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*4 AAAAAAAAAAA AAAAAAAAAA AAA AA AA AAA AAAA AAAA AAAAA

31293 017

”BMW“!
Michigan State
University

This is to certify that the

dissertation entitled

MINIMUM DISTANCE ESTIMATION FOR ARMA AND GARCH PROCESSES
presented by
Huimin Chung
has been accepted towards fulfillment

of the requirements for

Ph. D. degree in Economics

 

 

 

 

Major professor

Date (1/ Li lac/’7

 

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

MINIMUM DISTANCE ESTIMATION FOR ARMA AND GARCH PROCESSES

By
Huimin Chung

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Economics

1997

ABSTRACT

Minimum Distance Estimation for ARMA and GARCH Processes

By

Huimin Chung

This dissertation considers the estimation of the parameters of ARMA and
GARCH processes by the Minimum Distance Estimator (MDE). The estimator is
obtained by minimizing a quadratic distance function between the sample
autocorrelation and theoretical autocorrelation functions and has the advantage of
imposing very little in terms of distributional assumptions on the innovation process.
The asymptotic properties of the MDE for ARMA processes are discussed. The exact
asymptotic efficiency of using a block of sample autocorrelations is derived for some
low order MA and ARMA models. The MDE is surprisingly efficient for some parts
of the parameter space. We also investigate the properties of the MDE when used to
estimate linear GARCH models using autocorrelations of the squared process. Monte
Carlo results show that the MDE performs better than Quasi-MLE with certain
conditional densities which exhibit extreme departures from conditional Gaussianity.
An application of the MDE to estimate a GARCH(1,1) model fi'om high frequency

exchange rate data is provided.

ACKNOWLEDGEMENT

The completion of this dissertation would never have been possible without the
help and support of numerous. individuals. My greatest appreciation must go to my
dissertation committee chair, Professor Richard Baillie, who introduced me to this
dissertation topic, sacrificed countless hours in reading and correcting drafts, and
carefully guided and inspired me through each step of my research. I have benefited
greatly from his experience and wisdom. I will remain indebted to him throughout

the remainder of my career.

In addition I would like to express my appreciation to Professors Robert de Jong,
Peter Schmidt, and Jefl‘ Wooldridge, whose guidance, encouragement and help have
been invaluable in completing this dissertation. Many parts of this dissertation were
especially strengthened by their suggestions and contributions. I also thank Professor
Ching-Fan Chung for his help and encouragement in my early stage of writing the

dissertation.

I would never have completed my journey through graduate school without the
support of my family. Particularly crucial to this accomplishment is the unceasing
support of my wife, Wen-Hua. Words would not be enough for her love and
understanding. Thanks also go to my father and mother. Their encouragement and
support throughout the peaks and troughs of these past few years is invaluable. I owe

a special thank: too, to my younger brother Ching-Gee.

iii

I am also grateful to many colleagues and friends at Michigan State University. I
especially thank Jen-Je Su, Hailong Qian, Wen-Jen Tsay, and Yang-Seon Kim who

offered many valuable comments and discussions of this dissertation.

Finally, this dissertation is dedicated to my lovely daughter, Candice, who has

been giving me so much fun since she was born.

iv

TABLE OF CONTENTS

LIST OF TABLES .......................................................................... vi
CHAPTER 1 INTRODUCTION ........................................................ 1
CHAPTER 2 MINIMUM DISTANCE ESTIMATION FOR ARMA

PROCESSES ....................................................................... 4
1. Introduction ............................................................................. 4
2. The MDE .............................................................................. 5
3. The MDE Applied to AR(p) Processes ............................................... 8
4. MDE Applied to Moving Average Processes ...................................... 17
5. MDE Applied to ARMA( 1,1) Processes ............................................ 27
6. MDE Applied to Higher Order ARMA(p,q) Processes ............................ 34
7. Concluding Remarks ................................................................ 37
Appendix .................................................................................. 42
CHAPTER 3 MINIMUM DISTANCE ESTIMATION FOR SEASONAL MA

PROCESSES ........................................................................ 47
1. Introduction .............................................................................. 47
2. MDE of MA(l)-Seasonal MA(1)4 Processes ..................................... 49
3. MDE of MA(1)-Seasonal MA(1)12 Processes ....................................... 53
4. Estimation Results of MDE for Airline Model .................................... 59
5. Concluding Remarks .................................................................... 64
Appendix .................................................................................. 67
CHAPTER 4 MINIMUM DISTANCE ESTIMATION FOR GARCH

MODELS ............................................................................ 7O
1. Introduction .............................................................................. 7O
2. MDE of GARCH(1, 1) Process ....................................................... 72
3. Simulation Results of MDE ofARMA(1,1) Process with 1. i. d. Innovations" .74
4. Simulation Results of MDE of GARCH(1, 1) Models ............................. 77
5. Example : Estimation of the GARCH Model Applied to Hourly

Exchange Rate Data ............................................................ 98

6. Concluding Remarks ................................................................... 99
Appendix .................................................................................. 102
CHAPTER 5 CONCLUSION ........................................................... 103
LIST OF REFERENCES ............................................................... 105

LIST of TABLES

Table 1 Asymptotic variance of MDE for MA(l) processes .............................. 20
Table 2 Asymptotic variance of MDE for MA(2) processes .............................. 28
Table 3 Asymptotic variance of MDE for MA(2) processes .............................. 29
Table 4 Asymptotic variance of MDE for ARMA(1,1) processes ........................ 32
Table 5 Asymptotic variance of MDE for ARMA(1,1) processes ........................ 33
Table 6 Asymptotic variance of MDE for ARMA(2,1) processes ........................ 38
Table 7 Asymptotic variance of MDE for ARMA(2,1) processes ........................ 39
Table 8 Asymptotic variance of MDE for ARMA(1,2) processes ........................ 40
Table 9 Asymptotic variance of MDE for ARMA(1,2) processes ........................ 41
Table 10 Asymptotic variance of MDE for MA(l)-SMA(4) processes ................. 54
Table 11 Asymptotic variance of MDE for MA(l)-SMA(4) processes ................. 55
Table 12 Asymptotic variance of MDE for MA(1)—SMA(4) processes ................. 56
Table 13 Asymptotic variance of MDE for MA(l )-SMA(4) processes .................. 57
Table 14 Asymptotic variance of MDE for MA(l)—SMA(12) processes ................ 60
Table 15 Asymptotic variance of MDE for MA(l )-SMA(12) processes ................ 61
Table 16 Asymptotic variance of MDE for MA(1)—SMA(12) processes ................ 62
Table 17 Asymptotic variance of MDE for MA(l)-SMA(12) processes ................ 63
Table 18 Estimation Results of the MDE of Airline Model ............................... 65

vi

Table 19 Simulated mean and RMSE of MDE and MLE of ARMA(1,1) processes... 78
Table 20 Simulated mean and RMSE of MDE and MLE of ARMA(1,1) processes. .. 79

Table 21 Simulated mean and RMSE of MDE and MLE of GARCH(1,1)
processes ................. '. ................................................................. 87

Table 22 Simulated mean and RMSE of MDE and MLE of GARCH(1,1)
processes ................................................................................... 88

Table 23 Simulated mean and RMSE of MDE and MLE of GARCH( 1,1)
processes ................................................................................... 89

Table 24 Simulated standard deviation of MDE for GARCH(1,1) processes ......... 91

Table 25 Simulated mean and RMSE of MDE and QMLE of GARCH(1 ,1)
processes ................................................................................... 93

Table 26 Simulated mean and RMSE of MDE and QMLE of GARCH(1,1)
processes ................................................................................... 94

Table 27 Simulated mean and RMSE of MDE and QMLE of GARCH(1,1)
processes ................................................................................... 95

Table 28 Simulated mean and RMSE of MDE and QMLE of GARCH(1 ,1)
processes ................................................................................... 96

Table 29 Estimation Results of MDE and MLE of Exchange Rate Data ................ 99

Table 30 Sample ACF and the autocorrelations implied by the ML and
minimum distance estimates of the squared hourly exchange rate ............... 100

vii

CHAPTER 1
INTRODUCTION

This dissertation considers the estimation of the parameters of some linear time
series processes and GARCH volatility processes by the use of Minimum Distance
Estimator (MDE). The estimator is obtained by minimizing a quadratic distance
function between the sample autocorrelation and theoretical autocorrelation func-
tions. The MDE is very similar to Hansen’s (1982) GMM estimator. In the context
of this dissertation the moment conditions correspond to sample autocorrelations.
The MDE method has been previously applied to the problem of estimating the
Hurst coefficient, or order of fractional integration in ARFIMA models by Tieslau,
Schmidt and Baillie (1996) and Chung and Schmidt (1996). The MDE has the advan-
tage of imposing very little in terms of distributional assumption on the innovation
process. Hence in cases of extreme non—normality the MDE, which is straightforward
to compute, may have some distinct advantages.

The plan of this dissertation is as follows. Chapter 2 introduces the MDE and
derives the asymptotic properties of the MDE for some specific ARMA processes. We
derive the exact asymptotic efficiency of the MDE relative to MLE for some specific
ARMA processes. The application of the MDE to AR(p) process is first discussed.
The asymptotic variance of the MDE using the first p autocorrelation is compared
with that of MLE under NID innovations. The asymptotic variances of the MDEs
using different sets of autocorrelations are calculated for some MA(q) and ARMA(p,q)

processes for a variety of parameter values. Also the asymptotic variance of the MDE

are compared with that of MLE under NID innovations.

Chapter 3 investigates the asymptotic properties of the MDE for the MA(1)-
Seasonal MA(1), model, which is applied by Box and Jenkins to model the airline
passenger data and is also called ”airline model”, because of its widespread use in
time series analysis. This airline model has been applied to model many economic
time series. The asymptotic variance of the MDE are discussed for the cases that 3
equals to 4 and 12.

Chapter 4 deals with using the MDE based on the sample autocorrelations of the
squared process to estimate the parameters of the GARCH model. The GARCH
models and their extensions have been widely applied in characterizing the time de-
pendent heteroskedasticity present in many economic and financial economics series.
If the observed time series, y, follows a martingale with linear GARCH( 1,1) volatility
process, then y? has an ARMA(1,1) representation. However, despite the innovation
process being serially uncorrelated, it is not independent over time. When the inno—
vations are not i.i.d., it may not be valid to use the Bartlett’s formula to calculate the
weighting matrix. Nevertheless, the robust method of Domowitz and White (1982)
and White (1984) can be applied to obtain robust covariance matrix estimator of
the sample autocorrelation. We thus propose a Newey and West (1987) type covari-
ance matrix estimator of the sample autocorrelations of the squared process, which
is generated as a martingale GARCH(1,1). A Monte Carlo experiment is carried out
to compare the performance of the MDE using the Bartlett’s formula to construct

the weighting matrix and the MDE using the Newey-West method to calculate the

optimal weighting matrix for estimating the parameters of the GARCH(1,1) model.
Many studies have found evidence that the conditional density of GARCH models
for many speculative asset return series, such as exchange rates and stock prices, are
non Gaussian. The Quasi—MLE (QMLE) method is usually invoked to estimate such
GARCH models. A simulation experiment is performed at the end of this chapter

to compare the small sample properties of the MDE and the QMLE of GARCH(1,1)

models.

CHAPTER 2

Minimum Distance Estimation for ARMA processes

1 . Introduction

There is a long tradition in univariate time series analysis of approximating MA
and ARMA processes as infinite order AR processes, e.g. Durbin (1959, 1960). The
motivation for this approach was typically in terms of the ease of estimation of AR
processes by least squares and hence the avoidance of maximizing a likelihood which
was highly non linear in the parameters. Before the advent of modern computer
power such methods were the only realistic way of estimating ARMA models. There
is still some interest in the estimation of ARMA models by approximations in terms of
high order AR processes and Galbraith and Zinde Walsh (1994) consider the distance
in terms of Hilbert space between the true ARMA data generating process and the
AR(p) approximation. Also see Koreisha and Pukkila (1990), Saikkonen (1986), and
Galbrath and Zinde Walsh (1997).

This chapter considers the estimation of the parameters of some univariate time
series process by the Minimum Distance Estimator (MDE), which is closely related
to Hansen’s GMM estimator. In the context of this chapter the moment conditions
correspond to sample autocorrelations. The MDE method has been previously applied
to the problem of estimating the Hurst coefficient, or order of fractional integration
in ARFIMA models by Tieslau, Schmidt and Baillie ( 1996) and Chung and Schmidt

(1996). The MDE has the advantage of imposing very little in terms of distributional

CJ‘I

assumptions on the innovation process. Hence in cases of extreme non-normality, the
MDE, which is straightforward to compute, may have some distinct advantages.
The remainder of this chapter is organized as follows. Section 2 discusses the
general set up of the MDE. Section 3 then derives the exact asymptotic efficiency of
the MDE relative to MLE for some specific non-seasonal stationary AR processes. For
the AR(p) process the MDE using the first p autocorrelation can be asymptotically
as efficient as MLE under normality. Section 4 then derive the gain in asymptotic
efficiency from using higher order sample autocovariances when estimating the moving
average process. Section 5 deals with ARMA (1,1) process, while section 6 considers
the same approach for the higher order ARMA(p,q) process. Section 7 provides a

brief conclusion.

2. The MDE

The minimum distance estimation method is based on the following asymptotic
results for the sample autocorrelations of a stationary process. The sample autocor-

relation at lag k is given by

T '1‘
Pic = 2 (9t — g)(yt—k - m/ZQ/t — 17R
t=k+l t=1

where g is the sample mean. Bartlett (1946) and Brockwell and Davis (1991, p.221,
Theorem 7.2.1) show that if y, is a stationary process, yt = 28;“, wjet_j, 5, ~

i.i.d.(0,a2), where Z°° |¢j| < 00 and Ee‘,’ < oo , then

where fi’ = [61,fig,---,,o“g], p’ = [p1,p2,---,pg] and C is a g x 9 matrix with (i,j)th

element given by
00
Cij = [AX—2501:“ + pic—1' — 2P1Pk)(Pk+j + Pic—j — 210191).
Alternatively the assumption of finite fourth moment of 6, can be replaced with the
condition that Zfiwo 112,2] j | < 00.
The MDE is an alternative to the maximum likelihood estimator and has the
advantage of not invoking strong distributional assumption. On defining A as the

vector of parameter to be estimated, so that in the case of the ARMA(1,1) model

A’ = [45 6]. The MDE is obtained by minimizing the following criteria function

Min 3 = (,3 — p(A))'W(A5 — p(A)),

where W is a g x g symmetric, positive-definite matrix. Let A denotes the value of A

which solves the above minimization problem. Therefore, we have

as _ 0p(A) ’. ._
EXA=X_ [EV—1:1] W(p—p(A))—0.

 

 

Let A0 denote the true value of A. Using the mean value Theorem, we have A, such

that

x3 - p(A) = f) - p(Ao) + ————a(fi $15 W)

 

(A — A0). ('2)
AzA.

For convenience we have the following definition of notation :

Di 000)

BA’

 

A=A .
Similarly, DA, and D denote the partial derivative of p(A) with respect to A’ evaluated

at A. and A0, respectively.

Premultipling both sides of equation (2) with DEW, we have
D’j W05 — 0(3)) = D:\ W (P5 " P(/\o)) - D} W DA.(:\ — A0)-

The left hand side of the above equation is the first order condition and hence, equals

zero. Rearranging the above equation yields
.. -1 ..
A — AA = [Di-A W DA] 0’; WAp — poo»-

Notice that plim(;3 — p(Ao)) = 0. Given that plim ([Dg W DA,]"DS1 W) is finite, it
follows directly that A is consistent. We then have plimDA, = plimDA-A --= D. Let 3)

denote converge in probability. Then, it is easily verified that
m1 — AA) —”+ AD’WDt‘D’Wfi ( l5 — p(Ao) )
Using the results derived above, we obtain
A/Toi — 10) —+ N ( o, (D’WD)“D’WCW’D(D’WD)" )
On using the standard optimal weighting matrix of W = C", in which case the

limiting distribution of A is

A

fi(/\ — 10) —> N ( o, (D'C-lprl ). (3)

A consistent estimator of C is C, the g x 9 matrix of the sample counterpart of

C, with the (i, j)th element being estimated by

m
éij = ZU’HA + fik—t - Qfiifik)(fik+j + fik—j — 293m)- (4)
k=1

In practical applications the MDE is obtained by solving the following minimization

problem:

A

Min 5 = (fi - p(A))'0"’(fi - p(A))-

3. The MDE Applied to AR(p) Processes

The asymptotic efficiency of the MDE can be investigated for different sets of mo—
ment conditions. For the AR(I) process the MDE using the first autocorrelation is
asymptotically as efficient as MLE under normality. This is easily seen from investi-
gation of the redundancy conditions associated with the use of the first m1 moments,
as opposed to the use of the first m1 + m2 moments.

Let Eg(y¢,A) = 0 be the moments condition of the GMM estimator and §(A)
denote the sample average of the moment condition, i.e., §(A) = %Z;1 g(yt,A).
The GMM and MDE estimator is obtained by minimizing a quadratic function,
§(A)’W§(A). Let C denote the asymptotic covariance of figflA). The optimum
weighting matrix for the GMM estimator based on §(A) is W = C“. For the MDE
in this chapter §(A) corresponds to p‘ — p(A).

To investigate whether a subset of the moment conditions is redundant, 9(A)
is partitioned into two subsets, .i.e., g(A) = [g‘A(A) g'2(A)]’, and C is also can be

partitioned as

_ C11 C12
C‘ACAA C..]’

where C11 and C22 are the asymptotic covariance of {71930) and fig‘fiA), respec-

tively. The asymptotic variance of the MDE estimator using first m1 moment condi-

tions, 91(A), is
(DICI-IIDI)-1’

where D1 = MW]. Bruesch, Qian, Schmidt and Wyhowski (1997) show that if

D2 — 0:201:10. = 0, (5)

with D; = E [92%], then the extra m2 moment conditions are redundant and MDE
based on the first m1 moments is as efficient as MDE based on first m1+m2 moments.
Based on the above result, it can be shown that for the AR(l) process, MDE

using first 2 autocorrelations has no improvement in efficiency over MDE using just

the first autocorrelation. The AR(I) process is

3/: = (Wt-1 + 5t, (6)

where at is iid (0, 02). The autocorrelation function of the AR( 1) process is ph = (M

for h = 1, 2, - - -. The asymptotic variance of sample autocorrelations of the AR(l)
process is
' 1 2(1) 3¢2 4453
245 1 + 3d)2 2¢ + 4:153 391)2 + 5&1
C=(1—¢2) 3¢2 2¢+4¢3 1+3¢2+5¢4 2¢+4¢3+6¢5 , (7)

44” 36" + 545’ 2¢ + W + 6435 1 + 34>2 + W + 7466

A; s s s--.A

with the P, ch element of C can be written as

 

 

Q
CPQ = (1— <1?) ZAP — Q — 1 + amt-“”1“” . for P 2 Q.

i=1

10

Also,

D = —A1 245 343’ A (8)

It is straightforward to show that the asymptotic variance of MDE using the first
autocorrelation is 1 - (V, which is equal to the asymptotic variance of MLE under
normality. A direct calculation shows that the asymptotic variance of MDE using
first two autocorrelations is also equal to 1 — (122. Furthermore, it can be verified that
the redundancy condition of equation (5) holds for g 2 2. When 9 2 2, we have
D2 = -[2¢ 3452 ° " g¢9"’]’,C'12 = (1 - $2)[2¢ 3432 ' '° 9¢9_’]AC'11 = (1 - (1’2), and
D1 = —1 . Hence D2 — CizCfi’Dl = 0.

For AR(l) process the MDE using first 9 autocorrelations for g 2 2 does not have
any improvement in efficiency over the MDE using just the first autocorrelation. Al-
ternatively, we can estimate 41 from just the hth autocorrelation, 5),, for h. = 3, 5, 7, - - -.
Notice that we are not able to identify (11 if only p2 is used in the MDE, because the
sign of d) can not be determined. Similarly, we can not identify the AR parameter if
only p2]- is used for j = 2, 3, - - -. Consider the estimation of d) from [13. On using the

results given in equations (7) and (8), we have

(1 - $2)(1+ 3<I>2 + 5¢")

Tl/2(/3i/3 _ (A5) —+ N [0, 9454 ].

 

Therefore, the estimator of ()5 based on 53 only is not asymptotically efficient since

W > 1 for |¢| < 1. In general, for h = 3, 5, 7- - ~, we have the following result

'

of the asymptotic variance of the MDE using just 5,, (V1505).

(Mi-”(22' — 1)
h2¢2(h-1)

 

h _ _ 2:?zl _ 2
VMDE—(l <15) >(1 ¢)for |d>|<1.

11

The last inequality comes from the facts that Z?=1(22' — 1) = 112 and (152("1’ Z ¢2("”)
for 2' = 1, - uh. Therefore, the estimator of <15 based on ,6), only for h = 3, 5, 7--- is
not asymptotically efficient.

Similarly, for the AR(p) process the MDE using the first p autocorrelations is
asymptotically as efficient as MLE under normality, while a formal proof of this

result is given below. The AR(p) process is

3/: = ¢1yt—1 + 49291—2 + ' ' ' + ¢pyt—p + 5t: (9)

where at is NID(0, 02). The sample autocovariance of y; at lag h is defined by ”Ah =
T“1 2L1 ytth where h = 0, 1, 2, - - -. The sample autocorrelation at lag h is defined
as {3). = “Am/’70. The vector of parameters of the AR(p) process is denoted as (I) =
[(111 --~¢,,]’ and the true value of (I) is denoted as ¢o- The log-likelihood function of

the AR(p) process can be written as

1

T
2 T02 2: (91'- ¢lyt—1 - °"— ¢pyt—p)2'

t=p+1

 

—1 T 1 1 2
L=T Z l,=—§10g27r——loga —

t=p+l 2

Without loss of generality, assume that 02 is known. The MLE is asymptotically
equivalent to the GMM estimator using the following population moment conditions:

E(yt6t_j) = 0, j = 1 ~ - -p. Let the p x 1 score vector of the MLE be denoted as S:

T-l irzp+1(yt - 45191—1 - ‘ ' ‘ " ¢pyt-p) lit-1
S = '

T-l Z'trzp-l-lu/t " 45191-1 — ' ' ' - ¢pyt-p) yt-p
The GMM based on the above moment conditions is asymptotically as efficient as

MLE under normality.

Let 6y = [7‘0 7‘1 - .. 7“,, ]’. Note that S can be rewritten as

S=As+c

12

where A is a p x (p + 1) matrix given by

 

 

( —¢1 1 - 4’2 -¢>3 —¢4 —¢p-2 -¢p—1 “1% 0 \
-¢2 -(¢1 + ¢3) l — 454 -¢5 '" -¢p—1 -¢p 0 0
~¢3 —(¢2+¢4) —(¢1+¢5) 1-¢>6 *¢p 0 0 0
A :
‘4’p-2 "(4512—3 + ¢p—1) ‘(4’p-4 + ¢p) "' "' -¢1 1 0 0
-¢p—1 -(¢p-—2 + ¢p) -¢p-3 "' °'° -¢2 -¢1 1 0
"(pp -¢p—1 -¢p-2 "' "' -¢3 -¢2 -¢1 1 J

and, C is a p x 1 vector given by
f ‘ 2L2 sew-1 + ¢1( 5’3 1!? + xii) + MET}; yij—A + yryrn) + - - - + cpyTyT_(,,_1, A

—1
- 2L3 win—2 + ¢1(Z§=2 Aim-1 + yTyT-l) + - - - + ¢p(yT—1yT-p+l + grin—m)

 

 

\ ¢1yTyT—(p—1) + ¢2(yT-—lyT—p+l + WAIT—1&2.) + ‘ ' ' + ¢p ELI yI—pfl' }
Since p is fixed, it can be verified that the limiting variance of 711/24 is equal to 0 by
using the triangular inequality. Therefore, C is asymptotically negligible.

Now the moment conditions of MDE based on first p autocorrelations can be

written as

where T T 2
thz ytyt-l — 101(0)) thl 11:
M=T'l 5
211‘sz ytyt-p " pp(¢) {:1 Qt2

Similarly, we have

M=Bfi

13

where B is a p x (p + 1) matrix given by

 

 

{—m] 00 ~0)
‘P2010"°0
\—%()00o~1)

The MDE is asymptotically equivalent to the GMM estimator based on M, since

’yf," is not a function of (b. The following result demonstrates that the MDE is

asymptotically as efficient as MLE under normality.
Let 7 be a (p+ 1) x 1 vector denoted by [70 71 - - - 7,, ]’, where 7,, is the hth order

autocovariance function. It is obvious that

37:0
Multiplying equation (9) by y¢_,- for j = 1, - - - p, and taking expectation obtains

A 7 = 0. (10)
Hence,

and

M

BW-fl.

The GMM estimator based on moment conditions S is asymptotically equivalent
to the GMM estimator based on moment conditions M if there exists a px p, invertible
matrix ‘1! such that

B=WA (u)

14
Let —p denote the vector [—pl — p2 -- - — pp]’. We can partition B as
B = I -p I 1P ]1

where Ip is a p x p identity matrix.

Let —¢ denote the vector [—¢1 — 432 -- - — ¢p]’. Hence, A can be partitioned as
A = [ —¢ I L ]i

where L is a p x p matrix containing the 2th to pth columns of A.

Let 7. = [71 7p ]’. Rewriting equation (10) we have
L7. - (#70 = 0.
Dividing the above equation by 70 yields
Lp = d). (12)

For stationary AR(p) process, p can be uniquely determined by using the above
equation, so that L is invertible.

We thus argue that ‘I' = L“, i.e.,

{ 1— d» -a —¢. —¢._2 -¢.-A -¢. 0 A ‘1
—(¢: + 453) 1 — «:54 —¢:. —¢,,-A —¢, 0 o
-(¢2+¢4) —(¢1+¢5) 1-¢6 -¢p 0 0 0
\II =
-(¢p.—a + ¢p—l) —(¢p-4 + 41,.) --- -~ —¢A 1 0 0
-(¢p—2 + ¢p) -¢p-—3 ' ' ‘ ' ‘ ' ‘4’? ’4’1 1 0
K —¢._A —¢._2 —¢. —¢. —¢A 1 )

 

 

Also, ‘1! (b = p, which is implied by equation (12).

15

Using the result given in equation (11), we now show that the GMM estimator
based on moment conditions S is asymptotically equivalent to the GMM estimator
based on moment conditions M.

Let 435 and 6M be the GMM estimators based on the moment conditions S and

M, respectively. Under suitable regularity conditions, we have
77(3), — a0) —+ N( o, [D'o-IDA-l ),

where Q is the asymptotic variance of x/T S and D = plim{ % ¢=¢o}° Similarly, we
have

fi(‘£M - (be) ‘9 N( 0: [DIWQMIDMi—l ),
where OM is the asymptotic variance of x/T M and DM = plim{ 65:4 ¢=¢o}' Equation

(11) implies that T1/2M = T1/2\IIS + 0,,(1). Thus,

QM = \IIQ‘II'
and

DM = ‘I’D.
Hence, we have
[DAMQAQIDMFI = [ (\IID)’(\IIQM\II’)‘1\IID)]"l = [D’Q‘ID]‘1 = Q".

The last equality comes from the fact that under the assumption of at being i.i.d., the
usual information matrix equality holds and D = (2. Thus, [ D’ Q‘ID ]‘1 = Q“.
The above result demonstrates that (135 and 43,14 are asymptotically equivalent.

Obviously, 455 is asymptotically as efficient as MLE under normality, so is the MDE

based on the first g autocorrelations.

16

Example: MDE for AR(2) Process

The first order and second order autocorrelation functions of the AR(2) process

are

$1
(1 - $2),

 

and
as?
(1 — $2).

 

102=$2+

The score vector of the log-likelihood function of the AR(2) process is

( T" Zszp+1(llt — $1yt-1 " $2yt-2) 111—1 )
S :
T_l Z'tr=p+l(yt — $1yt-1 - $2yt—2) 111—2
( 71 - $170 — $271 ) 1 ( ‘92111 + $1(I/i + 3177+ $2(yTyT—1) )
= + T-
72 — $171 - $270 ¢1(yryr_1) + ¢2(y% + 11511-1)
70
_ -$1 1 - $2 0 .
_ (-$2 ‘$1 1)(YI)+C’
72
where,

1 ( —y2y1 + $1(yi + 11%) + $2(l/T3/T-1) )
C = T—

$1(yTyT—1) + $2(yi‘ + Ilia-1)
It can be verified that lianoo Var(T‘/2C) = 0, so that C is asymptotically negligible.

The moment conditions in MDE can be expressed as

71 — P170 “P1 1 0
= 71
72 — P270 '92 0 1

17

For stationarity, the roots of 1 — (1318 — $282 = 0 must lie outside the unit circle,

which implies that |q§2| < 1. Hence, L is invertible and

(1—¢2 0)-1 1 (I O )
‘1]: =
—$1 1 1—¢2 $11—$2

Clearly, \II‘l exists and
1 1 0 $1 (l—tlfi‘z) P1
Q¢=1—@ = w = .
¢1 1 ‘ $2 ¢2 (152 + (:3; P2

4. MDE Applied to Moving Average Processes

 

 

4.1 MDE Applied to MA(l) Process

The MA(l) process is,

yt = (1" 01/151, (13)

where at is iid(0, 02) and the autocorrelation function of the MA(l) process is p1 =
IAI—gf and pk = O for k 2 2. The limiting distribution of MLE of 0 when ct is iid(0, a2)
is,
fi(éMLE — 0) —) N(0,I — 02).
To calculate the asymptotic variance of MDE, it is necessary to obtain the asymp-

totic variance of sample autocorrelations, i.e, C. It is easily verified that for the MA(I)

18

 

 

process
' 1-3pf+4pi 21110-11?) pi 0 0
2p1(1 - pi) 1 + 20? 2P1 pi '
C = pi 2101 1 + 2p? 2m 0 .
0 pi 2m 1+2pi pi
E . . 2P1
1 0 - - - 0 pf 2m 1 + 2p? 1
and D = [D1 0---0]’, where D1 = (5%. For convenience, let the number of

autocorrelations used in MDE be denoted by 9. When g is small, e.g., g = 1 or 2,
it is very straightforward to obtain analytical results for the asymptotic variance of
{TWA — 0), i.e., (DC-ID)". From direct calculation the asymptotic variance of the

MDE based on only the first g autocorrelations for 9 =1, 2 and 3 are

1+02+404+06+03

 

 

 

g=1 =
V (6’2 -1)2 ’
V9=2 1+92+(9"+€519"+03+61°+912
(A92 —1)2(1+492+94) ’
and
V9=3 _ 1+02+04+66+808+01°+012+014+016

(02 —1)‘2(1 + 402 + 10194 + 496 + 198)
The increase in asymptotic efficiency from using two rather than one moment is

(V9=1— VP?) 2 402(1 + 202 + 304 + 206 + 08)
(6‘2 -1)2(1+ 402 + 04)

 

Similarly,

9(0" + 266 + 308 + 4610 + 3012 + 2914 + 916)
(92 -1)2(1+ 402 + 04)(1+ 492 + 10194 + 4196 + 98)’

 

(VF? — W3) =

which is always positive. Therefore, adding the second order autocorrelation to the
MDE improve the asymptotic efficiency, so does adding the third order autocorrelation

to the MDE.

19

Table 1 shows that asymptotic efficiency is increased as the number of moments
increases and presents the asymptotic variance of x/T (61M D 3 ~19) for the MA(l) model
for 0 = 0.1,0.2, 0.3, - . ~ , and 0.9. Calculations reveal that as 0 increases, the asymp-
totic variance of the MDE increases. Table 1 only reports positive values of 0 since
the results are symmetric around zero. If the absolute values of the MA(l) coefficients
are the same, the asymptotic variance of the MDE is the same.

Durbin (1959) has suggested estimating MA(q) models by their AR approxima-
tion and shows that if the number of AR coefficients is large enough, this estimator
is asymptotically as efficient as MLE. Analogously, it is necessary to have more au-
tocorrelations for the MDE in order to achieve relative efficiency for large value of 0.
Galbraith and Zinde-Walsh (1994) suggest estimate the MA models based on mini-
mizing the Hilbert distance between the MA model and its AR approximation and
have similar conclusions.

In many cases it is surprising to note that a remarkably small number of auto-
correlations are necessary for the MDE to be asymptotically efficient to two decimal
places when compared to that of the MLE under normality. For example, when 9 = 5
and 9 E (0, 0.4) the asymptotic variance of MDE is very close to that of MLE. The re-
sult implies that if enough autocorrelations are used, the MDE can be asymptotically
as efficient as MLE. Apart one extreme case when the moving average parameter, 0
is .9, the MDE is seen to be remarkably efficient.

While the result in Table 1 indicates that the MDE appears to be as efficient '

as MLE given that g is large enough, we now provide a theoretical justification of

20

Table 1: Asymptotic Variance of MDE of MA(l) Processes: y, = (1 — 6L)e,

 

 

0

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

g: 1 1.031 1.135 1.356 1.796 2.701 4.741 10.095 28.614 149.482
2 0.991 0.973 0.973 1.030 1.217 1.705 3.046 7.710 37.999

3 0.990 0.961 0.919 0.885 0.899 1.041 1.541 3.394 15.526

5 0.990 0.960 0.910 0.842 0.767 0.717 0.776 1.247 4.693

10 0.990 0.960 0.910 0.840 0.750 0.641 0.526 0.472 0.934

20 0.990 0.960 0.910 0.840 0.750 0.640 0.510 0.363 0.280
VMLE 0.990 0.960 0.910 0.840 0.750 0.640 0.510 0.360 0.190

 

Note: g=1 means only the first autocorrelation are used in MDE, while g=2 represents the
case that first 2 autocorrelations are used in MDE, etc. VMLE represents the asymptotic
variance of x/TWMLE - 0).

this result. The basic idea is to rewrite the score of the log-likelihood function as

a weighted sum of the sample autocovariances at various lags. It is found that the

weight for each sample autocovariance decreases exponentially. The lower bound of

the asymptotic variance of the MDE is equal to the asymptotic variance of MLE

under normality. Given that g is large enough, it will be shown later that the upper

and lower bounds of the asymptotic variance of MDE are the same when T —+ 00.

Therefore, if the number of autocorrelations used in the MDE is large enough, the

MDE for the MA(l) process is asymptotically as efficient as MLE under normality.

21

First, note that the log-likelihood function of the MA(l) model is
1 T 1 1 2 1 T t—l . 2
= _ = —— — — _ —' 0] -' ‘
L T g1, 2 log 27r 2 logo 21.02 g 1;) yt J

Without loss of generality we assume 02 is known. The score of the log-likelihood

function is

T t-l t-l

S = T”: (2 911/1— j) (Z 101-13114)

1:2 j=0 i=1
T t- 1
= T_l Z Z 01' yt—jjgj‘lyt—j
1:2 j=0
T— l T t—l h
+ 71-12: 2 9j yt-j (j +h)9"+ lyt-j— 11+: 9"” yt—j— 1179];1 yt-j
h=1t=2 '=0 j=0
T t—l . T-l T t—l
= T422392! 1y,_ ,+T- £2 (2j+h) )621'” 1y.-.- hyt j
t: 2j==0 h: 2j=0
T-l T-l ._
= j0211 7"0+Z[9”‘Z%(2j+h) 9217,.)—
where
T—l . T T-l T—l . T
C1 = T421702]—1 E y: + Z 0”" 2(23' + MO” 2 y.,y.,_h,
j:1 T:T—j+1 h=1 j=1 r:T—j+1

and the sample autocovariance of yt at lag h is defined by 7,, = T"Zf=h+1y,y,_h.
where h =0,1,2,---

The score of the log-likelihood function can be rewritten as

T-l
S = 21923‘7o+ :2319“ Z<<h+2m)92m1h1—
m=0

= 3;: wt 71 - C1,
where mo 2 1201 j 021 1,and 111),: 0" 12",-0 (h + 2m) 02'" for h- — 1, 2, .The
variance of TV 241 approaches zero as T —+ 00, while a formal proof of this result will

be given later. Hence, (1 is asymptotically negligible.

22

When T is very large, we is approximately equal to 205, where

0

105:

Similarly, for h 2 1 11);, is approximately equal to 11);, where

h — (h — 2W
(1 — 02)?

 

wg=6h-l[ ], forh=1,2,-~.

Hence, given that T is very large, we can rewrite S as

T-l
5: 2 10:23—41-

i=0

(14)

(15)

This result demonstrates that the score of the MLE of MA(l) process is approximately

equal to a weighted sum of the sample autocovariance at various lags. The weight

(21),?) for each sample autocovariance decreases exponentially.

Let 8(9) be defined as the summation of only the first 9 + 1 terms of S, so that

9
5(9) = Z w: ’AYi-

i=0

(16)

Now the moment conditions of MDE based on the first g autocorrelations can be

written as

where A .
r 71 — P1(0)’Yo 7
‘72

 

 

- 79 .4

23

Let R be a g x 9 matrix given by

'10; w; w; w;q

0 1 0 0

O 0 1 0

R: . . .
0 0 0
L0 0 0 0 1,

 

 

Notice that the MDE is asymptotically equivalent to the GMM estimator based on
moment conditions RM, because the GMM estimator minimizing M’Q‘IM also
minimizes (RM)’(RQR’)‘1RM, where Q is the asymptotic variance of x/T M.

The first element of the vector R x M is equal to w; x (—p1) ‘yo + 2le 10:3,.

Using the result given in equation (14) yields

1+W
(1—my'

t

I:

we have

Because p1 = fig—A,

. 9 .
w1x("/91)= — = “’0-

(1—my

It follows directly that the lst element of R x M is equal to 3(9). Hence,

P wf X (‘Pl) “"0 + 2?:1 wf‘AYi 1 rS(9) 1
‘12 5’2
R x M = . =
L 19 .J l. 3'9 .l

 

 

 

 

For convenience let the asymptotic variances of GMM estimator based on RM
and 8(9) be denoted as VRM and V59, respectively. Also VMLE denotes the asymptotic
variance of MLE. Since 3(9) is only a subset of moment conditions of RM, we have

VMLE S VRM S ng-

24

It will be demonstrated below that when 9 is very large, the asymptotic variances
of 5(9) and S are the same, so that 5(9) and S are asymptotically equivalent. If g is
very large, V59 = VMLE. It also implies that VRM = VMLE. Therefore, the MDE is
asymptotically as efficient as MLE given that the number of autocorrelations is large
enough.

The following result demonstrates that when 9 is very large, the asymptotic vari-
ances of 3(9) and S are the same. By the Cauchy-Schwarz inequality, it is sufficient
to show that

lim limsup (El T1/2(S— S( 9)) I2)”2 = 0.

g-+oo T—mo

First note that

T-l T
T1/2(S - 3(9)) == T1/2[ 2 (wh T-l 2 ytyt—h) ] - T1/2C11
h=g+1 t=h+l

where lim7~_,oo Var( TI/2 (1 ) = 0. Using the triangular inequality gives

glim limsup (El T1/2(S— S(g )) l2)1/2

T—mo
T— 1 T 1/2
= glim limsup ( E | Z (w; T—l/2 2: 31131141) — 7km C1 '2)
h=g+1 t: h+l
T— 1 T
3 11m limsup || 2 112,, T 1/2 2 ytyt-hllz
T—mo h=g+l t=h+1
T—1 T
S 11m11msup Z w; H T ”2 2 ytyt-hll2
T-*°° h: 9+1 t=h+l
T- 1
3 c1 lim limsup Z 10;,

9—100 T_’°° h=g+l

|/\

C2 91151.1(> card-Cs 9)

=0.

25

By using the result given in Brockwell and Davis (1991, p.227) it follows directly that
Cl is a finite positive constant which does not depend on h. Note that 02 and c3 are
some positive constants which are functions of 0.

The following result demonstrates that limeoo Var( T 1/ 2 C1 ) = 0.

Tum (E I T”? c1 1211/2

T—1 T

_ - —1/2 -2j-—l 2

— 1.1130(EIT {EM 2 yr
J=1 r=T-J+1

T-l T
“’1 2(21' +h)92’ Z yryr—h } l2 )1/2

+
M?
Q

 

11:1 1:1 r=T—j+l
T—1 T
= . _1/2 . 21' h— 1 2]
715“... < EIT {23mm 2. 113+?!) 2219 2 my. 1.
1:1 r=T-J+1 r=T-j+l
+:::::0h- ”1292). Z yryr- hl2 )1/2
r=T-J+l
T— 1 . T T— 1 T
S Tlim T‘W {2(2j+1)92’ Z ll:/3l|2+§:6’h ‘22:?” Z llyryr-nllz
-+OO j=l r=T—j+1 T=T—j+l
T-l T—1 . T
+ z: 0"“): Z 02’ Z ”yryr—hll2 }
h=l j=l r=T—j+l
T-1 . T—1 T-1 _ T—l T-1 .
s 713nm T-l/2 { 2(2j+1)621ja1+ 2 9H 2 2j02’ja1-1- 1:1 oh-lh Z; 921ja1 }
= J:
- —1/2 h— 1 h- 1
g qlemT al(b1+2:19 b2+E9 hb3)
11:1 [1: 1
b
< . -1/2 b2 3
-2@&T ““h+1—o+u—mfi)

:0,

where «21 = 11113112: (Es/1)”? and um. 1112-4131313 111/2 s (Ex/$111-1)“ =a1.

Also, it can be verified that ()1, b2 and b3 are finite constants such that

6+?
._ 2j—1]
b1 - 1122.219 =(,_—,—-.,.,
. 202(1+02)
_ - -2J-____
52 —1!1_{§°§2J9 J— (1_92)3A

26

and
T—l 2_ 02
= ' J . : -—————,
b3 rill—{301.29 3 (1—02)2

4.2 MDE Applied to MA(2) Process

The asymptotic variance of the MDE for estimating the parameters of an MA(2)
process can be investigated in an analogous manner. The MA(2) process is defined
33,

yt = (1 — 01L — 02L2)6t = (1 — 61L)(1— 62L)€t,

where at is iid(0, a2) and the second expression is in terms of the two invertible roots
of the MA polynomial. Let the maximum likelihood estimator of the MA(2) model be
denoted as XMLE, where A corresponds to [01 021' . Then, we have #01 MLE — A0) -—>

N(0, VMLE): where

V _ 1-0; 01(1-92)
MLE" 01(1—92) 1—03

Hence, the asymptotic variances of MLE of 01 and 02 depend only on the parameter

02. The autocorrelation function is

 

—01(1—02)
’01 1+6¥+0§’
p2 " 1+9¥+0§’

and

pk = 0 for [£23.

27

In the appendix, we provide the analytical results of the asymptotic variance of the
sample autocorrelations calculated by the Bartlett’s formula.

For the MA(2) process with i.i.d. innovations, and given a set of parameter
values the asymptotic covariance matrix C, of the sample autocorrelation is calculated
from Bartlett’s formula, while the matrix of partial derivatives, D, is analytically
straightforward to calculate. The asymptotic variance of flaw»; — A0) for the
MA(2) model are then the diagonal elements of the matrix (D’C‘ID)'1, and are
reported in Tables 2 and 3, for four different values of 61 and eight values of 62 which
give a total of thirty two points of the parameter space. The theoretical asymptotic
variances of the parameter estimates from the MDE are calculated from the use of g
autocorrelations, where g = 2,3,5,10,15 and 20. As the number of autocorrelations,
g, increases, the asymptotic variance of the MDE parameter estimates decreases and
approaches that of the MLE. The first panel of Table 2 presents the asymptotic
variance of MDE of MA(2) processes with 61 fixed at 0.1, whereas in the second panel
61 = 0.3. Table 3 presents the results for 61 = 0.5 and 0.7. When 62 increases, the
number of autocorrelations used in the MDE has to be increased in order for the
MDE to be asymptotically as efficient as MLE under normality. If both 61 and 62 are
not too large, the asymptotic variance of the MDE using only 5 autocorrelations is

very close to that of MLE under normality.

5 MDE Applied to ARMA(1,1) Process

In this section, we discuss the asymptotic properties of MDE of the ARMA(1,1)

28

Table 2: Asymptotic Variance of VT (61 — 01) and WM} .- 02) of
MA(2) processes, y, = (1 — 01L — 92L2)6¢ = (1 — 61L)(1- 62L)ct
with 61 = 0.10 and 0.30

 

61 0.10
62 0.20 0.40 0.60 0.80 -O.20 -0.40 ~O.60 -0.80

 

Asymptotic variance of x/TMI — 91)
g: 2 1.00 1.10 2.45 27.00 1.00 1.04 1.53 9.35
3 1.00 1.00 1.16 5.18 1.00 1.00 1.11 3.13
5 1.00 1.00 1.00 1.41 1.00 1.00 1.01 1.34
10 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.01
15 1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00
20 1.00 1.00 1.00 0.99 1.00 1.00 1.00 0.99
VMLE 1.00 1.00 1.00 0.99 1.00 1.00 1.00 0.99
Asymptotic variance of fi(ég — 02)
g: 2 1.38 2.09 3.14 3.78 1.04 1.36 2.09 3.73
3 1.04 1.25 1.70 2.22 1.00 1.09 1.41 2.22
5 1.00 1.02 1.14 1.45 1.00 1.00 1.08 1.43
10 1.00 1.00 1.00 1.08 1.00 1.00 1.00 1.07
15 1.00 1.00 1.00 1.01 1.00 1.00 1.00 1.01
20 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
VMLE 1.00 1.00 1.00 0.99 1.00 1.00 1.00 0.99
61 0.30
62 0.20 0.40 0.60 0.80 -0.20 -0.40 -0.60 -0.80

 

 

Asymptotic variance of x/Tw] — 01)

g: 2 1.06 1.65 8.12 117.59 1.02 1.06 1.40 5.43

3 1.00 1.01 1.54 15.07 1.00 0.99 1.07 2.54

5 1.00 0.99 0.98 1.77 1.00 0.99 0.98 1.30

10 1.00 0.99 0.97 0.95 1.00 0.99 0.97 0.97

15 1.00 0.99 0.97 0.94 1.00 0.99 0.97 0.95

20 1.00 0.99 0.97 0.94 1.00 0.99 0.97 0.94

VMLE 1.00 0.99 0.97 0.94 1.00 0.99 0.97 0.94
Asymptotic variance of VT (01» - 02)

g: 2 2.11 3.26 4.00 1.71 1.05 1.08 1.50 3.08

3 1.23 1.63 2.15 1.70 1.02 1.06 1.31 2.27

5 1.01 1.06 1.26 1.42 1.00 0.99 1.03 1.39

10 1.00 0.99 0.98 1.05 1.00 0.99 0.97 1.02

15 1.00 0.99 0.97 0.97 1.00 0.99 0.97 0.96

20 1.00 0.99 0.97 0.95 1.00 0.99 0.97 0.95

VMLE 1.00 0.99 0.97 0.94 1.00 0.99 0.97 0.94

 

29

Table 3: Asymptotic Variance of VT (61 — 01) and W176} - 62) of
MA(2) processes, y, = (1 — 01L —02L2)€t = (1 —61L)(1 -62L)e¢ with
61 = 0.50 and 0.70

 

61 0.50
62 0.20 0.40 0.60 0.80 -0.20 -0.40 -0.60 -0.80

 

Asymptotic variance of {1:091 — 01)

g: 2 1.86 6.59 48.76 714.85 1.14 1.19 1.52 4.65

3 1.04 1.33 5.04 80.51 1.02 0.98 1.01 2.22

5 0.99 0.96 0.99 4.63 0.99 0.96 0.93 1.21

10 0.99 0.96 0.91 0.85 0.99 0.96 0.91 0.88

15 0.99 0.96 0.91 0.84 0.99 0.96 0.91 0.85

20 0.99 0.96 0.91 0.84 0.99 0.96 0.91 0.84

VMLE 0.99 0.96 0.91 0.84 0.99 0.96 0.91 0.84
Asymptotic variance of Jim} — 02)

g: 2 3.20 4.15 2.38 53.73 1.39 1.18 1.41 3.03

3 1.65 2.16 2.16 4.19 1.16 1.16 1.39 2.66

5 1.09 1.23 1.44 1.15 1.01 0.98 0.99 1.36

10 0.99 0.96 0.95 1.00 0.99 0.96 0.91 0.91

15 0.99 0.96 0.91 0.89 0.99 0.96 0.91 0.85

20 0.99 0.96 0.91 0.85 0.99 0.96 0.91 0.84

VMLE 0.99 0.96 0.91 0.84 0.99 0.96 0.91 0.84

 

61 0.70
62 0.20 0.40 0.60 0.80 -0.20 -0.40 -O.60 -0.80

 

Asymptotic variance of @091 — 01)
g: 2 12.64 63.91 478.11 7168.56 2.39 2.12 2.42 5.41
3 2.29 7.08 49.72 816.60 1.38 1.27 1.18 2.01
5 1.04 1.11 2.59 42.59 1.05 1.00 0.92 1.12
10 0.98 0.92 0.83 0.89 0.98 0.93 0.83 0.76
15 0.98 0.92 0.83 0.70 0.98 0.92 0.82 0.69
20 0.98 0.92 0.82 0.70 0.98 0.92 0.82 0.69
VMLE 0.98 0.92 0.82 0.69 0.98 0.92 0.82 0.69
Asymptotic variance of WW} — 02)
3.81 2.08 33.90 1872.12 2.28 1.84 2.04 4.16
3 2.14 2.00 2.58 187.19 1.64 1.64 2.01 4.05
5 1.33 1.44 1.27 6.98 1.17 1.10 1.09 1.56
10 1.01 0.98 0.98 0.86 0.99 0.93 0.83 0.76
15 0.98 0.93 0.85 0.79 0.98 0.92 0.82 0.70
20 0.98 0.92 0.83 0.73 0.98 0.92 0.82 0.69
VMLE 0.98 0.92 0.82 0.69 0.98 0.92 0.82 0.69

on
H
to

30

process, which is

(1 - ¢L)yt = (1 - 9L)6¢,

where as, is i.i.d(0, 02). Let pk denote the kth order autocorrelation of y,. We then

have

(1-¢0)(¢-9)
1+62—2¢6’

 

Pl
and

PI: = Pic—1(1) for 1622-

Let A: [ ¢ 0 ]’ and AM“; denote the maximum likelihood estimator of ARMA(1,1)

model. Then, we have \f’fOMLE — A0) —> N(0, VMLE), where

V —_— 3:151 (1 - ¢2)(1- (M) (1 _ 452)“ _ 92)
MLE (¢ _ g)2 (1 _ ¢2)(1 __ 92) (1 _ 02)(1_ (I59) .

Given a set of parameter values the matrix C is calculated from Bartlett’s formula,
while the matrix of partial derivatives, D, is analytically straightforward to calculate.
The asymptotic variance of x/TOlMDE — A) for the ARMA(1,1) model are then the
diagonal elements of the matrix (D’ C “‘D)“, and are reported in Tables 4 and 5 , for
four different values of 43 and nine values of 0 which give a total of thirty six points of
the parameter space. Identification requires that at least 2 autocorrelations are used
in MDE. We present the cases that first 2, 3, 5, 8, 10, 15 and 20 autocorrelations are
used in MDE.

In general, as the number of autocorrelations used in the MDE increases, the

asymptotic variance of the MDE decreases. The MDE appears to be asymptotically

31

as efficient as MLE under normality when the number of autocorrelations is large
enough.

The first portion of Table 4 presents results for ARMA(1,1) models with the
AR coefficient being set to -0.5 and 0 being varied from 0.1 to 0.9 with steps of
0.1. The second part of Table 4 reports cases whose AR coefficients are fixed at
0.8, while MA coefficients are set to 0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, and 0.6.
Calculations reveal that if 0 is less than 0.3, MDE using first 5 autocorrelations can be
asymptotically as efficient as MLE under normality, while the asymptotic variance of
MDE using first 10 autocorrelations is very close to that of MLE when 0 lies between
0.4 and 0.6. Phrthermore, this result seems not to be affected by the value of $-

Some further investigation is presented in Table 5, which shows the results for
ARMA(1,1) models whose AR coefficients are fixed at 0.6 and 0.3. In Part I of Table
5, the MA coefficients are set to both positive and negative values, i.e., 0.2, 0.3, 0.4,
0.8, 0.9, -0.2, -0.4, -0.6, and -0.8. The results in Table 5 are very similar to those in
Table 4. If the absolute value of 0 of the ARMA(1,1) process is small, MDE using first
5 autocorrelations can be as efficient as MLE. In sum, when the absolute value of MA
coefficient is less than 0.3, asymptotic variance of MDE using first 5 autocorrelations
is very close to that of MLE under normality. If |0| E (0.3, 0.7), asymptotic variance
of MDE using first 10 autocorrelations is very close to that of MLE. When |0| is close

to one, g should be higher than 20.

32

Table 4: Asymptotic Variance of x/T(:\MDE — A) of ARMA(1,1) pro-
cesses, (1 - ¢L)y¢ = (1 — 0L)e¢ with ¢=-0.5 and 0.8

 

-0.5
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Q‘s-

 

Asymptotic variance of Wm; — d2)

 

 

g: 2 2.36 2.02 1.83 1.72 1.65 1.61 1.58 1.57 1.56

3 2.30 1.87 1.59 1.42 1.32 1.26 1.22 1.19 1.18

5 2.30 1.85 1.55 1.34 1.19 1.09 1.03 1.00 0.98

8 2.30 1.85 1.55 1.33 1.17 1.05 0.97 0.92 0.89

10 2.30 1.85 1.55 1.33 1.17 1.05 0.96 0.90 0.86

20 2.30 1.85 1.55 1.33 1.17 1.05 0.95 0.87 0.82

VMLE 2.30 1.85 1.55 1.33 1.17 1.05 0.95 0.87 0.80
Asymptotic variance of \fl—Té - 0)

g: 2 3.16 2.79 2.76 3.07 3.94 6.01 11.48 30.18 151.32

3 3.03 2.40 2.01 1.81 1.85 2.26 3.57 8.25 38.61

5 3.03 2.37 1.88 1.51 1.25 1.12 1.24 2.10 8.15

8 3.03 2.37 1.88 1.49 1.17 0.91 0.75 0.81 2.18

10 3.03 2.37 1.88 1.49 1.17 0.90 0.68 0.60 1.23

20 3.03 2.37 1.88 1.49 1.17 0.89 0.65 0.42 0.32

VM L E 3.03 2.37 1.88 1.49 1.17 0.89 0.65 0.42 0.20

d) 0.8

6 0.10 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60
Asymptotic variance of flu; — 4))

g: 2 0.63 0.78 0.90 1.09 1.39 1.88 2.72 4.25 13.95

3 0.62 0.71 0.78 0.87 1.00 1.20 1.53 2.11 5.53

5 0.62 0.71 0.76 0.83 0.92 1.05 1.23 1.51 2.95

8 0.62 0.71 0.76 0.83 0.92 1.04 1.20 1.44 2.48

10 0.62 0.71 0.76 0.83 0.92 1.04 1.20 1.44 2.44

20 0.62 0.71 0.76 0.83 0.92 1.04 1.20 1.44 2.43

VMLE 0.62 0.71 0.76 0.83 0.92 1.04 1.20 1.44 2.43
Asymptotic variance of x/Tw — 0)

g: 1.78 2.22 2.58 3.08 3.81 4.90 6.58 9.34 23.73

2
3 1.71 1.91 2.05 2.24 2.52 2.94 3.57 4.61 9.84
5 1.71 1.88 1.98 2.11 2.26 2.46 2.74 3.18 5.30
8 1.71 1.88 1.98 2.10 2.25 2.43 2.67 3.01 4.42
10 1.71 1.88 1.98 2.10 2.25 2.43 2.67 3.00 4.34
20 1.71 1.88 1.98 2.10 2.25 2.43 2.67 3.00 4.33
VMLE 1.71 1.88 1.98 2.10 2.25 2.43 2.67 3.00 4.33

 

33

Table 5: Asymptotic Variance of x/T(AMDE - A) of ARMA(1,1) pro-
cesses, (1 — ¢L)y¢ = (1 — 0L)e¢ with ¢=0.6 and 0.4

 

 

 

 

a3 0.6

0 0.20 0.30 0.40 0.80 0.90 -0.20 -0.40 -O.60 -0.80
Asymptotic variance of VT (<13 - 4))

g: 2 3.57 6.85 18.48 75.83 52.75 1.35 1.20 1.15 1.13

3 3.13 5.07 11.11 27.97 19.00 1.26 1.04 0.95 0.91

5 3.10 4.79 9.35 11.14 7.12 1.25 0.99 0.85 0.79

8 3.10 4.78 9.24 6.35 3.65 1.25 0.98 0.83 0.74

10 3.10 4.78 9.24 5.35 2.86 1.25 0.98 0.82 0.73

20 3.10 4.78 9.24 4.36 1.75 1.25 0.98 0.82 0.72

VMLE 3.10 4.78 9.24 4.33 1.50 1.25 0.98 0.82 0.72
Asymptotic variance of x/flé - 0)

g: 2 5.47 9.87 23.88 98.68 201.91 2.22 2.68 5.65 29.75

3 4.71 7.25 14.59 30.09 53.40 1.91 1.57 2.10 8.11

5 4.65 6.81 12.27 9.14 12.08 1.88 1.31 1.03 2.05

8 4.65 6.80 12.13 4.18 3.56 1.88 1.29 0.84 0.78

10 4.65 6.80 12.13 3.28 2.12 1.88 1.29 0.83 0.58

20 4.65 6.80 12.13 2.46 0.65 1.88 1.29 0.82 0.41

VMLE 4.65 6.80 12.13 2.43 0.45 1.88 1.29 0.82 0.40

d) 0.40

0 0.10 0.20 0.60 0.70 0.80 -0.20 -O.40 -O.60 -0.80
Asymptotic variance of x/Tw; - 4))

g: 2 8.93 20.88 50.14 30.39 23.56 3.02 2.40 2.18 2.10

3 8.61 18.01 23.83 13.51 10.09 2.75 1.91 1.62 1.51

5 8.60 17.78 14.36 7.09 4.86 2.72 1.78 1.36 1.21

8 8.60 17.77 12.36 5.30 3.20 2.72 1.77 1.30 1.09

10 8.60 17.77 12.17 5.00 2.83 2.72 1.77 1.29 1.05

20 8.60 17.77 12.13 4.84 2.44 2.72 1.77 1.29 1.02

VMLE 8.60 17.77 12.13 4.84 2.43 2.72 1.77 1.29 1.02
Asymptotic variance of x/TMA - 0)

g= 2 10.55 23.89 49.22 36.81 50.36 3.66 3.60 6.47 30.70

3 10.15 20.59 20.70 12.84 14.80 3.15 2.14 2.46 8.43

5 10.14 20.31 11.29 5.14 4.25 3.11 1.79 1.23 2.16

8 10.14 20.31 9.44 3.34 1.85 3.11 1.77 1.01 0.84

10 10.14 20.31 9.28 3.07 1.42 3.11 1.77 0.99 0.62

20 10.14 20.31 9.24 2.94 1.05 3.11 1.77 0.98 0.44

VMLE 10.14 20.31 9.24 2.94 1.04 3.11 1.77 0.98 0.44

 

34

6. MDE Applied to Higher Order ARMA(p,q) Processes

This section provides further investigation of the application of MDE to some
higher order ARMA(p,q) processes. The results for the ARMA(2,1) process are pre-

sented in Table 6 and 7. The ARMA(2,1) process is
(1— 45114 — ¢2L2)y¢= (1‘ 91062,

where e, is NID(0, 02). To calculate the asymptotic variance of MDE of ARMA(2,1)
processes, the first step is to obtain explicit representations of autocorrelation func-
tions in term of (1)1, 62, and 0. The autocorrelation functions of ARMA(2,1) process

are

p = 9(1—¢§)-¢1(1—0¢1+02)
1 ($2 — 1)(1— 9¢1+62)+¢10(1+¢2)’

 

and

PI: = ¢1Pk-1 + dam—2 for k 2 2.

Letting AMLE denote the maximum likelihood estimator of ARMA(2,1) model.

“CT—(XMLE -- A) -—> N“), VMLE), where

 

 

 

 

 

f 1‘452 0i _ 1 ‘
(1+¢2)[(1—¢>2)’—¢’{] iI+o-.»)[(1—<z>2)'2—¢?] 1—¢10-¢29§
V : 4’1 l—cb'z -0
MLE (1+¢2)[(1—¢2)2—¢fi (I+¢2)[(1—¢2)2-¢?] lama—«>295 ,
_ 1 —o 1
. 1-¢19-¢20§ 1—¢19—¢29§ 1-92

and A’ = [(131 452 0]. Table 6 presents the numerical calculation results of the asymptotic
variance of the MDE and MLE for ARMA(2,1) process. The AR coefficients, 451 and

(252, are equal to —1.00 and -0.16, respectively, whereas 0 is varied from 0.1 to 0.90 by

35

steps of 0.1. Alternatively, we can rewrite the ARMA(2,1) process as
(1 — alL)(1 — agL)y¢ = (1 — 9L)6t.

Thus, the ARMA(2,1) process with (151 = —-1.00 and $2 = —0.16 corresponds to
01 = —0.2 and a2 = —0.8. Clearly, identification requires at least 3 autocorrelations
to be used in MDE. The asymptotic variance of MDE using first 3, 5, 8, 10, and 20
autocorrelation are reported. Table 7 presents the results for the cases that 431 and
(p2 are equal to 1.00 and -0.21, while 0 is varied from -0.1 to -0.9 by steps of -0.1.

In sum, the results given in Table 6 and 7 indicate that the efficiency loss in
MDE appears to diminish as g increases. In particular, the results of the MDE for
ARMA(2,1) processes are very similar to those of ARMA(1,1) and MA(l) processes.
When the absolute value of 0 of the ARMA(2,1) process is small, MDE using first 5
autocorrelations is as efficient as MLE under normality. Furthermore, given that (151
and (1)2 are the'same, the higher the value of 6, the higher the number of autocorre-
lations are needed to guarantee that the asymptotic variance of the MDE is close to
that of MLE under normality.

We also investigate the asymptotic variance of MDE for the ARMA(1,2) process.

The ARMA(1,2) process is
(1" ¢L)yt = (1‘ 01L — 92L2)€t,
where q is i.i.d(0, 02). An alternative representation of the ARMA(1,2) process is

(1 - ¢L)yt = (1 - 51L)(1- 52106:-

36

The autocorrelation function of the ARMA(1,2) process is

p = ¢[—1+01(¢— 01) +(92(¢2 -—91¢— 62)]+01+02(¢—01)
‘ 6.(¢—6.)+62(¢2-01¢—02)+¢[6.+02(¢—0.)1—1 ’

p2: W _ 02019-1) .
1 ems—9,)+92(¢2—01¢—02)+¢[91”zap-01)]-1’

 

 

and

P1: = (Wk—1 for k 2 3-

We also compare the asymptotic variance of MDE to that of MLE. For the MLE
of ARMA(1,2) processes, we have the following result of the asymptotic variance of

x/T (A M [,3 — A), while a detail derivation is presented in the appendix.

- 1 -1 : ‘1

 

 

 

1—¢§ 1—¢01—¢592 l-Wi-¢ 92
v, __ —-1 l-OL 91
MLE — 1—¢01-¢202 (1+92)[(1-92)’-0¥] (1+02)[(1-92)L9f1

-¢ 01 ML
.1-¢91-¢292 (1+02)l(1—02)’-€fi (1+92)[(1-92)2-9f11

 

 

The calculation results of the asymptotic variance of MDE of MA(2) processes
might give some implications for ARMA(1,2) processes. In particular, it is expected
that given that d), 62 and number of autocorrelations (g) used in MDE are fixed, the
higher the value of 61, the higher the asymptotic variance of MDE for ARMA(1,2)
processes. Table 8 presents the ARMA(1,2) processes whose 62 =-0.35 and <15 =-0.60,
while 61 are equals to 0.10, 0.20, 0.30, 0.40, 0.50, 0.70, 0.80 and 0.90. Table 9 presents
the results of the cases that 62 =0.1 and 45 =-0.60, while 61 are equals to 0.20, 0.30,
0.40, 0.50, 0.60, 0.70, 0.80 and 0.90. The results in Table 8 and 9 reveal that the
asymptotic variance of the MDE is very close to that of MLE under normality if g is

large enough.

37

7. Concluding Remarks

This chapter investigates the asymptotic properties of the minimum distance es-
timator for a variety of ARMA processes. The results show that as the number of
autocorrelations used in the MDE increases, the asymptotic variance of the MDE
decreases. Numerical calculation results show that if the number of autocorrelations
used in the MDE is large enough, the MDE appears to be asymptotically as efficient
as MLE under normality. A formal proof of this result is provided for the MA(l)
process. Interestingly, for the MA(l) and ARMA(1,1) models, if the absolute value
of the moving average coefficient is not too large, the asymptotic variance of MDE
based on first 3 to 5 autocorrelations is very close to that of MLE under normality.
Furthermore, the higher the value of the moving average coefficient, the higher the
number of autocorrelations is needed for the asymptotic variance of MDE to be close
to that of MLE. The MDE is surprisingly efficient for some parts of the parameter

space.

38

Table 6: Asymptotic Variance of fi(AMDE — A) of ARMA(2,1) processes:

 

(1 -— ¢1L - ¢2L2)yt = (1 — 0L)et ; (1 — alL)(1 — 02L)y¢ = (1 — 0L)et
with (11 =-0.20 and (12 = -0.80. ( (:51 =-1.00, and (132 =-0.16 )
0
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
Asymptotic variance of x/flqgl - «#1)

g: 3 16.80 10.10 7.28 5.88 5.12 4.68 4.43 4.29 4.22
5 16.19 8.81 5.54 3.88 2.98 2.49 2.22 2.07 2.00

8 16.19 8.80 5.50 3.76 2.75 2.13 1.77 1.58 1.49

10 16.19 8.80 5.50 3.76 2.74 2.10 1.70 1.47 1.37

20 16.19 8.80 5.50 3.76 2.74 2.09 1.66 1.36 1.18
VMLE 16.19 8.80 5.50 3.76 2.74 2.09 1.66 1.36 1.14

Asymptotic variance of #052 - 452)

g: 3 13.38 8.44 6.31 5.24 4.64 4.30 4.09 3.98 3.92
5 12.91 7.41 4.87 3.53 2.79 2.37 2.14 2.01 1.95

8 12.91 7.40 4.83 3.43 2.58 2.05 1.73 1.55 1.47

10 12.91 7.40 4.83 3.42 2.57 2.02 1.66 1.45 1.36

20 12.91 7.40 4.83 3.42 2.57 2.01 1.62 1.35 1.18
VMLE 12.91 7.40 4.83 3.42 2.57 2.01 1.62 1.34 1.13

Asymptotic variance of \fT—(é - 0)

g: 3 17.15 10.27 7.56 6.59 6.82 8.61 14.02 32.88 154.38
5 16.48 8.74 5.25 3.46 2.53 2.17 2.41 4.16 16.30

8 16.48 8.73 5.20 3.30 2.16 1.47 1.12 1.20 3.27

10 16.48 8.73 5.20 3.29 2.15 1.42 0.96 0.81 1.69

20 16.48 8.73 5.20 3.29 2.15 1.40 0.89 0.52 0.37
VMLE 16.48 8.73 5.20 3.29 2.15 1.40 0.89 0.51 0.22

 

 

Table 7: Asymptotic Variance of fi(AMDE — A) of ARMA(2,1) processes:

39

(1 - ¢1L — ¢2L2)yt = (1 — 9L)€t ;
with al =0.30 and 02 = 0.70. ( ¢1 =1.00, and (fig =-0.21 )

(1 - 01L)(1 - 02L)y¢ = (1 — 0L)c¢

 

U100

20
VMLE

 

0
-0.10 -0.20 -0.30 -0.40 -O.50 -0.60 -0.70 -0.80 -0.90
Asymptotic variance of V7051 - 431)
11.72 7.80 5.95 4.98 4.43 4.10 3.91 3.80 3.75
11.31 6.84 4.59 3.35 2.65 2.25 2.03 1.91 1.85
11.31 6.84 4.55 3.25 2.45 1.95 1.65 1.48 1.41
11.31 6.84 4.55 3.25 2.44 1.92 1.58 1.39 1.30
11.31 6.84 4.55 3.25 2.44 1.91 1.55 1.29 1.14
11.31 6.84 4.55 3.25 2.44 1.91 1.55 1.29 1.10
Asymptotic variance of 77705.2 - (#2)
8.84 6.25 4.99 4.31 3.92 3.68 3.54 3.46 3.42
8.54 5.52 3.91 2.99 2.44 2.12 1.94 1.84 1.79
8.54 5.52 3.89 2.90 2.27 1.86 1.60 1.46 1.39
8.54 5.52 3.89 2.90 2.27 1.83 1.54 1.37 1.29
8.54 5.52 3.89 2.90 2.26 1.83 1.51 1.28 1.14
8.54 5.52 3.89 2.90 2.26 1.83 1.51 1.28 1.10
Asymptotic variance of «T09 — 0)
12.22 8.14 6.40 5.85 6.27 8.15 13.60 32.45 153.90
11.74 6.93 4.44 3.06 2.32 2.04 2.32 4.09 16.23
11.74 6.92 4.40 2.91 1.98 1.38 1.07 1.17 3.25
11.74 6.92 4.40 2.91 1.96 1.33 0.92 0.79 1.67
11.74 6.92 4.40 2.91 1.96 1.31 0.85 0.50 0.36
11.74 6.92 4.40 2.91 1.96 1.31 0.85 0.49 0.22

40

Table 8: Asymptotic Variance of fi(AMDE — A) of ARMA(1,2) processes:

 

(1 — ¢L)yt = (1 - 91L - 92L2)6t ; (1 — ¢L)yt = (1 — 51L)(1 - 52L)€t
with 62 =-0.35 and (b =-0.60
01: -0.25 -0. 15 0.05 0.05 0.15 0.35 0.45 0.55
02: 0.03 0.07 0.10 0.14 0.17 0.24 0.28 0.32
51
0.10 0.20 0.30 0.40 0.50 0.70 0.80 0.90
Asymptotic variance of x/fiq; — 4:)
g: 3 19.39 14.39 11.80 10.39 9.60 8.89 8.74 8.67
5 14.83 12.61 11.02 9.85 9.00 8.07 7.87 7.77
8 14.66 12.53 10.99 9.83 8.93 7.74 7.43 7.28
10 14.65 12.53 10.99 9.83 8.93 7.67 7.30 7.11
20 14.65 12.53 10.99 9.83 8.93 7.63 7.15 6.82
VMLE 14.65 12.53 10.99 9.83 8.93 7.63 7.14 6.74
Asymptotic variance of WW] — 01)
g: 3 20.87 16.00 13.52 12.16 11.35 10.34 10.78 24.50
5 16.05 14.01 12.58 11.54 10.80 9.82 9.41 10.67
8 15.87 13.92 12.54 11.52 10.76 9.74 9.38 9.13
10 15.87 13.92 12.54 11.52 10.75 9.71 9.36 9.05
20 15.87 13.92 12.54 11.52 10.75 9.69 9.31 9.03
VMLE 15.87 13.92 12.54 11.52 10.75 9.69 9.31 9.01
Asymptotic variance of x/Tw} - 92)
g: 3 4.35 4.53 4.94 5.67 6.80 10.77 14.56 24.59
5 3.46 4.07 4.66 5.27 5.97 8.24 10.28 14.49
8 3.42 4.04 4.65 5.24 5.84 7.28 8.51 10.73
10 3.42 4.04 4.65 5.24 5.83 7.10 8.06 9.79
20 3.42 4.04 4.65 5.24 5.83 7.00 7.60 8.47
VMLE 3.42 4.04 4.65 5.24 5.83 7.00 7.58 8.17

 

 

41

Table 9: Asymptotic Variance of «T014013 - A) of ARMA(1,2) processes:

(1 " ¢L)y¢ = (1 — 01L - 9211295: ;

(1 " ¢L)yt = (1 — 511M1 - 52105:

with 62 = 0.10 and (b =-0.60

 

 

01: 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
92: -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09
61
0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
Asymptotic variance of fiw - ¢)
g: 3 3.33 3.14 3.03 2.96 2.91 2.88 2.87 2.86
5 2.88 2.54 2.31 2.16 2.06 2.00 1.97 1.95
8 2.88 2.52 2.26 2.06 1.91 1.81 1.75 1.72
10 2.88 2.52 2.26 2.05 1.89 1.77 1.70 1.66
20 2.88 2.52 2.26 2.05 1.88 1.75 1.64 1.57
VMLE 2.88 2.52 2.26 2.05 1.88 1.75 1.64 1.55
Asymptotic variance of x/Tw] — 91)
g: 3 4.21 3.90 3.56 3.16 2.89 4.19 17.23 167.31
5 3.83 3.47 3.21 3.01 2.83 2.64 2.78 10.35
8 3.82 3.45 3.16 2.94 2.77 2.62 2.49 2.95
10 3.82 3.45 3.16 2.93 2.76 2.61 2.49 2.51
20 3.82 3.45 3.16 2.93 2.75 2.60 2.48 2.37
VMLE 3.82 3.45 3.16 2.93 2.75 2.60 2.47 2.36
Asymptotic variance of fi<ég -— 02)
g= 3 3.63 4.37 5.38 6.64 8.15 9.72 10.79 8.79
5 2.64 2.63 2.70 2.89 3.23 3.71 4.27 4.52
8 2.63 2.57 2.52 2.49 2.52 2.65 2.93 3.27
10 2.63 2.57 2.52 2.47 2.45 2.48 2.65 2.95
20 2.63 2.57 2.52 2.47 2.43 2.39 2.36 2.44
VMLE 2.63 2.57 2.52 2.47 2.43 2.39 2.35 2.31

 

APPENDIX

APPENDIX

A1 Asymptotic variance of sample autocorrelations of MA(2) process

For MA(2) process, we have pk = 0 for 1: >= 3. Similarly, by using the Bartlett’s

formula we derive the following result for the asymptotic variance of J70“) — p) :

where

Cu

012

013

014

051

622

023

024

 

 

rcn 012 013 014 615 0 0 .
C21 022 023 624 625 015
631 C32 €33 C34 C35 C25
C41 C42 C43 033 C45 035 0
CAI/1(2): 051 652 653 654 033 C45 €15 ,
0 051 C52 653 054 633 625
035
C45
_ 0 0 051 052 053 054 C33.
(1+p2 — 2p?)

621 = (1+ 92 - 2Pi)(P1 - 2p1p2)+ (Pl - 2p1p2)(1 - 203) + 91/22
C31 = (1 + p2 - 210?)!» + PI(P1 - 201102) + .02

C41 = (m — 272192102 + mm

651 = pg

(121 — 2mm)2 + (1 - 29%)” + 123+ pf

032 = 101(2 + 2102 '— 4.03)

on = p2(1- 293) + pf) + pg)

42

43

025 = 652 = 2P1P2
C33 = 1 + 2p? + 2p§
034 = C43 = 2PIP2 + 2P1
035 = 053 = Pf + 2P2

_ _ 2 2
C44 — 033 — 1 + 2P1 + 2P2
C45 = 654 = 2/’1(1 ‘1‘ P2)

655 = 033 = 1 + 2P1"' 2P3-

Recall that the autocorrelation functions of the MA(2) process are p1 = —61(1 —

62)/(1+ 0% + 03), p2 = —02/(1 + 0? + 0%), and pg: 0 for k 2 3. Hence,

 

 

r (02-1)(1—0§+o§) 01(1+o§+202—0g) *
(144(44):)? (1+0f+9§)?
29 a —1—02-+-02
D (1+0?+0,’)5 (1+9,’+9,’)5
0 0
. 0 0 1

 

 

A2 Asymptotic variance of MLE of ARMA(2,1) processes

To derive the asymptotic variance of MLE for ARMA(2,1) process, we apply a
result presented by Brockwell and Davis (1991, p.258). The ARMA(p,q) process can

be written as

(1- ¢1L - ' ' °¢pr)yt = (1 " 01L _ ' ' ° — 0<1Lq)€ta

44

where q is i.i.d. (0,02). Let A denote the vector of parameters of the ARMA(p,q)
model, i.e., A’ = [cpl ~ --¢,, 01 - ~09], and AM“; denote the maximum likelihood es-
timator of an ARMA(p,q) model. Brockwell and Davis (1991, p.258) show that

fiO‘MLE — A) ‘2 N0), VMLE) With

—1
_ 2 EUtU,’ EU,V,'
VMLE-U [EWUt' Envy a

where U, = (714“ -ut+1_p)’, V, = (v, . ' - v¢+1_q)’ and at, v, are AR processes defined by

(1—¢1L—-~¢pL”)ut = 6;, and

(1 — 61L — ..._ 9mm = —e..

For the case of ARMA(2,1) process, we have U; = (u; ut_1)’ and V, = 22,. Denote
the autocovariance of u, at lag k as '7}: and the autocovariance of v, at lag k as 7:.

Hence, we have

“/6 Vi‘ E(u¢v¢) _
VMLE = 02 71‘ 75‘ E(ut-lvt) - (17)
E(utvt) E(Ut—1’Ut) ’75
Since at follows an AR(2) process and 2), follows an AR(l) process, we have the

following results of the autocovariance functions of u, and 22,.

1-¢2

 

 

u 2
1° (1+¢2)[(1— 462)? — ¢¥J° ’
71‘ — (151 02 and
(1+ ¢2)[(1 - ¢2)2 - ¢fl ’
v __ 1 2
70 _ 1_ 020 '

In order to derive results for E (utvt) and E(ut_1vt), we assume that (1 — ¢1L —

62H) = 0 has two different roots and (1 — ¢1L — 62L?) can be factorized as (1 —

45

61L)(1 — 62L), where $1 = 61+ 62 and 452 = —6162. Since

(1‘ ¢1L - (152142)-l = 1 ( 61 62 ),

 

61—62 1—61L—1—62L

we have the following results for E(utv,):

Emu.) = E [(1 — ¢1L —— ¢2L2)-le,(1 —- 9L)-1(—e.)]

 

 

= -E ‘51 (1+61L+6fL2+~--)— ‘52 (1+62L+6§L2+-~)et><
61—62 61—62

(1 +49L+¢92L2 +-~)e,

 

_ -0’2 (11 _ 62
‘ 61-62 1—510 1—620

 

 

= -1 0’2
1 _ (51+ ago + 5,5292

_ ‘1 02

‘ 1 —- «me — 46202 '

Similarly,

E(u,_1v,) = E[(1—¢1L—¢2L2)‘lc,_1(1—0L)’1(-—e,)]

 

 

 

 

 

 

 

61 62
= —E L 6L2+-~ — L+6L2+~~ x
[61—62( +‘ ) 61—62( 2 )1“
(1+9L +02L2 + - - ')6¢
_ —0 61 _ 62 02
‘ 61—62 1-610 1—520
_ ’9 02
_ 1-(61+ 6M + 616202
_ ‘9 02
- 1 — ¢10 — (1)202 '
Substituting the above results into (17) yields
1-¢2 4n __ 1 ’1
(1+¢2)[(1—¢2)2—¢¥] (1+o2)[(1—¢2P—¢¥l 1-¢10—¢20§
_ ¢ 1-¢2 -0
VMLE - (1+¢2)[(1—1¢2)2—$fi (1+¢2)[(1-¢2)2-¢¥l W
-o 1
_1—¢lol—¢205 1—¢10—¢20§ 1:717f

 

 

46

Asymptotic variance of MLE of ARMA(1,2) processes

To obtain the asymptotic variance of MLE of ARMA(1,2) processes, we can apply

similar method we use above. Notice that for the ARMA(1,2) processes, at and 2);

are defined as (1 — ¢L)ut = q,

and (1 — 61L — 02L2)vt = —q. For the ARMA(1,2)

process, we have Ut = 21, and V, = (v, v,-1)’. The asymptotic variance of MLE is

VMLE = 02

76‘
13011111) '75
E(u¢v¢_1 7i)

E(u¢vt) E(u¢v¢_1)

-1

W
76’

Since at is a AR(I) process and v, is a AR(2) process, we have

By using the same method in previous appendix, we obtain E (utvt) = W0

and E(ut_1vt)mfiL¢59—202. Thus,

VMLE =

 

1 2
1—¢20
1—492

2

 

(1+ 92)[(1- 92)2

01

0
- 9f]

2

 

(1+ 92)[(1 — 9212

—1

0'
- 9?]

1-¢9:-¢502

 

 

 

 

1-4,2 1-4’91-45502
-1 1—02
l—wi-Wz (1+02)[(1—02)’-91‘1
_¢ 91
1—M1—¢292 (”92110-921240

91
(1+02)[(1-92)2-9f1

1-02
(1+92)[(1-92)7-9f] 1

 

2

Chapter 3

MDE for Seasonal ARMA Processes

1 . Introduction

The previous chapter discussed the asymptotic properties of the MDE for a variety
of ARMA models. Many economic time series exhibit periodic behavior. For example,
monthly observations that are 12 periods apart might behave very similarly. As noted
by Hylleberg (1992), the seasonality observed in the economic data might be caused
by changes of weather, the calendar, and the production and consumption decision
made by economic agents. In this chapter we discuss the asymptotic properties of
MDE of seasonal ARMA models. Following the notation of Box and Jenkins (1976),

the general multiplicative model is

¢p(L)¢P(L8)VdV?3/t = 6q(L)eQ(L3)Et

where 6; is i.i.d. (0,02), ¢p(L) = 1 — ¢1L - — (prP, (Dp(L’) = 1 — <I>1L‘ -
-— <I>pLP‘, V =1- L, V, =1— L3, 0,,(L) =1—01L — -—0qu, and,
GQ(L‘) = 1 — GIL‘ — - - - — equ‘. As noted by Box and Jenkins (1976), the general

multiplicative model is said to be order (p, d, q) x (P, D, Q) ,. For most applications,
3 is equal to either 4 or 12. The multiplicative model is an appropriate model for de-

scribing seasonal pattern observed in many data series. An example of the application

47

48

of the general multiplicative model is the ”airline model” provided by Box and Jenkins
(1976). International airline passengers data that are 12 months apart behave very
similarly. Box and Jenkins modeled the differenced and seasonal differenced monthly
data of international airline passengers by the MA(1)-Seasonal MA(1)12 model. Simi-
larly, Hillmer and Tiao (1982) fit a MA(1)-SMA(1), model to the regular and seasonal
differenced monthly data of US unemployment males aged 16 to 19 from January 1965
to August 1979. This airline model has been applied to model many economic time
series. See for example Abraham and Ledolter (1983): chap. 6; Granger and Newbold
(1986): chap. 3; and Hanses (1996): chap. 3.

This chapter presents the properties of MDE of MA(1)-SMA(1), processes. The

MA(1)-Seasonal MA(l), process is
y: = (1 - 9L)(1- GUM. (18)

where e; is i.i.d. (0,02). The autocorrelation function for the seasonal MA(1)-

SMA(1), process is

Pl = ‘9/(1+92),
P2 = P3='°'=Ps-2=0,
p3_1 = HEB/(1+ 02)(1+ 92),
p3 = -9/(1+92),
Ps+1 = 09/(1+62)(1+62),

and

pk 0 for k 2 3+2.

49

The first order autocorrelation function of MA(1)-SMA(1), process is the same as
simple MA( 1) process and is not affected by the presence of the seasonal MA fac-
tor.

Also we have the following result of the maximum likelihood estimator of MA(1)-
SMA(1), processes. Similar to the notation defined in chapter 2, we denote A ML E as
the ML estimator, where A’ = [0 9]. Without loss of generality, we assume that 02

is known. The limiting distribution of MLE is x/T(AMLE — A) —+ N (0, VMLE) , where

—1

 

 

 

(1__ 02)-l 0.9-1(1 _ 038)-1
VMLE = [ 1 (19)
93-1(1— 6‘6)“ (1 - 92H
or,
1 (1 — 02)(1— 0.9)2 —03-1(1 - 02)(1 - e?)
V =
MLE (1‘ 9’9)” ‘ 92“? [ _0._1(1_ 92)(1— e2) (1 — e?)(1 — we)2
(20)

If 6 is small enough or s is large, the asymptotic variance of «TWAMLE — 0) will be

very close to 1 — 02 and x/T(CMLE — G) will be very close to 1 — 82.

2. Asymptotic Variance of MDE for MA(1)-SMA(1)4 Processes

A seasonal MA(1)-SMA(1)4 process is
.111: (1" 9L)(1" GL4)€t1 (21)

Given that 3 equals to 4, we then have the following results for the autocorrelation
function of the seasonal MA(1)-SMA(1)4 process: p1 = —6/ (1 + 02), p2 = 0, p3 =

68/(1+ mm + 92), p4 = -e/(1+ 92), p5 = 06/(1+ o2)(1+ 92), and p,c :-

50

0 for k _>_ 6. The second order autocorrelation function of MA(1)-SMA(1)4 process
equals to zero, while its first order autocorrelation function is not affected by the
seasonal MA coefficient. Similarly, the autocorrelation at lag 4 is not affected by the
MA coefficient.

Given a set of parameter values, the asymptotic variance of the MDE for MA(1)-
SMA(1)4 process can be numerically calculated by the same method described in
chapter 2. Table 10 presents the asymptotic variance of MDE for some MA(1)-
SMA(1)4 models. The nonseasonal moving average coefficient is fixed at 0.15 for all
cases, while the seasonal MA coefficient (6) is varied from 0.1 to 0.9 by steps of 0.1. In
each case, the asymptotic variance of MDE using the first 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
16, 24, and 32 autocorrelations are reported. Given that 0 are all equal to 0.15, we find
that when the seasonal MA coefficient is small, e.g., 6 = 0.1 and 0.2, the MDE using
first 8 autocorrelations can be asymptotically as efficient as MLE under normality.
As the number of autocorrelations used in MDE increases, the asymptotic variance
of fi(é — 9) and x/T(61 — 0) decrease. Hence, if the number of autocorrelations is
large enough, MDE can be as efficient as MLE. Given that 0 = 0.15, if the absolute
value of the seasonal MA coefficient of MA(1)—SMA(1)4 process is less than 0.2, MDE-
using first 8 autocorrelations can be as efficient as MLE.

We also investigate the asymptotic variance of MDE for MA(1)-SMA(1)4 process
for the cases of negative 9, e.g., G = —O.1, —0.2, - - - , —0.9. If 0 is not very large and
[GI are the same, the asymptotic variances of MDEs are almost the same. In general,

the results of negative 6 is very similar to those of positive 6.

51

Obviously, identification requires at least two autocorrelations be used in MDE.
A very simple estimator is the MDE based on the autocorrelation at lag 1 and 3.
However, if only the first 2 autocorrelations are used in the MDE, we are not able to
identify the model. When only first 3 autocorrelations are used in MDE, asymptotic
variance of x/T(GMDE -— 9) is very large comparing with the asymptotic variance
of MDE using the first 4 autocorrelations. Consider the example that 0 = 0.15 and
G = 0.6 given in Table 10, the asymptotic variance of fi(éMDE — G) of MDE using
the first 3 autocorrelations (g=3) is 321.77, while that of MDE using the first 4 auto-
correlations (g=4) is 4.88. For other cases we also find that the asymptotic variance
of x/T(éMDE — 8) reduced a lot when g is increased from 3 to 4. This result sug-
gests that MDE using p1 and p3 is not an appropriate estimator for MA(1)-SMA(1)4
models. Phrthermore, this result also implies that the autocorrelations at lag 4, 8.
and 12 are very important for MDE estimation in MA(1)—SMA(1)4 models when 6
is large. For the case that 6 = 0.15 and 6 = 0.8, asymptotic variances of MDEs
using first 7 autocorrelations and first 8 autocorrelations are 27.42 and 7.87, while
asymptotic variance of MDE using first 9 ACF is 7.71. A similar result is found when
we compared the asymptotic variances of MDEs using first 11 and first 12 autocorre-
lations. This result demonstrates that the 4th, 8th, and 12th order autocorrelations
are important moments for MDE estimation in MA(1)-SMA(1)4 models. Besides.
neglecting the autocorrelations at lag of multiples of 4 can cause a large efficiency
loss. This is particularly important to the asymptotic variance of the MDE estimator

of seasonal MA parameter.

52

Given that MA coefficients are the same, it is found that the higher the seasonal
MA parameter, the higher the number of autocorrelations is needed to guarantee that
MDE is efficient. For the casesof large seasonal MA parameters, we find that the
number of autocorrelations should be higher than 32.

Table 11 reports results of the MA(1)—SMA(1)4 models whose moving average
coefficients all equal to 0.35, while Table 12 and 13 present asymptotic variance of
MDE for the MA(1)-SMA(1)4 models whose moving average coefficients equal to 0.55
and -0.25, respectively. The seasonal MA coefficient (6) is varied from 0.1 to 0.9 by
steps of 0.1. The asymptotic variance of MDE using first 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 16, 24, and 32 autocorrelations are reported. Given that 0 = 0.35 and 6 = 0.20,
the asymptotic variance of fi(é — 9) of MDE using first 12 autocorrelations is very
close to that of MLE, whereas the asymptotic variance of x/T(é — 6) of MDE using
only first 8 autocorrelations can be as efficient as MLE.

The results in Table 12 show that given that 0 = 0.55, if G S 0.3, the asymptotic
variance of x/TQ — 0) for MDE using first 12 autocorrelations is very close to that
of MLE. All the cases reported in Table 13 have their MA coefficients equal to -
0.25. If both |0| and IS] are small, the asymptotic variance of MDE using first 8
autocorrelations is very close to that of MLE.

In general, for the MA(1)-SMA(1)4 process the efficiency loss in MDE appears to
diminish as the number of autocorrelations used in MDE increases. The MDE using a
small number of autocorrelations is surprisingly efficient for a subset of the parameter

space. Calculations also reveal that the 4th, 8th, and 12th order autocorrelations are

53

important moments of MDE in estimating MA(1)-SMA(1)4 models.

3. Asymptotic Variance of MDE for MA(l)-SMA(1)12 Processes

In this section, we present the results for the asymptotic variance of MDE of

MA(1)-Seasonal MA(1)12 process. The MA(1)-Seasonal MA(1)12 process is
y. = (1 — 0L)(1 — eL‘2)e,, (22)

where c, is i.i.d. (0 ,02). For the seasonal MA(1)-SMA(1)12 process we have p1 =
-9/(1+02), p11 = 99/(1+92)(1+92), 912 = -9/(1+92), p13 = 99/(1+02)(1+92),
and pk = 0 for k = 2, 3, - - - 10, and k 2 14. The first order autocorrelation function
of MA(1)-SMA(1)12 process is the same as that of MA(I) process and is not affected
by the presence of the seasonal MA factor.

Equation (19) provides a general result of the asymptotic variance of MLE of
MA(1)-SMA(1), process. Setting 3 = 12 yields the following result of the maxi-
mum likelihood estimator of seasonal MA(1)-SMA(1)12 processes: \/T (A MLE — A) —>

N(0, VMLE), where

V _ (1_ 92)—1 011(1 _ 012e)-1 "1
MLE — 011(1 __ 912e)-1 (1_ e2)-—1 1

and A’ = [0 6)]. Since the 3 equals 12 in this case, the off-diagonal elements of the
asymptotic variance matrix will be very close to zero given that IQ] is not close to
unity.

It is interesting to investigate the asymptotic properties of MDE of MA(1)-SMA(1)12

process. The asymptotic variance of x/T(6 — 0) and x/CI—‘(é — 9) of MDE of MA(1)-

54

Table 10: Asymptotic Variance of \/T(61— 0) and x/T—(é — 9) of MA(1)-Seasonal
MA(1)4 models, y, = (1 — 0L)(1-— 9L4)e,, with 0 = 0.15

 

6
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

 

Asymptotic variance of Jfié — 9)

g = 3 1.01 1.08 1.17 1.26 1.35 1.42 1.48 1.51 1.53
4 0.99 1.02 1.08 1.15 1.23 1.30 1.36 1.40 1.42

5 0.98 0.98 1.00 1.03 1.08 1.13 1.19 1.23 1.25

6 0.98 0.98 1.00 1.02 1.07 1.12 1.18 1.22 1.24

7 0.98 0.98 0.99 1.00 1.04 1.08 1.13 1.18 1.20

8 0.98 0.98 0.99 1.00 1.04 1.08 1.13 1.18 1.20

9 0.98 0.98 0.98 0.99 1.00 1.04 1.08 1.12 1.15

11 0.98 0.98 0.98 0.98 0.99 1.02 1.06 1.10 1.13

12 0.98 0.98 0.98 0.98 0.99 1.02 1.06 1.10 1.13

16 0.98 0.98 0.98 0.98 0.98 0.99 1.02 1.06 1.09

24 0.98 0.98 0.98 0.98 0.98 0.98 0.99 1.01 1.05

32 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.99 1.02

VMLE 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98

Asymptotic variance of x/T(é - 6)
g: 46.91 54.57, 70.75 103.02 169.90 321.77 723.84 2123.5 11293

1.05 1.16 1.39 1.84 2.78 4.88 10.40 29.50 154.13
1.03 1.14 1.36 1.80 2.70 4.74 10.10 28.62 149.54
1.03 1.14 1.36 1.80 2.70 4.74 10.10 28.61 149.48
1.03 1.13 1.35 1.77 2.64 4.59 9.72 27.42 142.88
0.99 0.98 0.98 1.04 1.23 1.73 3.10 7.87 38.80
0.99 0.97 0.97 1.03 1.22 1.71 3.05 7.71 38.01
11 0.99 0.97 0.97 1.03 1.21 1.68 2.99 7.54 37.08
12 0.99 0.96 0.92 0.89 0.90 1.05 1.56 3.44 15.77
16 0.99 0.96 0.91 0.85 0.80 0.82 1.02 1.93 8.07
24 0.99 0.96 0.91 0.84 0.76 0.68 0.66 0.91 3.07
32 0.99 0.96 0.91 0.84 0.75 0.65 0.56 0.60 1.55
VMLE 0.99 0.96 0.91 0.84 0.75 0.64 0.51 0.36 0.19

CDWNIODU‘AW

Note: g=3 indicates that p = [p1 p2 p3]’. Hence, g=3 represents that first 3 autocorre-
lations are used in MDE, while g=4 represents that first 4 autocorrelations are used in

MDE, and etc. VMLE is the asymptotic variance of JCTKOAMLE -— 0).

55

Table 11: Asymptotic Variance of VTM—O) and fi(é—6) of MA(1)-Seasonal
MA(1)4 processes,yt = (1 — 0L)(1 — 6L4)e¢ : 0 = 0.35

 

9
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Asymptotic variance of s/T(é - 0)

g: 3 1.08 1.20 1.33 1.47 1.59 1.68 1.75 1.80 1.83
4 0.90 0.95 1.03 1.12 1.22 1.31 1.39 1.44 1.47

5 0.89 0.91 0.94 1.00 1.07 1.14 1.20 1.25 1.28

6 0.88 0.89 0.92 0.96 1.03 1.10 1.16 1.21 1.24

7 0.88 0.88 0.89 0.92 0.96 1.02 1.08 1.13 1.17

8 0.88 0.88 0.89 0.91 0.95 1.01 1.07 1.13 1.16

9 0.88 0.88 0.88 0.90 0.93 0.97 1.03 1.08 1.11

10 0.88 0.88 0.88 0.89 0.92 0.96 1.01 1.06 1.10

11 0.88 0.88 0.88 0.88 0.90 0.93 0.97 1.02 1.06

12 0.88 0.88 0.88 0.88 0.90 0.93 0.97 1.02 1.06

16 0.88 0.88 0.88 0.88 0.88 0.90 0.93 0.97 1.01

24 0.88 0.88 0.88 0.88 0.88 0.88 0.89 0.92 0.96

32 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.90 0.93

VMLE 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88

Asymptotic variance of \fflé — 6)
g: 9.37 10.82 13.85 19.82 32.10 59.78 132.73 385.95 2042.56

3
4 1.16 1.28 1.53 2.04 3.08 5.42 11.58 32.88 171.91
5 1.05 1.15 1.37 1.82 2.73 4.79 10.20 28.91 151.02
6 1.03 1.14 1.36 l.80 2.71 4.75 10.12 28.67 149.78
7 1.03 1.13 1.31 1.68 2.43 4.09 8.41 23.27 120.09
8 1.00 0.99 1.01 1.08 1.30 1.85 3.33 8.48 41.94
9 0.99 0.98 0.98 1.04 1.23 1.72 3.07 7.78 38.36
10 0.99 0.98 0.98 1.03 1.22 1.71 3.05 7.72 38.07
11 0.99 0.97 0.97 1.02 1.18 1.61 , 2.80 6.93 33.76
12 0.99 0.96 0.93 0.90 0.93 1.09 1.64 3.64 16.75
16 0.99 0.96 0.91 0.86 0.81 0.83 1.05 2.01 8.47
24 0.99 0.96 0.91 0.84 0.76 0.68 0.67 0.94 3.17
32 0.99 0.96 0.91 0.84 0.75 0.65 0.56 0.61 1.59
VMLE 0.99 0.96 0.91 0.84 0.75 0.64 0.51 0.36 0.19

56

Table 12: Asymptotic Variance of VT (6—0) and \fflé—e) of MA(1)-Seasonal
MA(1)4 processes,yt = (l — 0L)(1 -— 9L4)e¢ : 0 = 0.55

 

9
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Asymptotic variance of Jflé — 0)

g: 3 1.64 1.91 2.19 2.46 2.69 2.87 3.01 3.09 3.14
4 0.87 0.98 1.13 1.29 1.46 1.61 1.73 1.81 1.85

5 0.78 0.84 0.93 1.04 1.16 1.27 1.37 1.45 1.49

6 0.74 0.78 0.85 0.93 1.04 1.15 1.25 1.33 1.37

7 0.72 0.73 0.76 0.82 0.89 0.99 1.09 1.16 1.21

8 0.72 0.73 0.75 0.79 0.85 0.94 1.03 1.10 1.14

9 0.71 0.72 0.73 0.76 0.81 0.89 0.96 1.03 1.08

10 0.71 0.72 0.72 0.75 0.79 0.85 0.93 1.00 1.04

11 0.71 0.71 0.71 0.72 0.75 0.79 0.86 0.92 0.97

12 0.71 0.71 0.71 0.72 0.74 0.78 0.85 0.91 0.96

16 0.71 0.71 0.71 0.71 0.72 0.73 0.77 0.83 0.88

24 0.71 0.71 0.71 0.71 0.71 0.71 0.72 0.76 0.80

32 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.73 0.77

VMLE 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71

Asymptotic variance of x/T(é - 9)
g: 4.88 5.59 7.06 9.93 15.78 28.88 63.24 182.17 959.13

1.31 1.44 1.73 2.30 3.48 6.13 13.09 37.19 194.53
1.11 1.21 1.44 1.90 2.85 4.99 10.61 30.06 157.02
1.06 1.17 1.39 1.84 2.76 4.84 10.29 29.16 152.30
1.06 1.15 1.31 1.61 2.21 3.54 7.00 18.87 96.03
1.03 1.03 1.06 1.16 1.41 2.01 3.66 9.35 46.33
1.01 1.00 1.01 1.08 1.27 1.79 3.20 8.10 39.96
1.01 1.00 1.00 1.06 1.25 1.74 3.11 7.87 38.81
11 1.01 0.99 0.99 1.03 1.17 1.54 2.59 6.24 29.92
12 1.01 0.98 0.95 0.93 0.97 1.16 1.76 3.95 18.22
16 1.01 0.98 0.93 0.88 0.84 0.87 1.11 2.15 9.10
24 1.01 0.98 0.93 0.86 0.77 0.69 0.69 0.98 3.35
32 1.01 0.98 0.93 0.85 0.76 0.66 0.57 0.63 1.66
VMLE 1.01 0.98 0.93 0.85 0.76 0.65 0.52 0.36 0.19

gochacns-w

57

Table 13: Asymptotic Variance of fi(6 — 6) and fi(é - 8) of MA(1)-Seasonal
MA(1)4 processes,y¢ = (1 — 6L)(1 - 8L4)6t : 6 =-0.25

 

9
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Asymptotic variance of J11 (6 — 6)

g: 3 1.02 1.11 1.21 1.32 1.42 1.50 1.56 1.60 1.62
4 0.95 0.99 1.05 1.13 1.22 1.30 1.36 1.41 1.43

5 0.94 0.95 0.98 1.02 1.07 1.13 1.19 1.23 1.26

6 0.94 0.94 0.96 1.00 1.05 1.11 1.17 1.21 1.24

7 0.94 0.94 0.95 0.97 1.00 1.06 1.11 1.16 1.19

8 0.94 0.94 0.95 0.97 1.00 1.05 1.11 1.15 1.18

9 0.94 0.94 0.94 0.95 0.97 1.01 1.06 1.10 1.13

10 0.94 0.94 0.94 0.95 0.97 1.00 1.05 1.10 1.13

11 0.94 0.94 0.94 0.94 0.95 0.98 1.02 1.07 1.10

12 0.94 0.94 0.94 0.94 0.95 0.98 1.02 1.07 1.10

16 0.94 0.94 0.94 0.94 0.94 0.95 0.98 1.02 1.06

24 0.94 0.94 0.94 0.94 0.94 0.94 0.95 0.97 1.01

32 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.95 0.99

VMLE 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94

Asymptotic variance of \fflé — 6)
g: 17.28 20.04 25.83 37.33 61.06 114.79 256.74 750.29 3981.56

1.10 1.21 1.45 1.93 2.91 5.11 10.91 30.95 161.77
1.03 1.14 1.36 1.80 2.71 4.76 10.12 28.69 149.90
1.03 1.14 1.36 1.80 2.70 4.74 10.10 28.62 149.53
1.03 1.13 1.33 1.73 2.54 4.37 9.13 25.55 132.61
0.99 0.98 0.99 1.06 1.26 1.78 3.20 8.13 40.14
0.99 0.97 0.97 1.03 1.22 1.71 3.05 7.73 38.09
0.99 0.97 0.97 1.03 1.22 1.71 3.05 7.71 38.01
11 0.99 0.97 0.97 1.02 1.19 1.65 2.91 7.27 35.61
12 0.99 0.96 0.92 0.89 0.91 1.07 1.59 3.53 16.18
16 0.99 0.96 0.91 0.85 0.81 0.82 1.03 1.96 8.23
24 0.99 0.96 0.91 0.84 0.76 0.68 0.66 0.92 3.11
32 0.99 0.96 0.91 0.84 0.75 0.65 0.56 0.60 1.57
VMLB 0.99 0.96 0.91 0.84 0.75 0.64 0.51 0.36 0.19

Scoooucamhw

58

SMA(1)12 processes are reported in Table 14-16. In each table, the MA(1) coefficients
are all the same, while the seasonal MA coefficients (6) are varied from 0.1 to 0.9 by
steps of 0.1. The moving average coefficient is fixed at 0.25 for all cases in Table 14,
whereas in Table 15 and Table 16 the MA coefficients are 0.45 and 0.65 respectively.
In each case, the asymptotic variance of MDE using first 11, 12, 13, 22, 23, 24, 25,
26, 35, 36, 37, and 48 autocorrelations are reported. Note that we are not able to
identify the parameters if the first 10 autocorrelations are used in MDE. The asymp-
totic variance of x/T (61402 — 6) of MDE using first 25 autocorrelations is very close
to that of MLE when both of 6 and 9 are small.

In general, if the number of autocorrelations used in MDE is large enough, MDE
appears to be asymptotically as efficient as MLE. Given that MA coefficients are the
same, it is found that the higher the seasonal parameter, the higher the number of
autocorrelations is needed to guarantee that MDE is efficient.

The results in Table 15 are very similar to those in Table 14. The asymptotic
variance of MDE is very close to that of MLE given that the number of autocorre-
lations used in MDE is large enough. One set of the parameter values are set very
close to those found in the airline model of Box, Jenkins, and Reinsel (1994). Given
that 6 = 0.45, if g=25, asymptotic variance of fi(6MDE — 6) is very close to that
of MLE when 6) is small, e.g., 8 =0], 0.2, and 0.3. But g needs to be higher than
25 in order to guarantee that the asymptotic variance of seasonal MA estimators is
very close to that of MLE for the seasonal models that G = 0.2, and 0.3. Table 16

reports the results for the seasonal models which all have 6 = 0.65, while 6 changes

59

for all cases. When 8 = 0.1, MDE using first 25 autocorrelations can be as efficient
as MLE.

For the MA(1)-SMA(1)12 model MDE using first 24 autocorrelations shows a lot
of efficiency improvement than the MDE using first 23 autocorrelations. This result
demonstrates that the 12th, 24th, and 36th order autocorrelation might be very
important for MDE in estimating the seasonal models with 12 periods. If G is very
large, adding autocorrelations at higher lags such as p43, p60, p72, - 1 -, might give more
efficiency gain than adding other autocorrelations.

Table 17 provides a comparison of the asymptotic variances of MDE using differ-
ent sets of autocorrelations. It is found that the MDE using first 1-13 autocorrelations
and autocorrelations at lag 23, 24, 25, 35, 36, and 37 has some advantage over the
other methods. In Table 17 method 1 indicates the case that the MDE using autocor-
relations at lag 1, 11, 12, 13, 23, 24, 25, 35, 36, and 37, whereas method 2 represents
the case of MDE using autocorrelations at lag 1-13, 23, 24, 25, 35, 36, and 37. The
results in Table 13 show that method 2 is very useful in reducing the asymptotic
variance of the MA coefficient. Consider the case that 6 = 0.30 and G = 0.1. It is

clearly that the asymptotic variance of 6 significantly improved by using method 2.

4. Estimation Results of MDE for Airline Model

The MDE method is applied to estimate the airline passenger data, which is also
analyzed by Box and Jenkins (1976). The total number of observations (T) of this

data set after the regular and seasonal difference is 131. We present the estimation

60

Table 14: Asymptotic Variance of VT (6 — 6) and Jflé — O) of MA(1)-Seasonal
MA(1)12 processes,y¢ = (1 — 6L)(1 — 9L12)€¢ : 6 = 0.25

 

9
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Asymptotic variance of (TM — 6)

g: 11 0.96 1.01 1.08 1.16 1.24 1.30 1.35 1.38 1.40
12 0.95 0.98 1.03 1.10 1.17 1.24 1.29 1.33 1.34
13 0.94 0.95 0.97 1.00 1.05 1.10 1.14 1.18 1.20
22 0.94 0.94 0.95 0.98 1.01 1.06 1.11 1.14 1.17
23 0.94 0.94 0.95 0.96 0.99 1.04 1.08 1.12 1.14
24 0.94 0.94 0.95 0.96 0.99 1.04 1.08 1.12 1.14
25 0.94 0.94 0.94 0.95 0.97 1.00 1.04 1.07 1.10
26 0.94 0.94 0.94 0.94 0.96 0.99 1.03 1.06 1.09
35 0.94 0.94 0.94 0.94 0.95 0.97 1.01 1.05 1.07
36 0.94 0.94 0.94 0.94 0.95 0.97 1.01 1.05 1.07
37 0.94 0.94 0.94 0.94 0.94 0.96 0.99 1.02 1.05
48 0.94 0.94 0.94 0.94 0.94 0.95 0.97 1.01 1.04

VMLE 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94

Asymptotic variance of fi(é - 9)

g: 11 16.80 19.52 25.28 36.80 60.72 115.13 259.30 761.46 4051.71
12 1.10 1.21 1.45 1.93 2.91 5.12 10.92 30.99 161.98
13 1.03 1.14 1.36 1.80 2.71 4.76 10.14 28.75 150.18
22 1.03 1.14 1.36 1.80 2.70 4.74 10.09 28.61 149.48
23 1.03 1.12 1.32 1.72 2.53 4.35 9.09 25.45 132.14
24 0.99 0.98 0.99 1.06 1.26 1.78 3.20 8.13 40.16
25 0.99 0.97 0.97 1.03 1.22 1.71 3.06 7 .74 38.13
26 0.99 0.97 0.97 1.03 1.22 1.71 3.05 7.71 38.01
35 0.99 0.97 0.97 1.02 1.19 1.65 2.90 7.24 35.48
36 0.99 0.96 0.92 0.89 0.91 1.07 1.59 3.53 16.19
37 0.99 0.96 0.92 0.89 0.90 1.04 1.54 3.40 15.57
48 0.99 0.96 0.91 0.85 0.81 0.82 1.03 1.96 8.23

VMLE 0.99 0.96 0.91 0.84 0.75 0.64 0.51 0.36 0.19

Note: g=11 indicates that p = [ p1 p2 - .. p3]’. Hence, g=11 represents that first 11 au-
tocorrelations are used in MDE, while g=12 represents that the first 12 autocorrelations

are used in MDE, and etc. VMLE is the asymptotic variance of x/T(6MLE — 6).

61

Table 15: Asymptotic Variance of x/T(6 - 6) and x/T(O - 9) of MA(1)-Seasonal
MA(1)12 processes,yt = (1 — 6L)(1 — 6L12)€¢ : 6 = 0.45

 

9
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

 

Asymptotic variance of x/T(6 — 6)

g: 11 0.81 0.86 0.92 0.99 1.05 1.11 1.15 1.18 1.19
12 0.81 0.84 0.89 0.95 1.02 1.07 1.12 1.15 1.16
13 0.80 0.82 0.85 0.89 0.94 0.99 1.04 1.07 1.08
22 0.80 0.80 0.81 0.83 0.86 0.90 0.94 0.97 0.99
23 0.80 0.80 0.81 0.82 0.85 0.89 0.93 0.96 0.98
24 0.80 0.80 0.81 0.82 0.85 0.89 0.93 0.96 0.98
25 0.80 0.80 0.80 0.81 0.83 0.87 0.90 0.93 0.95
26 0.80 0.80 0.80 0.81 0.82 0.85 0.88 0.92 0.94
35 0.80 0.80 0.80 0.80 0.81 0.83 0.86 0.89 0.92
36 0.80 0.80 0.80 0.80 0.81 0.83 0.86 0.89 0.91
37 0.80 0.80 0.80 0.80 0.81 0.82 0.85 0.88 0.90
48 0.80 0.80 0.80 0.80 0.80 0.81 0.83 0.86 0.88

VMLB 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

Asymptotic variance of s/T(é — 9)

g: 11 5.17 5.96 7.64 10.98 17.93 33.71 75.48 220.88 1173.21
12 1.23 1.36 1.64 2.18 3.29 5.80 12.39 35.18 183.93
13 1.07 1.18 1.41 1.86 ' 2.81 4.94 10.52 29.83 155.89
22 1.03 1.14 1.36 1.80 2.70 4.74 10.09 28.61 149.48
23 1.02 1.10 1.26 1.57 2.21 3.64 7.36 20.16 103.50
24 1.00 1.00 1.02 1.11 1.34 1.92 3.48 8.88 43.96
25 0.99 0.98 0.98 1.05 1.24 1.75 3.13 7.94 39.16
26 0.99 0.97 0.98 1.03 1.22 1.71 3.06 7.76 38.23
35 0.99 0.97 0.96 1.00 1.14 1.52 2.60 6.35 30.70
36 0.99 0.96 0.93 0.91 0.94 1.11 1.69 3.77 17.35
37 0.99 0.96 0.92 0.89 0.91 1.06 1.57 3.47 15.89
48 0.99 0.96 0.91 0.86 0.81 0.84 1.07 2.06 8.71

VMLE 0.99 0.96 0.91 0.84 0.75 0.64 0.51 0.36 0.19

 

62

Table 16: Asymptotic Variance of fi(6 — 6) and fi(é — 9) of MA(1)-
Seasonal MA(1)12 processes,yt = (1 — 6L)(1 - GL12)6¢ : 6 = 0.65

 

9
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

 

Asymptotic variance of fi(6 - 6)

g: 11 0.60 0.65 0.70 0.76 0.81 0.86 0.89 0.91 0.93
12 0.60 0.63 0.68 0.73 0.79 0.83 0.87 0.89 0.90
13 0.59 0.62 0.66 0.70 0.75 0.80 0.83 0.86 0.87
22 0.58 0.58 0.59 0.61 0.64 0.67 0.70 0.73 0.74
23 0.58 0.58 0.59 0.60 0.63 0.66 0.69 0.72 0.73
24 0.58 0.58 0.59 0.60 0.62 0.66 0.69 0.71 0.73
25 0.58 0.58 0.58 0.60 0.62 0.65 0.68 0.71 0.72
26 0.58 0.58 0.58 0.59 0.61 0.64 0.67 0.69 0.71
35 0.58 0.58 0.58 0.58 0.59 0.61 0.63 0.66 0.68
36 0.58 0.58 0.58 0.58 0.59 0.61 0.63 0.66 0.68
37 0.58 0.58 0.58 0.58 0.59 0.60 0.63 0.65 0.67
48 0.58 0.58 0.58 0.58 0.58 0.59 0.61 0.63 0.65

VMLE 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58

Asymptotic variance of Jflé - 6)

g: 11 2.48 2.82 3.54 4.98 7.97 14.72 32.55 94.50 499.92
12 1.37 1.51 1.81 2.40 3.63 6.38 13.63 38.69 202.28
13 1.15 1.27 1.52 2.02 3.05 5.35 11.42 32.40 169.36
22 1.03 1.14 1.36 1.80 2.70 4.74 10.10 28.62 149.49
23 1.01 1.06 1.17 1.38 1.84 2.86 5.54 14.77 74.77
24 1.00 1.01 1.05 1.15 1.41 2.04 3.71 9.50 47.11
25 1.00 0.99 1.00 1.08 1.29 1.84 3.31 8.41 41.57
26 0.99 0.98 0.99 1.05 1.25 1.76 3.16 8.00 39.50
35 0.99 0.97 0.95 0.97 1.06 1.36 2.22 5.26 24.96
36 0.99 0.96 0.93 0.92 0.96 1.15 1.76 3.96 18.31
37 0.99 0.96 0.92 0.90 0.92 1.09 1.63 3.62 16.64
48 0.99 0.96 0.91 0.86 0.82 0.86 1.11 2.14 9.10

VMLE 0.99 0.96 0.91 0.84 0.75 0.64 0.51 0.36 0.19

 

63

Table 17: Asymptotic Variance of x/T (A M D E — A) of Seasonal MA(1) models,
3]; = (1— 0L)(1- 6L12)€t

 

 

 

 

 

 

 

0.10 0.30 0.37 0.30

0.30 0.10 0.57 0.70

1-13' 1.01 0.00 0.91 0.00 1.03 0.07 1.13 0.07
0.00 1.36 0.00 1.04 0.07 4.01 0.07 10.19

1-25 0.99 0.00 0.91 0.00 0.92 0.02 1.01 0.02
0.00 0.97 0.00 0.99 0.02 1.52 0.02 3.07

1-37 0.99 0.00 0.91 0.00 0.88 0.01 0.96 0.01
0.00 0.92 0.00 0.99 0.01 0.98 0.01 1.55

Method 11 1.03 0.00 1.35 0.00 1.65 0.00 1.42 0.00
0.00 0.92 0.00 1.00 0.00 1.00 0.00 1.56

Method 21 0.99 0.00 0.97 0.04 0.97 0.04 1.03 0.03
0.00 0.92 0.04 1.00 0.04 1.00 0.03 1.56

VMLE 0.99 -0.00 0.91 -0.00 0.86 -0.00 0.91 -0.00

-0.00 0.91 -0.00 0.99 -0.00 0.68 -0.00 0.51

 

‘ 1-13 represents the case that first 13 autocorrelations are used in MDE, while 1-25
represents MDE using first 25 autocorrelations.

1 Method 1 represents the case of MDE using autocorrelations at lag 1, 11, 12, 13, 23, 24.
25, 35, 36, and 37.

3 Method 2 represents the case of MDE using autocorrelations at lag 1-13, 23, 24, 25, 35.
36, and 37.

64

result of the MDE using the first 48 autocorrelations. The MD (minimum distance)
estimates are 6114135 2 0.399 and OM03 = 0.523 with standard error 0.0893 and
0.0982, respectively. The estimation results of approximate MLE are also reported as
a comparison. The MD estimates of airline model are very close to those of MLE. The
asymptotic variance of AMDE is calculated as (1/T)(D’-\C‘1D;\)‘1, where D; denotes
the partial derivative of p(A) with respect to A’ evaluated at A M DE and C is calculated
by using equation (4). The standard deviation is calculated by taking squared root
of the diagonal elements of the variance matrix. We also calculate the residuals of
the MA(1)-SMA(1)12 model. By calculating the ACF of the residuals, we find that

the Box-Pierce LM test from 25 degree of freedom is 22.29.

5. Concluding Remarks

This chapter discusses the properties of the MDE for MA(1)-seasonal MA(1),
processes, where s is equal to either 4 or 12. This airline model has been applied to
model many economic time series data. Although the main focus of the estimation of
the seasonal ARMA models is the MLE method, the MDE is an attracting alternative
to the estimation of seasonal ARMA models. The MDE has the advantage of imposing
very little in terms of distributional assumptions on the innovation process. Also, the
MDE is relatively simple to compute.

Calculations reveal that if the number of autocorrelations used in MDE is large
enough, the MDE for the airline model appears to be asymptotically as efficient as

MLE under normality. It is also found that when the MA parameter is fixed, the

65

Table 18: Estimation Results of MDE of Airline Model: y, = (1 — 6L)(1 — OL12)et

 

91405 91401-2

0.399 0.523

(0.089)‘ (0.098)

The acf of the residuals is:

1 ' 2 3 4 5 6 7 8 9
-0.0077 0.0242 -0.1331 -0.0787 0.0853 0.0740 -0.0433 -0.0025 0.1376
10 11 12 13 14 15 16 17 18
-0.0545 ~0.0027 -0.1226 0.0004 0.0220 0.0782 -0.1197 0.0588 0.0145
19 20 21 22 23 24 25

-0.0922 -0.0937 -0.0150 -0.0280 0.2156 -0.0287 -0.0626

The Box-Pierce LM test from 25 df, Q(25) : 22.29

 

éMLE éMLE
0.377 0.572
(0.085) (0.070)

The Maximized value of the log likelihood :-385.576.

The acf of the residuals is:

1 2 3 4 5 6 7 8 9
-0.045 0.042 -0.1 17 -0. 165 0.038 0.061 -0.067 -0.043 0.108
10 11 12 13 14 15 16 17 18
-0.127 -0.012 0.017 0.006 0.060 0.089 -0.l73 -0.019 -0.002
19 20 21 22 23 24 25

-0.104 -0.070 -0.010 -0.068 0.177 0.017 0.019

The Box-Pierce LM test from 25 df, Q(25) : 21.040

 

‘Standard deviation is in the parenthesis.

66

higher the value of the seasonal MA parameter, the higher the number of autocorre-
lations is needed to guarantee that MDE is efficient. For the MDE of MA(1)-SMA(1),
processes, adding autocorrelations at lags of multiples of s, such as p,, p2,, p3,, ~--,
to the MDE might have a larger improvement in efficiency than adding other auto-

correlations.

APPENDIX

APPENDIX

Asymptotic Variance of MLE of MA(1)-SMA(1), Processes

The log-likelihood function of the MA(1)-SMA(1), process is

L = "7310391) — IHIOS(U%EZ[(1 — 6L)‘ 1(1 — avg-1,31,]?

2

where T denotes the total number of observations. The first order conditions of the

maximization of log-likelihofbd function of the MA(1)-SMA(1), process are

and

BL :22

BL

_ = __ e (1 — 6L)‘1(1— GL’)‘2y _..
66 021:;(11 t 1
3L

_ = _z€2__
802 20",:1 ‘ 202

The second order conditions are

62L
79—0?

22
862

62L

6669

62L

8?

62L

00206

and

_i

..1

02 ([(1 — 6L)"(1 - eL')“1u_1]2 + 2640 — 6L)‘3(1 — eL')-1y,_2])
¢=1

T
-312. z ([(1 — 6L)‘1(1 - £9L‘)"1u_..]2
t=1
+2 6; [(1 — 0L)-1(1 — 6L‘)‘3y1—2.])

T
—513 {Z [(1 " 9111-10 — 911.1-23’1-11 [(1 " “6-20 ‘ 9L‘)"y1-1]
t=1

T
+ Zeta - 6L)‘2(1 - eL')-2y,_,_.}

t=1
T
— T
2
37225173

22540—01.) 2(1—eL') 111-1]

t=1

67

68

62 L

T
1
53556 = E? 2541 — 6L)‘1(l— eL')-2y,-.
t=1

Using the result that e, = (1 — 6L)‘1(1 — OL’)‘1yt, we have

(92L 2
0—02- : —— 2::[H1—0L1 ft. 1] +2:T:€t(_261—0L)
t=1
62L 1 T 2 T
56? = -:-:{:Z [(1—OL’)'lct_,] +2Ze¢(1—9L’)_:€c—2a}
_ t=1
62L 1 T
3756 = --2Z(l(l()1-6L‘et.n<1-0L> .,_ 11

+6. (1 — 9L)-1(1— eL:)-le,_,_,)

Because E(etyt_J-)- —- 0 for j_ > 1, we have E(%’§)— - 0 and E(5—;—)— - 0. To get
the results for E(a—;’§), E(g—2e%), and Egg—7’5), we define u, and v; as (1 — 6L)ut = q,
and (1 — GL’)v, = 6,. Hence, u, follows an AR(I) process and 1), follows an AR(s)
process.

Finally, the following results is obtained by using the result that E (6,6,- 1') = 0 for

62L 1 T T
—E __ = _ 2 :—
(662) 02 gEut‘l 1- 62’

(PL 1 T 2 T
‘E (as) a E, E's—s — —1 _ 92
62L 1 T
E (3%) 35 2:3, 1201-12.-.)

T
= g 2 E[(1 + 6L + 62L? +--.)e,_,(1+ GL‘ + 62L” + . - -)et_.]
i=1

= iT(gs—1 + 023—193 + 633-1923 + _ _ .)02
0-2

= T0"1(1+ 036‘ + 62392” + . - -)

69

 

_ 7163‘1

_ 1— 90
_E (2213.) .. T_"2 _ _T_ _ _T_
004 06 204 20‘"

We then have x/T(AMLE — A) —> N(0, VMLE), where

._ —1
r 1 08 1 -
1-6"E 1-96 0

93—1

_ 1
WW“ 1:797 179! 0

 

 

0 0 fi; 4

CHAPTER 4

Minimum Distance Estimation for GARCH Models

1 Introduction

The use of GARCH processes and their extensions to represent the time dependent
heteroskedasticity present in many economic and financial economics series is now a
widespread econometric procedure. Bollerslev, Chou and Kroner (1992) describe
many of the applications of the methodology. Inference in the GARCH class of
models is usually based on MLE or QMLE assuming a Gaussian conditional density
and Bollerslev, Engle and Nelson (1994) and Hansen and Lee (1994) discuss many of
the inferential issues involved in likelihood based procedures. For example, one of the
most important models in empirical work is when the observed time series, y, follows

a martingale with linear GARCH(1,1) volatility process, so that

yt = 0111:, ('33)

where u; is i.i.d. N (0,1) and at is a positive, time varying and measurable function

with respect to the information set which is available at time t-l, and

0‘2 =w+ayt2_1 +B0t2_11 (.24)

where w > 0,0 2 0, 6 Z 0, and a + 6 < 1. It is then straightforward to maximize
the Gaussian conditional density and obtain robust standard errors from the QMLE

method of Bollerslev and Wooldridge (1992).

70

71

This chapter is concerned with using the Minimum Distance Estimator (MDE)
based on the sample autocorrelations of the squared process to estimate the pa-
rameters of the GARCH model. For example, if y; follows the above martingale -

GARCH(1,1) process, then y? has the ARMA(1,1) representation of

3112 = w ‘1' (a + @3134 + 1’t— 18111-1- (25)

where v, = y? — 0,2 and the expectation and variance conditional on information

available at time t—1 are E,_1'ut = 0 and Vart_lv¢ = 20:. Hence despite the
innovation process, 0,, being serially uncorrelated, it is not independent over time.

The MDE of the GARCH process parameters are very simple to compute from
the first g sample autocorrelations of the squares of a realization of the process. The
MDE is particularly attractive to use as an estimator in situations where the true
underlying data generating process has extreme non-normality. When estimating a
GARCH model with data exhibiting extreme kurtosis, the maximization of a Gaussian
density and the subsequent use of QMLE to obtain robust standard errors, is not
necessarily going to realize asyptotically efficient parameter estimates. Monte Carlo
evidence presented in this chapter provides evidence that with certain conditional
densities, and over certain regions of the parameter space, the MDE can compare
very favorably with QMLE in terms of parameter estimation bias and mean squared
error. In cases where difficulty is experienced in estimating GARCH models from
extreme non Gaussian densities, the MDE can be recommended as an attractive
alternative which can avoid problems of convergence.

The remainder of this chapter is organized as follows. Section 2 defines the MDE

72

for the GARCH model. Section 3 illustrates the application of the MDE procedure to
estimating the parameters of the ARMA(1,1) process with i.i.d. innovations. Monte
Carlo results are presented. Section 4 then applies the MDE to estimating the param-
eters of the GARCH(1,1) process. The asymptotic efficiency of the MDE is found for
various points in the parameter space for the conditional normal densities. Section
5 considers the MDE applied to estimate the parameters of the GARCH model for

hourly exchange rate data. Section 6 provides a brief conclusion.

2. The MDE of GARCH Models

It is noted in chapter 2 that when the innovations in the ARMA process are
i.i.d., the asymptotic variance of the sample autocorrelation is given by the Bartlett’s
formula. Hence, a consistent estimator of C is C, with (2', j )th element given by

00
éij = £05k“ + [31—7 — Qfiifik)(fik+j + file-j - 263151;) (26)
In practical application the MDE is obtained by solving the following minimization
problem:

Min S = (73 - p(A))'C“(/3 - p(A)). (27)

While the computation of the Optimal weighting matrix is straightforward in the case
of i.i.d. innovations, the estimation of GARCH process parameters to be discussed
later requires estimation of an ARMA process with non i.i.d. innovations. When the

innovations are not i.i.d., the robust covariance matrix estimator of Domowitz and

White (1982) and White (1984) can be implemented. On letting

73

71: = T_IZT=9+1(33¢ - P)($t—k - P) and PI: = 7k/‘70 , where 53 = 714232117: and
E (:r,) = 11. For the model described in equation (25), :12, corresponds to yf. The
robust covariance matrix estimator of the sample autocorrelation can be obtained by

first noting that

T
m; — p) = T”? (1/70) 2 2.,

t=g+1

where Z, is a g x 1 vector defined by

(“It " P)($t—1 " P) " Pl()‘)($t " 102
Z, = '

(3:, — [00131-9 — P) " P900073: ‘ 102
Clearly, E (Z,) = 0 and under suitable regularity conditions, T"1/2 2;, Z, —+ N (0, V,),
where V, = 2” F], and F,- = E(Z,Z,’_J-). Since fig; - p) = "0‘1 T‘l/2 23;, Z,

j=-oo

and ’70 —+ '70, it then follows that,
7703-10) -> N( 0, 70‘2 V2 ),

In many practical applications Vz can be consistently estimated by the Newey and

West (1987) procedure by using,
‘ ‘ j ‘ “I
z = I‘ 1— — F- I‘- 28
V 0+:( 1+,1>(.+,) (1

2 . . . . . A _ l T ; ¢ I '
where 1‘]- IS a covariance matrix estimator at lag ], F,- — T 21:,- “ Z, Z,_J. w1th

($1 — 53)($t—1 — 53) ‘ P1(:\)(17t - 573)2
Z? = s ,

(x: - i)($c—g - i‘) - pg( )(z, - :7)?

)

and A is a consistent estimator of A. The value of q in equation (28) can be determined

by a data dependent automatic rule provided by Newey and West (1994).

74

The MDENW denoted by A, can be first obtained by setting the weighting matrix
equal to the identity matrix. Hence the optimal weighting matrix for the MDE can

be consistently estimated by

CVNW = (7)2 ‘22, (29)
70

where 7“,, = T‘1 23;,(32, - 07:)2 and CNW is a consistent estimator of C, given that 7“,,
and V, consistently estimate 70 and V,, respectively. The resulting estimate of C, Le.

CNW can then be used in the quadratic form,
Min 5 = (73 — 7201mm — p(A)) (30)

to obtain the feasible MDE of the parameter vector A in the case of non i.i.d. inno-

vations.

3. Simulation Results of MDE for ARMA(1,1) with i.i.d. Innovations

In the next section, MDE of the parameters of the GARCH(1,1) process from
its ARMA( 1,1) representation in the squared variable will be considered. The main
additional complication of this model is the presence of non i.i.d. innovations. Before
considering the non i.i.d. case, it is convenient to first consider the MDE applied to
estimating the parameters in a classic time series setting of an ARMA process with

i.i.d. innovations. In the ARMA(1,1) process,

311—11: ¢(y1_1-u)+et—66¢_1, (31)

where e, is i.i.d.(0, 02), the vector of structural parameters, neglecting 02, is A =

75

[<25 01’-

The asymptotic variance of MDE for the ARMA(1,1) model are reported in chap-
ter 2. As the number of autocorrelations, g, increases, the asymptotic variance of
the MDE parameter estimates decreases and approaches to that of the MLE. If the
absolute value of 6 is not close to one, the MDE is seen to be remarkably efficient. In
many cases, it is surprising to note that a remarkably small number of autocorrela-
tions is necessary for the MDE to be as asymptotically efficient to two decimal places
as that of the MLE under normality.

While our results in chapter 2 show that the MDE appears to be asymptotically
efficient, it is also necessary to investigate its small sample performance. Table 19
presents some simulation results based on 1,000 replications, to evaluate the bias and
MSE of the MDE applied to estimating the parameters of the ARMA( 1,1) model, for
the two points in the parameter space of d) = 0.8, 6 =0.4 and 45 = 0.3, 6 =0.6. The
total number of observations (T) is equal to 500 and there are 1,000 replications for
each design. The parameters of the ARMA(1,1) process are estimated by:

(i) MDE using Bartlett’s method to calculate the weighting matrix, where the
number of autocorrelations used in computing the MDE are either 2, 5, 10, 20, or 30.

(ii) MDE using a weighting matrix estimated by the Newey and West (1987)
method where the number of autocorrelations used in computing the MDE are again
either 2, 5, 10, 20, or 30.

(iii) MLE assuming Gaussian disturbances.

Table 20 presents the simulation results for the cases that 05 = 0.5, 6 =0.2 and

76

05 = 0.1, 6 20.3, while g is set to either 2, 5, 10, 15, or 20. From the results in
Table 19 and 20, it is apparent that both the MDEs have a very small parameter
estimation bias for both designs. There are no obvious departures from randomness
and consequently no clear biases in either direction.

The RMSE of the parameter estimates from the sample size of T = 500 are close,
but not quite efficient when compared with the MLE. The MDE based on only g
= 2 and particularly g = 5 autocorrelations are remarkably efficient for both the
parameter designs considered. The RMSE of the MLE presented in the table are
calculated from the 1000 replications. Alternatively these RMSE could be compared
with those derived analytically from the theoetical limiting distribution using the
formula given in chapter 2. For example, in the case of 43 = 0.8 and 6 = 0.4 the
theoretical RMSE are .0456 and .0697 compared with .0494 and .0743 for the RMSE
of the estimates of 03 and 6 respectively. From Table 19 it appears that the MDE
based on Bartlett’s formula performs slightly better than the MDE using the Newey
and West method of estimating the C matrix in equation (30). When the d.g.p. is
d) = 0.8 and 6 = 0.4, the MDE based on the first 5 autocorrelations performs better
than that using more than 10 autocorrelations. For the case that d) = 0.3 and 6 = 0.6,
the MDE using 10 autocorrelations performs better than when more autocorrelations
are used.

Another interesting feature of Table 19 and 20 is that the MDE based on g=5
autocorrelations, for the design of d) = 0.8 and 6 = 0.4, have lower RMSE than the

MDE based on g = 2, 10, 20, and 30 autocorrelations. Althought large sample theory

77

suggests that it is better to include as many moments as possible in the estimation
procedure, the simulation results show that it seems not to be true at the sample size
analyzed here. This apparent trade-off occurs between the information used in the
estimator as defined by the number of moments being used, and the quality of the
objective function, as measured by the precision of the estimated weighting matrix. A
similar deterioration in the quality of the GMM estimation method has been reported
in cases where the number of identifying moment restrictions is increased beyond a
certain level. This result is also noted by Chung and Schmidt (1996) and Andersen

and Sorensen (1996).

4. Simulation Results of MDE for GARCH(1,1) Process

As previously noted, the innovation sequence 0, in the ARMA(1,1) model for
y? is clearly not i.i.d. and hence estimation of the parameters by MDE will require
formula (30) where the Newey West method is used to estimate the weighting matrix,
or equivalently the covariance matrix of the sample autocorrelations of a realization
of the process. In this section, we first discuss the autocorrelation function of the
squared GARCH(1,1) process and then present some simulation results of the MDE
using the two different methods of estimating the weighting matrix.

The autocorrelation function of yf, given the data generating process of a GARCH(1,1)
process, has been derived by Bollerslev (1988) and Ding and Granger (1996), who also
derive the autocorrelation function of the IGARCH process. We now briefly summa-

rize the autocorrelation function of the squared process. In particular, the derivation

78

Table 19: Simulated mean and root mean square error (RMSE)
of MDE and MLE for ARMA(1,1) model: y, = ¢y,_1+e,—6e,_1,

 

 

6, ~ iid N (0, 1). T: 500 and total number of replications
=1000.
o = 0.8 6 = 0.4 .
(b 6
g mean RMSE mean RMSE
2 0.7865 0.0662 0.3911 0.1045
5 0.7890 0.0516 0.3780 0.0776
M DEB 10 0.7973 0.0534 0.3750 0.0829
20 0.8109 0.0573 0.3709 0.0896
30 0.8215 0.0633 0.3653 0.0976
2 0.7865 0.0662 0.3911 0.1045
5 0.7825 0.0546 0.3797 0.0782
M BEAM 10 0.7813 0.0575 0.3774 0.0830
20 0.7785 0.0624 0.3768 0.0901
30 0.7756 0.0655 0.3748 0.0975
MLE 0.7917 0.0494 0.3905 0.0743
03 = 0.3 6 = 0.6
(p 6
g mean RMSE mean RMSE
2 0.3133 0.1499 0.6230 0.1540
5 0.3068 0.1308 0.6075 0.1139
M DEB 10 0.2939 0.1309 0.5974 0.1070
20 0.2836 0.1488 0.5953 0.1190
30 0.2768 0.1612 0.5981 0.1311
2 0.3122 0.1504 0.6211 0.1540
5 0.3091 0.1327 0.6089 0.1162
M DENw 10 0.2987 0.1340 0.5981 0.1127
20 0.2932 0.1430 0.5939 0.1221
30 0.2899 0.1502 0.5913 0.1332
MLE 0.2945 0.1149 0.5977 0.0971

 

Note: The MDE using Bartlett’s Formula to calculate the weighting matrix
is denoted as M DEB, whereas M DENw denotes the cases that asymptotic
variance of sample ACF is calculated by Newey and West method. The
number of autocorrelation used in MDE is denoted as g.

79

Table 20: Simulated mean and root mean square error (RMSE)
of MDE and MLE for ARMA(1,1) model: y, = ¢y,_1+e,—6e,_1,
6, ~ iid N (0, 1). T: 500 and total number of replications

=1000.

 

 

¢ = 0.5 6 = 0.2 .
d) 6

g mean RMSE mean RMSE

2 0.4927 0.1356 0.1982 0.1526

5 0.4848 0.1295 0.1818 0.1393

M DEB 10 0.4886 0.1394 0.1792 0.1496

15 0.4933 0.1429 0.1785 0.1523

20 0.4973 0.1454 0.1766 0.1568

2 0.4927 0.1356 0.1982 0.1526

5 0.4815 0.1310 0.1826 0.1415

M DENW 10 0.4828 0.1347 0.1845 0.1473

15 0.4842 0.1433 0.1876 0.1548

20 0.4854 0.1430 0.1904 0.1578

MLE 0.4941 0.1225 0.1951 0.1350
(b = 0.1 6 = 0.3 - -
4’ 6

g mean RMSE mean RMSE

2 0.1312 0.2624 0.3381 0.2673

5 0.1049 0.2288 0.3042 0.2189

M DEB 10 0.1005 0.2405 0.3030 0.2290

15 0.0915 0.2528 0.2978 0.2402

20 0.0914 0.2639 0.3008 0.2485

2 0.1312 0.2624 0.3384 0.2681

5 0.0965 0.2347 0.2952 0.2284

M DENW 10 0.0945 0.2491 0.2915 0.2402

15 0.0886 0.2549 0.2858 0.2453

20 0.0889 0.2640 0.2867 0.2513

MLE 0.0984 0.2264 0.2978 0.2202

 

80

of the autocorrelation function here does not rely on any distributional assumption.

In general, only assumption 1 and 2 given below are required.

Assumption 1 u, is i.i.d.(0,1) with E(u‘,‘) = n < oo.

Recall that the ARMA(1,1) representation of y? is

y? = w + (a + 6)y,2_1+ ’Ut — ,B’U¢_1.

where v, = y? — of. Let E,-1(-) denote mathematical expectation of the process
conditional on the information available at time t-l. Clearly, E,_,v, = 0 and the
conditional variance of v, is E,_ 1(1)?) = (17 — l)a:. ’1

Since E(y§ ) = E(u :E‘) (0,) = 17E (0 ), it follows that the fourth moment of y, is
finite given that E(0,4) is finite. Let 02 denote the unconditional expection of y}, so
that 02 = E(y,2) = E(o,2). Given that 0+6 < 1 , we have 02 = (1 - a — 6)02. Taking

the square of equation (24) yields

E(0?) = E[(1 - a - PM" + 0313.1 + 6072—112

= a4[1 — (a + 6?] + E[az2y;‘_1 + 6203., + 206213—1034]

Therefore,

E(a;’) =a4[1— ()a+62]+(na2 +62+2a6)E(0',_1). (32)

It follows that if 170:2 + 2076 + 62 < 1 , E(af) exists, so does the fourth moment
of y,. Under the normality assumption of 11,, we have n = 3 and the condition for

the fourth moment of y, being finite corresponds to 3a2 + 206 + 62 < 1, which is

Ware—1(1):) = EM? -0?)’ Int-11'; 131(16- 2v¢20¢ +0?) I01- 11- (fl- 1)”:

 

81

given in Bollerslev (1986). A very general result of the necessary conditions for the

existence of the 3;?" moments for k = 1, 2, - .. is given in Terasvirta (1997).

Assumption 2 1102 + 205 + E < 1.
If n 2 1, assumption 2 also implies that 0 + fl < 1. Under assumptions 1 and 2

we have
_ a4 [1 — (0+ m2]
- 1— (1702 +32 +20fl)°

 

E (01‘ ) (33)

The following results of the autocorrelation and autocovariance functions of y? are
based on the above two assumptions. First note that 70 = E (y? - 02)2 = E(y;‘ —
231,202 + 04), so that

70 = nE(02‘) - 0‘4 (34)

Using the result given in equation (33) yields

 

_ (n-1)(1-32-203) 01
0—1—(1702+fl2+205)

Rearranging the ARMA( 1,1) equation of y? yields
y? - 02 = (0 +,8)(y3.1- 02) + 221— fivH.
Therefore,
71 = (a + flho + E [(213.1 - 02M] - MHz/12.1 - 02)?)1—11-
Using the result that E[ (y,2_j — 02) v, ] = 0 for j 2 1, we obtain

71 = (a + (3)70 — fl [ 775110?) - 19(01‘) l (35)

and

82

% =(a+m%4,brk22

Substituting the result given in equation (33) into the above equation we have

__m-1M0-afi-W)1
71 — 1-(n02+fl2+20fl) 0' (36)

 

It follows that under assumptions 1 and 2, the autocorrelation function of y? from

the GARCH(1,1) process is

 

 

_ 1125
and
Pk = (0 + 023 )(0 + fi)"‘1 for k > 2. (38)
1 - 203 — fl2 "

Therefore, similar to the autocorrelation function of the standard ARMA(1,1) process,
the autocorrelation function of y? from the GARCH( 1,1) process decreases exponen-
tially. For the standard ARMA model: (1 — ¢L)y¢ = (1 — 0L)et, where e, is white
noise, we have p1 = W, and pk = 14-14) for k 2 2. By substituting ¢ = 0+ ,3
and 0 = B into the above equation, we have the same results of the autocorrelation
functions as given in equations (37) and (38), given that E(y;‘) is finite.

In sum, the results given in equations (37) and (38) do not rely on any spe-
cific distributional assumptions. In particular, if the standardized innovation (at) of
GARCH(1,1) process is iid(0, 02) with finite fourth moment and E(y;‘) is finite, the
autocorrelation functions of the squared process are defined by equations (37) and
(38).

In much high frequency data it is unclear if the unconditional fourth moment of

y, exists. In the appendix we present some results of the autocorrelation function of

83

y? when E (y?) does not exist. 2

In order to assess the relative small sample performance of the MDE applied to
estimating the GARCH(1,1) model, a detailed Monte Carlo study was carried out.
The data generating process was a GARCH(1,1) model with NID(0,1) standardized
innovations. The parameter values are required to satisfy the constrain that 302 +
205 + 32 < 1, so that the fourth moment of y, is finite. The total number of
observations, T, was set at 1,000. In each of the 1,000 replications 10,000 initial
values were generated to avoid startup problems. For each replication, the following
statistical procedures were calculated:

(i) Equation (25) was estimated by MDE through minimizing the criteria function
given in equation (27). The weighting matrix was formed from W = 6'“ where the
(2', j)th element of C was computed from equation (26). Hence this simple version of
the MDE neglects the non i.i.d. nature of the disturbances and is known to be sub
optimal. However, this version of the MDE is extremely easy to compute and there
is some interest in knowing its properties in this non standard situation.

(ii) Equation (25) was again estimated by MDE through using the criteria function
given by equation (30). The implementation of this method requires the weighting
matrix, W to be set to C‘Nw A and is most easily done by initially estimating 51
through using an arbitrary weighting matrix such as the identity matrix and the

resulting new estimate used to provide a new update of C'Nw. For the various repli-

 

2Under the assumption of normality of at, Ding and Granger (1996) also show that if the process
starts at a very long time ago and 302 + 200 + 32 _>_ 1 but a + 0 5 1, then the autocorrelation
function of y? can be approximated by, pk = [0 + (1/3)fl](0 + fi)""l. They also show that for the
integrated GARCH model, p), = (1/3)[l + 20](1 + 202)"‘/2.

84

cations and the parameter values considered in this study, the number of iterations
required to estimate the weighting matrix was quite small, and generally less than
four. Although this practical version of the MDE requires the use of iterative proce-
dures, the total number of computer operations is still much less than that of MLE.

(iii) Equations (23) and (24) were estimated by assuming a Gaussian density.

4.1 Monte Carlo results of MDE of GARCH(1,1) with NID innovations

Table 21 reports the results of the simulation study and are based on 1,000 repli-
cations for each quantity. Both versions of the MDE are calculated from using the
first 2, 5, 10, 20,30, and 40 autocorrelations.

The parameter values are set very close to the estimation results of hourly ex-
change rate data reported in Baillie and Bollerslev (1991). The MDE using weighting
matrix calculated by Bartlett’s formula is denoted as M DE 3, whereas the MDE using
weighting matrix calculated by Newey and West method is denoted as M DE NW- The
only difference in these two methods is the way of calculating the optimal weighting
matrix.

We find that the standard deviation of MDE using the Bartlett weighting matrix
is higher than that of MDE using the Newey and West covariance matrix estimator.
It might be caused by the use of subOptimal weighting matrix. This result is expected
since the errors in the ARMA(1,1) process for the squared observation are not i.i.d.

For the cases of M DENW, when 0202 and B = 0.6, the root mean square error

(RMSE) of MDEs using 20 and 30 autocorrelations are lower than those of MDE

85

using 5 and 10 autocorrelations. The improvement in efficiency from g=5 to g=10
is quiet large. Similarly, when g is increased from 10 to 20, the RMSE is decreased
for both 0MDE and 31.403. However, there is little change in the RMSE when the
number of autocorrelations are more than 20.

Table 22 presents the results for two sets of parameter values: 0 = 0.1, fl = 0.8
and 0 = 0.05, 0 = 0.92. This simulation design corresponds to the cases that 0 + 0 is
close to 1, but the fourth moment of y still exists. These cases represent the GARCH
model with strong persistent in volatility. These two sets of parameter values are
very close to the estimation results of the exchange rates of US. dollar versus the
British pound and Deutschmark for a total number of 1,245 observations reported
in Bollerslev (1987). The Monte Carlo results indicate that the optimal number of
autocorrelations at the sample size of 1000 is around 30 or 40. Hence, it suggests when
0 + 0 is close to 1, a larger number of autocorrelations is required to be used in the
MDE. Also, both duos and BMDE have small downward biases, which diminish as
the number of autocorrelations increases. Table 23 presents the simulation results for
the case of T=5,000. When 0 = 0.2 and fl = 0.6, the RMSE of 0MDE using the first 5
autocorrelations is almost the same as that of using either 10 or 20 autocorrelations.
The RMSE of 33033 (the MDE using the first 20 autocorrelations) appears to be
lower than that of 33% The RMSE of MLE is apparently lower than that of MDE
for both 0 and fl.

For comparison, we also present simulation results of MLE of 0 and fl in Table

21 and 22. The small sample properties of MLE for the GARCH models have been

86

investigated by Lumsdaine (1995) and Bollerslev and Wooldridge (1992). Bollerslev,
Engle, and Nelson (1994, p.2983) provide a summary of the small sample properties of
MLE for the GARCH(1,1) process. In particular, Monte Carlo evidence suggests that
the ML estimate of 0 + 6 is downward biased. This bias comes from a downward bias
in 6, whereas 0 is upward biased. Some results for the case of T=200 are presented in
Table 1 of Bollerslev and Wooldridge (1992). Our results of the MLE are consistent
to previous Monte Carlo studies of MLE of GARCH(1,1) process. When the sample
size is equal to 1000, 6114”; is slightly downward biased. Clearly, the bias diminishes
as the sample size increases. The simulation results reported in Table 23 show that
this bias in 6M“; disappears when total number of observations is equal to 5,000.

The asymptotic standard error of MDE of GARCH(1,1) process can be approx-
imated in a simulation environment. The approach utilizes the fact that D can be
determined analytically, while the variance of sample autocorrelations given a certain
sample size may be estimated from a set of arbitrage large simulated samples.

The data generating process is the usual GARCH(1,1) with the standardized inno—
vations being NID(0,1), while the total number of observations is equal to 1,000. The
GARCH(1,1) data series is simulated 50,000 times. For each replication the sample
autocorrelations up to 40 lags are calculated. The variance of the sample autocorre-
lation is estimated and denoted as V‘. Then the asymptotic variance of MDE when
T=1000 is calculated as ( D’V"lD )‘1.

Table 24 reports the simulation results of the standard deviation of the MDE when

T=1,000. The number of moments are set to 2, 5, 10, 20, 30, 40. We present results

87

Table 21: Simulated mean and root mean square error (RMSE)
of various forms of MDE and MLE for GARCH(1,1) process:
5, = 0121;, 21, ~ i.i.d. N(0,1) and of = w + 0634 + 60,24.
T=1000 and total number of replications = 1000.

 

0:02 6:06

Q)
Q

g mean RMSE mean RMSE

5 0.1968 0.0703 0.5711 0. 1582

10 0.2059 0.0683 0.5599 0.1467

M DEB 20 0.2123 0.0731 0.5566 0.1495
30 0.2164 0.0753 0.5550 0.1538

40 0.2208 0.0786 0.5532 0.1555

5 0.1727 0.0687 0.5741 0.1689

10 0.1745 0.0590 0.5521 0.1462

M DE Nw 20 0.1730 0.0567 0.5604 0.1283
30 0.1715 0.0585 0.5779 0.1207

40 0.1732 0.0573 0.5832 0.1 190

MLE 0.2015 0.0433 0.5843 0.0914

 

0:0.15 6

0.7

a B
g mean RMSE mean RMSE

5 0.1460 0.0634 0.6714 0.1657
10 0.1597 0.0568 0.6560 0.1411
M DEB 20 0.1655 0.0602 0.6553 0.1363
30 0.1703 0.0642 0.6517 0.1417
40 0.1738 0.0676 0.6512 0.1424

5 0.1284 0.0596 0.6694 0.1737

10 0.1316 0.0474 0.6573 0.1380

M DENw 20 0.1319 0.0453 0.6699 0.1232
30 0.1326 0.0441 0.6845 0.1118

40 0.1327 0.0445 0.6949 0.1102

MLE 0.1496 0.0359 0.6853 0.0884

 

Note: w is equal to (1 — 0 - 6) at 0.02. The MDE using Bartlett’s Formula
to calculate the weighting matrix is denoted as M DE 3, whereas M DENw
denotes the cases that asymptotic variance of sample autocorrelation is
calculated by Newey and West method. The number of autocorrelation
used in MDE is denoted as g.

88

Table 22: Simulated mean and root mean square error (RMSE)
of various forms of MDE and MLE for GARCH(1,1) process:
at = 0111‘, u; ~ i.i.d. N(0,1) and 0,2 = w + 06f_1 + 60,24.
T=1000 and total number of replications = 1000.

 

0:01 6:08

 

0 6

g mean RMSE mean RMSE

5 0.0972 0.0572 0.7562 0.1910

10 0.1103 0.0454 0.7542 0.1419

M DE 3 20 0.1170 0.0457 0.7515 0.1227

30 0.1202 0.0481 0.7508 0.1213

40 0.1232 0.0508 0.7479 0.1296

5 0.0847 0.0536 0.7514 0.1936

10 0.0902 0.0379 0.7549 0.1345

M DENW 20 0.0939 0.0331 0.7600 0.1137

30 0.0955 0.0316 0.7687 0.1025

40 0.0968 0.0325 0.7737 0.1058

MLE 0.1032 0.0292 0.7763 0.0803
0 = 0.05 6 = 0.92

0 6

g mean RMSE mean RMSE

5 0.0461 0.0444 0.8322 0.2349

10 0.0504 0.0366 0.8705 0.1670

M DEB 20 0.0581 0.0301 0.8834 0.1149

30 0.0622 0.0290 0.8856 0.0939

40 0.0655 0.0306 0.8828 0.0983

5 0.0392 0.0430 0.8307 0.2417

10 0.0390 0.0355 0.8777 0.1558

M DE Nw 20 0.0428 0.0275 0.8969 0.1041

30 0.0479 0.0222 0.8961 0.0802

40 0.0505 0.0208 0.8932 0.0827

MLE 0.0503 0.0180 0.9024 0.0636

 

89

Table 23: Simulated mean and root mean square error (RMSE)
of various forms of MDE and MLE for GARCH(1,1) model:
y, = mug, u; ~ i.i.d. N(0,1) and 0,2 = w + 031,24 + 60,24.
T=5000 and total number of replications =1000.

 

 

0 = 0.2 6 = 0.6
0 6

g mean RMSE mean RMSE

5 0.1993 0.0398 0.5889 0.0792

10 0.2018 0.0390 0.5862 0.0714

M DEB 20 0.2031 0.0394 0.5859 0.0712

30 0.2042 0.0398 0.5859 0.0711

40 0.2054 0.0403 0.5855 0.0716

5 0.1858 0.0340 0.5863 0.0716

10 0.1809 0.0337 0.5837 0.0635

M DENw 20 0.1771 0.0349 0.5874 0.0596

30 0.1753 0.0362 0.5947 0.0586

40 0.1744 0.0371 0.6016 0.0574

MLE 0.2002 0.0184 0.5962 0.0373
0 = 0.2 6 = 0.5

a 3

g mean RMSE mean RMSE

5 0.1994 0.0371 0.4924 0.0766

10 0.2005 0.0371 0.4908 0.0736

MDEB 20 0.2014 0.0372 0.4911 0.0738

30 0.2023 0.0376 0.4911 0.0743

40 0.2032 0.0381 0.4909 0.0749

5 0.1878 0.0347 0.4856 0.0729

10 0.1837 0.0321 0.4802 0.0690

M DENW 20 0.1779 0.0342 0.4883 0.0650

30 0.1757 0.0359 0.4971 0.0655

40 0.1751 0.0369 0.5026 0.0675

MLE 0.1996 0.0200 0.4994 0.0439

 

Note: The MDE using Bartlett’s Formula to calculate the weighting
matrix is denoted as M DEB, whereas M DENw denotes the cases
that asymptotic variance of sample autocorrelation is calculated by

Newey and West method. The number of autocorrelation used in
MDE is denoted as g.

90

of three sets of parameter values : 0 = 0.2, 6 = 0.5, 0 = 0.2, 6 = 0.6 , and 0 = 0.15,
6 = 0.7. In general, as the number of autocorrelations used in MDE increased the
asymptotic variance of MDE decreases. When g is increased from 2 to 5, the standard
deviation of MDE reduced a lot. It is found that for the cases that 6 = 0.5 and 0.6,
the asymptotic variance of the MDE using the first 20 autocorrelations is very close to
that of the MDE using the first 30 autocorrelations. For the third case with 6 =07,
the improvement in efficiency is also very small when the number of autocorrelation
is increased from 20 to 30. Hence, for the three cases examined here the incorporation

of more than 30 autocorrelation in MDE is not likely to be very beneficial.

4.2 Monte Carlo results for GARCH(1,1) with leptokurtic errors

Many empirical applications of the GARCH models report that the assumption
of conditional normality for the standardized innovation is usually not valid. Boller-
slev (1987 ) provides evidence showing that the simple GARCH(1,1)-t model fit many
of the speculative asset return series better than the GARCH models with condi-
tional normal errors. Similarly, Nelson (1991) uses the generalized error distribution
(GED) as the density function of the MLE in estimating the Exponential GARCH
(EGARCH) model for stock returns data. Baillie and Myers (1991) find that the
GARCH model with a conditional student t density provides a better description
of commodity price changes than the GARCH model with conditional normality.
However, the true conditional density is usually not known for many economic and

financial time series. Alternatively, the Quasi-MLE (QMLE) method can be applied

91

Table 24: Simulated results of asymptotic standard deviation
of MDE of GARCH(1,1) process for T=1000.

 

 

0 = 0.2 , 0.2 0.15
6 = 0.5 0.6 0.7
standard deviation of 01mm
g: 2 0.0711 0.0918 0.0979
5 0.0591 0.0601 0.0527
10 0.0582 0.0560 0.0452
20 0.0582 0.0557 0.0442
30 0.0581 0.0557 0.0442
40 0.0581 0.0557 0.0442
standard deviation of 6MDE
g: 2 0.2821 0.3238 0.3913
5 0.1456 0.1328 0.1348
10 0.1358 0.1077 0.0936
20 0.1356 0.1059 0.0876
30 0.1356 0.1058 0.0873
40 0.1356 0.1058 0.0873

 

92

to estimate the GARCH models. Bollerslev and Wooldridge (1992) investigate the
properties of the QMLE and related test statistics, when a normal log-likelihood is
maximized but the assumption of normality is violated and show that if the first two
conditional moments are correctly specified, the QMLE is generally consistent and
has a limiting normal distribution.

To compare the performance of the MDE and the QMLE for the GARCH model
of data series exhibiting extreme departures from conditional Gaussianity, a simple
simulation study very similar to those in section 4.1 is performed. The true data
generating process (DGP) is the GARCH(1,1) process with innovations being either
standardized t or chi-square distributed. The total number of observations is equal
to 1,000 and there are 1,000 replications for each design.

Table 25 and 26 provide simulation results of the MDE and the QMLE for the
GARCH(1,1) with the standardized innovations being a standardized t distribution
with degree of freedom 5 (t,,=5), i.e. u, = 51/ «573, where 5, are i.i.d. tu=5 variate.
Provided that u > 2, the student t variable has the population mean zero and variance
given by u/(u — 2). If u > 4, the p0pulation fourth moment of a t variable is
311/ [(u — 2)(l/ -— 4)]. The variance and kurtosis of the tu=5 are 5 / 3 and 9, respectively.
The first portion of Table 25 reports the results of the MDE and QMLE when 0 = 0.1
and 6 = 0.6. In general, when the number of autocorrelations used in MDE increases,
the RMSE of MDE decreases. However, the decrease in RMSE is not significant when
g is increased from 30 to 40. It is also found that the MDE of 0 when 9 Z 20 ( 5mm:

) has smaller RMSE than the QMLE of 0 ( aMLE), while the RMSE of 8mm and

93

Table 25: Simulated mean and root mean square error (RMSE)
of various forms of MDE and QMLE for GARCH(1,1) process:
y; = 0,11,, 0? = t12+0y;"_l +60,2_1. where u, is i.i.d. standardized
tv=5. T=1000 and total number of replications 2: 1000.

 

020.1 6:0.6

 

6: 6

g mean RMSE mean RMSE

5 0.1001 0.0773 0.5234 0.3139

10 0.1076 0.0762 0.5135 0.2810

M DEB 20 0.1106 0.0780 0.5079 0.2815

30 0.1133 0.0809 0.5060 0.2843

40 0.1153 0.0834 0.5107 0.2822

5 0.0840 0.0601 0.4909 0.3272

10 0.0885 0.0543 0.4907 0.2745

M DE NW 20 0.0925 0.0549 0.5207 0.2530

30 0.0959 0.0567 0.5420 0.2444

40 0.0992 0.0575 0.5463 0.2438

QMLE 0.1101 0.0652 0.5419 0.2369
0 = 0.2 6 = 0.6

d 6

g mean RMSE mean RMSE

5 0.1838 0.1160 0.5450 0.2525

10 0.1996 0.1157 0.5292 0.2303

M DEB 20 0.2078 0.1223 0.5234 0.2302

30 0.2123 0.1265 0.5236 0.2326

40 0.2165 0.1287 0.5215 0.2341

5 0.1488 0.0933 0.5584 0.2327

10 0.1555 0.0834 0.5502 0.1963

MDENw 20 0.1600 0.0814 0.5679 0.1740

30 0.1633 0.0813 0.5792 0.1729

40 0.1658 0.0817 0.5796 0.1763

QMLE 0.2044 0.0780 0.5719 0.1305

 

94

Table 26: Simulated mean and root mean square error (RMSE)
of various forms of MDE and QMLE for GARCH(1,1) process:
y, = mm, 0,2 = w+0y,2_1 +6011, where at is i.i.d. standardized

t.,._.5. T=1000 and total number of replications = 1000.

 

 

0 = 0.1 6 = 0.5 ,
d 6

g mean RMSE mean RMSE

5 0.0930 0.0743 0.4631 0.3026

10 0.0979 0.0746 0.4496 0.2834

M DEB 20 0.1011 0.0767 0.4428 0.2808

30 0.1026 0.0791 0.4473 0.2829

40 0.1045 0.0806 0.4453 0.2839

5 0.0808 0.0585 0.4113 0.3167

10 0.0851 0.0559 0.4101 0.2821

M DENw 20 0.0893 0.0535 0.4498 0.2662

30 0.0923 0.0539 0.4682 0.2609

40 0.0957 0.0557 0.4759 0.2601

QMLE 0.1041 0.0625 0.4484 0.2562
0 = 0.1 6 = 0.8 .
0 6

g mean RMSE mean RMSE

5 0.0916 0.0870 0.7321 0.2675

10 0.1109 0.0825 0.7301 0.2143

M DEB 20 0.1 198 0.0860 0.7248 0.2052

30 0.1237 0.0901 0.7260 0.2006

40 0.1270 0.0933 0.7239 0.2018

5 0.0670 0.0680 0.7529 0.2493

10 0.0754 0.0522 0.7694 0.1698

M DENW 20 0.0858 0.0486 0.7642 0.1456

30 0.0905 0.0471 0.7680 0.1343

40 0.0922 0.0521 0.7729 0.1346

QMLE 0.1083 0.0586 0.7635 0.1242

 

95

Table 27: Simulated mean and root mean square error (RMSE)
of various forms of MDE and QMLE for GARCH(1,1) model:
y, = 0121;, a? = w +0yf_1+60,2_1, where u, is i.i.d. standardized
chi-square variable with degree of freedom 1, i.e. u, = (5; —
1)/\/§, where g, is i.i.d. xii) variates. T = 1000 and total
number of replications = 1000.

 

 

0 = 0.1 6 = 0.55

0 6

g mean RMSE mean RMSE

5 0.0882 0.0807 0.4845 0.3350

10 0.0950 0.0803 0.4680 0.3114

M DE 3 20 0.0979 0.0822 0.4675 0.3073

30 0.1007 0.0839 0.4624 0.3071

40 0.1034 0.0858 0.4585 0.3112

5 0.0702 0.0650 0.4002 0.3572

10 0.0775 0.0612 0.4315 0.2948

M DENw 20 0.0839 0.0601 0.4808 0.2687

30 0.0891 0.0610 0.5016 0.2649

40 0.0932 0.0620 0.5097 0.2603

QMLE 0.1157 0.0949 0.4858 0.2705
0 = 0.1 6 = 0.65 .
61 6

g mean RMSE mean RMSE

5 0.0881 0.0911 0.5606 0.3462

10 0.0980 0.0907 0.5514 0.3127

M DE 3 20 0.1033 0.0912 0.5454 0.3056

30 0.1055 0.0931 0.5443 0.3037

40 0.1079 0.0950 0.5416 0.3045

5 0.0677 0.0715 0.4970 0.3704

10 0.0750 0.0646 0.5274 0.3025

M DENw 20 0.0832 0.0615 0.5693 0.2561

30 0.0882 0.0615 0.5879 0.2452

40 0.0923 0.0621 0.5970 0.2399

QMLE 0.1166 0.0928 0.5816 0.2502

 

Note: The QMLE represents the cases that the simulated data series
are estimated by usual MLE method while assuming normality of 11;.

96

Table 28: Simulated mean and root mean square error (RMSE)
of various forms of MDE and QMLE for GARCH(1,1) model:
y, = (nut, 03 = w +0y,2_1+ 60,2_1, where u, is i.i.d. standardized
chi-square with degree of freedom 2, i.e. u, = (5, -2) / 2, where 51
is i.i.d. X72) variates. T = 1000 and total number of replications
= 1000.

 

 

0 = 0.1 6 = 0.55 .
0 6

g mean RMSE mean RMSE

5 0.0939 0.0764 0.4729 0.3193

10 0.1002 0.0759 0.4633 0.2914

M DEB 20 0.1031 0.0775 0.4572 0.2873

30 0.1058 0.0797 0.4571 0.2904

40 0.1080 0.0818 0.4565 0.2959

5 0.0765 0.0623 0.3845 0.3602

10 0.0842 0.0568 0.3957 0.3011

M DE Nw 20 0.0925 0.0565 0.4613 0.2689

30 0.0988 0.0580 0.4966 0.2539

40 0.1041 0.0594 0.5080 0.2452

QMLE 0.1099 0.0673 0.4691 0.2656
0:015 6:065 .
61 6

g mean RMSE mean RMSE

5 0.1308 0.0988 0.5859 0.2805

10 0.1482 0.0988 0.5720 0.2471

M D133 20 0.1554 0.1058 0.5630 0.2494

30 0.1592 0.1102 0.5629 0.2515

40 0.1623 0.1137 0.5625 0.2539

5 0.1007 0.0832 0.5831 0.2796

10 0.1114 0.0685 0.5689 0.2232

MDENw 20 0.1190 0.0633 0.6121 0.1828

30 0.1239 0.0624 0.6258 0.1774

40 0.1288 0.0642 0.6328 0.1719

QMLE 0.1626 0.0773 0.5998 0.1792

 

Note: The QMLE represents the cases that the simulated data series
are estimated by usual MLE method while assuming normality of u.

97

6141.3 are very close. However, when 0 = 0.2 and 6 = 0.6, the QMLE appears to
have smaller RMSE than the MDE for both 0 and 6.

Table 27 presents simulation results of MDE, where the innovations in the GARCH(1,1)
process is assumed to have a standardized chi-square distribution with degree of free-
dom 1 (fin), i.e. ut = (5, — 1)/\/2, where ’5 is iid X(1)- As given in Evans, Hast-
ings, and Peacock (1993 p. 45), a chi-square variable with degree of freedom 11 has
skewness = 3‘}; and Kurtosis = 3 + 1—3, so that the error distribution of x?” variable
is asymmetric and has the coefficients of skewness and kurtosis equal to 2w/2 and 15,
respectively. The first part of Table 27 presents the results for the case that 0 = 0.1
and 6 = 0.55, while in the second part 0 and 6 are set to 0.1 and 0.65, respectively.
It is found that for both of the cases presented in Table 27 the MDE appears to be
a better estimator than the QMLE. In particular, the MDENW with g=20, 30, or 40
have smaller RMSE than the QMLE.

Very similar results are found in Table 28, where u, is assumed to follow a stan-
dardized fo), i.e. u, = (5, - 2) / x/Z where 5 is iid x222). This error distribution has
variance =4, skewness=2 and kurtosis: 9.’ The x?” and tv=5 variates have the same
first, second and fourth momoents, while the skewness coefficients are different. The
results for 0 = 0.1, 6 = 0.55 and 0 = 0.15, 6 = 0.65 are reported. The MDENW using
either the first 20, 30 or 40 autocorrelations have smaller RMSE than the QMLE. For
the two cases presented in Table 28 the MDE appears to perform better than the
QMLE. In particular, the MDE using the first 30 autocorrelation has lower RMSE

than the QMLE.

98

5. Example: Estimation of the GARCH Model Applied to Hourly Ex-

change Rate Data

In this section we apply the MDE to estimate a hourly deutschmark (DM) vs.
US. dollar exchange rate data from 0:00 am. January 2, 1986 through 11:00 am.
July 15, 1986, which was also analyzed by Baillie and Bollerslev (1991). From Table
III of Baillie and Bollerslev (1991 p. 573), it is found that the DM exchange rate
data series has kurtosis equal to 8.171 and skewness equal to -0.3120, so that the
conditional density of the GARCH model is very unlikely to be Gaussian normal. The
estimation result of the MDE is given in Table 29. It seems that 6M“; and 6mm are
very close, while the difference between 51141.5 and é‘MDE is pretty large. Note that
tZJMDE = (% 23;, yf)(1 -— 0MDE — 6,1103). Many studeis apply the MLE to estimate
the GARCH model of speculative asset return data and find that the autocorrelations
at various lags implied by the ML estimates are not conformable with the sample
autocorrelations. See for example Jacquier, Polson, and Rossi (1994, Figure 1.).
Table 30 reports the sample autocorrelations and values of autocorrelation function
implied by the ML and minimum distance (MD) estimates. The results also indicate
that the values of autocorrelation function implied by the ML estimates appear to
be much higher than the sample autocorrelations of the squared observations. A
similar result is reported by Jacquier, Polson, and Rossi (1994). This result serves to

illustrate that the MLE put different weights on the moments conditions of the MDE.

99

Table 29: Estimation result of MDE and MLE of
GARCH model of hourly exchange rate data

 

MLE:
QMLE C“11141.19: 6MLE
0.0679 0.2291 0.5125
(0.0133) (0.0463) (0.684)
MDE:

0:1an @1403 611405

0.0914 0.1317 0.4885
(0.0205) (0.904)

 

Note: The standard errors are given beneath the param—
eter estimates in parentheses. The total number of obser-
vations is 3189. The number of autocorrelations used in
MDE is equal to 10.

6. Concluding Remarks

This chapter investigates the properties of the minimum distance estimator for
GARCH(1,1) processes. The MDE is applied to estimate the parameters of a GARCH(1,1)
model from the autocorrelations of the squared process which is known to follow an
ARMA(1,1) process, but with non i.i.d. innovations.

As a benchmark, a simulation experiment is carried out to compare the small

sample properties of the MDE using the Bartlett formula to calculate the weighting

100

Table 30: Sample ACF and the autocorrelations implied by the ML and minimum
distance (MD) estimates of the squared hourly exchange rate

 

lags

 

l 2 3 4 5 6 7 8 9 10
SACF 0.143 0.097 0.030 0.054 0.033 0.020 -0.019 -0.015 -0.024 —0.007
ACFMLE 0.283 0.210 0.155 0.115 0.085 0.063 0.047 0.035 0.026 0.019

ACFMDE 0.145 0.090 0.056 0.035 0.021 0.013 0.008 0.005 0.003 0.002

 

Note: SACF represents the sample autocorrelations. ACFMLE denotes the ACF fitted
by ML estimates (amp and amp). ACFMLE is calculated by plugging 0 = 0.2291
and 6 = 0.5125 into equation (37) and (38). Similarly, ACFMDE denotes the ACF fitted
by MD estimates (aMDE and 311403).

matrix (M DEB) and the MDE using the Newey and West covariance matrix estimator
to estimate the weighting matrix (MDENW), when the DGP is the ARMA(1,1)
process with NID innovations. It is found that both M DEB and M DENW perform
quiet well in the case of ARMA(1,1) with NID innovations. On the other hand, for
the GARCH(1,1) process the simulation results show that the RMSE of M DEB is
higher than that of M DENW. This might be caused by the use of a suboptimal
weighting matrix since the innovations in the ARMA(1,1) model for the squared
GARCH observations are not i.i.d.

The relationship between the asymptotic standard deviation of MDE and the
number of autocorrelations used are investigated in a simulation environment. The

results show that as the number of autocorrelations used in the MDE increases, the

101

asymptotic standard deviation of the MDE decreases.

The small sample properties of the MDE using first 5, 10, 20, 30 or 40 autocor-
relations are investigated for the sample size of 1,000 based on 1,000 replications.
The small sample results suggest that if 0 + 6 is close to one, a larger number of
autocorrelations is needed to be used in the MDE.

For the ARMA(1,1) process with NID innovations, the simulation results reveal
a deterioration in_ the quality of the MDE when the number of moment is increased
beyond a certain level. Even thought asymptotic theory suggests that it is optimal
to include as many moments as possible in the estimation procedure, this seems to
be not true for the sample size analyzed here. We document that these results arise
because of a fundamental trade-off between the number of moments used in the MDE
and the precision of the estimated weighting matrix.

Many studies find that the conditional densities of the GARCH models for asset
return series are usually not Gaussian normal and apply the QMLE method to esti-
mate the GARCH models. A comparison of the small sample properties of the MDE
and QMLE for the GARCH model is also provided. Monte Carlo results find that the
MDE can be an attractive alternative to QMLE in terms of bias and RMSE, partic-
ularly for high frequency financial economic series which exhibit extreme departures

from conditional Gaussianity.

APPENDIX

APPENDIX

The autocorrelation function of y? when E‘(y,‘) does not exist

The autocorrelation function of y? under assumption 1 and that the fourth moment
of y, doesn’t exist, i.e. 1702 + 206 + 62 > 1 can also be derived by using the similar
method in Ding and Granger (1996).

Assumption 3 7702 + 206+ 62 > land 0 +6 < 1.
We now briefly discuss the autocorrelation function of the squared GARCH(1,1)

process when the fourth moment of 3;, does not exist. First note that

E(crf) = o‘[1 — (0 + 6)2] + (1702 + 62 + 206)E('af_1)
= a‘[1 -— (0 + m2] [1 + (1702 + 62 + 206) + . - - (1702 + 62 + 206)’E(03)l

= a4 A.

Thus, A = [1 - (0 + 6)""] [1 + (1702 + 62 + 206) + - - - (1702 + 62 + 206)‘E(03)]. Under
assumption 3, 1702 + 206 + 62 > 1, E(o,‘) -) co and A —> 00 if t is very large.

Substituting equation (34) into equation (35) yields

L/

E(01’)
17E (01‘) - 0“

.. _(’7_’ll_(l‘_ll_—1
—cr+6617 61707141)

 

p1 = a+6-6(n-1)

If the process starts at a very long time ago (17A — 1)"1 —-+ 0. Hence, given assumption

1 and 3, the autocorrelation function of y? can be approximated by,
1 k—l
Pk ‘3 (01+;6X01 + 6) 1
which is also given by Ding and Granger (1996).

102

CHAPTER 5

CONCLUSION

This dissertation discusses the properties of the minimum distance estimator for
ARMA processes and GARCH processes. The MDE has the advantage of imposing
very little in terms of distributional assumptions on the innovation process. Under
certain suitable regularity conditions, the MDE is \/T consistent and asymptotically
normal. The MDE of AR(p) process using the first p autocorrelations is asymptoti-
cally equivalent to the MLE under normality. For the MA processes calculations show
that as the number of autocorrelations (9) used in the MDE increases, the asymptotic
variance of the MDE decreases. The MDE appears to be asymptotically as efficient
as MLE under normality if g is large enough. A theoretical justification of this result
is provided for the MA(1) process. Very similar results are found in the numerical
calculations of the asymptotic variance of MDE for the ARMA(p,q) processes and
seasonal ARMA processes. Interestingly, for the MA(1) and ARMA( 1,1) processes, if
the absolute value of the moving average coefficient is not too large, the asymptotic
variance of MDE based on first 3 to 5 autocorrelations is very close to that of MLE
under normality. For the MA(1)-seasonal MA(1), processes, given that the MA pa-
rameter is fixed, the higher the value of the seasonal MA parameter, the higher the
number of autocorrelations is needed to guarantee that MDE is efficient. Calcula-
tion results also reveal that adding the autocorrelations at lags of the multiples of s,
e.g. p,, p2,, p3,, -- o, to the MDE might have a larger improvement in efficiency than
adding other autocorrelations.

Under the assumption that the innovations in the ARMA(p,q) process are iid,

103

104

the Bartlett’s formula can be applied to obtain the feasible estimates of covariance
matrix (C3) of the sample autocorrelation. However, C3 is not a valid estimator
of covariance matrix of the sample autocorrelation for the ARMA(1,1) model of the
squared GARCH(1,1) process. We thus propose a robust covariance matrix estimator
for the variance-covariance matrix of the sample autocorrelations based on Newey
and West’s (1987) method and is denoted as CNW. Two simulation experiments are
conducted to compare the small sample properties of the MDE using CB (M DEB)
and the MDE using CW (MDENW ). The simulation results show that MDENW
and M DEB are quiet comparable to each other when the data generating process
is the ARMA(1,1) process with NID innovations. However, Monte Carlo results
indicate that for the GARCH(1,1) process M DENW performs better than M DEB.
This result is expected since M DEB uses a suboptimal weighting matrix for the’case
of GARCH(1,1) process.

The Quasi-MLE (QMLE) method is usually invoked to estimate the GARCH mod-
els of many speculative asset return series, such as exchange rates and stock prices,
because the assumption of conditional normality for the standardized innovations is
difficult to justify in many empirical applications. Monte Carlo evidence shows that
with certain conditional densities and over certain regions of the parameter space, the
MDE can compare very favorably with Quasi-MLE in terms of parameter estimation
bias and mean squared error. An application of the MDE to estimate a GARCH(1,1)

model from high frequency exchange rate data is provided.

10.

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