the forward premium anomaly is not pronounced for this particular

year. Thus, the results available from the analysis of the high frequency data during the

124

year 1996 and the study of Baillie and Bollerslev (2000) at the monthly frequency suggest

that if it does exist, the risk premium may be not so large as we expected especially for
the year 1996.

F ama's inequality (1984) implies that

COV[Et(St+l,n — St,n)P t,n] < 0 and Var (Pm) > var [Et(St+l,n — S t,n)]>

where pm is the risk premium and p t,“ = Et(st+1,n - 5 Ln ) - (f t,n,l - s t," ). Thus, if we can

construct the data such as the risk premium empirically from the high frequency data and
then check the Fama's inequality on the high frequency perspective, it may also help to
explain the puzzling forward premium anomaly. I leave further work along these lines

for future study.

125

(I-Lfytn = u + 02.1,. + a...

0.3 = co + Bed + [1-BL- (1 - <pL)(1-L>5]e.n2.

Table 6.1: Estimated ARFIMA-FIGARCH Models for Filtered 30 Minute
Differenced Interest Rate Differentials

where ym, = Z, =(n-1)k+1, “1.an for t=l, .., 262, and for n = 1, ..,24/k,

k = 1,2,3,4,6,12 and if, = R.,/ a..-

 

 

30min. 1 hour 1 1/2 hour 2 hour 3 hour 6 hour
1 2 3 4 6 12
6288 3144 2096 1572 1048 524
-0.0050 -0.0111 -0.0162 -0.0195 -0.0305 -0.0588
(0.0011) (0.0024) (0.0039) (0.0063) (0.0109) (0.0220)
-0. 1776 -0. 1329 -0. 1049 -0.0576 -0.0271 -0.0220
(0.0233) (0.0292) (0.0364) (0.0532) (0.0681) (0.0809)
-0.1017 -0.1733 -0.2026 -0.2428 -0.2423 -0.1435
(0.0460) (0.0462) (0.0562) (0.0819) (0.0792) (0.0882)
0.3067 0.3057 0.4344 0.5808 0.5419 0.6011
(0.0683) (0.0833) (0.1449) (0.3126) (0.1378) (0.1595)
0.0014 0.0043 0.0044 0.0031 0.0046 0.0094
(0.0005) (0.0021) (0.0024) (0.0023) (0.0028) (0.0066)
0.9544 0.8820 0.8550 0.8946 0.8695 0.8385
(0.0159) (0.0430) (0.0438) (0.0663) (0.0389) (0.0568)
0.8979 0.7687 0.5910 0.5390 0.4648 0.3684
(0.0347) (0.0674) (0.0904) (0.2174) (0.0955) (0.1194)

 

126

Table 6.1 (continued)

 

 

30min. 1 hour 1 1/2 hour 2 hour 3 hour 6 hour
ln(L) 4570.567 4511.417 -1325.046 -1056.241 -877.713 -547.779
Skewness -O.568 -0.528 -0.298 -0205 -0.178 0.110
Kurtosis 18.204 11.195 8.306 7.215 6.001 5.108
Q(SO) 75.501 60.245 54.002 50.762 43.073 37.902
02(50) 70.175 67.712 44.296 67.497 51.904 24.026
W3=0 20.141 13.464 8.984 3.453 15.459 14.207

 

Keys: ygn is the filtered 30 minute interest rate differential series. The rest of table is
the same as table 2.1 except that the Q(SO) and Q2(50) statistics are the Ljung-Box
test statistics for 50 degrees of freedom to test for serial correlation in the
standardized residuals and squared standardized residuals.

127

Table 6.2: Local Whittle Estimators of Long Memory Parameters for High

Frequency Forward Premium

 

30min. 1 hour 1 1/2 hour 2 hour

3 hour 6 hour

 

sample size 6288 3144 2096

d -01254 -01435 -0.1401
(0.0396) (0.0565) (0.0693)

5 0.1142 0.1841 0.2414
(0.0346) (0.0510) (0.0601)

1572

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(0.0825)

0.2453
(0.0804)

1048 524

-0.0656 0.0796
(0.1077) (0.1700)

0.2964 0.1756
(0.0995) (0.1358)

 

Key: The data description is the same as table 6.1. M = n/32 is selected.

128

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Figure 6.13: Daily DM-$ Spot Returns from January 4, 1984 through
December 31, 1998

 

 

 

. - r . .
50887 92989 22692 72294 1 12996

Figure 6.1b: Daily Forward Premium from January 3, 1984 through
December 31, 1998

 

 

 

 

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A man
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Figure 6.4a: Averaged 30 Minute Absolute DM-$ Spot Returns

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Figure 6.41: Averaged 30 Minute Absolute DM-$ Spot Returns from 08:00 GMT to 20:00 GMT
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