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THESIS

 

L58RAR"
Michigan State
University

 

 

This is to certify that the

dissertation entitled

Estimation of the GARCH Model:
Improving the Normal Quasi-MLE by Augmented GMM

presented by

Yi-Yi Chen

has been accepted towards fulfillment
of the requirements for

 

 

Ph .D . degree in Economics

em we

Major professor

Date Apr. 12, 2000

 

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

MAR 3 1 2007

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11.00 W.“

 

"2;. ',r

ESTIMATION OF THE GARCH MODEL: IMPROVING THE
NORMAL QUASI—MLE BY AUGMENTED GMM

By

Yi-Yi Chen

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Economics

2000

ABSTRACT

ESTIMATION OF THE GARCH MODEL: IMPROVING THE NORMAL
QUASI—MLE BY AUGMENTED GMM

By

' Yi-Yi Chen

The standard estimator for ARCH models is the normal quasi maximum
likelihood estimator (NQMLE). We interpret the NQMLE as a GMM esti-
mator whose moment conditions are the normal score function, and we seek
to improve it by adding more moment conditions based on autocorrelations
of squares and on the score function for a rescaled t distribution. These aug-
mented GMM estimators are asymptotically more efficient than the N QMLE
when the data are non-normal. We evaluate the efficiency gain and find that
it can be large, especially when the data are skewed. Simulations indicate
that achieving these gains in practice will require a rather large sample size,
such as 1,000 or more. Finally, we estimate a model for the DM/$ ex-
change rates, and find that the augmented GMM estimator performs largely

as asymptotic theory and our simulations would predict.

ACKNOWLEDGEMENTS

Life is unpredictable. During these years of pursuing the Ph.D. degree,
I have experienced some of the most dramatic moments in my life. Thank
God for giving me the strength and courage to walk through them. I’ll
always remember my mother as a kind and loving woman who dedicated
all her life to the family. I only wish she could share the enjoyment of my
accomplishment with me.

For the completion of this dissertation, first I would like to thank my
advisor Professor Peter Schmidt. The breadth of his research experience and
the depth of his knowledge in economics have provided me with tremendous
help throughout the course of writing this dissertation. His financial sup-
port in the last year and the editing on the dissertation are also gratefully
acknowledged. I also thank Professor Jeffery Wooldridge who gave insight-
ful comments on the dissertation and clarified some assumptions used in
this work. The part of empirical work uses the data provided by Professors
Richard Baillie, Tim Bollerslev and Charles Goodhart. This dissertation
cannot be finished without their assistances. A

Throughout these years of studying in the States, I never confronted
financial problems. For this I sincerely thank my parents-in-law for their

financial supports. My father’s encouragement, both on the emotional and

iii

spiritual sides, are powerful support to me. In addition, I thank my sisters,
Fang-Yi, Ang—Yi and Chu-En, for their unconditional love.

I would also like to thank my fellow classmates and friends for helping
me in different ways. I specially thank Bih-Shiow, Te—Fen, Chiung—Ying, and
Chung-Jung for sharing the happiness and life experiences with me, and also
for driving me to shOpping. At last, I thank my husband Hung-Jen; without

his love and support, I cannot complete this dissertation.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................. vii
CHAPTER 1
ARCH-TYPE MODELS AND ESTIMATION METHODS .......... 1
1.1 Introduction ................................................. 1
1.2 ARCH and GARCH Models .................................. 2
1.3 MLE for the GARCH Process ................................ 8
1.4 QMLE for the GARCH Process ............................. 10
1.5 Improved Normal QMLE and Motivation .................... 16
1.6 Plan of the Thesis .......................................... 18
Appendix 1 ..................................................... 20
CHAPTER 2
EXTRA MOMENT CONDITIONS
AND ASYMPTOTIC ANALYSIS .................................. 24
2.1 Introduction ................................................ 24
2.2 Moment Conditions Based on Autocorrelation of e? .......... 24
2.3 Moment Conditions Based on the Score Function from
the Rescaled Student’s Distribution ......................... 27
2.4 Asymptotic Variance for the Augmented-GMM Estimator . . . 28
2.5 Evaluation and Comparison of Asymptotic Variances ....... 29
2.6 Conclusion .................................................. 33
CHAPTER 3
FINITE—SAMPLE PROPERTIES .................................. 40
3.1 Introduction ................................................ 40
3.2 Design of the Experiment ................................... 41
3.3 Results of the Experiments .................................. 43
3.3.1 Different Distributions ................................ 43
3.3.2 Different Values of Sample Size ....................... 46
3.3.3 w Doesn’t Matter ..................................... 47

3.3.4 Effects of Changing a and fl .......................... 48

3.3.5 Accuracy of Inference ................................. 49

3.4 Conclusion ........ _ .......................................... 50
CHAPTER 4

AN EMPIRICAL STUDY ........................................ 62

4.1 Introduction ................................................ 62

4.2 Estimation of the GARCH(1,1) Model ...................... 63

4.3 Diagnostics ................................................. 65

4.4 Conclusion .................................................. 68
CHAPTER 5

CONCLUDING REMARKS ....................................... 75

Bibliography ........................................................ 78

vi

LIST OF TABLES

CHAPTER 1
Table A.1: Test for the expected value of the score function of the

standardized chi-square distribution ........................ 23
CHAPTER 2
Table 2.1: m2 + 2afl + 62 ............................................. 35
Table 2.2: Asymptotic standard error, w = 1.5, a = 0.15, 6 = 0.7 ........ 36
Table 2.3: Asymptotic standard error, w = 1.5, a = 0.1, fl = 0.75 ........ 37
Table 2.4: Asymptotic standard error, w = 1.0, a = 0.1, ,8 = 0.8 ......... 38
Table 2.5: Asymptotic standard error, w = 2.0, a = 0.2, fl = 0.6 ......... 39
CHAPTER 3
Table 3.1: Monte Carlo simulation result, standard normal distribution

T = 2000, R = 500,w = 1.5,a = 0.15, 3 = 0.7 ................ 51
Table 3.2: Monte Carlo simulation result, standardized t5 distribution

T = 2000, R = 500,w = 1.5,a = 0.15, H = 0.7 ................ 52
Table 3.3: Monte Carlo simulation result, standardized x3 distribution

T = 2000, R = 500,w = 1.5,a = 0.15, [3 = 0.7 ............... 53

Table 3.4: Monte Carlo simulation result, standardized gamma(2)
distribution T = 2000, R = 500,w = 1.5, a = 0.15, 6 = 0.7.. . 54
Table 3.5: Monte Carlo simulation result, standardized t5 distribution

T = 500, R = 500,w = 15,01 = 015,6 = 0.7 ................. 55
Table 3.6: Monte Carlo simulation result, standardized t5 distribution

T = 1000, R = 500,w = 1.5, a = 0.15,,8 = 0.7 ................ 56
Table 3.7: Monte Carlo simulation result, standardized t5 distribution

T = 2000, R = 500,w = 0.15,a = 0.15, 6 = 0.7 ............... 57
Table 3.8: Monte Carlo simulation result, standardized t5 distribution

T = 2000, R = 500,w = 1.5,a = 0.1,5 = 0.75 ................ 58
Table 3.9: Monte Carlo simulation result, standardized t5 distribution

T = 2000, R = 500,w = 1.0, a = 0.1, fl = 0.8 ................. 59

Table 3.10: Monte Carlo simulation result, standardized t5 distribution
T = 2000, R = 500,w = 2.0, a = 0.2, fl = 0.6 ................. 60

vii

Table 3.11: Size of the 5% Wald test ................................... 61

CHAPTER 4

Table 4.1: The first four unconditional moments of the distribution of 6? 70
Table 4.2: Estimation result of DM/$ exchange rate .................... 71
Table 4.3: The conditional distribution of 6th,— ”2 ...................... 72
Table 4.4: SACF and ACF ............................................ 73
Table 4.5: Overidentification test and conditional moment test .......... 74

viii

Chapter 1
ARCH-TYPE MODELS AND ESTIMATION METHODS

1. Introduction

It is an empirical regularity that many data, such as stock returns, commodity prices,
and foreign exchange rates, exhibit ”volatility clustering” — periods of low volatility
(variance) tend to cluster together followed by periods of high volatility. For example,
French et a1. (1987) show that in the period between 1928 and 1990, daily capital
gains have a larger variance during the 1930’s than during the 1960’s. In the case
of the Deutschmark/U.S. Dollar exchange rate during the period of 1981 to 1992,
Baillie and Bollerslev (1989) also show that the daily exchange rates have periods of
turbulence followed by periods of tranquillity.

This volatility clustering property was first documented by Mandelbrot (1963) and
Fama (1965), and was modeled econometrically by Engle (1982) as an Autoregressive
Conditional Heteroskedasticity (ARCH) process. The idea of ARCH modeling is
to allow the conditional variance depend on the history of the series. This is very
different from the traditional model in which the conditional variance is assumed to
be independent of the past information.

The modeling of time varying variances has important implications for dynamic
economic theory and modern finance theory. Recognizing the temporal pattern of
time-varying heteroskedasticity can help to impose the accuracy of forecasts and of

econometric inference. Furthermore, because risk and uncertainty play importance

roles in finance theory, the ARCH model is useful to incorporate risk and uncertainty,

as measured by variances and covariances, into the analysis of asset and option pricing.

2. ARCH and GARCH Models

Suppose that 6,, t = 1,2,...,T is an observable series with E(et) = 0 and unconditional
variance Var(et) = 02. Define \Ilt as the information set at time t. We assert that
E(et|\Ilt_1) = 0. The conditional variance, ht, can be expressed as Var(et|\IIt_1). How

to model this conditional variance is what we are interested in.

The basic model of ARCH type is
6; = hi/z - ut, with at iid D(0,1), (1)

where D is a specified distribution, such as the standard normal distribution or the
standardized t distribution with a certain number of degrees of freedom, etc. The

information set can be expressed as
‘Ilt : {Eta 6t—l, €t-21 ' ' ' ; ht: ht—l: ° ° '}i

or in principle just as
‘Ijt = {61) Ct-—la €t-2: ° ° '}:

since ht ultimately is a function of past observations.
It may be noted that the representation in equation (1) is stronger than some pos-
sible definitions of an ARCH model. For example, we could define the model simply

by the assumptions that E(et|\Ilt_1) = 0, and Var(ct|\IIt_1) = ht. The representation

in equation (1) implies these results for the first two conditional moments, but also
places restrictions on the higher conditional moments. Specifically, it implies that
E(ef|\Ilt_1) = hfflpk where pk E E(uf). The results in this thesis generally require
the validity of the representation in equation (1), not just the correctness of the first
two conditional moment assumptions.

Different specific models are defined depending on how ht is related to \Ilt_1. We
discuss a number of the formulations below.
A. ARCH
A.1 First-order linear ARCH

In the simplest ARCH model, ARCH(1), which was introduced by Engle (1982),

the conditional variance depends only on the latest past squared innovation,

ht =w+a1£f_1, w > 0,011 2 0.

If a, = 0, 6; would be white noise. Otherwise, ct will be dependent through higher
order moments.

A.2 General ARCH (ARCH(q))

The qth-order linear ARCH model is
IL, 2 w + iaicffl. = w + a(L)ef, with w > 0, oz,- 2 0 for each i,
i=1

where L is the lag operator so that Let = et_1 and a(L) = 3:1 aiL’. If and only
if the sum of the a,- is less than one, the process is covariance stationary, in which
case the unconditional variance is a2 = w/ (1 — a1 — a2 — - - - — aq). Engle (1982) uses

this specification to model the uncertainty of the inflation rate. Bodurtha and Mark

(1991) employ the ARCH(3) specification to model monthly NYSE stock returns,
and the same formulation is adopted by Attanasio (1991) to model monthly excess
returns on the S&P 500 index.

Although the ARCH model does imply volatility clustering, there are some dif-
ficulties in empirical applications. For example, without the restrictions on the lag
structure, estimation may result in negative parameter estimates and fail the non-
negative constraints. Also, one may need a large value of q in ht in order to model
the conditional variance correctly.

To avoid these two problems, Bollerslev (1986) proposed the Generalized Autore-
gressive Conditional Heteroskedasticity model, or GARCH model.

B. GARCH(p, q)

The GARCH process modifies the ARCH process by extending the AR process
for c? to an ARMA process, potentially permitting a more parsimonious pararneteri—
zation.

In the GARCH(p, q) model of Bollerslev (1986), ht is defined as follows:
a :2
ht = w + Z (156:4 + Z ,Biht-i, (2)
i=1 i=1
where
p20,q>m
w> 0, (1,20, 2': 1,...,q, 6,- 20, i=1,...,p.
The GARCH(p, q) process is covariance stationary if and only if 2le a¢+Zf=1 6.- <

1, in which case the unconditional variance is 02 = w / (1 — 3:10“ — 2le 3,). For

p = 0 the process reduces to the ARCH(q) process.

4

The leading case of the GARCH model is the GARCH(1,1) model, with p = q = 1.

The conditional variance for the GARCH( 1,1) model is given by
ht = w + (1634+ flht_1,

where w > 0, a 2 0, fl 2 0. This model is covariance stationary if and only if
a+fl<L

Many researchers have found that the GARCH(1,1) model performs well with
high frequency data (Diebold, 1987). For example, Baillie and Bollerslev (1989)
conclude ”the conditional heteroskedasticity in daily spot rates is well represented by
a GARCH(1,1) process with near unit roots” and the GARCH effects remain very
significant for weekly data. Also, Hsieh (1988) demonstrates that the GARCH(1,1)
model is suitable to describe daily data on nominal US. dollar return rates.
C. EGARCH

In the GARCH(p, q) model, the conditional variance only depends on the size
of the past innovations and not their sign. Nelson (1991) points out that such a
model cannot capture the ”leverage effect” (the tendency for changes in stock prices
to be negatively correlated with changes in stock volatility) which was first observed
by Black (1976). Nelson (1991) proposed the Exponential GARCH(p, q) model, or
EGARCH(p, q), which is related to the log-GARCH model proposed by Pantula
(1986) and Geweke (1986). In the EGARCH model, ht depends on both the sign

and the magnitude of at :

q P
111014) = (.0 + z a,(¢ut_.- + Vllut—il — E|ut_.-I]) + Z flgln(ht_,~).

i=1 i=1

There are no restrictions needed on a.- and 6.- to ensure the nonnegativity of the
conditional variance. If u, is assumed iid normal, then 6: is covariance stationary if
5:16.- < 1. Empirical evidence for stock returns supports this specification with
0,45 < 0, which implies that the conditional variance tends to increase (decrease)
when €t_,' is negative (positive), see Nelson (1991).
D. N ARCH
As another variant of the ARCH family, Higgins and Bera (1992) pr0pose the

non-linear ARCH (NARCH) model:
q p
h? = w + Z oz;(cf_,~)7 + Z fi,h;’_,-,
i=1 i=1

where w 2 0, a.- Z 0, 6.- _>_ 0, and '7 > 0. The reason for proposing a nonlinear form
for the conditional variance is that this specification is more flexible than the linear
GARCH model. This formulation is equivalent to the general GARCH(p, q) model
when '7 = 1. With 7 = 1 / 2, the conditional standard deviation hj/z is a distributed
lag of absolute residuals as proposed by Taylor (1986) and Schwert (1989). Higgins
and Bera (1992) suggest that the NARCH model does better for modeling the weekly
exchange rate series than the linear GARCH model.
E. TARCH

The threshold ARCH model, or TARCH, is designed to take into account that
the market’s responses to good and bad news may be asymmetric. The conditional
variance is defined as

q p
h,”2 = w + £10? I(et_,- > 0)|6t_.-|7 + a,— I(e¢_,- g 0)|ct_.-|7] + Z flihifi,

z=l i=1

where I(.) is the indicator function. Zakoian (1990) uses the model with 7 = l.
Glosten, Jagannathan and Runkle (1993) use this model with '7 == 2 for describing
the nominal excess return on stocks. The model is attractive because it allows for
more flexible responses of volatility to shocks of different signs and magnitude.

F. ARCH in mean

Consider a stochastic process, say yt, where
y: = f(‘I’t—1; b) + ft,

and f (\Ilt_1; b) is a function of \Ilt_1 and the parameter vector b.
In the ARCH-in-mean model, or ARCH-M, which was introduced by Engle, Lilien,
and Robins (1987), the conditional mean is an explicit function of the conditional

variance,
Ht = fort—l, ht; b),

where at is the mean of the stochastic process yt. Some finance theories predict a
tradeoff between the expected returns and the variance, or the covariance among the
returns. This model is capable of explaining the relation between the conditional
variance and the conditional mean provided the sign of the first derivative of f (ht, b)
with respect to ht is positive. The ARCH-M model has been applied to different
stock index returns, such as the daily S&P index by French, Schwert, and Stambaugh
(1987), and quarterly US. stock indices by Friedman and Kuttner (1988).
G. IGARCH

The Integrated GARCH model, or IGARCH, proposed by Engle and Bollerslev

(1986), is the special case of the general GARCH(p, q) model in (2) with 2le a +

7

{:1 = 1. It has the property that shocks to the conditional variance in the
model persist permanently, so sometimes it is also characterized as having ”persistent
variance” or ”integrated variance”.

The unconditional variance for the IGARCH(p, q) model does not exist, since

f=1a+ 2le ,6 = 1. In the IGARCH model, the process is not covariance stationary,

but Nelson (1990) shows that it is strictly stationary and ergodic.

3. MLE for the GARCH Process

To discuss maximum likelihood estimation, we presume that the distribution of at in
equation (1) is known. That is, we presume that the distributional assumption made
for at is correct. Let f(ut) denote the density function for u; E et/hj/z, normalized
to have mean zero and variance one. For the general model given in equation ( 1), the

log likelihood function can be written as

LT(0) = ‘21 MO), (3)

where the contribution of the tth observation is:

me) = —§log we» + log {f[u¢(0)]}. (4)

Here the notation ht(0) indicates that h, depends on some parameters 0, so that
ut(0) = ct/ht(0)1/2, but for simplicity we will hereafter just use the notation h, and ut.
The MLE of 0 is obtained by maximizing LT(0), as given in equation (3), with respect

to 0. For example, in the GARCH(p,q) model, 0 = (w, 01, a2, - - ~ , aq, [31, fig, - - - , flp)’.

Now consider the commonly-assumed case in which the distribution of at is stan-
dard normal. Then the log-density function for the tth observation, apart from an

irrelevant constant, is

“(0) = —-2—log ht —‘ —€t 2ht—l. (5)

The first and the second derivatives with respect to 6 are

(91 1 16h
Qu(€ 9): t‘ t(£—t - 1),

60 211—, 60 h,
021; _ (6? 1)_6_[1 1 6,11] 1 1 6h, Bht 6?
60’

8660’ _ ' Eh—gfififit’

(6)

ht

2ht06

where

6 = (60:61: ' ' 'aét)"

In addition, if we consider the GARCH(p,q) model, we have:

6h; aht— —i
g— “ Z: + gfli—B—O— ,
where zt=(1, et_1, - ,c?_q,ht_1,---,ht_p).

Following Weiss ( 1986), the maximum likelihood estimators 0M”; of the parame-
ters are consistent and asymptotically normal with mean 00 (the subscript 0 represents

the vector of true parameters) and covariance matrix A“, where
A = —E[021t/6069'] = E[(az./aa)(azt/aa)].

This covariance matrix equals the Cramer-Rao lower bound.
It is difficult to establish asymptotic theory for the estimation of the IGARCH

model. Hong (1987) provides Monte Carlo evidence for the IGARCH(1,1) model,

and suggests that the sample size must be very large (m 5,000 observations) for the
asymptotic distributions to be good approximations. On the other hand, Lumsdaine
(1996) proves that the Normal MLE of the parameters in the IGARCH(1,1) model is
still consistent and asymptotically normal.

An interesting sidelight, not previously noted in the literature, is that the MLE
will not be consistent for certain distributions of at. For example, let u; be standard
chi-square with u degrees of freedom, so that at 2 (fit — (1)/J2; where {it is xfi. Then
ii, 2 0 is equivalent to u; 2 -V/\/$ and at _>_ —h:/2u/\/21;, so that the range of 6;
depends on 0 ( through ht). This violates one of the standard regularity conditions
for the consistency of the MLE. More detail is given in Section A of Appendix 1.
For an example of an article that misses this point, see Engle and Gonzalez-Rivera

(1991).

4. QMLE for the GARCH Process

The likelihood function basedon a distributional assumption provides a criterion
function whose maximization defines an estimator. In the case that the distribu-
tional assumption is not correct, this estimator is called a quasi-maximum likelihood
estimator, or QMLE. For example, the normal QMLE is simply the estimator that
maximizes the normal log likelihood function.

The properties of the QMLE will depend upon the ”assumed” distribution, which
dictates the form of the estimator, and the ”true” distribution, which is a character-

istic of the data generating process. In general, when the assumed distribution is not

10

the same as the true distribution, the QMLE will be inconsistent. However, for some
assumed distributions, notably the normal, the QMLE is consistent and asymptoti-
cally normal, so long as the true distribution satisfies some regularity conditions. We
now discuss two important types of QMLE.
A. Normal QMLE

The Normal QMLE was investigated by Weiss (1986) and Bollerslev and Wooldridge
(1992). Bollerslev and Wooldridge use it in estimating a multivariate GARCH model.
They find that even if normal distributional assumption does not hold, estimates
based on the normal log-likelihood function are still consistent and asymptotically
normal, provided that both the mean and variance equations are correctly specified
and that some regularity conditions are satisfied.

The consistency and asymptotic normality of the Normal QMLE are given in the

following theorem, which is proved by Bollerslev and Wooldridge (1992):

THEOREM(B&W) : If (1) The regularity conditions in Section B of Appendix 1
are satisfied, and (2) For some 0 6 int 9, E(yt|‘Il¢_1) = pt(00) and Var(yt|\Ilt_1) =
Qt(00), then

(A‘lBA"’)‘1/2\/T(§T — 00) —> N(0, I),

where ET is the QMLE

T
B = T—1 : E[3t(00)3t(00)’]t

i=1

3,090) is the score function of lt(0),

lt = -1/210g|0¢(9)|- 1/2 (y: - u(0))'9¢(9)“’(y¢ — #(9)),

11

and

T
A = T‘1 Z: E[dt(00)], dt(90) is the hessian function of lt(0),

t=l

In addition,

AT—A—w and BT—B—>O,

where

T T .. A ’
AT = T_1 2 dt(6T), and BT = T-1 Z 3t(0T)3t(0T) .
t=l

t=l

The matrix AFBTAF is the robust covariance matrix of White (1982). If the true
distribution is normal, (or, more generally, if the third conditional moment is zero and
the fourth conditional moment is three times the square of the conditional variance;
that is, the first four conditional moments are the same as for the normal distribution),
A = B, and therefore the asymptotic covariance matrix of the QMLE is simply A“.

In the GARCH(1,1) case, the (3 x 1) score function, q1t(e, 0), can be represented
as in equation (6), with 0 = (w,a, 6),. The conditional expected value of the score
function, E(qlt|‘IIt_1), is zero when the first two conditional moments are correctly
specified as

E(€t|‘Ilt_1)-_—' 0,
VGT(€t|‘IIt_1) = ht = w + (16:1 + flht_1.

Lumsdaine (1996) provides different regularity conditions to prove asymptotic nor-

mality for the normal QMLE of the GARCH(1,1) model. There are two assumptions

12

for the true parameters and the distribution of ut made by Lumsdaine (1996); see
Section C of Appendix 1.
B. Non-Gaussian QMLE

Non-Gaussian densities have become increasingly pOpular for the estimation of
GARCH models. Examples include the student’s t (Bollerslev (1987)) and the ex-
ponential power distribution (Nelson (1991)). The consistency and asymptotic nor-
mality of the QMLE based on non-Gaussian distributions has been investigated by
Newey and Steigerwald (1997). Newey and Steigerwald show that consistency holds
when a particular identification condition is satisfied. The identification condition
is that there exists a unique maximum of the quasi-likelihood function at the true
conditional mean and relative scale parameters. This condition is essential for the
consistency of the QMLE. Newey and Steigerwald (1997) conclude that the identi-
fication condition holds if the conditional mean is identically zero, or a symmetry
condition (the true and assumed densities are both unimodal and symmetric around
zero) is satisfied. When the symmetry condition does not hold, one additional location
parameter should be included to establish the identification condition for consistency.

According to Newey and Steigerwald’s setup, the basic model can be represented

e. = 113/2” + 0.21.), (7)

where 'y is the location of the innovation distribution and a, is the scale parameter
for the density of at. The conditional variance H, is rescaled by a constant term. For

example, in the GARCH(p, q) model, H; is ht in equation (2) rescaled by the constant

13

term, w. That is, h, = wH, and correspondingly

H,(0) =' 1 + i (3)534 + 2p: flth-ia (8)

i=1 i=1

where (g).- is the relative scale parameter and 0 = ((3)1, (3)2, - - - , (5h, 61, 62, - - - , fip, <p’)’,
where cp is the parameter vector in mean function.

Thus we hope for consistent estimation of (a/w)i and ,B,, as opposed to the entire
set of parameters (w, 01,-, fl.) In particular, we generally do not obtain a consistent
estimate of scale (on).

Given the assumed density a(ut,u) with :1 representing nuisance parameters in
the density (e.g. degrees of freedom), the non-Gaussian QMLE is the value of 6 =

(0', 7,03,11)’ that maximizes the log-likelihood function,
T
ma) =T-12u6), (9)
t=l
where lt(6) is given by:
1
1.05) = —In a, — 5111 H, + 1n a([a.H,1/2(9)]-1[y, — f.(0) — 7Ht1/2(0)],u),

where ft(0) is the mean function. Notice that the additional parameter 7 enters the
conditional mean but not the conditional variance.

The identification condition for consistency as given by Newey and Steigerwald
( 1997) is stated in the following theorems. The first theorem applies to the case where
the additional location parameter does not need to be included. The log-likelihood
function is the same as equation (9) but with 7 E 0 and at = Ill/208m. The second

theorem applies to the case where the location parameter is added to the model.

14

THEOREM (N&S, 1) : If Assumptions 1, 2, 3 in Section D of Appendix 1 are satisfied,
and either Assumption 4 is satisfied or the conditional mean equals zero, then the
expected log-likelihood EM) = E [lt(6)] has a unique maximum at some (I with 5 = 00.
THEOREM (N&S, 2) : If Assumptions 1, 5, 6 in Section D of Appendix 1 are satisfied,

then the expected log-likelihood L(6) = E[lt(6)] has a unique maximum at some 5

Also, for the ARCH and GARCH models, they prove that there is no asymptotic
efficiency loss for the QMLE even if the location parameter is included in the case
where the symmetry condition holds.

As previously stated, when the conditional mean is identically zero, the identifica-
tion condition is satisfied without imposing the symmetry condition or the additional
location parameter. The martingale-GARCH(1,1) model we used in this dissertation
satisfies the condition that the conditional mean equals zero. Thus, the symmetry
condition is not an issue in our model. The rescaled conditional variance of the

GARCH(1,1) process, Ht, has the form of
a 2
Ht = 1 + a€t_1 + flHt_1.

We will consider cases in which the student’s t density is used. Many researchers
find that the thick tail property of the student’s t distribution describes financial data

relatively well. The student’s t QMLE, following the setup of Newey and Steigerwald

15

(1997), is the value of 6 = (0,, g, [3) that maximizes equation (9), where lt(6) is

))-ln(I‘(g—))—-;-ln(7r(u—2))—ln as—é—ln Ht—V—‘2i11n(1+Hta,:;_ 2) ).

(10)

u+1
2

 

M6) = ln(I‘(

 

Here u, the number of degrees of freedom, is taken as given.
The score functions of the likelihood function contributions for observation t in

equation (10) with respect to as, g, and H are

slim) = ’— ("“ ' ”"3“ ‘ 2’), (11)

 

 

 

 

a, 6% + Hta§(u — 2)
T 9- .. ll— aHt V6? — Ht0'3(l/ — 2)
3’ (w) — 2Ht0(g) (a? + Htaflu — 2) ’ (12)
T _ ILBHt V6? — Htaflu — 2)
3‘ (fl) — 2 H, 83 e? + Htaflu — 2) ’ (13)

where

6H; 2 6H¢_1
“—0— = 6 — + HT,
5(5 t 1 8(;)
6H; aIft—l

53—:Ht-l+fl 6,6 .

 

5. Improved Normal QMLE and Motivation

Although the Normal QMLE provides consistency and the asymptotic normality, its
efficiency property is in doubt when the true distribution is far from normality. Engle
and Gonzalez-Rivera (1991) show that the efficiency of NQMLE is low when the true
density follows the student’s t distribution with degree of freedom less than 12. One

way to improve efficiency is to try to discover the true distribution of the innovations

16

u, and use the MLE based on this true distribution. However, this procedure runs
the risk of inconsistency if we get the ”true” distribution wrong. So, instead, we may
wish to improve upon the efficiency of the normal QMLE, but in such a way that we
retain its robustness property.

The contribution of this thesis is to propose an augmented GMM estimator to
improve the efficiency of the NQMLE. We interpret the normal QMLE as a GMM
estimator, where the moment conditions it uses simply state that the score of the
normal quasi-likelihood has expected value zero. Thus, the first-order conditions
from the normal log—likelihood function are the base set of moment conditions with
which we start. We then find other moment conditions to add on to the base set
of moment conditions. In doing so we rely on the general result that adding extra
valid moment conditions can never decrease the asymptotic efficiency of estimation.
Of course, if the true density is Gaussian, the extra moments will be redundant.
However, these extra moment conditions will generally improve efficiency if the true
density is non-Gaussian.

There are two sets of extra moment conditions which we are particularly interested
in. The first set of extra moments is derived from the relationship between the sample
autocorrelations and the population autocorrelations of the squared observations. The
use of such moments has been proposed by Baillie and Chung (1999). The second set
of extra moment conditions is from the score of a non—Gaussian quasi log-likelihood
function. Specifically, we consider the score function from the student’s t quasi-

likelihood function.

17

6. Plan of the Thesis

The thesis is organized as follows. Chapter 2 discusses in details the two extra sets
of moment conditions that we will use. Combining all moment conditions, we show
how to calculate the asymptotic variance of the augmented GMM estimators. To see
the asymptotic efficiency gains from adding these extra moment conditions, we report
asymptotic variances for specific parameter values and different assumptions on the
true distribution. The results show that the augmented GMM estimators improve on
the efficiency of the normal QMLE in a non-trivial way when the true density is far
from normality.

Chapter 3 gives Monte Carlo simulation results for finite samples to investigate the
finite sample pr0perties of the normal QMLE and the augmented GMM estimators.
We wish to see whether the asymptotic efficiency gains identified in Chapter 2 can
actually be attained in reasonable sized samples. We find that rather large sample
sizes, say T = 2,000, are required to do so. We also investigate the finite sample
reliability of inferences based on asymptotic theory for the NQMLE and augmented
GMM estimators.

Chapter 4 presents an empirical application to the high frequency data using the
GARCH(1,1) model. The data we analyzed is the hourly exchange rate of the West
German Deutschmark versus the US. dollar (DM/$) from 0:00 am. January 2, 1986
through 11:00 am. July 15, 1986. The same data has previously been analyzed
by Baillie and Bollerslev (1990). Compared to the normal QMLE, the augmented

GMM estimates appear to be considerably more precise, with conventionally calcu-

18

lated standard errors reduced by a factor of about two.

Chapter 5 gives our concluding remarks.

19

Appendix 1
A. MLE under Asymmetric distribution

Suppose that it; follows a x2 distribution with degrees of freedom 11:

 

m» = .2.) (;)%(..)<-s--l>...( — Em 2 0,
2

This X2 variable has mean u and variance 21/. Now we standardize {it to obtain
at with mean zero and variance one: at = (fit — V) / \/ 21/. After this standardization,

the density of u; is

K—

m.) = “pm—(r) W- L) 2 —

5|“
Y

Transforming to the conditional distribution of 6; (ct = Iii/221‘),

(Q’s—X 21.1.“) %‘1’..p(_ ;( 2y%+u)),

“fill/1:4) = ht? ht

where 6; 2 —T‘;;h§.

We note that the natural constraint fit 2 0 corresponds to at 2 —1// J27 and
at Z —h:/2u/\/2—u. Thus the range of 6; depends on the parameter 0, because ht
depends on 0. This dependence of the range of 6, on 0 violates one of the standard
regularity conditions for the consistency and asymptotic normality of the MLE1,and
suggests that the MLE will be inconsistent.

More specially, let sT(e,0) = 9%}91 be the score function. The fundamental

condition for the consistency of the MLE is that E[3T(e,00)] = 0, where 00 is the

 

1The conditions for asymptotic normality of MLE are stated as Theorem 5.2 of Wooldridge
(1994).

20

true value of 0. This condition fails for the case that u, is standardized x2. In Table
A.1 we evaluate the expected value of the score function, with T =2,000 and r 5
number of replications = 500, for the xf and xg cases and two different parameter
values. In each case the mean appears to be nonzero. More formally, the hypothesis
of zero mean is rejected by the usual asymptotic t test, where the standard error is

calculated with the usual Newey-West formula with m = number of lags = 50.

B. Regularity Conditions -— Bollerslev and Wooldridge

1. 6 is compact and has a nonempty interior.

2. The conditional mean and the conditional variance are measurable for all 0 and

twice continuously diflerentiable on 9.

3. (a){lt(et, 6),t = 1, 2, ...} satisfies the uniform weak law of large numbers. (b) 60

is the identifiably unique .maximizer of E [2,21 lt(0)].

4. (a) 612/6660 and E [61? / 60' 60] satisfy the uniform weak law of large numbers.

(b) A = T‘1 231:1 E[at(00)] is uniform positive definite.
5. st(0)st(0)' satisfies the uniform weak law of large numbers.

6. (a) B = T‘1 2;] E[s¢(00)st(00)'] is uniformly positive definite.
(b) B-1/2T-1/2 23:13.09) —+ N(0, I).
C. Regularity Conditions — Lumsdaine

1. The true parameter vector 00 is in the interior of O, a compact, convex pa-

rameter space. For any vector (w,a,6) E O, assume that m s w s M,

21

m g a S (1 — m), and m g 6 S (1 — m) for some constant m > 0, and

0104—3030-

2. utis iid drawn from a symmetric, unimodal density, bounded in a neighborhood
of 0, with mean 0, variance 1, and E(uf’2) < 00. In addition, assume that ho, is

independent of {ut,u¢+1, . - }.

D. Identification conditions for consistency - Newey and Steigerwald

1. E[|l¢(6)|] < 00 for all 9 E <I>, V 6 N and as > 0, where <I> and N are feasible sets

for 6 and V respectively.

2. The function Ht1/2(60) > 0, and if 6 7é 60 then either Ht1/2(0)/Ht1/2(00) is not

constant or P[ft(0) 7e ft(60)] > 0.

3. The function Q(a,, V) = —ln 0, + E [In dugout/as, V)] has a unique maximum

at somea, anduovera,>0andV€N.

4. The innovation at is symmetrically distributed around zero with unimodal den-
sity k(u) satisfying k(u1) S k(u2) for lull 2 lugl. For each V, a(u, V) is symmet-

ric around zero and a(u1, V) < a(u2, V) for |u1| > |u2|.

5. The function rig/2(a) > 0, and if 0 ee 00 then either H,‘/2(0)/H,‘/2(00) or

[ft(9) — ft(00)]/Htl/2(00) is not constant.

6. The function Q(7,a,,V) = —ln 0, + E[ln a((030ut + 7)/0,,V)] has a unique

maximum in (7, 0,, V)’.

22

Table A1
Test for the expected value of the score function of the standardized chi-square distribution

T=2,000, r = 500
a) = 0.2, a = 0.15, I3 = 0.65
True distribution: Standardized chi-square with d ree of freedom 1

 

 

 

 

 

 

 

 

 

 

 

 

I mean I s.e.( m = 50) t-value I

to -0.54 0.18 -2.93

a I -0.30 0.10 -3.06

[3 -0.46 0.15 -3.04

True distribution: Standardized chi-square with degree of freedom 5

I mean I s.e.(m = 50) t-value I

m -0.35 0.1 1 -3.27

a I -0.25 0.09 -2.88

B -0.32 0.1 1 -2.78
T=2,000, r = 500

co=0.1, a=0.1, B=0.8
True distribution: Standardized chi-square with degree of freedom 1

 

 

to
a
l3

 

 

 

 

 

 

I mean I s.e.( m = 50) I t-value I

-Q.96 0.33 -2.87

-0.56 0.18 -3.06

-0.81 0.27 -3.02

True distribution: Standardized chi-square with degree of freedom 5
I mean I s.e.(m = 50) I t-value I

-0.60 0.23 -2.59

-0.46 0.16 -2.90

-0.55 0.20 -2.76

0)
0.
l3 -

 

23

 

 

Chapter 2

EXTRA MOMENT CONDITIONS AND ASYMPTOTIC
ANALYSIS

1. Introduction

The normal QMLE is widely used in estimating ARCH type models. As we mentioned
in chapter 1, the normal QMLE is consistent and asymptotically normal, but the loss
in efficiency may be large when the true distribution is far from normality. The
normal QMLE is a GMM estimator, based on the fact that the expectation of the
score function of the normal log likelihood equals zero, whether the true distribution
of the innovations is normal or not. We want to improve the normal QMLE by adding
other moment conditions that also do not require normality to be valid.

In this chapter we give details of two extra sets of moment conditions; one is
based on the autocorrelations of 6? and the other is based on the rescaled student’s
t distribution. Then we show how to calculate the asymptotic variance of the aug-
mented GMM estimator. The asymptotic variances for specific parameter values and
different true distributions are reported and compared to see how much efficiency we

can gain from adding these extra moment conditions.

2. Moment Conditions Based on Autocorrelations of 5?

Baillie and Chung (1999) apply a Minimum Distance Estimator (MDE) to the GARCH

model. This estimator minimizes the distance between the population and sample

24

autocorrelations of the squared observation (6?). They perform Monte Carlo simu-
lations and find that for non-normal innovations, especially in the asymmetric case,
the MDE can be more efficient than the normal QMLE.

The MDE is defined by solving the following minimization problem:

Min [f2 — p(9)]'W[/5 — p(0)1. (1)

where W is a positive definite weighting matrix, and p“ and p(0) are (9 x 1) vectors,
which contain 9 sample autocorrelations and 9 population autocorrelations respec-
tively. We briefly explain how to calculate the three components in the objective

function, ,6, p(9), and W. First, the sample autocorrelation fihk = 1, - - - , g, is de-

fined by
T _ _ T _
in. = Z (6? -«52)(62"_;c -62)/ Z (6? -62)2,
t=k+1 t=k+l

where 6—2 is the sample mean of the 6?. Second, the population autocorrelation func-
tion of 6?, p09), depends on the model used in describing the series. We will consider
the case of the GARCH(1,1) model. For this case an explicit formula for the au-
tocorrelations is available. Specifically, the autocorrelation function of 6? for the
GARCH(1,1) process has been derived by Bollerslev (1988) and Ding and Granger

(1996). If 01+ [3 < 1 and the fourth moment of e, existsz, the autocorrelation function

for the GARCH(1,1) process is.

(12,8
p1=(a+1_2afl_fl2),

 

and

 

2That is, m2 + 200 + 32 < 1, where n is 3 plus the coefficient of excess. In the normal case, n
is equal to 3.

25

025
1 - 2010 — fl2

Finally, the optimal weighting matrix W is the inverse of the asymptotic covariance

 

pk = (a + )(a + fl)"—l, for k 2 2.

matrix of fi, a result given by Chiang (1965) and Ferguson (1958).
Because we want to use standard GMM results to evaluate potential efficiency
gains, we want to consider a GMM estimator rather than a MDE estimator. To do

so, we note that

7'
T1/2(fi-p(0)) = T‘“ Z Zt/io,

t=g+1

where

- T
’70 = T’1 Z (6? - 62)2,
t=k+1

(6f - €2)(ei..1 — 6—2) - P1(9)(€? — 62?
Zt = 5 - (2)

(.3 — 32M. — £2) — pgwxe? — 6'2)?

Thus the MDE should be asymptotically equivalent to a GMM estimator based on
the moment conditions E(Zt) = 0.

The augmented GMM estimator combines these moment conditions with those
based on the score of the normal likelihood. Explicitly, it is based on the moment

conditions: E[q¢(00)] = 0, where

q¢(0) = [qlz(9)', (12t(9)']', (3)
where q1¢(0) is given in equation (6) of chapter 1, and where q2t(0) = Z as in equation
(2) above.

26

A standard result of GMM estimation is that adding more valid moment conditions
cannot decrease the asymptotic efficiency of estimation. Thus the augmented GMM
estimator is asymptotically at least as efficient as the normal quasi-MLE. If the data
are normal, the moment conditions based on qgt must be redundant, whereas otherwise
the augmented GMM estimator would be expected to be strictly more efl‘icient than
the normal quasi-MLE.

A relevant detail of estimation is that, whereas qu is uncorrelated over t, qgt is
correlated over different t. The estimation (or evaluation) of the weighting matrix

needs to recognize this fact. Thus we would use a Newey-West type estimator:

F j )1 (4)
where F,- = %:tQt(é)qt_j(é)’, .where 5 is an initial consistent estimator of 0 , and

where m grows with T at a suitable rate. Then W = Q“.

3. Moment Conditions Based on the Score Function from the
Rescaled Student’s t Distribution

In section 4.B of Chapter 1, we discussed the fact, proved by Newey and Steigerwald
(1997), that the QMLE based on the rescaled student’s t distribution yields a con-
sistent estimator for the parameters (0,, a/w, ,6) of the GARCH model. Since we are
interested in the parameters (a, fl,w) in the natural GARCH parameterization, we
simply will use the two moment conditions that correspond to the scores with respect

to a/w and fl. That is, we conSider

(1349) = [Sad/w), 3f(fi)]', (5)

27

where sfla/w) and sflfl) are given in equations (12) and (13) of chapter 1. Then our
set of extra moment conditions is :EIq3¢(00)] = 0. We can then use these conditions to

augment the moment conditions based on the score of the normal likelihood function.

4. Asymptotic Variance for the Augmented-GMM Estimator

Putting together the moment conditions based on the score functions from N QMLE,
the autocorrelations of 6?, and the score functions from the rescaled t distribution,
there are 9 + 5 moment conditions, where g is the number of autocorrelations of 6?

that are considered. Explicitly, we have

Qlt(90)
E [(1400)] :- E [( «12400) )1 = 0- (6)
‘ q3t(00)

Here qu, qgt and (13,, are as defined in the preceding two sections.

A

The augmented-GMM estimator, 0, minimizes
qT(0)’WqT(0)a

where 67(0) 2 T ’1 2, qt(0) and W is the weighting matrix. Under some standard

regularity conditions, 5 is consistent and has the following asymptotic distribution:
Tia} — 0) —> N[0, (D’WD)‘1D’WQWD(D’WD)‘1],

where D = EIBqt(c,0)/00’] and x/TQTWO) —> N (0, (2) Since qgt is correlated over
time, (2 not only includes the expected square of qt(9), but also involves the cross

products over time of qt(0). Thus

o = 111’." Var(\/Tq'T) = P0 + 2m + 1‘2),
00 1:1

28

where I}, = EIqt(0)q¢(0)'], and I‘; = E[qt(0)qt_;(0)'].
The optimal GMM estimator is obtained when W is a consistent estimator of Q“ ,
as shown in Hansen (1982). In this, the asymptotic variance simplifies to (DR-ID)“.
In practice, given an initial consistent estimator 5, D can be consistently estimated
by

T
D = T-1 Z aq.(e, wank, .

t=1

Q can be consistently estimated by the method of Newey and West (1987), as in

equation (4) above.

5. Evaluation and Comparison of Asymptotic Variances

In order to assess the relative asymptotic efficiency gain of the augmented-GMM
estimator over the normal QMLE, we simply wish to calculate and compare the
asymptotic variances of the estimators. These depend on expectations that we cannot
take analytically. Thus, we will rely on simulation to give us numerical results. That
is, the asymptotic variance is of the form (DR—ID)“, and we use simulation to
evaluate D and Q. We stress that this is still an evaluation of the asymptotic variance
of the estimates. We do not calculate any estimates of 6; we simply use simulation to
replace the expectations in the definitions of D and (2 (evaluated at the ”true value”
00).

The model is a GARCH(1,1) process,
ft = héut,

29

where u, is drawn from a distribution with mean 0 and variance 1. Thus
ht=w+acf_l+flht_1, t=1,...,T,

where w > 0, a 2 0, ,8 2 0. For sample size equal to T and number of replications

equal to R, the asymptotic variance is evaluated as (DD-ID)“, where

_1 0q('a)( (e, 0)

63.25——

Hrzl t=l

and

(1 — —)(r, +r;)], with
r=1[ [=1

_X;(1t(7‘)((6 0)q(r)(6, 0”)

=2 gm. 0 )q§:’(e,o),

t=l+1

where r = 1,2, - - -,R, and t = 1,2, - . - ,T. All evaluations are for 0 = 00.

For large R and T, and for m picked suitably, (D'Q—1D)-1 should be close to
the asymptotic variance (D'QTIDYI. We calculate our results with R = 500 and
T = 2, 000, so that we are averaging over 1,000,000 different observations to evaluate
D and 52. After some experimentation, we picked m = 50 as the number of lags to
use in evaluating Q.

The parameter values we choose satisfy the following constraints. For stationarity
of the process, a + 3 < 1 must hold. In addition, the fourth-order moment of 5, must
exist in order for asymptotic theory to hold for the moment conditions based on the
autocorrelations of cf. In Table 2.1, we tabulate the value of no? + Zafl + 02, which

must be less than one for the fourth moment to exist, for various values of n, a and 0.

30

Since different distributions have different 19, Table 2.1 shows all the cases which we
will analyze in this paper. The variance of the unconditional density, 01/ (1 — a — ,6), is
chosen to equal to 10. Also, to minimize the effect of startup problems, we generate
a sample of size equal to T + 200, and discard the first 200 data points; the remaining
T data points are then used in simulation.

We first report the results with a = 0.15 and fl = 0.7, in Table 2.2. We compare
results from five different GMM estimators based on different sets of moment con-
ditions. The N QMLE uses the moment conditions Qu, the score from the normal
log-likelihood function. The MDE is based on the moment conditions (12:, which
equate sample and population autocorrelations of 6?. The estimator using moment
conditions qu and qzt is called GMMl. GMM2 refers to the estimator that uses
the moment conditions qu and (13¢, Where q3t is based on the score for the rescaled
student’s t distribution. Finally, GMM3 uses all g + 5 moment conditions: qu, qgt,
and 43t-

When the true distribution is normal, the asymptotic standard errors are reported
in column 1 of Table 2.2. The NQMLE is the MLE in this case and is therefore
efficient. Adding more moment conditions should not improve efficiency. The MDE
should be inefficient relative to NQMLE, and it is. All of the other GMM estimators
(GMM2 and all variants of GMMl and GMM3) should have asymptotic standard
errors equal to that of the NQMLE, and this is a check on the accuracy of the
calculations. There are only very minor differences, usually no more than 3%, and

this indicates that our calculations are reasonably accurate.

31

Column 2 of Table 2.2 gives the asymptotic standard errors when the true distri-
bution is a standardized student’s t distribution with the degree of freedom 5. We
consider the same estimators as before, but we also consider the MLE based on the t5
distribution (TMLE), which should be efficient. The asymptotic standard errors for
NQMLE are larger than those for TMLE by about 50%. The MDE still has larger
asymptotic standard errors than those of NQMLE, even for g as large as 20. For
GMMl with g = 5, the asymptotic standard error of a is smaller than for NQMLE
by about 20%, but there are only very small differences for w and ,6. Increasing g
reduces the asymptotic standard errors in all three parameters, with larger decreases
for w and 3 than for a. With g = 20, the GMMl asymptotic standard errors are about
20% smaller for a and about 10% smaller for w and fl, compared to NQMLE. The
efficiency gains from GMM2 are of comparable magnitude. It is perhaps surprising
that GMM2 is not more efficient, since it uses the score function from the rescaled
student’s t distribution; but it uses only two moment conditions based on student’s
t, not three. Finally, GMM3 results in further efficiency gains. For estimation of a it
essentially reaches the TMLE lower bound, whereas for w and ,6 its standard errors
are roughly midway between those for NQMLE and TMLE.

The results for the x2 distribution with two degrees of freedom are given in column
3 of Table 2.2. The MDE does better here than previously. For MDE with g = 10, the
asymptotic standard error of a is smaller than that of NQMLE, and when g increases
to 20, the asymptotic standard error of H is close to that of NQMLE. For GMMl,

the asymptotic standard error of a is almost 30% less than that of NQMLE. The

32

improvements for w and fl are. still minor. For GMM2, the efficiency improvement
for a is comparable to those for GMM], while for 0 and w the results are similar to
those for GMM] with g = 5 but not as good as for GMMl with g larger. GMM3 is
not much better than GMMl.

In column 4 of Table 2.2, the true distribution is the standardized gamma distri-
bution with two degrees of freedom . These numbers show roughly the same pattern
as the standardized t distribution with the degree of freedom 5.

We also analyze three different sets of values for a and fl. Tables 2.3, 2.4 and 2.5
correspond to { a = 0.1, fl = 0.75 }, { a = 0.1, ,6 = 0.8 } and { a = 0.2, fl = 0.6}
respectively. We will not discuss these results in detail because they are fairly similar
to those in Table 2.2. The augmented GMM estimators improve on the NQMLE, in
terms of asymptotic standard error, by an amount that varies over parameters and

distributions, but is perhaps typically in the range of 10% ~ 20%.

6. Conclusions

When the true distribution is Gaussian, using the first set of moment conditions qu
based on the normal score function is enough to obtain the Cramer-Rao lower bound.
The extra two sets of moment conditions are redundant. The augmented GMM has
no gain in efficiency when the data series actually follows the normal distribution.
When the true distributionis non-Gaussian, such as student’s t distribution, X2
distribution, or gamma distribution, we gain asymptotic efficiency by using extra

moment conditions. GMMl uses g moment conditions based on the autocorrelations

33

of the 6?, while GMM2 uses the score for the rescaled student’s t distribution. GMM3
uses all of these and must be most efficient. The efficiency gains that are achieved
typically amount to reduction of the asymptotic standard error by 10% ~ 20%.
This benefit does not come without some costs. The augmented GMM estimators
are computationally more complicated than the N QMLE. Furthermore, GMMl and
GMM3 especially are potentially heavily overidentified and there may be worries
about their finite-sample properties. We will present Monte Carlo evidence on this

point in the next chapter.

34

Table 2.1
r<r12+2<IB+I32

(x = "degree of excess" plus 3. Fourth moment exists if Ka2+2aB+B 2< I).

K=3

standard normal distribution

a

 

 

 

 

 

 

0.1 0.15 0.2 0.25 0.3 0.35
0.4 0.2700 0.3475 0.4400 0.5475 0.6700 0.8075
0.45 0.3225 0.4050 0.5025 0.6150 0.7425 0.8850
0.5 0.3800 0.4675 0.5700 0.6875 0.8200 0.9675
0.55 0.4425 0.5350 0.6425 0.7650 0.9025 1.0550
8 0.6 0.5100 0.6075 0.7200 0.8475 0.9900 1.1475
0.65 0.5825 0.6850 0.8025 0.9350 1.0825 1.2450
0.7 0.6600 0.7675 0.8900 1.0275 1.1800 1.3475
0.75 0.7425 0.8550 0.9825 1.1250 1.2825 1.4550
0.8 0.8300 0.9475 1 .0800 1 .2275 1 .3900 1 .5675
K = 6
standardized gamma distribution with degree of freedom 2
a
0.1 0.15 0.2 0.25 0.3 0.35
0.4 0.3000 0.4150 0.5600 0.7350 0.9400 1.1750
0.45 0.3525 0.4725 0.6225 0.8025 1.0125 1.2525
0.5 0.4100 0.5350 0.6900 0.8750 1.0900 1.3350
0.55 0.4725 0.6025 0.7625 0.9525 1.1725 1.4225
[3 0.6 0.5400 0.6750 0.8400 1.0350 1.2600 1.5150
0.65 0.6125 0.7525 0.9225 1.1225 1.3525 1.6125
0.7 0.6900 0.8350 1.0100 1.2150 1.4500 1.7150
0.75 0.7725 0.9225 1.1025 1.3125 1.5525 1.8225
0.8 0.8600 1.0150 1.2000 1.4150 1.6600 1.9350
x = 9
standardized twith degme of freedom 5
standardized chisquare with degree of freedom 2
a
0.1 0.15 0.2 0.25 0.3 0.35
0.4 0.3300 0.4825 0.6800 0.9225 1 .2100 1.5425
0.45 0.3825 0.5400 0.7425 0.9900 1.2825 - 1.6200
0.5 0.4400 0.6025 0.8100 1.0625 1 .3600 1.7025
0.55 0.5025 0.6700 0.8825 1.1400 1.4425 1.7900
8 0.6 0.5700 0.7425 0.9600 1.2225 1.5300 1.8825
0.65 0.6425 0.8200 1.0425 1.3100 1.6225 1.9800
0.7 0.7200 0.9025 1 .1300 1.4025 1 .7200 2.0825
0.75 0.8025 0.9900 1 .2225 1.5000 1 .8225 2.1900
0.8 0.8900 1 .0825 1 .3200 1 .6025 1 .9300 2.3025
Note:
standard normal distribution , K = 3
standardized twith degree of freedom v1, x=( 6/(ol—4)) + 3
standardized chisquare with degree of freedom v2, 1: = (12/02) +3
standardized gamma distribution with degree of freedom v3, 1: = (6/03) +3

35

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39

Chapter 3
FINITE—SAMPLE PROPERTIES

1. Introduction

In Chapter 2 we found that, for cases in which the true distribution is far away from
the gaussian, augmented GMM gives a non-trivial gain in asymptotic efficiency. The
gains were on the order of 10% ~ 20% for a and smaller percentages for w and fl
in terms of asymptotic standard error. In this section, we try to investigate whether
these efficiency gains can be realized in finite samples.

An issue of concern in finite samples is the problem of weak instruments. Some
studies find that when the instruments are weak, i.e. the partial correlation between
the instruments and the included endogenous variable is low, the conventional asymp-
totic results fail even if the sample size is quite large. In applications of two—stage
least squares (ZSLS), Nelson and Startz (1990) and Bound, Jaeger and Baker (1995)
find that the 2SLS estimator is biased away from the true value and in the direction
of the ordinary least squares (OLS) estimator. Also, the asymptotic distribution is a
poor approximation to the true distribution in finite samples when the instruments
are weak, so that inference based on asymptotics is inaccurate.

Since some of the augmented GMM estimators employ many moment conditions,
we have a high degree of overidentification. A highly overidentified problem is par-
ticularly vulnerable to the problem of weak instruments. In such cases, the estimates
may be biased and the asymptotic standard errors may be overly optimistic in finite

samples. Correspondingly the efficiency gain may not be as large as it is asymptoti-

40

cally, and in fact may not even exist. Moreover, inference may not be accurate, e.g.
t tests may overreject the null hypothesis. Monte Carlo experiments will be carried
out to check the quality of the asymptotic approximation to the finite sample distri-
butions, and to see whether augmented GMM is really better than the normal QMLE

in finite samples.

2. Design of the Experiment

We generate data from the GARCH(1,1) model in the same way as in section 2.5.
An experiment is defined by a choice of the sample size (T), the parameter value (0,
a, 5), and the true distribution (e.g. normal, t5). For each experiment, we estimate
the model by NQMLE, MDE, GMMl, GMM2 and GMM3. There are three sets of
results for MDE, GMMl, and GMM3 corresponding to g = 5, 10, 20. In addition, the
non-Gaussian MLE based on the true distribution will be reported when the MLE is
available.

For the estimators other than NQMLE, we use the estimated weighting matrix
proposed by Newey and West (1987), with the lag length determined by an automatic-
lag selection criterion suggested by Newey and West (1994)3. We use NQMLE as the
initial consistent estimator for estimation of the weighting matrix. I

For each experiment and estimator, we obtain the following statistics; mean, stan-

dard error, root mean square error (RMSE) and the average estimated asymptotic

 

3m = 0T5, where o = 1.1447{§<1)/§(°>}2/3,§<1> = 22);, jfi,,§<°> = a. + 22;, (1,, and
n = [4(T/100)2/9] . fl; is defined as equation (4) in chapter 2.

41

standard error (using the usual GMM formula)‘. We want to check (i) whether the
bias is significant, (ii) whether the standard error and RMSE are lower for the aug-
mented GMM estimators than for the NQMLE, and (iii) how large is the difference
between the finite sample standard error and the average estimated asymptotic stan-
dard error.

The Monte Carlo simulations are performed using Gauss (Windows version 3.2.32)
on a Pentium II 350 PC. We use the CML (Constrained Maximum Likelihood) and
CO (Constrained Optimization) modules to do the optimization. The NEWTON
(Newton-Raphson) method is the main optimization algorithm, but in a few cases
we also used the BFGS (Broyden, Fletcher, Goldfarb, Shanno) algorithm because of
convergence problems using NEWTON. We use 500 replications (R = 500).

We organize our experiments around a ”base case” with T = 2,000, a = 0.15,
H = 0.7, unconditional variance = 10, and data generated from the t5 distribution.
We then perform four sets of experiments. First, holding all other features of the DGP
constant, we consider different distributions: standard normal, student’s t distribution
with five degrees of freedom, standardized chi-square with two degrees of freedom,
and standardized gamma with two degrees of freedom5. In the optimization process,
we found that some replications did not satisfy the constraint that a must be greater
than or equal to zero. Such results would cause the asymptotic standard error to be

incorrectly calculated. This happens only in the smallest sample size (T = 500), and

 

‘For each replication, we apply asymptotic theory to calculate the standard error evaluated at
the parameter estimates. These standard errors are then averaged over all replications.

5The density of the standardized gamma distribution, at, with 11 degrees of freedom is f (ut) =
[(u/2)1/2/l‘(u/2)] - ((u/2)1/2ut + u/2)(‘”2)/2 - e$p(-(u/2)1/2ut - u/2), where u; > —(u/2)1/2. Its
mean and variance are equal to u.

42

happens in less than 10 out of every 500 replications. We discard these replications.
Keeping them would have little effect on bias or RMSE. Second, again holding all
other features of the DGP as in the base case, we vary T: we consider T = 500 and
1,000 in addition to the base value of 2,000. Third, we verify that changing 0 does
not change the results, in a sense to be made precise later. Fourth, again holding all

other features of the GDP as in the base case, we vary a and ,6.

3. Results of the Experiments

3.1 Different Distributions

In this set of experiments we consider different distributions. In all cases we use the
base case parameter values: T = 2, 000, a = 0.15, ,6 = 0.7, unconditional variance 2

10 (w 21.5).

A. The flue Distribution is Gaussian

Table 3.1 shows the results for the case in which the true distribution is standard
normal. Asymptotically, augmentation makes no difference but intuition suggests
that redundant moment conditions may be harmful in finite samples, so that NQMLE
might be best. This is generally true but the differences are not too large. The results
for GMM2 are very similar in all respects to those for NQMLE. The GMMl estimates
of w are severely biased. Apart from that, however, GMM2 and GMM3 show some
bias but not very much, and their RMSE is larger but not much larger than that

of NQMLE. The main disadvantage of GMMl and GMM3 is that the asymptotic

43

standard errors understate the finite sample standard errors, especially when g is

large.

B. The True Distribution is Standardized t5

For the student’s t distribution with V degrees of freedom, the mean is zero and the
variance is u/(u — 2) for u > 2. The fourth moment is equal to 3u2[(u — 2)(V — 4)]-1
for u > 4. When u = 5, the degree of excess6 equals 6, which indicates that the t
distribution has a thicker tail than the normal distribution (whose degree of excess
equals zero).

Table 3.2 provides the Monte Carlo simulation results when the true distribution
is student’s t with five degrees of freedom. We first note that the NQMLE performs
adequately, in the sense that there is little bias and the average asymptotic standard
error is only a little smaller than the finite sample standard error. GMM2 also shows
little bias, and its RMSE is considerably smaller than that of the NQMLE, for w
and B at least. So GMM2 does achieve finite sample efficiency gains over NQMLE.
However, its asymptotic standard errors are somewhat less reliable than NQMLE’s.
The highly overidentified estimators (MDE, GMMl, and GMM3) all have noticeable
bias for at least some of the parameters, and their asymptotic standard errors are
not very reliable, especially when g is large. However, GMM3 does have smaller MSE
than N QMLE, and the' differences are non-trivial, especially when g is not too large.

TMLE is the MLE with the standardized student’s t likelihood function. The

6The degree of excess is measured by the fourth central moment normalized by the squared
variance minus 3.

 

44

RMSE for all three parameters is quite small: The RMSE of NQMLE is more than
35% larger than that of TMLE. So, if the true distribution were known, there would

be a fairly large efficiency gain from using the true MLE.

C. The True Distribution is Standardized xg

The x2 distribution with 11 degree of freedom is asymmetric, and from Patel, Kapadia,
and Owen (1976) the mean and the variance are equal to u and 21/. Its coefficient of
skewness and coefficient of excess are equal to 23/2 /1/1/2 and 12/ V, respectively. The
x3 distribution has a fat tail and a ”long tail” in the right direction; this is seen from
the fact that the coefficient of excess (6) and the coefficient of skewness (2) are both
larger than zero.

Table 3.3 presents the Monte Carlo simulation results when the true distribu-
tion follows the standardized xg distribution. These results show many of the same
patterns as in the case of the standardized t5 distribution, but they are much more
favorable for the augmented GMM estimators. The NQMLE performs adequately in
the same sense as before — little bias, and relatively reliable asymptotic standard er-
rors. In terms of RMSE, GMM2 and GMM3 are better than NQMLE, while GMMI
and even MDE are sometimes better. MDE, GMMl and GMM3 give biased esti-
mates, especially of a, but GMM2 is essentially unbiased. Its asymptotic standard
errors are somewhat less reliable than those of NQMLE, but much more reliable than
those of GMMl and GMM3. Overall, GMM2 seems best, since it achieves consid-

erable finite sample efficiency gains relative to NQMLE, without being badly biased

45

and without having very unreliable asymptotic standard errors.
As mentioned before, MLE is not valid in this case. The efficiency loss of the

NQMLE cannot be shown through the comparison with the x3 MLE.

D. The True Distribution is Standardized Gamma Distribution with Two
Degrees of freedom

The gamma distribution with V degrees of freedom has mean and variance both equal
to V, while the coefficient of skewness equals 211‘”2 and the coefficient of excess is 6/11.
With two degrees of freedom, the gamma distribution has coefficient of skewness equal
to 1.414, and coefficient of excess equal to 3. This means that the gamma distribution
has a fat tail and is asymmetric.

The Monte Carlo simulation results for this distribution are shown in Table 3.4.
We will not discuss these results in detail because they are relatively similar to those
for the xg case. Both NQMLE and GMM2 are essentially unbiased, and both have
fairly reliable asymptotic standard errors, but GMM2 is better in the sense of smaller
RMSE. GMMl and GMM3 are generally good in terms of RMSE, but they are biased
and their asymptotic standard errors are not reliable, especially when g is large.

Overall GMM2 seems like the best choice.

3.2 Different Values of Sample Size

We now consider the effects of changing the sample size (T). Our base case (Table
3.2) had T = 2, 000, with the standardized t5 distribution and with a = 0.15, 6 = 0.7,

w = 1.5. We now give results for the same parameter values and distribution, but for

46

T = 500 (Table 3.5) and T = l, 000 (Table 3.6). We wish to check our intuition that
the augmented GMM estimators will do better relative to NQMLE when T is larger,
and worse when it is smaller.

This intuition is supported by our results. When T = 500, GMM2 offers little
efficiency gain over NQMLE, while it is more biased and its asymptotic standard errors
are less reliable. GMMl and GMM3 do offer reductions in RMSE over N QMLE, but
their asymptotic standard errors are quite unreliable. When T = 1, 000, the results
are (unsurprisingly) between those for T = 500 and T = 2, 000, and in particular the
comparison between NQMLE and GMM2 depends on which parameter (a, 6 or w)

you look at.

3.3. w Doesn’t Matter

Up to now, we assume the unconditional variance is equal to 10. In this section we
verify that its value does not affect any of our substantive results (e.g. comparisons
of estimators).

In fact, a change in the unconditional variance only rescales the whole series.
That is, the series with the unconditional variance (1) is the same as the series with
the unconditional variance 1 times J3, if the same basic random numbers are used.
Since these two data series are identical in the above sense, the simulation results for
estimates of a and 6 should be the same. For w, the estimates (and their standard
errors and RMSE) should all change by the proportion 4).

Table 3.7 gives the results for a case that is the same as the base case except that

47

the unconditional variance equals 1 (w = 0.15) instead of 10 (w = 1.5). Comparing
these results to those for the base case (Table 3.2), we see that the results do match
in the sense described above: for a and 6 the results are the same, while for no they
decrease by a factor of 10. This correspondence is exact (to the number of decimal
places reported) for NQMLE, TMLE and GMM], while there are some very minor
differences for GMM2 and GMM3. Apart from these minor computational differences,

the results verify that the choice of w doesn’t matter.

3.4 Effects of Changing a and ,8

We have assumed a = 0.15 and fl = 0.7 in the previous sections. In this section, we
analyze the effects of changing the parameter values of a and 6 while we maintain
the assumptions that T = 2, 000, and that the true distribution is a student’s t
distribution with 5 degrees of freedom.

Table 3.8, 3.9 and 3.10 give the results when {a = 0.1, fl = 0.75}, {a = 0.1,
fl = 0.8} and {a = 0.2, B = 0.6} respectively. If we compare these results to those
for the base case {a = 0.15, 6 = 0.7} in Table 3.2, we see that the new results are
broadly similar. NQMLE and GMM2 are fairly similar, but GMM2 generally has
smaller RMSE. The more highly overidentified estimators often are reasonably good
in terms of RMSE, but bad in terms of bias and the reliability of their asymptotic

standard errors.

48

3.5 Accuracy of Inference

In previous sections we have compared the finite sample standard error to the aver-
age estimated asymptotic standard error, to see whether the estimated asymptotic
standard errors are reliable. For example, when the asymptotic standard error is on
average smaller than the finite sample standard error, we would expect that tests
based on the asymptotic standard errors would overreject. In this section we pro-
vide direct evidence on this point, by giving the rejection frequency (true size) of the
normal 5% Wald tests of the hypotheses that w, a and 6 equal their true values.

These results are given in Table 3.11. We assume the base case except that we
consider four different choices of the true distribution, as in section 3.1 above.

For the NQMLE, the simulation size is less than 0.05 when the true distribution
is standard normal; for the other (non-Gaussian) cases the simulation size is larger
than 0.05. For GMM2, the size for w and 6 are below 0.05, while the size for a is
over 0.05 for all distributions. Size is roughly in the range from 0.02 to 0.10. In that
sense inference is relatively reliable.

For the MDE, GMMl, and GMM3 estimators, the size distortions are quite seri-
ous, especially for a. The frequency of rejection for a is often over 30%. The over-
rejection problem is generally worst for the highly overidentified estimators (MDE,
GMMl and GMM3 with large values of g). This pattern is exactly what we would

expect from our earlier comparisons of finite sample and asymptotic standard errors.

49

4. Conclusions

Recall that the results in Chapter 2 indicate that, in large samples, the augmented
GMM estimators provide an asymptotic efficiency gain when the distribution of the
innovations is not normal, and the only price to pay for this gain is computational
time and effort. In this chapter, we turn our attention to the finite sample properties
of N QMLE, MDE and the augmented GMM estimators, to see whether an efficiency
gain can be realized in samples of reasonable size.

The simulation results show that, although for some estimators there is an ef-
ficiency gain in terms of smaller MSE, the gain comes with non-trivial cost: the
estimates may be biased, and inference based on the asymptotic distribution of the
estimates may be very inaccurate. These problems are particularly serious for MDE,
GMM], and GMM3, even for rather large sample sizes. Therefore, doubts can be
raised regarding these estimators’ usefulness in empirical studies. Fortunately, not
all of the augmented GMM estimators perform badly. GMM2 does not suffer from a
serious bias problem, and inference based on its asymptotic distribution is reasonably
reliable if the sample size is large enough. Also, if the sample size is large enough,
GMM2 does provide non-trivial efficiency gains over NQMLE. This is especially true
when the data are asymmetric. 4

In practice, our results support the use of GMM2 if the sample size is as large as
1,000 and there is evidence of asymmetry, or if the sample size is as large as 2,000 and
the data are symmetric but with fat tails. For smaller sample sizes, N QMLE would

remain the preferred method.

50

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61

Chapter 4
AN EMPIRICAL STUDY

1 Introduction

In this chapter, we apply the augmented GMM estimators to a GARCH(1,1) model
of the the hourly foreign exchange rate series of the West German deutschmark versus
US. dollar (DM/$). The hourly data cover a six-month period in 1986, from 0:00
am. January 2, 1986 through 11:00 am. July 15, 1986. The data set contains a total
of 3,190 trading hours. The exchange rate is taken from the average of the last five
bid rates recorded at the end of each hour by the fifty largest banks in the foreign
exchange market; for additional information, see Baillie and Bollerslev (1990).

This data set has been analyzed in a number of studies, including Baillie and
Bollerslev (1990). They test the null hypothesis of a unit root in the logarithm of the
exchange rate series, based on the methodology of Phillips (1987) and Phillips and
Perron (1988), and find that it cannot be rejected. Therefore, our estimation is based

on the first difference of the exchange rate,

at = 100 - [ln‘(st) — ln(st_1)], t = 2, 3, ..., 3189.

where 3; is the exchange rate before the transformation.

We calculate the first four unconditional moments and the correlogram of the se-
ries, Q, as shown in Table 4.1. The mean is -0.004 and the variance is 0.035. The
skewness and the kurtosis are, respectively, 0.171 and 10.027. The normal distri-

bution would have skewness equal to zero and kurtosis equal to three. The series

62

is clearly non-normal; it is an asymmetric distribution with thick tails. These are
features of the unconditional distribution of the series, whereas the potential advan-
tages of augmented GMM derive from non-normality of the conditional distribution.
Nevertheless, the clear non-normality of the series should make the augmented GMM

estimators potentially useful.

2. Estimation of the GARCH(1,1) Model

Before we begin estimation, we first ask what kind of model specification is suitable
for this series. The correlogram in Table 4.1 shows that there is no linear correlation
in the mean, but there is linear correlation in the second moments. Therefore an
ARCH-type model would seem appropriate. Baillie and Bollerslev ( 1990) analyzed
these hourly exchange rates using an MA(1)-GARCH(1,1) model. They found that
the variables in the conditional variance of the GARCH(1,1) model were significant
but the MA(1) coefficient was insignificant. This result, and the lack of correlation
in the levels of the variable which we found in Table 4.1, suggest that the martingale-
GARCH(1,1) model is appropriate. This is the specification later adopted by Baillie
and Chung (1999). On the other hand, unlike Baillie and Bollerslev (1990), Baillie
and Chung (1999) do not include hourly dummy variables and vacation day dummy
variables in the model. We follow Baillie and Chung (1999) and adopt a martingale-
GARCH(1,1) model without hourly or vacation dummy variables.

The model to be estimated is

Etl‘I’t—l N D“), ht),

63

ht = w + aۤ_1 + ,Bht_1.

NQMLE, MDE, and three types of augmented GMM estimators are used. The
augmented GMM estimators are GMMl, GMM2 and GMM3 as previously defined
in chapters 2 and 3. The estimation was performed using the CML and CO pro-
cedures in GAUSS for Optimization. The NEWTON (Newton-Raphson) method is
the optimization algorithm. The weighting matrix has the specification suggested by
Newey and West (1987), with the lag length selected with the automatic-lag selection
criterion of Newey and West (1994). For NQMLE, we tried different initial values
for the optimization, and obtained the same results. The initial values used in the
estimation of the augmented GMM estimators are from the results of NQMLE.

Table 4.2 shows the estimation results. For NQMLE, the estimates of w, a and 3
are significant and the sum of a and B is less than one. There is no evidence of inte-
grability in variance. For MDE with at least 10 autocorrelations of 6?, the estimates
of fl are close to those of NQMLE but with smaller standard errors. However, MDE
gives a smaller estimate of a. For GMMl, the estimate of a is still less than those
of NQMLE, while the estimates of w and ,8 are close to those of NQMLE. For all of
the parameters, the GMM] standard errors are smaller than those of N QMLE. For
GMM2, all estimates are close to the results of NQMLE, but with smaller standard
errors. For GMM3, the estimate of a is less than the NQMLE estimate, while the
estimate of ,6 is slightly larger than for NQMLE, and again the standard errors are
smaller.

In conclusion, NQMLE and GMM2 are similar in terms of parameter estimates

64

and standard errors. For MDE, GMMl, and GMM3, we get smaller estimates of a;
there is no clear pattern on ,6. The various augmented procedures do give smaller
standard errors. For example, the t-statistic of the estimated a is 4.74 for NQMLE

but is 8.88 for GMM3 with g = 20.

3. Diagnostics

It is interesting to check the sample moments of the conditional distribution of the

/2, which are reported in Table 4.3. The innovations are skewed

innovation u, = cthfl
and have severe excess kurtosis. Thus the conditions for possible efficiency gains from
augmented GMM appear to exist.

Table 4.4 gives the sample autocorrelations of 6? and the theoretical (p0pulation)
autocorrelations evaluated at the various estimates of a and ,8. The NQMLE and
GMM2 values for the theoretical autocorrelations do not match the sample values very
well. For MDE, GMMl and GMM3, we should have closer agreement because the
criterion minimized by the estimator includes the distance between the theoretical
and sample autocorrelations. Still the discrepancies seem fairly large. This casts
doubt on the validity of the model.

A formal test of the model can be carried out using the overidentification test of
Hansen (1982). Since the augmented GMM estimators are overidentified (the number

of moment conditions is larger than the number of parameters), the over-identifying

restrictions can be tested to examine the validity of the moment restrictions. The

65

test statistic is the minimized value of the GMM criterion function:
8(9)’Wria~(é), (7)

where 6 and W are valued at the corresponding augmented GMM estimates. This
statistic asymptotically has a x2 distribution with degrees of freedom equal to the
number of moment conditions minus the number of parameters. Table 4.5 gives the
values of the test statistic. For MDE, GMM2, GMM], and GMM3 with g up to
10, we cannot reject the null hypotheses. For GMMl and GMM3 with g = 20, the
statistic exceeds the 5% critical value for the relevant x2 distribution. Thus, in these
two cases, the null hypothesis that all moment conditions are satisfied is rejected.

We next apply the conditional moment test of Newey (1985). This test assumes
the validity of one set of moment conditions, say qu, and tests the validity of a second
set, say qzt. In our case Q1: is the score function from the normal log likelihood and
qgt is the extra set of moment conditions for augmented GMM.

Suppose that the optimal GMM estimate 0 is obtained using the first set of mo-

ment conditions, qu. Suppose also that 0 is a GMM estimate by adding the extra

moments q2t. The test statistic proposed by Newey (1985) is
mT = Hq’rQ_1HT,

where HT 2 fizflé) — flglflfilijlflé), and Q is the asymptotic covariance matrix of

mT, which can be consistently estimated by

{222 — 0210:1162” + (D2 — QQIQI—IIDI)(D’1QBID1)(D2 — 62191-11130,.

66

Here f2 and D are as defined in chapter 2, and D1, 152,911,ng and {222 are the ap-
propriate submatrices. Asymptotically mT is distributed as chi-squared with degrees
of freedom equal to the number of moment conditions being tested. In our case, this
is the same as the degree of overidentification (total number of moment conditions
for augmented GMM, minus the number of parameters). Note that in our case the
added conditions ”qgt” above can include the moment conditions called qgt (GMMl),
q3t (GMM2) or both (GMM3) in chapter 2 and 3.

In our case, the reason why this is an interesting test to consider is that the
extra moment conditions used by the augmented GMM estimators rely on stronger
assumptions than the moment conditions used by the NQMLE. The validity of the
NQMLE moment conditions requires only that the first two conditional moments be
specified correctly (plus some regularity conditions). However, the validity of the
extra moment conditions requires the representation (1) of chapter 1, which implies
restrictions on the higher conditional moments. So, we are interested in testing these
extra restrictions.

The test results are shown in Table 4.5. For GMMl and GMM2, the values of
mT do not exceed the chi-squared critical value, and the hypothesis that E[q2T] = 0
cannot be rejected. For GMM3, this hypothesis is rejected no matter how many
moments based on autocorrelations of 6? are included.

Thus, for our empirical study, both types of specification test (overidentification
test and conditional moment test) give mixed results. There is some doubt about the

model but it is not decisively rejected. Perhaps adding the dummy variables used by

67

Baillie and Bollerslev (1990) would help to improve the model’s conformance to the

data.

4. Conclusions

We applied our augmented GMM estimators to a GARCH( 1,1) model for the hourly
DM/$ exchange rate. Using GMM2, the model passes our diagnostic tests, but both
the estimates and their standard errors are very similar to N QMLE. The similarity
is so strong that there is not very much point in using GMM2 instead of NQMLE.
For GMMl and GMM3, the results of the diagnostic tests are mixed. The estimates
are only modestly different from the NQMLE estimates, but the standard errors are
considerably smaller. For example, for GMM3 with g = 20, the asymptotic standard
errors are about half as large as for NQMLE.

This raises the question of whether these efficiency gains are genuine, or just a
reflection of an unreliably small asymptotic standard error. The simulations of chapter
3 give evidence on this point. In our empirical example we have T = 3, 190, a = 0.18,
,6 = 0.55, and innovations that are slightly skewed and that have approximately the
same degree of kurtosis as a t distribution with 5 degrees of freedom. The closest
match in our simulations is in Table 3.10, where we have T = 2,000, a = 0.2,
fl = 0.6, and the t5 distribution. Here it was also true that GMM] and GMM3
with large values of g had asymptotic standard errors that were about half as big
as for N QMLE. The finite-sample (simulation) standard errors for GMM3 were also

smaller than for N QMLE, but only 10 ~ 20% smaller. Thus we might guess that, for

68

our empirical example, the GMMl or GMM3 estimates really are more precise than
the NQMLE estimates, but not by as much as the asymptotic standard errors would

indicate.

 

69

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NQMLE

MDE
$5

g=10
9=20

GMM1
g=5
g=10
9=20

GMM2

GMM3
$5
g=10
g=20

Semiparametric

Table 4.2

Estimation result of ow: exchange rate

Y! = 8!! a(I‘I't-‘I ~ 0(0, h!)

ht=C°+a€MZ+BhM

 

0)

a

[3

 

 

estimate

0.01028

0.00990
0.01130
0.01112

0.01056
0.01035
0.01024

0.00997
0.00937
0.00943
0.00942

0.00995

0.00262

nla
nla
nla

0.00167
0.00137
0.00125

0.00253

0.00138

0.00119
0.00114

estimate

0.17711

0.08783
0.13056
0.13662

0.13868
0.15292
0.15935

0.17852
0.14727
0.14417
0.15514

0.17147

 

8.6.

0.03736

0.02476
0.02424
0.02270

0.02322
0.02208
0.02073

0.03499

0.01916

0.01835
0.01748

 

estimate

0.54582

0.62687
0.54372
0.54286

0.54825
0.53549
0.53085

0.55343
0.57000
0.55819
0.55852

0.55626

0.09157

0.09514
0.05659
0.04909

0.05507
0.04498
0.04119

0.08598

0.04896

0.04138
0.03904

 

GMM1-the score from Normal distribution & MDE
GMMZ— the scores from Normal distribution 8. rescaled t distribution
GMM3- the scores from Normal distribution 8. rescaled t distribution and MDE

71

 

Table 4.3
The conditional distribution of on.“

 

 

mean variance skewness kurtosis
NQMLE -0.023 0.999 0.189 10.932
MDE
g = 5 -0.022 0.989 0.183 10.416
9 = 10 -0.023 1.008 0.177 10.652
9 = 20 -0.023 1.012 0.178 10.692
GMM1
g = 5 -0.023 1.036 0.181 10.728
9 = 10 -0.024 1.058 0.183 10.830
9 = 20 -0.024 1.066 0.183 10.877
GMM2 -0.023 1.001 0.192 10.952
GMM3
g = 5 -0.024 1.060 0.191 10.824
9 = 10 -0.024 1.092 0.188 10.821
9 = 20 -0.024 1.071 0.191 10.877

 

 

 

 

skewness -- Eff/cs3
kurtosis - Eq‘lo‘ .
GMM1—the score from Normal distribution 8 MDE

GMM2— the scores from Normal distribution 8 rescaled t distribution
GMM3- the scores from Normal distribution 8 rescaled t distribution and MDE

72

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74

Chapter 5
CONCLUDING REMARKS

The normal quasi maximum likelihood estimator (NQMLE) is the MLE under
the assumption of normality, but this assumption is likely to be violated in empirical
data. Although in such cases the NQMLE is still consistent and asymptotically
normal, it is inefficient. Therefore it is worthwhile to find more efficient estimators.
We suggest augmented GMM estimators to improve the efficiency of the NQMLE
under non-normality. We interpret the NQMLE as a GMM estimator, where the
moment conditions represent the normal score function, and then we augment this
set of moment conditions with other moment conditions that also do not depend on
the validity of any particular distributional assumption.

We consider two sets of extra moments to augment the GMM (NQMLE) estimator.
The first set of extra moments is from the autocorrelations of the squared innovations.
The second set of extra moments is from the rescaled student’s t distribution. The
student’s t distribution is of particular interests because its property of leptokurtosis
(fat tail) is consistent with many empirical financial data sets. We consider three
combinations of these extra moments, and the resulting augmented GMM estimators
are called GMMl, GMM2, and GMM3.

We compare the performance of these different estimators by calculating and com-
paring the asymptotic standard errors. When the true density is non-Gaussian, our

results show that the augmented GMM estimators have moderate improvements in

75

efficiency compared with NQMLE. The efficiency gain is mostly in estimation of the
parameter a; for w and 8 there is very little improvement.

Monte Carlo results show that the augmented GMM estimators have an efficiency
gain over the N QMLE when the true distribution is non-Gaussian, but this requires a
rather large sample size, such as T = 2, 000. For smaller sample sizes, adding moment
conditions from the autocorrelations of of helps, especially for a, but extra moments
from the rescaled t distribution do not seem to improve efficiency when the sample
size is small (e.g. 500). The GMM3 estimator, which uses both extra sets of moment
conditions, shows more of a gain in efficiency for all of sample sizes we consider.

Our simulation results show that the augmented GMM estimators that use a large
number of moment conditions (GMMl and GMM3) can be biased in finite samples,
even when the sample size is rather large (e.g. T = 2, 000), and inference based on
the asymptotic distribution can be inaccurate. For example, tests suffer serious size
distortions. GMM2 does not suffer from these problems. One possible reason for the
poor finite-sample performance of GMMl and GMM3 is that some of the moment
conditions based on the autocorrelations of the squared data may be ”irrelevant” (or
nearly irrelevant), as discussed by Hall and Peixe (1999). They show that including
irrelevant moment conditions may lead to bias and size distortions.

We give an empirical application to the DM/$ exchange rate to illustrate the use
of the augmented GMM estimators and to compare the results with those of NQMLE.
GMM2 is not very different from NQMLE, while for GMMl and GMM3 the estimate

of a is somewhat smaller and the standard errors are smaller.

76

A promising line of future research would be to investigate the use of moment
conditions from rescaled asymmetric distributions. This is motivated by the fact that

many financial data are asymmetric as well as fat-tailed.

77

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83

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