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This is to certify that the

dissertation entitled

Modified Cox Tests for Time Series and Panel Data

presented by

Donggeun Kim

has been accepted towards fulﬁllment
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Ph. D Economics

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Modiﬁed Cox Tests for Time Series and Panel Data

By

Donggeun Kim

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Economics

2002

Abstract

Modiﬁed Cox Test for Time Series and Panel Data
By

Donggeun Kim

It has long been one of the main interests among econometricians to test
nonnested models between two different families. But computational difficulties of
the nonnested testing have restricted its application to rather simple linear or non-
linear regression models. This dissertation proposes a new approach based upon
the conditional mean and the conditional variance speciﬁcation in order to solve the
computational difficulties and to extend its application to more complicated cases
including time series and dynamic panel data. The ﬁrst chapter of this disserta-
tion proposes a modiﬁed Cox test under normality, examines its application to two
different nonlinear error equation models with three different time series data sets,
performs Monte Carlo experiments to investigate the potential applicability of our
proposed test. Chapter two extends its applicability under nonnormality and de—
velops a robust modiﬁed Cox test under nonnormality. Chapter three presents its
application to the nonlinear dynamic panel data models with the US. patents and

R&D expenditures data.

To my parents

iii

ACKNOWLEDGEMENTS

During my time at MSU, I have been helped and supported by many good

people. I thank you all.

I would like to thank my committee members, Richard Baillie, Anna-Maria

Herrera, Christine Amsler, and Ernest Betts for their suggestions and support.

I must express my fullest respect to my mentor, Jeffrey M. Wooldridge. This
dissertation could not have started without his insightful guidance, invaluable ad-

vice, constant encouragement, and extreme patience.

I would like to thank Ernest Betts for his support. I have been able to learn

many important things while working for him as a tutorial coordinator.

I would like to thank Korean economics fellow students, especially my peer,
Byungrae Choi and his family for their affection and special friendship. Also, I

would like to thank my pastor Rick Erickson for his steadfast support and prayer.

I must thank my parents for their endless love and unlimited support. With-

out them, I would not be here today.

Even though my dissertation could not have been written without the helps

and support of these special people, any remaining error is of my own responsibility.

iv

Contents

1

A Modiﬁed Cox Test for Dynamic Models of Conditional Means
and Variances 1
1.1 Introduction ............................... 1
1.2 A Modiﬁed Cox Test ........................... 6
1.3 Empirical Application .......................... 21
1.4 Simulation Experiments ......................... 34
1.5 Conclusions ................................ 41
A Robust Version of the Modiﬁed Cox Test 43
2.1 Introduction ................................ 43
2.2 A Robust Modiﬁed Cox Test ....................... 44
2.3 Monte Carlo Experiments ........................ 49

2.4 Empirical Application under Nonnormality ............... 81

2.5 Conclusions ................................ 83

An Application of a Quasi-Modiﬁed Cox Test to Nonlinear Panel

Data Models 85

3.1 Introduction ................................ 85

3.2 Two Competing Count Panel Data Models with the Unobserved Effects 88

3.3 An Empirical Application ........................ 96
3.4 Conclusion ................................. 103
Modiﬁed Cox test 104
Regularity Conditions 108
Bibliography .................................. 110

vi

List of Figures

Figure 1.1 Cox test values of T1 when GARCH(1,1) is true .........

Figure 1.2 Cox test values of T2 when Bilinear( 1,1) is true .........

Figure 2.1 Robust Coxt Test when GARCH(1,1) is true with chi—dist.

Figure 2.2 Nonrobust Coxt Test when GARCH(1,1) is true with chi-dist

Figure 2.3 Robust Coxt Test when Bilinear(1,1) is true with chi-dist . . . .

Figure 2.4 Nonrobust Coxt Test when Bilinear(1,1) is true with chi-dist . .

Figure 2.5 Robust Coxt Test when GARCH(1,1) is true with t-5 dist

Figure 2.6 Nonrobust Coxt Test when GARCH(1,1) is true with t-5 dist . .

Figure 2.7 Robust Coxt Test when Bilinear(1,1) is true with t-5 dist . .

Figure 2.8 Nonrobust Coxt Test when Bilinear(1,1) is true with t-5 dist . .

Figure 2.9 Robust Coxt Test when GARCH(1,1) is true with t-10 dist . . .

Figure 2.10 Nonrobust Coxt Test. when GARCH(1,1) is true with t-lO dist

vii

39

39

67

68

69

70

71

72

73

74

76

Figure 2.11 Robust Coxt Test when Bilinear(1,1) is true with t—10 dist

Figure 2.12 Nonrobust Coxt Test when Bilinear(1,1) is true with t—10 dist .
Figure 2.13 Robust Coxt Test when GARCH(1,1) is true with chi-dist . . .
Figure 2.14 Robust Coxt Test when Bilinear(1,1) is true with chi-dist . . .

Figure 2.15 Robust Coxt Test when Bilinear(1,1) is true with t—5 dist . . .

viii

77

78

79

80

80

List of Tables

1.1 Summary statistics: S&P 500 ...................... 28
1.2 Summary statistics: British pound ................... 28
1.3 Summary statistics: IP .......................... 29
1.4 Estimated GARCH Models: S&P 500 .................. 30
1.5 Estimated GARCH Models: British Pound ............... 30
1.6 Estimated GARCH Models: IP ..................... 30
1.7 Estimated Bilinear Models: S&P 500 .................. 31
1.8 Estimated Bilinear Models: British Pound ............... 32
1.9 Estimated Bilinear Models: IP ...................... 32
1.10 Test resultszHozGARCH vs. H1: Bilinear ................ 33
1.11 Test resultszHozBilinear vs. H1: GARCH ................ 34
1.12 Simulation results when GARCH(1,1) is true ............... 36

ix

1.13

1.14

1.15

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

2.13

2.14

Simulation results when Bilinear(1,1) is true ................ 37

Simulation results when GARCH(1,1) is true ............... 40
Simulation results when Bilinear(1,1) is true ................ 41
Robust. Cox test. results when GARCH(1,1) is true ............ 52
Nonrobust Cox test results when GARCH(1,1) is true ........... 53
Robust Cox test results when Bilinear(1,1) is true ............. 54
Nonrobust Cox test results when Bilinear(1,1) is true ........... 55
Robust Cox test results when GARCH(1,1) is true ............ 56
Nonrobust Cox test results when GARCH(1,1) is true ........... 56
Robust Cox test results when Bilinear(1,1) is true ............. 58
Nonrobust Cox test results when Bilinear(1,1) is true ........... 58
Robust Cox test results when GARCH(1,1) is true ............ 59
Nonrobust Cox test results when GARCH(1,1) is true ........... 60
Robust Cox test results when Bilinear(1,1) is true ............. 60
Nonrobust Cox test results when Bilinear(1,1) is true ........... 61
Robust Cox test results when GARCH(1,1) is true ............ 62
Robust Cox test results when Bilinear(1,1) is true ............. 64

2.15 Robust Cox test results when Bilinear(1,1) is true ............. 65

2.16 Test resultszHozGARCH vs. H1: Bilinear ................ 82
2.17 Test resultszHozBilinear vs. H1: GARCH ................ 82
2.18 Test resultszHozGARCH vs. H1: Bilinear ................ 82
2.19 Test results:H0:Bilinear vs. H1: GARCH ................ 83
3.1 Summary Statistics: the Patents and lnR&D Data ........... 96
3.2 Estimation Results for the Patents Model: Linear Time Trend . . . . 97

3.3 Estimation Results for the Patents Model: Full Set of Year Dummies 98

3.4 Estin'iation Results for the Patents Model: Linear Time Trend . . . . 98

3.5 Estimation Results for the Patents h-‘Iodel: Linear Time Trend Only . 99

3.6 The quasi-modiﬁed Cox Test Results .................. 100
3.7 The quasi-modiﬁed Cox Test Results .................. 100
3.8 The quasi—I'I'iodiﬁed Cox Test Results .................. 101
3.9 The quasi-modiﬁed Cox Test Results .................. 102

xi

Chapter 1

A Modiﬁed Cox Test for Dynamic
Models of Conditional Means and

Variances

1 . 1 Introduction

Since Cox (1961, 1962) devised a speciﬁcation testing based upon a modiﬁcation
of the Neyman-Pearson maximum-likelihood ratio, testing nonnested models has
been one of the main interests among econometricians. However, the application
of the nonnested Cox test has been restricted to rather simple linear or nonlinear

regression models mainly due to its complicated and, in many cases, intractable

1

derivation of the pseudo-true value in the second part of the Cox test. (See, for
example, Pesaran and Deaton (1978), Gourieroux, Monfort, and Trognon (1983),
and Mizon and Richard (1986).] In general, the quasi-maximum likelihood estimate
(QMLE) of a nonlinear model does not have a closed form, so it may not be possible
to obtain the analytical derivation of its pseudo-true value and its ﬁnite sample

estimation of the pseudo-true value in the Cox test.

To avoid these computational difﬁculties some authors developed alternative
approaches. Davidson and Mackinnon (1981) combined the two nonnested models
as an artiﬁcial nesting model and replaced the nuisance parameter under the null
hypothesis with the estimated value. under the alternative hypothesis to avoid the
identification problems. For example, suppose there are two competing speciﬁcations

All and 11/12,

11/11 :yt : mt(Xt,’y) + ut,’where at | Xt ~2'.z'.d(0,02),t:1,-~,T (1.1)

Mg : yt : nt(Xt,6) + vt,where w | Xt ~ i.i.d(0,T2),t : 1,. - - ,T (1.2)
then Davidson and Mackinnon (1981) transformed these two nonnested models as

yt : (1*- /\)mt(Xt,7) + /\,1L1(Xt,6) + whwhere wt I Xt ~ le(0,T]2),t = 1,: ' - ,T
(1.3)

They replaced 6 with an OLS estimate, 6, under Mg instead of the pseudo-true value

of 6 undeer and tested if A20. The DM test can be written as

A

M = mt(Xt,70) + )‘(Mt(Xta 5) — mt(Xt,70)) + wt (1-4)

A

Now we can regard the DM test as an omitted variables test of ( ,ut( X t, 6)—mt(Xt, 7))
in the nonlinear model yt = mﬁX?) +89. If there are nonnormality, heteroscedastic-
ity, or serial correlation, their test statistics becomes invalid. Wooldridge (1990,1991)
suggested a robust version of Davidson and Mackinnon test (DM test) by modify-
ing the misspeciﬁcation indicator of his conditional Mean Encompassing test (CME
test). Under heteroskedasticity, a robust version of DM test is derived by simply set-
ting the misspeciﬁcation indicator A E (pt(Xt, 6) —mt(Xt, 7)) and applying the CME
test procedure. For the weighted nonlinear least squares (WLN S) estimator, a robust
DM test is obtained by setting A E (lit/ﬁt)(,ut(Xt, 6) — mt(Xt, 6)) (See, Wooldridge
(1990)). For possible nonzero correlation between the residuals ét = yt —- m(Xt, ’y)
and a particular weighting of the difference in the estimated regression functions, set
the misspeciﬁcation indicator A E CileglUMXt, 6) — mt(Xt,"y)), where C“ is the
estimated variance function for the model under the null and Cftg is the estimated
variance function for the model under the alternative (See, Wooldridge 1991). On
the other hand, Pesaran and Pesaran (1993) offered another approach to deal with
the computational difficulties of obtaining the pseudo-true value of the Cox test by

a method of stochastic simulation.

Let Hf : f(yt, a | Xt) and Hg : 9(yt,6 | Xt) be the two nonnested competing

models. Then the Cox test (1961,1962) is based upon

Q)

Tf = {Lf(d) — L903» — Edam) — Lg( )} (1.5)
-—- Lf(d)—Lg(8)+0(a.a), (1.6)

where C(&,[§*) = Ed{Lf((3) '" 149(3)}

Lf(d) = T_IZ$=110gf(yt,a | Xt),Lg(/3) =— T‘IZthllogm/aﬁ l Xt) are the
maximized log likelihood functions under H f, Hg respectively, and 001,6...) is the
unconditional expectation of the log likelihood ratio when the null is correctly speci-
ﬁed. To obtain [3... by simulation method, a T x 1 vector of independent observations
of yt is artiﬁcially generated under H f and then the ML estimate of 6 is derived
by using these artiﬁcially generated observations under Hg. This procedure is repli-

cated R times to obtain
(3* = E 2 Bi (1'7)

Then, the same procedure is applied to obtain C(d,6*) by the same simulation

method
R A
C(aﬁQ£321{Lf(yj,é-Lg(yjﬁ*)} (1-8)

Even though Pesaran and Pesaran(1993) argues that these estimators ob-

tained by the simulation method converge to the pseudo-true values consistently and

4

fairly quickly with a relatively small number of replications, this simulated method
is not such a favorable approach to the practitioners. Besides, it is very difﬁcult
to use the original Cox test if the given models contain the lagng dependent vari-
ables: ft(yt,a | art,yt_1,:rt_1, . . .) and gt(yt,ﬁ I rt,yt__1,:1:t_1,...) because poten-
tially very severe con'iputational difﬁculties arise from computing the unconditional
expectation of the differenced log likelihood functions. Bera and Higgins (1997) pre-
sented nonnested Cox test results using the stochastic simulation method proposed
by Pesaran and Pesaran (1993) between two nonlinear equation error models, the
autoregressive conditional heteroscedasticity(ARCH) and the bilinear models with

three time series data sets.

The difﬁculty in applying the original Cox test in time series applications
and possibly dynamic panel data is that it requires computing the unconditional
expectation of the differenced log likelihood functions when the null is correctly
speciﬁed. In this paper, we propose a new approach to solve the computational
difficulties of the Cox statistic by using conditional mean and conditional variance.
Our approach here is to conumte, for each t, the conditional expectation. In some
important applications including ARCH and GARCH models in time series, this
approach leads to substantial simplifications. Another attractive feature of our
approach is that we can test other distributional features because our approach uses

the first two conditional moments while the DM test is for the conditional mean,

E (yt I rt), only. In section 2, we describe our new modiﬁed Cox test procedure;
in section 3, we present an empirical result with three time series data sets; in
section 4, we provide simulation experiments of this modiﬁed Cox test and we draw

conclusions in section 5.

1.2 A Modiﬁed Cox Test

1.2.1 Motivation and General Concepts

Suppose that there are T individually, identically distributed random variables
yt,t : 1, - - - ,T. f (yt,a) is the probability density function under the null hy-
pothesis, H f, and g(yt, ,3) is the probability density function under the alternative
hypothesis, Hg, where (1,6 are unknown parameters, and f (yt,o) and 9(yt, 6) be-
long to separate families. If H f is not nested in H9, and Hg is not nested in H f, then
it is said that the two hypotl‘ieses, H f and Hg, are nonnested with each other. If
one model can account for the results from the other model, then the former is said
to encompass the latter. [see Mizon and Richard (1986), and Hendry and Richard
(1990).] This means that a correctly speciﬁed model can explain the results of its
competing model and the pseudo-true value is the probability limit of the alternative

model under the null hypothesis. Thus, the nonnested test statistic devised by Cox

(1961,1962) is an example of encompassing test. The Cox test, Tf,of H f against Hg

is based on

Tf = {Lf(d') — 139(6)} — EafLﬂd') — L902» (1-9)

A

where L f(ci), Lg(6) are the maximized log likelihood functions under H f and Hg
respectively and c1, 6 are maximum log likelihood estimators. The test statistic is
based upon the difference between the log likelihood ratio and its expected estimate
under the null hypothesis, Hf. If E5,{Lf(ci) —Lg(6)} : 0, then the Cox test statistic
is just simpliﬁed to the form of log likelihood ratio statistic, but, in general, this
term is nonzero under nonnested hypotheses. So the Cox test takes the deviation
between the maximum log likelihood ratio and its expected value under the null
hypothesis. Under the correctly specified null hypothesis, Tf should be close to zero
while a large deviation from zero constitutes evidence against the null hypothesis.
The standardized Cox test statistic, \/T £1175, where Vf is a consistent estimator of
f
the asymptotic: variance of Tf, is asymptotically distributed as unit normal. White
(1982) provided general regularity conditions and the asymptotic normality of the

Cox test statistic.

Despite its theoretically refined feature, the derivation of pseudo-true value

of E(-,{Lf(d) — Lg(6)} is not straightforward, and even analytically intractable.

 

To solve these computational difﬁculties, we offer a new approach based upon the

conditional mean and conditional variance method.

What makes difﬁcult to apply the Cox test is that it requires computing
the unconditional expectation of the differenced log likelihood functions that is not
significantly ai‘ialytical or tractable in many cases. The observations are assumed
independent in case of Cox (1960,1961) and it reduces the computational difﬁcul-
ties in some degree but it still requires signiﬁcant computational effort. Besides,
for this reason, it becomes very challenging to apply the original Cox test to time
series applications that ccmtain the lagged dependant variables as the explanatory
variables. But these difﬁculties can be avoided by computing the conditional expec-
tation, for each t. And this leads to substantial simplifications in some important
applications including ARCH and GARCH models. White (1994) showed the com-
putational simpliﬁcation of the second part of the Cox test using conditional densi-
ties ft and gt given It_1 where It_1 is the information set (a-algebra generated by

{a}, yt-1,;rt_1, - - }) available at time t.

Let Hf : ft(:rt, a) and Hg : g(:rt, 6) be the two nonnested competing models,

then their maximized log likelihood functions are, respectively,

. 1 T .
Ln(o') : T 2 log ft(-l?t, a) (1.10)
t=l
. 1 T -
[471(5) : ‘7: Z 10% {Ida/1,5) (1-11)

W

00

and let f : 3?” x a —> §R+ and g : if?” x 6 —> §R+ be conditional densities and d and
6 be the QMLEs under H f and Hg respectively. An estimate of the expected value

of the average log likelihood ratio when the null is correctly speciﬁed is

- - 1 - . .
E}:[Lf,, — Lyn] E /—(log f"(.1:" ,0”) — log 9 "(1' ",7n))f"(1:",an)dvn(r") (1.12)

TI,

: /(n 1:] log ft(£L‘ t,an) - 10g 9;:(13 3sz )) )H ft( (ft 011))dvf’14’3)

t 1

= n” Zl/(1()gft(+l7t,dn)—10g9t<$t,61’z))ft($ti51161021391014)
t=1

where f ”

Ill
7.1::
he

The ut- fold integration in the equation(1. 14)c causes the severe difﬁculties of comput-
ing the unconditional expectation. It is assumed that the observations are indepen-
dent in the case of Cox (1960,1961), so we can reduce the integral above equation

as 'U-fold integral

= % ZI/(logfdl'uél —10g9t($t,6))ft(1‘t,éldvdl‘tll (115)
£21

White (1994) argues that it still requires computational effort, even though this
is much more tractable than before. In addition, the analytical intractability still
remains when we apply Cox test to time series applications that include the lagged
dependent variables as explanatory varial:)les. To avoid these difﬁculties, we suggest
an approach using the conditional densities of ft and gt given It_1. By computing
the conditional expectation for each t, we can achieve some substantial simpliﬁca-

tions of the Cox test in some important applications. Now we can rewrite the second

9

part of Cox test as

D:
D
'e'
K.”
8;
I
h
‘2
3)

ll
'ﬂlH
Me

[/(103‘ ft(l't,d I It—I) — 10g9t(l't,6 l It—1))ft($tad l It—lld’Ut(17f:1116)

@0-
||
t—l

II
’ﬂlt‘
Mrs

[Eftlbgfdl'tﬁ l It—l) —10g9t(1’t,6 l It—l) l It—ll 0-”)

(s.
H
p—I

Equation (1.17) is the conditional expectation of the differenced log likelihood func—

tions.

1.2.2 A Modiﬁed Cox Test

Let (gt, Zt; t = 1, ~ - - ,T} be a sequence of observable random vectors with yt 1 x J,
and Zt 1 X K ;yt is the vector of endogenous variables, and Zt is the vector of
explanatory variables. We assume the regularity conditions in White (1982) held.
Suppose the two competing nonnested parametric models ft under the null and gt

under the alternative respectively, then
Mli MIN | 6—1390), 90 E 9 Q 3119,15 = 1,2,°°'T (1-18)
1112: gt(yt |1t_1,(50), 60 E A Q 3iq,t : 1,2, - - ~T (1.19)
where It_1 is the information set (o-algebra generated by {yt_1, Zt, - - - , Z1}) avail—
able at time t.

lf6 is VT-consistent estimator of 60 under the null hypothesis when 60 is a

10

true value of 6, and 6 is \/_-c0nsistent estimator of 60 under the alternative hypoth-
esis when 60 is a true value of 6, then x/T(6 — 60) and x/T(6 — 60) are distributed
as asymptotic normal. The null hypothesis is that A! 1 is correctly speciﬁed and the
alternative hypothesis is that .Mg is correctly speciﬁed. Now we write the modiﬁed

Cox test as

T
TM, : T—IZUngtf’yt l It—1;90)—1089t(yt lIt—1§6*)}
t=1

T
“T—1:{EMl(logft(3/t | It—1;90) —10g9t(yt | It—1;5*) | It-llGl-20)
t=1

where 6* : plimri when M 1 is true. It is important to note that 6* ¢ 60 in general.
60 and 60 are the true parameters which are unknown, so we replace them with
6 and 6; the ML estimators of 60, and 60 respectively. This modiﬁed Cox test is
composed of two parts, the averaged log likelihood ratio of the null hypothesis to
the alternative hypothesis and its expected value under the null hypothesis. Now

we analyze these two terms one after another.

i) The ﬁrst term: Let the log likelihood functions of log ft(yt | It_1;6) and

I"

10s 9(yt l It—1;6) be

 

 

- 1 1 - 1 z —mt6 2
logft(yt|1t_1;6’) : —§log27r-§loght(6)——2—(Jt h (6; )) (1.21)
t
- 1 1 . 1 yt—uti 2
waterless) = ~§10827T’§10g77t(5)—'2‘( 72(6)) (1.22)
t

11

Plug these two log likelihood functions into the ﬁrst part of the Cox test and

we can rewrite this term as

T
Z{108ft(ytIIt—1;9)—10391(ytIIt—1;5)}

i=1
T T (311 — mt(9))2 (yt — #10962}
: —— 10 h( 1 — —- - — - .23
2{ g M 097” 9)} 9.26 11(6) 721(6) (1 )

 

 

ii) The second term: The conditional expectation of the log likelihood ratio

0f10gftIytI1t— 1; HO 10g9(ytI1t 1; 9N5

T
Z [EMIUngd yt I It— 1; 9)— 10g9t(yt I It—1;5) I [if—1}]
t=l

3"

. T 6) .
t 9 +12: mt(9 MM) (124)
t( (9) 2t=1 76(9)

 

 

NI’S

T 1 T
—log77t(6) — —2-loght(6) —+-2-Z
i=1

:2

i

Now we combine these two terms together and rewrite the modiﬁed Cox test

 

 

d _ 1 T_1T (gt—7711(9))2 (t/t-#t(9))2 (11(9)
TM 5‘72 — -

 

 

 

 

t=1 ht(9) 721(9) 771(9)
T ‘ 2 ‘
: T—l { _ 6 ( l—A/itw) 111(6) —ht(9)
g (Ut t( )) WM) 2
1 1 _
x (7171—61 _ W6} “'26)

A more detailed derivation of this result is given in the appendix.

The equation (1.26) above is the modiﬁed Cox test statistic when we assume

M 1 is correctly Speciﬁed. If we exchange the role of the hypotheses, i.e. Mg becomes

12

the null hypothesis and [W 1 becomes the alternative hypothesis, then under Hg, this

modiﬁed Cox test has a different form as

TMQ :- T_1{10g9t(yt I I11—1;<50I ‘10gftfyt I 11—5917}

T
71-1}: EMQUOggtI'l/t I I11—1;(50I — log ft(yt I It—1§9*I I It—lX}-27I
i=1

 

lit(90I — mt(9*I €t(50I2 — Tlt(50I
_/J't ()60 I ht(6*) + 2

 

 

T
:T’M:

t=1
1 1
XI’HWI _ nt(50)>I (128)

Now we consider the asymptotic distribution of the modiﬁed Cox test under

 

 

the null hypothesis that Ail is correctly speciﬁed. Since we do not know the true
values of parameters, 60 and 6*, we use the consistent estimates, 6 and 6 instead,

so the test statistic is based upon

 

T . -
T ,1 : T_1mt(6)){mt(0)-M(6)}
M gm:

 

 

1 T)t(9I
+UtI9I2—(lt(9_71ntyl(I 1 __i_\)] 129
21(9) 111(9) (1 )

We can expand T111 by the mean-value theorem as

 

 

 

 

 

 

T *
_ —1 mt(90I — Ht(9 I Ut(90I ht(90I
— T g (yt .- mt(60){ 7]t(6*) } 2
p T — 77 F — 6
x (W959 — h't(190I)I “LT-1;; 939;. IQ” — mt(6))I ”(6669“ )}
'ut(6—)2 — ht(6) 1 1 .. .
+ 2 (77t(9I — 1115(9) ] (6 —— 60) (1.30)

where 6 and 6 lie on the segment between (6, 60) and (6, 6*).

Now we 111111tiply x/T on both sides

 

 

T * 9
—1/2 1 _ m ’mt(90I - M65 I} Ut(90I‘ * ht(90I
T t; [(Jt t(90I{ ”“6,” +

x 1 __1__ -1? T—a—II _ _ {mt(6)—m(6)}
(WW) M90) +T Z .(Jt mt(9II —

1t
+ut(9I‘ *llt(9I ( 1‘ _ 1 )] ﬁ(é_90) (1,31)

 

 

 

 

 

2

 

 

T_1/2 i [m _ WU, {mm — mm} + Ut(90I2 — ht(00)

 

 

:21 mm 2
x 1 _ 1 -1T [ _m {mime—mun}
(77th?) ht(90I) +T Eve (gt t(90II 7}t(6*)

 

 

 

ut(60I2—ht(60)( 1 _ 1 )J A-
+ 2 7746*) moo) WW 90) (1.32)

under the null hypothesis and (6 —-> 60) and (6 —> 6*) by the mean-value property

and V9 is the gradient operator.

 

 

 

 

 

 

 

Define
we 6*) = T‘liv [o —mt<60>){7”‘(60)““(6*)}
‘0’ “ ,2, 9 ‘ 7746*)
+Ut(9OI2—ht(90I< 1 _ 1 N
2 77t(5*I ht(90I
Then
T ‘ ‘ , *2 “
T—1/2 , _m 0* {7711(9Iil1'th} “4(9) -ht(9I( 1, _ 1, )J
1; M t( I) 71M) + 2 WW him)
T *
__ _1/2 , —m "IIIOOI—Htw I}
T g [(3% t<00){ 7/t(6*)

14

 

2 ‘VIt(5*I htI90I

—\1:t(60,5*)\/T(é — 60) —”—> o (1.33)

+Ut(90I2—ht(90I( 1 _ 1 )J

The asymptotic distribution of \/TTMl is equivalent to the asymptotic distribution

m 6 (5 l 6 2—h 6 ——5
off[yt—mt(00>{ “#21553“ M} m (1)2 i<o)(m(16*)_ht(160)]+

 

 

1

 

 

 

we), (mm—60> and mé-eo) = (—%z$:1At<oo>)” 71—? 2;; v9 logfet, 60>.
_ m 6) (6 u 6 2—h. 6 1
Note that f Zthl ((3/1 — mt(60){ tI1))1I(6‘I)I(D}+ 1C OI 2 z( 0) (7”(16‘) _ h1(60))I

-1
—\Ilt(60, 6*) (E [71~ZT:1A1(60)]) —\/1T 231:1 V9 log f(yt, 60) is martingale difference

sequence random variable with mean zero and variance, V(60, 6*), under the null

PG — W60, )) (1
t_.1
Define

= u mtIgOI’/lt(5*I ’ut(90I2—ht(60)( 1 — 1 )I
Dt _ ItI90)( 7W“) I+ 2 7246*) Mao)

Atwo) _ 82 1 meow/00186}

,; * Z mt(9()I-Ht(5*I ui(60)2—ht(00)( 1 _ 1 II
”(009) ‘ V9I"(60)< 7M5“) I+ 2 7M5”) MW)

1 TT 1
Therefore, \/TTMl ~ N((),V(60,6*)) and _v_"1~a N(0, 1) if V is a con-

hy1')othesis, where V (60, 6*) is
2

T —1
221M910) V9 101% f (yt,(903}4)

:EMH

T
V(60,6*) = T’1 ZD
t:1

"l I

 

 

 

 

 

 

sistent estimator of V(60,6*). Under the regularity conditions, it is easily shown

that
2

A 1 T A A 1 T A _1 A
Dt — (:7 Z M9,”) (f 2 At(9I) V9 log ft(9I (1-35I
1:1 1:1

15

A T
VT : T4}:

i=1

 

Thus, T — V(60, 6*)

 

 

1 T A 1 T A A 1 T —1 A
:: T; Dt — (That/466)) (TE/12(6)) Valogfdm
1 1 T 1 T A 2
_T Z 02 _ (T Z w(00,6*)) (if: 2 242090)) V0108 f(yt,90)
2:1 2:1 2:1
1, 0 (1.36)

Under the null hypothesis, the statistic of the modiﬁed Cox test is asymp-

totically normally distributed with mean zero and variance, VMl- Thus, the stan—
. . . ﬁTAI . . . . .

dardized modified Cox test, 77%, IS asymptot1cally distributed as unit normal

Ml

N(0, 1) under the null hypothesis.

We now consider a time series a lication as an exam le. Su ose t t =
’
1, 2, - - - , T is a sequence of 22'. d observable random variables. Two competing models

are given as
M1 3’62 = mt(90) + 112, WWW Ut ~ N(0,ht(90)), (1-37)

E(:Ut | 12—1) 2 7712(90), (1'38)

Va'r(y, | It_1) : ht(60), where (21(60): 020 + al'ut2-1-~ARCH(111.39)

and A212 : yt : [12(50) + at, (1.40)
ELI/2 | 12—1) : #2610) + ()1152—152—1 (1-41)
Va7‘(yt I It—l) = 0?, (1.42)

where 52 = 19152—152—1 + £2, and it ~ Nata?)

16

- - Bilinear model

Under the null hypothesis that All is correctly specified, we can write the

modified Cox test as

 

 

 

T . “ ‘ 2 ” -* “ 2
. _ ~ mt(6 — 11¢ 6) 11(9) — ((10 + al-u__ )
T1111 : T 12 [112(9) ).2 ( + 2 2 2 1
2:1 Ge
1 1
X“? _ . - 2 )J (1.43)
(76 00 + 01111-1

Define

"22(90) - 212(6’“)

0...

 

b
N
H

”I
\D

?

and

.9
III

1
0—2 (12(90)’

 

Ut(90)2 — h2(90)

 

 

 

02 E 11-2(90)D21 + 2 022
- , 62—h é .
then ﬁTAIl : 1/2tzl{ut(6)Dt1+ Ut( ) 2 t( )Dt2} (l. 44)
292 — h (9
Therefore, T_1/2 Z (Uta?) )Dtl + ut< ) 2 ( )D22}
t=T1

 

6 —} 6
_ T—1/2 Z {112(90) 021 + U2( 0) 2 It( 100”}
2: 1

T
+ (l 2,1,,(90,5*))\/f(é— 019—73 0 (1.45)

11460)? — h2(90)
2

 

T T
where T—1 : wt(60,6*) : T—1 : V9 [221(60)Dt1 + 022]

t=l tzl

. 1 T —1 1 T
and 6776—00) = -( ZAt(90)> it: 010gf(I/2 90)

The asymptotic distributions of the modified Cox test are as follows

T
ﬁTJ)[1 71—1/2:
t=1
X V0 10% f(y2, 90)]
1T 1T
Let (12 D2— —Z'U’22(0025*) —ZA2(0
Th1 n:
Then,
T T
1 1
"_ (It : “—
m2, 211:1
1T"
: —— Dt— 11111 — 22% (6,6
ﬁg pT—ongt 0
V0105f62190)l+0p(1)
1i,
: — (1,
Vﬂﬂt
where
* : D 1 _
(1‘ t (pTI—IfOT
T
d l — (6 :- E 2 6 ,6*
an p TinéoTtZZl:t( 0,6 ) [Hf 0 )l
pTlgréo—11:LZIA2()90 : ElAt(90)l
Therefore,
Efollt—l) — 0

E(q{2I12_1)

 

 

1T 1T 4
0—21 — (er ,(._ A6
2 pﬂﬂTgwm,)leT§tM)

T—->oo

6%

—1
0)) V9102; f(‘y2,90)

P 1T 1T 4
D2 - (T 2: 2/2’2(90,5*)) (f Z A2(90)> V0108 ffytfpo‘) )
2:1 2:1

1T 4
*O@ﬁgf;m%ﬂ

um

(1.49)

XWW%2)

—1
x<pT limoO — T: At( (60) > X VQIng(3/ta‘90)

18

(1.50)

E [(02 — (E [Warm (E [Atwonrl

XVe 2022222220»? I 22-1] (1.51)

 

= V1.21(60,6*) (162)
Therefore,
1 1 T 2 1 ~ 2 1 T ‘1 1
V1111 = .— 2 D2 — 202 — 2 WWW - Z At(9) V010gf(y2,9)
T1521 T t=1 T t=1
1 T , 1 T A T1 - 1
+ — Z WW5) — 2 MO) Valogf(yt,9)v2910gf(y2,9)’
T t=l T t=1
1 T ‘1 1 T ’
— 2 242(6)) (— 2 1114625)) (153)
(T 2:1 T 2:1
_. __ ,, _.. . x/TTm.
The mod1ﬁed Cox test stat1st1c under the conditional mean and var1ance,—W2—l, 1s
Ml

standard normal, N (0, 1).
0 Proposition
Assume that the following conditions are satisfied under the null hypothesis,

1. Regularity conditions1 hold [see White (1982).]
2. T1/2(é — 220) —2 0pm) and T1/2(5 — 50) —> 0,,(1)

3. Conditional mean and conditional variance exist and are finite.

1 2222(6) - 222(5)

 

T
Then, T1141 2 T—IZ

 

 

 

2:1 (222 — mt( )) 772(5)
2222(6)2 — (12(6) ( 1 _ 1 >] 154
+ 2 222(5) M9) ( I )

 

1The regularity conditions are given in the appendix.

19

and the standardized Cox test statistlc, 77%{1 IS asymptotically dlstrlbuted as umt
MI

1

normal, N (0, 1), where 17M, is the consistent asymptotic variance of x/TTMI.

Note that the equation (1.54) is a function of 6 and 6, the ML estimators
of 60 and (50 respectively. Comparing to the Cox test (1961,1962) and the sim-
ulation method by Pesaran and Pesaran (1993), the modiﬁed Cox test does not
require pseudo-true parameters or estimators from artiﬁcially generated data. This
approach, based upon conditional mean and conditional variance specifications, is a
more convenient method for a computational purpose. Now we apply this proposi-

tion as follows;

0 Procedure 1.1

1. Obtain 6 and 6, the ML estimators of 60 and 6*, save residuals, ut(6), and
the conditional variance, ht(6) from the log likelihood function log f (yt |

It_1;6) and et(6), and 71(6) from the log likelihood function log gt(yt |

A

It—l;5)-

A A

2. Compute D21, D22, '¢’t(62(§)a 311d V0102; 6(6) Deﬁne D“ 5 MW’
‘ t

D” E Eta—lull”, and 7122211 #226925) 5 i533; —ngntbtl — 26111522 2

 

212 — In

3. Compute \/TTM1 : T"1/2 21:1 [2221521 + 472—122 —(71-ZtT:12/1(6,5))
x<7l~th=1A(é))-1vglog 22(2)] and
- - . 22. 22 . ~ ~ » _
VM, = $2:le [0.2122(6) + #032 + 2% 2;; 2222, 6))(+ 2;; A26» 1

20

XVe log wave log 222622722211 A<é>r1<71~ 11229222

A

—2(71~Z;I:164625))(71-2211A(6))_1(D21V97222(6)+ Tax—Vmw»

4ht(0)
and use the standardized Cox test statistic, —.,1/—‘1,li, as asymptotic unit
Ml

normal under the null hypothesis.

1.3 Empirical Application

Bera and Higgins (1997) took generalized autoregressive conditional heteroscedastic-
ity (GARCH) by Bollerslev (1986) and bilinearity by Granger and Anderson (1978)
as two competing models for nonlinear dependence in time series data and showed
the nonnested Cox test results using a stochastically simulated method by Pesaran
and Pesaran (1993) with three time series data sets; S&P 500 stock index, the daily
pound/ dollar exchange rate, and the rate of growth of the monthly U.S. index of
industrial production. In this section we compare our modiﬁed Cox test results to

those results from Bera and Higgins (1997).

1.3.1 GARCH and Bilinearity

Forecasting as well as estimating a model are very substantial components in econo-
metrics. These components play a very important role in the analysis of time series
data. If a series is assumed as a white noise (this is very common assumption in

21

econometrics), the process is independent of its own past and it becomes very difﬁ-
cult to forecast this series because we cannot get any information from its own past.
But most of the financial and macroeconomic data in time series shows evidence of
a dependency upon the past. Granger and Anderson (1978) suggested that a white
noise process could be forecastable from its own past in a nonlinear manner and
introduced a bilinear process that allowed dependence 011 the past realization of the
series.

Suppose Xt : 65t_1Xt_1 + 5;, where Q is white noise with mean zero and
variance 0? and X t and 51 are uncorrelated. The conditional mean Xt is 6Xt_1€t_1

while the unconditional mean is zero because

E(Xt|12—1) : 6X2--1€2—1+E(52|12-—1) (1-55)
I 2'3X2—152—1 (1-56)
while E(Xt) -— E(,6X,~-.15)_1+€t) (1.57)
: 2’3E(X2—1)E(€t—1)+E(€t) (1-58)
—. 0 (1.59)

where It_1 is an information set (o—algebra) available at time t. Next, the condi-

2
. . . . . . . . 0'
tional variance of yt is a? while the unconditional variance lS £73523? because
6

VaT(Xt | 12-1) : 1112(5) | 12—1) (1-60)

(“‘1 M

(1.61)

22

while Var(Xt) = 62Var(Xt)Var(5t_1)+Var(5t) (1.62)

= 62Var(Xt)0§ + 052 (1.63)
27?
Therefore, Var(Xt) 1————3—2——9 (1.64)
_ ‘2 0-5

If 620? < 1, then this process is non-explosive and becomes stationary. So 23%? < 1

is a very important condition for stationarity.

Engle (1982) further developed the idea of nonlinearity in his model, the
autoregressive conditional heteroscedasticity (ARCH), which is very close to the

bilinear process. Suppose y) : X;,13+ (it, then yt ~ N(Xt’6,ht) where ht 2

h(5t_1, - - - ,et-p, a). The conditional mean and the unconditional mean are both
Xt’ﬁz

E(’yt I 12—1) = Xiﬁ + E(€t l 12—1) (1265)

: Xt’ﬁ (1.66)

and E(yt) : E(X£ﬁ + at) (1.67)

: X£6+E(et) (1.68)

: X,’,B (1.69)

The conditional variance is ht : 020 + 02152-1 + - - - + apet_p but the unconditional

variance is 03;

VOW/t l 12—1) 2 Va"‘(€t I 12—1) (1-70)

23

: ht (1.71)

and Var(yt) = E(€t2) (1.72)

('1 to

(1.73)

N ote that the conditional variance, ht, contains the current and lagged values
of independent variables through information set available at time t because 5t :

gt — Xéﬁ. Thus, we can decompose the ht as follows2

ht : h{(5t_1,€t_2,'",Ef-p,O’,Xt,Xt_1,'",Xt_p) (174)

: ht(€t—115t—27 ' ' ° igt—p) O)ht(Xt,Xt_1, ' ° ' ,Xt—p) (1'75)

Bollerslev (1986) extended the ARCH process to the generalized autoregres-
sive conditional heteroscedasticity (GARCH) process allowing for a longer memory
and a more ﬂexible lag structure. The GARCH(p,q) process includes the lagged con—
ditional variances as well as the linear function of past variances of the ARCH(q)
process so, it corresponds to and forecasts from its own past in an adaptive expec-
tation fashion . Suppose y) = X {,6 + at where y) is the dependant variable, Xt is a
vector of independent variables, and ,6 is a vector of unknown parameters, then the

GARCH(p,q) process is given as
2See p.3 on ARCH selected reading, Engle, 1995

 

24

52 | 12—1 N N(0,ht) (1-76)
q I?

where ht :- 020 + Z a,-2:,2_,- + 2 61h“, (1.77)
i=1 i=1

and p20, q>0,

Therefore, yt ~ N(Xt’6,ht) where Var(yt I It_1) = ht, while Var(yt) =

Var(€t) = 0?.

The bilinear process and the GARCH(p,q) process as well as the ARCH(q)
process have forms of nonlinearity and provide more information for forecastability
from their own past realization. Although it is hard to ﬁnd the true speciﬁcation
between the bilinear and the GARCH processes due to the similarity between them,
there are some remarkable differences between these two processes. The main and
fundamental difference between the bilinear process and the GARCH (or ARCH)
process is the conditional moments condition. The conditional distributions of a
dependant variable between these two processes are pretty distinguishable. Suppose
a dependant variable yt is generated by yt : X56 + at where 22) is a stochastic error.

Under the null hypothesis, at is speciﬁed as

25

.Ml :ut I It_1 ~ N(O,ht) (1.78)

where ht = 00 + (1121,24 + 1’31ht_1 - - -GARCH(1, 1) process (1.79)

and under the alternative hypothesis, at is speciﬁed as

612 : 'Ut : bllut—lgt—l + Q (1.80)

where 52 ~ N(0,o§)~-Bilmear process (1.81)

In the GARCH(1,1) model, E(yt l It_1) : Xt’ﬁ and Var(yt | It_1) = ht and

in the bilinear model, E(yt | It_1) = Xgﬁ + bllut—lft—l and Var(yt | It_1) = 03.

The conditional mean of bilinearity shows that the bilinear process does augment the

adaptive information between its past errors and innovations in a nonlinear manner

while the conditional variance of the bilinear model is constant. This nonlinearity

in the conditional mean of the bilinear model may increase the forecastability of the

dependant variable while the GARCH(1,1) process does not bring any augmented

information from its own past and innovation from the unconditional or conditional

mean. On the other hand, the conditional variance of the GARCH(1,1) process

provides augmented adaptive information from its own past realization while the

conditional variance of the bilinear process is constant. Although the conditional

26

distributions between the bilinear process and the GARCH process are fundamen—
tally different, the unconditional distributions between these two are very similar,
as shown earlier. Due to the nonlinearity and similar unconditional distributions
between the bilinear process and the GARCH process, it is more difﬁcult to ﬁnd
the true speciﬁcation. In the next section we do the modiﬁed nonnested Cox test

between these two nonlinear speciﬁcations with three time series data sets.

1.3.2 Empirical Application

In this application, we consider three time series data sets: the daily percentage
changes of the S&P 500 stock index, the daily log price changes of the British
pound in terms of the US. dollar (£/$), and the annualized growth rate of the US.
monthly index of industrial production (IP). Note that the ﬁrst two data sets are
high frequency ﬁnancial time series and the third data set is a non-financial time
series. These three data sets are the same ones that Bera and Higgins (1997) used.3

We consider that the stochastic error equation follows the nonlinearity and

specify the GARCH model as the null hypothesis and the bilinear model as the

alternative hypothesis. The exogenous variables are considered as autoregressive

 

3They retained the last 10 per cent of the observations to compute root mean squared errors
for the one-step-ahead forecastability from each of models and we used the same data samples as

they did for nonnested test between GARCH and bilinear models.

27

Table 1.1: Summary statistics: S&P 500

 

 

Mean s.d. Skew Kurt Max Min sample size

 

Bera&Higgins .060 .820 -.651 8.759 3.468 -5.877 1138

 

Kim .042 .925 —.711 8.796 3.455 -7.008 1138

 

 

Table 1.2: Summary statistics: British pound

 

 

Mean s.d. Skew Kurt Max Min sample size

 

Bera&Higgins -.023 .477 .032 4.758 1.959 -2.252 1210

 

Kim .0260 .692 -.202 4.632 2.990 -2.784 1210

 

 

AR models4. Now the model s )ecifications are iven as
I

All : yt : X;3 + ut
at I It_1 ~ i.i.d(0, ht)
where ht = 00 + 012234 + 6ht_1
A1223” : Xéﬁ + at
where at = (2112224524 + at,

and 5t ~ i.i.d(0,o§)

(1.82)
(1.83)
(1.84)
(1.85)
(1.86)

(1.87)

First, we take the daily S&P 500 stock index (SP) from January 4, 1978 to

May 28, 1993 and compare our statistics summary to that of Bera and Higgins ( 1997)

 

4In modeling of exogenous variables, we take autoregressive (AR) models and the order of the

autoregression following Bera and Higgins (1997).

28

Table 1.3: Summary statistics: IP

 

 

 

Mean s.d. Skew Kurt Max Min sample size
Bera&Higgins 3.357 10.604 - .645 5.653 37.699 -51.732 359
Kim 2.728 8.823 -.623 5.649 33.483 -42.364 359

 

 

in Table 1.1. Next, we take the daily log exchange rate of the British pound to the
US. dollar (fl/SB) in a sample period from December 12, 1985 to February 28, 1991
and present the statistics results in Table 1.2. As Bera and Higgins (1997) considered
in their paper, we also take the annualized growth rate of the US. monthly index
of industrial production (1P), a non-financial time series data set, from January,
1960 to March, 1993 for the third empirical application and present the summary
statistics in Table 1.3. Note that the summary statistics between Bera and Higgins
(1997) and our ﬁndings given in Table 1.1 to 1.3, are similar but not exactly the
same, even though we used the same data sets with the same sample periods that
Bera and Higgins (1997) considered. There are a couple of things to be noted from
the summary statistics. First, as given in Table 1.1 through 1.3, all the series are
of high kurtosis, especially S&P 500 stock index series. Another is that we have
different signs of the mean values in the British Pound series; -0.023 of Bera &

Higgins and 0.026 of us.

Table 1.4 through 1.6 present the estimation results of the GARCH(1,1)

model. Again our estimation results, using the British Pound series, reveal the

29

 

 

 

Table 1.4: Estimated GARCH Models: S&P 500
Bera & Higgins Kim
yt=.052 +.066yt_1+ at 21122041 +.010yt_1+ at
(025) (031) (023) (030)
22):.011 +.013u§_1+ .96822.._1 22):.013 +.01222§_1+ .97222._1
(.006) (.005) (.013) (.004) (.004) (.006)
2(6): -1367.67 2(6): -1511091

 

Table 1.5: Estimated GARCH Models: British Pound

 

 

Bera & Higgins Kim
yt:-.024 +222 yt=.032 +ut
(.014) (.018)
26 : 010 +4059u§_1+- .897h._1 ht: 017 +n065u§_1+- .902ht_1
(.004) (.002) (.017) (.009) (.022) (.036)
2(6): —785.72 2(6): -1231.208

 

Table 1.6: Estimated GARCH Models: IP

 

 

Bera & Higgins Kim
3).:268 +.2793,2,_l +.114y._2+u. y.:2.135 +2702).1 +.122y2_2+u.
(.303) (.033) (.013) (.766) (.100) (.054)
25::6024 5235261 +1012)“1 hf:47114. 22332261 + 040h2_1
(5.42) (.034) (.021) (35.128) (.116) (.524)
2(6): -1301327 2(6): -1247462

 

30

Table 1.7: Estimated Bilinear Models: S&P 500

 

 

Bera & Higgins Kim
yt=.017 +.102yt_1 + 222 yt:—.0004 +.045yt_1 + 222
(.030) (.017) (.029) (.031)
at: .053ut_1€t_1+€t 'ut 2-047Ut—152—1
(.017) (.011)
2( “): -1368.884 2(6): 4517.299
63: .651 63: .844

 

difference in sign; -0.024 of Bera 82: Higgins vs. 0.032 of our estimation in Table 1.5.
Table 1.6 shows that the two estimations results are very close and the GARCH
effects in the IP series are rather small in both estimations (0.101 from Bera and
Higgins vs. 0.040 from out estimation) compared to the two other GARCH effects

in S & P and £/$ data sets.

Table 1.7 through 1.9 present the estimation results of the bilinear model and
there are some signiﬁcant differences between Bera & Higgins and our estimation
results. First, the bilinear effects in the British Pound series are different in sign;
(0.039, 0.083) from Bera & Higgins vs. (-0.016, 0.025) from our estimation. Second,
the estimated variances of the bilinear model(65) are also different between Bera &

Higgins and our results.

Table 1.10 and 1.11 present the modiﬁed Cox test results. Table 1.10 reports
the test results when the GARCH model is the null hypothesis and Table 1.11

reports the test results when the bilinear model is the null hypothesis. Note that

31

Table 1.8: Estimated Bilinear Models: British Pound

 

Bera & Higgins Kim
3112.024 +222 yt:.034 +ut
(.020) (.022)

u,:.039'ut_1€t_1
(.021)

l( )=

63

)

20832224524 + 52
(.024)

—819.62

.226

Ut:-.016U.(_1€t_1

(.031)

+.025ut_262_1 + 52
(040)

-1271.357

.478

Table 1.9: Estimated Bilinear Models: IP

 

 

Bera & Higgins Kim
y.:2.34+ .32iy._.+ .125y,_2 + u. y,:2.057+ .30792-1+ 13322-2 + u.
(.634) (.053) (.030) (.566) (.069) (.055)
12,: -.006'u¢_1€¢_1 +52 22,: -.008ut_15,_1 +52
(.003) (.005)
2( °): -131115 2( ‘): 4255.231
63: 90.69 63: 63.651

 

32

Table 1.10: Test resultszHozGARCH vs. H1: Bilinear
Bera & Higgins modified Cox test

 

 

 

S&P 500 .023 .322
British Pound .196 -.033
Industrial Production .533 .021

 

the absolute values of our test results are bigger than those of Bera 82 Higgins when
the bilinear model is the null hypothesis in Table 1.11. When the GARCH model
is the null hypothesis, our test results are close to zero for all three series, so we
cannot reject the null hypothesis in those three data sets at any signiﬁcance levels.
In Table 1.11, Bera & Higgins reject the British Pound series as the null at 1 ‘70 of
significance level and reject the IP series at 10 % of signiﬁcance level when the null
hypothesis is the bilinear model, but all three series are rejected in our test results,
which produces much greater test values in absolute value than those from Bera and
Higgins (1997). In Table 1.9, the estimated bilinear effect is -0.008 and the standard
deviation is 0.005 in our estimation results, it is marginally signiﬁcant and indicates
the bilinear effect is very trivial for the IP series. For the IP series the test value is
~23.724, which is almost 15 times bigger than that of Bera and Higgins and rejects
the bilinear model as the null hypothesis at any signiﬁcance levels. It is shown that
the error equation does not follow the bilinear model in the IP series and this is

consistent with the test result in Table 1.11 for the IP series.

33

Table 1.11: Test results:HO:Bilinear vs. H1: GARCH

 

 

Bera & Higgins modiﬁed Cox test

 

S&P 500 -.910 —6.775
British Pound -2.797 —8.300
Industrial Production -1.643 -23.724

 

1.4 Simulation Experiments

In this section we perform some simulation experiments to investigate the potential

applicability of the modiﬁed Cox test.

We consider a linear regression model with two different nonlinear error equa-

tions as competing models. We specify an AR(1)-GARCH(1,1) model as the null

hypothesis and a AR(1)-ﬁrst order bilinear model as the alternative hypothesis.

Thus, the nonnested model speciﬁcations are

Mi 161,2 = 00 + 0191,2—1 + U2.
U2 |12:1 ~ N(0,ht).
ht = 22 + 21222—1 + 5722—1.
and rut : \/h.t222, vt ~ N(0, 1)
M2 3 92,2 1‘ 230 + 6192,2—1 + 82.

52 = biiEt—iét—iJrEt,

34

(1.88)
(1.89)
(1.90)
(1.91)
(1.92)

(1.93)

and Q ~ N(0,1)

We generate the artiﬁcial data in the following way. F irst,we generate the
normally distributed random variables from RNDN GAUSS program to calculate the
AR(1)-GARCH(1,1) model, y”. Then again we generate the normally distributed
random variables from RNDN GAUSS program for the AR(1)—ﬁrst order bilinear
model, ygy. The pseudo-true population parameters for 111 1 are given as y” 2 015+
0.85y1,t_1+ at with a strong GARCH effect; ht = 0.1+ 0.2u§_1+ 0.75ht_1. For M2,
the pseudo-true population parameters are given as ”62,2 2 O.19+0.8y2,t_1+5t where
at = 0-38552—152—1 + 52- The parameter values chosen for both models correspond
to the empirical estimates of the time series. Next we combine these two data sets
with weight A to generate a new data set yt = Ath + (1 — A)y2,t. Using this new
generated data, we perform the testing experiments by setting different values of
A; A:0, and 1. If A:1, then :02 : y”, so Ml becomes the correctly speciﬁed one,
while Mg is correctly speciﬁed if A20. The QMLES of these two speciﬁcations are
calculated based on BHHH algorithm and the simulation results are calculated from
200 replications and with a sample size of 1000, 2000, 3000, and 5000 and 250, 500,
and 750 for the small sample size properties. T1 is the modiﬁed Cox test when MI
is correctly speciﬁed and T2 is the modiﬁed Cox test when Mg is correctly speciﬁed.

When the null is true, the test value(T) should be approximately zero.

35

Table 1.12: Simulation results when GARCH(1,1) is true

 

 

 

 

 

 

 

 

 

 

sample size N=1000 N: 2000 N=3000 N=5000
T1 T2 T1 T2 T1 T2 T1 T2
mean 0.056 -19.498 0.129 -34577 -0027 -44.458 0.054 -59.661
s.d 0.846 7.045 0.986 3.986 0.945 2.353 1.019 0.678
skew -0.002 1.770 -003 5.692 -0.369 6.129 0.449 -O.276
kurt 2.866 4.423 2.368 39.269 3 782 44.359 3.630 3.138
R.F.(a:.05) 0.020 0.960 0.040 0.995 0.045 1.000 0.055 1 000
toohigh 0.005 0.000 0.030 0.000 0.010 0.000 0.040 0.000
toolow 0.015 0.960 0.010 0.995 0.035 1.000 0.015 1.000

 

 

 

 

 

 

 

 

 

 

 

two-tailed test with 02 = 0.05 and A = 1

In Table 1.12, we report the simulation results when the null is the GARCH(1,1)
model with A z 1. The four moments of the unconditional probability distribution
of the simulated test. are very close to normal for all four sample sizes. The actual
size is very close to the nominal size for all sample sizes except for N=1000, in which
the actual size is little bit understated.

Table 1.13 reports the simulation results when the null is the bilinear model
for N=1000, 2000, 3000, and 5000. The distribution of the simulation results appear
to be very close to the standard normal distribution for all sample sizes. And the
actual size is very close to the nominal size.

Figure 1.1 and 1.2 present the empirical density functions (edfs) of the mod-
iﬁed Cox test against the cdf of N(0,1) for N: 1000, 2000, 3000, and 5000 and 200

replications. Figure 1.1 shows that the edfs of the simulation results of T1 appear

36

Table 1.13: Simulation results when Bilinear(1,1) is true

 

 

 

 

 

 

 

 

 

 

sample size N=1000 N=2000 N=3000 N=5000
T1 T2 T1 T2 T1 T2 T1 T2
mean -9236 -0.063 -13045 -0145 -15.883 -0039 -20.811 -0.136
s.d 3.125 1.029 4.050 0.923 4.631 0.969 5.394 0.905
skew 2.002 0.022 2.203 -01 18 2.369 0.022 2.548 -0241
kurt 5.949 2.794 6.851 2.439 8.069 2.745 8.720 2.625
R.F.(a = .05) 0.945 0.050 0.945 0.045 0.995 0.050 0.980 0.030
toohigh 0.000 0.020 0.000 0.005 0.000 0.025 0.000 0.000
toolow 0.945 0.030 0.945 0.040 0.995 0.025 0.980 0.030

 

 

 

 

 

 

 

 

 

 

two-tailed test with a = 0.05 and A = O

to be normal for all sample sizes. Figure 1.2 shows that the edfs of the simulation

results of T2 appear to be approximately normal.

37

 

6:: 6. 2.35640 SE; C do 6639, :62 So

 

 

  

 

 

 

 

 

 

m N — O _. I N I W I
d 1.. lq - ‘ i 4
A
2 1
T
F

J
fl. I
T l
4
IA
4

0000.2 .0 .3. O I I
COOﬁIz .0 _u. i... A
DOON'Z .0 vDU 0000000 I

OOOPIZ .0 .00 I II
a n—vaz no .00 l .4

.1031...
nigh...- » — .
p p 1P » b P —2 n . b p

 

2.82 do :8 6:: .m> :62 .80 8:608 9: :o 6:6 9:, :83:

1'0 0'0

[0 9‘0 9'0 170 CO 20

6'0 8'0

01

uoiiouni Kusuap

38

mu: m_ C._Y.omc_:m CME; NH *0 mm3.o> “mm: x00

m. N F O

 

 

4 q q d J

....»tﬁ‘h‘qx

n3u“...“o‘to\

 

 

 

 

 

 

,1 «I I
C m. 0
I O
O
.O
4 Z
L
O
C;
O
J .
.V
O
C4
A
O
4 9
O
/_
O
coonnz .0 :3 .ll. 8
GOOﬁIz .0 .U. Iii! L
OOONIZ.O :quoIoooo 1.0
0009'). .0 ‘0. I I 6
A~.sz .0 .00 I 1
P P F b p

 

Cdvz “.0 you mr: _m> “mm: x00 omEDoE

9: B 38 9: memSoC

O‘l

uoqaun’; Mgsuap

39

Table 1.14: Simulation results when GARCH(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=250 N=500 N=7 50
T1 T2 T1 T2 T1 T2

mean -0.277 -5.507 0.024 -11.027 -0.007 -15.509
s.d 0.760 3.466 0.875 4.973 0.936 6.106
skew -0.446 0.411 -0.223 1.107 —0.177 1.519
kurt 3.015 1.485 2.871 2.667 3.104 3.736
R.F.(a:.05) 0.030 0.725 0.020 0.880 0.040 0.935
toohigh 0.000 0.000 0.005 0.000 0.020 0.000
toolow 0.030 0.725 0.015 0.880 0.020 0.935

 

 

two-tailed test with a = 0.05 and /\ : 1

Table 1.14 reports the simulation results with small sizes for N = 250, 500,
and 750. The mean and standard deviation for N = 250 slightly deviate from the
standard normal N (0,1) but close to normal for other sample sizes. The simulation
results undersize for all three sample sizes and the rejection frequency of T2 is lower

than 0.95 for N = 250 and 500.

In Table 1.15, the means are little bit greater than zero in absolute value
for all three sample sizes but this deviation is getting smaller as the sample size
increases. The actual size and the rejection frequency are approximately equivalent

to the nominal levels.

40

Table 1.15: Simulation results when Bilinear(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=250 N 2500 N =7 50
T1 T2 T1 T2 T1 T2
mean -4.952 -0.417 -6.806 -0.348 -8.274 -0.256
s.d 1.493 1.047 2.017 1.028 2.530 0.968
skew 2.406 -0.015 2.198 -0.325 2.076 -0.004
kurt 9.508 2.730 7.411 2.573 6.253 2.560
R.F(a=.05) 0.935 0.070 0.940 0.070 0.945 0.055
toohigh 0.005 0.010 0.000 0.000 0.000 0.005
toolow 0.930 0.060 0.940 0.070 0.945 0.050
two-tailed test with a = 0.05 and A = 0

1 .5 Conclusions

A new approach based upon the conditional mean and the conditional variance
speciﬁcations has been proposed in order to solve the computational difﬁculties of
the Cox test. This modiﬁed Cox test has some attractive features. The major
attraction of the modiﬁed Cox test is its computational conveniency because it
does not require computing the pseudo-true values. As this proposed test is based
upon the specification of the ﬁrst two conditional moments, we can also test other
distributional features unlike the DM test is for the conditional mean property only.
Furthermore, it can be easily extended to the more complicated nonlinear models.
Monte Carlo experiments indicate that this proposed test seems to perform well for
all different sample sizes. The actual size from this proposed test is almost always

close to the nominal size but the actual size is slightly different from the nominal size

41

for N = 250, and 500. Further study needs to be done to examine the applicability

to the ﬁnite-sample properties.

42

Chapter 2

A Robust Version of the Modiﬁed

Cox Test

2. 1 Introduction

In the previous chapter we proposed a modiﬁed version of Cox test under speci—
ﬁcation of the ﬁrst two conditional moments. We examined its applicability with
three different data sets: S&P 500 stock index, the £/ $ exchange rate, and U.S
monthly IP data sets, and we also did some Monte Carlo simulation experiments.
Both, empirical and simulated, test results are quite convincing the applicability of
the modiﬁed Cox test and the actual size from the simulation results is very close

to the nominal size regardless of sample size. But these empirical test results and

43

simulation performances are derived under normality assumption. In this chapter
we relax this normality assumption and extend our model in the univariate case
to the robust version of the modiﬁed Cox test under nonnormality. In section 2,
we reexamine our modiﬁed Cox test under nonnormality and derive the robust and
nonrobust versions of the modiﬁed Cox test. Section 3 summarizes some Monte
Carlo simulation experiments under nonnormality. In section 4, we compare the
test results in the previous chapter assuming normality to the test results from the
robust modiﬁed Cox test under nonnormality. Then we follow with a summary and

conclusion in section 5.

2.2 A Robust Modiﬁed Cox Test

Assume there are two competing nonnested parametric models under the conditional

mean and variance speciﬁcations.

Allzyt : mt(60)+ut (2.1)
E(yt|1t—1) = mt(90) (22)
V(3/t I It—l) = ht(90) (2-3)

and

44

Mziyt = Mt(50)+€t (2-4)
Hm I It—l) = Ht(50) (2-5)

V(yt|1t-1) = 7It(50) (2-6)

Following the previous chapter when M 1 is correctly speciﬁed, the modiﬁed Cox test

 

 

 

 

is
A 1 T M{mt -,Ut()(§ )}
T = — 2
[”1 T;( Jt—
ut(6)2 —ht(6) (61 1 ))J 27
+ 2 w) W) H

And the asymptotic distributions of the modified Cox test are as follows1

 

 

 

 

 

~ T mt(6
ﬁTMl = -71:Z1( yt- mt(9 ”(7: $05)}
ut(6)2 —ht(6) ( 1 1 “6»)
+ , — -
2 WW) (M9)
T
—(pTlgréo%Zwt((6, 6))(pT1imOO—tZlAtwl)
XVQ log ft(6)] + 0p(1) (2.8)
1 T 1 T 1
*ED—-1°— ”66*'1'— A6"
1.36wa (It t (IJngTL—ZIWI 0, ”Wiring t( 0))
Xvologft(90)
Then, E((12"|1t—1) = 0, (2-9)
—1
EU]? I It—l) = E (0t —EI(T —Z¢/H( 90,5 )I) (BI—$24M 90) I)
t 1
XValogft(90)) IIH] (2.10)

 

1We follow these from the previous chapter. See section 1.2.2 for more details.

45

= V612k I It—l) (2-11)

Under conditional normality when [I11 is correctly speciﬁed,

 

 

E(Ut(ut2 — ht) I [t—I) ’—‘ 0, (2-12)
and E(u§|1,_1) : 211460)? (2.13)
. h. 6 2
50, EU]:2 I It—l) = Dtglh't(60)+ M40) Dt22
1 T a: 1 T —l 1 T at I
+(— Z wt(9o,6 ))(- Z At(90)) (- Z M9045 ))
thl Tt=1 Tt=1
1 T 1 T 1
-2(— Z U’Jt(90,<5*))(— Z At(00))—
thl Ti=1
><(DVm(6)+ D” Vh(9)) (214)
A T A A A 2 A
Thus, VT1 : %Z Dglfrt(6)+h—t(ﬂ—Dt22

F.-

1 t

,3 M ii A
—2< Zw.<e,6))<;ZAt(6>)—l
t=1

X(Dt1V6mt(6) + Vahtlé» (2-15)

 

. . . . . ﬁT‘ 1/2
Under conditional normality, the modiﬁed Cox test, Tl ~ N (0, 1),
T1

but if the conditional normality does not hold, then the limiting distribution of

ﬂy: 1/2

V is not standard normal in general. Under nonnormality the modiﬁed Cox
T1

test derived from the previous chapter is not valid and the actual size from this

nonrobust modiﬁed Cox test can be different from the nominal size. The robust

46

modiﬁed Cox test under nonnormality is

Under nonnormality,

EIUtWZ? — ht) I It—1)

EIIu? — ht)2 I 1H)

and

extended from equation(2.10)and (2.11).

ECU? — Utht I It—1)

= EIu'z‘IIIm (2.16)
: E(II;?-2u§ht+h?|1,_1)
= Em? IIt_1)—h? (217)

Under nonnormality, E01? | It_1) and E (u;1 | It_1) are generally unspeciﬁed

but we can derive the conditional variance of the robust modiﬁed Cox test using the

Law of Iterated Expectation (LIE):

DtlDt2
2

D2
_QE
4

 

EI

and E [

E (Ut(90)3 I It—

(Ut(60)4 I It—

'DtlDt2
_U

 

 

1) = E_ 2 4903] (2.18)
‘ ’ 2
1) z: E sz'ut(60)4] (2.19)

So the conditional variance of the robust modiﬁed Cox test under nonnor-

mality is

-, 1T - - 0,132 -. D2 -
V2? = 7.; 031W) 412—‘2 <6>3+T’2<u.(6>4-m<a>)
l T , . . 1 T . -1 1 T I
+(— Z Wt(9,5))(— Z At(9)) (— Z WW5»
Tt=l T121 Tt=1

 

T i=1 t=l
x(D V m (6) + DI? (1— —1—)V 11(6)) (2 20)
t1 6 t 4h¢(6) h¢(é) 6 t -

Now we apply these properties as follows:

0 Procedure 2.1

1. Obtain 6 and 6, the QML estimators of 60 and 60, save residuals, ut(6),

and the conditional variance, ht(6), from the quasi-log likelihood func-

A A

tion log f (gt | It_1; 6) and 61(6) and 77t(6) from the quasi-log likelihood

function log g(yt, | It_1; 6) .

A

2. Compute Du, 122. r} 2.11 we”, 8), (I 23;. Min-1, V9 log W“), W)?

. , . . 6— 8
u)(6)4, and ht(6)2. Deﬁne D“ E W, Dtg E £7205} — hi6?

A 2 ‘ A A A ~2__“ A
At(0) E W, and mum) 2 Vacant, — Ljﬂptg).
A A *2_‘ A A A
3. Compute 6?er 2 TV2 237:, [awn + 13720.2 — (71. 23;, we, 6))
(%Zf:1AI(9))‘1V91<>g me]

‘R_1T *2 * 1')D_*3D2 ‘4 2
and VT1 — T thl Dtlht(9) + 412—”ut(6) + j‘2(ut(6) — ht(6) )
+(1I‘ 2;:1 WtIéI glll’} Zthl Atlé))_1(7l‘ 5.:le WM» 5)),

‘2(’}‘ 2?:1 'I(-’t(éa 8))(71‘ 2?:1 At(é))—l

A

X(Dt1ngt(6) + ﬁVghtw» and use the robust modiﬁed Cox test
It

statistic, W721 as asymptotically unit normal under the null hypothes1s.
Ti

48

This robust modiﬁed Cox test has some appealing characteristics. First, as
Bollerslev and Wooldridge (1992) showed in their robust version of LM test, this
approach is also valid under normality and can be applied to the case where normal-
ity assumption does hold. Second, this procedure requires only the ﬁrst derivatives
of the conditional mean and variance functions, it is relatively easy to compute.
Finally, even though this robust inference procedure requires the conditional third
and fourth moments, this is not a restrictive condition and we can calculate the

third and fourth moments for the robust modiﬁed Cox test using L.I.E.

2.3 Monte Carlo Experiments

To investigate the applicability of the robust modiﬁed Cox test, we perform some
simulation experiments for different sample sizes. Following the previous chapter,
we consider a linear regression model with two different nonlinear error equations:
we specify the AR(1)-GARCH(1,1) model as the null hypothesis and the AR(1)-the
ﬁrst order bilinear model as the alternative hypothesis. Thus, these two nonnested

model speciﬁcations are

M15l/t = 00+alyt—1'I‘uta (2-21)

at I It__1 N i.i.d(0,ht), (2.22)

49

ht = K + 7U%_1 + (Sht_1, (2.23)

and at = \/h7tz/t (2.24)
MQ i yt = 50 + ﬁiyt—i + 5t, (225)
and 6t I It—l = bii€t—iEt—1 + {t (226)

As seen often in time series analysis, high frequency financial time series are
of leptokurtosis and the unconditional distribution of many ﬁnantial time series typ—
ically shows fatter tails than a normal distribution. But as shown in Engle (1982)
and Bollerslev (1986), unconditional error distribution could be leptokurtic even
though the conditional error distribution is normal. Bollerslev (1987) proposed that
if the error distribution is not normal, for example the conditionally t-distributed
errors, then it permits a conditional leptokurtic distribution and it also accounts for
the unconditional kurtosis. To investigate the valid inference from the robust mod-
iﬁed Cox test under nonnormality, we generate the error terms from two different
nonnormal distributions. First, we have considered that Vt (ft as well) is condi-
tionally distributed as a Student’s t-distribution with 5 and 10 degrees of freedom.
The mean and variance of t-distribution are 0 and ﬁn f v 2 3, respectively,
where v is degree of freedom, so the variance from It). distributed random variables

is bigger than the variance from the random variables generated by the standard

50

normal distribution, if v is relatively small number. But the t-distributed random
variables still contain symmetricity on the distribution. Next, in order to examine
the effect of asymmetric error distribution, we generate the error terms from two
i.i.d x? distribution, i.e. Vt (ﬁt as well) is formed from w where $t1I$t2 are i.i.d
x? variates, respectively. Thus, the distribution of Vt (ﬁt as well) is i.i.d(0,1) per-
taining to asymmetric property. The tv distributed random variables were formed
as (v — 2) times a N(0,1) random variable divided by the square root of X3. variate.
The normal variate was generated by the RNDN GAUSS program and X3, with 2)
df by the RN DCHI GAUSS program. Beside the error generating procedures under
nonnormality, we proceed in a similar way for the pseudo-true population parame-
ters for M1 and Mgzand for the data generating procedure in the previous chapter
(see 1.4 Simulation Experiments). The QMLEs for both models are found through
BHHH algorithm and the simulation results are based upon 200 replications and a

sample size of 500, 1000, 2000, and 3000.

Table 2.1 and 2.2 report the simulation results under nonnormality: the error
terms were generated from x? distribution. In Table 2.1, the four moments of the
unconditional probability distribution of T1 are approximately close to normal but
the means are a little bit larger than zero in absolute value for N = 500, and 1000.

The actual size is very close to the nominal size for all sample sizes.

 

2We change the bilinear parameter value to b“ = 0.085 for a computational conveniency.

51

Table 2.1: Robust Cox test results when GARCH(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N = 500 N = 1000 N =2000 N=3000
T1 T2 T1 T2 T1 T2 T1 T2
mean -0373 -14.269 -0207 -23.380 -0075 -35.692 0.105 -46.313
s.d 0.966 3.341 1.069 4.390 1.008 5.043 1.050 2.436
skew -0272 2.023 -0031 3.080 0.003 4.011 0.170 10.404
kurt 4.868 6.470 4.361 11.810 3.148 19.776 2.599 127.053
R.F.(a=.05) 0.050 1.000 0.060 1.000 0.060 1.000 0.055 1.000
toohigh 0.010 0.000 0.015 0.000 0.020 0.000 0.045 0.000
toolow 0.040 1.000 0.045 1.000 0.040 1.000 0.010 1.000
max3 0.148 0.076 0.015 0.060
min4 -0.096 -0075 -0.056 -0047
mean5 0.004 0.001 0.001 -0000

 

Data are generated from X“)

Table 2.2 reports the simulation results from the nonrobust modiﬁed Cox
test under nonnormality: the error terms were formed from the x? distribution.
Nonrobustness indicates that we apply the modiﬁed Cox test derived from the nor-
mality assumption to the situation where this normality assumption does not hold
any more. The means and standard deviations are overstated in absolute value and

the actual size is more than twice as large as the nominal size varying from 0.095 to

2

 

3The maximum value of correlations for a sample size N = 500, 1000, 2000, and 3000, and with

200 replications

4The minimum value of correlations for a sample size N = 500, 1000, 2000, and 3000, and with

200 replications

5The mean of correlations

52

distribution and R:200

 

Table 2.2: Nonrobust. Cox test results when GARCH(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=500 N=1000 N =2000 N=3000
T1 T2 T1 T2 T1 T2 T1 T2
mean -0495 -13930 -0439 -22590 0.059 -36.517 0.008 -45741
s.d 1.211 3.821 1.262 5.108 1.383 3.139 1.240 3.822
skew 0.165 1.669 0.029 2.337 0.168 6.017 0.106 6.561
kurt 3.582 4.636 2.875 7.194 2.879 48.729 3.354 54.734
R.F.(a=.05) 0.120 0.995 0.160 1.000 0.135 1.000 0.095 1.000
toohigh 0.035 0.000 0.035 0.000 0.085 0.000 0.045 0.000
toolow 0.085 0.995 0.125 1.000 0.050 1.000 0.050 1.000
max 0.117 0.083 0.072 0045
min -0102 -0.069 -0052 -0055
mean -0001 -0004 -0003 0.003

 

 

 

 

 

 

 

Data are generated from X21) distribution and R2200

A

0.160. Under nonnormality, the robust modiﬁed Cox test performs far much better

than the nonrobust modiﬁed Cox test.

Figure 2.1 and 2.2 show the empirical density functions (the edfs) of the
robust and the nonrobust modiﬁed Cox tests against the cdf of N (0,1). In Figure
2.1, the empirical density functions appear to be equivalent to the cdf of N(0,1) for
all the sample sizes except for N : 500. In Figure 2.2, the edfs are distorted and

are far from normal for all four sample sizes.

Table 2.3 reports the simulation results of the robust modiﬁed Cox test under
x? distribution when the bilinear model is correctly speciﬁed. The actual size is a

little bit overstated than the nominal size but approaches the nominal size as sample

53

Table 2.3: Robust Cox test results when Bilinear(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=500 N=1000 N=2000 N=3000
T1 T2 T1 T2 T1 T2 T1 T2
mean -3.736 -0404 -4.898 -0090 -6.865 -0.106 ~8.278 0.152
s.d 1.930 1.326 2.711 1.108 3.563 1.182 4.448 1.060
skew 0.215 -0.811 0.405 -0294 0.556 -O.328 0.589 -0.469
kurt 2.689 4.066 2.389 2.479 2.268 2.549 2.182 3.276
R.F.(a=.05) 0.840 0.120 0.820 0.090 0.865 0.080 0.845 0.060
toohigh 0.000 0.005 0.000 0.020 0.000 0.010 0.000 0.030
toolow 0.840 0.115 0.820 0.070 0.865 0.070 0.845 0.030
max 0.151 0.101 0.101 0.088
min -0. 126 -0077 -0051 -0044
mean -0002 -0000 -0002 0.001

 

 

 

 

 

 

 

Data are generated from x?” distribution and R2200

size becomes larger. The four moments of the unconditional probability distribution

of T2 are close to normal but, again, the mean and standard deviation are slightly

different from N (0,1) for N = 500.

Table 2.4 reports the simulation results of the nonrobust modiﬁed Cox test
under x? distribution. As expected, the four moments of the unconditional prob-
ability distribution of T2 are far from normal and the actual size is very different

from the nominal size and overstated varying from 0.150 to 0.260.

Figure 2.3 and 2.4 show the edfs of the robust and nonrobust modiﬁed Cox
test of T2. In Figure 2.3, the edfs are very close to the cdf of N (0,1) for all four
sample sizes but the edfs are severely distorted and far from the cdf of N(0,1) in

54

Table 2.4: Nonrobust Cox test results when Bilinear(1,1) is true
sample size N =500 N 21000 N=2000 N =3000

T1 T2 T1 T2 T1 T2 T1 T2

 

 

 

 

mean -3.736 0.172 -4.898 0.309 -6.865 0.358 -8.278 0.322

 

s.d 1.930 2.080 2.711 2.820 3.563 2.912 4.448 2.877

 

skew 0.215 1.138 0.405 1.610 0.556 0.450 0.589 0.337

 

kurt 2.689 6.646 2.389 9.842 2.268 4.106 2.182 3.376

 

R.F.(a=.05) 0.840 0.290 0.820 0.350 0.865 0.450 0.845 0.475

 

toohigh 0.000 0.150 0.000 0.220 0.000 0.260 0.000 0.255

 

tOOlOW 0.840 0.140 0.820 0.130 0.865 0.190 0.845 0.220

 

 

 

 

 

 

 

 

 

max 0.135 0.074 0.079 0.049
min -0107 -0.068 -0054 -0045
mean -0002 0.001 0.001 0.000

 

 

 

 

 

Data are generated from the x?” distribution and R2200

Figure 2.4.

Table 2.5 and Table 2.6 report the simulation results for the robust and
the nonrobust modiﬁed Cox test under t5 distribution. The means and standard
deviations are approximately normal but the actual size is understated than the

nominal size for N = 500 and slightly overstated for other three sample sizes in

Table 2.5.

Table 2.6 reports the simulation results of the nonrobust modiﬁed Cox test
under t5 distribution. The actual size and the unconditional four moments of the

probability distribution of T1 are far from normal for all sample sizes.

55

Table 2.5: Robust Cox test results when GARCH(1,1) is true
sample size N=500 N =1000 N=2000 =3000
T1 T2 T1 T2 T1 T2 T1 T2

 

 

 

 

mean -0.112 -13.739 0.101 ~23.109 0.026 -35.247 0.147 -45.876

 

s.d 0.965 3.003 1.065 3.504 1.115 5.193 1.105 2.170

 

skew 0.693 1.657 0.277 3.435 -0.016 3.866 0.469 8.670

 

kllI‘t 3.748 4.852 2.937 15.485 3.410 18.060 3.251 96.825

 

RF. 0025 1.000 0.070 1.000 0.075 1.000 0.075 1.000

 

toohigh 0.020 0.000 0.055 0.000 0.055 0.000 0.050 0.000

 

 

tOOlOW 0.005 1.000 0.015 1.000 0.020 1.000 0.025 1.000
Data are generated from the t-distribution with 5 degrees of freedom and R2200.

 

 

 

 

 

 

 

 

 

 

Table 2.6: Nonrobust Cox test results when GARCH(1,1) is true
sample size N=500 N=1000 N 22000 N=3000

T1 T2 T1 T2 T1 T2 T1 T2

 

 

 

 

mean -O.754 -13.255 -0.407 -22.793 -0.102 -35.999 0.019 -45.765

 

s.d 1.912 3.338 2.309 4.134 2.228 3.470 2.232 2.439

 

skew -1327 1.352 -0971 2.741 -0531 5.113 -0354 5.532

 

kurt 7.495 3.704 4.700 10.334 2.963 31.989 2.981 39.032

 

R.F. 0.250 1.000 0.325 1.000 0.370 1.000 0.385 1.000

 

toohigh 0.035 0.000 0.125 0.000 0.180 0.000 0.200 0.000

 

 

toolow 0.215 1.000 0.200 1.000 0.190 1.000 0.185 1.000
Data are generated from the t-distribution with 5 degrees of freedom and R=200.

 

 

 

 

 

 

 

 

 

 

Figure 2.5 and 2.6 Show the edfs of the robust and the nonrobust modiﬁed
Cox test. The edfs in Figure 2.5 are very close to the cdf of N(0,1) while the edfs in

Figure 2.6 are far from the cdf of N(0.1).

Table 2.7 and 2.8 report the simulation results of the robust and the nonro-
bust modiﬁed Cox test under t5 distribution when the null is the bilinear model.
The simulation results show that the robust modiﬁed Cox test performs far better
than the nonrobust modiﬁed Cox test generally under nonnormality except for the
simulation results from the robust modiﬁed Cox test for N = 500. They are very
similar to the results from the nonrobust modiﬁed Cox test for the same sample size.
The edfs in Figure 2.7 are slightly deviated from the cdf of N (0,1). In Figure 2.8,
the edfs are more distorted from the cdf of N(0,1). We suspect that this relatively
poor performance may be mainly due to the parameter value chosen for the bilinear

effect.6

Table 2.9 and 2.10 report the simulation results under 7310 distribution. In

Table 2.9, the simulation results are very close to normal and the actual size is also

 

6We could not compute the Hessian matrix for the GARCH(1,1) model when we used the
same parameter value(0.385) for the bilinear effect in the previous chapter, so we chose a different
parameter value(0.085) that managed to ﬁt in the GARCH(1,1) model. But this parameter value
chosen for the bilinear effect is rather small and dose not provide a strong bilinear effect. Thus,
the error terms generated from this bilinear parameter value do not ﬁt in well enough to perform

the simulation experiments and the performance is particularly worse in the small sample size.

57

Table 2.7: Robust Cox test results when Bilinear(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=500 N=1000 N 22000 N 23000
T1 T2 T1 T2 T1 T2 T1 T2
mean -4052 -0.569 -5.386 -0375 —7.823 -0151 -8.496 —0.209
s.d 2.208 1.529 2.828 1.193 4.206 0.932 4.693 0.967
skew -0429 -1332 0.167 -O.678 0.275 -0323 -0011 -0277
kurt 4.425 5.217 2.316 3.979 1.931 2.752 1.908 3.141
R.F.(a2.05) 0.880 0.145 0.865 0.080 0.900 0.040 0.905 0.050
toohigh 0.000 0.005 0.000 0.000 0.000 0.000 0.000 0.005
toolow 0.880 0.140 0.865 0.080 0.900 0.040 0.905 0.045

Data are generated from the t-distribution with 5 degrees of freedom and R2200

Table 2.8: Nonrobust Cox test results when Bilinear(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=500 N 21000 N=2000 N=3000
T1 T2 T1 T2 T1 T2 T1 T2
mean -4052 -0101 -5.386 -0222 -7.823 -0174 -8.496 -0320
s.d 2.208 1.587 2.828 1.577 4.206 1.630 4.693 1.840
skew -0429 1.126 0.167 0.824 0.275 0.050 -0011 -0.783
kurt 4.425 7.683 2.316 4.716 1.931 4.170 1.908 7.390
R.F.(a2.05) 0.880 0.145 0.865 0.175 0.900 0.210 0.905 0.265
toohigh 0.000 0.080 0.000 0.070 0.000 0.080 0.000 0.105
toolow 0.880 0.065 0.865 0.105 0.900 0.130 0.905 0.160

Data are generated from the t—distribution with 5 degrees of freedom and R2200

58

Table 2.9: Robust Cox test results when GARCH(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=500 N=1000 N=2000 N=3000
T1 T2 T1 T2 T1 T2 T1 T2

mean -0.083 -13.378 0.041 -22550 0.118 -35794 0.079 -45379
s.d 0.882 2.643 0.967 2.523 1.050 1.394 1.021 0.709
skew 0.130 1.826 0.246 3.332 0.085 6.267 0.051 0.630
kurt 2.532 5.476 3.438 16.512 2.723 62.431 3.496 5.046
R.F. 0.015 1.000 0.050 1.000 0.055 1.000 0.060 1.000
toohigh 0.005 0.000 0.035 0.000 0.030 0.000 0.045 0.000
toolow 0.010 1.000 0.015 1.000 0.025 1.000 0.015 1.000

 

 

 

 

 

 

 

Data are generated from the t-distribution with 10 degrees of freedom and R2200.

very close to the nominal size for all four sample sizes. The simulation results of the
nonrobust modiﬁed Cox test in Table 2.10 are far from normal. Figure 2.9 and 2.10
show evidence that robust modiﬁed Cox test performs better than the nonrobust
modiﬁed Cox test under nonnormality and that the robust modiﬁed Cox test is also

very accurate.

Table 2.11 and 2.12 report the simulation results of the robust and the nonro—
bust modiﬁed Cox tests under 1510 distribution. As expected, the simulation results

are very similar to those from Table 2.7 and 2.8.

Figure 2.11 and 2.12 Show the edfs of the robust and the nonrobust modiﬁed
Cox test. Again, the edfs in Figure 2.11 slightly deviate from the cdf of N (0,1) but

they are relatively close to the cdf of N(0,1) compared to the edfs in Figure 2.12.

So far, we have illustrated the applicability of the robust modiﬁed Cox test

59

Table 2.10: Nonrobust Cox test results when GARCH(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=500 N: 1000 N 22000 N=3000
T1 T2 T1 T2 T1 T2 T1 T2
mean -0030 -22592 -0.096 -22347 0.120 -35.860 0.129 -45420
s.d 1.309 2.622 1.315 2.969 1.443 1.179 1.523 0.755
skew -O.158 3.182 0.226 2.914 -0117 4.036 0.269 0.377
kurt 3.131 14.538 2.921 12.381 2.859 29.950 2.813 2.816
RF. 0130 1.000 0.110 1.000 0.160 1.000 0.210 1.000
toohigh 0.050 0.000 0.030 0.000 0.090 0.000 0.125 0.000
toolow 0.080 1.000 0.080 1.000 0.070 1.000 0.085 1.000

 

 

 

 

 

 

Data are generated from the t-distribution with 10 degrees of freedom and R2200.

Table 2.11: Robust Cox test results when Bilinear(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample Size N: 500 N=1000 N=2000 N=3000
T1 T2 T1 T2 T1 T2 T1 T2
mean -3885 -0510 -6.280 -0.632 -8.182 -O.261 -10227 -0.189
s.d 2.195 1.407 3.550 1.232 4.962 0.992 6.202 1.005
skew -0.596 -1532 0.060 -0453 0.024 -0293 -0072 -0204
kurt 2.951 6.808 2.097 3.329 1.865 2.914 1.700 2.499
R.F.(a2.05) 0.830 0.120 0.850 0.160 0.880 0.070 0.875 0.065
toohigh 0.000 0.000 0.000 0.010 0.000 0.010 0.000 0.010
toolow 0.830 0.120 0.850 0.150 0.880 0.060 0.875 0.055

 

Data are generated from the t-distribution with 10 degrees of freedom and R2200

60

 

 

Table 2.12: Nonrobust Cox test results when Bilinear(1,1) is true
sample size N 2500 N=1000 N=2000 N=3000

T1 T2 T1 T2 T1 T2 T1 T2

 

 

 

 

mean -4.052 -0.101 ~5.864 -0.285 -8.324 -0.153 -9.712 0.167

 

s.d 2.208 1.587 3.370 1.215 4.763 1.219 6.019 1.420

 

skew -0429 1.126 -0.075 0.619 0.043 0.343 -0.063 0.356

 

kurt 2.076 4.195 2.076 4.195 1.778 2.880 1.783 4.015

 

R.F.(a2.05) 0.870 0.120 0.870 0.120 0.915 0.120 0.885 0.125

 

toohigh 0.000 0.080 0.000 0.045 0.000 0.065 0.000 0.050

 

toolow 0.880 0.065 0.870 0.075 0.915 0.055 0.885 0.075
Data are generated from the t-distribution with 10 degrees of freedom and R2200

 

 

 

 

 

 

 

 

 

 

 

from the procedure 2.1 but in order to use this proposed test we have to calculate
each term in the conditional variance in equation (2.15). This might be a little
cumbersome. As an alternative way, we suggest E [qz‘ 2 I It_1] from equation (2.10)
as the conditional variance of the robust modiﬁed Cox test. Both the robust and
nonrobust modiﬁed Cox test are originally derived from the equations (2.10) and
(2.11). An attractive feature of this robust modiﬁed Cox test is that it does not
require computing every term in equation (2.15). If the error terms do not follow
normality, then the third and fourth moments are automatically calculated in equa-
tion (2.10). We did the simulation experiments with some selective cases. But these

simulation results strongly suggest that this alternative robust modiﬁed Cox test

would perform properly for other cases as well.

Table 2.13 reports the simulation results under nonnormality when we use

61

Table 2.13: Robust Cox test results when GARCH(1,1) is true

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sample size N=500 N2 1000 N=2000 N 23000
T1 T2 T1 T2 T1 T2 T1 T2
mean -0355 -14130 -0192 -22715 -0059 -35.198 0.043 -45845
s.d 0.899 3.597 0.927 4.937 0.951 6.516 0.9745 3.761
skew 0.313 1.840 0.285 2.222 -0.668 3.498 -0043 5.766
kurt 3.848 5.427 2.791 6.731 5.060 14.637 2.834 37.900
R.F.(a2.05) 0.030 0.995 0.025 1.000 0.040 1.000 0.045 1.000
toohigh 0.010 0.000 0.010 0.000 0.010 0.000 0.020 0.000
toolow 0.020 0.995 0.015 1.000 0.030 1.000 0.025 1.000
max 0.096 0.061 0.047 0.060
min -0.096 -0077 -0051 -0.056
mean 0.004 0.001 -0000 -0002

 

Data are generated from x2 distribution, R2200, and the conditional
(1)

variance from equation (2.10) is used

62

 

equation (2.10) as the conditional variance for the robust modiﬁed Cox test. These
simulation results are very similar to those in Table 2.1. Note that the actual size
in Table 2.13 is slightly understated than the nominal size while the actual size in
Table 2.1 is slightly overstated than the nominal size. But these differences from

both cases are trivial and very close to the nominal size for all sample sizes.

Figure 2.13 shows the edfs of the alternatively proposed robust modiﬁed Cox
test. The edfs are also very close to those in Figure 2.1 and they approach the cdf

of N (0,1) as sample size increases.

Table 2.14 reports the simulation results under x? distribution when the
bilinear model is correctly speciﬁed. The means and standard deviations of this

simulated results appear to be approximately normal and the actual size is lower

than that in Table 2.3 for N 2 500, and 1000.

Figure 2.14 shows the edfs of this proposed test. The edfs are very close to

the cdf of N (0,1) for all sample sizes as in Figure 2.3.

Table 2.15 reports the simulation results of the robust modiﬁed Cox test
using the conditional variance from equation (2.10) under tlo distribution. The
simulation statistics are outperformed compared to Table 2.7. The four moments
of the probability distribution of T2 are very close to normal and the actual size is

also very close to the nominal size for all sample sizes. Note that the actual size

63

Table 2.14: Robust Cox test results when Bilinear(1,1) is true
sample Size N =500 N=1000 N=2000 N=3000

T1 T2 T1 T2 T1 T2 T1 T2

 

 

 

 

mean -3.736 -0.250 -4.898 -0.076 -6.865 0.091 -8.278 0.078

 

s.d 1.930 1.139 2.711 1.026 3.563 1.196 4.448 1.063

 

skew 0.215 -0433 0.405 -0380 0.556 -0473 0.589 -0074

 

kurt 2.689 2.651 2.390 2.900 2.268 3.068 2.182 2.944

 

R.F.(a2.05) 0.840 0.085 0.820 0.060 0.865 0.100 0.845 0.060

 

toohigh 0.000 0.005 0.000 0.010 0.000 0.040 0.000 0.025

 

tOOlOW 0.840 0.080 0.820 0.050 0.865 0.060 0.845 0.035

 

 

 

 

 

 

 

 

 

 

max 0.094 0.099 0.071 0.066
min -0122 -0088 -0.063 -0044
mean -0001 -0001 -0001 0.000

 

 

 

 

Data are generated from x?” distribution, R2200, and the conditional

variance from equation (2.10) is used

64

 

Table 2.15: Robust Cox test results when Bilinear(1,1) is true
sample size N=500 N=1000 N=2000 N=3000

T1 T2 T1 T2 T1 T2 T1 T2

 

 

 

 

 

mean -4.052 -0.235 -5.386 -0.254 -7.823 -0.119 -8.496 -0.197

 

s.d 2.208 0.920 2.828 1.021 4.206 0.914 4.693 0.914

 

skew -0.429 -0.095 0.167 -0.051 0.275 -O.143 -0.011 -0.180

 

kurt 4.425 2.538 2.316 2.609 1.931 2.553 1.908 3.065

 

R.F.(a2.05) 0.880 0.040 0.865 0.050 0.900 0.015 0.905 0.045

 

toohigh 0.000 0.000 0.000 0.010 0.000 0.000 0.000 0.005

 

 

 

 

 

 

 

toolow 0.880 0.040 0.865 0.040 0.900 0.015 0.905 0.040
Data are generated from tlo distribution, R2200, and the conditional

 

 

 

 

variance from equation (2.10) is used

for N 2 500 in Table 2.15 is 0.040 which is almost equivalent to the nominal size of
0.05 and it is more than three times lower than the actual size of 0.145 from Table
2.7 for N 2 500.

Figure 2.15 shows the edfs of the robust modiﬁed Cox test. The edfs are
almost equivalent to those in Figure in 2.7 and they appear approximately to be the
cdf of N(0,1).

These simulation results, in Table 2.13 through 2.15, exhibit that the actual
size is very close to the nominal size and that the simulation statistics are very
similar to those from Table 2.1, 2.3, and 2.7. But the main difference between this

type of robust modiﬁed Cox test and the previously proposed one is that the actual

size of the alternative way of the robust modiﬁed Cox test is slightly understated

65

while thehactual size of the robust modiﬁed Cox test is slightly overstated.

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2.4 Empirical Application under Nonnormality

As noted earlier, high frequency ﬁnancial time series ususally exhibit leptokurtosis
and the distribution from these time series typically shows fatter tails than the
normal distribution. Applying the modiﬁed Cox test, assuming normality to the
situation under nonnormality, generally leads to invalid test inferences. As seen in
Table 1.1 through 1.3 in section 1.3, all the empirical data sets reveal higher kurtosis
than the normal distribution, so we suspect that the distributions of these time series
follow the normal distribution. To investigate this, we perform the robust modiﬁed
Cox test for these three time series data sets and compare the test results from

the robust modiﬁed Cox test to the test results from the modified Cox test in the

previous chapter.

Table 2.16 and 2.17 are the test results from the robust modiﬁed Cox test
and Table 2.18 and 2.19 are the test results from an alternative way of the modiﬁed
COX test. First, when the null is the GARCH(1,1) model, the test values from the
r Obust modiﬁed Cox test are smaller than those from the nonrobust test but both
test results from the robust and the nonrobust modiﬁed Cox test could not reject
the null hypothesis at any signiﬁcance levels. Second, when the null is the bilinear
model, the test values from two types of the robust modiﬁed Cox test are much

S'Irlaller than those from Table 1.11, especially the test value of IP series but the test

81

Table 2.16: Test resultszHozGARCH vs. H1: Bilinear

stochastically simulated Cox test modiﬁed Cox test

S&P 500 .023 .114
British Pound .196 -.011
Industrial Production .533 .017

Table 2.17: Test results:H0:Bilinear vs. H1: GARCH

stochasticly simulated Cox test modiﬁed Cox test

 

 

 

S&P 500 -.910 -2.896
British Pound -2.797 -6.289
Industrial Production -1.643 -3.802

 

values between these two robust modiﬁed Cox tests are very similar: Test results
in Table 2.16 vs. Table 2.18 and Table 2.17 vs. Table 2.19. And both test results
reject the bilinear model as the null at any signiﬁcance levels. The test results, in
Table 2.16 through 2.19, are different from those in Table 1.10 and 1.11 and show

evidence that these time series do not follow the normal distribution and the test

Statistics from the robust modiﬁed Cox test are more valid.

Table 2.18: Test resultszH0:GARCH vs. H1: Bilinear

 

 

 

: stochasticly simulated Cox test modiﬁed Cox test
S&P 500 .023 .179
British Pound .196 -.051

Nindustrial Production .533 .020

 

82

Table 2.19: Test resultszfﬁyzBilinear vs. H1: GARCH

stochasticly simulated Cox test modiﬁed Cox test

 

S&P 500 -.910 -2.896
British Pound ~2.797 ~7.058
Industrial Production -1.643 -2.282

 

2.5 Conclusions

As noted earlier, most ﬁnancial time series do not show evidence of the conditional
normality. Thus, applying the modiﬁed Cox test, assuming normality to the non—
normal situation, yields invalid test inferences. In this chapter, we have pr0posed

the robust modiﬁed Cox test under nonnormality.

Monte Carlo simulation experiments suggest that the robust modiﬁed Cox
test performs fairly well and can improve the validity of the test statistics and the
actual size over the nonrobust modiﬁed Cox test under nonnormality. In comparison
With the simulation results in the previous chapter, the means and standard devi-
ations are slightly deviated from the standard normal distribution for some sample
Sizes such as N = 500. It is also shown that the robust modiﬁed Cox test oversizes
but not signiﬁcantly and the actual size approaches the nominal size as sample size
increases. Evidence from Lumsdain(1995) suggested that the robust modiﬁed Cox
t(Est would perform relatively well under nonnomality. She compared the robust tra-

C1itional test statistics to the nonrobust traditional test statistics under normality

83

and showed the actual size and the test statistics are not close enough to the nominal
levels even under normality. We infer that these results would be even worse under
nonnormality. We also performed an alternative way of the robust modiﬁed Cox test
with the conditional variance from equation (2.10). The simulation results are very
similar to those from the robust modiﬁed Cox test in general but the actual size is

usually understated while the robust modiﬁed Cox test usually slightly oversizes.

In some situations, as shown in Table 2.15, an alternative way of robust

modiﬁed Cox test that we proposed here preforms very well and the simulation

results are very close to normal. It is emphasized that this robust modiﬁed Cox test
has computational advantage because it does not require computing every term in
the conditional variance in equation (2.15).

We also have summarized the nonrobust Cox test with three time series data

sets in the previous chapter. The robust test results are far different from the

nonrobust test results, especially when the null is the bilinear model: the test values

are much smaller than those from the nonrobust modiﬁed Cox test in absolute value

fOr all three data sets.

84

Chapter 3

An Application of a
Quasi-Modiﬁed Cox Test to

Nonlinear Panel Data Models

3 . 1 Introduction

Irl many instances, the dependent variable takes on nonnegative integer values: for
eJv<€-.imple, number of hospital visits in a given year, number of alpha particles emitted
fI‘Om a radioactive source during a given period of time or number of patents applied
for and received by a ﬁrm during a year. When a variable takes on nonnegative

iIlteger values, it is referred to as a count variable. With the nonnegative property

85

of count data, the most p0pular functional form for the conditional mean is the
exponential function: E (y | 9:) = exp(:rﬁ), where y is a count variable and :1: is a
vector of explanatory variables. When there are unobserved effects in count panel
data models, we cannot simply apply the standard linear unobserved effects model
if we want to impose nonnegativity of the conditional mean. Hausman, Hall, and
Griliches (1984) (hereafter HHG) is a pioneering work that deals with the unobserved
effects in count panel data analysis using the conditional maximum likelihood (CML)
approach of Anderson (1970, 1972). HHG also presented an application to the
patents and R&D expenditures relationship. Wooldridge (1999) showed the QCMLE
is consistent and asymptotically normal just under the conditional mean assumption
in the multiplicative models. He also showed that Poisson QMLE is robust if the

conditional mean is correctly speciﬁed. But it will be inefﬁcient, in general, unless

the conditional varience is also correclty speciﬁed.

The most popular distributional assumption for count data is the Poisson dis-
t'I‘ilzmtion. To remove the unobserved heterogeneity or ﬁxed effects in the nonlinear
Count panel data analysis, the Fixed Effects Poisson (FEP) model was developed
by HHG. But one of the shortcomings of the Poisson model is that the ﬁrst two
1rl'loments are the same. But in many applications the variance of a count variable
is larger than the mean of it and we encounter overdispersion of the data. To solve
t'11is problem, the Fixed Effects Negative Binomial (FENB) model was developed

86

as an alternative to the FEP. As shown in Wooldridge (1999), both the FEP and
the FENB models have the same form of the conditional mean and are estimated
by the multinomial QCML methodology. Like the GARCH and the Bilinear mod-
els in the previous chapters, the F EP and the F ENB are two competing models in
nonlinear count panel data analysis. It is worth while to note that the QCMLE of
the FEP is consistent and asymptotically normal if the conditional ﬁrst moment is
correctly speciﬁed but, in general it is inefficient. On the contrary, the QCMLE of
the FEN B is not consistent unless the ﬁrst two conditional moments are correctly
speciﬁed because negative binomial is not in LEF but the F ENB is usually more
efficient than the FEP. Therefore, there is a robustness and efﬁciency trade-off be-
tween these two models. In principle, we could try the Cox test when we consider
the specification testing between these two models because the Cox test applies to
any two distributions and it is derived from the difference between log likelihood
ratio and its expected value under the null. But using the original Cox test or even
1311s modiﬁed Cox test from the previous chapter is very challenging task in this case.
The log likelihood function of the FEP model is derived from Poisson distribution
and the log likelihood function of the FENB model is derived from negative binomial
distribution and these two 10g likelihood functions take very different forms from
tlle normal log likelihood function. Therefore, to get the difference between these

two different log likekihood ratio and its expected value is very complicated and

87

computationally very difﬁcult too. Instead, we want something computationally
simpler and we derive a new Cox test (quasi-modiﬁed Cox test) using the property
of normal quasi log—likelihood as shown, for example in Bollerslev and Wooldridge
(1992). In this case the quasi-modifed Cox test is based only upon the implied con-
ditional varience because the conditional means of these two models are the same.
In section 2 we briefly explain these two models and the quasi-modiﬁed Cox test. In
section 3 we present the applications of these models to the US. patents and R&D
expenditures panel data, and then apply the quasi-modiﬁed Cox test to see if either

model is rejected. Conclusions follow on section 4.

3.2 Two Competing Count Panel Data Models

with the Unobserved Effects

Developed ﬁrst by HHG, the FEP and the FENB models have been used as two
cOmpeting counterparts in the nonlinear count panel data analysis. In this section

We brieﬂy discuss these two models.

We assume ramdom sampling from cross section and let {(yit:$z‘t, 43,-),2' :-
1 3 2, . . .,N,t 2 1,2,...,T} be a sequence of i.i.d random variables across i, but

not t, where y.” denotes the discrete observable count variable, fit is a vector of

88

explanatory variables and d, is an unobserved random scalar. For the F EP model

we assume that

yit l alum ~ P01880n(¢m($z't,ﬂ0)),t = 1,2, - - - ,T (3-1)

yihyis are independent conditional on email, tyés, (3.2)

and the conditional mean of Mt is

E(yit I LEM/>1) = EQ/z't | 13mm)

= ¢i#($z't, 30) (3-3)

HHG took the functional form of the conditional mean of Hit as an exponential
function: E(y,-t l 23,-, (bi) = oierpCvl-tﬁo). Under assumptions (3.1) and (3.2), HHG
used the CML techniques of Anderson(1970,1972) to estimate ,6, conditioning on

the sum of the dependent variable across time, ELI yit : ni. HHG showed that

M l Ramada ~ multinomialmupiim,50),---,P2'T(Iz',50)) (3-4)
T T

where Pit = €$P($it,50)/ Z€$P($ir,ﬁo) and 2P1t=1 (35)
1‘21 t=1

Eq(3.4) reveals that the distribution of (yihyz-g, - - - ,yz-T) given (WW1) does

not, depend upon the unobserved effects 4),. Therefore, the log likelihood function

89

of the Poisson CML methodology by HHG can be written as

T T T
li(l3)FEP = Z Wyn + 1) — Z 3127th8 2 (KM—(5132': - 5%)5), (3-6)

3:1
where F(.) is gamma function.
Gourieroux, Monfort, and Trognon (1984) (hereafter GMT) showed that the

multinomial QCMLE of the FEP is consistent even though the multinomial distri-

bution is not correctly speciﬁed if

E(yz't I Nari) = Prdhﬁolni

T
where n,- = Zyit
t=1

However, Wooldridge (1999) argued that this is too restrictive and showed that,
while the FEP estimator is derived under assumptions (3.1) and (3.2), it is consistent

and asymptotically normal only under the conditional mean assumption (3.3).

On the other hand, E(y.,-t | 1.2, (15,-) = Var(y,-t | $1,451) 2 A.“ where A“ is the
Poisson parameter from assumption (3.1). But it is not difﬁcult to ﬁnd, empirically,
t1lat the conditioanl variance of Hit is not the same as the conditional mean of yit-
More likely, the conditional variance of Mt is larger than the conditional mean of Mt
01‘ it is increasing with yit in many cases. To solve this overdispersion problem, HHG
11Sed the negative binomial distribution and developed the Fixed Effects Negative

Binomial (F ENB) model as an alternative to the FEP. To derive the FEN B by HHG,

90

we assume that1

yit | (L‘i, ¢i ~ NegativeBinomial(/r(:r.it,ﬁg),1/qbi) (3.8)
where <15,- is the unobserved effect and ab,- > 0
Hit, y.,-,. are independent conditional on (xit, (152‘), t # r (3.9)

E(yit|$i,¢r) = ¢i/t(ivit,ﬂ30) (3-10)

Interestingly, under (3.8) to (3.10), the conditional mean is E (yit | ni, 33,) = pit(:1:i, ﬁo)n,-,

which is the same as Eq. (3.7).

The conditional log likelihood function2 for the FENB by HHG is

T
li(ﬁ)FENB = 2008 F(#it + yit) —103F(Mit) — 103m?!“ + 1))
t=1
T T
+ log IX: ﬂit) + log F(n,~ + 1) — log N: 1111+ n,) (3.11)
t=1 t=1

Under assumptions (3.8) to (3.10), the strict exogeneity of grit, the CMLE of the

FEN B is consistent and asymptotically normal.

Now, we compare the Possion model and the Negative Binomial model ana-

1y 28d in Wooldridge (1999). In Possion model,

50121 l 13mm) = <f>i#(l‘it,50) (3.12)
\ = VaT(yit|$z't,¢i) (3.13)

1 We follow the notation from Wooldridge (1999)
2see HHG p.924 for more details

91

and the variance to mean ratio of the Possion model is unity. In Negative Binomial

model,

E(yz't I xitaCDi) = ¢i#($it,ﬁo) (3-14)

Va?‘(yz't | 113mm) : E(yit l mac/100+ 452') (3-15)

and the variance to mean ratio of the NB model is (1 + d.,-) > 1. The NB model
shows the overdispersion and also allows the variance to mean ration to be different

from each i.

The conditional mean of both Poisson and NB models conditional on the
sum of dependant variable across time is E(y,-t | n,,a:,-) : Pit($i,50)ni- Next,
we consider the conditional variance of both the F EP and the FENB models.
Following HHG (1984), we ﬁrst construct the conditional variance of the F EP
model from the multinomial covariance matrix, Q,- : diag(pi) — pfipz- where Pit =
”(cribﬁoﬂ XIII ,u(a:.,-.,.,ﬂ0). From the ﬁxed effects assumption, 9,- is singular by
construction. Therefore, we remove the ﬁrst row and column to construct 92-, which
iS (T — 1) x (T — 1) matrix.3 We derive the conditional variance of the F EP from
the diagonal elements of 92': V(y,-t | ni, sci) : (1 — Pit)Pit- Next, we derive the con—
ditional variance of the FENB as we did that of the FEP but an extra term added

En the NB assumption: 92‘ = (221% + ZT=1H($it:80))/(1+ Lil/4132150)) and
3We can remove any time period from 0,. We take the ﬁrst row and column for convenience.

92

QFENB = g.i(diag(pz-) — pgpi) and the conditional variance of the FENB is the
diagonal terms of Q : V(y.,;t | nhxi) = gip.,;t(1 — Pit)-

The original Cox test is Tf : {L f(a)—Lg(.é)}—Ed{L f(a)—Lg(a“) }. The test
statistic of the Cox test is based upon the difference between the log likelihood ratio
and its expected estimate under the null. In principle, we can try the original Cox
test but this may cause very severe computational difficulties. Instead, we use the

ﬁrst two conditional moments from the QCML methodology and construct a quasi-
Inodiﬁed Cox test using the normal quasi-log likelihood framework. Bollerslev and
Wooldridge (1992) showed that the normal log-likelihood and its expected values are
Inaximized when the correct conditional mean and variance are used, even though
the normality assumption is violated. Using this property, we now construct a quasi-

modiﬁed Cox test. Let Ml denote the model deﬁned by Eqs. (3.1) and (3.2). Under

 

M1,
Hm I n-iiivi) = PitIx-iﬂohz' (3-16)
VaTI’yit I Nanci) = (1 —p-it($i,ﬁo))Pit($i,ﬁo) (3-17)
T
where ”it = Z yit,
t=1
d . “($21,180)
an pit T ,
Zrzl Mmir, ﬁo)
Let Mg denote the model by Eqs. (3.8) and (3.9), so that
EIyit I niixi) = Pit($i,90)ni (3-18)

93

Va?“(yit I n2313i) = gi(1 - Pit($i,90))19it(33i,90) (3-19)

where m. = m...

”(xiii 60)
2L1 #(irir, 60)

T T
and 92' = (2 ”it + Z #(IritiBOD/(l + Z #Iévitaao»
— t=1

 

Pit Z

We use these two conditional mean and variance from QCML methodolody and put
them into the modiﬁed Cox test that we derived from previous chapter to get the

test statistic. Now, the quasi-modiﬁed Cox test has a form of

 

 

N T— 1 ‘ ‘

. . . Pit($i,i3)ni-Pit($i,9)ni

TM 54);.21 I“ p“ WII <1-p..<z.,é>>p.-t(z..é>I
MB)? —(1 — I)'it($iaﬁ))Pit($iiB)

 

+ 2

x( 1- . — 1 . )] (3.20)
91(1—Pit($i,9))P-it($ia9) (1—PitI1‘ii P))Pit($i,5)

 

 

Following the previous chapter4, we can derive the asymptotic distribution of the

quasi-modiﬁed Cox test:

 

«rm, 2 —\/1=§[(g, p,i.,a)n){’(

i (
ﬁ(ﬁ)2—(1-pi(rvi,ﬁ))ﬁixi(
2

23%” pl($2)0)n’l}
Pi ($1,6))Pi($i,6)
)

 

+

 

 

l l
x (92(1— PiIIiignﬁd-Tiaé) — (1 “Pi($iiP))Pi($iiP))
A A N _ A —l _
- (reﬁt/W) (wait—.Eﬂi‘”) V”““”””I

+op(l) (3.21)

4see section 2.2 for more detail

 

94

And the variance of the robust quasi-modified Cox test, VTI, is

A 1 N A -
VT1 = N Z [012210—I3i($i,ﬁ))ﬁi($i,ﬁ)+—

 

 

><(1— 1i _ A )
(1 - ﬁi($i,5))Pi(-’Ei,ﬂ)

A

XVﬁ ((1 — 15i($i,3))13i($i,3)))] (3.22)

 

8
2"
Cb
‘3
Cb
b
3

H

l I

 

‘ _ _1__t:11((19it(33it, W) pit(1itaé))nz‘)

gzp1t(miti 0)(1— pit($iti 6)
A 1

1
_1t“

:(gzpzt($ita 9)(1 - Pit($ita 9)

b
E,
III

 

ll :MEEH

K3

 

l
Pit($itaB)(]: — pit($it,B)) ,
— . _ 1 3212i(ﬁ)FEP
— T—l aﬁaﬁ’

1 “l < “
—— v3 a-tmwnl
T_1t=1 z

_ait(6)2 - Pit($ita8)(1_ Pit($it,3))D 2)]
2 n

 

3?
§
I

’

 

 

D
3
9..
ﬁn
:Q’
52>
II

 

And the robust quasi-modified Cox test statictics, 16—1351, follows aymptotically unit
T1

normal under the null hypothesis.

95

Table 3.1: Summary Statistics: the Patents and lnR&D Data

 

 

mean s.d median Min Max proportion of zeros
Patents 37.133 72.642 6.000 0.000 515 0.000
lnR&D 1.415 1.947 1.196 -3.849 7.034

 

 

 

 

3.3 An Empirical Application

3.3.1 An Application to US. Patents and R&D Data

In this section, we estimate the FEP and the F ENB models under the CML frame-
work using the U.S. patents and R&D expenditures data and apply the quasi-

modiﬁed Cox test between these two competing models.

We examined the dynamic speciﬁcation properties from the data on US.
patents and R&D expenditures from 1970 to 1979. We obtained this US. patents
and R&D spending data set, patrhghtxt, from the data directory in the NBER
website. This data set is a subset of the patents and R&D data used in HHG
(1986), ”Patents and R&D: Is there a Lag?”, IER 27: 265-283. There are a total
of 346 ﬁrms and 22 ﬁrms (about 6.4% of all the ﬁrms) have zero patents during all
time periods and we deleted these ﬁrms from our data because these observations

do not contribute to the estimation.

Table 3.1 presents the summary statistics of the dependent variable, patents,

and the explanatory variable, lnR&D.

96

Table 3.2: Estimation Results for the Patents Model: Linear Time Trend

 

 

 

Parameter the Fixed Effects Poisson the Fixed Effects Neg Bin
lnR&D 0.428 (0.038) 0.261 (0.090)
lnR&D_1 -0.159 (0.048) —0.112 (0.115)
lnR&D_2 0.021 (0.044) 0.042 (0.103)
lnR&D_3 0.174 (0.041) 0.114 (0.098)
lnR&D_4 0.090 (0.039) 0.178 (0.092)
lnR&D_5 0.259 (0.030) 0.224 (0.068)
time -0.083 (0.003) —0.080 (0.010)
Sum of lnR&D 0.813 0.707

log likelihood -6069.156 -3935.991
Skewness of residuals 0.141 0.207
Kurtosis of residuals 7.430 7.559
Probability of Normality 0.000 0.000

 

 

 

* The standard errors are in the parentheses.

Table 3.2 presents the estimation results for a patents model with linear time
trend using the FEP and the FENB estimators. In this table, both estimation
results indicate that the contemporaneous effect of lnR&D is signiﬁcant. The sums
of lnR&D are similar but the sum of lnR&D of the FEP is slightly bigger than that
of the FENB. In the FEP estimator, the sum of lnR&D is 0.813 but it is 0.707 in
the F ENB estimator. And the time coefﬁcient is -8.3 per cent per year in the FEP
estimator and -8 per cent per year in the FENB estimator. Table 3.3 presents the
estimation results for patents model with a full set of year dummies by the FEP

and the FENB estimators.

97

Tablciiﬁ: Estimation RESILtS for the Paten_t:s Modelszull Set of Year Dummies

 

 

Parameter the Fixed Effects Poisson the Fixed Effects Neg Bin
lnR&D 0.407 (0.039) 0.245 (0.090)
lnR&D_l -0.115 (0.058) -0.087 (0.120)
lnR&D_2 0.061 (0.045) 0.063 (0.108)
lnR&D-3 0.111 (0.042) 0.084 (0.099)
lnR&D_4 0.073 (0.040) 0.165 (0.094)
lnR&D_5 0.279 (0.030) 0.238 (0.069)
year76 -0.044 (0.014) -0.052 (0.038)
year77 -0.077 (0.014) -0.105 (0.040)
year78 -0.238 (0.015) -0.233 (0.041)
year79 -0.320 (0.015) -0.309 (0.042)
Sum of lnR&D 0.816 0.708

log likelihood -6042.707 -3934.719
Skewness of residuals 0.229 0.259
Kurtosis of residuals 7.290 7.437
Probability of Normality 0.000 0.000

 

 

 

 

* The standard errors are in the parentheses.

Table 3.4: Estimation Results for the Patents Model: Linear Time Trend

 

 

 

Parameter the Fixed Effects Poisson the Fixed Effects Neg Bin
lnR&D 0.826 (0.009) 0.694 (0.020)

Time -0.065 (0.009) -0.079 (0.020)
Time*1nR&D -0.008 (0.002) -0.005 (0.005)

Sum of lnR&D 0.826 0.694

log likelihood -6273.494 -3981.590
Skewness of residuals -0.245 -0.295

Kurtosis of residuals 36.014 35.173
Probability of Normality 0.000 0.000

 

 

 

* The standard errors are in the parentheses.

98

Table 3.5: Estimation Results for the Patents Model: Linear Time Trend Only

 

 

 

 

Parameter the Fixed Effects Poisson the Fixed Effects Neg Bin
lnR&D 0.800 (0.006) 0.679 (0.014)

Time -0.097 (0.003) —0.096 (0.010)

Sum of 1nR&D 0.800 0.679

log likelihood —6280.996 -3982.094
Skewness of residuals -0.392 -0.394

Kurtosis of residuals 38.446 37.159
Probability of Normality 0.000 0.000

 

 

 

* The standard errors are in the parentheses.

In this table, only the contemporaneous effect of lnR&D is signiﬁcant in both
models except the last lag of lnR&D. In the FEP estimator, the sum of lnR&D is
0.816 but it is 0.708 in the FENB estimator. Table 3.4 presents the estimation
results including only current lnR&D, time trend and the multiplication of these
two variables. The coefﬁecients of current lnR&D in both models are much higher
than those in Table 3.2 and 3.3 but the sum of lnR&D are very similar. Table
3.5 presents the estimation results including current lnR&D and time trend only.
The time trend coefﬁcient for the F EP is -9.7 per cent and -9.6 per cent for the
FENB. These coefﬁcients are bigger in absolute value than those in Table 3.4. Not
surprisingly, the standard errors in the F ENB are much larger than those in the F EP

and it is expected from the increased noise in the Negative Binomial Specification.

99

Table 3.6: The quasi-modiﬁed Cox Test Results

 

 

HozFEP vs. leFENB

H0:FENB vs. H1 :FEP

 

Nonrobust Cox test

-8.740

-6.318

 

 

Robust Cox test

 

-7.570

 

-6.531

 

 

:4: Test results from the patents model with Linear Time "ﬁend

Table 3.7: The quasi-modiﬁed Cox Test Results

 

 

HozFEP vs. leFENB

HozFENB vs. H1 :FEP

 

Nonrobust Cox test

-0.725

-2.170

 

 

Robust Cox test

 

-0.706

 

-2.181

 

 

 

 

at Test results from the patents model with a full set of year dummies

3.3.2 The Quasi-Modiﬁed Cox Test Results

Our quasi-modiﬁed Cox test has been used to compare the correct speciﬁcation

between the FEP model and the F ENB model.

Table 3.6 presents the quasi-modiﬁed Cox test results for the patents model
with liner time trend. In this table, the nonrobust test results indicate that both
models are rejected against the correctly speciﬁed model at any signiﬁcance level.
The Jarque—Bera test (probability of Normality) reveals that the residuals of both
models are not distributed as normal in Table 3.2. And the robust quasi-modiﬁed

Cox test results also reject both models to be correctly speciﬁed.

Table 3.7 presents the quasi-modiﬁed Cox test results for the patents model
with a full set of year dummies. In this table, the nonrobust quasi-modiﬁed Cox test

results show that the FEN B model is rejected against the correct speciﬁcation at the

100

Table 3.8: The quasi-modiﬁed Cox Test Results
HozFEP vs. leFENB HozFENB vs. leFEP
Nonrobust Cox test ~28.417 -6.678

Robust Cox test -10.621 -7.043

* Test results from the patents model with linear time trend

 

 

 

 

 

 

 

 

 

 

5 per cen signiﬁcance level but we fail to reject both models at 1 per cent signiﬁcance
level. The probability of normality in Table 3.3 indicates that the residuals from

both models are not normally distributed.

The robust quasi-modiﬁed Cox test results are very close to the nonrobust
test results and the F EN B is rejected at 5 per cen signiﬁcance level but both models
failed to reject the null at a 1 per cent signiﬁcance level. Table 3.8 presents the quasi-
modiﬁed Cox test results for the patents model including only current lnR&D, time
trend and the multiplication of these two variables. Both nonrobust and robust test
results reveal that both models are rejected against the correctly speciﬁed model at
any signiﬁcance level. Table 3.9 presents the quasi-modiﬁed Cox test results for the
patents model including current lnR&D and time trend only and shows that both
models are 8810 rejected at any signiﬁcance level. Interestingly, including a full set
of year dummies seems to play an important role to correct the model speciﬁcation.

We can suggest the role of time dummy with an example below. Suppose

Var(yit l 3i,¢i,7t) = 7t¢i€xp(17it5)

: [Vt/1'0“}: $2)

101

Table 3.9: The quasi-modiﬁed Cox Test Results

 

 

HozFEP vs. leFENB

H0:FENB vs. [-11 :FEP

 

N onrobust Cox test

-13.345

-10.088

 

 

Robust Cox test

 

-9.773

 

-11.437

 

 

* Test results from the patents model with linear time trend only

If at is time dummy, then there is no more overdispersion problem when we include

this time dummy in the model.

Varfyz‘t I 332', 4%)

What we can infer from this example and possibly from the test results is that the
overdispersion problem may be caused not by the distributional misspeciﬁcation
but by the parametric misspeciﬁcation. And including this time dummy can correct

the overdispersion problem and lead the Poisson model to be the correctly speciﬁed

model.

= ¢~i€$P(Izt,3 + at)

where 7t 2 exp(ozt)

Further, suppose 7t is independent of (451,351), then

= E[Var(y)-t l WNW/7t) | $23451]

+Var [E(yz‘t | 331,618,711) | $23M

= madam) + 0?, [emexpmm]?

> Cb'ieirpmtﬁ) = VaT(yz't I (”bah/7t)

102

 

3.4 Conclusion

In the count panel data models with unobserved effects, the F EP and the FENB
models are frequently compared as two competing counterparts. The QMLE of the
F EP is consistent if only the conditional mean is correctly speciﬁed but it is generally
inefficient. On the contary, the QMLE of the FENB is not consistent unless the ﬁrst
two moments are correctly speciﬁed but it is more efﬁcient than that of the F EP.

Therefore, there is robustness and efficiency trade-off between these two models.

We applied the FEP and the F ENB models to the US. patents and lnR&D
expenditures relationship. The quasi-modiﬁed Cox test results indicate that includ-
ing a full set of year dummies plays a major role for the correct model speciﬁcation.
When we include a full set of year dummies, the quasi-modiﬁed Cox test results
become different from those without a full set of time dummies and both the FEP
and the FEN B models fail to be rejected against the correctly Speciﬁed model at 1
per cent signiﬁcance level while both models are rejected at any signiﬁcance level if
we include the linear time trend instead. We can conjecture from these test results
that the overdispersion problem may not be a matter of the distributional misspec-
iﬁcation but a matter of parametric misspeciﬁcation. The further study is needed
to ﬁnd more correctly speiciﬁed model for the nonlinear count or continuous panel

data model with unobserved effects.

103

Appendix A

Modiﬁed Cox test

Here we derive the modiﬁed Cox test. The original Cox test statistic can be written

in terms of the information set available at time t as

:r
- 1 ..
TM1 = ftzlflogfdytIIt—1;9o)-1089t(ytIIt—1;6)}

T
1 *
“EMllf ZflOBftfl/t IIt—1;90)—1089t(yt |1t—1;5 )} | It—1l(A-1)
t=1
where 60 is the MLE under M1 and 6* is the MLE under M2 when M1 is correctly

speciﬁed. Now we decompose the equation (A.1) by two terms and rewrite these

two terms as

i) The ﬁrst term;

T
Zﬂogftlyt |1t—1;90)—1089t(yt I It—1§6*)} (A?)
t=1

104

 

 

1 1 1 —m 6 2
108ft(ytIIt—1;90) = -§108271—'2-108ht(90)—'2'(yt ht(9t(I)O)) (A3)
* 1 1 , 1 -— 6* 2
1380.14-19 ) = ~§Iog2vr—§Iogm(6 )— 5(1), 77.13:) )) (AA)

From HOW 011 we deﬁne 108ft(yt I It—1§60) = 10813:, 1089t(yt I It—1;6*) = loggt,

ht(60) : ht, 77t(6*) : 0t, mt(60) 2 rm, and MW“) = M for notational simplicity.

1 1 1 —m 2 — 2
logft — loggt = -10877t — — loght — — [Iii—1 — (946%)”)

 

2 2 2 ht
T T 2 2
1 (1% — mt) (yt - Mt)
80,2{108ft-1089t} Z _"Z _——_——
t=1 2 t=1 ht 7”
T
—§{108 ht —1080t} (A-6)

ii) The second term;

T T 1T — 2

 

T
EM] Zilogft-loggtHIt—l] = EM1

t=1 2 2 t=1 "t
“(gt—2?): I lit—1}] (A.7)
= 310877: — I loght +132": EM {_(y‘ _ ”)2

2 2 2 H I 7%
Jig-17793 | It—l} (A8)

T T T
— glognt— '2‘108ht— 5

1 T _ 2
+§ Z Ell“1 {—(yt ﬁt) I It--l} (A9)
t=1 ’72

Because

( EM1{(3/t — mt)2 I It—1} = Ele’ui? I It—1}
= ht

T T T
— 510877: " 5108M — ‘2-

105

 

T 2 2
1 yt — mt + mt — Ht
+5 E E1111 {( > ( ) I It_1} (A.10)
t=1 Tit

Because
( (yt — #02 = (Ll/t — W + mt - #02

= (M - mt)2 + (mt - #02

+2(’yt - mt)(mt - Mt)

= (M - mt)2 + (mt - #02 + 2Ut(mt - lit)
EMIIIyt - #02 I It—l} = EM1{(l/t - mt)2 I It—l}

+EM1{(mt - #02 I It-1}

+2EM1{(?/t - 77%th - Mt) I It—1}v and

 

EMIIIE/t - mt)(mt - #t) I It—l} = EM1{ut(mt - ﬁt) I It—l}
= Eleut I It—IIEMIImt - ﬁt I It-I}
= O
50, EMIIIUt — #02 I It—I} = EM1{(?/t - mt)2 I It—I}
+EM1{(mt - #02 I It—I}
I = ht + (W — (0)2
= glognt—gloght—§—+ éé:—+ éém; M)2 (A.11)

Now we combine these two terms back together, and we produce

T 1T yt-mt2 1T Kit—W2
= —§{loght—log77t}——Z(——h—)—+ ZL—l-

21:1 2 2t=1 '72
T T 1 ht 1 T (gt—m2
— —lo ——10 h}+———§j———§:——— A.12
{2 8m 2 8t 2 2t=177t 2t21 m I I

 

 

= I _ l {i Eur T (mt “.1102 + T: (yt—mt)2 _ i (yt—ﬂt)2§A 13)
2 2 t=1 ’72 t=1 7” t=1 ht t=1 7”
T 2 T _ 2 2
z Z_1zﬂt_+:{ht+(mt #t) (9t #0} (A14)
2 2t=1 t t=1 77‘
T 2 T 2 _ 2
Z I _12 fit. + Z (9: m) ht (mt 74) (A15)
2 2 t=1 t t=1 2’72

106

On the other hand, we can rewrite the numerator of the third term as

( (yt — 202

mt + Ht #02

=I

=Imt- #t + Ut)2

=Imt- #02 + “t + 2utImt - Ht)
=Imt

80, Iyt — Mt)2 — ht - (mt — #02 #02 + 11%- 2UtImt - Ht)

 

 

 

 

 

 

 

 

— — (mt - #02
I — —ut— ht- zutImt - Mt)
T 1 T U? T “at-(mt Mt) Tu? - ht
= ———Z—+Zut +2 (A16)
2 2 t=1 ht t=1 7” t=1 2’72
T T u? — ht + ht T ut(mt - pt) T u? - ht
= '— — + + A.l7
2 t; 27% t; 7h t; 2m ( )
T T 2 __ T T T 2 _ h
2 5‘2“ch ht_ glziJrZUtImt #t)+mz):2 t (A.18)
t=1 t t=1 t t=1 WT t=1 '72
T T 1 T 2 (11) ,1)
= ———+— u—h —-—— + A.19
2 2 2t=1( t t) 77t ZI yt— Tit I )
Therefore, the modiﬁed Cox test is derived as
A T
TM; = T ZlflogftIyt I [t—1§60) loggtIyt I It—1§5*)}

—EM1ITtZ:1{108ftI3/t I11-1;90)-1089tIytIIt—1;5*)} I It—ll

_ 1T (mt-pt) 02—h) 1 1
T Tt=1I(yt—mt) 7h + t2 (——_)I (A20)

 

107

Appendix B

Regularity Conditions

Suppose yt,t = 1, - - - , T is a vector of i.i.d observations and we wish to compare when
yt has the density function f (gt, 6) for some 6 in 9 under the null hypothesis, H f,
and when yt has the density function g(yt, 6) for some 6 in A under the alternative
hypothesis, Hg. Then let 60 denote the true value of 6 under H f, let 6 be the MLE
of 60 and let 6* denote the value that 5, the QMLE of 6, converges to. For notational
brevity, we state the regularity conditions in terms of f (y, 6) but these conditions
are also applicable to g(y,6) as well. Below are the regularity conditions for the

existence and the consistency of QMLE(White, 1982).

1. The sequence of i.i.d observations yt,t = 1, - - - ,T have common joint distrib-

ution function G on Q with measurable Radon-Nikody’m density 9 : dG/dv.

108

2. Radon-Nikody’m density f(y,6) = dF(y,6)/dv where F(y, 6) is the family of
distribution function is measurable in y for every 6 in O, a compact subset of

a p—dimensional Euclidean space, and continuous in 6 for every y in Q.

3. a) | log f(y, 6) IS m(g) for all 6 in 9, where m is integrable with respect to G.

b) E(log f(yt, 6)) has a unique maximum at 6 in O.

4. 610g f(y,6)/06,—,z' = 1, ~ - - ,p, are a measurable function of y for each 6 in O

and a continuously differentiable function of 6 for each y in Q.

5. | a2 log f(y,6)/66,- . 89,- | and | Blog f(y,6)/86,- - 010g f(y,6)/86j |,z',j 2
1, - - - , p, are dominated by functions integrable with respect to G for all y in

Q and6 in O.

6. Deﬁne 21(9) 2 {19(02 log f(y, 6) we, . 00,-»,
and 8(6) E {EIBIOE f (9.0/80. ~ 6109. f (90)/09j)}.
a) 6 is interior to O,

b) 24(6) and 8(6) are nonsingular.

Under these conditions, /T (6 — 60) is asymptotically normally distributed.

109

 

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III[IIIIII[IIIIIIIIILIIIII '