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THESIS

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1688 0530

This is to certify that the

dissertation entitled

Long Memory and Asymmetry
In Conditional Variance Models

presented by
Yeongil Hwang

has been accepted towards fulfillment
of the requirements for

PhD. degree in Econormcs

 

 

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TO AVOID FINES return on or before date due.

 

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use muss-p14

LONG MEMORY AND ASYMMETRY IN CONDITIONAL VARIANCE MODELS

By

Yeongil Hwang

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Economics

1998

ABSTRACT

LONG MEMORY AND ASYMMETRY IN CONDITIONAL VARIANCE MODELS

By

Yeongil Hwang

The dissertation introduces a new family of models for the conditional variance of
economic time series. The new models allow for both asymmetries and long memory,
whereas previous models had allowed for one or the other but not both. These models are
applied to two different kinds of data, on stock returns and exchange rates. In each case,
there is strong evidence of both asymmetry and long memory, and correspondingly the

new models fit the data better than other simpler models.

Dedicated to my parents

iii

ACKNOWLEDGEMENTS

This dissertation would never have been begun, nonetheless finished, without the con-
stant encouragement, invaluable advice, and strong help of Peter Schmidt, my dissertation
committee chairman. Robert H. Rasche and Richard T. Baillie also read the paper and
made valuable suggestions. I also wish to thank Subbiah Kannappan through entire my
school life.

The research has been helped out by other numerous gracious scholars inside or out-
side school. I wish to thank Mark Wohar at University of Nebraska, Ramond P.
DeGennaro at University of Tennessee, Thierry Roncalli at University of Bordeaux, Rich-
ard Quandt at Princeton University, Fallow Sowell at Carnegie Mellon University, William
G. Schwert and Ludger Hentschel at Rochester University to name a few. Peter Schmidt
has kindly suggested numerous excellent ideas throughout the dissertation. I wish to thank
his warmest kindness and dedicated supervision in every aspect, academically or nonaca-
demically. Robert H. Rasche and Richard T. Baillie have offered many precious com-
ments. Kenneth Boyer, Jack Meyer, Christine Amsler, and Warren J. Samuels helped me
out whenever I was in the water. I must express the full respect to Jefferey M. Wooldridge
to help me especially left alone in the wet battle field. Without their helps, I might have
had the more idling time.

I must thank my family for their invaluable support.

Any remaining errors are of course my own responsibility.

iv

TABLE OF CONTENTS

LIST OF TABLES .......................................................................................................... vii
LIST OF FIGURES ........................................................................................................ ix
CHAPTER 1. INTRODUCTION ....................................................................................... 1
CHAPTER 2. NEW FAMILY OF FRACTIONALLY INTEGRATED
VOLATILIT Y MODELS
I. INTRODUCTION ..................................................................................................... 4
II. BASIC MODEL ......................................................................................................... 6
HI. SPECIFIC MODELS FOR 0,2 ................................................................................ 9
IV. AUTOCORRELATIONS OF 2,2 AND of ............................................................... 19
V. CONCLUSION ..................................................................................................... 24
APPENDIX 1 ............................................................................................................ 29
APPENDIX 2 ............................................................................................................ 38
APPENDIX 3 ............................................................................................................ 47
CHAPTER 3. ASYMMETRIC LONG MEMORY IN VARIANCE OF US. STOCK
RETURNS
I. INTRODUCTION .................................................................................................... 54
II. METHODOLOGY .................................................................................................. 58
III. EMPIRICAL RESULTS ........................................................................................ 64
IV. CONCLUSIONS .................................................................................................. 71

CHAPTER 4. LONG MEMORY IN VARIANCE OF EXCHANGE RATES

I. INTRODUCTION ............................................................................................... 8O

II. THE ASYMMETRIC LONG MEMORY FIFGARCH MODEL ........................ 80

III. RESULTS AND ANALYSIS .............................................................................. 81
IV. CONCLUSIONS ............................................................................................... 87
LIST OF REFERENCES ................................................................................................ 96

vi

LIST OF TABLES

CHAPTER 2
TABLE 1 Various short memory GARCH models with special names .................... 26
CHAPTER 3
TABLE 1 Family GARCH and Family FIGARCH: 17,582 daily returns of

the Standard & Poor Index, 1928/01/03-1993/09/30 ............................. 72
TABLE 2 Forecasts of sf from symmetric and asymmetric models ....................... 73
TABLE 3 Comparison of sample and theoretical autocorrelations of sf ................ 74
TABLE 4 Likelihood ratio tests for asymmetry in volatility .................................. 75
TABLE 5 Likelihood ratio tests of functional form in long memory models ......... 76
TABLE 6 Likelihood ratio tests of long memory and asymmetry ........................ 77
CHAPTER 4
TABLE 1 Family GARCH and Family FIGARCH: 6, 241daily exchange rate

returns of German Mark per US. Dollar,

1973/04/02-1998/02/13 .......................................................................... 88
TABLE 2 Forecasts of sf from symmetric and asymmetric models in

exchange returns of German Mark per US. Dollar,

1973/04/02-1998/02/13 .......................................................................... 89
TABLE 3 Comparison of sample and theoretical autocorrelations of 8,2

in German exchange returns .................................................................. 90
TABLE 4A Family GARCH and Family FIGARCH: 6, 241daily exchange rate

returns of Japanese Yen per US. Dollar,

1973/04/02-1998/02/ 13 ......................................................................... 91
TABLE 4B Family GARCH and Family FIGARCH(1,d,O): 6, 241daily exchange

returns of Japanese Yen per US. Dollar,

vii

TABLE 5A

TABLE 5B

TABLE 6

1973/04/02- 1998/02/13 ......................................................................... 92

Forecasts of 3,2 from symmetric and asymmetric models in exchange

returns of Japanese Yen per US. Dollar,
1973/04/02-1998/02/ 13 ......................................................................... 93

Forecasts of sf from symmetric and asymmetric Family
FIGARCH(1,d ,0) models in exchange

returns of Japanese Yen per US. Dollar ............................................... 94
Comparison of sample and theoretical autocorrelations of 2,2
in Japan exchange returns ..................................................................... 95

viii

LIST OF FIGURES

CHAPTER 2
Figure 1 THE ASYMMETRIC TRANSFORMATION OF ((87) ............................ 27
Figure 2 THE TRANSFORMATION OF RC?) ..................................................... 28
CHAPTER 3

Figure] AUTOCORRELATIONS OF SQUARED ERRORS ................................ 78
Figure 2 AUTOCORRELATIONS OF ERRORS ................................................... 79

CHAPTER 1

CHAPTER 1

Introduction

This dissertation proposes new models for the conditional variance of an economic
time series. The first conditional variance model was the ARCH model of Engle (1982),
which was followed by the GARCH model of Bollerslev (1986) and a large number of
other models. These models are surveyed in Bollerslev, Engle and Nelson (1994).

Chapter 2 provides a general discussion of ARCH and GARCH models. It focuses on
two distinct strands of this literature. First, in some empirical applications there is evi-
dence of long memory in variance, in the sense that volatility is persistent. Standard
ARCH, GARCH and related models cannot deal satisfactorily with long memory. The
FIGARCH (Fractionally Integrated GARCH) model of Baillie, Bollerslev, and Mikkelsen
(1996) was the first conditional variance model to allow long memory in variance. It was
constructed by analogy to models of fractionally integration (long memory) in mean,
which date back to Granger (1980) and Hosking (1981). Second, in some empirical appli-
cations there is evidence of asymmetry, which is usually taken to mean that negative inno-
vations imply a different effect on variance than positive innovations of equal magnitude.
A comprehensive treatment of asymmetry is given by Hentschel (1995), who defines the
FGARCH (Family GARCH) models, a set of models that add parameters to represent
asymmetries and also different power transformations in the basic GARCH equation. This
family includes most previous models as special cases, but it does not allow for long mem-

ory. The main contribution of Chapter 2 is to combine these two strands of the literature.

We define FIFGARCH (Fractionally Integrated Family GARCH) models that basically
combine the FGARCH models Of Hentschel with the FIGARCH model of Baillie et al., so
as to allow simultaneously for both asymmetry and long memory in variance.

Chapter 2 also derives analytical results for the autocorrelations of the squared errors
and of the conditional variances, for the special case of the asymmetric GARCH( 1,1)
model. These results generalize results of Ding and Granger (1996B), who derived the
autocorrelations of the squared errors for the symmetric GARCH( l, 1) model.

In Chapter 3 we apply the FIFGARCH model to data on daily stock returns. We use a
very long data set of 17,582 daily returns from January 3, 1928 to September 30, 1993.
With this abundance of data, there is hope of supporting a fairly intricate model. Prelimi-
nary analyses reveal evidence of both asymmetry and long memory, so the FIFGARCH
model is a reasonable choice for these data. The model was consistently found to be better
that other simpler models, according to standard statistical criteria including the value of
the likelihood function, measures of the accuracy of prediction of squared errors, and
closeness of the sample and theoretical correlations of squared errors. Simple models are
also convincingly rejected by likelihood ratio tests. Thus we regard our application of the
FIFGARCH model to these data as successful.

Chapter 4 is similar to Chapter 3, but now the FIFGARCH model is applied to daily
data on exchange rate returns. For the case of the DM/$ exchange rate, there is evidence of
long memory but not of asymmetry, while for the case of Yenl$ exchange rate, there is evi-
dence of both asymmetry and long memory. A restricted version of the FIFGARCH
model, which equates certain exponents in the power-transformed GARCH equation, fits

the data well, and is superior to other simpler models.

The research in this dissertation could be continued in several ways. Further empirical
work will be needed to understand how widely applicable the new models suggested here
are. Theoretical research is also needed to establish rigorously the asymptotic properties of
the estimates and inferences based on quasi-maximum likelihood estimation. We have fol-
lowed much of the literature in simply assuming that the usual asymptotic properties of the
quasi-MLE’S apply. While there is no specific reason to doubt that this so, a rigorous

investigation of this equation is called for, and remains to be done.

CHAPTER 2

CHAPTER 2

New family of fractionally integrated volatility models

I. Introduction

This chapter makes two contributions. The first is to propose a family of asymmetric,
long-memory models for conditional variances. The second is to provide results on the
correlations of squared observations and of the conditional variance for symmetric and
asymmetric GARCH models.

There has recently been a large amount of econometric work on long-memory, frac-
tionally integrated processes. These processes are associated with hyperbolically decaying
impulse response weights, and therefore with long-memory persistence of shocks and of
autocorrelations. They have been applied both to the level (mean) and to the volatility
(variance) of economic time series. There is substantial evidence that long memory pro-
cesses can be used to describe financial or macroeconomic data such as excess returns,
inflation rates, forward premiums, interest rate differentials and exchange rates. Most
recently, long memory models have been applied to the volatility of asset prices and to
power transformations of returns. Specifically, the FIGARCH model of Baillie et al.

( 1996) allows for fractional integration of the conditional variance and thus provides a
useful model for series for which the conditional variance is very persistent.

Another recent development has been the development of asymmetric models for con-

ditional variances. There has long been evidence of asymmetries in financial data; for

example, negative returns may have a different effect on volatility than positive returns of
equal magnitude. Hentschel (1995) has defined a family of GARCH models that allow for
such asymmetries. However, his models do not allow for long memory. This chapter
defines models that allow for both long memory and asymmetry, thus joining two strands
of the literature that had previously been largely separate. In a recent paper, McCurdy and
Michaud (1997) combined the FIGARCH model with the asymmetric power ARCH
model of Ding, Granger and Engle (1993). The basic idea is very similar to the idea of this
chapter, but our models are more general than theirs, because Hentschel’s model is more
general than the model of Ding, Granger and Engle.

The second contribution of this chapter is to derive expressions for the autocorrela-
tions of squared observations and conditional variances from symmetric and asymmetric
GARCH models. Ding and Granger (1996B) have given the autocorrelations for the
squared observations for the symmetric GARCH(], 1) model. In this chapter we provide
similar expressions for the case of asymmetric GARCH. Note that, in discussing the
notion of persistence in models of this type, one could focus on the degree of persistence
either in the squared errors or in the conditional variance itself. The conditional variance is
a random variable and it is perfectly reasonable to consider its autocorrelations. We derive
expressions for the autocorrelations of the conditional variance for symmetric and asym-
metric GARCH models.

The plan for the rest of the chapter is as follows. Section 2 establishes notation and
presents a generic model of conditional heteroskedasticity. Section 3 discusses specific
models, and proposes the F IF GARCH model, which allows for asymmetry and long mem-

ory in conditional variance. Section 4 presents results for the correlations of squared errors

and conditional variances. These results are derived in detail in Appendices 1-3. The final

section provides a brief review.

II. Basic model

Let y t be an observed series. We specify its first two conditional moments:

E(ytlQI-1) : 8(Qr_1961)’ (1)

etEyt—E(yt|Qt—l)=yt-g(Qt-1’el)’ (2)
‘ 2 2

VAR(yt|Qt_1) = E(et lop Jae, = h(Qt_1,92), (3)

where Q“ 1 is the set of information available at time t— 1 , and 91 and 92 are sets of
unknown parameters to be estimated. It is often assumed that g“)! _ 1.61) = x, _ 1B ; i.e.

linearity is usually found adequate, assuming x t _ 1 e (2 _ 1 . Different models correspond

t
to different functional forms of g and h.
For constructing likelihoods, or predictions of anything other than mean and variance,

we must assume a distribution for at. We can write

at = Otcut, wt~i.i.d. D(0,l), (4)

where D(O,1) represents some Specific distribution with mean zero and variance one.

Examples include normal, student’s t, lognorrnal, or more flexible distributions. Although
models of this form generate fat-tails in the unconditional distribution even under condi-
tional normality, they do not fully account for excess-kurtosis present in many financial
data. The student t-distribution with the number of degrees of freedom to be estimated has
been used by several authors such as Bollerslev (1987) and Baillie and DeGennaro (1990).
Other densities which have been used are the normal-Poisson mixture distribution of
Jorion (1988) and Nieuwland et al. (1991), the normal-lognormal mixture distribution of
Hsieh (1989), the generalized error distribution of Nelson (1991), the Bernoulli-normal
mixture of Vlaar and Palm (1993), the power exponential of Baillie and Bollerslev (1989),
and the stable distribution of De Vries (1991). The more flexible distributions include the
stable, Pearson, generalized beta, exponential generalized beta of the second kind, and
generalized t families of distributions. Each includes many common distributions as spe-
cial cases.

SO called “ARCH-M models”, in which the conditional variance affects the mean of

the series, can be specified as follows:

2
“Qt—1’91) =xt—IB+TOI‘ (5)

Engle, Lilien and Robins (1987) introduced the ARCH in mean (ARCH-M) model in

which the conditional mean is a function of the conditional variance, and the conditional

variance follows an ARCH process. This model generalizes easily to more complicated

models for 0,2. It arises in a natural way in mean-variance analysis where Tot2 could

denote the risk premium for some asset with 0,2 being a measure of risk. Pagan and Ullah

(1988) refer to these models as models with risk returns. For the usual ARCH model, the
information matrix is block diagonal, with blocks for the mean and variance parameters.
Therefore the regression coefficients and the ARCH parameters can be estimated sepa-
rately without loss of asymptotic efficiency. Also, their variances can be obtained sepa-
rately. These results do not hold for the ARCH-M model as the parameters of the
conditional variance process affect the conditional mean of the series.

Given the density of a) , the likelihood can be formed. Suppose that the density of (”t
. . . -1 -1 . . —1 —1 _
lS d(mt) . Then the densrty of 8: 1S at d(°t at) and the densrty of yt 18 at d(°t (yt—ut)),

as before, we assume it! = g(Qt_ 1, 61) and a? = h(Qt_1,92). So the log likelihood func-
tion is
T T

—1
lnL = —% Z ln6t2+ Z 1nd(ot (yr—[.19). (6)
t = l t = 1

The maximum likelihood estimates (MLES) of 61 and 92 are then typically found by

numerical maximization.

Consistency of the MLES generally requires that the density of m, be specified cor-

rectly. An important exception is the assumption of normality. The (quasi) MLE obtained

by maximizing the normal log likelihood is consistent even if the normality assumption is

2

violated, as long as at and at

are correctly specified. This point is discussed in more

detail in section H of chapter 3.

111. Specific models for of

1. ARCH process

Engle (1983) that considers the discrete ARCH process, {at}.

0’2 = k+or(L)et2 (7)

8:0)0’ (8)

where (0‘ is iid (0, 1), Et_ 1(8t/O't) = O, VARt_1(et/ot) = 1, Ldenotes the lag or back-

ward shift operator, a(L) a alL + 0:sz + + 0:qu , o is a positive time-varying and mea-

t
surable function with respect to the information set available at time t— 1 , and Er _ l(...)

and VAR t _ 1(...) refer to the conditional expectation and variance with respect to this same

information set. {at} is serially uncorrelated with mean zero, but the conditional variance

of the process, 02 is changing over time.

t9

2. GARCH process

The symmetric GARCH(p,q) specification of Bollerslev (1986) added flexibility to the

ARCH(p) model of Engle(l983). The model is defined by

0‘2 = k + or(L)et2 + 8(L)o?' (9)

where or(L) a alL + 0:sz + + (1qu , and 8(L) a 81L + 82L2 + + SPLP. An important spe-

cial case is the GARCH(],l) model in which p = q = 1 , so that

2_ 2 52
at — K+aet_l+ ot_1.

A main attraction of the GARCH model is that low-order models, like the (1,1) model,

have often been found to be empirically adequate.
3. FGARCH process

There have been a large number of efforts to study asymmetries in ARCH and
GARCH models. In the standard (symmetric) ARCH and GARCH models, only squared
values of 8 effect the conditional variance, so the Sign of e is unimportant. Models that

allow negative errors to have different effects than positive errors will be called asymmet-

10

ric models. Examples include Pagan and Schwert (1990), Campbell and Hentschel (1992),
Nelson (1991), Zako‘i'an (1994), Rabemananjara and Zako'r'an (1993), Ding et al. (1993),
Glosten, Jagannathan, and Runkle ( 1993), Harvey et al. (1994), Harvey and Shephard
(1993), Hentschel (1995), and Fomari and Mele (1997). Negative equity returns are
thought to be followed by larger increases in volatility than equally large positive returns,
due to leverage effects. The economic explanation for this asymmetry, given by Black
(1976), is that negative excess returns make the equity value less, increasing the leverage
ratio of a given firm, thus raising its riskiness and the future volatility of its assets. This is
called the leverage effect. For example, models such as exponential GARCH of Nelson
(1991), quadratic GARCH of Sentana (1991) and Engle (1990), and threshold GARCH of
Zako'r'an (1994) allow for asymmetry. Volatility switching was added by Fomari and Mele
(1997) to the Sign switching developed by Granger and Terasvirta (1993). The latter allows
the drift term in the GARCH equation to change according to the sign of previous shocks,
while the former captures asymmetries via the impact of past shocks on the level of the
volatility.

A systematic attempt to capture asymmetry in the GARCH model is given by

Hentschel (1995). He defined a family of asymmetric GARCH models (Family GARCH,
or F GARCH) by allowing functions of at other than 8’2 in the GARCH equation, and by

considering power transformations. The FGARCH model is given by

o?‘=x+ao?‘_1fv(et_l)+80tx_l, (10)

ll

E
t
E'b
I

f(£,) =

 

 

—c[§£—b],lcl$l. (11)

t

where eq. ( l l) is the news impact curve introduced by Pagan and Schwert (1990). Here

fv(et_ 1) = [flat _ l)]v , and A and v are parameters. Equation (10) essentially gives a Box-

Cox transformation of the GARCH equation. The usual GARCH(1,1) model corresponds
to b = c = 0 and A = v = 2. Many other models in the literature are special cases of the
FGARCH model. Table 1 lists some of these, along with the corresponding restrictions on
b , c , v , A. Exponential GARCH of Nelson ( 1990), Threshold or Absolute GARCH of
Zako'i'an (1990), symmetric GARCH of Bollerslev (1986), Absolute Power GARCH of
Engle and Ng (1993), and Family GARCH of Hentschel (1995) are representative exam-

ples. The asymmetry of eq. (11) is displayed in Figures 1 and 2.
4. FIGARCH process

Baillie et al. (1996) proposed the symmetric long memory fractionally integrated

GARCH model for the long memory of the squared innovations. The ARMA(m,p) repre-

sentation of 3‘2 for the GARCH(p,q) process is:

(l -(l(L)-5(L))£t2 = K+(1 -6(L))(et2—ot2) (12)

12

where m a max{p, q} , and V: a 2,2 - 02 is mean zero and serially uncorrelated. The fraction-

t
ally integrated GARCH, FIGARCH, is defined by introducing the fractional differencing

operator into the AR polynomial. Thus we obtain:

¢(L)(1 -L)det2 = x+(1_5(2))(ef—ef), (13)

where 0 < d < 1 , and ¢(L) and 8(L) are polynomials in the lag operator of orders p and q

respectively. The fractional differencing operator, (1 - L)d , has a binomial expansion

which is most conveniently expressed in terms of the hypergeometric function,

(1 —L)d = F(-d, 1,1;L)

= X I‘(k-d)l"(k+l)-ll‘(—d)_1Lk
k = o

= 2 nkLk, (14)
k=0

where F (.) denotes the Gamma function. The GARCH equation (9) for the

FIGARCH(p,d,q) model is rewritten as:

13

[1 — 5(L)]O’t2 = K+[l—5(L)-¢(L)(1-L)d]£t2. (15)

Thus, the conditional variance of at can be expressed as follows:

2_ x l q>(L)(l—L)d 2
Gt ‘ 1—8(1)+[ " 1-S(L) 12,

17%.) ”(2)92, (16)

where ML) = AIL + 1.sz + . We call this the reduced form or infinite ARCH version. It

should be noted that the coefficients f‘k decay hyperbolically (Ak is proportional to kd_ l

for large k) rather than exponentially, as is true for the usual GARCH process. This slow

decay generates long memory in 0,2. For the FIGARCH(p,d,q) model in eq. ( 15) to be

well-defined and for the conditional variance to be positive almost surely for all I, all the
coefficients in the infinite ARCH representation in equation (16) must be nonnegative. As
for the GARCH(p, q) process analyzed by Nelson and Cao (1992), generalized conditions
to ensure nonnegativity of all the lag coefficients in ML) have proven elusive. Sufficient
conditions are fairly easy to establish for low-order special cases.

The FIGARCH(p,d,q) process is strictly stationary and ergodic for 0 < d s 1 with the
roots of ¢(L) and 6(L) outside the unit circle. Even though the cumulative impulse

response function converges to zero for 0 s d < 1 , the fractional differencing parameter

14

provides important information regarding the pattern and the speed with which Shocks to
the volatility process are propagated.

In most practical applications relatively simple low-order models such as
GARCH(],I) or GARCH(],Z) have often been found to be adequate. Similarly, the
FIGARCH( Let, 1) model may often be adequate to capture long memory in variance. The

GARCH( 1 , 1) model,

2 2 2
ct =x+aet_1+60t_l, (17)

is rewritten in ARMA(1,1) form as

(1—aL—5L)et2=K+(l—6L)(ef—ot2). (18)

Similarly, the FIGARCH(],d,1) is written as

_ x _(l-—¢L)(l-L)d 2

63—m'i'll l-OL ]£t, (19)

 

where 0 < d < 1 .
Under the assumption of conditional normality, the Maximum Likelihood Estimates

(MLES) for the parameters of the FIGARCH(p,d,q) process based on the sample

{81,82 ..... 27.} may be obtained by maximizing the expression

15

T
LogL((-9;el,£2 ..... eT) = —o.5 ~ T-log(27t)-0.5 )3 [log(ot2)+etzo_2], (20)
I: 1

where 9' a (K,81,...,5p,¢1,...,¢q) . The QMLE obtained by maximizing (20), say ’61, is

consistent and asymptotically normally distributed,

Tl/2(

A —1 -1
OT—Ooj—iN(O,A(GO) B(GO)A(G)0) ), (21)

where A(.) and B(.) represent the Hessian and the outer product of the gradient respec-

tively, both evaluated at the true parameter, 90. This so whether or not the normality

assumption is correct. For different distributional assumptions, the likelihood can be con-
structed using the general expression (6) above.

For further discussion of Quasi MLE, see Bollerslev and Wooldridge (1992) and
Brock and Lima (1996). The latter suggest that the asymptotic properties of the (quasi)-
maximum likelihood estimation rely on the verification of a set of regularity conditions

and that it is not yet known whether those are satisfied for FIGARCH.
5. FIFGARCH process

The Family FIGARCH, or FIFGARCH, model is a combination Of the FGARCH and

16

FIGARCH models. Like the FGARCH model, it allows for asymmetric effects of 81 on

the conditional variance. Like the FIGARCH, it allows for long memory in the conditional
variance process.
The FIFGARCH model modifies the FIGARCH model in the same way that

2

FGARCH modifies GARCH. Thus at

is replaced by (stifle!) and 0,2 is replaced by of”.

Making these changes in the GARCH version of the FIGARCH model (equation (15)

above), we obtain:

(1—8L)ot" = k+[l—5L—(l—¢L)(l —L)d]o}f"(et), (22)

8
t
3—1)
I

where f(et) =

 

 

—c[;—b], IcISl. (23)

t

Alternatively, we can rewrite (22) as

 

d
631' = '§3+[1-(1-¢II)—(81L—L) ]0?‘f’(et). (24)

This model nests existing short memory or long memory GARCH models in a general
Specification. It highlights the relations between those models and offers valuable opportu-
nities for testing sequences of nested hypotheses regarding the functional form for condi-

tional second order moments. Some special cases are discussed below.

17

6. Some special cases

Here we assume p = q = 1 , for simplicity. The asymmetric short memory models can

be embedded by a Box-Cox transformation of the absolute GARCH (AGARCH) model as

follows
Oil-1 v A div—1’1
A = 1H0‘f(fit—1)Gt—l+8 X ’ (25)

E
t
5“”
t

 

where f(et) =

 

at
-c[—--b], IcISl.
at

The exponential GARCH model arises from eq. (25) when A = 0 , v = 1 , and b = O.
For A = v = 1 and [cl 5 1 , eq. (25) specializes to the AGARCH model. The model for the
conditional standard deviation suggested by Taylor (1986) and Schwert (1989) arises
when A = v = 1 and b = c = O. Zako’r’an’s (1994) TGARCH model for the conditional
standard deviation is obtained when A = v = 1 , b = 0 and [cl 3 1 . The GARCH model
arises if A = v = 2 and b = c = 0. Engle and Ng’s (1993) nonlinear asymmetric GARCH
corresponds to the values of A = v = 2 and c = 0 , whereas the GARCH model proposed
by Glosten-Jagannathan-Runkle (1993) is obtained when A = v = 2 and b = 0. The non-
linear ARCH model of Higgins and Bera (1992) sets A = v with b = c = 0. The asymmet-
ric power ARCH (APARCH) of Ding, Granger, and Engle (1993) sets A = v with b = 0
and [cl 5 1 . The log likelihood and QMLE are assumed to follow eq. (20) and eq. (21)

respectively. For the more complex details we include Table 1.

18

The models just listed are short-memory models that are special cases of Hentschel’s
FGARCH model. We could also modify them to allow for long-memory; that is, we could
consider the corresponding special cases of the FIFGARCH model. For example, the FIA-
PARCH model of McCurdy and Michaud combines the FIGARCH model with the
APARCH model of Ding, Granger and Engle (1993), and thus corresponds to b = 0 and

A = v . We will consider some of these special cases in the empirical work in chapters 3

and 4.

IV. Autocorrelations of a? and 0'2

We derive the autocorrelations of 8’2 and 0‘2 in both the symmetric and asymmetric
GARCH( 1,1) models. Ding and Granger (1996B) gave the expression for the correlations
of the 6‘2 for the stationary symmetric GARCH(1,1) model. These autocorrelations may

be useful for a variety of purposes. For example, a reasonable check of model specification
would be to compare the sample autocorrelations of the 2‘2 with the autocorrelations
implied by the fitted model. We extend their results to the asymmetric GARCH(1,1)
model. Also, we consider the autocorrelations of the conditional variance of . Especially
when considering questions of persistence or long memory, it is reasonable to ask whether
one should think in terms of the autocorrelations of the .22 or the 02 and it is useful to

t t’

have expressions for both.

19

1. Correlations of 5‘2 for the symmetric GARCH(1,1) model

The correlations of 2’2 for symmetric GARCH(1,1) were derived by Ding and Granger

(1996B). For the stationary (or + 8 < 1 )GARCH(1,1) model, with 30:2 + 2018 + 62 < 1 so that

the fourth moment of at exists, they show that

p

2
= [or + ——oi—2:l(a + 6)k _ l. (26)
k,e

1-2a8-8

"N

In the case that the finite fourth moment condition does not hold, they derive the result:

1 _
p 22(a+§8)(a+8)k 1forlarge k. (27)

k,£t

2. Correlations of a? for the symmetric GARCH(],l) model

The autocorrelations p 2 of the conditional variances 0,2 for the stationary (or + 5 < 1 )
k 0
’ t

GARCH(],l) model are as follows:

p 2 = (or+5)k. (28)

k,ot

20

This simple result does not depend on normality or on the finite fourth moment condition.

Its derivation, which is straightforward, is given in Appendix 1.
Comparing equations (26) with (28), we see that the autocorrelations of £2 and of

decay at the same rate. Both are proportional to (a + 6)k. However, the factors of propor-

tionality are different. From equations (26) and (28),

2
=5[ 1-(or+8) 2](or+6)k-l>0 (A.ll)

1—(a+8) +01

 

For equal values of a and 5 , the conditional variance is more strongly autocorrelated than

the squared error in the symmetric model.

The situation is more complex when the condition a + 5 < 1 is relaxed. See Appendix 1

for details.

3. Correlations of 5,2 for the asymmetric GARCH(1,1) model

We first concentrate on the asymmetric GARCH (1,1) model with b -shift (A = 2 ,
v = 2 , and c = 0).

In Appendix 2 we derive the result

21

k — l
(a + 8’) , (29)
1 - 2016’ - 8’2 + 2a2b2]

 

2, 2
p(b) [01+ a6(1+2b)

2 =
k,et

(b)
2
k, at

where 8’ = 8 + orb2 and p = c0rr(£t2, 2‘2_ k) for the model with b-shift but 6 = 0.

We next consider the asymmetric GARCH (1,1) with c -rotation (A = 2 , v = 2 , and
b = 0).

We now obtain

 

,2 2 2
p(c)2 = [05+ or 5+6121c 52 2](or'+8)k-l, (30)
1&8, 1-2a’5-8 +601 c

where a’ = 01(1 +c2).

Finally, for the asymmetric GARCH(1,1) model with both b -shift and c -rotation, the

correlations of the 8,2 depend in a complicated way on the nuisance parameter

2 2 2
g = ZacE(b-wt_1)|b—(ot_1|ot_1(8 _1-0)

and no useful expression is derived. In contrast, in the next section we will see that a use-

ful expression can be derived, in this model, for the autocorrelations of 0,2.

22

4. Correlations of 0,2 for the asymmetric GARCH(],l) model

We derive the autocorrelations of 0‘2 for the asymmetric GARCH(], l) in Appendix 3.

We first consider the asymmetric GARCH (1,1) model with b -shift (A = 2, v = 2 , and

c=0):

9“”, = <a+a'>". (31)

k, at

(b)
k,o

where 8’ = 8(1 + abz) and p j for the model with b-shift but c = 0.

R‘-

2 = c0rr(6 ,
t

Comparing this result to the corresponding result for the symmetric GARCH( 1,1) model,
as given in equation (28) above, we have 8’ = 8(1 + abz) > 8 if a > 0 , and hence

p(b) 2 > p 2. For equal values of a and 8 , the conditional variance is more strongly

k, at k, 6t

autocorrelated in the asymmetric model than in the symmetric model.
Next we consider the asymmetric GARCH(1,1) with c -rotation (A = 2, v = 2, b = 0).

For this model we Obtain the correlations:

p(C)
k, c

2 = (Ot’+6)k, (32)
t

23

where a’ = 01(1 + c2). Since a’ > a for c at 0 , we clearly have p(c) 2 > p 2. Once again,

’0: k, at

for equal values of a and 8 , the conditional variance is more strongly autocorrelated in the
asymmetric than in the symmetric model.

Finally, we consider the asymmetric GARCH( 1 , 1) model with both b -shift and c -rota-

tion (A = 2, v = 2). We obtain

p(17.62) = (a,+5,,,)k,
k,o

t
I 2 II I 2 ~

where a = 0t(1+c ) asabove, 8 = 8+ab , <p(wt;b) = (b—mt)lb-mtl,

(paEcb(wt;b) = E(b—mt)|b—wtl , and 8’” = 8"+ Zaccp.

For a>0,wehave a’>a and 8”>8’>8. (p>0 for b>0 and (p(b) = -tp(—b) underthe

normality assumption. Therefore, for b > 0 and c > 0 or b < 0 and c < 0 ,

b,C b b,C C
p( 2)>p()2>p 2am“; 2)>p()2>p
k,ot k,ot k,ot k,ot k,or k,ot

V. Conclusion

A new type of long memory Family GARCH, called the fractionally integrated family
GARCH or FIFGARCH, has been proposed. It combines previous models that allowed for

asymmetry or long memory, so as to allow for both at once. This model will be applied to

24

stock returns and exchange rates in chapters 3 and 4.
The autocorrelations of squared errors and conditional variances were derived for
symmetric and asymmetric GARCH(1,l) models. The autocorrelations of conditional

variances are different from those of squared errors.

25

Table 1. Various Short memory GARCH models with special names

 

 

 

 

 

 

A v b c Model

0 1 0 free Exponential GARCH (Nelson)

1 1 0 |c| s 1 Threshold GARCH (Zakoian)

l 1 free Icl s 1 Absolute value GARCH (Taylor/Schwert)

2 2 0 O GARCH (Bollerslev)

2 2 free 0 Nonlinear-asymmetric GARCH (Engle, Ng)

2 2 0 free GJR GARCH (Glosten, Jagannathan, Runkle)

free A 0 0 Nonlinear ARCH (Higgins, Bera)

free A O Icl s 1 Asymmetric power ARCH (Ding, Granger, Engle)
free free free [c1 5 1 FGARCH (Hentschel)

 

1. Exponential GARCH (Nelson)

“103: K+a[f(€t_ l)-E(f(€t_ I)” '1’ 51"0,2_1

2. Threshold GARCH (Zakoian) and Absolute value GARCH (Taylor/Schwert)
o = 1c+orot_ 1f(.st_ 1) +8ot_1

t

3. GARCH (Bollerslev), Nonlinear-asymmetric GARCH (Engle, Ng), and GJR GARCH
(Glosten, J agannathan, Runkle)

of = K+a012-lf2(£t— l)+15<52,_1

4. Nonlinear ARCH (Higgins, Bera) and Asymmetric power ARCH (Ding, Granger,
Engle)

a} = “0103: 111mb 1) +803:l
5. FGARCH (Hentschel)
a} = moot); 1f"(et_ l)+50’?"_1
Note that the following is assumed throughout

8!
-c[——b], MS].
of

E

t
—-b
0'

flat) =
t

 

 

26

 

 

Figure l. The asymmetric transformation of /{

(b)

(d)

 

 

 

 

(a)

 

 

(C)

 

 

 

 

 

 

3

3 up}? 3 - .quqexé

81/61

Eli/Or

27

 

U.)

f V(st/<3.)=(le,/ot - bl -c(s,/ot - b))"
-— N

O

O

 

 

 

 

28

 

 

(b) b = 0.50
C = 0.00

 

 

 

 

 

(d) b = 0.50
c = -0.25

 

 

Appendix 1.

We first derive the autocorrelation functions of conditional variances for the covari-

ance-stationary GARCH (1,1) model. For notational simplicity we will write p k a p 2

k, at

and p}: a p 2, and similarly for 7k and 7;.

k, of

When a + 8 < 1 , the GARCH (1,1) process can be represented as follows:

2 2 2
—K+a8t—l+8°t—l’ (A.1)

“Q
I

0
ll

2 K/(l -or-8), (A.2)

where 02 is the unconditional variance of 81' Substituting (A2) to (A1) one gets

O2 = (52(1—01-8)+0te2

: i- l + 50,2. i - (A3)

Rearranging the above equation one gets

2 2_ 5 2 2 2 2 2
ot-o —(or+ )(ot_1-o +aOt—1mt—l—Ot—l

29

= (a + 8)(ot2_ 1 - 02) + aotz_ [(0113: 1 - I). (Pt-4)

where as before 81 = Uta)! and the “’1 are iid (0, 1). Now, for k > 0 , multiply both sides by

(oi k - 02) and take expectations to obtain

y;=(or+8)y;_l,k21. (A.5)

In evaluating the required expectation, we note that

Eotz_ 1(mtz_ 1 - 1X01 k — 02) = E(m3_ l - 1)[012— 1(6?_ k — 62)] = 0 (A6)

since, for k 2 1 , of_ 1 and 612- k are functions of (“t-j , jz 2 , which are uncorrelated with

m2_ 1 in light of the iid assumption on the a)

t t'

Clearly (A.5) implies that
s k
pk = (n+8) (A.7)

which is equation (32) of the main text.

To obtain an explicit expression for 7;, we can further assume that 30:2 + 2018 + 82 < 1 ,

30

so that the fourth moment of at exists. From the conditional variance equation one can get

4 _ 04(1-01—8)(1+0r+8)

E0 —
r-1
1— (3012 + 208 + 82)

 

(A8)

2
Substituting (A.8) into 7:) = B(otz - 02) = E0: — o4 and doing some simple algebra shows

 

s 201204
Y0 = 2 2 ° (A9)
l—(30t +2018+8)
s s k
Then 7k = 70(01 + 8) for k 2 l . (A.10)

We compare the autocorrelation functions of squared errors p k with those of conditional

variances pi. Note that equations (30) and (31) below for the autocorrelation functions of

squared errors refer to those in the main text.

For the stationary (a + 8 < 1 ) GARCH(],I) model, with 3012 + 20:8 + 82 < 1 so that the

fourth moment of 81 exists,

2
pk = [a+————a 6 2](01+8)k_1. (30)
l-2a5-8

31

In the case that the finite fourth moment condition does not hold,

pica-(01+§l$-8)(0r+8)k‘l forlarge k. (31)

From equations (A.7) and (30),

1—(or+8)2

 

pic-pk: I: 2:l(01+8)k-1>0 (A.ll)

l-(0r+6) +01

since 01 + 8 < 1 so that both denominator and nominator are positive.

From equations (A.7) and (31) in the main text, obviously

2 k—l
pi-pk=§8(a+8) >0. (A.12)

1—(01+8)2

l-(a+8) +012

 

Of course 3012 + 2018 + 82 <1: 8[ ](01 + 8)k —1>§8(a + 8)k- 1 . This sim-

plifies some other comparisons.
The situation is quite different when the covariance-stationary assumption is removed.

We consider the IGARCH(1,1) case which 01 + 8 = 1 . Assume

mt~iid N(O,l), 02 = (182_1+(1-a)0't2_1, (A.l3)

8:00) I t

t It’

32

1 , a constant. Then

and 00

_ 2 1 2 2
e! — aet_l+( -01)ot_l)wt

and it is not difficult to Show that:

E02=1,

K

t t
E6? (1 +2012) E03 = (1+2012) ,

2 2 4 2t—k
EEtEt—k=(l+2a)E°r—k=(1+2a)(1+2a ) ,
2
2 2 2
EeIEet_k = (Eot_k) =1,

2 t
2 4 2 2
V2=3(1+201 )Eot—k—(Eor-k) =3(1+201)—1,
81
2 t—k
4 2 2
V 2 = 3EOt—k_(E°r-k) = 3(1+20t) —1,
er—k

33

(A. 14)

 

W)

pl

2 4 zt-k
[—

2
Eoto k=E°t—k=(l+2a) ,

2
4 2 2'
V 2=Eot—(Eot) =(1+201 ) -1,
0t

2 r—k
(1+20t)(1+20t) -l

pk"=J 2! J 2t-k
3(1+20r)—13(1+201) -l

1

2 L—k
s (1+201) —1

pk,t=J 2t J 2t—k .
(1+201)—1 (1+201) —1

When t» k > 0 and a at 0, one has approximately

2 —k/2
pk'~'1+3 a(.l+2a2) ,

 

—k/2
P:=(1+2a2) .

34

(A.lS)

(A.l6)

(A.l7)

(A.l8)

 

 

and

pi>pk. (A.l9)

It is seen that the autocorrelation function decreases exponentially. In the extreme case

01 = 0, so that 0’2 = “3-1 = = of = 02, i.e., the variance is constant over time and

there is no heteroskedasticity, then (A.15) and (A.l6) give p k = p; = 0. On the other

extreme, if a = 1,50 that 0’2 = 82_

r 1 , then (A.15) and (A.l6) give

 

p,“ = ~/(3""“—1)/(3”'—1), (A20)

 

p1,: J<3“’"—1>/(3’-1). (A.21)
When 1» k > 0 , (A20) and (A.21) become
pk = p}: z 4‘” (A22)

and again it is exponentially decreasing.

Similar results can be derived for the IGARCH (1,1) process with a drift. Assume now

that

35

2 _ 2 1 2
0t — K+aet_1+( —01)0t_l
and 03 = K, aconstant. Then

E0,2 = (t+ l)1<.

When or at O and t is large, E0? is approximately as follows

 

2
4 2
(1+201)Eot__ k' -__(Eot k)
o4
6—t

[3... #121351 1463—le

36

(A23)

(A.24)

(A.25)

(A.26)

(A.27)

(A.28)

4 2 2
s Eot_k—(Eot_k)
pk = . (A.29)

lie?41031214114103- If]

 

When 1» k > 0 , one has approximately

 

—k/2
pkz l +3201“ + 2012) , (A30)
3 2 -k/2
pk==(l+201) . (A.31)

Comparing (A. 17) with (A.30), it is seen that the autocorrelation functions for IGARCH

(1,1) models with or without a drift are the same.

37

Appendix 2.

We derive the autocorrelation functions of squared innovations for the covariance-sta-
tionary asymmetric GARCH (1,1) model under the assumption of conditional normality.
The algebra is very similar to that in Ding and Granger (1996B, Appendix).

When a + 8 < 1 , the asymmetric Family GARCH (1,1) with b -shift and c -rotation can

be represented as follows:

{145.311, (A.1)

0)V = 1c+01fv(et_ 1)“:

t

where f(st) = let/ot—bl —C(£t/Ot_b) , —1<c<l. (A.2)

We first concentrate on the asymmetric GARCH (1,1) with b -shift (A = 2 , v = 2 , and

c=0).

t2: K+af2(Et—l)63—1+5612—1’ (A.3)

0’

where f(et) = let/ot—bl , (or = et/ot, wt~iid D(0,1).

We rewrite (A.3)

38

02 = K+ 0182_ l +(8-i-01b2)ot2_1 —201b(ot_ 162

t t t- 1 . (A4)

Define 8’ = 8 + 01b2 , and 02 = x/ (1 - a — 8’) , where 02 is the unconditional variance of at.

Substituting 1: = 02(1 — a — 8’) to (A4) one gets

2 2 , 2 , 2 2
at = o (1—01—8)+01£t_1+80,_1—20rb0)t_lot_1. (A.5)
Rearranging the above equation one gets
2 2 , 2 2 , 2 2 2 2
et—o =(a+5)(€t—1"°)+(1‘5L)(°tmt‘01)‘20‘bw1-101—1‘ (A.6)

Multiplying both sides of the above equation by (8’2_ 1 - 02) and taking expectations one

has

4

t_ 1 , (A.7)

71 = (01 + 8378 — 25’EO

where 71 = E(812 - 02)(et2_ 1 — 02) is the covariance between 8‘2 and etz_ 1 while
2

2
2 . . 2
70 = 15(81-1’0 ) IS the variance of et_1.

Dividing both sides of (A.7) by 70 , which is finite, gives

39

 

p1 = or + 8’ - 2850;; 1 . (A8)
Also by definition

70 = E(et2_1—02)2 = 3150:: 1—04 , (A.9)

E011 2 %(yo + 04). (A.lO)

If it is further assumed 3012 + 2a8’ + 8’2 + 4012b2 < 1 , so that the fourth moment of 81 exists,

from the conditional variance equation one can get

4 04(1—01-8’)(1+01+8’)

E0 = .
t- 1
l - (3012 + 2118’ + 8’2 + 4012b2)

 

(A.ll)

Substituting this into (A9) and some simple algebra shows

4 , ,2 22
Y = 20' (1-205-5 +201b) . (A12)

0
l- (30:2 + 2018’ + 8’2 + 4012122)

 

40

)k—l

Combining (A8) with p k = p 1 (01 + 8’ , k 2 2 , one has the autocorrelation function for

the asymmetric GARCH (1,1) process as follows:

k“. (A.l3)

 

2 , 2
pk = [01+ 01 8(14-22b )2 2](01+8’)
l-2018’-8’ +201 b

We next consider the asymmetric GARCH(1,1) model with c ~rotation (A = 2 , v = 2 ,

and b = 0):
2 2 2
at = n+nfl(e,_1)et_l+50,_1, (AM)
where flat) = let/otl -C(Et/ot) , —l < c<1.

We rewrite (A. 14) as

2 , 2 2 2
at = 1c+ast_]+8ot_]—201cltot_llmt_lot_1, (A.15)

where 01’ = 01(1 +c2).

02 = K/(l—Ot’-8), (A.l6)

41

where 02 is the unconditional variance of 81' Substituting (A.l6) to (A. 15) one gets

2 2 , , 2 2 2
at = 0(1-01 -8)+01 Et-l+801—1-2aclwt—llmt-lOt—l' (A.l7)
Rearranging the above equation one gets
2 2 , 2 2 2 2 2 2
81 —o = (a +8)(€t—1'° )+(1—8L)(otwt -ot)-201clwt_1|wt_lot_ 1. (A.18)

Multiplying both sides of the above equation by (s?_ 1 — 02) and taking expectations one

has

71 = (a' + 8m, - 28156;: 1 . (A. 19)

where y] = B(etz — 02)(€12- l - 02) is the covariance between 2‘2 and £1 1 while
2

2
2 . . 2
70 = B(et_1-o ) is the vanance of £1- 1.

Dividing both sides of (A.l9) by 70 , which is finite, gives

p1: 01’+5-28E “1. (A20)

 

42

Also by definition

2 2 2 4 4
70 = E(Et_1—O’ ) = 3E0, —o , (A.21)
4 1 4
E0, = 3(yo+o ). (A.22)

If it is further assumed 301’2 + 201’8 + 82 + 12012c2 < 1 , so that the fourth moment of at exists,
from the conditional variance equation one can get

4 , 1
E011 = a (12-01 —8)(l +01 +8) (A23)

1— (301’ + 201’8 + 82 +12012c2)

Substituting this into (A.21) and some simple algebra shows

204(1—201’8-82 + 6012c2) (A 24)

22'

YO =
1-(301’2+201’8+82+1201 c )

Combining (A.20) with p k = p 1(01’ + 8)k ’ 1 , k 2 2 , one has the autocorrelation function for

GARCH (1,1) process as follows:

43

 

[ , 01’28 + 6012c28
01 + 2

2 2](01’ + ii)" ‘ 1 . (A25)
1-201'8—8 +601 c

pk:

Finally we consider the asymmetric GARCH (1,1) with b -shift and c -rotation (A = 2 ,

v=2),

2 2 2
at = K+af2(et-l)°t—l+561-l’ (A.26)

where f(et) = let/ot-bl _C(Et/Gt_b)’ —l <c< 1.

We rewrite (A.26) as

2 ’2 ”2 I 2
o =1c+01£_1+8ot_l-201btot_lo_

2
t t 1 1‘20“le 1 ‘bl(‘°:— 1 “b)":- 1’ (A-27)

where 01’ = 01(1+c2), 8” = 8+01’b2. Define
¢(w,;b) = (b—w,)|b—co,|,

(p a 58mph) = EU’ ‘ “91b - (”ti

= b2W0-2le + W2.

where Wi = If” wif(w)dm—I:wif(w)dw , i = 0, 1,2. Under the assumption of normality,

(p = (b2 +1)(2<D(b)—l)+ 2b¢(b)

where ¢(.) and ¢(.) are the p.d.f. and c.d.f. under the normality assumption respectively.

Set

6’” = 8”+2acq> andfi = (3)—(p.

Define

02 = K/(l — a’- 8’”) , (A.28)

where 02 is the unconditional variance of 8r Substituting (A.28) to (A.27) one gets

2 2 I ”I I 2 ”I 2 I 2 ~ 2
0 =0 (l-a -6 )+ae 1+8 ot_l-Zabwt_lot_l+2acnot_l. (A.29)

t t-

Rearranging the above equation one gets

45

£2 — 02 = (a’ - 8”’)(e;2_ 1 - oz) + (l - 8”’L)(o;2wt2

t 2_ 1 + Zacfioi 1 .(A.30)

2 ,
—o )—2a bwt_ 1°:

t

Multiplying both sides of the above equation by (8’2_ 1 - 02) and taking expectations, one

has

71 = (a’ + 8"’)yO — 25”’Eo;1_1+§, (A.31)

where 71 = E(et2 — 02)(e;7'_ 1 — 02) is the covariance between 62 and 8,2- 1 while

I

2 2 2 . . 2 - 2 2 2
70 = E(€t_ l -o ) IS the variance of et_1 and§ = E2acnot_ 1(st_ l —o ).The presence

of the nuisance parameter g prevents us from obtaining a useful expression for higher-

order autocovariances or autocorrelations.

46

Appendix 3.

We derive the autocorrelation functions of conditional variances for the covariance-
stationary asymmetric GARCH (1,1) model.
When a + 6 < 1 , the asymmetric Family GARCH (1,1) process with b -shift and c -rota-

tion can be represented as follows:

1.1 +So?_l, (A.1)

oA = K+afv(et_ l)ot

t
where f(e,) = |e,/o, - bl — c(e,/o, — b), -1 < c < 1 . (A.2)

We first concentrate on the asymmetric GARCH (1,1) process with b -shift (A = 2 , v = 2 ,

and c = 0):

of = K+ af2(€t_ l)(I;7'_ l + 50’2_ 1 , (A.3)
where flat) = let/ot-bl , tot = et/ot, mt~iid D(O,l).
We rewrite (A.3) as

0‘2 = K+a82_l+(5+ab2)0t2_1—2abwt_10’2 (A4)

I t—l'

47

Define 8’ = 8 + ab2 and 02 = K/(l — a — 5’) , where 62 is the unconditional variance of 8r

Substituting K = 02(1- a- 8’) to (A.4) one gets

02 = (12(1-01-25’)+oze2 l+5’ot2_ 1 —2abmt_ 1°?- 1. (A5)

t [—
Rearranging the above equation one gets

2

2
t—1)-2abwt—l°t—l‘ (A.6)

2 2_ 8’ 2 2 2
ot-o —(oc+ )Ot-l_6 +aet_l-o

Multiplying both sides of the above equation by (oi k - <52) , for k 2 1 , and taking expecta-

tions one has

7:3 = (on + 6372": (A.7)

where 725 = E(°t2 - 02)(ot2_ k - 02) is the covariance between 02 and oi k' Superscript

t

“ 9,

“b ” indicates b -shift, while 5 represents a covariance for the conditional variance. This

implies

pies = (a + 5’)k. (A.8)

48

2

To proceed further, assumed that 3a2 + 2a8’ + 8’2 + 40: b2 < 1 , so that the fourth moment of

at exists. From the conditional variance equation one can get

 

 

4 I I
E0211: 0 (12—0t—8)(1;or+82)2 . (A.9)
l—(3a +2a6’+8’ +4a b)
Substituting this into 73s = Ea:l — o4 and doing some simple algebra show
4 2 2
7’83 = 20‘ a (1+2b ) (A.lO)

1- (30:2 + 2a6’ + 8’2 + 4a2b2)

b b b b , k
Then “(ks = 76-ka = 708(a+8) .

For the symmetric GARCH( l ,1) model, we showed in Appendix 1 that p: = (a + 8)k.

In the asymmetric case, p23 = (a + 8)" where 8’ = 8 + abz. If a > o , 8’ > 8 and p23 > p2;
the conditional variance is more strongly autocorrelated (for equal values of a and 8) in
the asymmetric case.

Next we consider the asymmetric GARCH (1,1) process with c-rotation (2» = 2 ,
v = 2,andb = 0):

0,2 = x+af2(et_l)ot2_l+80t2_l, (A.ll)

49

where flat) = let/otl _c(£t/Ot) , -l < c< 1.

We rewrite (A.1 l) as:

2_ ,2 6 2 2 2
at — K+a€t-l+ Ot-l- aclwt—llwt-lOt-l’

where a’ = a(1 + c2) and o2 x/(l - a’ -8) , where o2 is the unconditional variance of

81' Substituting 1: = o2(1 — 0t’ - 8) into this equation one gets

2

2 2 , , 2 2
or = o (l —a —8)+a et_1+80t_1—2aclmt_ Ila)“ lCt—l' (A.12)
Rearranging the above equation one gets
2 2 , 2 2 , 2 2 2
o, -o = (a +8)(6t-1_0 )+a(ehl—ot_1)-2aclcot_llmt_lot_l. (A.13)

Proceeding as before, we now multiply both sides of the equation by (oi 1 — o2) , for

k 2 1 , and take expectations. We obtain

, 2 2 2
7:3 = (a +8)yZS_I—2acEmt_llwt_llot_1(ot_k—o ), (A.14)

50

2

where 7:5 = E(o3‘ - oZXot _ k —o2) for the model with c-rotation. As before, mt- llwt- 1'

is uncorrelated with of_ l(o,2_ k-oz) by virtue of the iid assumption. However, for

Emt_ 1"”: _ II = 0 we require the symmetry (e.g. normality) of (0. Under this further

assumption, we obtain:
7:3 = (a’ + SHE”: l .
which implies
p23 = (a’+ 8)k.

We note that a’ = a(1 + c2) > a for c¢0 , so that pzs > p; for equal values of a and 8.

If it is further assumed that 3a’2 + 2a’8 + 82 + 12a2c2 < 1 , so that the fourth moment of

at exists, then from the conditional variance equation one can get

4 , ,
EG?_1 = o (12-0: -8)(l+a +8) (A.15)

1— (3a’ + 2a’8 + 82 +12a2c2)

 

Substituting this into 783 = 15o:t — o4 and doing some simple algebra show

51

4 2 2 2
cs_ 20 (a’ +6ac)
70 ' 2 2 2 - (A.l6)

l—(3a’ +2a’5+82+12a c)

 

Then 7:5 = ygspzs.

Finally, we consider the asymmetric GARCH (1 ,1) model with b -shift and c -rotation

2 2 2
or = K+af2(£t—l)ot—l+50t—l’ (A.l7)

where f(et) = let/ot—bI—dat/ot-b), —l <c< 1.

We rewrite (A. 17) as (A29) in Appendix 2

2 2 I III I 2 III 2 I 2 ~ 2
o =o(1—a-8 )+a£ l+8 6,_1-2abw,_16,_1+2acn0,_1 (A.18)

t t-

where as before a’ = a(1+c2), 8" 8+a’b2, ¢(wt;b) = (b—wt)|b-mt| ,

¢EE¢(wt;b) = B(b-wt)lb_°)tl , 8’” = 8”+2accp, and 1'1 = (ii—(p.

Rearranging the above equation one gets

52

2 2 I III 2 2 I 2 2 , 2 ~ 2
at —o = (a +8 )(ot_l-o )+a(et_l—ot_l)-2a bwt_lot_l+2acnot_l.(A.19)

Now multiply both sides of this equation by (a: k - 02) and take expectations. This yields

bcs , ,,, bcs
Yk =(a‘l'5 )yk—l-ék

where 7:“ 5 5(03 - 02)(o;2_ k - 02) and g k = ZacEfiot2_ 1(otz_ k - 02). Therefore

bs , ,,, bcs
me = (a +8 )Yk_1

which implies

pi“ = (a’ + 8”’)k.

53

CHAPTER 3

CHAPTER 3

Asymmetric long memory in variance of US. stock returns

1. Introduction

In the previous chapter we introduced a new family of GARCH models that allow for
both asymmetry and long memory in variance. In this chapter we apply these newly devel-
oped models to data on daily stock returns.

The remainder of the chapter is organized as follows. In this section we briefly
describe the data and the mean equation specification. Simple examples of the need to
allow for asymmetry and long memory are presented. Section 2 lists the models employed
and discusses measures of fit. Section 3 discusses the estimation of the asymmetric family
short memory or long memory GARCH models and gives the empirical results. The final

section concludes.

1. Data

The stock returns are daily returns of Standard & Poor’s 500 index, defined as the first

differenced logarithms of the index:

yt = ln(pt/pt_l)°

The data span the period from January 3, 1928 to September 30, 1993, and contain 17,582

observations. The source of the data is Compustat.

54

2. Mean equation specification
The conditional mean equation is an MA(1) Model:

yt = C"St+8t+elet-l' (1)

As in chapter 2, let 9: be the set of information available at the time t. Then we

assume B(stlnt_ 1) = 0 so that at is a martingale-difference process. However, it may be

conditionally heteroskedastic. As in chapter 2 we assume

8,: 6,03,, wt~i.i.d. D(O,l), (2)

where at is a positive time-varying and measurable function with respect to the informa-

tion set available at time t— 1 , and VAR(et|Qt_ ]) = 0,2. We considered higher-order

ARMA specifications for the mean equation, but the MA(1) process seemed adequate.
This is also reasonable theoretically, since we do not expect levels of returns to be very

predictable.

3. Simple evidence of the need for asymmetric models

55

The results of this section are based on the residuals at of an MA( l)-symmetric

GARCH( l , 1) model. There has not been much discussion in the literature of how to easily
detect the existence or measure the degree of asymmetry in stock returns without actually
fitting an asymmetric specifications (even though the importance of asymmetries is widely

acknowledged). We consider the very simple method of computing the average values of

2‘2 following positive and negative shocks, respectively. These results are given below.

 

 

 

 

 

 

 

2 Average of
Frequency Est + 1 82
H 1
Negative at 8,769 1.251553 0.000143
Positive at 8,810 1.029605 0.000117
Difference -41 0.221947 0.000026

 

There are 8,769 negative at , and the average value of 83+ 1 given at < 0 is 0.000143. Simi-

larly there are 8,810 positive 8: , and the average value of 23+ 1 given 8: > O is 0.000117.

That is, negative shocks tend to be followed by larger squared errors than positive shocks.
4. Simple evidence of the need for long memory models

Persistent autocorrelations of absolute returns have been much discussed in the litera-

ture. For example, Ding and Granger(1996) and Ding, Granger, and Engle (1993) describe

56

the long memory property of S&P daily 500 stock market absolute returns. The absolute
values of returns have significantly positive serial correlations up to 2,700 lags. In our
example, we are more interested in persistent autocorrelations of the squared errors. This
would be in line with the stylized fact that volatility (variance) shows mean-reverting long
memory while returns are stationary.

As before, let 8: be based on the residual from the fitted MA(l)-symmetric

GARCH( 1,1) model, based on 17,582 daily observations as above. The table below gives

the autocorrelations of our at and a? . The sample autocorrelations of the squared errors

are consistently positive until 2,683 lags. They decrease rather quickly for very small lags,
but then decay very, very slowly. For example, the 10—period autocorrelation of the
squared errors is 0.115, and by 300 periods it has decreased only to 0.075. This very slow
decay at long lags is persuasive evidence of long memory and suggests the applicability of
a fractionally integrated model for the variance.

In contrast, autocorrelations of the at are very small at all lags, and are sometimes pos-

itive and sometimes negative. This is as expected since returns themselves should be

unforecastable.

2

Figures 1 and 2 give a graphical display of the autocorrelations of at

and at , and also

support the conclusions given above.

57

 

II. Methodology

1. List of Models

The conditional variance equation is assumed to follow one of the following short or

long memory family GARCH models:

03‘ = K + aogfl lf"(st _ 1) + 50?: 1 , for the asymmetric family GARCH; (3)

 

d
0,)“ = 5-5 + [1 1141141911- L) ]otlf"(et) , for the asymmetric family FIGARCH; (4)

where in either case

58

8
t
3"”
t

 

 

—c[2—b],lcl$l. (5)

t

f(£,) =

As in chapter 2, we have 8: = otwt , to ~ i.i.d. D(0,1) . The members of the short memory

t
family of eq. (3) are listed in Table l of the previous chapter. For the “short memory” fam-
ily members we have d = 0 , whereas the “long memory” models have d at 0. Similarly, the
“symmetric” models have b = c = 0 , whereas “asymmetric” models have b and/or c ¢ 0.

Some special symmetric models that are of interest are as follows:

0’2 = K+aet2_l +6otz_l,forGARCH(l,l) (A. = v = 2, d = 0) (6)

l

a} = “but, et_1/ot_l +5cf~_l,forNGARCH(1,1)(x = v, d = o) (7)

/ot_1v+86?'_1,forFGARCH(1,1)(d = 0) (8)

a}: max

t—lE

t—l

6’2 = K/(1—5)+[l—(l-¢L)(l—L)d/(l-8L)]£t2,f0rFIGARCH(1,d,l)(2» = v = 2, d

unrestricted) (9)

a?” = x/(1— 5) + [1 — (1 — ¢L)(1— L)d/( 1 — mule/0410? , for FINGARCH(1,d ,1)

(x = v , d unrestricted) (10)

59

a} = K/(l —8)+[1—(1—¢L)(1—L)d/(1-8L)]|et/otlvoi”,forFIFGARCH(1,d,1) (x, v,

d unrestricted) (1 1)

We can also consider the asymmetric versions of each of the models. These are models
of the general form of equation (4) above, but with the various restrictions given above.
We will denote these with the same names just used, preceded by “ASYMMETRIC”, so

that, for example, ASYMMETRIC NGARCH(1,1) corresponds to A. = v , d = 0, but b at 0

and/or c at 0 .
2. Measures of Fit for Comparing Different Models
(1) Log likelihood values

One measure of fit is simply the maximized value of the log likelihood. If we compare
two different models with the same number of parameters (e.g. NGARCH (1,1) vs.
FIGARCH(1,d ,1)), it is fair to say that the model with the higher log likelihood value fits
the data better. If we compare different models with different numbers of parameters,
comparisons are less clear because extra parameters will tend to improve the fit. However,
nested models can be compared simply using likelihood ratio tests. All of our models are
special cases of the ASYMMETRIC FIFGARCH model, and we can use likelihood ratio
tests to test the restrictions that they impose. This statement assumes that the regularity

conditions necessary for standard inference from the QMLE are satisfied, so we will now

60

give a brief review of the literature on the QMLE in GARCH-type models.

Maximum likelihood (ML) or quasi-maximum likelihood (QML) are often employed
for the estimation of various GARCH models, while the generalized method of moments
(GMM)l is generally utilized for stochastic volatility models. Several authors have
recently developed Bayesian methods for GARCH2 and stochastic volatility models.3 The
simplicity of MLE is an attractive advantage over other methodologies. Under the assump-
tion of conditional normality, the log likelihood is as given in equation (20) of chapter 2. It

is well known that under certain regularity conditions, the normal (Q)MLE of the

GARCH(],I) model, say 07, is consistent and asymptotically normally distributed:

11/2051- 90) —> MD, We“). (12)

Lumsdaine (1992) provides a proof of the consistency and asymptotic normality of the

ML-estimator for the GARCH(],I) and IGARCH( 1 , 1) models under the condition that

E[ln (are? + 8)] < 0. Unlike models with a unit root in the conditional mean, the ML estima-

tor has the same limiting distribution in models with and without a unit root in the condi—
tional variance. Bollerslev and Wooldridge (1992) and Gouriéroux (1992) showed that the
quasi-MLE of 9 for the GARCH model, obtained by maximizing the normal log likeli-
hood function even though the true probability density function is non-normal, is consis-

tent and asymptotically normal. Weiss (1986) showed this earlier for the ARCH model.

 

1. See Glosten et al. (1993).
2. See Jacquier et al. (1994).
3. See Geweke (1994).

61

Lee and Hansen (1994) prove the consistency and asymptotic normality of the QMLE of

the Gaussian GARCH( 1 ,1) model, where (”t = et/Gt need neither be normally distributed

nor independent over time. A simulation study by Bollerslev and Wooldridge (1992)
found that the QMLE is close to the exact MLE for symmetric departures from conditional
normality, in finite samples. However, for nonsymmetric conditional distributions, both in
small and large samples the loss of efficiency of QML compared to exact ML can be quite
substantial. Palm (1996) argues that semi-parametric density estimation as proposed by
Engle and Gonzalez-Rivera (1991) using a linear spline with smoothness priors would be
an attractive alternative to QMLE. Palm (1996) also suggests that indirect inference as in
Gouriéroux and Monfort (1993) and the efficient method of moments of Gallant et al.
(1994) would be good alternatives when (Q)MLE is difficult to apply.

How good the normal QMLE is depends firstly on the distribution assumptions of the
errors. Although GARCH combined with conditional normality generates fat-tails in the
unconditional distribution, it does not fully account for the excess-kurtosis present in
many financial data. The t-distribution, normal-Poisson mixture, the normal-lognormal
mixture, generalized error distribution, Bemoulli—normal mixture, and stable distribution
are used by numerous authors in this context.

Secondly, asymmetric errors raise questions that have not been addressed rigorously in
the literature. Not much is known about the properties of the normal QMLE in the pres-
ence of asymmetry, nor has anyone checked whether Hentschel’s FGARCH models satisfy
standard regularity conditions for MLE.

Thirdly, the long memory mean reverting features present in financial data are not

included in short memory GARCH, while the FIGARCH model has no terms to represent

62

asymmetry. Brock and Lima (1996) suggest that the asymptotic properties of the (quasi)-
maximum likelihood estimators discussed by Baillie et al. (1996) rely on the verification
of a set of conditions put forward by Bollerslev and Wooldridge (1992) and that it is not
yet known whether those conditions are satisfied for FIGARCH or FIEGARCH processes.
Proceeding to the ASYMMETRIC FIFGARCH model (and its special cases), it is simi-
larly true that, while there is no specific reason to doubt that the required regularity condi-

tions hold, this has not been verified.

(2) Fit of 0,2 to 2,2

Since 423'!!! _ 1) = 0'2 , any measure of the closeness of 3:2 to 0,2 can be a reasonable

measure of the fit of the model. For example, the normal log likelihood value is propor-

tional to 2[log (a?) + 23/03]. A more direct measure of fit might be the sum of squared
t

differences:

2
SSD = 2(812 — of)
t

or the sum of absolute differences:

SAD = flag—03'.
t

63

The SAD measure puts less weight on extreme observations.

(3) Comparison of actual and theoretical autocorrelation of 8‘2

Let p k be the correlation between 2,2 and etz_ k (k = 1,2,... ) implied by a particular

model. The form of these correlations was derived in the previous chapter for some of the

models we consider. For more complicated models, these correlations could be calculated

by simulation, given specified values of the parameters. Now let p k be the sample autocor-

relation between 22

I and of_ k. If the model is correct, the 5,, should be close to the p k,

and this is a basis for measures of model adequacy. We will consider the measure

It
.. 2
SSRHO = Z (Pi-Pi)

i = l
where “n ” indicates the highest order autocorrelation that we seek to match.

[11. Empirical Results

In this section and Tables 1-6, we report our empirical results.

Table 1 gives the quasi-MLE estimates of the parameters, with robust standard errors
in parentheses; the log likelihood values, denoted “Likelihood”; and the measure

2

I, of the quality of the one-period-ahead forecasts of 2,2.

SAD = Elsi-0’
I

1. Log likelihood values

The ASYMMETRIC FIFGARCH model has the highest log likelihood value, as it
must because it nests all of the other models. The differences in log likelihood values are
quite large. There is very strong evidence of asymmetry, since each asymmetric model (b ,
c unrestricted) has a log likelihood value that is more than 100 larger than the value for the
corresponding symmetric model (b = c = 0 ). There is also strong evidence of long mem-
ory. For example, the log likelihood for FIGARCH is about 80 larger than the log likeli-
hood for GARCH; and similarly the log likelihood for ASYMMETRIC FIGARCH is
about 80 larger than the log likelihood for ASYMMETRIC GARCH.

Broadly speaking, the results in Table I favor the ASYMMETRIC FIFGARCH
model. All of its parameters except the intercept are statistically significant at usual levels.
Furthermore, likelihood ratio tests would reject the restrictions that lead to any of the sim-
pler models. We will discuss these tests in more detail below, but for now we simply note
that the ASYMMETRIC FIFGARCH model may be said to have a “significantly” higher

log likelihood value than the other models.

2. Fit of of to of

65

We consider SAD = 2's? — otZI , given in the last line of Table l.
t

(l) Symmetric family

More complicated models generally have smaller values of SAD than simpler models.
The exception is that NGARCH has a larger value of SAD than GARCH. The FIFGARCH

model has the smallest value, 2.4217, and therefore fits best in the sense of SAD.
(2) Asymmetric family

The SAD values of 2.3823 of FIFGARCH indicates that it has the best predictive fit
among all of the models, symmetric or asymmetric, long memory or short memory, power
transformed or not. ASYMMETRIC FIGARCH is not favored over symmetric
FIGARCH. Otherwise, models with more parameters are better in prediction than simpler
ones. This is further evidence that relatively complicated models, such as ASYMMETRIC

FIFGARCH, are supported by the data.

In Table 2 we present some additional information on the prediction of 2,2. Notably,

2
we present SSD = 2(8? — of) as well as SAD = 2's? — 03' . The comparison of SSD is
t t

much the same as the comparison of SAD. The ASYMMETRIC FIFGARCH model has

the smallest value of SSD among all models considered. More generally, asymmetric mod-

66

els fit better than the corresponding symmetric models, and in fact more complicated mod-
els essentially always fit better than simpler ones. As before, we conclude that a data set of

over 17,000 observations will support a fairly complex parameterization.

3. Comparison of actual and theoretical autocorrelations of a?

In this section we compare the sample and theoretical autocorrelations of 2,2 , as a

measure of the adequacy of our fitted models. For a given model, let p j be the jth sample

autocorrelation of the 8’2 and p]. be the jth theoretical autocorrelation, evaluated at the

estimated parameter values. Our summary statistic is

m

, 2
2(w-w)
i=1

where “m ” is the highest-order autocorrelation considered. Because we have long memory
in variance, large values of m may be relevant. We display results in Table 3 for values of
m from 1 to 5,000.

We provide this measure for these models: GARCH, ASYMMETRIC GARCH, and
ASYMMETRIC FIFGARCH. For GARCH and asymmetric GARCH the theoretical auto-
correlations were calculated using results from chapter 2. For ASYMMETRIC FIF-

GARCH, they were obtained from a simulation. The simulation used T = 17, 582 (as in the

67

sample), with artificially generated data from the fitted ASYMMETRIC FIFGARCH
model, with conditional normality assumed. 6,000 observations were generated and dis-
carded (for purposes of initialization) before each artificial sample was drawn. The num-
ber of replications was 100; given the large sample, more replications did not change the
results perceptibly.

We see in Table 3 that, according to this criterion, ASYMMETRIC GARCH is better
than GARCH, and ASYMMETRIC FIFGARCH is better than ASYMMETRIC GARCH.

This is more evidence of the relevance of allowing for both asymmetry and long memory.
4. Tests of hypotheses and further discussion of results
(1) Introduction

In this section we test various hypotheses concerning the parameters in our models.
We are especially interested in testing the restrictions that convert our more complicated
models into simpler ones.

We can test hypotheses in two ways. First, we can construct tests based on the (asymp-

totic) standard normal or chi-squared distributions, using the estimated variance matrix of
the estimates. The usual t -tests fit this category of tests. An advantage is that the estimated
asymptotic variance matrix is robust to violation of the assumption of conditional normal-
ity. Second, we can use likelihood ratio tests based on the maximized values of the likeli-

hood functions. These tests are very simple, but depend on conditional normality for their

validity. We will consider both types of tests.

68

We begin with some general remarks about the results for the ASYMMETRIC FIF-
GARCH model, our most general and best-fitting model. All of the parameter estimates,
except for the scale parameter x , are significantly different from zero by the usual t statis-
tics. The exponents x and v are significantly different from zero, one, two, and each other.
The value of the long memory parameter d , 0.279, is significantly different from zero and
from one-half, so that it supports the finding of long-memory in variance, but does not
lead to nonstationarity or a failure of mean reversion. The values of b and c are signifi-

cantly different from zero, and support the relevance of asymmetry.

(2) Asymmetry tests

The significant estimates of b and c support the idea that asymmetry is an important

feature of daily US. stock returns. For the ASYMMETRIC FIFGARCH model, the “shift ”
parameter b is significantly different from zero based on its asymptotic t-statistic of 5.97.

The “rotation ” parameter c is significantly different from zero based on its asymptotic t-

statistic of 2.3 l. The joint hypothesis b = c = O is decisively rejected with a value of x: of

393.
Similar results are obtained in other models. In virtually every model, the null hypoth-

esis of symmetry (b = c = 0) is rejected. See Table 4 for the relevant likelihood ratio sta-

tistics.

(3) Functional form tests

69

Family models provide easy step-by-step hypotheses test procedures because they

imply sequentially nested functional forms. We begin by discussing the most general
model, ASYMMETRIC FIFGARCH. Here we have I = 1.295 , 9 = 1.629. The ASYM-

METRIC FINGARCH model imposes the restriction A = v , and yields A = 9 = 1.548.
This restriction is rejected by the likelihood ratio test, with a test statistic (x?) of 5.60.
Similarly, the ASYMMETRIC FIGARCH model imposes A = v = 2 , and this model is
decisively rejected by the likelihood test, with a test statistic (7(3) of 86.3. The restriction

A = v = 2 is also rejected in the ASYMMETRIC FINGARCH model, with a statistic

(x?) of 80.71.

Similar results hold for the symmetric long memory models. See Table 5 for more

details.

We conclude that the power transformations in the FIFGARCH and ASYMMETRIC

FIFGARCH model are clearly supported for these data.
(4) Tests of short versus long memory

In the ASYMMETRIC FIFGARCH model, the long memory parameter d is very sig-
nificantly different from zero. With a = 0.279 and an asymptotic standard error of 0.034,

we have an asymptotic t-statistic of 8.20. The estimate of d is in fact very significantly

different from zero in every model, symmetric or asymmetric, in which it is estimated.

70

Long memory in variance is obviously a strong feature of these data.
Table 6 gives the results of more likelihood ratio tests, in which short memory models
(symmetric and asymmetric) are tested against long memory alternatives. In all cases,

short memory is rejected decisively.

VI. Conclusions

In this chapter we have applied the conditional variance models of chapter 2 to a long
series of daily stock returns. The results support the empirical relevance of both asymme-
try and long memory, as embodied in the ASYMMETRIC FIFGARCH model. This model
was consistently found to be better than other simpler models according to log likelihood
values, predictions of squared errors, and closeness of sample and theoretical autocorrela-
tions of squared errors. Simpler models are also clearly rejected by likelihood ratio tests.

Further research will be needed to see whether the ASYMMETRIC FIFGARCH

model performs well in other similar problems.

71

 

 

 

 

 

 

 

 

 

 

 

 

 

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Table 3. Comparison of sample and theoretical autocorrelations of 5,2

m
Entries in the table are Z (‘31- pj)2
i=1

 

 

 

 

 

 

m GARCH ASYMM. ASYMM.
GARCH FIFGARCH

1 0.0495 0.0275 0.0030
10 0.8735 0.5256 0.0501
20 2.0823 1.2150 0.1152
30 3.1669 1 .7394 0.1564
50 5.0761 2.4692 0.2179
100 8.4068 3.1920 0.341 1
200 10.8461 3.2827 0.4844
300 1 1.4038 3.41 18 0.5972
1,000 1 1.6789 4.0499 1.3233
5,000 12.1544 4.5390 2.9024

 

74

 

Table 4. Likelihood ratio tests for asymmetry in volatility (with significance levels)

 

 

 

 

 

 

 

 

 

HA

Maintained
Hypothesis Ho c = 0, b free b = 0, c free b and c free
Short memory
A = 2, v = 2 b = c = 0 310.50 (0.5%) 253.36 (0.5%) 312.06 (0.5%)
GARCH

c = 0, b free 1.56

b = O, 6 free 58.07 (0.5%)
A = v b = c = 0 315.68 (0.5%)
NGARCH

c=Qbfiw

b=QcMw
2., v free b = c = 0 316.98 (0.5%)
FGARCH

c = O, b free

b = O, c free
W
2. = 2, v = 2 b = c = 0 106.01 (0.5%) 94.15 (0.5%) 113.96 (0.5%)
FIGARCH

c = O, b free 7.85 (1.0%)

b = 0, c free 19.81 (0.5%)
2. = v b = c = 0 394.67 (0.5%)
FINGARCH

c=Qbfiw

b=Qchw
A, v free b = c = 0 393.41 (0.5%)
FIFGARCH

c=Qbfiw

b = 0, c free

 

 

75

Table 5. Likelihood ratio tests of functional form in long memory models (with signifi-

cance levels)

 

 

 

 

 

 

Ho HA
Maintained Hypothesis 2. = v A, v free
Symmetric
FIGARCH
A = 2, v = 2 0.46 6.85***
(<0.05)
FINGARCH
A = v 6.39**
(<0.025)
Asymmetric
FIGARCH
2. = 2, v = 2 80.71*** 86.30***
(<0.005) (<0.005)
FINGARCH
A = v 559’”
(<0.025)

 

76

Table 6. Likelihood ratio tests of long memory and asymmetry (with significance levels)

 

 

 

 

HA
Ho
d free, symmetric d free, asymmetric with b and c
Maintained Hypothesis
FIG FIN FIF FIG FIN FIF
Symmetric
GARCH
A = 2, v = 2 162.40* 162.86* 169.25* 47636“ $57.07* $62.66*
(<0.005) (<0.005) (<0.005) ((0.005) (<0.005) (<0.005)
NGARCH
A = v 149.66* 156.05* $43.87* $49.46*
(<0.005) (<0.005) (<0.005) (<0.00$)
FGARCH
A, v free 142.98* $42.78*
(<0.005) (<0.005)

 

Asymmetric with b and c

 

GARCH
A:2,v=2

NGARCH
A = v

FGARCH
7. , v free

 

 

164.30* 235.01* 250.60*
((0.005) (<0.005) (<0.005)
228.19* 233.80*

(<0.005) (<0.005)

22$.80*
(<0.00$)

 

77

Figure l. Autocorrelations of squared errors

 

‘14 18 .22

ACF of squared errors

0.02

 

—-0.02

0 500 1000 1500 2000 2500 3000
Time

78

      

 

 

l

3500 4000 4500 5000

l l

Figure 2. Autocorrelations of errors

 

 

 

ACF of errors

—0.04—

—0.06

 

 

I I l I l l l l I

 

—0.08

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Time

79

CHAPTER 4

CHAPTER 4

Long memory in variance of exchange rates

I. Introduction

The long memory or persistence of autocorrelations of volatility has been documented

in numerous articles on exchange rate returns as well as on stock returns. We apply the

asymmetric long memory process in the conditional variance to exchange rate returns. We

use daily data on DMI$ and Yenl$ exchange rates. In both cases, there is strong evidence

of long memory, but asymmetric models were not really necessary for the DM/$ case.

11. The asymmetric long memory FIFGARCH model

The asymmetric FIFGARCH(1,d,1) model is defined as in chapter 2:

 

d
a?“ = 17511 -“ "1‘30”“ Wadi“, (I)
K V A. 2)
= 1——‘6 + A(L)f (spot , (

E

where f(et) = 81-h
t

 

 

8t
—c[——b], IcI_<.l.
0t

80

We name the following special cases of the FIFGARCH model:
FIEGARCH: x = 0,v =1,FITGARCH:X =1,v =1,FIGARCH:A. = 2,v = 2,

and FINGARCH: x = v but otherwise unrestricted.

III. Results and analysis

We consider daily data on both German Mark and Japanese Yen spot exchange rate

returns from April 2, 1973 through February 13, 1998; which realizes a sample of

T = 6, 241 observations. The returns are defined as ln(pt/pt_ l) where pt is the exchange

rate in terms of foreign currency per dollar (e. g. DM/$).

1. Some relevant descriptive statistics

 

 

 

 

DM-$
Data\Stat. Average Std. dev. Skewness Kurtosis
p t 2.055 0.453 0.517 -0.745
In ( pt/ Pt _ 1) -7E-5 0.007 0.0738 3.553

 

 

 

 

 

 

 

81

Yen-$

 

 

 

 

 

 

 

 

Data\Stat. Mean Std. dev. Skewness Kurtosis
p t 190.16 67.55 0.1507 -1.429
1,. (pt/pt _ 1) -0.00012 0.00623 5.6267 14.281

 

 

As in chapter 3, we calculate the average squared errors 53+ 1 (from a fitted MA(l)

model), conditional on at being positive or negative, as a simple measure of the need for

an asymmetric model. The results are given below.

DM-$

 

 

 

 

 

 

 

2 Average of
Frequency 25‘ + 1 82
t + 1
Negative at 3,089 0.135891 0.000044
Positive 8: 3,149 0.141719 0.000045
Difference -60 -0.005828 -0.000001

 

82

 

Yen-S

 

 

 

 

 

 

 

2 Average of
Frequency Xe, + 1 82
H 1
Negative at 2,960 0.123525 0.000042
Positive at 3,278 0.118329 0.000036
Difference -3 18 0.005 196 0.000006

 

Comparing the respective averages of squared errors for the DM/$ exchange rate (0.00044
vs. 0.00045) following negative and positive shocks, we see that they are very similar.
There is certainly no strong evidence of the need for an asymmetric model. For the Yenl$
exchange rate, there is a larger difference (0.000042 vs. 0.000036, or about 17%), which is
a slightly smaller percentage difference than was found in chapter 3 for stock returns.

Thus there is some evidence of the need for an asymmetric model.

2. Germany (1973/04/02 - 1998/02/13 - 6,241 observations)

Table 1 gives the results from various models fitted to the DM/$ exchange rate return

series. It gives the quasi-MLE estimates, robust standard errors, log likelihood values, and

the sum of the absolute values of the one-period forecast errors of 2,2 (SAD). Table 2 gives

more detail on the forecast errors, including the sum of squared forecast errors (SSD).

83

( 1) Short memory GARCH models

In the GARCH(],I) model, the constant term cnst (-0.000034) is insignificantly nega-
tive. The MA term 9 (0.024) is insignificantly positive. The sum of 5 + a is 0.991, which
is very close to unity, implying considerable persistence of volatility. The asymmetric
GARCH(],I) model gives similar results. This is not surprising, since the asymmetry
parameters b (0.104) and c (—0.020) are small and statistically insignificant. The symmet-

ric GARCH model fits better than the asymmetric GARCH model by the SAD criterion.

(2) Long memory GARCH models

Consider first the symmetric FIGARCH model. The differencing parameter d is esti-
mated as 0.520 and is very significantly different from zero, so there is strong evidence of
long memory. The FIGARCH model is also better (though not much better) than the
GARCH model in terms of SAD and SSD.

Power transformation also seem to be supported by the data. The FIFGARCH model

gives a significantly higher likelihood value than the FIGARCH model (x; = 6.20 is sig-

nificant at the 5% level) and also yields smaller values of SAD and SSD. The FINGARCH

model (FIFGARCH with the restriction A = v) yields an insignificantly smaller log likeli-

hood value (x% = 1.36) than FIFGARCH, and still smaller values of SAD and SSD; there-

fore FINGARCH may be preferred to FIFGARCH or FIGARCH. The FITGARCH model

84

(2. = v = 1 ) is more controversial. It yields a much lower log likelihood value than
FIGARCH, FIFGARCH or FINGARCH, but is superior in terms of SAD and SSD.

The results from asymmetric long memory models are generally quite similar to those
from the corresponding symmetric models. For example, in the asymmetric FINGARCH
model the asymmetry parameters b and c are individually and jointly insignificant. The
same is true for the asymmetric FIFGARCH and FITGARCH models. The asymmetric
models typically have very slightly smaller values of SAD and SSD than the correspond-

ing symmetric models. Overall, there is just not much evidence of asymmetry.

Comparing the sample and theoretical autocorrelations of 5,2 (Table 3), the best mod-

els are FITGARCH and asymmetn'c FITGARCH.

(3) General conclusions for Germany

General conclusions for the DMI$ exchange rate returns are as follows. There is strong
evidence of long memory in variance, but little evidence of asymmetry. In the general FIF-
GARCH model, the FINGARCH restriction 2. = v is supported by the data. The FIN-
GARCH model (with I. = v = 1.774) is better than the FIGARCH model (A = v = 2), and
is better in some ways and worse in other ways than the FITGARCH model (A = v = 1 ).
Measures of fit other than log likelihood value would lead to the choice of the FITGARCH

model as “best”.

3. Japan (1973/04/02 - 1998/02/13)-Dai1y 6,241 observations

85

Tables 4A, 5A and 6 give the same results for the Yenl$ exchange rate as Table 1, 2
and 3 gave for the DMl$ rate. In addition, Tables 4B and 5B give some results for the FIN-

GARCH and FIFGARCH (1,d , 0) models

(1) Short memory GARCH models

The symmetric GARCH( 1,1) model yields a + 6 = 0.993 , again indicating strong per-
sistence of the variance. The MA(l) term 9 is 0.046, which is higher than for Germany
and significantly different from zero. In the ASYMMETRIC GARCH model, the asym-
metry parameters b and c are very significant, and the log likelihood value is much higher

than for the symmetric GARCH model.

(2) Long memory GARCH models

There is strong evidence of long memory in variance. For example, the differencing
parameter d is estimated as 0.363 in the FIGARCH model and 0.312 in the ASYMMET-
RIC FIGARCH model. Alternative power transformations are possibly helpful. For exam-
ple, the ASYMMETRIC FIFGARCH (2. , v unrestricted) and ASYMMETRIC
FINGARCH (A = v but otherwise unrestricted) models do not seem to improve on the
ASYMMETRIC FIGARCH model; their log likelihood values are insignificantly higher
and their SAD and SSD values are not much better. Interestingly, the ASYMMETRIC
FITGARCH model (A = v = 1) offers a clear improvement in terms of SAD and SSD

even though its restriction would be rejected by a likelihood ratio test. The ASYMMET-

86

RIC FITGARCH model is also favored in terms of closeness of the theoretical and sample

autocorrelations of the 8?".

(3) General conclusions for Japan

There is strong evidence of long memory in variance. There is reasonably strong evi-
dence also of asymmetry. As was the case for Germany, the FIN GARCH restriction 2. = v
is supported by the data, and the FIT GARCH model (A = v = 1 ) is favored in terms of

measures of fit other than the likelihood value.

VI. Conclusions

To capture the long memory properties and asymmetry features in exchange rate
returns, the asymmetric long memory FIFGARCH model was employed. The Yenl$
exchange rate returns are found to have significantly negative asymmetries, while the DM/
3 returns are not. In both cases there was significant evidence of long memory in variance.

The full ASYMMETRIC FIFGARCH model was not necessarily required in either
case. The FINGARCH restriction (A = v) seemed useful, and not rejected by the data. The
FITGARCH restriction (x = v = 1 ) typically led to better measures of fit, even though
this null hypothesis could be rejected by a likelihood ratio test. Thus the ASYMMETRIC
FIFGARCH family proved useful here, as it did in chapter 3 for stock returns, even though

the most general member of the family was not needed.

87

 

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Table 3. Comparison of sample and theoretical autocorrelations of sf in German exchange
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Entries in the table are 2 (13,-- 1),-)2
j -l

 

 

 

m GARCH ASYMM. ASYMM. FITGARCH ASYMM.
GARCH FIFGARCH F ITGARCH
1 0.0762 0.0859 0.0173 0.0075 0.0078
10 0.8977 1 .0071 0.2074 0.0838 0.0908
20 1 .8339 2.0647 0.3939 0.1378 0.1497
30 2.6423 2.9929 0.5346 0.1657 0.1790
50 3.9094 4.4847 0.7423 0.1927 0.2058
100 5.7362 6.7484 1 .1 142 0.2348 0.2469
200 6.6327 8.0173 1 .4974 0.2922 0.3081
300 6.7658 8.2436 1 .7324 0.3535 0.3689
1 .000 6.9022 8.4234 2.4547 0.6262 0.6477
5,000 7.5065 9.0278 5.1250 2.1360 2.1641

 

 

 

 

 

 

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Table 6. Comparison of sample and theoretical autocorrelations of sf in Japan exchange

returns

Entries in the table are 2 (pj— pj)2

 

i'l

 

 

 

 

 

 

 

 

m GARCH ASYMM. ASYMM. FITGARCH ASYMM.
GARCH FIFGARCH F ITGARCH
1 0.1216 0.0584 0.0225 0.0183 0.0198
10 1.5083 0.7865 0.3191 0.1141 0.1127
20 2.8803 1.4785 0.5317 0.1336 0.1372
30 4.1302 2.0921 0.7029 0.1439 0.1494
50 6.1310 3.0416 0.9416 0.1555 0.1633
1 00 9.2555 4.5076 1 .3387 0.1796 0.1 876
200 1 1.51 12 5.7681 1.7793 0.2209 0.2324
300 12.0518 6.2766 2.0497 0.2557 0.2681
1 .000 12.2353 7.0891 2.7518 0.4962 0.5168
5,000 12.8318 10.6355 5.5943 1.8165 1.8408

 

95

 

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