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THESIS

9
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This is to certify that the
dissertation entitled

Dynamic Unobserved Effects Model

for Continuous and Binary Response

presented by

Chung-Jung Lee

has been accepted towards fulfillment
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Dynamic Unobserved Effects Model for Continuous and

Binary Response
By

Chung-Jung Lee

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

2000

ABSTRACT

Dynamic Unobserved Effects Model for Continuous and

Binary Response
By

Chung-Jung Lee

In this thesis I consider estimation of dynamic, unobserved effects panel data
models for both continuous and discrete outcomes. In order to handle correlation
between the unobserved heterogeneity and the initial condition, I use the method of
conditional maximum likelihood estimation (CMLE). This method turns out to be
tractable for nonlinear binary response models as well as for dynamic linear models
when the unobserved heterogeneity interacts with the lagged dependent variable. The
CMLE performs well compared with various competitors that have been proposed in
the literature.

The thesis is in four chapters. Chapter l surveys the existing literature for es-
timating dynamic linear models with unobserved effects, with attention to various
assumptions that have been made on the initial conditions. In Chapter 2 I study the
CMLE for the linear, dynamic model with an additive unobserved effect. I show how
to construct the conditional likelihood function~ which uses an assumption about the
distribution of the unobserved effect given exogenous variables and the initial condi-

tion. Monte Carlo evidence is provided, with and without the normality assumption

in the conditional distribution for heterogeneity, and I include an empirical applica-
tion to wage dynamics for employed men.

Chapter 3 considers the CMLE for a useful extension of the basic linear model.
Namely, I allow the unobserved heterogeneity to interact with the lagged dependent
variable. Apparently, this model has not been treated in the literature. Conditional
MLE is especially useful for obtaining consistent. estimators. Chapter 4 studies the
dynamic logit model with an unobserved effect. Even in this case, where the con-
ditional mean function is nonlinear, the CMLE is feasible and produces interesting

results in an application to union membership.

To my father, Fu-Lz' Lee and my mother, Yue-Xz'a Lee- Wu

iv

ACKNOWLEDGMENTS

It is impossible to express adequately my gratitude for the assistance and encour-
agement which many persons have given me in the preparation of this thesis. The
first person to whom I must much owe is my thesis advisor, Professor Jeffrey M.
Wooldridge. For his timely recommendation and continued instruction about this
thesis, it can smoothly come out at. last. Especially for the original idea about the
construction of the dynamic panel model he came up with and numerous associated
suggestions and valuable advice, I, here, want to acknowledge special debt to my
thesis advisor. I also thank the other members of my committee: Professors de Jong
and Strauss. I must thank my parents and the rest of my family members, Chung-yi,
Chung-chuan, Chung-Liang, Li-juan and Li-mei. Without their continual support
and encouragement, I cannot go though with the Ph.D program. I also acknowledge
my intimate friends, Chang-shun, Rei-yuan, Feng-yue, Hong-yao and shang-yu for
providing the software and timely PC maintenance during the preparation of the the-
sis. Last, not the least, there are still a lot of persons who help me behind, but I have

not mentioned in the list. l always appreciate all of their help with gratitude.

TABLE OF CONTENTS

LIST OF TABLES viii
1 Overview Of The Linear AR(1) Model With Unobserved Effects 1
1.1 Introduction .................................. 1
1.2 The Inconsistency of the LSDV Estimator ................. 3
1.3 Estimators of error components model ................... 10
1.4 Properties of the ML estimator ....................... 14
1.5 The efficiency of GMM estimator ...................... 16
1.6 Conclusion ................................... 23
2 Conditional Maximum Likelihood Estimator For The AR(1) Model 26
2.1 Introduction .................................. 26
2.2 General CMLE ................................ 30
2.2.1 Conditional Likelihood Function ..................... 30
2.2.2 Asymptotic Properties of the CMLE ................... 33
2.3 Linear AR(1) Model With Unobserved Effects ............... 35
2.3.1 Linear AR(1) Model ............................ 35
2.3.2 Simulation Evidence ............................ 38
2.4 Linear AR( 1) Model With Unobserved Effects And Exogenous Regressors 41
2.4.1 Linear AR(1) Model With Exogenous Variables ............. 41
2.4.2 Conditional Mean and Variance ...................... 42
2.4.3 Simulation Evidence ............................ 44
2.5 Empirical Example .............................. 46
2.6 Comparison With The Other Estimators .................. 49
2.7 Conclusion ................................... 53
3 Models Where State Dependence Depends On Unobserved Hetero-
geneity 97
3.1 Introduction .................................. 97
3.2 AR(1) Models With Unobserved Heterogeneity, State Dependence . . . . 101
3.2.1 AR(1) Model Without Exogenous Variables ............... 101
3.2.2 AR(1) Model With Strictly Exogenous Variable ............. 104
3.3 Simulation Evidence ............................. 107
3.3.1 Model without Exogenous Variable .................... 107
3.3.2 Model With Strictly Exogenous Variable ................. 111

vi

3.4 Empirical example .............................. 113

3.5 Conclusion ................................... 117
4 CMLE For Logit Model With Individual Heterogeneity 134
4.1 Introduction .................................. 134
4.2 The CMLE for Dynamic Logit Model with Unobserved Heterogeneity . . 136
4.2.1 Estimation of Fixed Effects Model .................... 136
4.2.2 Conditional Maximum Likelihood Estimator .............. 140
4.3 Simulation Evidence ............................. 143
4.3.1 The Model Without Exogenous variables ................ 144
4.3.2 The Model With Exogenous Variables .................. 147
4.4 Empirical Example .............................. 148
4.5 Conclusion ................................... 151
A Conditional mean and variance 175

B IV estimator for average autoregressive coefficient across population
of unobserved heterogeneity 181

BIBLIOGRAPHY 186

vii

1.1
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

3.1
3.2

3.3
3.4

3.5
3.6

LIST OF TABLES

Consistency Properies of the MLEs for Dynamic Unobserved—effect Models 25

y“ = pyi,t_1+a0+al yio+ci+eu under Normality Assumption; H0 : 6 = 60,

where p =0 ~ 0.95 ............................ 56
The P-Value of p Under Normality Assumption;H0 : 6 = 60, where p

=0 ~ 0.95 ................................. 60
yit = p y,,t_1 + (1’0 + a1 yio + C1: + 8,, under Non-normality Assumption . . 64
The P-Value of p under Non-normality Assumption; H0 : 6 = 60, where p

=0 ~ 0.95 ................................. 68
CMLE for the dynamic panel data of log-wage with unobserved hetero-

geneity, period21980 ~ 1987 ....................... 72
ya = p yi,t—1 + [3 113a + 010 + at 3.00 + $02 + e + 52': under Normality

Assumption; H0 : 6 = 60, where p : 0 ~ 0.95 ............. 73
The P-Value of p under Normality Assumption; H0 : 6 = 60, where p =

0 ~ 0.95 .................................. 77
y“ = p yi,t_1 + ,6 33,, + 00 + a1 3,1,0 + Eng + c, + an under Non—normality

Assumption ; H0 : 6 :2 60, where p = 0 ~ 0.95 ............. 81
The P-Value of p under Non-normality Assumption; H0 : 6 = 60, where

p = 0 ~ 0.95 ................................ 85
CMLE for the dynamic panel data of log—wage with unobserved hetero-

geneity period21980 ~ 1987 ........................ 89
Performance of DIF, GMM, GLS and CMLE (a) .............. 90
Performance of DIF, GMM, GLS and CMLE (b) ............. 93
Performance of DIF, GMM, GLS and CMLE (c) .............. 96
Performance of CMLE (d) .......................... 96
ya = pythl + a, + 7 aiyiysl + 5,1 under Normality Assumption ...... 119
The P-Value of p under Normality Assumption. ya : py,,t_1 + a,- +

’yaiqu + 51‘; ............................... 121
ya = pyi,t_1 + a,- + 7 dig/LA] + 5,", under Non-normality Assumption . . . 123
The P-Value of p under Normality Assumption. y“ = py,,t_1 + a,- +

’Y nigh-J4 + Eu ............................... 125
y“ : py,,t_1 + 52:“ + a, + 7 aiyi,t_1 + 5,, under Normality Assumption . . 127

The P-Value of p under Normality Assumption. ya : pythl + 6:13,, + a,- +
’7 air/“-1 + 5“ ............................... 128

viii

3.7
3.8

3.9

3.10

3.11

3.12

3.13

3.14

4.1

4.2
4.3

4.4
4.5

4.6
4.7

4.8
4.9

ya = pygtq + 7351:“ + a, + 7 aim-J4 + 5,; under Non-normality Assumption 129
The P-Value of p under Non-normality Assumption; y.” = py.,-,t_1 + [3.1“ +

a, + ’7 a,y,-y_1 + 51'; ............................ I30
Empirical Evidence: Log Hour Wage, Case a: lnwageit : p l'v'zrwagei‘tq +

a,- + 7 a,- -lnwage,,t_1 + 5a ........................ 131
Empirical Evidence: Log Hour Wage, Case bzlnwageu = 6 lnrwagem_1 +

a,- + 7 (a, — pa) - ln'wagei,t_1 + en .................... 131
Empirical Evidence: Log Hour Wage, Case czlmuageu = p lnwagemq +

(it dt + a,- + 7 a,- - lnwagemq + 5,, .................... 132
Empirical Evidence: Log Hour Wage, Case (1 :lnu'ageu : 19 lnwage,‘,_1 +

6; d; + (11+ '7 (C11 - pa)-lnwage,~y_1 + 51'; ................ 132
Empirical Evidence: Log Hour Wage, Case ezl'nwagm = p lrzxwagemq +

6 union“ + a,- + 7 a,- - lnwagethl + Sit ................. 133
Empirical Evidence: Log Hour Wage, Case f: lnwage“ = 19 ln'wage,y_1 +

6 union“ + a,- + 7(a, —— pa) - l'n.'wage,,t_1 + e“ .............. 133
Model a :<I>(y,~t) = A(p y,‘t_1 + (1,), a,- is Normal; H0 : 6 = 60, where p

=0 ~ 0.95 ................................. 167
The P-Value of p for Model a: H0 : p = po, p =0 ~ 0.95 .......... 168
Model b :<I>(yit) = A00 yi,t—1 + (1,), ai is Non-normal; H0 : 6 = 60, where

p =0 ~ 0.95 ................................ 169
The P-Value of p for Model b: H0 : p = p0, p =0 ~ 0.95 ......... 170
Model 0 :<I>(y,-t) = A(p y,,t_1 +6131 + ai), a,- is Normal; H0 : 6 = 60, where

p =0 ~ 0.9 ................................. 171
The P-Value of p for Model c: Ho : p = p0, p =0~ 0.95 .......... 172
Model d :<I>(y,~t) = AU) yet—1 + 13.1",“ + (1,), a, is Non—normal; H0 : 6 = 60,

where p =0 ~ 0.9 ............................. 173
The P-Value of p for Model d: HO : p = p0, p =0~ 0.9 ........... 174

Empirical Evidence for Labor Union Membership, Period:l980 ~ 1987 . 174

ix

CHAPTER 1

Overview Of The Linear AR(1)
Model With Unobserved Effects

1. 1 Introduction

The AR(1) panel data model with an additive, unobserved effect has received
much attention in recent years. Nickell (1981) noted that the usual within, or fixed
effects, estimator was inconsistent with fixed time series dimension (T) as the cross
section dimension (N) gets large. Further, maximum likelihood approaches that
either treat the initial condition as nonrandom - in particular, independent of the
unobserved heterogeneity ~ are also inconsistent for fixed T. As most panel data sets
on individuals, families, and even firms are characterized by small T and large N,
the interest in obtaining a consistent estimator of the autoregressive root with fixed
T has become an important problem.

Anderson and Hsiao (1982) (AH for short) show how a simple instrument variables
(IV) estimator, obtained from the first-difference equation, is consistent for fixed
T. Subsequently, the AH estimator was shown to have poor properties when the

autoregressive root is large ( see Arellano and Bond [1991], Sevestre and Trognon

[1990]). More recent work has proposed additional moment conditions, often based
on further assumptions, that can be used in generalized method of moments (GMM)
estimation to improve upon the basic IV estimator( e.g. Arellano and Bond, [1991],
Arellano and Bover [1995], and Ahn and Schmidt, [1995]). Furthermore, Blundell
and Bond (1998) and Hahn (1999) recently show that the gain of efficiency of GMM
over a certain range of parameter space is significant.

The current chapter is to give an overview of the prevailing estimators for a linear
dynamic panel data model with an additive, unobserved effect. A commonly used

dynamic model for panel data in the AR(1) model:
ya = P flu—1 + 3W5: + Um 'i=1,--oaN; t:1,~--,Ta “-1)

where u“ = a,- + Ea and a, is unobserved heterogeneity. Here, yit is a scalar and 13,-, is
a K—vector random variable. Most available panel data sets contain a large number of
observations on individuals (N) over a limited number of periods (T), which means
that a sensible asymptotic analysis treats N —+ 00 with fixed T. With T fixed, the
stationarity assumption p < 1 is not necessary for usual inference procedures, but
p < 1 is relevant for most of empirical applications. Because we just consider the
case of N —+ 00 with fixed T, we do not distinguish semi-consistency with consistency
and just adopt the consistency term (or inconsistency) standing for semi-consistency
(or semi-inconsistency)(Nerlove and Balestra [1966]). We use the setup of (1.1) as
a standard model to discuss different approaches to estimate the parameters in the
following sections. Sometimes, we use full matrix notation to express equation (1.1)

as follows:

Y=pY_1+X§+ Da+e, (1.2)

with

K 3111 \l K 910 \ K 113]] ‘Tfl \

 

 

 

 

 

 

 

 

= Z 2 1 J:
Y 311T a Y-1 3111—1 ’ X 331T 1171‘ ’
K yNT ) K y;\",'I‘—l ) K frivr ‘var /
NTXI NTXI ‘ NTXK
{ €11
( a1 )
[7’1
02
E: ELT 7a: aé: I 3D=IN®1T2a
BK ,
K ”N / le
le

 

 

KEMT/ ,
NTxl

where [T is a (T x 1) unit vector.

For convenience, we define two matrixes used often in this chapter:

W, = 1N <3; (1T — £1), and “B, = IN a is,

where [T is T-order identity matrix and JT = (1, . . . , UixT'

The plan of this chapter is as follows. Section 2 considers the inconsistency of
LSDV estimator when T is finite. Section 3 considers several estimators from the
setup of dynamic error component models when the unobserved effects is assumed
to be random. Section 4 consider the MLE estimator in consideration of the initial
conditions. Section 5 show the gain of efficiency from the CMM estimator by imposing
the extra moment conditions and the restrictions on the initial conditions. Section 6

gives some concluding summary.

1.2 The Inconsistency of the LSDV Estimator

In the static case in which all the explanatory variables are exogenous and are

uncorrelated with the effects, the OLS estimator, although possibly less efficient, is

3

still unbiased and consistent. But in the dynamic case the correlation between the
lagged dependent variable and individual—specific effects would seriously contaminate
the property of OLS estimator. We will show the bias of the least-squares dummy-
variables (LSDV) estimator for a dynamic fixed-effects model and then see how to
treat with the problem. We assume that the disturbances satisfy the conditions as

follows:

E(51t[y:,t—la---anOeXi) : O (1.3)

V(e,-t[y,-,t_1, . . . ,y,0, X,) = 03 for all i. and t
i.e., the disturbances have a zero conditional mean ( which implies they are serially
uncorrelated), and are homoscedastic. In the traditional fixed effects approach, a,- is

treated as a scalar parameter to be estimated. l\r’Iultiplying the equation (1.2) by W",
WnY : anY_1 + WnXfi + Wne, (1.4)

Because Wn is a symmetric idempotent matrix, the LSDV estimators for p, 73 can be

expressed as the within estimator are as follows:
—1

p Y11WnY_1 Y_’l lit/’nX Y: lWn 1C] (1 5)
B X'W, Y_1 X’W,,X X'W, K,
The estimator of unobserved effects is as follows:
Oizgi—[Algi’_1—Ti3, 2:1,...,N, (1.6)

where 92‘ = %Zf=1yit and ii = %Zf=1$n-
When N —-+ 00 with fixed T, given the above assumption of (1.3) on the distur-

bance, we write the equation of (1.5) in the probability limit as follows:

plim fiYfill/VnY4 plim FIT—.yLII/VnX

N—ooo

IQ >b)

plim ( ) = ( f3) ) +
N.... plim 7.?7X'Wnic. plim 1:37X ’WnX

N—ooc N—ooo

° 1 I ,r
eefififle

' 1 I
Iii—I‘m W X WnE

00

We prove plim Wiijll/Vne # 0 in the following. The inconsistency of this esti-
N—too
mator rely on the fact that, given the assumption about the disturbances, one has

plim WifX’Wne = 0 under the strict exogeneity assumption, but

 

B132 NITY/l W115 : R132 7v]? 2:12:1(yu—1 — 6i,—1)(€it — 5i)
Z — 11313.1: fizz; 3711-153 (1'8)
: _0'2(T—-I)—Tp+pT#0
T (1 - p)2

Equation (1.7) can be rewritten as follows:

~ _ plim —,—Y_’ Wne
. p p __ NM AT
plim ( 3 3 > — A x °°

N—voo
0

: l I ,r
A“ ‘ plim Vii/4%,“:

N-eoo

A21 - plim —,}—T-YL,W,,5

N—ooo
-—l

’ WnX

plim fiylli/VnY-l Plim fiy—l

where A = N"°° N"°°
plim 7—WX' W Y_1 plim N—le’ W X

N —~oo

When T is kept fixed, the LSDV estimator of an AR(1) fixed effects model is not
consistent. The inconsistency mainly comes from the fact that correlation exists be—
tween (yi‘t_1 — 3],) and (5,1 — 5—,). In other words, the individual means, y, and e“ are
correlated with each other, although the past of y” and 5a are uncorrelated. As it
is clear from equation (1.8), when N and T ——> 00, this estimator is consistent since

[plim "NIT L 1Wne‘ = 0. Unfortunately, most panel data sets of interest contain small
number of time-periods. Therefore, we should look for estimation methods that are
consistent when T is fixed. A traditional way to tackle the problem of within esti-
mator is to use an instrumental variables estimation method after a transformation

to estimate a,. To be more precise, using an appropriate transformation and then IV

can implemented to consistently estimate the parameters.

5

Balestra and Nerlove (1966) have shown that Two-Stage Least Squares which
uses current and lagged values of :13“ as instrument variables is available. Based on
the model of (1.2), let us define the complete set of instruments as Z” = (D, Z),
where D can is the set of dummy variables accounting for the individual effects. An

appropriate transformation for (1.2) is as follows:
Pz-szpz-Y_1+PZ’Xé+ PztDQ-fpz-E, (1.10)

where P2. = Z *(Z “Z *)"‘Z *', which is projector onto the space spanned by Z *. By
the Frisch—Waugh theorem, the solution to the problem amounts to applying the OLS

to the equation as follows:
WnPZJ/ = anPZ—Y_1 + WnPZ.Xé + WnPZ.e, (1.11)

where Wn = I — PD = I — D(D’D)“D’ = [N ® (IT - 11.1). The fixed effects a,- can
be ”estimated” as ti = PD(Y — Y_1p — X g) If we add the assumption on the error
terms as follows:

2

e“ are independently and identically distributed with mean 0 and variance 05,

(1.12)
the property of x/N-asymptotic normality is valid, i.e.,
‘ 2 - 1 “r I _1 I ~ _1
x/N(6 — Q) ~ Normal(0, 0€( pllm EX WnZ(Z wnz) z WnX) ), (1.13)
N—-oo i

where .9. = (Mg) and X = (Y_1,X).
Anderson-Hsiao ( 1982) have proposed to use as instrument variables the lagged
first-difference of dependent variable or the level of dependent variable lagged two or

more periods after first-difference transformation into equation (1.2) as follows:
AY=pAY_1+AXé+Ae, (1.14)

It is obvious that the variable y,,t_2 (or lagged more periods) and Ay,,t_2 are valid
instruments since they are correlated with Aqu but uncorrelated with the distur-

bance A5“.

Arellano (1988) considers a specific model allowing for only one exogenous vari—
able which follows a stationary AR(1) process plus a lagged endogenous variable and
he has shown that the variance of the estimator using Ay,,t_2 as instrument variable
can be very high due to near—singular matrices entering its definition. Arellano (1988)
proposed yi,t__2 instead of Ay,,t_2 as the instrumental variable. Given the assumptions
(1.3) and (1.12), the property of JN-asymptotic normality is valid. We write the

asymptotic distribution as follows:

x/MQ _ Q) N Normal (0, 03( plim %((ZAX’)“Z\IJZ(AX’Z)‘1))) , (1.15)

where z = [21,...,ZN]’ , X =[Y_1,X]

and

WZIN®ZD=IN® , (1.16)

1

(0 —12)

since the disturbance in model (1.14) MA(1).

 

 

There exists estimators more efficient than that of (1.15) since the disturbance
in model (1.14) is MA(1). Transforming the equation (1.14) by multiplying \Il—Tl, we

have an equation as follows:
t%AY=pt%AKJ+t%AXQ+t%Ae (LN)

Sevestre (1992) suggested using as instruments y,t_2 or Agata, plus the current and
lagged values of AX or \IlzzlAX, provided that X is strictly exogenous. Such an esti-
mator is more efficient than the one using the same instruments on the untransformed

equation (1.14) ( see White [1984]). Given assumptions (1.3) and ( 1.12), the property

of W-asymptotic normality is as follows:
\/N(6 — Q) ~ Normal (0, 03] plim N((A)~(\IJ"1Z)’IZ’\II’]Z(Z'\IJ"1A)~()‘1)]). (1.18)

Nevertheless, \Ilzz'lAe means that the disturbances are linear combination of e“ and
hence the lagged values of yl't__2 or Agata are no longer valid instruments except for
yio, but nothing can be said about the relative performance of this estimator and the
ones suggested by Anderson-Hsiao (1982), since the instrument is different.

Based on the differenced equation (1.14), Arellano-Bond (1991) proposed another
way to find a more efficient estimator, generalized instrumental variables estimator,
which contains all the orthogonality conditions that exist between lagged values of

the endogenous variables and the disturbances. Assumption (1.3) implies
E(yi5AEit):0,f:2,...,T,8:0,...,t—2. (1.19)

At period of t, y,0,y,1, . . . ,y,,t_2 are valid instruments for Ay,,t_1, respectively.
Because X is assumed to be strictly exogenous variables, A22“ is a valid instrument

for itself. Then the complete set of instrument variables can be defined as

 

 

(yio 0 0... 0 Am 0 0 )
Z = 0 (y,0,y,-1) 0 0 0 A3723 0
I (910, 3111,3112)
K 0 (y,0,y,-1,...,y,-,T_2) 0 Amy-T}
(1.20)

The generalized IV estimator is defined as
. ~ ~ —1 -
Q = (AX’PZAX) (AX’PZAy_1), (1.21)

where

N
. ~, . ~ 1
PZ=ZFZ wzthP= (quz (N12120:

Actually, the estimator is to apply GLS to the model (1.13) multiplied by 2’ as

follows:

Z’AY .—. Z'AY_1,0 + Z’AXQ + Z’Ae. (1.22)

Since the e“ is not autocorrelated in this model, the estimator is the most efficient
within the class of instrumental variables estimators using lagged value of ya as instru—
ments. Given assumption (1.3) and (1.12), Its property of VN-asymptotic normality

is as follows:

- 1 2 2 —1
\/N(Q _ Q) ~ Normal (0 , plim N (AX’PZAX) ) . (1.23)

N -¢oo

We can write GMM estimator by replacing P2- with PZ“, where P2 = ZI‘Z’ with
F = (1%,- 211:1 Z,14,-1_/§Z)“1 where 1_/, is the vector of disturbances of the differenced
equation (1.14). Nevertheless, to ensure that the instrument of (1.20) is valid, the
order of autocorrelation of disturbances is required to be not greater than one.

The choice of these various estimators in estimating an AR(1) fixed model depends
on two main criteria: the degree of serial correlation of the e“ disturbance terms and

the exogeneity of X. We end the section with some conclusion as follows:

1. When the values of e” are correlated and 2“,, is strictly exogenous the Two-Stage
Least Squares estimator which use the current and lagged values of the 22,-; as

IV (Balestra—Nerlove, 1966) is better one.

2. If the values of en are correlated while :ru is still strictly exogenous then the
generalized IV estimator proposed by Arellano-Bond (1991) is better than oth-

ers .

3. If the values of e“ are correlated and at“ is not strictly exogenous then using

lagged values of the Ar“ as IV for estimating the model (1.14) is preferred.

1.3 Estimators of error components model

The section considers estimation of the AR(1) model under the assumption that.
the unobserved effects are always random. The model of (1.1) or (1.2) is adopted in

this section and we use the following assumptions:
E(a,) = E(5,,) = 0, for all '1' and t,
E(a,.r,,) = 0, for all i and t,
E(a,5,,) = 0, for all 2' and t,

(1.24)
03 12]}
E(a,aj) =
0 «1% j.
E(5, 5;)— — 051T, where 5,: (5 5,1 ..... 5,7~)’, for all i.

(1.24) implies that the special second-order structure of disturbances in model (1.2),

11,, = a, + 5,, or u, = a + 5,. as follows:
Var(u,) = E('u.,u;) = QT = JEWR +(052 + T02 a)B,,= USU/IQ, +—B n),

with 62 = 03/(03 + T03). Under the specific assumption we can obtain the GLS

estimator by simply imposing OLS on the equation (1.2) multiplied (W, + 6B,):
(W, + 6B,)Y : p012, + (may.1 + (W, + 03,)Xg + (W, + 6B,)U, (1.25)

where U = (111, . . . ,uN)’. Nevertheless, the GLS estimator is not the most efficient
estimator because it does not impose any restrictions on the relation between 31,0 and
a, or 5,,. To shed light on the importance of the initial value, we write substitution
recursively in equation from (1.1) to obtain:

yit :pty10+:pI—1~Tz,—t j+lf31 +_:ppaq+:p7_l Egg—131.1. (1.26)

j: —1

Each observation on the endogenous variable can be expressed as a linear combination

of four variables: py,0, 22—11074 .r,,_j+1_'3,, —£’p:a,, and 21.41274 5,, 1+1. The first

10

term p‘yw depends on the initial value. At this stage, it is clear that the initial values
do influence the asymptotic behavior of estimator as long as T is finite and p is not
zero. Theoretically, the first date of the sample is arbitrarily chosen and we cannot
easily justify a different treatment. of the first and the subsequent observations. For
example, we assume y,, = f(a,, 5,,, 5,,,_1, . . ..) This means that the outcome on y in
time t depends on the individual effects a, and on a serially uncorrelated disturbance
5,,. Therefore, if unobserved effects are non-random, then the initial observations are
also non-random; and on the contrary, if the unobserved effects are random, then the
initial observations are random.

As a practical matter, the assumption that the unobserved effects are non-random
means that the initial observations are independent of the exogenous variables and
the unobserved effects, usually an untenable assumption. The interpretation of the
relation between the initial observation and the unobserved effects characterize the
dynamic panel data with random-effect formulation. We assume that 3,1,0 is identically
and independently distributed variables characterized by the second order moment
E(y,-20) and the correlation with a,, E (y,0a,~). Replacing 6 with \/X, Maddala (1971)
proposed A-class estimator and have shown that all usual error—component estimators
belong to such an estimator for a AR(1) framework. For example, the within estimator
has A = 0; OLS if /\ = 1; GLS estimator if A = 62 ; If p =0, then, obviously, all these
estimators are consistent, while almost all A-class estimators are not consistent if p #
0.

Under the above assumption on the distribution of y,0 the asymptotic bias of any
A-class estimator is dependent on E (11,20) and E (y,0a.,). This shows that assumptions
on the initial observations do influence the magnitude of the bias of these estimators.

The main result is as follows: whatever E(y,20) and E(y,0a,) are, plim p()\) is an

N—¢oo

11

increasing function of /\ and hence the relation is assured as follows:

plim pm) < p < plim pm?) < plim ,3(1) < plim p(oo). (1.27)

N—voo N—ooc N—.oo Naoo
It is obvious that there exists a value X“ E [0, 62] such that plim p().*) = p. Sevestre-
N—ooo
Trognon (1983) have given the value of X" as follows:

1 " [JT 130/1001)
1— p 03 + 052

 

x :K(1—6)/( +K(1—6+T6)), (1.28)

with K = (T — 1— Tp+ pT)/T(1— p)2, (5 = 03/03 + 062.

When E(y,0a,) = 0, /\* is equal to 62, which means that the consistent estimator
A-class estimator is the GLS estimator. Usually, A“ 74 62, which confirms GLS is not
consistent in such a mode].

It is worth noticing that sometimes the A-class estimator cannot be thought of as
an estimator because of unknown parameter A, which leads to two—stage estimation;
and hence the property of x/N—asymptotic normality heavily depends on the asymp-
totic property of )1. p(/\*) and 3(x\*) with X‘ defined as in (1.28)are derived from the
AR(1) model and such an estimator cannot be extended to AR(p) models.

GMM procedures are thus possible to impose some restrictions to find more ef-
ficient estimators. We assume ,3, = 13 for all 1'. By adding the assumption that
3110 = C + 01101 + 012510

proposed by Anderson-Hsiao (1982) into the system (1.2), the system is a triangular

model with T+1 endogenous variables y,0, . . . ,y,7~ and T+1 exogenous variables (C
and 513,1, . . .,2:,T). We can express the T+1 equations by compact matrix form as
follows:

A31- 3;, = 17.1" (1.29)
where

912(91039119- ' '1y17‘),a 32,-:(1,$2‘1,. - '1xiT)’9

I
21' : (0101+ (125,0, (1.,- + 5,1], . . . ,a, + 51.731

12

and

{C 0 .H 0)
0 f3
0

(0 .H 0 0 1) )0 ”"” fl)

where A and B are (T+1) >< (T+1) matrix. The structure of disturbances are defined

 

 

 

 

as E(77,) = 0 and Var(gi) = fl and hence

wr’

Q: , VZO'EJT-i-UEIT,
'r V

2

E, r’ = 0102(1,...,l).

w = ego: + (130 a
In the simplified case, we find that the IV estimator is consistent. If the disturbance
have an error component structure, the 3SLS estimator is not as efficient as full in-
formation maximum likelihood (FIML) estimator. If the variance-covariance matrix
Q is unconstrained, then 3SLS and FIML are fully efficient. Based on the differ-
enced model, several IV or GMM suggest consistent and efficient estimator, but their
relative efficiencies are hard to determine. For example, the GMM estimator with
asymptotic efficiency may not often perform better than the Balestra-Nerlove estima-
tor in finite samples ( See Sevestre—Trognon [1990]). We found the interpretation of
the initial conditions make it possible to obtain more efficient GMM estimators. The
subsequent section we discuss the MLE estimators with the different treatments of

11,0. And then we introduce the initial conditions into GMM to obtain more efficient

estimators.

13

1.4 Properties of the ML estimator

According to (1.10), the two-stage estimators of Balestra and Nerlove (1966) is OLS
applied to equation (1.11). Such an estimator is equal to MLE assuming that y,0 is
non—random. Because this assumption implies that the initial observation is discarded
from the system, the ML estimator is not the unconditional ML estimator. The
asymptotic correlation of 31,0 and a, contaminate the consistency of Balestra-Nerlove
estimator when the nonrandom assumption of the initial observation is dropped.
A natural solution is to construct the full likelihood function which includes the
initial observation to obtain the unconditional ML estimator. Barghava-Sargan (1983)
proposed an unconditional estimator by considering the framework (1.1) into which

the observed individual variables 2, will be introduced. We write the model as follows:

921 = P 3111—1 + 51311.3 + 217 + U11, U11 = (11 + 511,
' (1.30)

The initial values are assumed to follow:
y,0 = 2,92 +140. (1.31)

Such a formulation has been adopted by Chamberlain (1984) and Blundell and Smith

( 1991) among others. The unobserved effects is assumed to be as follows:

a, = 6211,10 + 0,, (1.32)
where c, is independent of 14-0. In this model (1/,0,c,, 5,1, . . . ,5,7~) are distributed as
Normal(0,diag(030,0§, 031“) and the log—likelihood function is:

£(p.I3.7.t/2,0§o,0390§) = —NT10527T — 1% 10s It?! — 42V- 10s 03

_ I N IQ—l , _ 1 N 2 (133)
Q 21:171. T2 202 21:1 ”101
y0

14

with
Ti, = (9:1 — P’yio — $113 — 2:"? — W140, - . . a y-rr — Pyi/F—l “ $2773 - 2W — W40)
V10 = 3120 — 21¢,

f2 = aan + (a? + T0395”.
The ML estimators solve the normal equations:

aL<pafii 7971/),0309 0:, 03/80), (31 ’77 1(1), 0503 039 052) : 0 (134)

The ML estimators of <15 and 030 are solved by the equation from (1.34) as follows:
N

1
85/30 = 7—— : uwztrr, = o

(05 + T03) 1:1

and

N 1 N

85/6030 = ——2— + ,———.— Zufo = .

03/0 310 i=1
These imply that these two estimators are OLS estimators on the equation (1.31).
The other estimators can be solved by replaced the residual 11,0 in T,- on equation
(1.34). This approach turn out to be two—stage estimation from which we can solve
the ML estimation of (1.30) where the unobserved uio is replaced by the residual 0,0

obtained from the OLS estimation on (1.31)in advance. Obviously, if we add another

term 33,001 into the equation (1.31), the initial observations are defined as follows:
yiO : 6021' ‘i' 1300’ ‘l‘ V10 (1.35)

and (1-30) are unchanged, i.e., the variable 33,0 does not enter the autoregressive
equation and hence the simple split ion between OLS and MLE disappears. Sevestre-

Trognon (1990) proposed an auxiliary model:
yit : ,0 3124—1 + :r,,,z3 + zn + 517.05 + u“. (1.36)

The ML estimation of p, 6, 'y, o, ‘l/J, (5, 050, 03, and 03 are asymptotically equivalent to

the ML estimators of (1.30). If (i is OLS estimator of a on (1.35), it is shown that. (71* :

15

d + 7%; is asymptotically efficient if 5* and 1])" are the ML estimators of 6 and 2/2 in the
auxiliary equation (1.36) (See Sevestre-Trognon [1990], and Blundell-Smith [1991]).
Such a ML estimation suggests the asymptotically most efficient estimators when the
disturbances are normal. The consistency of MLE depends on the assumptions on
the initial observations. The other less restrictive assumption on yw is to leave the
correlation of initial observations and unobserved effects E (y,0a,) free and consider
it as a parameter to be estimated. By specifying a distribution of 31,0 with mean ayo
and variance 0,2,0, as well as cov(y,;0,a,-) = 92030, Anderson-Hsiao (1982) solved out
the unconditional ML estimators from the model (1.30) and studied the consistency
properties of the MLES for dynamic model with a random-effect formulation. We
give a summary in Table(1.1). It is obvious that the properties of ML estimators for

dynamic random-effects models depends on the assumptions on the initial conditions;

so do those of GLS and GMM.

1.5 The efficiency of GMM estimator

The previous IV methods for estimating the dynamic panel data model (e.g.
Anderson and Hsiao [1981]; Hsiao [1986]; Arellano [1988]; Arellano and Bond [1991])
first-difference the equation to remove the unobserved effects, and then use instru-
mental variables, using as instruments values of the dependent variables lagged two
or more periods. More recent papers (Ahn and Schmidt [1995]; Arellano and Bover
[1995]; Blundell and Bond [1998]; Hahn [1999]) proposed additional moment condi-
tions, often not exploited by these estimators, that can be used in generalized method
of moments (GMM) estimation to improve upon these IV estimators.

For expository purpose, we consider the simple dynamic panel data model, which

does not contain any additional regressor beyond the lagged dependent variable, will

16

be used often to express the available moment conditions.1 We write the simple model

as follows:
git = py,,t_1 +11“, 11,-, = a,- + 5,). (1-37)

Ahn and Schmidt (1997) assume that all variables across individual are indepen-
dent. Various subsets of assumption about initial conditions and the errors are made
in the following:

(i) For all z', 5,, is uncorrelated with 31,0 for all t.
(ii) For all i, 5,, is uncorrelated with c,- for all t.
(iii) For all i, 5,, are mutually uncorrelated.

(iv) For all 2', var(5,-,) is the same for all t.

Given the four assumptions (i)- (iv), there are several plausible cases of 16 possible
combinations corresponding to imposing or not imposing each of then. We just dis-
cuss the case imposed by assumption (i)-(iv) to explain the exploitation of additional
moment conditions. Let E be the covariance matrix of (5,1, . . . ,5,7~, yio, (1,) (see Ahn
and Schmidt [1995b] Eq.(6)). Let A be the covariance matrix of (11,1, . . .,u,-T,y,-0)
(see Ahn and Schmidt [1995b] Eq.(7)). Assumptions (i) - (iv) imply that we have
[(T — 1) + (T — 1) + (T(T — 1)/‘2 — 1)] restrictions on 2 as follows:

00,, E E(5,«ta,-) : 0, for all t,
a —:—E ,5, =0, forallt,
0‘ (yo ‘) (1.38)
at. E Meta.) = 0, t 74 8,
(

5&8“) Z 0, TOT all t.

 

1 The moment conditions implied by exogeneity assumptions on additional regressors have been
identified by Shmidt, Ahn, and Wyhowski (1992), Ahn and Schmidt (1995b), and Arrelano and
Bover (1995)

17

(1.38) implies restrictions on A in three types:
A0, E E(y,-0’u.,:t), t = 1,. . .,T,
At, E E(u,tzt,5), t 55 s; t, s = 1,. . . ,T, (1-39)
A” E E(u,2,) = 03, t: '1,...,T.
These restrictions corresponding to the moment conditions are as follows:
E(y,-0Au,:t) : 0, for t 2: 2, . . . , T,
Exug-—u§)==0, fort:=2,u.,7§ (L40)
E('u,~tu,-, — 21,221,“), for t = 3,. . . ,T,3 < t.
The conditions of (1.40) are algebraically equal to the following:
E(y,-5Auz‘t) =0, fort=2,...,T, s=0,...,t—2,
E(u,-TAu,t) = 0, for t = 2,. . . ,T — 1, (1-41)
E(E,Au,t) = 0, fort: 2,. . .,T,
Where H,- = T"1 2:111“. The first condition of (1.41) is derived from the differenced
equation of
Ayn = part-H + A5,, fort = 2,. . . ,T (142)
Holtz-Eakin (1988), Holtz—Eakin, Newey, and Rosen (1988), and Arellano and
Bond (1991) have shown that the valid instrument variables for equation (1.42) is
(yio, . . . ,y,,t_2) which implied by the first condition of (1.41). Given assumptions (i)
- (iv), Ahn and Schmidt (1995b) show that the second condition of (1.41) can be
replaced by
E(y,-‘t_2Au,,t_1 — y,,t_1Au,,) : 0, t = 3,. . .,T, (1-43)
which are linear in p.
To derive the general result, Ahn and Schmidt consider the model including ex—

ogenous variables. We write its compact matrix form as

Y:Y_1p+X,8 +27 + U=W§+U, (1.44)

18

where U 2 0+ 5, X is time—varying explanatory variables, Z time—invariant explana-

tory variables and Y_1 lagged dependent variables. We write the T observation for

individual 2' as u,(§) = y, — W,£ to emphasize the dependence of u,- on (E. Exogeneity

assumptions on :13,- and z, generate linear moment conditions of the form

E[R;u,-(§)] : 0,

(1.45)

Where R,- is a function of the exogenous variables and g is (p, (3’,7’)’, E(y,sAu,-t)

leads to the moment conditions being a linear function of 5 and can be written as

E(A§u,-(€)):0, where A, is the T x T(T — 1)/2 n'1atrix

 

*3/10
’5/40
0
0

0
0

0
—(yi09 ya)
(910131211)
0

0
0

O 0
0 O
-(y109 3111,3112) ... 0

(1110-. 31111 3112)
0 ‘ " —(yi09yi17"'7yiT-2)

0 (y101yila°°~1yiT—2)

 

(1.46)

and the moment conditions in the first condition of ( 1.42). The moment condition

in (1.43) leads to the moment condition being linear in g and can be written as

E( [,u,(£)):0, where Bl,- is the T x (T — 2) matrix defined by

 

P —y.-. 0 0 l
(3111 + 3112) _yi2 0
*1/12 (3112 + .7113) 0
0 _y13 0
0 0 . . . —yi,T—2
0 0 — - . (yiT—2 + 911—1)
_ 0 0 . . . —yi,T—1

 

19

(1.47)

The third of moment condition (1.42) will lead to moment condition being quadratic
in 6. Let S, (T x 1,) be made up of columns of 12,, A,, and B,, so that it represents
some or all of the available linear instruments. The corresponding linear moment

conditions are E(f,(£))=0, with

fi(€) : Si'liz'fél : flz' + f'Ziée fli = 5:311, f'Zi = —S;W’,. (148)

Let the dimension of g be k, 11 Z k is assumed such that f, can identifyé. The rest
of IQ = l — 11 moment conditions will be written as E(g,(§))=0. g,(§) = g], + 92,6 +
(I®€’)gg,§, where 91,, 92,, and 93., are L; x 1, lg x k and lgk x k matrices in respective
and the dimension of the identity matrix is [2. {AGMM can be obtained by GMM based

on all of the moment conditions:

f1(€) )

£30714“)le :0
91(5)

VIN —=- N_] Z, "11'an With fN, ftNa fay, 9N» 911v, 921v and 931v are defined similarly;
MN E BmN/Bé’ E (Ffv,G]V) Let [W E plim M, , F E plim MN, F E plim FN
N—ooo N-+O° N—’°°

and G E plim GN. Define the optimal weighting matrix:

N—ooo

[o o ,'
Q E H M [ E plim c011 (WZmJ .
N—aoo '
ng 999] i

Let Q be a consistent estimate of Q of the form

. 1 . - 1

(2 Z N Z m,(§)m,(§) ,
where 6 is an initial consistent estimate of g. The efficient GMM estima-
tor EGMM minimizes N mn(§)’f2‘lrrz.N(§). The asymptotic covariance matrix of
N1/2(€GMM - 6) is [MD—IMF]. Ahn and Schmidt showed that the statistic

JN : N mN(£~£;M MQ-lmwvécw) can be used to test the validity of the moment

condition E (m,(§)):0. The statistic is asymptotically chi—squared with (l-k) degrees

20

of freedom under the joint hypothesis that all the moment conditions are valid.

It is worth noticing that the assumptions of uncorrelatedness (i) - (iv) is not
sufficient to imply the asymptotic. equivalence of IV and GMM estimates when the
moment conditions of the first condition on (1.42) because the asymptotic equivalence
of IV and GMM includes the restrictions on the fourth moment condition (such as
cov(y§0, 5%)). Ahn (1990) has shown that the asymptotic equivalence of IV and GMM
based on the first condition on (1.42) is ensured when (i) - (iii) are strengthened by
independence assumption while (iv) is maintained. Wooldridge (1996) proposed more
general treatment of case in which the asymptotic equivalence of IV and GMM holds

if we replace the assumptions (i) - (iv) with conditional expectations as follows:

E(€1 [11.11.611.81- We — ) = 0
1 1 1 1 (1.49)

E(5121[7J1020i.15112- . - aEi,t—l) I 055-

The conditions of (1.49) will be employed in the study of the conditional maximum
likelihood estimator in later chapters.

Blundell and Bond (1998) consider two alternative estimators that impose further
restrictions on the initial conditions process, designed to improve the properties of
the standard-difference GMM estimator. The one is extended linear GMM estimator
that uses lagged differences of 31,, as instruments for equations in level, in addition to
lagged levels of y,, as instruments for equations in first difference. The other is the
use of the error components GLS estimator on an extended model that conditions
on the observed initial values. Both estimators require restrictions on the initial
conditions process. Asymptotic efficiency comparisons and Monte Carlo simulations
for the simple AR(1) model demonstrate the dramatic improvement in performance
of the proposed estimator compared to the usual first-difference GMM estimator,
and compared to non-linear GMM. They use stationarity-like assumptions to show

that system GMM estimators work better than the other estimator for ,0 close to 1

21

by exploiting all moment conditions: E(y,,,_,A5,,) = O for t=3,. . ., T and s 2 2,
E(u,,,Ay,,,_1)-—-0 for t=4,5,. . ., T and E('u,,3Ay,2)=0 as well as E(y,,u,, — y,,,_1u,,,_1)
coming from homoscedasticity restrictions on 5,, for t=3 ,. . ., T.

Hahn (1999) consider the AR( 1) panel model with fixed—effects. He investigate
the estimation method developed by Blundell and Bond (1998), which makes use of
the stationarity of the initial levels. By semi-parametric methods, he investigate an
alternative linear GMM estimator based on additional moment restrictions, which are

valid if we have

Cl a 00
yo=——-+u=—-—+ #56.

where 5, are i.i.d. mean-zero random variables. By numerically comparing the semi-
parametric information bounds for the case that incorporates the stationarity of the
initial condition and for the case which does not, it is found that the efficiency gain
is potentially.

Im, Ahn, Schmidt and Wooldridge (1999) showed that with panel data, exo-
geneity assumptions imply many more moment conditions than standard estimators
use. However, many of the moment conditions may be redundant can not increase
efficiency. They propose to establish the standard estimators’ efficiency. The redun-
dancy of moment conditions in GMM depends on relationships between the matrix
of expected derivatives of the moment conditions and the optimal weighting matrix.
They established results under assumption of no conditional heteroscedasticity, which
implies a simple and tractable from for the optimal weighting matrix. They prove
efficiency results for GLS in a model whit unrestricted error covariance matrix, and
for BSLS in a models where regressors and errors are correlated, for example the
Hausman-Taylor model. For models with correlation between regressors and errors,
and with unrestricted error covariance structure, they provide a simple estimator

based on a GLS generalization of deviations from means (see Im, Ahn, Schmidt and

22

Wooldridge 1999).

1.6 Conclusion

The presence of correlation between the initial observations and unobserved effects
contaminates the consistency of within estimator and related methods. Many ap-
proaches suggest a transformation to remove the unobserved effects, and then choose
instruments based on sequential conditional moment assumptions.( for example, An-
derson and Hsiao [1981]; Hsiao [1986]; Holtz and Eakin [1988]; Holtz, Eakin, Newey,
and Rosen [1988]; Arellano and Bover [1990]; Arellano and Bond [1991].) While this
treatment leads to the consistent estimators, the estimators are not efficient under
standard assumptions because it. does not make use of all of the available moment
conditions. The random—effects formulation raises the interpretation of initial values.
The traditional ML estimator (Balestra and Nerlove [1966]) is not generally consis-
tent when y,0 is allowed to be random.

This question will not occur when y,0 is included in the joint density function
as in unconditional ML estimation. When the other exogenous variables are intro-
duced into the form of y,0, the ML estimation of model (1.30) can be very complex.
Sevestre and Trognon (1990) proposed another auxiliary autoregressive regression to
tackle this problem. These treatments of initial conditions appears inflexible. Espe-
cially, when the non-linear model is necessary, it is difficult. to find or approximate a
proper distribution of y,0. The different treatment of initial conditions will be intro-
duced in subsequent chapters.

Although the A-class estimation is often not practical, the asymptotic bias of A-
class estimator sheds some light on the existence of more efficient estimators. By
containing the additional moment conditions identified in the Ahn and Schmidt’s pa-

per, the extended GMM leads to nontrivial gains in asymptotic efficiency. Several

23

literatures suggest that imposing moment restrictions in GMM estimation to obtain
more efficient estimators over a certain range of parameter space. Blundell-Bond
(1998) and Hahn (1999) have shown the improvement on the efficiency is significant
by the inclusion of the initial conditions.

Wooldridge (2000b) proposes a different treatment of initial conditions. His sug-
gestion is to model D(a,[y,0,X,q~) and then construct the density of (yiT,...,y,-1)
given (yiO,Xz‘T)- This allows 31,0 to be random and does not require us to find, or
even approximate D(y,0|a,, X17“). Further, we need not specify an additional model
for D(a,|X,T), or assume that a, and X,T are independent and then model D(a).
According to this framework, we can easily construct the conditional log likelihood
function to obtain the consistent conditional maximum likelihood estimator. Accord-
ing to this framework, we can easily construct the conditional log likelihood function
to obtain the consistent conditional maximum likelihood estimator. It pays to in-
efficiency that the method, conditional maximum likelihood estimator (CMLE), for
handling the initial conditions problem appears to be novel, and offers a flexible, rel-
atively simple alternative to the previous ones. The following chapters applies the
CMLE to deal with some topics of dynamic panel data models: linear AR(1) dynamic
panel data model with unobserved effect, the state dependence interacting with the

unobserved effects, and the binary choice model.

24

Table 1.1: Consistency Properies of the MLEs for Dynamic U nobserved-elfect Models

 

 

 

 

Case (A) (B)

p, 1'3, 03 Consistent Consistent

yiO fixed 2 . .
7, 0,, Inconszstent Conszstent

y,0 random

. p, ,3, 03 Consistent Consistent
(a)y,0 independent of a, , 2 2 [I , ,- p C ,- t
nyo, ’y, 00, ago nconszstent .zonszsten

, 2 . .
. .110. Conszstent Conszstent

(b)y,0 independent of a, p ‘ “ .2 2 I _ C .
11,0, 7, on, aye, c5 nconszstent .,onszstent

 

 

 

 

 

 

Case (A) is N fixed , T —> 00
Case (B) is T fixed , N ——> oo

25

CHAPTER 2

Conditional Maximum Likelihood

Estimator For The AR(1) Model

2.1 Introduction

A panel data model allows us to study the dynamics of economic behavior at an
individual level in which the individual heterogeneity is taken into consideration. As
discussed in Chapter 1, fixed effects approach does not lead to a consistent estimates
for the parameters. The inconsistency mainly comes from the fact that the within
transformation induces a correlation of order 7}. between variable and the error. The
estimator of the conditional MLE explored by Balestra and Nerlove (1966) is not.
consistent as well because they treat the initial observations as nonrandom and such
estimators, for a wide of combinations of the parameters, are equal to the within
estimator and thus they are not consistent (see 'Iiognon [1978] ). Moreover, it is an
untenable assumption to treat the first observation as nonrandom, since that implies

it is independent of any other exogenous variables and any unobserved heterogeneity.

26

It is a natural solution to the estimation problem is to use maximum likelihood
principle when the disturbances are assumed to be normal. The assumption of fixed
initial observations can be relaxed when the likelihood function takes into consid-
eration the density function of the initial observations, that is, the likelihood func-
tion is ”unconditional” ( Barghava and Sargan [1983] ). In such an approach, we
first describe the distribution of the dependent variables (yT, . . . ,yo) conditional on
(xT, . . . , 221, a), where as, is strictly exogenous variables and (1 individual heterogene-
ity, t=1,...,T. We can specify the distribution of yo given (XT, a) to obtain the
distribution,D(yT, . . . ,y0|XT,a) and then integrate out a by specifying D(a|XT) or
more typically just assuming that a is independent of XT. This leads to a parametric
density function f (w, . . . , yOIXT; 90), which allows us to obtain the conditional max-
imum likelihood estimation with conditioning on XT. Traditionally, this is viewed as
”unconditional” MLE because the X 7‘ are treated as nonrandom. Unfortunately, such
an approach is made possible only provided that we have a steady state distribution
for yu- ‘The inclusion of XT makes matters more complicate. Barghava and Sar-
gan (1983) treated the initial observations as random accounted for by time-constant

variables and random errors in (1.31) as follows:
910 = 9522: + 14:0,

where z, is time-constant variable. The unobserved effects is assumed to be (1.32):
a.- = VII/1'0 + Ci,

where c, is independent of 1/.,~0. Such a framework has been used, for example by
Chamberlain (1984) and Blundell and Smith (1991). This can be solved by two-step
way we have discussed in Chapter 1. However, this setup for initial observation do not

include the exogenous variables. Sevestre and Trognon (1990) added another term

27

21:,oa into (1.31) meanwhile he need do estimate the autoregressive auxiliary model
(1.36) beforehand. This method leads to a two-step estimation.

In the non-linear case (e.g. dynamic probit model), Heckman (1981) first make
approximation to D(y0[XT, a) and specify a D(a) with assuming that a and XT are
independent. This method is flexible but it is more complicated and more restrictive
than necessary. The misspecification of the distribution of 31,0 would result in the
inconsistency of the resultant estimator. It is obvious that the consistency properties
of various error component estimators for the dynamic models with unobserved effects
depends on the treatment with the initial value. Different assumption on the initial
value induce more moment conditions needed to be exploited to gain more efficiency
(e.g. Ahn and Schmidt [1995, 1997], Blundell and Bond [1998]).

The important drawbacks of unconditional MLE do not occur when we consider
the distribution of (m, . . . ,y1) given (yO,XT,a) and then specify D(a[y0,XT). This
leads directly to a density for (yr, . . . ,y1) given (311), XT). Moreover, we do not treat
yo as nonrandom variable and it in not necessary to assume the independence between
X; and a. Our suggestion is to model D(alyo.XT) and then construct the density
of (yT,. ..,y1) given (y0,XT,a). This allows us to avoid the problem of having to
find or even approximate, D(y0[XT, a) and specify an auxiliary model for D(c[XT) or
assume that a and XT are independent and then model a marginal distribution of a
(See Wooldridge [2000b]).

In this chapter, I first show how to construct the conditional MLE for

yit=nyi,1—1+ai+€m i=1,---,N1
(2.1)

where the a, is the individual effect and is assumed that a, = (10 + olym + c,.

5,, and c, are assumed to be normally distributed. Later on I consider the case with

28

exogenous variables in which equation (2.1) will be added by the term 33,,6 and that
of a, will be altered by adding one more term, $7,012.

The approach of CMLE keeps us away from understanding the exact form of the
distribution of the first observation because it is conditioned on the initial observa-
tion. To specify an auxiliary conditional distribution for the unobserved heterogeneity
has inherent drawback of all parametric methods: misspecification of this distribu-
tion generally results in inconsistent parameter estimates. Nevertheless, Wooldridge
(2000b) has shown that in some leading cases the method leads to some remark-
ably simple conditional maximum likelihood estimators ( especially for the non-linear
case: partial effects on the mean response, averaged across the population distribu-
tion of the unobserved heterogeneity). For example, it is easy to obtain estimated
average probability response across the population distribution of the unobserved
heterogeneity discussed in Chapter 4. The plan of this chapter is as follows. Sec—
tion 2 considers the general conditional MLE for the dynamic model. In this section I
construct the conditional likelihood function to obtain the conditional maximum like-
lihood estimators and discuss the consistency of CMLE. Section 3 applies the CMLE
to basic AR(1) model with unobserved effects. I examine the asymptotic properties
of the CMLE as N ——> 00 with fixed T. Beginning with normality assumption on
the unobserved effects and the random noises, I examine the AR(1) regression of
dependent variables without exogenous variables and conduct a Monte Carlo stud-
ies to investigate the performance of the conditional maximum likelihood estimator.
Theoretically, non—normality is known not to cause inconsistent in Gaussian CMLE.
I proceed with the same studies with the replacement of normality by non-normality
assumption. Section 4 examines the same model except that we include the strictly

exogenous variables and employs the same procedure as that of section 3 to build up

29

a simulation for the CMLE with inclusion of exogenous variables. Section 5 studies
some empirical example for the previous two case. Section 6 makes the comparison of
CMLE with the estimators discussed by Blundell and Bond (1998). Section 7 contains

some concluding remarks.

2.2 General CMLE

2.2.1 Conditional Likelihood Function

In this section I will construct. a generic likelihood function for the conditional max-
imum likelihood estimator in dynamic, unobserved effects models where the lagged
value of dependent variable is included in the list of explanatory variables. The AR(1)
model is a good choice to describe such a dynamic process. The primary principle on
which estimation will be based is maximum likelihood. Let 6 denote the vector of pop-
ulation parameters. Suppose we have observed a sample of size T+1, (y0,y1, . . . ,yT).
We need find a joint distribution of D(yT, . . .,y1|y0,XT,a) where a is unobserved
heterogeneity and its relevant parameterizing joint density function conditional on
(y0,XT,a) is f(yT,...,y1[y0,XT,a;6) and thus the MLE estimate of 6 is the value
for which this sample is most likely to have been observed. Because a is unobserved,
we need try to remove it out of the function. Typically, a distribution D(a[y0, XT) is
required and hence we can integrate a out of the joint density function with condi-
tioning on XT and yo by the usual product law. We make some assumptions in the

following.

D(yt[$t1)/t—laa) : D(yt[XTi)/t—19a)9 (22)

The assumption of (2.2) can be thought of as a basis for a standard dynamic unob-

served effects analysis with strictly exogenous variables that means that, once current

30

:13,, past 31, and a are controlled for,:1:, , s aé t, has no effect on the distribution of Y,.

Therefore, we can define a parameterizing density for the conditional distribution of
(2.2) as follows:

f,(y,[Y,-1,:1:,,a;60), t=1,...,T. (23)

According to (2.3). the joint density of first t observations can be described as

the prOduct of f(y,|Ys_1, .175, a; (50) over 1 to t. It follows that the parametric density

Of (yTi ' - ° 1 yl) given (y09XT1 0) is
T
ffyrs . - wyllf/Os X'raaé 50) = Hft<ytlyt_1,fl‘.,, a; 60). (2.4)
1:1

To integrate (1 out of the density function, Wooldridge (2000b) suggests modeling

D(a[y0, XT). We define the parametric density function as follows:
h(a[XT13/O§ A0) (2-5)

corresponding to D(a|XT, yo), where A is a vector of parameters. For example, we can
assume that E (alyo, XT) : ;1(y0, W,), where W, might be some linear combination of

XT. The simple case is that W, = .13, = %Z,:1 1,,. By the usual product law for

conditional densities, the joint parametric density of yr, . . .,y1 given (yo, XT, a) is

T
Pf‘yrs - - - . 1J1 [91% XT- 0: 90) = H ft(.l/t[Yt—la $1.0: 5olhfaly0, XT; )10)- (2.6)

t=l

where 60 = ((50, /\0). Once we have specified h(a[y0, XT; A0), we obtain the log density

of (yT, . . . ,y1) given (y0,XT) by integrating out a.

108 f(yT1'° - ,y1ly0.XT§50)}II(CI|XT,;U0;Aol'U(da)a (2-7)
[Rm

where m is the dimension of a and v(-) is a suitable measure. We let m = 1 and
begin with the normality where the conditional mean, and possibly the conditional

variance, are flexible functions of (yo, XT). For example, we assume that

a, = on + olym + c, (2.8)

31

when we consider the standard dynamic panel data without exogenous variables.

It is usually assumed that c,[y,0 ~ N(0,o§). It means that h(c,[y,0,)\)

1

1 .
WWW-#36192), where c, = a, — ao — my“) and /\ = (010,511,03). Given (2.6)
7f0’a a

 

and (2.7) without XT, we can build up the log—likelihood function for the model

without exogenous variables for cross section 17 is
((3117:, - « - 291019)
= 108/ “3117‘, - - - ayil [1110, (1; 5ihfclyz'0; A) dc (2.9)
R

108/ 1,1,1, ft(yz't[Y1,t—1,aiidihfclyioiAldc
1R

According to (2.9), we maximize the sum of 1 (31,7, . . . ,y,0; 6) across 1' from 1 to N.

The log-likelihood is as follows:

N T
maax Zlog/ Hf,(y,,|Y,,,_,,a,;6)h(c[y,0;/\)dc. (2.10)
1:1 1Rt:1

To extend the model to include strictly exogenous variables, the simple case is to

specify an equation as follows:
a, = avo+oly,o+i,o2+c,, (2.11)

where c,|y,0,.r, is Normal(0, 03). The setup of the model with strictly exogenous

variables is as follows:

 

N T
1 ’1Y1—wi. 85h i~'i;)‘da 2'12
Igg§§i=1:os/Rt1:[ft(ytl 1 )(clyo,:r ) c < >
I 1 2'
where A = (oo,ol,og,o§) and h(c,[y,0,T,;)\) : l , exp(——(—C—)2), where c, =
27mg,2 2 0,,

ai — 00 — 013/10 — 5202-
A different description of the likelihood function for a sample of size T from a
Gaussian AR(1) with unobserved effects is sometimes useful. Let (y,|y,0,.r,.a) 2

(31,7, . . . ,y,, |y,0, (13,, a,) could be viewed as a single realization given (y,0, 517,, a,) from a

32

T—dimensional Gaussian distribution. Viewing the observed sample y, as a single draw
from a Normal(u(y109$i)a “(910,350) where lift/10,331) :E(yilyi0a$i,ai) and nysz')
= Var(y,|y,g, 33,, a,), the sample likelihood function could be written down from the

formula for the multivariate Gaussian density:

_—.1_ 1 , _
“0(9109372‘” 2 BXPl—§(yi — 111-(91095130) aft/10,5134) 1(yi — #(yioaxilll-

—_7.‘
2

ff‘yz'; 5) = (271)
(2.13)
By specifying a Gaussian distribution h(a,[y,0, 513,), the individual log likelihood func-

tion can be written as follows:

log/1R (27f)? (lflf‘yz‘o, Lil—7] CXPl—é'f’yz' - #(yan 331)),Q(y10:$i)_1(yi _. lift/210,331)”

h(a|y,0, a3,)d a.
(2.14)

Expression (2.13) is algebraically equal to (2.7). We can maximize the sum of (2.14)

with respective to 6 across 2' from 1 to N to obtain the CMLE estimators.

2.2.2 Asymptotic Properties of the CMLE

In the current setting, the conditional maximum likelihood estimator is gener-
ally consistent ~ with fixed T and N goes to infinity — if the conditional density of
(11,1, . . . ,y,T) given (the, X ,T) is correctly specified. This follows from standard results
on maximum likelihood estimation with conditioning variables because we are assum-
ing random sampling in the cross section.( See, for example, Manski( 1988, Chapter
5)), Wooldridge (2001, Chapter 13).) In the present application to linear, dynamic
unobserved effects models, the log-likelihood function satisfies all smoothness require-
ments, and the sufficient moment conditions are likely to be met. Practically, the key
issue is parameters are identified under weaker assumptions based only on certain

moment conditions, so identification holds when we specify a full conditional distri-

33

bution.)

It is useful to sketch the consistency of the CMLE for general dynamic mod-
els where the likelihood function is conditional on the initial value. The density
in 2.7)is correctly specified if there are values 60 and A0 such that the density of
(31,1, . . . ,y,T)given (yiO,X,T) is given by the integral in (2.7). Under this assump—
tion, the conditional Kullback-Leibler information inequality holds (see, for example,

Manksi (1988, Section 5.1)):

EUR/2:; 90) [171-.1910» Z E(l(y,; 0) [Tel/20)), (2-15)

for all 6 in the parameter space. By the law of iterated expectations and (refeq2-10)
we have

E [lit/1,1324%” 2 E “(f/1,5132? all - (2-16)

Therefore, 60 is a solution to the population maximization problem:
I '. ii 6 . .1
151313 I (yum )l (2 7)

This shows that the CMLE is Fisher consistent for 60. Under identification, 60 is
the unique solution to (2.15). Then, we can use the usual analogy principle and the
uniform weak law of large numbers to conclude that the CMLE is generally consistent
for 60 as N —> 00.

In rare situations, the log-likelihood function can be shown to be globally concave.
Unfortunately, this does not appear to be the case for dynamic panel data models.
As a practical matter, this means we may locate local extrema. In practice, several
different starting values should be used in estimation to try to uncover a global max-
imum.

Under sufficient differentiability assumptions - which, as mentioned earlier, are

34

satisfied by the models of this and the remaining chapters - the CMLE is \/N asymp-
totically normal. Newey and McFadden(1994) and Wooldridge(2001) show that a

consistent root to the maximization problem is also asymptotically normal:
\/N(6N — 60) —+ N(0, A(90)-‘B(00)A(00)-1),

where

14(60) = EI(321(y2:;0)/0900')aol.
and
13(90) = E[(01(yi;0)/80)60 >< (Ultyi:6)/30')aol-
Under correct specification of the conditional density, A(60) : —B(60), that is, the

information matrix equality holds. This simplifies estimation of the asymptotic vari-

ance and computation of test statistics.

2.3 Linear AR(1) Model With Unobserved Effects

2.3.1 Linear AR(1) Model

The conditional MLE approach is one method for making the initial condition
problem tractable. We begin with the linear case without additional explanatory
variables. The model is

911 : P Uta—1 + (12' + 521, (2-18)

and we make the following assumptions.

 

[Assumption 2.1]5,,[y,,,_1,...,y,0,a, ~ Normal(0,o€2).

 

 

[Assumption 2.2] a,|y,0 ~ Normal(oo + alym, 03).

 

According to Assumption 2.1 and Assumption 2.2, the distribution for

35

(3,1,43,11,14, . . . , y,1) conditioning on y,0 is as follows:

(yiTayz’T—laH-ayill .610) N N(#()’10),Q(Y10)), i=1,--.,N,

where

lift/2'0) Z E(y,|g,0),
9(3/10) :- V(y10ly10)
= E ((y, — E(y1ly10))(f/i — E(y,|yz-o))’|y.~o)

i:1,....N,y,=(mp-Wyn)-

Assumption 2.2 implies that

a, : 010 + my“, + c.,, where c,[y,0 ~ N(0, 0:)

(2.19)

(2.20)

Equation (2.1) can be rewritten as y,, = pit/1o + 23:, p7‘1(o'0 + (113/,0 + 5,) +

2;:1pj‘15,,,_,+1. The E(y,|y,0) can be obtained by replaceing E(a,[y,0) with 00 +

(113/,0 in (2.19). The conditional mean of y, is as follows

__ 1 — ‘ — ‘
#(3/2’0) — ( 0‘0 + (01 + 101610, ,T:%0‘0 +(i1—_%01+ Pill/2‘0,

, ) (2.21)

The conditional variance of y,, can be obtained by calculating the form as

E(€(yiol5(yiol’l 3110), where 5(610) 2 y, — E(y1ly10)' In the same manipulation as

that of conditional mean, the conditional variance can be written as

(.011 W1T\

WT1 WTTJ

1 ‘21

— t _. ,
wt1=(i[:%)203 + (7521:; t=1,...,r.

where

1_81_t —91__28 .
wst=(T:‘%jt‘%)03+/J” "(_1f%7)0§§ Sift, s,t=1,...,T.

36

(2.22)

The jointly parametric density function of y,[y,0:

“MIL/40:9) I (—\/——12:7r)—T/2([Q(yi0)[)_1/2exp(———é—1— (5(yio)'Q(y10)—I€(y,o))) (2-23)

where 5(y,0) = y, — E (y,|y,0). We can directly construct the log-likelihood function

across 2' from 1 to N as follows:

—T

N
to; 9) = 2 (710g V2? + $10,, meal-1 — é{5(.Uzol,9(yz'0)_15(yio)l) , (2.24)

where 6 = (p, 00,01, 02, 03). The CMLE estimators can be obtained by maximizing
the likelihood function (2.24).

Another approach to calculate the CMLE estimators, according to equation (2.10)
and Assumption 2.2, is in the following. We specify the distribution of a, conditioning

on y,0 as follows:

‘1 (1'2: — 0'0 — 01910

l
I 2' t ,A Z 1' — 2 . 2.25
7“(a [y 0 0) m Ckp< 2 ( 0,0 ) ) ( )

 

 

By employing (2.13), the joint density of y, given (y,0, a,) is the product of
f(y,T, . . . ,y,1 [11,, gm) and h(a,|y,0), where the f() and h() are the relevant conditional
normal density functions. It follows that the density of (3,1,7, . . . ,y,1) given (y,0; 6) is

lift/259) =
0° —T/2 —1/2 1 . _ , , ~1.. ..
log/_ < ) 0526.0)! cam—go. wetness) (y. wow»)-

1 itifi

e2 ”a dc,
(/27ro§

at
:3

 

(2.26)
where p(y,0) = py,,_, + a,lT. Therefore, the CMLE estimators are to solve out the

problem of maximization as follows:

N
méax£(Y; 6) = max: l,(y,, 6) (2.27)

121

37

where 6 is the vector of parameters. Because we place no restrictions on h(alyio, A0),
once we have specified h(a,|y,0; A0), we generally obtain the density of y, given y,0 by
integrating out a,.

While the log-likelihood function is ”consistent” under the normality for 5,, and
a, — and therefore, y, given y,0 is multivariate normal — the conditional MLE is robust
to ce’teris pa’ripus from the assumptions. In particular, the normal quasi-MLE is
consistently and asymptotically normal provided the first two conditional moments,
E (y,[y,0) and Var(y,[y,0) are correctly specified. this follows from the work of Gourier-
oux, Monfort and Trognon (1984) and Bollerslev and Wooldridge (1992). Without
normality, the information matrix equally does not hold and so the variance matrix

needs to be estimated in a robust way.

2.3.2 Simulation Evidence

In order to investigate the performance of maximum-likelihood estimators given
the initial value, we conducted Monte Carlo studies. We use the MLE software of
Gauss to do our simulation for the conditional maximum likelihood function. The

notations for the simulation are as follows:
1. 6* means the conditional maximum likelihood estimators in each iteration.
A __ 1 1200 :1:

3. 6 means true value of parameter, where 6:(p, do, 011, oa, 0,):

(p, 0.2, 0.4, \/1.2, 62.4).

38

Our true models were generated by

yit = ,0 yi,t—1 + 01+ Eu
1:1,...,250,t:1,...,5, (228)
p 2 0.0.05, . . . ,0.95,
where

a. = 0.2 + 0.4 gm + c., 2:: 1, . . .,250. (229)

We generated the c, and 8,, by two cases, one from in dependently normal distribu—
tion, 5,, ~ N(0,2.4) and c,- ~ N(0, 1.2) and the other from a t-distribution with the
freedom 6 and 10, in respective. In case where 31,0 are treated with being given, we
do not need pay attention to its distribution in our approach. For the simplicity, we
generate y” from a N(0, 1) or uniform distribution for convenience. The value of p
goes from 0 to 0.95 in an increment of 0.05. We use the individual likelihood function
(2.26) and (2.27) and then construct. the framework of maximization to solve out
estimators.

The specification for the distribution of (a,|y,~0) in the use of the framework (2.25)
is flexible. We can see the advantage of framework (2.25) in non-linear model, for
example logit with unobserved effects model will be discussed in chapter 4; it, never—
theless, is heavy time-consuming in the maximization of the likelihood function (2.26)
across 2' to N. We employ the Hermite integral formula as the approximation of the
integral ( see Butler and Moffitt [ 1982] ). It is a good idea in the use of framework
(2.26) to specify a more flexible distribution of the unobserved heterogeneity given
the growing speed of CPU.

Table 2.1 reports the simulation result for the power test of the conditional max-
imum likelihood estimators, H0 : 6 = (p, 0.2, 0.4, \/2._4, M). The true values of p

range from 0 to 0.95 with the increment of 0.05. We repeat the same procedure of the

39

CMLE for 1200 times and calculate the frequence of the p—value greater than a certain
level, 0.01, 0.05 or 0.10. Table 2.2 shows the simulation for the test of H0 : p 2 p0.
For example, under the p-value is 0.01 and the true value of p is 0, the second row of
Table 2.2 shows that the frequence of rejecting H0 : p = p0 in creases with p0, namely,
we can reject most of po, away from the true value, p. There are same results for the
p—value, 0.05 or 0.10.

It is crucial to see that it is more powerful to do the hypothesis H0 : p 2 p0 when
the true value of p closer to 0. For example, the true value of p is 0.75 and the power
0.01 in the second row of Table 2.2, the p-value of H0 : p : 0.90 is 0.6183. Comparing
with p = 0.75, the p-value of H0 : p = 0.15 is 0.9050 when the true value of p is 0; our
approach, obviously, for the hypothesis test of p performs well when the true value of
p is getting closer to 0.

To examine the simulation for the model under non-normality, we generate the 5,,
and c,- from the t-distribution with freedom 6 and 10, i.e. the parameters, 05 = a
and 0a 2 g, respectively. We report the simulation results for the conditional
maximum likelihood estimators in Tables 2.3 - 2.4. Table 2.4 shows the simulation
for the tact of H0 : p : p0, with p0 ranging from 0 to 0.95 for p = 0, 0.1, 0.05, . . . ,0.95.
We obtain similar results of the model with normality assumption. The simulation
support that the conditional maximum likelihood estimator perform very well. The
model of interest is a regression model in which the lagged value of the dependent vari-
able appears in the list of explanatory variables, it is crucial for the test of coefficient
of the lagged dependent variable, H0 : p = 0. Our approach supports that the CMLE
is a good estimator. When the true value is closer to zero, the test is more significant.
The sixth column of Table 2.3, the frequence of rejecting the H0 : 0a 2 m is larger,

namely it is likely to be rejected in comparison with the 06. By increasing N, the

40

power of testing 0,, will be increase. We have discussed the properties of CMLE for
dynamic models with individual-specific effects. In the next section, we study the
same linear AR(1) model with unobserved effects and strictly exogenous explanatory

variables.

2.4 Linear AR(1) Model With Unobserved Effects

And Exogenous Regressors

2.4.1 Linear AR(1) Model With Exogenous Variables

In this section, we add exogenous variable, :r,,, to model (2.1). The new model is

ya = Pym—1 + 3511/3 + Clzi + 5m ’i = 1, . - - , N,
(2.30)

t = 1, . . . , T,
where so“ is assumed to be strictly exogenous variable. The exogenous variables
might be the variables of discrete value, e.g. some policy variables, status variable

and the like, or variables of continuous value, e.g. years of education. We make some

assumption as follows in this case:

 

lfissumption 2.3J€,tly,:vt_1,. . . ,yl‘o, (BLT: . . . ,fL'z'] , a,- N Normal(0, 0'2).

 

 

[Assumption 2.4) a,|at,~t, . . . ,r,1,y,0 ~ Normal( 0’0 + my“) + Ti, 02, 0,2,).

 

and thus we have the equation
Haiti/10.51:.) = 0'0 + alyzto + 5.- a2, (2.31)

where T,- = 7:,- 23:1 17,, and :13,- 2 (23,7, . . . , 33,1). In the empirical study in section 2.5,
we let :3“ be a union status variable; then 5,» is the fraction of time in a labor union
over the sample period. For example, if a worker had been in labor union for three

years, e.g. 1981, 1983 and 1984 from 1981 to 1987, then the ratiofi, is %. We can

41

construct the conditional multivariate normal distribution for the model by adding

the exogenous variable into (2.1) and (2.19) as follows:

yiTayi,T—la-~ayill yiofli N N(/‘(YiO:Xi)a0(3‘7i0axi», i=1,...,N, (2.32)

where

Mylo-.5131) : Ef'yz‘lyioflil;
9(61021‘2‘) = nyilyi0~ffi)

(2.33)
: E (fl/2' — E(yllf/i03 Ii))(yi — E(f/1l.y103$i)),lyi03$i)
i: la°°°9N9 312‘ = (”yiTan-al/u), 332' = ($iT,---,113u)-
With Assumption 2.4, we rewrite (2.20) as follows
a,- = 00 + 01 31,0 + FE,- 0'2 + (7,, (2.34)

where c,|y,-0,a:,:T, . . .,:r,-1 ~ N(0,0§).

2.4.2 Conditional Mean and Variance
By iteration, equation (2.31) can be expressed as
t t t
3121 = #31204"; P7_1(00+a1yio+471G2+¢J+Z 01—1 517i,t—j+1 +2 p]_1€i,t—-j+l- (2-35)
j=1 j=1 j=1
The mean E(y,~|y,0, 23,) can be obtained by substituting E(a,|y,0, 2.3) with 00 + alyio +
5,02 . The conditional mean of y,- is as follows

K 00 + (011 + Plyz‘o + $21 \

my... a) = . . ‘ (2.36)
1 - _ 1 — _-
f(): p (00 + 5’31)+(T—fl_ p 01+ Pill/:0 + 23:1 pt J+1$i,t—j+l

K ‘ l

The conditional variance of y“ can be obtained by calculating the form as equation

 

 

(2.33) do. In the same manipulation as that of conditional mean, the conditional

42

variance can be described as follows

9(9109531) 2
um . .. qup
Actually, provided that .r,, is strictly exogenous, Q(y,0, 10,) is equal to Q(y,0). Equa-
tions (2.23) and (2.24) can be applied here. We parameterize the conditional densities
0f (yilyiOaxi):

l—T/2(lfl(yi0)|_l)l/2 exp(——1(€(y,0, $i))Q(yi0)—l(€(yi0y 5136),),

“gilt/20,3759) :( 2

1
v27r
(2.37)
where €(y,0,:r,) = y, — E(y,:ly,o,a:,). We can directly construct the log likelihood

function across 2 from 1 to N as follows:

N .
—T 1 1 , _
£(Y,X;6) = Z (~2— log v27r + i log |Q(y,0)|‘1 — 5 (5(y,0,ac,) Q(y,0) 15(y,0,x,))) ,
27:]
(2.38)
where 6 = (p, 6,00,01, 03, 0,3). The CMLE estimators can be obtained by maximizing

the likelihood function (2.38).
Another approach to calculating the CMLE estimators, according to equation
(2.10) and Assumption 2.4, is in the following. We specify the distribution of a,

conditioning on 31,0, :13, as follows:

1 -1 a, - (00 + 01 gm +3, 02)

eXp(—(

2W? 2 a. )2). (2.39)

 

h(ailyia 5132'; A0) :

 

The likelihood function of this case is similar to that of the previous model without
exogenous regressors except that the conditional mean, p(y,0) = E (y,|y,o,a,), must
be replaced with p(y,0,:r,) = E(y,|y,0,a:,,a,), so equations (2.26) and (2.27) can be

directly applied here. We write the likelihood function of interest as follows:

43

lie/2333216) Z

108/ ("J/TVT/Qflflfyme HUD—.1” exp[—%(yi — lift/2'0, fl?i))'Q(3/z‘0, $i)—1‘
-_1(_c_ .2 (2.40)
. . _ . 1 2 a d
ll 2 a 1 a ’
(J nyo 1 ”Image 0

:1

 

where p(y,0, 23,) = Why—1 + 513,6 + a,lT. Therefore, the CMLE estimators are to solve

out the problem of maximization as follows:

N

rn;1x£(Y,X; 6) : mélng,(y,,17,;0) (2.41)

where 6 is a vector of parameters. According to the framework discussed previously,

I set up a simulation for it in section (2.4.3).

2.4.3 Simulation Evidence

I conducted Monte Carlo experiment to examine the performance of the CMLE
model in which exogenous variables are included. I use the MLE software of Gauss to
do the simulation for the conditional maximum likelihood estimator. The notations

for the simulation are as follows:

* 1200 . . . . .
1. 6 = i21—00 21.2, 6; where 6; IS the estimates from the CMLE 1n each Iteration.

2. 6 means true value of parameter, where 6 = (p, ,3, do, 01, 0'2, 00, as) =

(p, 0.15, 0.2, 0.4. 0.35, \/1.2, «2.4). .

Our true model was generated by

yit : P Elm—1 + 015 2321+ Cl,‘ + Eu, i=1, . . . ,250,
t—_——1,...,5, (2-42)
p = 0,005,. . .,0.95.

where,

a, = 0.2+0.4 y.~o+0.35 me, z:1,...,250. (2-43)

44

I generated the c, and 5,, by two ways, one from independently normal distribution,
5,, ~ N(0,2.4) and c, N N (0, 1.2) and the other from t-distribution with the freedom
6 and 10, in respective. The value of p ranges from 0 to 0.95 with an increment of
0.05. I use the individual likelihood function (2.26) with replacement of ,u(y,0,:1:,) =
pithy—1 + 23,,"3 + 0,17 and then construct.
N
mgtxizzl l(y,; 6). (2.44)
Table 2.6 reports the simulation results for the power of tests of H0 : 6 =
(p, 0.15,0.2,0.4,0.35, V2.4, M). The true values of p range from 0 to 0.95 with
the increment of 0.05. I repeat the same procedure of the CMLE for 1200 times and
calculate the frequence of the p—value greater than a certain level, 0.01, 0.05 or 0.10.
Table 2.7 shows the simulation for the test. of H0 : p : p0 . I calculate the frequency
of p-value greater than a certain level, 0.01, 0.5 or 0.10. For example, under the
p-value is 0.01 and the true value of p is 0, the second row of Table 2.7 shows that the
frequence of rejecting H0 : p 2 p0 increases with p0. It means that most of po, away
from the true value, p can be rejected in the CMLE. There are same results for the
p-value, 0.05 or 0.10. Table 2.8 shows that the result of the simulation by replaceing
the normality assumption with t distribution. Table 2.9 shows that the frequence
of rejecting H0 : p 2 p0 increases with p0 even without the normality assumption.
It pays to notice the test of estimated standard deviation of unobserved effect when
we drop the normality assumption. The 8th column of Table 2.8 ~ Table 2.9, the
frequence of rejecting the H0 : 0,, = V1725 is larger, namely it is likely to be rejected
in comparison with the 05- The reason might be that we generate the unobserved
effects from the distribution from the t distribution, the variance will become larger
and the number of a, is much smaller than the number of 5,,. By increasing N value

, the power of testing 0,, will be increased.

45

It is crucial to see that the true value of p getting closer to 0 or 1, it is more
significant to reject H0 : p : p0 than the true value falling within the interval of 0
and 1 in which p0 deviates from the true value. For example,let us set the deviation
be three increment, 0.15, meaning the deviation is 0.05 x 3. The true value of p is
0.5 and the power 0.01 in the second row of Table 2.7, the p-ivalue of H0 : p = 0.65
is 0.7558; the p-value of H0 : p z 0.25 is 0.8833 when the true value of p is 0.1 (see
Table 2.7); the p—value of H0 : p = 0.8 is 0.9908 when the true value of p is 0.95 (See
Table 2.7).

The results of this simulation show that most of conditional maximum likelihood
estimators deviating away from the true value of the associated parameter will be

rejected, especially when the true value of parameter is getting closer to 0 or 1.

2.5 Empirical Example

I have discussed the properties of the conditional maximum likelihood estimators
for dynamic model with individual heterogeneity in previous sections. In this section,
I use the data from Vella and Verbeek (1998) to study the conditional maximum
likelihood estimator in estimating dynamic model using observations draw from a
time series of cross sections. These data are for young males taken from the National
Longitudinal Survey (Youth Sample) for the period 1980 - 1987. The dependent
variable is the log of hourly wage and the explanatory variable is labor union status.
Each of the 545 men in the sample worked in every year from 1980 through 1987.
We begin with the OLS for the empirical data, i.e. we run the OLS regression of
17200095, on 1, lnwage,,,_1. The OLS estimates of autoregressive is 0.627. The OLS
estimates cannot be identified with the effects of unobserved effects. It is necessary to

incorporate the effect of individual heterogeneity to study both the state dependence

46

in earnings as well as the effects of union status on wage. We assume that omitted
ability and other productivity factors can be accounted for by initial wage rate. The

model is set up as follows:

lr'zxwage,, : p lnwage,,,_1 + a, + 5,,, '1'. = 1,. . .,545,
(2.45)

t. : 1,. . . ,7,

where
a, : do + d, Inwagem + c,, i : 1...,545. (246)
Table 2.5 shows the conditional maximum likelihood estimates, (,3, do, d1):
(0.3405, 0.8784, 0.1839 ) are all significantly different from zero. The estimated
average effects of unobserved heterogeneity given initial log wage, d, is measured by
(0.8784 + 0.1839 lnwa.ge,o). This verifies that the higher is the initial wage rate,
the higher is the individual worker’s ability. Replaceing a, in (2.16) with the above
equation and taking the mean of lnrwage,, given the l'n,wag‘e,o, equation (2.25) can be

expressed as follows by iteration
E(lnwage,,|lnwage,o) : pt l'n.wage,o+(1+p+p2+. . .+pt‘1) (do+d1 ln’wage,o) (2.47)

From equation (2.47), the estimated response of the current wage rate change

 

—— t 0
into the initial wage, 8 lmffageu / 8 Ingagem is (0.3405‘ + 0.1839 - 11:%§3%%55) 1n-
stead of 0.3405‘. Specifically, when t=1,2,. . .,7, the estimated responses are 0.4884,
0.5444,. . .,0.5688, respectively.

Vella and Verbeek (1998) study the effects of union membership on wages in a

static model. Here I add union status to the AR(1) model with an unobserved effect.

Specifically, the model is

ln'wage“ = p lnwage,.,_1 + ,3 u-nxi0n,, + a, + 5,,, i = 1,. . . ,545, (2 48)

t=1,...,7,

47

where the unobserved effect is assumed to follow

45. (2.49)

O"

a, 2 do + d1 lnwage,o + (1’2 union, + 0,, 2' = 1 . . .,

Given past wage and controlling for unobserved heterogeneity, the return to union
membership is about 4.7 percent and it. is marginally statistically significant. The
estimates suggest that, once the initial wage is controlled for, there is no partial
correlation between individual heterogeneity and the propensity to belong to a union.

The analysis here assumes that union status is strictly exogenous. In the context
of model (2.48), this means that innovations in lnwage today, as measured by 5,,, do
not affect the decision to join a union in the future. This may not be true, although we
are controlling already for the most recent wage and an unobserved effect. One way
to test the strict exogeneity assumption is to put a lead of union, that is, run'idn,,,+1,
in the equation and test its statistical significance.

The W, is the ratio of periods staying in labor union to the periods outside
of labor union for a given periods of time. For example, if a worker had been in
labor union for three years, e.g. 1981, 1983 and 1984 from 1981 to 1987, then the
ratio, m is g. Table 2.10 shows (,3, .3 do, d1, 5,):(03380, 0.0474, 0.8721,
0.1745, 0.0488). 6 is marginally significant and do is not significantly different from
zero. A lot of empirical literatures are raised to explore the question of union effect
how equivalent workers’ wage differ in union and non-union employment. While the
unobserved factor that influence the sorting into union and non-union employment
may also affect wage, this makes endogeneity of union variable and thus we can not.
just assume that the status of union is strictly exogenous. In chapter 4, I will discuss
the logit model with unobserved heterogeneity to explore how the current status of
union respond to the union membership in the initial period in terms of the individual

workers’ characteristics.

48

2.6 Comparison With The Other Estimators

In the section I report the results of Monte Carlo simulations which compares
the conditional maximum likelihood estimator in finite sample with the GMM and
conditional GLS estimators (see Blundell and Bond [1998]). I follow the notations
and definitions of three GMM and CGLS estimators studied by Blundell and Bond
as follows :

DIF: The standard first-differenced GMM estimator, based on moment conditions,
E(y,,,_,A5,-,) = 0 for 1523,. . .,T and s 2 2.
SYS: The system GMM estimator, based on linear restriction.
ALL: The system GMIV’I estimator which also exploits the complete set of second-
order moment restrictions.
CGLS: The feasible conditional GLS estimator, which uses residuals from the one-
step GMM (SYS) estimator to estimate the required variance components.

I follow the data generation processes for y,, used by Blundell and Bond except

for the 3120 and 0,.
Mt = pyi,t—1+ai+€3t.i:1,--.,N,t=1,...,T. (2.50)

I use the same magnitude of N and T as that in Blundell and Bond paper (1998) to
make the comparisons. N is chosen as 100, 200 and 500 T = 4 and 11. The true
value of p is taken to be 0, 0.3, 0.5, 0.8, 0.9. Table 2.11 reports model (2.50) of N
= 100, 200, 500 with T = 4 and Table 2.12 further reports the same model of N =
100, 200, 500 with T = 11. All results of simulation are based on 1000 Monte Carlo
replications, with new values for the initial conditions drawn in each repetition.

The data generation of the first period in the model of CMLE is different from

the other models in this section. The true models of GMMs and CGLS consider the

49

generation of the initial conditions y,o as :

yiO = Tiff—p + U30. (2.51)
where u,o is an i.i.d N (0, 4/ 3) random variable and independent of both a, and 5,,,
The variance of u,o is designed to satisfy stationarity. The a, and 5,, are drawn as
mutually independent N(0,1) random variables. In the case of CMLE, the unobserved

effects are assumed to be conditionng on the initial observations y,o, so the true linear

projection is assumed to be:

(1,0 = 0.2 + 0.43/30 + Ci, (2.52)

where c, is assumed to be N(0, (il—p)2 +4/3) and y,o is generated from N(0, 1). The
magnitude of variance of a, in model (2.52) is designed to be equal to the variance of
y,o conditional on a, in model (2.51). The contribution of the individual effects of the
error terms becomes less important due to the fact that the variance increases with
the p.

As for the non-normality assumption of errors, the comparisons among various
estimators in this section will be limited on the case of p = 0.5, 02 = 1 and N = 200
with T=4 for the models studied by Blendell and Bond and the CMLE. Accordingly,
on the one hand, the true model of GMMs and CGLS turn out to be that M0 = 2
a, + u,o with u,o ~ N(0, 4/3), 0?, = 1 while 5,, = 91—27—1, where 5,, ~ X2(1); on
the other hand, the true model of CMLE generates from (2.52) in which 0, ~ i.i.d
N(0, 10/3) and 5,, = 91—211, where 5,, ~ X2(1). Table 2.13 presents a stationary
design but with non-normal errors for various GMMs, GLSs and that of CMLE with
non-normal errors. Table 2.14 presents the performance of the CMLE with different
value of N with fixed T = 4 under non-normality on errors.

As is well known, when p is close to zero, the influence of the initial conditions

50

becomes less important; therefore, the performance of the estimators is similar. The
more interesting case is high values of p, which is where the GMM estimators sug-
gested by Blundell and Bond (1998) show a clear advantage over the usual IV es-
timator. Table 2.11(a) shows the dramatic improvement resulting from using extra
moment conditions based on restrictions of the initial conditions. for true values of p
of 0.8 and 0.9, respectively, the lV'Ionte Carlo averages for the estimator of p are: pp [p
= 0.4844, 0.2264; p5y5_cMM = 0.8101, 0.9405; pAL,_G,,,M = 0.8169, 0.9422; pCGLS
= 0.8365, 0.9572; pm“; 2 0.8004, 0.8988.

The conditional MLE has the least amount of bias, whereas the standard first-
differencing IV estimator behaves very poorly. The GMM and conditional GLS esti-
mators work better, but not as well as the CMLE. The CMLE also has the smallest.
standard deviations and root mean squared errors. We can summarize the findings

in Table 2.11 for bias, standard deviation, and RMSE as follows:

lBiaS(f3CMLEll < lBiasffiSYS—GMMH <
lBiaS(laALL—GMAl)l < lBiEIS(/3C:GLS)| < lBiaS(/501F)l-
The ranking of corresponding standard deviations and RMSE of these estimators is

as follows:
SDUJCMLE) < SD(fi,1LL—GMM) <

SD(/35YS—GMM) < SD(:5('7(:LS) < SDU’UIF),

and
RMSEWCMLE) < RMSEffiALL—GMM) <

RMSElfiSYs—GMM) < RMSEWCGLS) < RMSEXfiDIF)‘
Table 2.11 shows that the performance of ALL-GMM , SYS-GMM and CMLE esti-

mators is getting close to each other with larger N.
When T increases to 11, the bias of all estimators decreases and the standard

deviations of all estimators significantly decrease. For example, from Table 2.11-

(b) and 2.12- (b), at the high value of p = 0.8, the means of pp”: changes from
0.4844 ( 0.5219) to 0.7373 ( 0.0742); the means of p3y3_GMM changes from 0.8050 to
0.8025; the means of pALL_GMM changes from 0.8112 ( 0.1195) to 0.8075 ( 0.0420);
the means of pCGLS changes from 0.8259 ( 0.1138) to 0.8039 ( 0.0423); the means
of pCMLE changes from 0.8004 ( 0.0684) to 0.8003 ( 0.0127), where the number of
bracket is standard deviation.

According to the ranking of /\-class estimators of (1.27):
[plim p(0) < p < plim M62) < plim {3(1) < plim 6(00).
-—-ooo N—voo N—«voo N—ooo
The means of estimators in Table 2.13 follow the ranking:

p.,..,..-..(= —0.0343) < am: 06659) < pom: 0.8740),

and the estimates of the other estimates fall the range {-0.0343, 0.6659]. The com-
parison of Table 2.11-(b) and 2.13 suggest that the assumption of non-normality has
little impact on the means and standard deviations of these estimators. At the true
value of p = 0.5, the means of pmp changes from 0.4828 ( 0.1821) to 0.4867 ( 0.1844);
the means of p3y3_GMM changes from 0.5098 ( 0.0936) to 0.4999 ( 0.1082); the means
of pALL_GMM changes from 0.5079 ( 0.0922) to 0.5067 ( 0.1109); the means of pCGLS
changes from 0.5135 ( 0.1006) to 0.5124 ( 0.1030); the means of pCMLE changes from
0.5068 ( 0.1036) to 0.5179 ( 0.1227), where the number of bracket is standard de-
viation. Obviously, the standard deviations of all estimators become larger and the
bias of all estimators enlarge a little. In Table 2.13, the standard deviations and the
bias of CMLE estimator is slightly greater than GMMs and CGLS in the absence of
normality assumption. Table 2.14 shows that at the true value of p = 0.5, the bias
of CMLE estimator decrease almost triple and the standard deviation decrease about

one and a half times to double when N increase by one time. When N is large enough

52

the estimator of CMLE perform well in the absence of normality assumption.

2.7 Conclusion

In this chapter I consider the CMLE for the AR(1) model with unobserved effects
which was proposed by Blundell and Smith (1991) in the case of no covariates. I
treat the initial value in different way. Balestra and Nerlove (1966) first explored the
conditional MLE, but he treat the initial value as nonrandom. It means the initial
value is independent of the unobserved effects. Such assumption is usually unten-
able assumption. Blundell and Smith (1991) consider a range of CMLE estimators is
equivalent to the ML estimator in Bhagarva and Sargan (1983), from the case with-
out the full error components restrictions, to the fully stationary error components
model. We need to care what about the restrictions on the initial value (, or distri-
bution of h(y,0|a,)) and the distribution of a,. The inclusion of :r,, make matters even
more complicate. Because we do not need impose restrictions on the 31,0 and specify
the distribution of (2,. Under the linear case the conditional ML estimators can be
worked out in a simple way. The inclusion of strictly exogenous variables 113,, will not
complicates matters. This approach can be easily applied in the more complicate
model, such as the state dependence model and the logit model considered in later
chapters in this thesis by using the approach proposed by \Nooldridge(2000b).

In practice, if we want to include the non-strictly exogenous variables, we need
to specify another conditional distribution for explanatory variables X on which we
do not impose strict exogeneity, D(:1:,|Y,_1, Z,, a) in constructing the CMLE model,
where Z, denotes the other strictly exogenous variables (see, Wooldridge [2000a]). We
can let

D($tl)/t—13Ztaa) : D(xtl}/t—1)Ztia) (2'53)

53

which means that once current z,, past y, and a are controlled for, 2,, 3 # t, has
no effect on the distribution of 33,. Practically, we can parameterize the conditional
density:

gt($t|Yt—1,Zt,a;‘ro) (2-54)

where 70 is finite dimensional parameter. By equations (2.3), (2.56) and the

usual product law for conditional densities, the joint density of (y,,:r,) given

(2T9 l/t—la Xt—la a) IS

pt(wth/t—Ii Zia a; 00) : ft(yil)/t—19‘Tta Cl; 60)gt(xtl)/t—la 217a; A/0) (255)

where w, = (y,, 33,), W, = (10,, . . .,wo) and 6 = (6,7). It follows that the density of

(wT,...,w1) given (ZT,x0,a) is

T
PUUT, - - - swllZTaw09 a; 90) = HPtI'lL’tht—l, 2t. a; 90) (256)

t=1

Similarly, we set up an log-likelihood function by the use of the joint conditional
density function (2.56) and conditional density function for a, similar to function

(2.5) to integrate out the unobserved effects. The question can be written as follows:

T
log/ Hp,(w,,|w,y,_1, z,,, a; 6)h(a|w,0, 2,; A) 'U(d a).

R t=1
If we have random sampling in the cross section dimension and standard regularity
conditions, with fixed T the CMLE for 290 will be consistent and \/N-asymptotically
normally distributed. (See N ewey and McFadden [1994] for sufficient regularity con-
ditions.) But it will be computationally difficult, especially in the wage-union appli-

cation: union would have to follow a dynamic probit or logit model, as in Chapter 4.

In the previous simulation, I employ the Hermite integral formula,

00
2 7 a g a v
/ f(z)e—Z d a: 2 2:72, f(z,)w,, but this computation IS costly. When we need to

’00

54

include‘the conditional density of non-strictly exogenous variables in the integration,
the problem of calculating the integration grow burdensome. In the model, although
we do not need full distributional assumption on the non-strictly exogenous variables
and the unobserved effects for consistent estimation, we need measure how sensitive
are the estimates of important quantities to the specifications of (2.5), e.g. the 00,

0'], OT 02.

55

Table 2.1: H0 : 0 = 60, where p =0 ~ 0.95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 = (p, 0.2, 0.4, ,/2.4, «1.2)
90 = (p0, 0.2, 0.4, «2.4, «1.2)
P\é 5x10-4 0.1990 0.3996 1.5482 1.0841 p0
0.01 0.0108 0.0075 0.0092 0.0150 0.0117
0.05 0.0692 0.0442 0.0458 0.0542 0.0558 0
0.10 0.1142 0.0933 0.0967 0.0958 0.1175
P\é 0.1012 0.2007 0.4012 1.5481 1.0840 p0
0.01 0.0133 0.0083 0.0092 0.0158 0.0108
0.05 0.0692 0.0442 0.0467 0.0533 0.0567 0.1
0.10 0.1008 0.0925 0.0950 0.1000 0.1550
P\é 0.1512 0.2006 0.4012 1.5481 1.0839 p0
0.01 0.0125 0.0083 0.0100 0.0158 0.0100
0.05 0.0683 0.0442 0.0483 0.0500 0.0550 0.15
0.10 0.1125 0.0925 0.1000 0.1025 0.1133
P\é 0.2013 0.2006 0.4011 1.5482 1.0837 p0
0.01 0.0117 0.0083 0.0117 0.0158 0.0092
0.05 0.0650 0.0442 0.0492 0.0500 0.0550 0.2
0.10 0.1108 0.0933 0.1025 0.1008 0.1117
Normality
Repetitions=1200, 0 = T2103 3:0," 6;, ,/2.4 2 1.5492, ,/1.2 2 1.0954

Continue (a)

56

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 = (p, 0.2, 0.4, m, \/1—.2)
6’0 2 (P0, 0-2, 0-4, 7571, M)
P\é 0.2517 0.2006 0.4011 1.5483 1.0835 p0
0.01 0.0125 0.0075 0.0117 0.0158 0.0083
0.05 0.0625 0.0442 0.0483 0.0508 0.0558 0.25
0.10 0.1125 0.0933 0.1033 0.1025 0.1125
P\é 0.3014 0.2006 0.4010 1.5484 1.0833 p0
0.01 0.0125 0.0075 0.0108 0.0158 0.0083
0.05 0.0625 0.0442 0.0483 0.0517 0.0550 0.3
0.10 0.1108 0.0942 0.1000 0.0992 0.1125
P\é 0.3514 0.2006 0.4009 1.5484 1.0830 p0
0.01 0.0117 0.0075 0.0100 0.0150 0.0083
0.05 0.0583 0.0433 0.0492 0.0508 0.0508 0.35
0.10 0.1092 0.0933 0.1025 0.0975 0.1108
P\é 0.3944 0.2022 0.4035 1.5473 1.0888 p0
0.01 0.0142 0.0067 0.0067 0.0150 0.0100
0.05 0.0508 0.0400 0.0500 0.0508 0.0492 0.4
0.10 0.1117 0.0858 0.1000 0.0992 0.1092
P\é 0.4517 0.2005 0.4006 1.5487 1.0822 p0
0.01 0.0125 0.0075 0.0092 0.0142 0.0092
0.05 0.0525 0.0433 0.0542 0.0550 0.0500 0.45
0.10 0.1083 0.0925 0.1017 0.0925 0.1075
Normality

Repetitions=1200, 0 = 72130 21:200 0; v2.4 2 1.5492, v1.2 2 1.0954

J=l

Continue (b)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 =(p,0.2,0.4,,/21,/fi)
6O :(p0,0.2,0.4,\/'2.—4,\/fi)
P\6 0.5021 0.2008 0.3998 1.5491 1.0811 p0
0.01 0.0100 0.0075 0.0108 0.0150 0.0100
0.05 0.0500 0.0425 0.0525 0.0525 0.0467 0.5
0.10 0.1050 0.0942 0.1071 0.0867 0.1017
P\6 0.5520 0.2005 0.4001 1.5490 1.0812 p0
0.01 0.0083 0.0092 0.0108 0.0150 0.0058
0.05 0.0467 0.0417 0.0525 0.0525 0.0442 0.55
0.10 0.1058 0.0967 0.0950 0.0842 0.0933
P\6 0.6022 0.2004 0.3998 1.5492 1.0806 p0
0.01 0.0083 0.0092 0.0117 0.0158 0.0050
0.05 0.0458 0.0408 0.0542 0.0533 0.0392 0.6
0.10 0.1017 0.0950 0.0958 0.0850 0.0925
P\6 0.6517 0.2012 0.4008 1.5490 1.0811 p0
0.01 0.0075 0.0092 0.0108 0.0150 0.0050
0.05 0.0475 0.0400 0.0558 0.0508 0.0350 0.65
0.10 0.0958 0.0900 0.1042 0.0825 0.0883
P\6 0.7023 0.2004 0.3996 1.5493 1.0801 p0
0.01 0.0100 0.0092 0.0108 0.0158 0.0042
0.05 0.0500 0.0425 0.0542 0.0517 0.0317 0.7
0.10 0.0933 0.0942 0.0950 0.0858 0.0833
Normality

Repetitions=1200, 6 2 121m 21.200 6; ,/2.4 2 1.5492, 4/12 2 1.0954

3:1

Continue (c)

58

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 = (p, 0.2, 0.4, «24, M)
60 :(p0,0.2,0.4.\/2—.4,\/1._2)
P\6 0.7525 0.2003 0.3993 1.5495 1.0791 p0
0.01 0.0108 0.0067 0.0108 0.0083 0.0067
0.05 0.0383 0.0350 0.0425 0.0383 0.0267 0.75
0.10 0.0750 0.0714 0.0742 0.0708 0.0633
P\6 0.8019 0.2005 0.3999 1.5491 1.0813 p0
0.01 0.0100 0.0100 0.0083 0.0117 0.0042
0.05 0.0467 0.0433 0.0542 0.0492 0.0383 0.8
0.10 0.0983 0.0983 0.0892 0.0875 0.0808
P\6 0.8521 0.2004 0.3995 1.5493 1.0802 {)0
0.01 0.0108 0.0100 0.0100 0.0108 0.0050
0.05 0.0442 0.0425 0.0508 0.0508 0.0367 0.85
0.10 0.0958 0.0975 0.0917 0.0942 0.0858
P\6 0.9018 0.2005 0.3998 1.5491 1.0809 p0
0.01 0.0100 0.0100 0.0108 0.0108 0.0050
0.05 0.0458 0.0442 0.0475 0.0525 0.0383 0.9
0.10 0.0967 0.0958 0.0933 0.0900 0.0875
P\6 0.9515 0.2005 0.4001 1.5489 1.0816 p0
0.01 0.0108 0.0092 0.0108 0.0108 0.0067
0.05 0.0467 0.0433 0.0483 0.0517 0.0433 0.95
0.10 0.0983 0.0933 0.0967 0.0892 0.0900
Normality

Repetitions:1200, 6 : 121% 212‘)" 6;, ,/2.4 2 1.5492, \/1.2 2 1.0954

1:1

(d)

59

Table 2.2: H0 : 0 = 90, where p =0 ~ 0.95
= (p, 0.2, 0.4, v2.4, v1.2)

9

60 = (p0, 0.2, 0.4, ,/2.4, «1.2)

 

 

 

P\ ”‘14: [I 0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.05 0.1 0.15 0.2 0.25 0.3 p
0.01 0.0108 0.1167 0.5400 0.9050 0.9925 1.0000 1.0000
0.05 0.0692 0.2883 0.7525 0.9608 0.9983 1.0000 1.0000 0
0.10 (0.1142 0.3825 0.8367 0.9817 0.9992 1.0000 1.0000
P\ p05 ll 0 0.05 0.1 0.15 0.2 0.25 0.3 p
0.01 0.5100 0.1000 0.0013 0.1108 0.5000 0.8817 0.9892
0.05 0.7450 0.2575 0.0692 0.2667 0.7117 0.9525 0.9975 0.1
0.10 0.8375 0.3767 0.1108 0.3758 0.8083 0.9717 0.9992
P\p94: 0 0.05 0.1 0.15 0.2 0.25 0.3 [lp
0.01 0.9100 0.4950 0.0950 0.0125 0.1100 0.4842 0.8617
0.05 0.9708 0.7250 0.2525 0.0683 0.2542 0.6967 0.9433 0.15
0.10 0.9883 0.8267 0.3633 0.1125 0.3667 0.7975 0.9667 1
P\ p05 0 0.05 0.1 0.15 0.2 0.3 0.35 II,»
0.01 0.9950 0.8967 0.4742 0.0900 0.0117 0.4700 0.8467
0.05 1.0000 0.9692 0.7117 0.2425 0.0650 0.6850 0.9400 0.2
0.10 1.0000 0.9842 0.8133 0.3542 0.1108 0.7825 1.0000;

- Normality 7
Repetitions=1200, 6 = 131,70 23:0,” 6;, ,/2.4 2 1.5492, \/1.2 2 1.0954

Continue (a)

60

 

6

z (p, 0.2, 0.4, ,/2.4, 61.2)

60 2: (p0, 0.2, 0.4, \/2.4, «1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\ ”1» 0.1 0.15 0.2 0.3 0.35 0.4 0.45 p
0.01 0.8833 0.4600 0.0842 0.1033 0.4542 0.8317 0.9708
0.05 0.9675 0.6950 0.2283 0.2542 0.6675 0.9325 0.9942 0.25
0.10 0.9783 0.8000 0.3417 0.3533 0.7650 0.9542 0.9967
P\p°—>= 0.15 0.2 0.25 0.35 0.4 0.45 0.5 p
0.01 0.8725 0.4358 0.0800 0.1042 0.4325 0.8125 0.9650
0.05 0.9625 0.6767 0.2192 0.2475 0.6450 0.9233 0.9933 0.3
0.10 0.9775 0.7817 0.3358 0.3400 0.7542 0.9500 0.9967
P\ ’00—? 0.2 0.25 0.3 0.4 0.45 0.5 0.55 p
0.01 0.8600 0.4117 0.0758 0.1025 0.4133 0.7933 0.9558
0.05 0.9567 0.6633 0.2083 0.2383 0.6358 0.9092 0.9900 0.35
0.10 0.9750 0.7750 0.3208 0.3325 0.7408 0.9442 0.9933
P\p94= 0.25 0.3 0.35 0.45 0.5 0.55 0.6 p
0.01 0.8333 0.3733 0.0625 0.1083 0.4183 0.7808 0.9533
0.05 0.9500 0.6333 0.1933 0.2458 0.6425 0.9050 0.9950 0.4
0.10 0.9742 0.7425 0.2942 0.3417 0.7425 0.9450 0.9967
P\ “L: 0.3 0.35 0.4 0.5 0.55 0.6 0.65 p
0.01 0.8333 0.3742 0.0625 0.0983 0.3858 0.7542 0.9383
0.05 0.9492 0.6450 0.1983 0.2292 0.6175 0.8825 0.9808 0.45
0.10 0.9742 0.7475 0.2983 0.3150 0.7192 0.9292 0.9917
Normality

Repetitions=1200, 0 z

1 1200
1.200 :1:

Continue (b)

1

61

6; ,/2.4 2 1.5492, ,/1.2 2 1.0954

 

6 = (p, 0.2, 0.4, «2.4, ,/1.2)
60 = (p0, 0.2, 0.4, ,/2.4, «1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P \p‘f 0.35 0.4 0.45 0.55 0.6 0.65 0.7 p
0.01 0.8200 0.3525 0.0575 0.7442 0.9317 0.9883 0.9975
0.05 0.9450 0.6225 0.1842 0.8667 0.9767 0.9950 0.9992 0.5
0.10 0.9733 0.7442 0.2992 0.9242 0.9900 0.9983 1.0000

P \p‘lf‘ 0.4 0.45 0.5 0.6 0.65 0.7 0.75 p
0.01 0.8183 0.3442 0.0517 0.1075 0.3783 0.7317 0.9225
0.05 0.9433 0.6192 0.1800 0.2258 0.6033 0.8625 0.9733 0.55
0.10 0.9742 0.7425 0.2933 0.3158 0.6992 0.9133 0.9867

P \ ”94: 0.45 0.5 0.55 0.65 0.7 0.75 0.8 p
0.01 0.8217 0.3408 0.0392 0.1067 0.3792 0.7208 0.9117
0.05 0.9425 0.6200 0.1758 0.2308 0.6008 0.8600 0.9675 0.6
0.10 0.9742 0.7433 0.2850 0.3125 0.6875 0.9058 0.9850

P W94: 0.5 0.55 0.6 0.7 0.75 0.8 0.85 p
0.01 0.8275 0.3417 0.0308 0.1117 0.3958 0.7283 0.9175
0.05 0.9450 0.6233 0.1758 0.2425 0.5933 0.8583 0.9667 0.65
0.10 0.9775 0.7383 0.2808 0.3300 0.6842 0.9075 0.9842

P \ ’90—? 0.55 0.6 0.65 0.75 0.8 0.85 0.9 p
0.01 0.8558 0.3633 0.0292 0.1 1. 92 0.4108 0.7350 0.9800
0.05 0.9550 0.6442 0.1758 0.2425 0.5983 0.8742 0.9917 0.7
0.10 0.9833 0.7725 0.2958 0.3292 0.6908 0.9150 0.9975

Normality

Repetitions=1200, 0 =

1 1200
1200 21:1

Continue (c)

62

0;, V2.4 2 1.5492, v1.2 2 1.0954

 

6

(,0, 0.2, 0.4, 4/24, ,/1.2)
00 2 (p0, 0.2, 0.4, V 2.4, V 1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P \ 5’3 0.6 0.65 0.7 0.8 0.85 0.9 0.95 p
0.01 0.7142 0.3317 0.0283 0.0975 0.3458 0.6183 0.7533
0.05 0.7867 0.5617 0.1567 0.2000 0.5067 0.7183 0.7900 0.75
0.10 0.7967 0.6500 0.2550 0.2750 0.5767 0.7517 0.7967

P \p‘L: 0.65 0.7 0.75 0.85 0.9 0.95 1 p
0.01 0.9300 0.4592 0.0442 0.1292 0.8058 0.9483 0.9908
0.05 0.9842 0.7442 0.2108 0.2767 0.9092 0.9817 0.9967 0.8
0.10 0.9942 0.8358 0.3350 0.3725 0.9392 0.9917 0.9992

P \p‘lf 0.65 0.7 0.75 0.8 0.9 0.95 1 p
0.01 1.0000 0.9617 0.5392 0.0625 0.1458 0.5333 0.8475
0.05 1.0000 0.9925 0.7958 0.2475 0.2942 0.7125 0.9342 0.85
0.10 1.0000 0.9975 0.8792 0.3650 0.3992 0.7958 0.9600

P \ ”of 0.65 0.7 0.75 0.8 0.85 0.95 1 p
0.01 1.0000 1.0000 0.9850 0.6558 0.0908 0.1658 0.5975
0.05 1.0000 1.0000 0.9975 0.8542 0.2858 0.3258 0.7717 0.9
0.10 1.0000 1.0000 1.0000 0.9258 0.4067 0.4392 0.8358

P \pof 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 1.0000 1.0000 0.9942 0.7583 0.1258 0.0108 0.1933
0.05 1.0000 1.0000 1.0000 0.9208 0.3392 0.0467 0.3725 0.95
0.10 1.0000 1.0000 1.0000 0.9208 0.3392 0.0467 0.3725

Normality
Repetitions=1200, 6 = 1.21% 3:0,“ 6;. $271 2 1.5492, «G 2 1.0954

((1)

63

 

Table 2.3: HO : 6 = 00, where p =0 ~ 0.95
6 = (p, 0.2,0.4,\/1.5, v1.25)

60 = (p0, 0.2, 0.4, ,/1.5, ,/1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 8x10—5 0.2015 0.3982 1.2226 1.1095 [)0
0.01 0.0092 0.0092 0.0133 0.0725 0.0200
0.05 0.0467 0.0575 0.0600 0.1442 0.0783 0
0.10 0.0858 0.0958 0.1108 0.2242 0.1433
P\6 0.0999 0.2044 0.3971 1.2225 1.1090 p0
0.01 0.0108 0.0100 0.0108 0.0700 0.0192
0.05 0.0467 0.0550 0.0600 0.1417 0.0742 0.1
0.10 0.0892 0.0942 0.1125 0.2192 0.1425
P\6 0.1499 0.2044 0.3970 1.2225 1.1090 p0
0.01 0.0108 0.0100 0.0108 0.0683 0.0192
0.05 0.0475 0.0558 0.0592 0.1442 0.0742 0.15
0.10 0.0858 0.0958 0.1092 0.2208 0.1433
P\6 0.2000 0.2044 0.3970 1.2226 1.1090 p0
0.01 0.0100 0.0108 0.0117 0.0667 0.0183
0.05 0.0475 0.0550 0.0608 0.1442 0.0742 0.2
0.10 0.0850 0.0967 0.1092 0.2208 0.1425

 

 

Non-normality

Repetitions=1200, 6 = ,gfi 21.200 6;, 4/15 2 1.2247,,/1.25 2 1.1180

F1
Continue (a)

64

6

(p, 0.2, 0.4, ,/1.5, 61.25)

60 = (60,0204, ,/1.5, 4/125)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.2500 0.2044 0.3970 1.2226 1.1090 p0
0.01 0.0117 0.0108 0.0117 0.0642 0.0158
0.05 0.0467 0.0550 0.0600 0.1450 0.0742 0.25
0.10 0.0833 0.0958 0.1050 0.2225 0.1392
P\6 0.3000 0.2044 0.3970 1.2226 1.1090 p0
0.01 0.0108 0.0108 0.0117 0.0625 0.0158
0.05 0.0467 0.0542 0.0600 0.1442 0.0700 0.3
0.10 0.0867 0.0958 0.1075 0.2217 0.1317
P\6 0.3500 0.2044 0.3970 1.2226 1.1090 p0
0.01 0.0125 0.0100 0.0117 0.0558 0.0167
0.05 0.0508 0.0533 0.0600 0.1450 0.0675 0.35
0.10 0.0892 0.0950 0.1050 0.2192 0.1250
P\6 0.4000 0.2044 0.3970 1.2226 1.1091 p0
0.01 0.0125 0.0125 0.0117 0.0558 0.0175
0.05 0.0525 0.0533 0.0575 0.1442 0.0683 0.4
0.10 0.0950 0.0967 0.1033 0.2150 0.1233
P\6 0.4500 0.2044 0.3970 1.2226 1.1091 p0
0.01 0.0117 0.0083 0.0125 0.0525 0.0183
0.05 0.0508 0.0517 0.0592 0.1417 0.0658 0.45
0.10 0.0983 0.0958 0.1067 0.2150 0.1217

 

 

Non-normality

 

Repetitions=1200, 6 = 7210—0 21200 6;, ,/1.5 2 1.2247,\/1.25 2 1.1180

1:1

Continue (b)

6

= (p, 0.2, 0.4, ,/1.5, 4/125)

60 = (W02, 04, 4/15, ,/1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.4999 0.2045 0.3970 1.2226 1.1093 {)0
0.01 0.0100 0.0100 0.0133 0.0550 0.0175
0.05 0.0558 0.0517 0.0608 0.1400 0.0658 0.5
0.10 0.1017 0.0967 0.1033 0.2133 0.1267
P\6 0.5499 0.2045 0.3970 1.2226 1.1094 p0
0.01 0.0125 0.0092 0.0125 0.0550 0.0167
0.05 0.0567 0.0525 0.0575 0.1367 0.0633 0.55
0.10 0.1017 0.0967 0.1075 0.2125 0.1308
P\6 0.5999 0.2045 0.3971 1.2226 1.1095 p0
0.01 0.0125 0.0100 0.0125 0.0533 0.0150
0.05 0.0575 0.0525 0.0575 0.1392 0.0608 0.6
0.10 0.1017 0.0967 0.1092 0.2150 0.1283
P\6 0.6498 0.2046 0.3972 1.2225 1.1097 p0
0.01 0.0142 0.0092 0.0117 0.0542 0.0150
0.05 0.0575 0.0533 0.0558 0.1408 0.0625 0.65
0.10 0.1017 0.0967 0.1100 0.2183 0.1242
P\6 0.6998 0.2046 0.3973 1.2225 1.1099 p0
0.01 0.0150 0.0092 0.0133 0.0533 0.0142
0.05 0.0575 0.0542 0.0550 0.1400 0.0642 0.7
0.10 0.1025 0.0967 0.1100 0.2208 0.1242

 

 

Repetitions=1200, 6 =

N on-normality

76—662

1200
i=1

Continue (c)

66

6; ,/1.5 21.2247,,/1.25 21.1180

 

6

= (p, 02,04, ,/1.5, 4/125)

60 = (p0,0.2,0.4, ,/1.5, ,/1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.7498 0.2047 0.3974 1.2225 1.1101 p0
0.01 0.0142 0.0092 0.0133 0.0517 0.0142
0.05 0.0600 0.0542 0.0542 0.1375 0.0692 0.75
0.10 0.1000 0.0958 0.1083 0.2200 0.1200
P\6 0.7997 0.2047 0.3976 1.2224 1.1103 [)0
0.01 0.0158 0.0100 0.0117 0.0508 0.0117
0.05 0.0617 0.0542 0.0550 0.1433 0.0683 0.8
0.10 0.1033 0.0967 0.1108 0.2258 0.1183
P\6 0.8497 0.2048 0.3977 1.2223 1.1105 p0
0.01 0.0175 0.0092 0.0125 0.0492 0.0133
0.05 0.0617 0.0542 0.0542 0.1408 0.0675 0.85
0.10 0.0975 0.0967 0.1125 0.2258 0.1208
P\6 0.8997 0.2048 0.3979 1.2223 1.1107 p0
0.01 0.0175 0.0108 0.0117 0.0508 0.0142
0.05 0.0617 0.0542 0.0558 0.1142 0.0650 0.9
0.10 0.1025 0.0983 0.1175 0.2225 0.1242
P\6 0.9496 0.2048 0.3980 1.2222 1.1108 p0
0.01 0.0175 0.0100 0.0117 0.0525 0.0167
0.05 0.0583 0.0533 0.0550 0.1475 0.0658 0.95
0.10 0.1058 0.1000 0.1200 0.2242 0.1258

 

 

Repetitions=1200, 6 =

N on-normality

1
mi:

1200
i=1

((1)

67

6;. \/1.5 2122476125 21.1180

 

Table 2.4: H0 : 6 = 60, where p =0 ~ 0.95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 = (p, 0.2, 0.4, 61.5, «1.25)
60 2 (p0, 0.2, 0.4, ,/1.5, ,/1.25)

P\ ”of 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.0092 0.1258 0.5900 0.9433 0.9992 1.0000 1.0000 1.0000
0.05 0.0467 0.3133 0.7808 0.9850 0.9992 1.0000 1.0000 1.0000 0
0.10 0.0858 0.4158 0.8733 0.9958 1.0000 1.0000 1.0000 1.0000

P \ ’00—»: 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.5592 0.0933 0.0108 0.1267 0.5583 0.9267 0.9975 1.0000
0.05 0.7833 0.2725 0.0467 0.2967 0.7633 0.9758 0.9992 1.0000 0.1
0.10 0.8800 0.3800 0.0892 0.3992 0.8583 0.9900 0.9992 1.0000

P \ 994: 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.9533 0.5442 0.0883 0.0108 0.1283 0.5500 0.9183 0.9967
0.05 0.9875 0.7717 0.2650 0.0475 0.2942 0.7575 0.9742 0.9992 0.15
0.10 0.9917 0.8700 0.3758 0.0858 0.3950 0.8483 0.9875 0.9992

P \pO—i: 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.9983 0.9492 0.5333 0.0825 0.0100 0.1250 0.5408 0.9108
0.05 1.0000 0.9867 0.7658 0.2642 0.0475 0.2875 0.7492 0.9683 0.2
0.10 1.0000 0.9908 0.8617 0.3650 0.0850 0.3925 0.8433 0.9833

Non-normality
Repetitions=1200, 6 = 516,—, 231.200 6; \/1.5 2 1.2247,\/1.25 2 1.1180

1:1

68

Continue (a)

 

6

2 (p, 0.2, 0.4, «1.5, ,/1.25’)
60 Z (10010-21 04, V 1-5, V 1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\p‘f 0.1 0.15 0.2 0.25 0.3 0.35 0.45 0.5 p
0.01 0.9408 0.5183 0.0817 0.0117 0.1233 0.5342 0.9058 0.9900
0.05 0.9867 0.7525 0.2517 0.0467 0.2825 0.7450 0.9675 0.9675 0.25
0.10 0.9917 0.8583 0.3658 0.0833 0.3858 0.8392 0.9808 0.9992
P\pO—f 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 p
0.01 0.9375 0.5100 0.0800 0.0108 0.1250 0.5242 0.8992 0.9858
0.05 0.9858 0.7467 0.2425 0.0467 0.2800 0.7375 0.9658 0.9983 0.3
0.10 0.9908 0.8467 0.3600 0.0867 0.3825 0.8367 0.9775 0.9992
P\""’_.= 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.9342 0.5083 0.0775 0.0125 0.1208 0.5208 0.8950 0.9850
0.05 0.9867 0.7442 0.2367 0.0508 0.2792 0.7375 0.9650 0.9983 0.35
0.10 0.9917 0.8458 0.3650 0.0892 0.3833 0.8350 0.9775 0.9992
P\‘O‘L= 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p
0.01 0.9317 0.5025 0.0758 0.0125 0.1167 0.5133 0.8908 0.9825
0.05 0.9850 0.7400 0.2317 0.0525 0.2825 0.7375 0.9617 0.9975 0.4
0.10 0.9925 0.8467 0.3633 0.0950 0.3850 0.8292 0.9758 0.9983
P\p‘lf 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 p
0.01 0.9383 0.5058 0.0767 0.0117 0.1142 0.5167 0.8833 0.9825
0.05 0.9875 0.7442 0.2292 0.0508 0.2858 0.7425 0.9583 0.9967 0.45
0.10 0.9925 0.8517 0.3642 0.0983 0.3908 0.8258 0.9758 0.9975

 

 

 

 

 

 

Repetitions=1200, 6 2 121—00 [go 6;, ,/1.5 2 1.2247,,/1.25 2 1.1180

N on-normality

69

Continue (b)

 

6

(p, 0.2, 0.4, ,/1.5, «1.25)

60 2 (p0, 0.2, 0.4, 4/15, «1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P \p0—42 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 p
0.01 0.9392 0.5125 0.0792 0.0100 0.1183 0.5217 0.8842 0.9842
0.05 0.9875 0.7542 0.2317 0.0558 0.2950 0.7450 0.9575 0.9950 0.5
0.10 0.9950 0.8558 0.3667 0.1017 0.3900 0.8308 0.9767 0.9975
P\p‘l? 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 p
0.01 0.9467 0.5350 0.0825 0.0125 0.1292 0.5408 0.8950 0.9842
0.05 0.9883 0.7700 0.2383 0.0567 0.3083 0.7500 0.9558 0.9950 0.55
0.10 0.9950 0.8675 0.3617 0.1017 0.3933 0.8375 0.9792 0.9975
P\p‘lf‘ 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 p
0.01 0.9550 0.5583 0.0867 0.0125 0.1383 0.5683 0.9042 0.9850
0.05 0.9908 0.7917 0.2475 0.0575 0.3142 0.7592 0.9583 0.9958 0.6
0.10 0.9950 0.8792 0.3700 0.1017 0.4150 0.8425 0.9800 0.9967
P\p‘f 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.9642 0.5917 0.1008 0.0142 0.1575 0.6000 0.9117 0.9892
0.05 0.9908 0.8167 0.2617 0.0575 0.3308 0.7892 0.9650 0.9958 0.65
0.10 0.9958 0.8992 0.3482 0.1017 0.4267 0.8517 0.9817 0.9975
P\6-L»: 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 p
0.01 0.9767 0.6425 0.1150 0.0150 0.1700 0.6400 0.9242 0.9917
0.05 0.9933 0.8508 0.2858 0.0575 0.3592 0.8067 0.9758 0.9975 0.7
0.10 0.9967 0.9175 0.4008 0.1025 0.4550 0.8650 0.9867 0.9975

 

 

 

 

 

Repetitions=1200, 6 = F100 21.200 6;, ,/1.5 2 1.2247,,/1.25 2 1.1180

Non-normality

1:1

70

Continue (c)

 

6

= (p. 0.2. 0.4. \/1.5, «1.25)
00 2 (p0, 0.2, 0.4, V1.5,V1.25)

 

 

P\pof

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 p
0.01 0.9858 0.7042 0.1308 0.0142 0.1850 0.6750 0.9408 0.9975
0.05 0.9950 0.8867 0.3083 0.0600 0.3767 0.8258 0.9833 1.0000 0.75
0.10 0.9975 0.9325 0.4283 0.1000 0.4842 0.8883 0.9892 1.0000
P \ ”94: 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 0.9900 0.7725 0.1533 0.0158 0.2175 0.7250 0.9608 0.9967
0.05 0.9975 0.9192 0.3425 0.0617 0.4042 0.8642 0.9892 0.9975 0.8
0.10 0.9992 0.9558 0.4708 0.1033 0.5208 0.9108 0.9942 0.9992
P \ ’09»: 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 1.0000 0.9933 0.8375 0.1792 0.0175 0.2492 0.7750 0.9775
0.05 1.0000 0.9992 0.9417 0.3908 0.0617 0.4608 0.9017 0.9942 0.85
0.10 1.0000 0.9992 0.9642 0.5200 0.0975 0.5625 0.9375 0.9967
P \“f 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 1.0000 1.0000 0.9992 0.8950 0.2208 0.0175 0.2908 0.8367
0.05 1.0000 1.0000 0.9992 0.9650 0.4442 0.0617 0.4933 0.9333 0.9
0.10 1.0000 1.0000 0.9992 0.9867 0.5833 0.1025 0.6000 0.9625
P V”? 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 1.0000 1.0000 1.0000 0.9992 0.9367 0.2833 0.0175 0.3425
. 0.05 1.0000 1.0000 1.0000 0.9992 0.9858 0.5200 0.0583 0.5600 0.95
0.10 1.0000 1.0000 1.0000 1.0000 0.9900 0.6533 0.1058 0.6517

 

 

Repetitions=1200, 6 = 12—60 2120," 6;, ,/1.5 2 1.2247,,/1.25 2 1.1180

N on-normality

J:

((0

71

 

Table 2.5: CMLE for the dynamic panel data of log-wage with unobserved heterogeneity,
period:1980 ~ 1987

 

 

coefficient. ,6 do (i, 65 6a

 

CMLE 0.3405 0.8784 0.1839 0.3511 0.2162

t-statistics (18.418) (25.328) (8.704) (77.425) (18.904)

 

 

 

 

 

 

 

72

Table 2.6: HO : 6 = 60, where p = 0 ~ 0.95

6 : (p, 0.15, 0.2. 0.4. 0.35. 4/24, ,/1.2)
60 2 (/)0, 0.15, 0.2, 0.4, 0.35, 4/24, ,/1.2)

{

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 1.7x10-3 0.1500 0.1988 0.4029 0.3528 1.5478 1.0801 p0
0.01 0.0100 0.0083 0.0100 0.0100 0.0142 0.0108 0.0142
0.05 0.0583 0.0517 0.0625 0.0492 0.0483 0.0533 0.0483 0
0.10 0.1058 0.1033 0.1067 0.1025 0.0917 0.1033 0.1067
P\6 0.1017 0.1500 0.1988 0.4028 0.3527 1.5479 1.0798 p0
0.01 0.0108 0.0083 0.0100 0.0100 0.0133 0.0108 0.0142
0.05 0.0567 0.0517 0.0617 0.0457 0.0508 0.0517 0.0500 0.1
0.10 0.1033 0.1033 0.1058 0.1033 0.0908 0.1058 0.0142
P\6 0.1571 0.1500 0.1988 0.4027 0.3527 1.5479 1.0797 p0
0.01 0.0100 0.0083 0.0108 0.0108 0.0133 0.0108 0.0142
0.05 0.0583 0.0517 0.0617 0.0483 0.0500 0.0508 0.0492 0.15
0.10 0.1025 0.1033 0.1058 0.1017 0.0900 0.1042 0.0142
P\6 0.2017 0.1500 0.1988 0.4027 0.3527 1.5479 1.0795 p0
0.01 0.0117 0.0083 0.0108 0.0108 0.0133 0.0117 0.0142
0.05 0.0567 0.0517 0.0617 0.0508 0.0492 0.0500 0.0492 0.2
0.10 0.1017 0.1025 0.1042 0.1000 0.0917 0.1050 0.1083
Normality

Repetitions=1200, 6_ _

73

17.10299; \/2 .4 2 15942612210954

Continue (a)

 

6 z (p, 0.15, 0.2, 0.4, 0.35, 62.4, \/1.2)
60 2 (p0, 0.15.0.2, 0.4, 0.35, 62.4, 61.2)

 

 

P\6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.2517 0.1500 0.1988 0.4026 0.3526 1.5480 1.0794 p0
0.01 0.0125 0.0083 0.0108 0.0108 0.0133 0.0117 0.0150
0.05 0.0550 0.0517 0.0617 0.0517 0.0500 0.0508 0.0467 0.25
0.10 0.1008 0.1025 0.1050 0.1000 0.0925 0.1050 0.1092
P\6 0.3017 0.1500 0.1988 0.4026 0.3526 1,5481 1.0792 p0
0.01 0.0117 0.0083 0.0108 0.0108 0.0133 0.0125 0.0133
0.05 0.0533 0.0517 0.0633 0.0508 0.0500 0.0500 0.0458 0.3
0.10 0.0992 0.1033 0.1050 0.1017 0.0883 0.1025 0.1075
P\6 0.3517 0.1500 0.1988 0.4025 0.3526 1.5481 1.0791 p0
0.01 0.0117 0.0083 0.0108 0.0100 0.0142 0.0125 0.0125
0.05 0.0517 0.0508 0.0625 0.0500 0.0525 0.0492 0.0458 0.35
0.10 0.0967 0.1033 0.1050 0.1017 0.0900 0.1000 0.1025
P\6 0.4017 0.1051 0.1988 0.4024 0.3526 1.5482 1.0789 p0
0.01 0.0125 0.0083 0.0108 0.0083 0.0142 0.0125 0.0100
0.05 0.0542 0.0508 0.0625 0.0492 0.0517 0.0450 0.0500 0.4
0.10 0.0925 0.1042 0.1050 0.1042 0.0892 0.0950 0.1017
P\6 0.4517 0.1501 0.1988 0.4024 0.3526 1.5482 1.0788 p0
0.01 0.0125 0.0083 0.0100 0.0092 0.0133 0.0117 0.0075
0.05 0.0525 0.0517 0.0633 0.0508 0.0508 0.0467 0.0508 0.45
0.10 0.0958 0.1042 0.1067 0.1025 0.0908 0.0958 0.0942

Normality

Repetitions=1200, 6—

1710022106; x/‘z ._4 ~ 1. 5942 ,\/1 .22 1. 0954

Continue (b)

74

 

6 = (p, 0.15, 0.2, 0.4, 0.35, 62.4, \/1.2)
60 = (p0, 0.15, 0.2, 0.4, 0.35, 62.4, ,/1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.5017 0.1501 0.1988 0.4023 0.3526 1.5483 1.0786 p0
0.01 0.0117 0.0083 0.0100 0.0083 0.0142 0.0117 0.0033

0.05 0.0500 0.0517 0.0642 0.0492 0.0517 0.0467 0.0492 0.5
0.10 0.0992 0.1042 0.1058 0.1075 0.0908 0.1025 0.0958

P\6 0.5517 0.1501 0.1989 0.4022 0.3527 1.5484 1.0785 p0
0.01 0.0133 0.0083 0.092 0.0083 0.0142 0.0100 0.0058

0.05 0.0483 0.0517 0.0633 0.0500 0.0508 0.0483 0.0492 0.55
0.10 0.0942 0.1042 0.1067 0.1083 0.0908 0.0992 0.0983

P\6 0.6016 0.150] 0.1989 0.4022 0.3527 1.5484 1.0784 p0
0.01 0.0125 0.0083 0.0092 0.0067 0.0150 0.0100 0.0042

0.05 0.0525 0.0517 0.0625 0.0500 0.0508 0.0483 0.0458 0.6
0.10 0.0933 0.1033 0.1067 0.1108 0.0917 0.0950 0.0983

P\6 0.6515 0.1502 0.1989 0.4022 0.3528 1.5484 1.0787 p0
0.01 0.0108 0.0092 0.0100 0.0067 0.0167 0.0100 0.0042

0.05 0.0500 0.0517 0.0625 0.0500 0.0492 0.0475 0.0450 0.65
0.10 0.0958 0.1025 0.1067 0.1117 0.0908 0.0950 0.0950

P\6 0.7016 0.1501 0.1990 0.4021 0.3529 1.5485 1.0779 p0
0.01 0.0133 0.0092 0.0092 0.0042 0.0175 0.0125 0.0050

0.05 0.0508 0.0508 0.0625 0.0533 0.0517 0.0492 0.0458 0.7
0.10 0.0975 0.1025 0.1050 0.1125 0.0900 0.0983 0.0925

Normality
Repetitions=1200, 6 : 1.2% 23:0,” 6;, m 2 1.5942,,/fi 2 1.0954

Continue (c)

75

6 = (p, 0.15, 0.2, 0.4, 0.35, 62.4, ,/1.2)
60 = (p0, 0.15, 0.2, 0.4, 0.35, 62.4, ,/1.2)

 

 

A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.7515 0.1501 0.1990 0.4022 0.3530 1.5485 1.0782 p0
0.01 0.0142 0.0092 0.0100 0.0042 0.0175 0.0142 0.0058
0.05 0.0558 0.0517 0.0600 0.0525 0.0492 0.0525 0.0417 0.75
0.10 0.0992 0.1025 0.1083 0.1100 0.0908 0.0950 0.0908
P\6 0.8013 0.1501 0.1991 0.4024 0.3533 1.5484 1.0790 p0
0.01 0.0125 0.0092 0.0100 0.0050 0.0617 0.0125 0.0050
0.05 0.0542 0.0517 0.0592 0.0533 0.0483 0.0508 0.0433 0.8
0.10 0.1050 0.1017 0.1083 0.1058 0.0892 0.0967 0.0892
P\6 0.8512 0.1501 0.1992 0.4026 0.3534 1.5483 1.0794 p0
0.01 0.0117 0.0083 0.0100 0.0033 0.0167 0.0133 0.0067
0.05 0.0542 0.0508 0.0592 0.0517 0.0492 0.0525 0.0450 0.85
0.10 0.1067 0.1008 0.1100 0.1075 0.0883 0.0992 0.0900
P\6 0.9011 0.1501 0.1990 0.4022 0.3530 1.5485, 1.0782 p0
0.01 0.0100 0.0083 0.0092 0.0042 0.0167 0.0133 0.0075
0.05 0.0583 0.0508 0.0583 0.0525 0.0492 0.0492 0.0458 0.9
0.10 0.1083 0.0992 0.1083 0.1058 0.0900 0.1033 0.0925
P\6 0.9509 0.1501 0.1993 0.4030 0.3538 1.5480 1.0804 p0
0.01 0.0092 0.0083 0.0108 0.0058 0.0167 0.0133 0.0092
0.05 0.0608 0.0508 0.0592 0.0517 0.0475 0.0475 0.0458 0.95
0.10 . 0.1125 0.0983 0.1075 0.1083 0.0892 0.0892 0.0942
Normality

Repetitions=1200, 9 = 171% Z

J=1

(d)

76

1200 03., \[2—4 9: 1.5942612 2 1.0954

 

{

Table 2.7:

6 = (p, 0.15, 0.2, 0.4, 0.35, \/2.4, \/1.2)
60 = (p0, 0.15, 0.2, 0.4, 0.35, 62.4, \/1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\ ”94: 0 0.05 0.1 0.15 ' 0.2 0.25 0.3 0.35 p
0.01 0.0100 0.1083 0.5283 0.9000 0.9892 1.0000 1.0000 1.0000
0.05 0.0583 0.2658 0.7617 0.9675 0.9975 1.0000 1.0000 1.0000 0
0.10 0.1058 0.3758 0.8358 0.9792 0.9983 1.0000 1.0000 1.0000

P\ “L: 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.5233 0.0983 0.0108 0.0958 0.4775 0.8833 0.9825 1.0000
0.05 0.7483 0.2450 0.0567 0.2525 0.7217 0.9558 0.9958 1.0000 0.1
0.10 0.8392 0.3692 0.1033 0.3617 0.8117 0.9758 0.9975 1.0000

P\ ”L: 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.9200 0.5033 0.0917 0.0100 0.0917 0.4675 0.8692 0.9800
0.05 0.9708 0.7317 0.2342 0.0583 0.2508 0.7125 0.9517 0.9942 0.15
0.10 0.9908 0.8267 0.3567 0.1025 0.3525 0.7992 0.9733 0.9975

P\ ’00—»: 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.9975 0.9075 0.4817 0.0867 0.0117 0.0892 0.4525 0.8508
0.05 0.9992 0.9675 0.7133 0.2325 0.0567 0.2550 0.6908 0.9450 0.2
0.10 0.9992 0.9883 0.8092 0.3442 0.1017 0.3508 0.7808 0.9708

Normality

Repetit.ions:1200, é : 121—00 21200

2:1

77

63‘, v2.4 9: 1.5942,\/1.2 2 1.0954

Continue ( a)

 

9 = (p, 015,02, 0.4, 0.35, «2.4, \/1.2)
90 2 (pg, 015. 0.2, 0.4. 0.35, 62.4, 61.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

P\ ”of 0.1 0.15 0.2 0.25 0.3 0.35 0.45 0.5 p
0.01 0.8950 0.4633 0.0792 0.0125 0.0867 0.4342 0.8308 0.9750
0.05 0.9642 0.6992 0.2250 0.0550 0.2542 0.6708 0.9367 0.9875 0.25
0.10 0.9833 0.7958 0.3367 0.1008 0.3425 0.7608 0.9650 0.9958
P\ ”L: 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 p
0.01 0.8725 0.4392 0.0767 0.0117 0.0875 0.4233 0.8117 0.9725
0.05 0.9617 0.6825 0.2200 0.0533 0.2492 0.6483 0.9258 0.9858 0.3
0.10 0.9817 0.7925 0.3333 0.0992 0.3350 0.7492 0.9608 0.9942
P\ p94: 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.8617 0.4167 0.0725 0.0117 0.0875 0.4175 0.7983 0.9675
0.05 0.9583 0.6683 0.2150 0.0517 0.2417 0.6350 0.9167 0.9842 0.35
0.10 0.9792 0.7800 0.3200 0.0967 0.3308 0.7375 0.9583 0.9892
P\ ”"5 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p
0.01 0.8450 0.3975 0.0617 0.0125 0.0883 0.4142 0.7775 0.9633
0.05 0.9558 0.6550 0.2017 0.0542 0.2283 0.6267 0.0.9075 0.9825 0.4
0.10 0.9775 0.7700 0.3150 0.0925 0.3317 0.7275 0.9517 0.9867
P\ ”of 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 p
0.01 0.8350 0.3825 0.0525 0.0125 0.0883 0.4050 0.7683 0.9550
0.05 0.9517 0.6533 0.1975 0.0525 0.2292 0.6167 0.8942 0.9817 0.45
0.10 0.9750 0.7633 0.3150 0.0958 0.3333 0.7092 0.9400 0.9850
Normality

Repetitions=1200, é : 17155 23:01" 0;. «2.4 2 1.5942,\/1.2 2 1.0954

Continue (b)

78

 

 

6 = (p, 0.15, 0.2, 0.4, 0.35, v2.4, v1.2)

60 2 (p0, 0.15, 0.2, 0.4, 0.35, V 2.4,V12)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\ ”1»: 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 p
0.01 0.8333 0.3725 0.0475 0.0117 0.0925 0.4067 0.7558 0.9483
0.05 0.9550 0.6342 0.1867 0.0500 0.2258 0.6117 0.8833 0.9808 0.5
0.10 0.9758 0.7550 0.3067 0.0992 0.3283 0.7042 0.9308 0.9833
P\p‘f 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 p
0.01 0.8308 0.3658 0.0408 0.0133 0.0992 0.4017 0.7475 0.9383
0.05 0.9517 0.6317 0.1833 0.0483 0.2375 0.6092 0.8808 0.9792 0.55
0.10 0.9767 0.7508 0.2925 0.0942 0.3350 0.7058 0.9233 0.9825
P\p‘lf 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 p
0.01 0.8367 0.3658 0.0375 0.0125 0.1067 0.3967 0.7492 0.9300
0.05 0.9525 0.6367 0.1833 0.0525 0.2450 0.6033 0.8800 0.9758 0.6
0.10 0.9783 0.7525 0.2833 0.0933 0.3358 0.7108 0.9267 0.9817
P\p‘l»: 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.8533 0.3708 0.0333 0.0108 0.1058 0.4008 0.7525 0.9308
0.05 0.9592 0.6458 0.1817 0.0500 0.2517 0.6133 0.8850 0.9725 0.65
0.10 0.9792 0.7600 0.2900 0.0958 0.3417 0.7133 0.9275 0.9833
P\pof 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 p
0.01 0.8867 0.3867 0.0375 0.0133 0.1150 0.4233 0.7683 0.9325
0.05 0.9675 0.6625 0.1792 0.0508 0.2625 0.6325 0.8892 0.9725 0.7
0.10 0.9817 0.7742 0.2975 0.0975 0.3492 0.7250 0.9292 0.9825
Normality

Repetitions=1200, 6 = % 231.2006}, 62.4 2 1.5942,\/1.2 2 1.0954

J=1

79

Continue (0)

 

6 = (p, 0.15, 0.2, 0.4, 0.35, 62.4, 61.2)

60 2 (p0, 0.15, 0.2, 0.4, 0.35, 62.4, 61.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Repetitions=1200, 6 = filo—0 21.200 6;, 62.4 2 1.5942612 2 1.0954

3:1

(d)

80

P\ ”of 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 p
0.01 0.9033 0.4208 0.0383 0.0142 0.1258 0.4517 0.9375 0.9808
0.05 0.9725 0.6842 0.1825 0.0558 0.2767 0.6458 0.9767 0.9942 0.75
0.10 0.9875 0.8017 0.3067 0.0992 0.3617 0.7425 1.0000 0.9958
P\pof 0.65 0.7 0.75 0.80 0.85 0.9 0.95 1 p
0.01 0.9333 0.4708 0.0425 0.0125 0.1392 0.4917 0.8267 0.9500
0.05 0.9825 0.7350 0.2008 0.0542 0.2867 0.6875 0.9050 0.9783 0.8
0.10 0.9925 0.8392 0.3233 0.1050 0.3742 0.7733 0.9408 0.9858
P\p‘L 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01. 1.0000 0.9575 0.5367 0.0567 0.0117 0.1458 0.5492 0.8558
0.05 1.0000 0.9917 0.7967 0.2217 0.0542 0.2967 0.7308 0.9250 0.85
0.10 1.0000 0.9967 0.8808 0.3458 0.1067 0.3983 0.7967 0.9550
P\poj 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 0.9992 0.9992 0.9758 0.6300 0.0742 0.0100 0.1617 0.6117
0.05 1.0000 1.0000 0.9967 0.8575 0.2583 0.0583 0.3258 0.7750 0.9
0.10 1.0000 1.0000 1.0000 0.9142 0.3858 0.1083 0.4400 0.8308
P \p‘L, 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 1.0000 1.0000 1.0000 0.9908 0.7217 0.1033 0.0092 0.1850
0.05 1.0000 1.0000 1.0000 1.0000 0.9025 0.2942 0.0608 0.3717 0.95
0.10 1.0000 1.0000 1.0000 1.0000 0.9500 0.4333 0.1125 0.4933
Normality

 

Table 2.8: H0 : 6 = 60, where p = 0 ~ 0.95

60 = (p0, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)

{ 6 = (p, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 7x111-4 0.1498 0.2017 0.3986 0.3490 1.2220 1.1070 [)0
0.01 0.0108 0.0125 0.0108 0.0125 0.0133 0.0733 0.0200
0.05 0.0433 0.0517 0.0558 0.0633 0.0558 0.1442 0.0850 0
0.10 0.0842 0.1058 0.0992 0.1125 0.0983 0.2283 0.0842
P\6 0.0998 0.1504 0.2045 0.3977 0.3450 1.2219 1.1064 p0
0.01 0.0100 0.0125 0.0100 0.0108 0.0117 0.0725 0.0200
0.05 0.0450 0.0517 0.0525 0.0625 0.0533 0.1467 0.0833 0.1
0.10 0.0842 0.1058 0.1000 0.1100 0.0992 0.2250 0.1425
P\6 0.1498 0.1504 0.2045 0.3977 0.3450 1.2219 1.1064 p0
0.01 0.0083 0.0125 0.0100 0.0108 0.0125 0.0725 0.0192
0.05 0.0483 0.0517 0.0517 0.0600 0.0533 0.1467 0.0808 0.15
0.10 0.0867 0.1058 0.0992 0.1117 0.0983 0.2242 0.1433
P\6 0.1998 0.1504 0.2045 0.3977 0.3450 1.2219 1.1065 p0
0.01 0.0100 0.0125 0.0092 0.0108 0.0125 0.0675 0.0183
0.05 0.0467 0.0517 0.0525 0.0600 0.0558 0.1475 0.0800 0.2
0.10 0.0908 0.1058 0.0992 0.1100 0.0992 0.2242 0.1450

 

 

 

 

 

 

 

Repetitions=1200, 6_ —

Non- normality

Continue (2 1)

81

1700 21-309" 61.5 21.2247,61.25 21.11870

6 = (p,0.15.0.2.0.4,0.35, 61.5, 61.25)

60 = (/)0, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)

 

 

A

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.2499 0.1500 0.2042 0.3983 0.3451 1.2220 1.1074 p0
0.01 0.0100 0.0125 0.0083 0.0108 0.0108 0.0575 0.0183
0.05 0.0417 0.0567 0.0492 0.0658 0.0633 0.1367 0.0792 0.25
0.10 0.0825 0.1083 0.0925 0.1042 0.1058 0.2092 0.1350
P\6 0.2998 0.1504 0.2045 0.3977 0.3451 1.2219 1.1180 p0
0.01 0.0117 0.0125 0.0117 0.0108 0.0117 0.0608 0.0183
0.05 0.0517 0.0517 0.0525 0.0608 0.0558 0.1475 0.0758 0.3
0.10 0.0950 0.1050 0.0983 0.1050 0.0.0975 0.2208 0.1383
P\6 0.3497 0.1504 0.2045 0.3977 0.3451 1.2219 1.1066 p0
0.01 0.0125 0.0125 0.0092 0.0125 0.0108 0.0575 0.0175
0.05 0.0508 0.0517 0.0533 0.0575 0.0608 0.1442 0.0717 0.35
0.10 0.0925 0.1058 0.0975 0.1092 0.0958 0.2208 0.1325
P\6 0.3997 0.1504 0.2046 0.3978 0.3451 1.2219 1.1068 p0
0.01 0.0108 0.0125 0.0100 0.0125 0.0100 0.0575 0.0158
0.05 0.0508 0.0517 0.0525 0.0575 0.0592 0.1433 0.0725 0.4
0.10 0.0983 0.1067 0.0975 0.1083 0.0950 0.2200 0.1325
P\6 0.4497 0.1504 0.2046 0.3978 0.3452 1.2219 1.1069 p0
0.01 0.0100 0.0125 0.0100 0.0133 0.0100 0.0550 0.0158
0.05 0.0508 0.0508 0.0508 0.0550 0.0600 0.1400 0.0692 0.45
0.10 0.1017 0.1058 0.0967 0.1067 0.0958 0.2192 0.1275

 

 

 

 

 

 

 

Repetitions=1200, 6:

N on-normality

12—‘(70 21:00 6; 61 1.5 21.2247,61.25 2 1.11870

Continue (b)

82

 

6 = (p, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)
60 = (p0, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.4994 0.1507 0.2043 0.3984 0.3456 1.2219 1.1160 p0

0.01 0.0050 0.0133 0.0092 0.0133 0.0108 0.0525 0.0158

0.05 0.0533 0.0550 0.0517 0.0500 0.0533 0.1292 0.0575 0.5
0.10 0.0942 0.1000 0.0950 0.1000 0.0992 0.2092 0.1142

P\6 0.5496 0.1504 0.2047 0.3979 0.3453 1.2218 1.1072 p0

0.01 0.0100 0.0125 0.0117 0.0133 0.0108 0.0558 0.0158

0.05 0.0550 0.0508 0.0508 0.0525 0.0542 0.1375 0.0667 0.55
0.10 0.1058 0.1042 0.1000 0.1100 0.0942 0.2158 0.1283

P\6 0.5995 0.1504 0.2047 0.3980 0.3453 1.2218 1.1074 p0

0.01 0.1050 0.1042 0.1000 0.1100 0.0950 0.2125 0.1242

0.05 0.0550 0.0508 0.0517 0.0517 0.0550 0.1400 0.0633 0.6
0.10 0.1050 0.1042 0.1000 0.1100 0.0950 0.2125 0.1242

P\6 0.6495 0.1504 0.2048 0.3982 0.3454 1.2218 1.1077 p0

0.01 0.0125 0.0125 0.0083 0.0125 0.0108 0.0542 0.0158

0.05 0.0567 0.0500 0.0508 0.0550 0.0525 0.1417 0.0658 0.65
0.10 0.1050 0.1033 0.0992 0.1108 0.0942 0.2125 0.1258

P\6 0.6995 0.1504 0.2048 0.3983 0.3455 1.2217 1.1079 p0

0.01 0.0142 0.0125 0.0083 0.0133 0.0117 0.0542 0.0142

0.05 0.0567 0.0500 0.0508 0.0542 0.0525 0.1458 0.0650 0.7
0.10 0.1033 0.1050 0.0983 0.1092 0.0967 0.2133 0.1242

 

 

Repetitions=1200, 6— —

N on-normality

7—260 231-1000; 611.5 2 1.2247,61.25 2 1.11870

Continue (0)

83

 

6 = (p, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)
60 = (p0, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.7494 0.1504 0.2049 0.3984 0.34 55 1.2217 1.1081 p0
0.01 0.0133 0.0125 0.0083 0.0117 0.0117 0.0533 0.0150
0.05 0.0550 0.0500 0.0525 0.0550 0.0525 0.1442 0.0633 0.75
0.10 0.1050 0.1050 0.0975 0.1067 0.0967 0.2167 0.1208
P\6 0.7994 0.1504 0.2049 0.3986 0.3456 1.2216 1.1083 {)0
0.01 0.0142 0.0125 0.0083 0.0125 0.0117 0.0542 0.0117
0.05 0.0575 0.0508 0.0533 0.0558 0.0542 0.1442 0.0658 0.8
0.10 0.1000 0.1042 0.0983 0.1092 0.0983 0.2200 0.1175
P\6 0.8494 0.1504 0.2050 0.3987 0.3456 1.2216 1.1084 p0
0.01 0.0158 0.0125 0.0083 0.0108 0.0117 0.0542 0.0142
0.05 0.0550 0.0508 0.0525 0.0550 0.0517 0.1450 0.0642 0.85
0.10 0.1017 0.1042 0.1000 0.1083 0.0975 0.2200 0.1225
P\6 0.8994 0.1504 0.2050 0.3988 0.3457 1.2215 1.1085 p0
0.01 0.0200 0.0125 0.0083 0.0133 0.1117 0.0550 0.0150
0.05 0.0583 0.0500 0.0517 0.0567 0.0508 0.1442 0.0675 0.9
0.10 0.0992 0.10-:12 0.1008 0.1117 0.0983 0.2217 0.1192
P\6 0.9494 0.1504 0.2050 0.3989 0.3457 1.2215 1.1085 p0
0.01 0.0200 0.0125 0.0083 0.0125 0.0117 0.0542 0.0158
0.05 0.0542 0.0508 0.0525 0.0558 0.0517 0.1508 0.0700 0.95
0.10 0.0958 0.1042 0.1008 0.1133 0.1000 0.2208 0.1175

 

 

Repetitions=1200, 6 z

N on-normality

1
mi:

1200
i=1

((1)

84

6;, 61.5 21.2247,61.25 21.1180

 

Table 2.9: H0 : 0 = 60, where p = 0 ~ 0.95

6 = (p, 0.15, 0.2, 04,035, 61.5, 61.25)
60 2 (p0, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)

{

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P \p9-f 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.0108 0.1433 0.6225 0.9558 0.9992 1.0000 1.0000 1.0000
0.05 0.0433 0.3242 0.8183 0.9900 1.0000 1.0000 1.0000 1.0000 0
0.10 0.0842 0.4342 0.8958 0.9958 1 .0000 1.0000 1.0000 1.0000

P \p9—f 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.5892 0.0992 0.0100 0.1342 0.5900 0.9117 0.9992 1.0000
0.05 0.8075 0.2825 0.0450 0.3058 0.7933 0.9825 0.9992 1.0000 0.1
0.10 0.8958 0.3492 0.0842 0.4167 0.8742 0.9917 1.0000 1.0000

P 6’9: 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.9667 0.5800 0.0950 0.0083 0.1300 0.5808 0.9383 0.9983
0.05 0.9883 0.8025 0.2742 0.0483 0.3067 0.7858 0.9808 0.9992 0.15
0.10 0.9950 0.8842 0.3875 0.0867 0.4092 0.8675 0.9900 1.0000

P \pO—T 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p
0.01 0.9992 0.9592 0.5675 0.0833 0.0100 0.1333 0.5650 0.9292
0.05 1.0000 0.9875 0.7925 0.2667 0.0467 0.3050 0.7783 0.9750 0.2
0.10 1.0000 0.9950 0.8775 0.3842 0.0908 0.4017 0.8617 0.9867

 

 

N on -normality

 

Repetitions=1200,6= 171—0021.:006; 615 . 21.2247,61.25 21.11870

Continue (1.)

6 = (p, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)
90 Z (p0,0.15,0.2,0.4,0.35, V1.5, V1.25)

 

%

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P 6’05 0.1 0.15 0.2 0.25 0.3 0.35 0.45 0.5 p
0.01 0.9617 0.5508 0.0900 0.0100 0.1342 0.5442 0.9175 0.9967
0.05 0.9892 0.7808 0.2533 0.0417 0.2993 0.7742 0.9733 1.0000 0.25
0.10 0.9942 0.8658 0.3958 0.0825 0.3925 0.8542 1.0000 1.0000
P\pO—C; 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 p
0.01 0.9992 0.9500 0.5458 0.0817 0.0117 0.1308 0.5417 0.9167
0.05 1.0000 0.9867 0.7725 0.2450 0.0517 0.2975 0.7642 0.9725 0.3
0.10 1.0000 0.9942 0.8625 0.3842 0.0950 0.3992 0.8508 0.9817
P\ ’00—»: 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.9475 0.5442 0.0758 0.0125 0.1300 0.5325 0.9117 0.9875
0.05 0.9883 0.7658 0.2408 0.0508 0.2917 0.7608 0.9717 0.9992 0.35
0.10 0.9942 0.8600 0.3858 0.0925 0.3983 0.8508 0.9783 0.9992
P\ ”94: 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 p
0.01 0.9983 0.9483 0.5358 0.0775 0.0100 0.1292 0.5308 0.9050
0.05 1.0000 0.9875 0.7542 0.2383 0.0508 0.3000 0.7592 0.9683 0.4
0.10 1.0000 0.9967 0.8617 0.3817 0.1017 0.4008 0.8500 0.9792
P\ ”94: 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 p
0.01 0.5375 0.0767 0.0108 0.1283 0.5283 0.9075 0.9883 0.9992
0.05 0.7608 0.2358 0.0508 0.2933 0.7583 0.9692 0.9975 1.0000 0.45
0.10 0.8592 0.3833 0.0983 0.3975 0.8533 0.9783 0.9992 1.0000

 

 

 

 

 

Repetitions=1200, 6 = 131—05 221.200 6; 61.5 :4 1.2247,61.25 9: 1.11870

N on-normality

3:1

86

Continue (b)

 

6 = (p, 0.15, 0.2, 04,035, 61.5, 61.25)
60 2 (p0, 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)

 

 

P\p9_.=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 p
0.01 0.9525 0.5425 0.0758 0.0050 0.1342 0.5425 0.9067 0.9875
0.05 0.9908 0.7542 0.2375 0.0533 0.3150 0.7642 0.9683 0.9908 0.5
0.10 0.9983 0.8633 0.3775 0.0942 0.4083 0.8583 0.9783 0.9983
196”? 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 p
0.01 0.9583 0.5642 0.0775 0.0100 0.1425 0.5608 0.9183 0.9892
0.05 0.9892 0.7825 0.2483 0.0550 0.3208 0.7658 0.9675 0.9967 0.55
0.10 0.9958 0.8775 0.3833 0.1058 0.4050 0.8550 0.9817 0.9975
P\p‘l»: 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 p
0.01 0.9683 0.5883 0.0842 0.0125 0.1517 0.5917 0.9250 0.9900
0.05 0.9917 0.8058 0.2583 0.0550 0.3300 0.7833 0.9692 0.9975 0.6
0.10 0.9942 0.8917 0.3842 0.1050 0.4192 0.8600 0.9825 0.9975
P\""’—): 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.9758 0.6308 0.1000 0.0125 0.1633 0.6250 0.9292 0.9925
0.05 0.9933 0.8317 0.2800 0.0567 0.3392 0.7950 0.9750 0.9975 0.65
0.10 0.9950 0.9075 0.3900 0.1050 0.4392 0.8742 0.9875 0.9975
P\ 90—1: 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 p
0.01 0.9833 0.6725 0.1183 0.0142 0.1817 0.6658 0.9425 0.9967
0.05 0.9950 0.8667 0.3008 0.0567 0.3525 0.8267 0.9808 0.9975 0.7
0.10 0.9967 0.9250 0.4150 0.1033 0.4692 0.8900 0.9917 0.9975

 

 

 

 

 

 

Repetitions:1200, 6 = 7100’ 21.200 6;, 61.5 c: 1.2247,61.25 '2 1.11870

Non-normality

1:1

87

Continue (0)

 

6 = (p, 0.15, 0.2. 0.4, 0.35, 61.5, 61.25)
60 = (p0. 0.15, 0.2, 0.4, 0.35, 61.5, 61.25)

 

 

P\L":

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 p
0.01 0.9900 0.7333 0.1367 0.0133 0.2042 0.7150 0.9975 1.0000

0.05 0.9958 0.9025 0.3325 0.0550 0.3817 0.8550 0.9958 1.0000 0.75
0.10 0.9983 0.9450 0.4483 0.1017 0.4975 0.9025 0.9983 1.0000
P\pof 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 0.9942 0.8033 0.1617 0.0142 0.2258 0.7567 0.9800 0.9975

0.05 0.9975 0.9300 0.3758 0.0575 0.4292 0.8817 0.9933 0.9983 0.8
0.10 0.9983 0.9633 0.5025 0.1000 0.5350 0.9400 0.9967 0.9992
P \p‘f 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p
0.01 1.0000 0.9975 0.8700 0.1883 0.0158 0.2725 0.8192 0.9875

0.05 1.0000 0.9983 0.9583 0.4175 0.0550 0.4675 0.9275 0.9967 0.85
0.10 1.0000 1.0000 0.9783 0.5500 0.1017 0.5808 0.9617 0.9983
P\p9—+: 0.65 0.70 0.75 0.80 0.85 0.9 0.95 1 p
0.01 1.0000 1.0000 0.9983 0.9925 0.2483 0.0200 0.3183 0.8725

0.05 1.0000 1.0000 1.0000 0.9792 0.4800 0.0583 0.5242 0.9575 0.9
0.10 1.0000 1.0000 1.0000 0.9900 0.6133 0.0992 0.6458 0.9775
P\pof 0.65 0.70 0.75 0.8 0.85 0.9 0.95 1 p
0.01 1.0000 1.0000 0.9625 0.3125 0.0200 0.3817 0.9342 0.9983

0.05 1.0000 1.0000 0.9925 0.5667 0.0542 0.5942 0.9800 1.0000 0.95
0.10 1.0000 1.0000 0.9958 0.6967 0.0958 0.7033 0.9867 1.0000

Non-normality
Repetitions=1200, 6 = 1.3% $200 6]., 6R 2 1.2247,6fi 2 1.1180

i=1 1

(d)

88

 

Table 2.10: CMLE for the dynamic panel data of log-wage with unobserved heterogeneity
period:1980 ~ 1987

 

 

A

coefficient ,6 13 do 021 612 65 (3a

 

CMLE 0.3380 0.0474 0.8721 0.1745 0.0488 0.3506 0.2148

t-statistics (18.330) (2.174) (25.251) (8.224) (1.253) (77.473) (18.897)

 

 

 

 

 

 

 

89

Table 2.11: Performance of DIF, GMM, GLS and CMLE (a)

 

 

 

 

 

 

 

 

 

(T:4,N:100)
GMM2 GMM2 GMM2
(DIF) (SYS) (ALL)
9 Mean RMSE Mean RMSE Mean RMSE
SD SD SD
00 —0.0044 0.1227 0.0.100 0.0994 0.0060 0.0970
' 0.1227 0.0990 0.0969
03 0.2865 0.1853 0.3132 0.1221 0.3100 0.1216
' 0.1849 0.1215 0.1213
05 0.4641 0.2693 0.5100 0.1333 0.5100 0.1356
' 0.2674 0.1330 0.1353
08 0.4844 0.8805 0.8101 0.1620 0.8169 0.1541
' 0.8824 0.1618 0.1533
09 0.2264 1.0659 0.9405 0.1615 0.9422 0.1415
' 0.8264 0.1564 0.1351
CGLS CMLE
D
[W can 17111 SE 1616077. Bil/I SE
SD SD
0.0157 0.0986 0.0054 0.0916
0.0
0.0974 0.0957
0.3188 0.1228 0.3054 0.1034
0.3
0.1215 0.1067
0.5182 0.1353 0.5068 0.1082
0.5
0.1342 0.1036
0.8365 0.1396 0.8004 0.0696
0.8
0.1349 0.0684
0.9572 0.1121 0.8988 0.0355
0.9
0.0964 0.0351

 

 

 

 

 

 

 

 

 

 

(a)

90

 

(1:4, N=200)

 

 

 

 

 

 

 

 

 

 

 

 

 

0111111712 GM M2 GM M2
(DIF) (SYS) (ALL)
p Mean RAISE Alean RAISE Mean RMSE
SD SD SD
—0.0037 0.0854 0.0051 0.0670 0.0028 0.0651
00 0.0854 0.0669 0.0651
0.2919 0.1272 0.3092 0.0838 0.3061 0.0812
03 0.1270 0.0833 0.0810
0.4828 0.1828 0.5098 0.0941 0.5079 0.0925
05 0.1821 0.0936 0.0922
0.6362 0.5468 0.8050 0.1196 0.8112 0.1143
08 0.5219 0.1195 0.1138
0.3731 1.1000 0.9235 0.1499 0.9308 0.1243
09 0.9661 0.1481 0.1205

CGLS CMLE

p

.Mean RMSE .Mean RMSE

SD SD

0.0083 0.0700 0.0007 0.0638

00 0.0696 0.0654

‘ 0.3120 0.0895 0.3001 0.0708

03 0.0887 0.0718

0.5135 0.1015 0.5068 0.1083

05 0.1006 0.1036

0.8259 0.1115 0.8004 0.0696

08 0.1085 0.0684

0.9431 0.1022 0.8999 0.0251

09 0.0927 0.0219

 

 

 

 

 

 

(b)

91

 

(T=4, N=500)

 

 

 

 

 

 

 

 

 

GM M2 0.17.172 GM M2
(DI F) (SYS) (ALL)
p AIean RAISE AIean RAISE AIean RMSE
SD SD SD
—0.0033 0.0557 0.0012 0.0434 0.0001 0.0421
0.0 0.0556 0.0434 0.0442
0.2936 0.0827 0.3025 0.0552 0.3008 0.0530
03 0.0824 0.0552 0.0530
0.4887 0.1177 0.5021 0.0632 0.5006 0.0612
05 0.1172 0.0632 0.0612
0.7386 0.3144 0.7939 0.0781 0.7942 0.0770
08 0.3085 0.0779 0.0769
0.5978 0.7081 0.9043 0.1000 0.9038 0.0884
09 0.6401 0.0099 0.0883

CGLS CMLE

p

Mean RAISE Mean RAISE

SD SD

0.0025 0.0462 0.0013 0.0406

00 0.0461 0.0406

0.3030 0.0607 0.3022 0.0592

03 0.0606 0.0588

0.5025 0.0710 0.5008 0.0441

0'5 0.0710 0.0438

0.8007 0.0853 0.7999 0.0303

08 0.0853 0.0309

0.9172 0.0880 0.8997 0.0158

09 0.0863 0.0162

 

 

 

 

 

 

 

 

 

 

(C)

92

 

Table 2.12: Performance of DIP, GMM, GLS and CMLE (b)

 

 

 

 

 

 

 

 

 

 

 

 

 

(1:11, N:100)
GMM2 GMM2 GM M2
(DIF) (SYS) (ALL)
’0 AIean RMSE AIean RMSE Mean RMSE
SD SD SD
0 0 —-0.0138 0.0483 —0.0183 0.0468 —0.0153 0.0467
' 0.0463 0.0431 0.0441
0 3 0.2762 0.0591 0.2728 0.0558 0.2795 0.0545
' 0.0541 0.0487 0.0506
0 5 0.4629 0.0725 0.4689 0.0618 0.4794 0.0592
' 0.0623 0.0535 0.0555
0 8 0.6812 0.1576 0.7925 0.0655 0.8043 0.0624
' 0.1036 0.0651 0.0623
0 9 0.6455 0.2996 0.9259 0.0522 0.9302 0.0523
' 0.1581 0.0453 0.0428
CGLS CMLE
p
AIean RAISE AIea-n RAISE
SD SD
—0.0071 0.0364 0.0011 0.0350
0.0
0.0358 0.0371
0.2832 0.0424 0.3005 0.0350
0.3
0.0389 0.0365
0.4761 0.0490 0.5001 0.0328
0.5
00428 0.0336
0.8025 0.0595 0.8002 0.0180
0.8
0.0595 0.0179
0.9422 0.0623 0.8999 0.0076
0.9
0.0459 0.0074

 

 

 

 

 

 

(a)

93

 

("1:11, N=200)

 

 

 

 

 

 

 

 

 

 

 

 

 

GMM2 GMM2 GMM2
(DIF) (SYS) (ALL)
p Mean RMSE AIean RMSE AIean RMSE
SD SD SD
—0.0070 0.0358 —0.0059 0.0310 —0.0057 0.0313
00 0.0352 0.0304 0.0307
0.2883 0.0427 0.2914 0.0345 0.2925 0.0348
03 0.0411 0.0335 0.0340
0.4815 0.0503 0.4899 0.0373 0.4922 0.0373
0'5 0.0468 0.0359 0.0365
0.7373 0.0971 0.8025 0.0421 0.8075 0.0430
08 0.0742 0.0420 0.0423
0.7256 0.2152 0.9231 0.0435 0.9263 0.0445
09 0.1261 0.0369 0.0359

CGLS CMLE

p

Mean RAISE Mean RMSE

SD SD

~00037 0.0272 0.0006 0.0248

00 0.0270 0.0262

0.2907 0.0318 0.3002 0.0248

03 0.0304 0.0258

0.4858 0.0369 0.5000 0.0232

0'5 0.0340 0.0238

0.8039 0.0449 0.8003 0.0127

08 0.0448 0.0127

0.9345 0.0506 0.8999 0.0053

0'9 0.0370 0.0053

 

 

 

 

 

 

(b)

94

 

(1‘21 1, N=500)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

GMM2 GMM2 GMM2
(DIF) (SYS) (ALL)
AIean RAISE AIean RAISE AIean RAISE
SD SD SD
—0.0025 0.0201 —0.0010 0.0172 —0.0012 0.0173
00 0.0200 0.0172 0.0173
0.2959 0.0237 0.2986 0.0182 0.2984 0.0183
03 0.0233 0.0181 0.0182
0.4934 0.0276 0.4984 0.0189 0.4983 0.0190
05 0.0268 0.0189 0.0190
0.7695 0.0536 0.8019 0.0244 0.8027 0.0249
0’8 0.0441 0.0243 0.0248
0.8110 0.1168 0.9120 0.0306 0.9135 0.0312
0'9 0.0757 0.0280 0.0282
CGLS CMLE
p
AIeaII RAISE AI ean RAISE
SD SD
—0.0016 0.0169 0.0003 0.0406
00 0.0168 0.0410
. 0.2960 0.0196 0.3003 00442
M 0.0192 0.0442
0.4937 0.0228 0.5000 0.0147
05 0.0220 0.0143
0.8008 0.0316 0.8003 0.0080
08 0.0316 0.0079
0.9206 0.0361 0.9002 0.0034
0'9 0.0296 0.0033

 

 

 

 

 

 

(C)

Table 2.13: Performance of DIP, GMM, GLS and CMLE (c)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[ Estimator 1] Mean Std. Dev. Mean ASE RMSE]
OLS 0.8740 0.0203 0.3746
Within -0.0343 0.0565 0.5373
GLS 0.6659 0.0965 0.1919
GMM(D1F) 0.4867 0.1844 0.1775 0.1848
GMM(SYS) 0.4999 0.1082 0.1068 0.1081
GMM(All) 0.5067 0.1109 0.1078 0.1111
CGLS 0.5124 0.1030 0.1037
CMLE 0.5179 0.1227 0.0769
N=200, T=4
Table 2.14: Performance of CMLE (d)
FN,T [1 p ,3 Std. Dev. MAE RMSE ]
100,4 0.0 0.0119 0.1100 0.0837 0.0931
100,4 0.3 0.3217 0.1430 0.1052 0.1102
100,4 0.5 0.5322 0.1700 0.1228 0.1148
100,4 0.8 0.8116 0.1067 0.0821 0.0709
100,4 0.9 0.9027 0.0465 0.0366 0.0354
200,4 0.0 0.0065 0.0711 0.0837 0.0644
200,4 0.3 0.3155 0.0910 0.0697 0.0736
200,4 0.5 0.5179 0.1227 0.0816 0.0769
200,4 0.8 0.8078 0.0741 0.0573 0.0491
200,4 0.9 0.9021 0.0330 0.0259 0.0251
500,4 0.0 0.0027 0.0450 0.0359 0.0403
500,4 0.3 0.3043 0.0558 0.0445 0.0447
500,4 0.5 0.5057 0.0637 0.0501 0.0458
500,4 0.8 0.0823 0.0470 0.0983 0.0305
500,4 0.9 0.9003 0.0215 0.0173 0.0158

 

 

 

 

 

96

 

CHAPTER 3

Models Where State Dependence
Depends On Unobserved

Heterogeneity

3. 1 Introduction

In the previous chapters we have discussed existing methods and the CMLE
suggested by Wooldridge (2000b) for the AR(1) model with unobserved heterogeneity.
The model is restrictive in that it assumes the amount of state dependence does not
depend on unobserved heterogeneity. A more general model is

ya = pyi,t—1+ai+7(0191,1—1)+€1t
13:1,...,N,t=1,...,T,

(3.1)

which means that the amount of state dependence depends on the heterogeneity. The
autoregressive coefficient for each 1' is given by p + 7a,, so that it is a function of the

individual heterogeneity. In (3.1) we clearly cannot estimate p + 70,: for each 7' with

97

a short time period. But we can hopefully estimate the average effect, 19 E p + 7,140,
where 11,, = E(a,-).

One interesting question surrounding model (3.1) is: Do standard IV methods
applied to the first difference equation consistently estimate interesting parameters?
To see that the answer is no, we consider the IV estimator that uses Ay,,t_2 as a IV

for Ay,,,_1. Differencing (3.1) gives
A3141 2 P Alla—1 + "/ (MAE/1,14) + A521 (3.2)

The IV estimator of p applied to the first differenced equation is, with fixed T,

asymptotically equivalent to:

plim 61v

N—coo
: plim NIT :1 :1 AyitAyi,t_2

N7” N17 Zr 21 A3/1‘,11—1A3/z',t—2
XII—77’ 22? EXP AMA—1 + ’7 (aiAy.,1_1) 'l' AfitlAyi¢—2

 

 

 

= plim
N7°° fi :27 21491344392314
7 1131131 537 Z,Z,a1Ayi,t_1A1/a-2

113,13”: Ni? 22' 2t Ayi,t—1Ayi,t—2
19,132 767 Z,- 21 AgitAyiJ—Z
REE} 6+7" 22' :1 491,1—1Ayi,t—2

7 glim F17" Z,Z.hiAy.,1_1A1/.~1_2

= 6+7xta+ ‘77 +
1,2122 NIT :1 2.211331134439814

B11111 fi Z,- 21 AEaAym—z

plim 1er 24 Zr 4.714.1-1Ayz'3—2

where h,- E a, — ,ua. To simplify the exposition, we make the following standard

 

 

 

assumptions:
E(e,-t|y,-,t_1, . . . ,y,0,a,) = 0 for all 7', t=1,. . .,T

Var(e,:t|y,-,t_1, . . . ,y.,-0,a,-) = 03 for all 2', t=1,. . .,T

98

Obviously, plim 3‘7 ZiztAe,tAy,,t_2 is zero under the above assumptions, but
N-Ooo

7 Rum A—lf 272th1A3/i,1—1A.Uz,t—2 is equal to:

’7 REE} 7717 Z,- :1 hi Ayi,t-—1Ayi,t—2
= 7( 113111;} 71; 2,771+ 113132 % Z,-hz‘C0V'1(A.I/i,t—1,AMI—2)) (35)
:- 77 Z, EU'lvz‘Aym—h Aux—2),
and this does not equal zero without some unusual assumptions. Therefore, Ay,,t_2
is not a valid IV for Ayah] because Ay,,t_2 and the error, 'yaiAyit_1 + A5“, are
correlated and we are doing fixed T asymptotics.
we can also ask whether the IV estimator consistently estimates the average

autoregressive coefficient, 6 = p + 7110. Unfortunately, the answer is no.

 

. N T
1' 19 1:11:10] 1671‘ 24:1 21:1 Ayi.tAyi,t—2
1m ' z z
1,3122 W 21:1 21:1 Ayn—14491.14
. N T
13112.1 N17 2421 21:10!) + 706491.21 + AEIt)Ayi.t—2
. N T
plim [VI—I Z,=1 27:1Ayi,1-1A;Ui,1—2
. N ‘ .
Pllm 767 27:1 277211119 + 7’1'2')Ayi,t—1 + AEItlAyi,t—2

N—ooo
. 1 N T
Rhm W 21:1 thl Ayi,t-1Ayi,t—2

 

 

. N T
4 1,3133; 5% >32. z.=.11.Ay.-,._.Ay.,._2
: 6 + N T +
113111: N]? 2,21 21:1 Ayi,t-1Ayi,t-2
. N T
18,1333 fi 2:121 2:121 AsitAyi.t—2

. 1 N T .
131mm 717 2.21 231:1 Alla—1431444
l'fim

 

 

where h,- = a,- — Ma. The consistency of 6“» depends on the second term of the last
expression of (3.6), equal to (3.5), which does not vanish even though we strengthen

the assumption (3.4) such that h,- and 6,, are independent for all 7', t = 1,...,T.

99

The denominator of (3.6) is easily proved to be nonzero. The probability limit of the
numerator of the third term on the right hand of (3.6) is zero, but the probability limit
of the numerator of the second term is not always zero and thus the asymptotic bias
of the IV estimator is given by the probability limit of the second term on the right
hand of (3.6). Because the presence of the state dependence which depends on the
unobserved effects, the unobserved effects are transmitted into the estimate of p. This
causes that p can not be identified from p + 7111. In addition, 61V cannot consistently
estimate 19 even if we further assume that c,- is independent of yw. Therefore, in the
model with an interaction effect, the IV estimator does not consistently estimate p or
the average autoregressive coefficient. It does not appear that differencing alone is a
reasonable strategy for estimating the model with the interaction term.
'Ifansformations other than differencing may work to estimate the parameters
of (3.1), but they do not immediately suggest themselves. Instead, we can apply
conditional MLE, as in the simpler model from Chapter 2. As in Chapter 2, we
directly model the distribution of a,- given yio and any strictly exogenous variables.

The most general model we consider is

yi = 10 311,14 + 3311.3 + (12' + ”Y 0191,1—1 + 511
t l l (3.7)

21: 1,...,N,t= 1,...,T.
We could consider a model where a,- also interacts with :15“, but the computational
requirements would be severe. For many purposes, the most interest is in a model
where unobserved heterogeneity interacts with the lagged dependent variable. The
plan of this chapter is as follows. Section 2 considers model (3.1) and then considers
model (3.7). Section 3 presents the simulation of CMLE for the model (3.1) and (3.7).
Section 4 applies the models to log hourly wage for the panel of working man used

in Chapter 2 in considering the interaction term, product of log hourly wage and the

100

unobserved heterogeneity. Finally, Section 5 contains some concluding remarks.

3.2 AR(1) Models With Unobserved Heterogene-
ity, State Dependence

3.2.1 AR(1) Model Without Exogenous Variables

This section clmracterizes the CMLE for model (3.1). When 13 is omitted, we
refer to a general cross-sectional observation. Using the general treatment in Chapter

2, the conditional densities corresponding to equation (3.1) is f (ytlyt_1,a; 60). We

assume D(€,~t|y,-,t_1,. . . , 31,0, (1,) = D(€,t) and thus the joint density of (yT,. . .,yl) given
(go, a) is
'r
per, . . . ,y11y0.a; 50) = H form... a; 60). (3.8)
t=1

We can not estimate 60 by directly using (3.8) because it. depends on a which is
unobserved. According to the discussion of Chapter 2, we can model D(a|y0) and
then construct the density (yrr,...,y1) given yo by integrating out a from the joint

density function. In practice, we can specify a parametric density :

h(alyo; A0), (3-9)

where A0 is a vector of nuisance parameters. Wooldridge (2000b) suggested that
we assume the a,- are from a conditional normal distribution, where the mean and
variance given yo are flexible functions of yo. Let us make assumptions on the 5,7 and

a,- as follows:

 

- , , . . 2
Assumptlon 3.1 Salt/7,74, . . . , $110, a, N Normal( 0, 0E ).

 

 

Assumption 3.2 ailyz’o N Normal(11.a(yi0),a§).

 

 

 

With a linear mean in Assumption 3.2 we can work a,- : (10+ (11 yio + Ci, where

101

a,- given ym is Normal(0, 0 3.) To characterize the conditional mean of 3],, given ym,

we use equation (3.1) to obtain

1——(p+'ya,~)’) t '—1
1+ + 1'] 82' _' . 3.10
1__ (p+'7' a.) a 2(p 7 a) ,1 9+1 ( )

1:1
Let us define a polynomial of t-order as 11(2) : ww + wmz + . . . + wuqz‘, where

 

yit 2 (,0+ 7 ailt’yz’o + (

130(2): _ 100.0, and 13.1 E 0. Then equation (3. 10)( can be rewritten as followzs
yit : [dt + Pt_1(C.lj)C-zil ' 3110 + (23:1(dj_1 + Pj_2(Ci)Cz‘)) (#0 + C7)

+ 23:1(dj_1 + Pj—2(Ci)("i) €i,t—j+la

where d = p + 7 pa. and U.,’t’t_] :2 7“]. From equation (3.11), the coefficient wm_1 is

(3.11)

a function of p, 00, a1, and 31,0 over t from 1 to T: 101,0 = d‘ and wt,t_1 = 7‘. Under
Assumption 3.1, Eu is independent. of 31,0 for all t. We obtain p(y,-0) E E(y,-t|y,0) as
follows:
I‘ll/2'0) = dtyz‘o + (fid‘) [ta + 3110 Z]; 1 E(Pj—1(Ci)cilyi0)+
Ma ZE=1E(Pj-2(Ci)cilyi0) + Zj=1E(Pj—2(Ci)63lyio)-

Because we assume that c,|y,~0 is Normal(0,0§), the t-th moment of c,- exists

(3.12)

and is a function of 00. Therefore, the last three terms of equation (3.12),

Zj=1E(Pj—1(Cilcil3/i0)~ Zj_1EU)j—2(Ci)C1lyiol and Z]: 1E(P j_ _2(c,-) cflyw) can be

concisely expressed as a function of p, 00, 011, yio, and 0a. Therefore, p(y,-0) = (fig/1.0+
(11— d?) ,ua + A(ao, 0] H10, (TO). The conditional variance matrix (Kym) E V(y,|yi0)

can be expressed as follows:

V(y.-ly.o) = E [((yi ‘l‘fy10)[1‘)(yi -xt(yz~o)€T)'|3/iol (3.13)

= Etyz-yilyz-o) - ”(y10)2€TgI"

Under Assumption 3.1 and 3.2, we can solve out the elements of the conditional

variance matrix

0(1/10): 3 3 , (3.14)

le . . . LOTT

102

where cast is a function of p, 00, (11, 31,0, Ga and 05. In principle, we can solve out
the formula for €(y,0) ( E y,- — p(y,0) 6p) and rust in terms of p, 0’0, (11, 31,0, 0,, and
05 based on (3.10) and (3.12). we set up the log-likelihood function away from the

constant for cross-section observation 2' is as follows:

z<y.; 6) = —-;-Iog<19<y.o>l) — % (e<y.o>n<y.o>-1e<y.o>') (3.15)

Therefore, we can obtain the CMLE estimators by solving out the following maxi-

mizing problem:
N N 1 1
11,311: la; 6) = rggxz [—§10g<le<y.o>l> — ,- (ety.o)0<y.o)-‘e<y.o>') (3.16)
i=1 i=1
In fact, the complexity of the Q(y,:0) will increase with the value of T, so it become
intractable for handling for equation (3.16). The general approach is that f (Yleo, a)

can be expressed as the product of f (ytlyt_1,a) over t from 1 to T by parameterizing

(3.8). And then we specify a conditional distribution of the unobserved effects h(alyo),

 

in particular the normal density function, \/217T—02 exp(—_Ql(a—;c—ly—a)2). We can set up

the log density function as
1(YT;9) =log/ f(Y7~|yo,a)h(a|yo)da

(3N)

= log [:0 [11.11 f(ytlyt—hall h(alyo)da

By equation (3.1) and Assumption 3.1, we obtain the following equation:

1 —1 1 - 2's— +ai + 02'. i. — 2
flyitlyz',t-1,a) = eXp — [311 (p y ’t 1 ’7 ( 31,1 0)] (3.18)
\/27ror,32 2 Us

Putting (3.18) into (3.17) gives the log-likelihood function for each i is as
[(31219) =
°° 1 T T . —-1 yn—(p 1111—1 +a+7(ayu—1)) 2
.7 («ml 1 <7— 1 .5 ’ 1 )1 -
< 1 >exp(:}<—i—la ‘ ’3: “'0 mm.

(3.19)

 

 

 

 

 

[\D
:1

Q
EN)

103

We can obtain the CMLE estimators by maximizing the sum of function (3.19) across

ifrom 1 to N.

3.2.2 AR(1) Model With Strictly Exogenous Variable

In this section we consider (3.8). Assuming strict exogeneity of :rt, we have
D(yt|Yt_1,XT,a) = D(y¢|yt_1,.rt,a),t = 1, . . . ,T. (3.20)

Equation (3.20) means that once current 112,, and yt_1 and a are controlled for, 512,, s 76 t,
do not affect the distribution of yt. The conditional distribution can be parameterized

as a conditional density,
f(ytlyt-ls$taa;60)a (3.21)

where the parameter 60 is finite dimensional parameters. In our application we assume
that f(ytlYt..1,:1:t,a;60) depends only on one lag of yt and the current cut. By the
usual product of law for conditional densities, the joint density of (yrp, . . . ,yo) given

(:cT, . . . , 3:1,a) is as follows:

f(yTa' ° ° ayllei ' ' wxlvyOa G160) : Hf:1f(ytlyt—ls$taa;60)a t Z 11- - - ,T-
(3.22)

As discussed in previous chapter, because the density of (HT,...,y1) given (:rT,

.., 21:1, yo, a) depends on a, which is unobserved, to consistently estimate 60, we
integrate a out of the density. The recommended solution ( Wooldridge [2000b]) is
that to model a conditional distribution D(a|XT, yo), and then construct the density
of (yT,...,y1) given (.137, . . . ,z1,y0). It is crucial that this allows yo to be random
and need not find, or even approximate, D(y0|XT,a). Further, we do not have to

specify an additional model for D(a|XT) or, assume that a and XT are independent

104

and then model D(a). In practice, we parameterize the conditional density : let.

h(alXT, yo; A0), (3.23)

be the density corresponding to D(a|XT, yo) , where /\0 is a vector of parameters. It
is convenient to assume normality with conditional mean and variance in terms of

(XT, yo). We make the following assumptions:

 

[Assumption 3.3] eith,t_1,X,-y7w,a, ~ Normal(0,a€2).

 

 

, . _ _ T
[ Assumption 3.4] a,|;r,~. ym ~ I\-ormal(pa(y,0. x,),0§), where 2:,- = %thl .73”.

 

According to Assumption 3.4, we can write the equation for a,- as follows
a, 200+ olyio+ Eag+ci,i=1,...,N, (3.24)

where 0,- given (:r,,y,-0) is Normal(0 , 03). Assumption 3.4 and (3.24) imply that

l- _ 2
exp [i (at — (00 + 01 910 + :13, a2)) ] , (325)

 

 

h(alXT, yo; /\0) =

1
V 271'0'2 2 0a

where A0 = ((10, (11.02, 07,). Once we have specified h(alzT, yo, 51:0; A0). we obtain the

log-likelihood function for each cross section 17 as follow:

[(1%) (File) = 10g] 113;] f(yit|$,t,y,,t_1,a ;60).
we — 2 3.26
1 exp [:21 (a _ (0'0 '1' 0'] 3110 'l‘ 5131' 02)) ] da’ ( )

V27r0‘f, I 0“
where 00 is a vector of all parameters of the model.

Under Assumption 3.3 and 3.4, we still can apply the procedures of the previous

section to solve the CMLE. (3.11) and (3.12) can be re—written respectively as follows:

yit = [dt + Pt—1(Cz')C-il ‘3/20 + (ijlldj_l + Pj-2(Ci)ci)) (#a + Ci)

+ Zj=1ldj—l + P1407001) ,3 fat—3+1 + Zj=1ldj_1+ P1403061) Eat—3+1,
(3.27)

105

and

_ t
E(yithit9 3120) = dtyz‘O + (——11—((1i) pa + yjo 23:1E(Pj-1(Ci)cileitayi0)+
23:1(dj—1 + E(PJ'—2(Ci)6ilxita 29/20)) 3 mat—341+

Ha 23:1E(Pj—2(Ci)C-ilXitayi0) + 23:1E(Pj_2(Ci)C22|Xit,y-io).
(3.28)

Because we assume that (c,|.r,~,y,0) is Normal(0,0§), the t-th moment
of 0,- exists and is a function of 00, and hence the last three terms
of equation (3.28): 23:1E(Pj_1(c,-)c,-IXz~t,y,-0), 23:1E(Pj_2(Ci)Ci|Xit,yiO), and
23:1E(Pj-2(c,-)c,-2|X,-t,y,0) can be compressed as a function of p, (10, a], 33,-”, gig,
and Ca. Therefore, the E(;y,,|X,t,y,-0) : dtyiO + (115—(ff) ”a + 23:1(dj—1 +
E(Pj_2(c,)c,~|X,-t,y,—0))g3 shay-+1 + A(a0, 01, 113—1’, 31,0, 00). If T is very small, say T
g 3, we can use the log-likelihood function (3.15) and replace s(y,0) and 90/20) with
€(yz-0,:c,) and 93,10,131. and then to maximize the sum of the log-likelihood function

across 2' from 1 to N. If T is not very small, we use the following as log-likelihood

function:

[(yia 1171'; 6) :

log /00( 1 )T [1'1le exp (:21 [Mt _ (,0 yi,t—1 + $210? + a + ’7 (ayi,t—1))]2)] _

( 1 )exp<:}(“‘“gff{:0’m)2)da,

t/2n03
(3.29)

 

 

 

 

Where ”aft/10,31): 00 + 01 MD + 502

By maximizing the sum of l(y,-,:z:,-;6’) over 2' from 1 to N, we can obtain the
CMLE estimator. According to the discussion of the consistency in Section 2.2.2,
the consistency of CMLE estimators that use (3.28) as the log—likelihood function
can be ensured. If we have random sampling in the cross section dimension and

standard regularity conditions, with fixed T, the CMLE for 00 will be consistent and

106

[IV-asymptotically normally distributed. ( See Newey and McFadden [ 1994] for

sufficient regularity conditions.)

3.3 Simulation Evidence

3.3.1 Model Without Exogenous Variable

The true model without exogenous variables of the simulation is as follows:

ya 2 p Elm—1 + ai + “r [02' yi,t—1l + 5m
i: 1,...,250, t=1,...,5,

(3.30)

where a, = (10 +01 yio +c,-. The true value of 6 : (p, (1'0, (11, 7, 05, aa)-—-(p, 0.2, 0.4,
'7, m, \/l._2), and we set p as different values equal to 0, 0.25, 0.5, 0.75, 0.9, 0.95
and 7 as values equal to 0, 0.1. The values of c,- and En are generated by N(O,1.2) and
N(0,2.4), respectively. Because this method allows yio to be random, we generate the
yio from the N(0,1). Using the generated data, we maximize the sum of log likelihood
function (3.19) over i from 1 to N

By the use of the data coming from the above rule, we apply the MLE procedure
of Gauss software to obtain the conditional maximum likelihood estimators. it is
difficult and time-consuming to directly maximize the objective function , the sum
of (3.19) over 2' from 1 to N, so we need find another easier numerical form for the
log-likelihood function (3.19). We might calculate the integral of equation (3.19)
by applying the formula for the evaluation of the necessary integral which is the

00

2
Hermite integral formula / e_Z g(z)dz = 2:11 1159(2)), Where K is the number
of evaluation points, wj is the weight given to the jth evaluation point, and g(zj) is
9(2) evaluated at the jth point of z (See Butler and Moffitt [ 1982]). Equation (3.19)

can be re-written as follows:

107

((31:99) 2

 

00 T
:22: log 27f + log /—00 (VI?) [11:19“) (é—O—lefyn — (P ’3/2‘,t—1

(3.31)
+\/§Ua3 + #afilz‘ol + ’7 (flaw? + /-1a(.Uz'0))._l/z,t—1ll)] 9“"P(_32)d 3-

We can let 9(2) to be

 

T T
1 —1
( ) [I I exp (70 (ya - (p yet—ma»? +l1-a(910)+ 7 (£an + #a(yi0))yi.t1)))]

2
0'5 1:21

and then the log—likelihood function away from the constant is
00 __ 2
log/ g(z)e Z dz (3.32)
—00
This formula is appropriate to our problem because the integration of equation
2
(3.19) can be transformed as the product of g(z) and e‘z equation. We can approx-

00

. . . . N _Z2
lmate the OijCthC functlon 2,21 log / g(z)e d 2 as follows

—00

N K
210g (2: mm») (3.33)

i=1 j=1
In the simulations we obtain the conditional maximum likelihood estimators from
equation (3.33). The feasible computation of the Hermite integral depends on the
number of evaluation points at which the integrand must be evaluated for accurate
approximation. Although the value of K determines the accuracy of the calculation of
integral, we do not discuss the relation of K and the evaluation of integral as Butter
and Moffitt ( 1982) did. Several evaluations of the integral using seven periods of
arbitrary values of data and coefficients on two right-hand-side variables shows that
the value of K is chosen to be 21 is highly accurate. We repeat the maximization of

(3.33) for 300 hundreds. We make the notations in the simulation as follows:

1. 0* means the conditional maximum likelihood estimators.

108

" 300 ...
232%: a

1:1 J'

3. 0 means true value of parameter, where 0 = (p, a0, 01,), aa, 05) =

mmamMmMM)

In each repetition, we proceed with the hypothesis Ho : p 2 p0, where p0 is 0, 0.25,
0.5, 0.75, 0.9, 0.95 when the true value of p is 0, 0.25, 0.5, 0.75, 0.9 for each value of pa
with keeping the true values of the other parameters unchanged, (a0, a1, 7, 0’0, 05)
= (0.2, 0.4, 'y, x/T.2, \/2_4) where 7 is 0 or 0.1. For each hypothesis test for estimates,
we calculate the numbers of occurrence that greater than 300 x 0.01, 0.05 and 0.1 in
respective and the result is divided by 300. In other words, we examine the p-value
of each estimates under the hypothesis test. Table 3.1 reports the results for true
value of p20, ..., 0.9, and 7': 0 and 0.1. The true value of (do, (11, 00, as) is always
set to be (0.2, 0.4, \/1—2, \/2._4). Table 3.2 reports the p-value when testing value p0
is different from the true value of p. To examine the non-normality, Table 3.3 and
Table 3.4 report the results of the same test as the previous while the Eu comes from
t-distribution with freedom 6 and the c,- frorn t-distribution with freedom 10.

In case where 7 is zero, the performance of the model in Chapter 3 is close to
the model in the absence of state dependence in Chapter 2. For example, given
00 = (0,0.2,0.4, J24, \/T_2) in Table 2.1-( i )_, the average of ,5 is 5 x 10‘4 and its
p-value is 0.0108, 0.0692, 0.1141 respectively at the corresponding sizes, 0.01, 0.05,
0.1.

Under the same data set in which the parameters is the same as the model in
Chapter 2, the average of ,5 is 2.5 x 10'3 and the p—values are 0.0133, 0.07, 0.1 re—
spectively at the corresponding sizes, 0.01, 0.05, 0.1 in Table 3.1-( i ). To check the
hypothesis H: p = p0 when the true value of p is zero, Table 3.2—( i ) and Table 2.2-( i

) show that given the power 2 0.01, the two sets of p-value are { 0.0133, 0.1, 0.5200,

109

0.9133, 0.9933, 1.000,...}, { 0.0108, 0.1167, 0.5400, 0.9050, 0.9925,...} in respective
when p0 is {0, 0.05, 0.1, . . .} at the step of 0.05. Although the bias of; is adequately
larger, the results gives the numerical evidence that both of models have close power
of rejecting wrong when p is close to zero.

To check the other extreme case where the value of p is getting close to 1, for
example p = 0.9, Table 3.1-( ii ) and Table 2.1—( iv ) show that E is 0.9004, 0.9018 in
respective and the corresponding p-value 0.01, 0.0433, 0.09 and 0.01, 0.0458, 0.0967
in respective at the power of 0.01, 0.05, 0.1 when p is 0.5. To check the hypothesis H:
p 2 p0 when the true value of p is 0.9, Table 3.2-( ii ) and Table 2.1—( iv ) show that
given the power 2 0.01, the two sets of p-value are { ..., 1.000, 0.9800, 0.5967, 0.0633,
0.0100, 0.1700, 0.5967 } and { ..., 1.000, 0.9850, 0.6558, 0.0908, 0.0100, 0.1658, 0.5975
} in respective when p0 is {..., 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1 } at the step of 0.05.

In case where p is not near either 0 or 1, say p = 0.5, (Table 3.1- ( i ) and Table
2.1- ( iii ) show )6 are 0.5004, 0.5021 and the relevant p-value is 0.0167, 0.0333, 0.09
and 0.0100, 0.0500, 0.1050 at the power of 0.01, 0.05, 0.1. To check the hypothesis
H: p 2 p0 when the true value of p is 0.5, Table 3.2-( i ) and Table 2.2-( iii ) show
that given the power : 0.01, the set of p-value are { ..., 0.04, 0.0167, 0.1000, 0.3733,
0.7300, 0.9300,... } and { ...,0.0575, 0.7442, 0.9317, 0.9833, 0.9975,... } when p0 is
{...,0.45, 0.5, 0.5506, 0.65, 0.7,... } at the step of 0.05. The numerical evidence
shows that the model considering the state dependence have good performance even
in the true model in the absence of state dependence.

Comparing Table 3.3 and Table 3.4 with Table 2.3 and Table 2.4 the non-normality
cases corresponding to the previous ones have good performance. For example, when
p is 0.5, Table 3.3- ( i ) and Table 2.3-( iii ) show p are 0.4982, 0.4999 in respective and

the relevant p-values are 0.0033, 0.0633, 0.1033 and 0.01, 0.0558, 0.1017 in respective

110

at the power of 0.01, 0.05, 0.1. To check the hypothesis H: p =2 p0 when the true value
of p is 0.5, Table 3.4-( i ) and Table 2.3—( iii ) show that at the power of 0.01, the
two sets of p—values are { 0.07, 0.0033, 0.1433, 0.5233, 0.8933, 0.9833,... } and {...,
0.0792, 0.0100, 0.1183, 0.5217, 0.8842, 0.9842,... } in respective when p0 is {...,0.45,
0.5, 0.55,0.6, 0.65, 0.7,... } at the step of 0.05 .

The numerical evidence support that the application of CMLE into model (3.1)
have a good performance even when the true model is in the absence of state depen-

dence.

3.3.2 Model With Strictly Exogenous Variable

In this section, the true model is as follows:

ya = P yi,t—l + 0-15 517a + (12‘ + 7 lat yi,t—ll + 521, (3.34)

i=1,...,250, t=1,...,5,
where a,- = 0.2 + 0.4 3,1,0 + 0.35 T,- + c,. The true value of p = 0, 0.25, 0.5, 0.75,
0.9, 0.95, and the true value of 7 : 0 and 0.1. 31,0 and 23,-, is generated from N(0,1).
We generate 5,, by two methods, one is N (0,2.4) and the other is t-distribution with
freedom 6. c, is also generated from N(0,1.2) and t-distribution with freedom 10. By

subtracting the constant from the (3.29) and rewrite it as follows:

 

 

l(y.-, 27.; 6) = log / 9(a) exp<—(” " ““(y‘o’ 1,) )2) da. (3.35)
—00 flag
where
1 T T 1
9(01) = ( 2) [H eXP (QB—(ya — (,0 yi,t—1 + 13 5181+ (12‘. + 7 (Gillan—HO]
05 t=1 E

and a,- 2 0'0 + 01 31,0 + 02 T,.

Let 2:,- = a,- — :L/‘gyio‘im. Equation (3.35) can be transformed into the form of
0a

 

111

Hermite integral formula as follows:

((4.416) = log/ gen—Z dz. (3.36)

—00

 

T
9(a) = ( 1 ) [TIL 6X1) (53311.. — (p 14-1—1 + L3 5131': + (720.12.: + #a(yiOa-ji))+
7 («20.124 + 14a(yz-o.?5i))))]-

By maximizing 211:1 log{Z]:1 wJ-g(zj)}, we proceed with the same procedure as that
of the previous section. The results are reported from Table 3.5 - 3.8.

Similarly, in each repetition, we proceed with the hypothesis H0 : p 2 p0, where
p0 is 0, 0.25, 0.5, 0.75, 0.9, 0.95 when the true value of p is 0, 0.25, 0.5, 0.75, 0.9
for each value of po with keeping the true values of the other parameters unchanged,
((10, al, 7, 0a, 05) = (0.2, 0.4, ’7', m, «23) where 7 is 0 or 0.1. For each hypothesis
test for estimates, we calculate the numbers of occurrence that greater than 300 x
0.01, 0.05 and 0.1 in respective and the result is divided by 300.

We make the comparisons between Table 3.5 - 3.8 and Table 2.6 - 2.10 to Show
that including the exogenous variables the performance of CMLE for the model with
sate dependence when the true model is in the absence of state dependence. In first
case where '7 is zero, theiperformance of the model in Chapter 3 has. the performance
similar to the model in the absence of state dependence in Chapter 2. For example,
given 90 = (0, 0.15,0.2,0.4, «T4 «13) in Table 2.6-( i ), 73 = 1.7 x 10-3 and its
p—value is 0.0100, 0.0583, 0.1058, respectively at the corresponding power, 0.01, 0.05,
0.1.

Under the same data set in which the parameters is the same as the model in
Chapter 2, p = 2.2 x 10‘3 and the p-value is 0.0067, 0.0400, 0.0867 respectively at

the corresponding power, 0.01, 0.05, 0.1 in Table 3.5. To check the hypothesis H:

112

p = p0 when the true value of p is zero, Table 3.6 and Table 2.7-( i ) show that given
the power 2 0.01, the two sets of p-value are { 0.0067, 0.0933, 0.5033, 0.8833, 0.9933,
1.000,...}, { 0.0100, 0.1083, 0.5283, 0.9000, 0.9892.1.000,...} in respective when p0 is {
0, 0.05, 0.1, 0.15, 0.20, 0.25,...} at the step of 0.05. We can check the cases where p is
0.9 or 0.5 by the same method used by model (3.30) to compare the current models
and the corresponding model in the absence of state dependence in Chapter 2. ( see
Table 3.5 and 3.6 , and Table 2.6- ( iii ), ( iv ) and Table 2.7- ( iii ), ( iv ).) In the
non—normality cases where 7 is zero, we can see the performance in comparison of
Table 3.8 and Table 3.9 with Table 2.8- ( i ), ( iii ), ( iv ) and 2.9- ( i ), ( iii ), ( iv ).

With regard to the inclusion of the exogenous variables or not, the performance
of the CMLE for models allowing for the interaction between the unobserved effect
and the lagged dependence is very well even in the true model which is in the absence

of effect of state dependence.

3.4 Empirical example

In this section, we use the data from Vella and Verbeek (1998) to study the
conditional maximum likelihood estimator in estimating the AR(1) model in which
the unobserved effects interact with the past dependent variable. These data are for
young males taken from the National Longitudinal Survey (Youth Sample) for the
period 1980-87. As in Chapter 2, we estimate a dynamic log wage equation. We
consider the data of log hour wage and the status of labor union. Each of the 545
men in the sample worked in every year from 1980 through 1987. We begin with a
single dependent variable, lnwage,,, to see what is the response of the current wage
rate change into the past one in consideration of individual heterogeneity.

It seems reasonable that the amount of state dependence could depend on unob-

113

served heterogeneity. We allow the interaction between the unobserved heterogeneity
and the lagged wage rate to account for the heterogenous autoregressive root (p+7a,).
An interesting parameter is the average effect, 19 E p + 7,170.

We parameterize the model in two ways. We allow for the interaction between the
unobserved heterogeneity and the lagged log hour wage as well as the unexplained
heterogeneity, a,- is assumed to be E ((1,) lnwagem, 6;,-t) = 070 + allnwagem + c, for all

i and t. The first case is set up as follows:

ln'wage“ : p lnwagemq + a,- + ’7 at lnwagei,t—l + Em

(3.37)
i=1,...,545, t=1,...,7,

where a,- 2 0'0 + a], lnwagew + c,, i = 1 . . . ,545.
In order to obtain a valid standard error for the estimated average effect, rearrange

equation (3.37) as follows

lnwageu : i9 l7‘i'wage,,t_1 + a,- + 7[a,- — pa] lnwageuq + a“,
(3.38)

27:1,...,545, t=1,...,7,
where p : 19 — 7 pa. Models (3.37) and (3.38) are the same model, but formulation
(3.38) is convenient because 19 is the average autoregressive coefficient across pop-

ulation of unobserved heterogeneity. The third case considers adding time dummy

variables into equation (3.37) as follows:

("waged I p 17210096714 + 5t dt + a, + ’7 (17171111098134 + 5a, (3.39)
i=1,...,545, t=1,...,7,
Similar to the manipulation of model (3.37) and (3.38), the amount of state de-

pendence through the average effect of unobserved heterogeneity, 7 pa lnwagetpl is

introduced into equation (3.39) and expressed as follows.

ln’wage“ = 19 lnwagethl + 6, dt + a, + 7[a,~ —- pa] lnwagemn + 8“, (3 40)

i=1,...,545, t=1,...,7,

In table 3.9 to 3.14, do and al is significantly greater than zero and [1,- has positive
value. The estimated amount of unobserved heterogeneity of a worker is do + dlyio
and thus the individual estimated amount of autoregressive coefficient is measured
by p + 7 61,. That is the response of the current log hourly wage rate into the
lagged log hourly wage rate is varied with individual worker, where the estimated
size of difference is measured by 7 61,-. We are more concerned with the average effect,
p + 7E(a,-), and its corresponding estimated value is f) + fi'fia. In equation (3.37),
we calculate the value of p + 7 x p}, 2 0.376. p], is calculated by do + 021 W0,
where W0 2 57113 2:51 lnwagem. The estimated amount of average effect can
be obtained from estimating the 19 in equation (3.38). Table 3.10 shows 19 is 0.3755
significantly greater than zero. The. same manipulation on equations (3.39) and (3.40)
give evidence that the estimated average effect is about 0.215, ( see Table 3.12). Table
3.12 shows 19 is significantly greater than zero. Empirically, we often consider the
model with exogenous variables, for example the labor union membership, once we
control for state dependence and unobserved heterogeneity, does union membership
matter? Under the assumption that the labor union membership is strictly exogenous
we add it to the basic equation. Because we assume that labor union membership is
strictly exogenous variable, once the past log hour wage rate, the current log hourly
wage rate is not affected by past or future labor union membership. The assumption
that reasonable because in general employers might just see if the employees have
the labor union membership at present in determining the level of hour wage. There
could be feedback from wage innovation to future union membership, although this

is possibly small.

115

We write the model with strictly exogenous variable as follows:

lnwageu : p lnwagemq +13 union.“ + a, + 7 a,- l7zxwage,,t_1 + 5,1,

 

(3.41)
17:1,...,545, t=1,...,7,
and we assume
a,- : oo + 011 lnwagem + (12 117211071..- + Ci, 2' = 1 . . . ,545,
and its corresponding average effect, p + 7/10, expression model is
lnwageu = 19171/ui'age,,t_1 + )8 117117012“ + a,- + 7 [(1, — pa] lnwagei,t_1 + 5:1, (3 42)

i=1,...,545, t=1,...,7,

where a,- = oo+o1 lnwage,o+og m, + c, i=1,. . .,545 and pa = ao+al WO+
(12 m, where W, is the fraction of employment membership. That is the
length of keeping the labor union membership more or less reflects the individual
preference of a worker. Empirically, the smaller is the ratio, the lower is willingness
to keep membership. As often as a worker with higher ability less intend to keep
union membership. We report the CMLE estimates of the model (3.41) in Table
3.13. The estimates of 6 is significant and do is not significantly different from
zero. Except for the 7, the other estimates is very close to those of model (3.37).
The estimated amount of individual autoregressive coefficient is p + 7 (2,, where
d,- = ao+aly,~o +ogm,. Table 3.13 shows that. (p, do, 611, do, 7) is (-0.4771, 0.0507,
0.943, 0.082, 0.0244, 0.7757).

The same manipulation as the model without exogenous variable the estimated

amount of average effect is measured by [3 + 7 x 1,2,, 2 0.3476. [1,, is calculated

 

. . —— ——.— —— 515

by 00 + a] lnwageo + Otg'lt‘n/LOIL, where lnwageo = 575 2.: 1lnwage,o and union —
545— . . .

5+5 Z,_ 1 union,- which means the average length of period for keeping the membership

for the workers we observed. Model (3.39) gives 19 = 0.3480, (see Table 3.14). The log

116

hourly wage rate is more or less influenced by the labor union membership. However,
the estimates of do is not significantly different from zero in the log hourly wage
equation. The linear relationship between labor union membership on the unobserved
heterogeneity is small. Similar to equation (3.38), we report model (3.42) in Table
3.14.

We report the empirical examples in Table 3.9 - 3.14. At last we report the range
of estimate of the response to the future (p+7 pa) by measuring (p+7 ([1,, istd.(a,;))).
The results for the range of models (3.37) and (3.38) are 0.376 :1: 0.829(0.090). The

range of models (3.39) and (3.40) are 0.215 d: 0.134(0219).

3.5 Conclusion

In this chapter we apply the CMLE in estimating a panel data model where un-
observed heterogeneity interacts with a lagged dependent variable. The IV estimator
for the coefficient of y,,t_1 is inconsistent even for the average effect. In other words,
the existing approach can not estimate the amount of average state dependence. Our
recommended approach for the model is flexible. Firstly, we just model the distribu-
tion of the unobserved heterogeneity, D(a,-|:1:,:, y,o), and then construct the density of
(y,1,...,y,-T) given (13,, y,o). To specify a conditional parametric density function of
a, in which the conditional mean and variance are flexible function of yo and 1:,. We
can easily define a log-likelihood function conditional on (55,, y,o). It is crucial point
that the conditional maximum likelihood function is valid no matter what we con-
dition on. Furthermore, the conditional maximum likelihood function is consistent,
«N-asymptotically normal, under standard regularity conditions. Secondly, we can
easily estimate p and 7 as well as (p + 7110).

The idea of specifying a distribution for the unobserved effects given the initial

117

condition to construct CMLE means we can easily estimate the average partial effect
across a,. If .73,- is not strictly exogenous, we can apply the suggestion of Wooldridge
(2000a) as follows. \Ne parameterize 17(11th 1—1, 21, a; /\o), where 2:: is strictly exoge—
nous and build up the joint density of (11:, 517:) given (2T, K-1,Xt_1, a) and then apply
the same procedure as discussed previously to set up a log—likelihood function. We
can use numerical methods to solve out the CMLE.

In the empirical example of hourly log wage rate, the interaction of lagged wage
and the unobserved heterogeneity is significant. This restricts the case where we add
union status to the model. Union status is marginally significant and is estimated to
increase wage by about 5 percent.

We need to add more explanatory variables and apply the more general method
allowing for non-strict exogeneity assumption to decrease the degree of the inter-
dependence between the unobserved heterogeneity and the error term. We propose
the basic model to illustrate conditional MLE. These models can be easily extended
to more complicate case. It is crucial problem that we need to find an appropriate
formula to approximate the integral to integrate out the unobserved heterogeneity.
The more complicated is the conditional parametric density of the unobserved hetero-
geneity, the more difficult it is to find a good formula for approximating the necessary
integral. The Gauss quadrature method seems to work well, is time-consuming. Fu-

ture research could focus on simulation methods of estimation, as in Keane(1993).

118

6 = (p, 0.2, 0.4, 7, «2.4, «1.2)

60 2 (p0, 0.2, 0.4, ’70, V 2.4,V12)

Table 3.1: Ho: 6 = 90, where p =0 ~ 0.9

 

 

K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 2.5x10-3 0.2022 0.3966 1.1x10-3 1.5502 1.0950 (p,’)')
3 x 10-3 0.2025 0.3964 0.1022 1.5503 1.0804

0.01 0.0133 0.0067 0.0100 0.0067 0.0167 0.0067 (0,0)
0.0167 0.0033 0.0133 0.0067 0.0167 0.0100 (0,0.1)

0.05 0.0700 0.0367 0.0433 0.0367 0.0633 0.0533 (0,0)
0.0600 0.0500 0.0467 0.0400 0.0567 0.0500 (0,0.1)

0.10 0.1000 0.0800 0.0867 0.0933 0.1000 0.0900 (00)
0.1033 0.0867 0.0800 0.1033 0.1100 0.0867 (00.1)

P\6 0.2515 0.2017 0.3970 1.3x10-3 1.5502 1.0804 (p,7)
0.2519 0.2022 0.3967 0.1025 1.5502 1.0811

0.01 0.0133 0.0067 0.0100 0.0033 0.0200 0.0033 (025,0)
0.0100 0.0033 0.0067 0.0033 0.0167 0.0000 (025,01)

0.05 0.0600 0.0467 0.0467 0.0433 0.0567 0.0433 (025,0)
0.0433 0.0533 0.0533 0.0433 0.0533 0.0433 (025,0.1)

0.10 0.0933 0.0867 0.0833 0.1033 0.1100 0.0867 (025,0)
0.0967 0.0933 0.1000 0.1000 0.1067 0.0833 (025,01)

P\6 0.5004 0.2014 0.3978 1.3x10-3 1.5500 1.0818 (p,7)
0.5004 0.2019 0.3979 0.1028 1.5496 1.0838

0.01 0.0167 0.0033 0.0100 0.0033 0.0167 0.0033 (05,0)
0.0133 0.0067 0.0200 0.0067 0.0133 0.0033 (05,01)

0.05 0.0333 0.0467 0.0500 0.0467 0.0667 0.0300 (0.5,0)
0.0333 0.0567 0.0600 0.0533 0.0533 0.0400 (05,0.1)

0.10 0.0900 0.0833 0.0900 0.1000 0.0967 0.0767 (0.5,0)
0.0833 0.0967 0.0967 0.0933 0.1000 0.0800 (050.1)

Normality
Rrepetm'ons = 300, 6 2 310—0 231016;, «Z4 2 1.5492, «T2 9: 1.0954

The content of bracket is (po , 7o)
( i l

119

 

 

 

 

 

 

 

 

 

 

P\6 07504 0.2015 0.3984 6x10'4 1.5500 1.0815 (p, 5,1)
07484 0.2010 0.3998 0.1025 '1 .5482 1.0893

0.01 0.0133 0.0067 0.0167 0.0067 0.0200 0.0067 (075,0)
0.0133 0.0100 0.0267 0.0067 0.0167 0.0067 (075,01)

0.05 0.0500 0.0533 0.0567 0.0333 0.0633 0.0200 (075,0)
0.0467 0.0600 0.0733 0.0533 0.0600 0.0200 (075,01 )

0.10 0.1000 0.0900 0.0900 0.1100 0.1067 0.0567 (075,0)
0.1000 0.1000 0.0967 0.1033 0.0900 0.0633 (075.01)

P\é 0.9004 0.2016 0.3987 2x10“4 1.5498 1.0822 (p, 7)
0.8980 0.2001 0.3993 0.1019 1.5472 1.0917

0.01 0.0100 0.0067 0.0167 0.0067 0.0200 0.0033 (0. 9 ,0)
0.0067 0.0067 0.09300 0.0067 0.0167 0.0033 (0. 9, 0 1)

0.05 0.0433 0.0533 0.0567 0.0367 0.0700 0.0267 (0 9,0)
0.0733 0.0600 0.0800 0.0500 0.0567 0.0200 (0 9 ,0 1)

0.10 0.0900 0.1000 0.1067 0.1033 0.1133 0.0500 (0. 9 0)
01100 0.0967 0.1233 0.1267 0.0967 0.0733 (09,01)

 

 

Rrepet'zit‘ions = 300, 6

3—0—0 2330, 63, «2 2.4 a: 1.5492, «1.2 2 1.0954

The content of bracket 18 (po , 70)

(ii)

120

 

Table 3.2: Ho : 6 2 60, where p =0 ~ 0.5

6 = (p, 0.2, 0.4, 7, «2.4, «1.2)

00 = (p0, 0.2, 04,70, «2.4, «1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\ ’00—.“ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (p,7)
0.01 0.0133 0.1000 0.5200 0.9133 0.9933 1.0000 1.0000 1.0000 (0,0)
0.0167 0.0833 0.5067 0.9033 0.9900 1.0000 1.0000 1.0000 (0,01)
0.05 0.0700 0.2567 0.7433 0.9533 0.9967 1.0000 1.0000 1.0000 (0,0)
0.0600 0.2433 0.7600 0.9567 1.0000 1.0000 1.0000 1.0000 (0,01)
0.10 0.1000 0.3633 0.8267 0.9833 1.0000 1.0000 1.0000 1.0000 (0,0)
0.1033 0.3700 0.8400 0.9833 1.0000 1.0000 1.0000 1.0000 (0,0.1)
P\ 60—7 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 (p, 7)
0.01 1.0000 0.8900 0.4567 0.0767 0.0133 0.0900 0.4167 0.8333 (025,0)
1.0000 0.9133 0.4500 0.0867 0.0100 0.0767 0.4400 0.8500 (025,01)
0.05 1.0000 0.9967 0.6667 0.2167 0.0600 0.2400 0.6767 0.9267 (025,0)
1.0000 0.9800 0.7033 0.2133 0.0433 0.2300 0.6800 0.9333 (025,01)
0.10 1.0000 0.9800 0.8033 0.3067 0.0933 0.3433 0.7767 0.9533 (025,0)
1.0000 0.9867 0.8100 0.3233 0.0967 0.3367 0.7733 0.9600 (025,01)
P\ ”—1" 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 (p, 7)
0.01 0.3267 0.0400 0.0167 0.1000 0.3733 0.7300 0.9300 0.9967 (0.5,0)
0.3900 0.0500 0.0133 0.1167 0.4300 0.8100 0.9767 0.9967 (05,01)
0.05 0.6300 0.1533 0.0333 0.2233 0.5967 0.8833 0.9867 0.9967 (0.5,0)
0.6867 0.2033 0.0333 0.2233 0.6567 0.9167 0.9967 0.9967 (05,0.1)
0.10 0.7233 0.2800 0.0900 0.3233 0.7067 0.9267 0.9967 0.9967 (0.5,0)
0.7867 0.3200 0.0833 0.3267 0.7267 0.9600 0.9967 1.0000 (05,0.1)
Normality

Rrepetitions : 300,6 2
The content of bracket is (po , 70)

333,—, 23:", 6;, «2.4 2 1.5492, «1.2 2 1.0954

(1)

121

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\ ’00—? 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 (p, 7)
0.01 0.8500 0.3533 0.0267 0.0133 0.1300 0.4500 0.7700 0.9433 (075,0)
0.9233 0.4767 0.0633 0.0133 0.1400 0.5433 0.8733 0.9833 (075,01)
0.05 0.9567 0.6667 0.1633 0.0500 0.2667 0.6233 0.8933 0.9767 (075,0)
0.9833 0.7567 0.2300 0.0467 0.3233 0.7433 0.9533 1.0000 (075,0.1)
0.10 0.9867 0.7733 0.2867 0.1000 0.3767 0.7233 0.9433 0.9900 (075,0)
0.9900 0.8467 0.3500 0.1000 0.4267 0.8267 0.9700 1.0000 (075,01)
P\ ’00—»: 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 (p,7)
0.01 1.0000 1.0000 0.9800 0.5967 0.0633 0.0100 0.1700 0.5967 (09,0)
1.0000 1.0000 1.0000 0.7467 0.1367 0.0067 0.2200 0.7500 (09,025)
0.05 1.0000 1.0000 0.9967 0.8200 0.2600 0.0433 0.3500 0.7667 (0.9,0)
1.0000 1.0000 1.0000 0.9033 0.3400 0.0733 0.4433 0.8900 (09,025)
0.10 1.0000 1.0000 1.0000 0.8900 0.3800 0.0900 0.4700 0.8367 (0.9,0)
1.0000 1.0000 1.0000 0.9367 0.4400 0.1100 0.5267 0.9200 (0.0025)
R'repetz'tz'ons = 300,6 = 3376 23:0 6;, «2‘4 2 1.5492,«1§ 1*: 1.0954

The content of bracket is (po , 70)

(ii)

122

 

Table 3.3: Ho :6 = 60, where p =0 ~ 0.9
= (p, 0.2,0.4,7, «1.5, «1.25)

60 2 (p0, 0.2, 0.4, ’70, V 1.5, V 1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 8x10"4 0.2015 0.3984 5x10” 1.2230 1.0985 (0,0)

-1.35x10‘2 0.1884 0.3870 0.0947 1.2245 1.0992 (0,0.1)

0.01 0.0067 0.0133 0.0167 0.0333 0.0700 0.0133 (0,0)

0.0233 0.0133 0.0100 0.0433 0.0667 0.0133 (00.1)

0.05 0.0533 0.0533 0.0467 0.0933 0.1200 0.0800 (0,0)

0.0700 0.0600 0.0500 0.1100 0.1133 0.0767 (00.1)

0.10 01067 0.0967 0.1100 0.1567 0.2000 0.1500 (0,0)

0.1167 0.0967 0.1067 0.1567 0.2000 0.1467 (0,01)
P\6 0.2498 0.2020 0.3988 2 x 1071 1.2228 1.0996 (025,0)
02364 01841 0.3863 0.0950 1.2244 1.0991 (025,01)
0.01 0.0067 0.0167 0.0133 0.0333 0.0567 0.0167 (025,0)
0.0200 0.0133 0.0133 0.0400 0.0533 0.0167 (025,01)
0.05 0.0467 0.0567 0.0433 0.0900 0.1167 0.0667 (025,0)
0.0567 0.0533 0.0567 0.1167 0.1200 0.0700 (025,01)
0.10 0.1000 0.0967 0.0900 0.1567 0.1933 0.1300 (025,0)
0.1067 0.0933 0.1067 0.1800 0.2067 0.1133 (025,01)

P\é 0.4982 0.2027 04033 -00000 1.2222 1.1030 (0.5,0)
0.4846 0.1794 0.3870 0.0944 1.2236 1.1036 (05,0.1)

0.01 0.0033 0.0100 0.0100 0.0433 0.0467 0.0233 (0.5,0)
0.0233 0.0100 0.0167 0.0367 0.0500 0.0200 (05,01)

0.05 0.0633 0.0500 0.0400 0.0967 0.1333 0.0667 (0.5,0)
0.0733 0.0533 0.0500 0.1233 0.1267 0.0667 (0.5,01)

0.10 0.1033 0.0933 0.0900 0.1733 0.2033 0.1000 (05,0)
0.1367 0.0867 0.1000 0.2033 0.2067 0.1233 (05,01)

 

 

Rrepctitions = 300.6 =
The content of bracket is (po , 70)

235—623

Non-normality

6]", «1.5 21.2247,\/1.2 131.1180

300
1:1

(1)

123

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\é 0.7469 0.2032 0.4028 —2x10-4 1.2213 1.1077 (075,0)
0.7338 0.1720 0.3855 0.0925 1.2217 1.1126 (0.7501)
0.01 0.0100 0.0100 0.0100 0.0333 0.0500 0.0200 (075,0)
0.0333 0.0267 0.0167 0.08670 0.0600 0.0267 (0.75.0.1)
0.05 0.0567 0.0533 0.0433 0.1133 0.1433 0.0567 (075,0)
0.1000 0.0467 0.0600 0.1800 0.1467 0.0567 (0.750 1)
0.10 0.1233 0.0833 0.0900 0.1600 0.2033 0.1100 (075,0)
0.1600 0.1133 0.1133 0.2567 0.2100 0.1267 (0.7501)
P\é 0.8971 0.2034 0.4039 .2x10-4 1.2210 1.1090 (0.90)
0.8852 0.1707 0.3833 0.0915 1.2224 1.1161 (0.901)
0.01 0.0133 0.0067 0.0100 0.0433 0.0467 0.0133 (0.90)
0.0420 0.0180 0.0260 0.1340 0.0600 0.0200 (0,9,0 1)
0.05 0.0567 0.0567 0.0533 0.1233 0.1500 0.0700 (09.0)
0.1140 0.0440 0.0860 0.2380 0.1400 0.0700 (0.90.1)
0.10 0.1133 0.0833 0.0967 0.1533 0.2167 0.1167 (0.90)
0.2000 0.1180 0.1320 0.3240 0.2180 0.1220 (0.9.0.1)
Rrepetttions = 3009— 3—00 233016;, \/._5 2 1.2247, m 2 1.1180

The content of bracket is (p0 , ’70)

(ii)

124

 

Table 3.4: H0 : 0 = 60, where p =0 ~ 0.9

6 = (p, 020.4,), «1.5, «1.25)

00 = (p0,0.2, 0.4.70, 4/15, «1.25)

 

 

 

 

 

 

 

 

 

 

 

 

P\ ’00—? 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (p, 7)
0.01 0.0067 0.1267 0.5767 0.9200 1.0000 1.0000 1.0000 1.0000 (0,0)
0.0233 0.2400 0.7067 0.9667 1.0000 1.0000 1.0000 1.0000 (0,0.1)
0.05 0.0533 0.2900 0.7733 0.9733 1.0000 1.0000 1.0000 1.0000 (0,0)
0.0700 0.4433 0.8800 0.9967 1.0000 1.0000 1.0000 1.0000 (0,0.1)
0.10 0.1067 0.4000 0.8533 0.9867 1.0000 1.0000 1.0000 1.0000 (0,0)
0.1167 0.5600 0.9300 1.0000 1.0000 1.0000 1.0000 1.0000 (00.1)
P\ p9: 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 (p, 7)
0.01 1.0000 0.9133 0.5133 0.0800 0.0067 0.1267 0.5200 0.8867 (025,0)
1.0000 0.8700 0.3800 0.0367 0.0200 0.2133 0.6467 0.9500 (025,0 1)
0.05 1.0000 0.9833 0.7433 0.2300 0.0467 0.2867 0.7233 0.9567 (025,0)
1.0000 0.9633 0.6167 0.1533 0.0567 0.4133 0.8333 0.9867 (0.2501)
0.10 1.0000 0.9933 0.8267 0.3800 0.1000 0.3800 0.8300 0.9700 (025,0)
1.0000 0.9800 0.7333 0.2433 0.1067 0.5200 0.9167 0.9933 (0.2501)
P\ ’00—.— 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 (p, 7)
0.01 0.4833 0.0700 0.0033 0.1433 0.5233 0.8933 0.9833 0.9967 (0,5,0)
0.01 0.3700 0.0300 0.0233 0.2367 0.6767 0.9533 0.9933 1.0000 (05,01)
0.05 0.7133 0.2067 0.0633 0.3100 0.7200 0.9533 0.9933 1.0000 (0.50)
0.05 0.6033 0.1300 0.0733 0.4500 0.8400 0.9833 1.0000 1.0000 (0.5,0. 1)
0.10 0.8267 0.3533 0.1033 0.4100 0.8267 0.9733 0.9967 1.0000 (0.5,0)
0.10 0.7333 0.2233 0.1367 0.5500 0.9033 0.9900 ‘ 1.0000 1.0000 (05,01)

 

 

 

 

Rrepettttm‘zs = 300,é = 3Tl)5 Z

Non-normality

6;, \/1.5 21.2247,\/1.2 21.1180

300
i=1

The content of bracket. is (p0 , 70)

(ii)

125

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The content of bracket is (p0 , 70)

(ii)

126

P\ ’00—»: 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 (p, 7)
0.01 0.9733 0.6467 0.0933 0.0100 0.1900 0.6733 0.9533 0.9900 (0.750)
0.9567 0.5100 0.0433 0.0333 0.3133 0.8000 0.9867 0.9967 (075,01)
0.05 0.9967 0.8333 0.2767 0.0567 0.3733 0.8400 0.9833 0.9967 (075,0)
0.9933 0.7167 0.1800 0.1000 0.5467 0.9233 0.9933 0.9967 (0.7501)
0.10 1.0000 0.8933 0.3900 0.1233 0.5000 0.8867 0.9900 0.9967 (075,0)
1.0000 0.8100 0.2800 0.1600 0.6300 0.9633 0.9933 1.0000 (075,01)
P\ ’00—? 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 (p,’7)
0.01 1.0000 1.000 1.0000 0.8233 0.1567 0.0133 0.2733 0.8367 (0.9,0)
1.0000 1.000 0.9900 0.7120 0.0840 0.0420 0.4620 0.9300 (0.90.1)
0.05 1.0000 1.0000 1.0000 0.9500 0.3800 0.0567 0.5233 0.9433 (090)
1.0000 1.0000 1.0000 0.8820 0.2740 0.1140 0.6520 0.9760 (0.901)
0.10 1.0000 1.0000 1.0000 0.9800 0.5200 0.1133 0.6067 0.9767 (0.9,0)
1.0000 1.0000 1.0000 0.9280 0.3920 0.2000 0.7560 0.9840 (09,01)
Rrepetitions = 300,6 = 533 23316;, £3 2 1.2247, m 2 1.1180

 

Table 3.5: H0 : 6 = 60, where p :0 ~ 0.9

0

: (p, 0.15, 0.2, 0.4, 0.35, 7, «2.4, 4/12)’
60 2 (p0, 0.15, 0.2, 0.4, 0.35, ’70, V 2.4, V 1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\6 0.0022 0.1530 0.2018 0.3956 0.3355 -0.0002 1.5471 1.0828 (pry)
0.0019 0.1529 0.2027 0.3962 0.3362 0.1001 1.5470 1.0841

0.01 0.0067 0.0100 0.0067 0.0167 0.0100 0.0033 0.0067 0.0067 (0,0)
0.0133 0.0100 0.0067 0.0100 0.0100 0.0067 0.0067 0.0067 (0,0.1)

0.05 0.0400 0.0367 0.0633 0.0567 0.0400 0.0500 0.0567 0.0433 (0,0)
0.0567 0.0333 0.0667 0.0567 0.0400 0.0467 0.0533 0.0467 (0,01)

0.10 0.0867 0.0867 0.0967 0.1167 0.0933 0.0800 0.1100 0.1133 (0,0)
0.0933 0.0900 0.0967 0.1233 0.1033 0.0967 0.1100 0.1033 (0,0.1)

P\6 0.5010 0.1530 0.2024 0.3960 0.3365 0.0003 1.5473 1.0838 (72,7)
0.5010 0.1529 0.2028 0.3958 0.3368 0.1009 1.5472 1.0841

0.01 0.0067 0.0100 0.0100 0.0133 0.0133 0.0033 0.0067 0.0033 (0.5,0)
0.0167 0.0100 0.0067 0.0100 0.0133 0.0100 0.0067 0.0067 (0.5,0.1)

0.05 0.0533 0.0367 0.0633 0.0533 0.0433 0.0433 0.0400 0.0433 (0.5,0)
0.0533 0.0267 0.0667 0.0567 0.0533 0.0433 0.0500 0.0467 (0.5,0.1)

0.10 0.1067 0.0867 0.1033 0.1100 0.0900 0.0733 0.0967 0.0867 (0.5,0)
0.1000 0.0967 0.0967 0.0967 0.0800 0.0867 0.1100 0.1133 (0.5,0.1)

P\O 0.9009 0.1533 0.2020 0.3966 0.3384 -0.0002 1.5472 1.0846 (pgy)
0.8997 0.1531 0.2015 0.3997 0.3365 0.1016 1.5463 1.0879

0.01 0.0133 0.0100 0.0133 0.0067 0.0167 0.0033 0.0100 0.0167 (0.9,0)
0.0167 0.0067 0.0100 0.0167 0.0200 0.0167 0.0133 0.0133 (0.9,0.1)

0.05 0.0533 0.0400 0.0600 0.0533 0.0467 0.0433 0.0433 0.0500 (0.9,0)
0.0833 0.0367 0.0600 0.0667 0.0500 0.0667 0.0533 0.0833 (0.9,0.1)

0.10 0.1167 0.0900 0.1033 0.1200 0.0833 0.0867 0.0933 0.1067 (0.9,0)
0.1400 0.1033 0.1067 0.1233 0.1000 0.1267 0.1133 0.1433 (0.9,0.1)

Normality

Rrepetttions : 300.6 :
The content of bracket 28 (p0 , 2,40)

300

1
56—022

127

10;, 4/15 21.5492,\/1.2 21.0954

 

Table 3.6: H0 : 6 = 60, p =0 ~ 0.5

6 = (p, 0.2, 0.4, 7, v2.4, v1.2)
90 : (p0, (12,114.70, V 2.4, V 1.2)

 

 

 

 

 

 

 

 

 

 

 

 

P\ ’00 “ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (p, 7)
0.01 0.0067 0.0933 0.5033 0.8833 0.9933 1.0000 1.0000 1.0000 (0,0)
0.0133 0.1133 0.5067 0.9000 1.0000 1.0000 1.0000 1.0000 (0,0.1)
0.05 0.0400 0.2700 0.7167 0.9800 0.9967 1.0000 1.0000 1.0000 (0,0)
0.0567 0.2667 0.7333 0.9733 1.0000 1.0000 1.0000 1.0000 (0,0.1)
0.10 0.0867 0.3767 0.8000 0.9833 1.0000 1.0000 1.0000 1.0000 (0,0)
0.0933 0.3567 0.8100 0.9867 1.0000 1.0000 1.0000 1.0000 (00.1)
P\po—f 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 (p,‘7)
0.01 0.8133 0.4000 0.0400 0.0067 0.1100 0.4000 0.7400 0.9367 (0.5,0)
0.8467 0.4567 0.0667 0.0167 0.1133 0.4033 0.8033 0.9733 (05,01)
0.05 0.9367 0.6367 0.2000 0.0533 0.2367 0.5733 0.8767 0.9900 (0.5,0)
0.9600 0.6667 0.2133 0.0533 0.2333 0.6067 0.9267 0.9933 (05,01)
0.10 0.9700 0.7333 0.3000 0.1067 0.3167 0.6900 0.9300 0.9933 (0.50)
0.9733 0.7767 0.3367 0.1000 0.3567 0.7567 0.9533 0.9967 (05,01)
P\ ’00—.— 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 (12,7)
0.01 1.0000 1.0000 1.0000 0.9733 0.5967 0.0767 0.0133 0.1733 (0.90)
1.0000 1.0000 0.9800 0.7333 0.1367 0.0167 0.2033 0.7167 (09,01)
0.05 1.0000 1.0000 1.0000 0.8200 0.2500 0.0533 0.3300 0.7533 (0.9,0)
1.0000 1.0000 1.0000 0.8900 0.3367 0.0833 0.3800 0.8467 (0.90.1)
0.10 1.0000 1.0000 1.0000 0.8200 0.2500 0.0533 0.3300 0.7533 (0.9,0)
1.0000 1.0000 1.0000 0.9233 0.4467 0.1400 0.4967 0.9033 (0.90.1)

 

 

 

 

 

 

Rrepetitions 2 300,6_ — 3—00 233016;, «2.4 2 1.5492, (/1.2 2 1.0954
The content of bracket is (p0 , '70)
Normality

128

Table 3.7: H0 : 6 = 00, where p =0 ~ 0.9

6 = (p, 0.15, 0.2, 0.4, 0.35,»), 4/15, «1.25)
60 = (p0.0.15, 0.2, 0.4, 0.35, )0, \/1.5, «1.25)

 

 

 

 

 

 

 

 

 

 

P\6 0.0004 0.1521 0.2017 0.3986 0.3451 0.0005 1.2223 1.0964 (p,’)’)
-00032 0.1521 0.2424 0.4141 0.3429 0.0170 1.2228 1.0996

0.01 0.0100 0.0133 0.0133 0.0167 0.0067 0.0367 0.0700 0.0133 (0,0)
0.0133 0.0133 0.0200 0.0133 0.0067 0.9533 0.0700 0.0133 (0,0.1)

0.05 0.0600 0.0600 0.0467 0.0433 0.0400 0.0933 0.1200 0.0867 (0.0)
0.0633 0.0600 0.0733 0.0533 0.0400 0.9767 0.1167 0.0800 (0,0.1)

0.10 0.1067 0.0933 0.1067 0.1100 0.0900 0.1467 0.1900 0.1400 (0,0)
0.1167 0.0933 0.1500 0.1233 0.0967 0.9900 0.1933 0.1500 (0.0.1)

P\6 0.4975 0.1519 0.2028 0.4009 0.3472 0.0001 1.2214 1.1015 (p,')')
0.4973 0.1518 0.1988 0.3997 0.3441 0.1021 1.2210 1.1006

0.01 0.0133 0.0133 0.0133 0.0133 0.0067 0.0333 0.0550 0.0167 (0.5,0)
0.0167 0.0133 0.0100 0.0133 0.0100 0.0567 0.0567 0.0167 (050.1)

0.05 0.0600 0.0633 0.0500 0.0333 0.0400 0.0933 0.1300 0.0533 (0.5,0)
0.0433 0.0667 0.0433 0.0533 0.0400 0.1033 0.1400 0.0700 (05,01)

0.10 0.0933 0.0867 0.1000 0.0867 0.0900 0.1533 0.2133 0.1100 (0.5,0)
0.0700 0.0833 0.0833 0.1000 0.1000 0.1700 0.2133 0.1133 (05,01)

P\6 0.8967 0.1500 2036 0.4045 0.3487 -0000] 1.2203 1.1073 (p,'7))
0.8954 0.1517 0.1962 0.4024 0.3468 1009 1.2178 1.1063

0.01 0.0167 0.0133 0.0067 0.0100 0.0067 0.033 0.0467 0.0100 (0.9,0)
0.0100 0.0067 0.0133 0.0267 0.0100 0.0400 0.0773 0.0167 (090.1)

0.05 0.0600 0.0633 0.0600 0.0567 0.0367 0.1033 0.1567 0.0600 (0,9,0)
0.0400 0.0700 0.0400 0.0933 0.0733 0.0900 0.1533 0.0633 (0.9,0.1)

0.10 0.1067 0.0867 0.0900 0.0833 0.0933 0.1533 0.2000 0.1067 (0.9,0)
0.1067 0.0900 0.0700 0.1633 0.1300 0.1333 0.2233 0.1133 (0.9.0.1)

 

 

 

 

 

. Non-normality
Rrepetittons = 300. 6 = 31% 230016;,M15 2 1.2247, 4/125 2 1.1180

J:
The content of bracket 178 (p0 , 70)

129

 

Table 3.8: H0 : 6 = 60, where p =0 ~ 0.9

6

= (p, 02,04, 7. «1.5. 71.25)

00 2 (p0, 0.2, 0.4, ’70, V 1.5, V 1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\ p9? 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (p. 4,)
0.01 0.0100 0.1433 0.6067 0.9333 1.0000 1.0000 1.0000 1.0000 (0,0)
0.0133 0.1767 0.6467 0.9467 1.0000 1.0000 1.0000 1.0000 (001)
0.05 0.0600 0.2967 0.7833 0.9767 1.0000 1.0000 1.0000 1.0000 (0,0)
0.0633 0.3400 0.8367 0.9867 1.0000 1.0000 1.0000 1.0000 (0,0.1)
0.10 0.1067 0.4733 0.8933 1.0000 1.0000 1.0000 1.0000 1.0000 (0,0)
0.1167 0.3700 0.8400 0.9833 1.0000 1.0000 1.0000 1.0000 (0,0.1)
P\ ”0 ‘ 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 (p,'7)
0.01 0.9300 0.5200 0.0700 0.0133 0.1600 0.5400 0.9000 0.9833 (0.5,0)
0.9900 0.7633 0.2567 0.0433 0.3633 0.8000 0.9867 0.9967 (05,01)
0.05 0.9867 0.7167 0.2200 0.0600 0.3100 0.7667 0.9667 0.9967 (0.5,0)
0.9900 0.7633 0.2567 0.0433 0.3633 0.8000 0.9867 0.9967 (0.501)
0.10 0.9933 0.8133 0.3633 0.0933 0.4233 0.8400 0.9733 0.9967 (0.5,0)
0.9967 0.8733 0.3800 0.0700 0.4500 0.8567 0.9900 1.0000 (05,01)
P\ ”—7 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 (p, 7)
0.01 1.0000 1.0000 1.0000 0.8800 0.1967 0.0167 0.3000 0.8933 (09.0)
1.0000 1.0000 1.0000 0.9233 0.2567 0.0100 0.3867 0.9533 (09,01)
0.05 1.0000 1.0000 1.0000 0.9733 0.4167 0.0600 0.5367 0.9600 (0.9,0)
1.0000 1.0000 1.0000 0.9900 0.4800 0.4400 0.6267 0.9967 (09,01)
0.10 1.0000 1.0000 1.0000 0.9900 0.5367 0.1067 0.6567 0.9867 (0.9,0)
1.0000 1.0000 1.0000 0.9933 0.6067 0.1067 0.7033 0.9967 (09,01)

 

 

 

 

 

Rrepet'it’ions = 300,6 =
The content of bracket is (p0 ,

Non-normality

1 300
300 2:321

"10 l

130

6;, ./1.5 21.2247,./1.2 21.1180

 

Table 3.9: Case a: lnwageu = p lnxwage,‘t_1 + a,- + 7 a,- - lnwagei‘tq + e“

 

 

 

ln'wagea = p lnwagem—l + a. + 7 a. - 172-14109651-] + 5,,,
a,- = (10 + 071 lnwageio + c,-
Coeffi cient p do d1 7“ 6. 6,,
CMLE -0.4951 0.9448 0.0758 0.8289 0.3493 0.0898
t-statistics (-1744) (35.986) (3.893) (2.754) (75.516) (4.148)
log—likelihood value 302.614

 

 

 

 

 

 

 

N=545, periods is 1980 ~ 1987

Table 3.10: Case bzlnwageit = 19 lnwagethl + a, + 7 (a,- — pa) - lnwagethl + an

17111109611 = 19 lnwageLt-l + 04' + ’7’ 102' — #al ' lnwageikl + 5“,

 

 

 

a, = 00 + 041 lnwageio + c,-
Coeflicient 6 do (i) '3' [75 {70
CMLE 0.3755 0.911418 0.0759 0.8285 0.3493 0.0898
t-statistics (17.046) (35.995) (3.843) (2.759) (75.531) (4.156)
log—likelihood value 302.614

 

 

 

 

 

 

 

N=545. periods is 1980 ~ 1987

131

Table 3.11: Case czlnwageit = p lnuiagemq + (51 dt + a, + 'y a,- - lnwageikl + 51-)

lnwageu = p l'nu1a.ge,-,t_1 + (5, d) + a, + 7 a, - lnwagethl + 5,),
a,- = 00 + (.11 lnwagem + c,-

 

 

pr Ir

Coefficient ,6 do d1 4,. 61 62 63

 

CMLE 0.0083 0.6565 0.1652 0.2567 0.2842 0.3093 0.3424

t-statistics (0.0067) (1.31) (7.389) (3.605) (0.563) (0.613) (0.678)

 

 

 

 

 

 

 

 

 

Coefficient 64 65 66 67 {75 60

 

CMLE 0.4023 0.4328 0.4815 0.5356 0.3309 0.1841

t-statistics (0.797) (0.858) (0.954) (1.061) (78.1) (10.089)

 

 

 

 

 

 

 

N=545, periods is 1980 ~ 1987

Table 3.12: Case dzlnwagefl = 6 lnwagemq +6) d) + a, +7 (a,- —pa) ~lnwage,-,t_1 +8“

lnwageit = 6 lnungethl + 6) d) + a, + '7' (a,- — pa] - lnwageikl + 5,),
a,- = (:10 + 01 lnwageio + c,-

 

 

F

Coefficient 6 do d1 '3' 61 62 63

 

CMLE 0.21545 0.50248 0.223801 0.134065 0.599911 0.6541561 0.397765

t-statistics (11.86) (0.2724) (8.9881) (2.5400) (0.3253) (0.3547) (0.2157)

 

 

 

 

 

 

 

 

 

v

COOl’fiClellt 6.; 65 65 67 (5'5 (2'0

 

CMLE 0.422917 0.457085 0.517636 0.549881 0.330714 0.21901

t-statistics (0.2293) (0.2487) (0.2807) (0.2982) (73.9500) (12.0017)

 

 

 

 

 

 

 

N=545, periods is 1980 ~ 1987

132

Table 3.13: Case e: lnwage“ = p l‘n»U,’(Lg€i’t_1 +6 union.) + a, + '7 a,- - lnwagetbl +5.)

 

 

 

 

lnwageu = p l'nrwa,ge,-,t_1 + 6 union“ + a,- + 7 a,- - ln,wage,-,t_1 + 51-),
a,- = (10 + 071 lnwageio + a2 union, + c,
Coefficient r3 6 do d1 67;. a 6. 6.,
CMLE -0.4771 0.0507 0.9430 0.0820 0.0244 0.7757 0.3518 0.0987
t-statistics (-1.861) (2.049) (33.093) (3.923) (1.248) (2.819) (67.366) (4.279)
log-likelihood value 85.6199

 

 

 

 

 

N=545, periods is 1980 ~ 1987

Table 3.14: Case 1: lnwageit = 6t'n‘wage,,t-1+6'avnton,t+a,-+7[a,-—pa]-lnwage,-,t_1+e,-t

 

 

 

[7111109811 2 19 ("100984.14 + ('3 unimfli + ’7 [at “ #al ' ln'wageikl + 541,
a, = 00 + 01 lnwageio + 02 union, + c,
Coefficient 6 8 do 6:, 52 ‘7 6, 6.,
CMLE 0.3480 0.0511 0.9425 0.0820 0.0240 0.7773 0.3518 0.0985
t—statistics (13.726) (2.065) (33.091) (3.909) (1.235) (2.814) (67.343) (4.265)
log-likelihood value 85.6199

 

 

 

 

 

N=545, periods is 1980 ~ 1987

133

 

 

 

 

CHAPTER 4

CMLE For Logit Model With

Individual Heterogeneity

4. 1 Introduction

In the previous chapters, we considered estimation of the AR(1) panel data model
with unobserved heterogeneity. In the standard model with additive heterogeneity,
transformations exist that can be combined with instrumental variables estimation
to produce consistent estimators. (As we discussed in Chapter 1, the usual within
estimator is not consistent with fixed T.) Even in this simple model , however, the
conditional maximum likelihood estimator has some advantages. For one, it is gener-
ally more efficient than method of moments estimators that do not make assumptions
on the distribution of the initial condition.

In Chapter 3 we considered estimation of the AR(1) model when the unobserved
heterogeneity and the lagged dependent variable possibly interact. As shown there,

the usual IV estimators that are consistent in the model with only additive hetero—

134

geneity are no longer consistent when the autoregressive coefficient depends on the un-
observed heterogeneity. Nevertheless, the conditional maximum likelihood approach
does produce consistent estimators ( under a normality assumption and, perhaps,
more generally). In the empirical application to a dynamic wage equation, the inter-
action between lagged log wage and the unobserved heterogeneity was statistically
and practically important.

In this chapter we turn to a model where conditional maximum likelihood meth-
ods are indispensable: the dynamic logit model with unobserved heterogeneity. The
fact that the logit model is nonlinear makes dealing with a lagged dependent variable
even much more difficult. than in Chapter 2 and 3. First, with small T, we cannot
simply treat the unobserved effects as parameters to estimate. Even without a lagged
dependent variable, the incidental parameter problem (Neyman and Scott [ 1948])
caused inconsistent estimation of the parameters. Secondly, as with the linear AR(1)
model, the inclusion of lagged dependent variable is very difficult to characterize the-
oretically, but the intuition is the same as for the linear model. Plus, for large N,
treating the unobserved heterogeneity as parameters to estimate is computationally
burdensome.

In this chapter I show how to implement conditional maximum likelihood esti-
mation, following the general treatment of Wooldridge (2000). As in the previous
chapters, this entails modeling the distribution of the unobserved heterogeneity con-
ditional on the initial condition and any strictly exogenous variables. Nevertheless,
we need not find a steady state distribution for initial condition, and we need not
approximate this distribution. An added benefit is that we can consistently estimate
the average partial effects - that is, the partial effect averaged across the distribution

of the heterogeneity - rather than just the parameter. Thus, while this approach is

135

almost fully parametric, it delivers estimates of interesting quantities that semipara—
metric approaches cannot.

The plan of this chapter is as follows. Section 2 applies the CMLE to a basic
dynamic logit model with unobserved effects. In this section we construct the condi-
tional likelihood function to obtain the conditional maximum likelihood estimators.
Section 3 we set up Monte Carlo studies to examine the performance of conditional
maximum likelihood estimator as N —> 00 with small T. Section 4 investigates em-
pirical example of union membership and calculate the average partial effects. Finally,

we make some concluding remarks.

4.2 The CMLE for Dynamic Logit Model with Un-

observed Heterogeneity

4.2.1 Estimation of Fixed Effects Model

I consider the dynamic logit model with unobserved heterogeneity as follows:

POM = 113/20: - - - Jim—1,5137, at)
E F(yit)
E Mpg/4,21 + fit/3 + at)
eXP(Pyi,t—1 + 51711.5 + Oi)

E 1+€Xp(pyi,t_1+$ut3+atl t=1,...,T;T>2,z=1,...,N,

where a,- is an individual-specific effect that may depend on the exogenous explanatory

(4.1)

 

variables :17,- E (1,1,...,.T,'t) in an arbitrary way and where yio is the initial value
of the response variable. If we assumed F (ya) is a linear function we would have
a dynamic linear probability model and we would apply IV methods discussed in
Chapter 1 to estimate the parameters. The LPM, however, has inherent defects

because the response probability might not be constrained between 0 and 1, and

136

the true response effect is probably not constant. Theoretically, the inability to
obtain the consistent estimator of a,- will not rule out the possibility of obtaining a
consistent estimator of 6 if equation (4.1) is in the form of static linear-regression
model, ya = 56,,6 + a, + 5,,, because the estimation of 6 and a,- are asymptotically
independent ( Hsiao [ 1986]). Even more, we can obtain an appropriate transformation
to remove the effect of a,- in regressive model provided that we properly interpret the
initial conditions. Unfortunately, the same things can not be said for the non—linear
case because of the estimation of 6 or p and a,- are not independent of each other. The
inconsistency of a,- is transmitted into the estimator 6 or p. For example, as discussed
in Chapter 3, when the lagged dependent variable interacts with the unobserved
heterogeneity, no transformations immediately suggest themselves to eliminate the
effect of unobserved heterogeneity.

Just as in the dynamic linear-regression model, the problem of initial conditions
for the dynamic logit model with unobserved heterogeneity must be resolved before
we can consistently estimate the parameters generating the stochastic process. The
effect is very difficult to characterize theoretically, but the intuition is the same as for
the linear model. Even for a static logit model, ( p = 0 in (4.1), P(yit = 1]:I:,-,a,-) =
A(:1:,t6 + a,), the assumption of non-random unobserved effect, a,, means we need to
estimate both 6 and a,- which are unknown parameters. When T tends to infinity,
the MLE is consistent. However, as we know, T is usually small for panel data, in
which case we have an incidental parameters.

Let us illustrate the inconsistency of the MLE for 6 in the static logit model in

the following (Hsiao ] 1992]). The log-likelihood function for the static model (4.1)

137

given p = O is

log L = — 2 2109.11 + exp(r..6 + 431+ 2 Z 941(176134—04). (4.2)
i t i t

For simplicity, we assume T = 2, one explanatory variable, with 56,1 =0 and 37,2 = 1.

Then the first—derivative equations are
810 L_ exp(6+a,) _
735’ ‘ 2"] (1+ exp(fi +fl + 6,2] 7 0’ (4'3)

BlogL _ 2 __ exp(62‘,” + a.,-) 0 _
”i _ 221 l 1+ exp(flmu + a.) + 91!] — 01 (4-4)

Solving (4.4), we have

 

 

04:00 ifyil+y12221
a,- = —oo if yil + M? = 0, (4-5)
‘6 ifyil+y72:1-
Inserting (4.5) into (4.3) and letting 71.1 denote the number of individuals with ya +

3142 = 1 and 71.2 denote the number of individuals with ya + 31,2 = 2, we have

N
exp(6 + 0,) , . _ exp(6/2)
2 (1+ exp(6 + a,-) + 11,2 72 1+ exp (6/2)+ 2:: 31.2 (4.6)

 

 

Therefore,
6 = 2008(2612 - 62) -108(n4 + 62-)}-

By a law of large number

A

plim 6 = 26,

N—ooo

which is not consistent because

. _1_ N . _ _ L N exp(6 + a,)
BEE: 5421:1922 "2) _ N 21:1 (1+ exp(a.))(1+ exp(6 + at»,

 

 

1 _ N ,f _ 1 N exp(fi + 0.)
plim,V (m + 112 2,21%» “ N 22:1 (1+ exp(a,))(1+ exp(t3 + ail),

Thus we have incidental-parameter problem in that there is only a limited number

of observations to estimate a, ( Neyman and Scott [ 1948]). It is meaningless that any

138

estimation of a,- if we intend to judge the estimators by the large-sample properties
(N —> 00).

The nonlinear panel data model with individual heterogeneity may be estimated
by the semiparametric approach which allows us to make use of the linear structure
of the latent variable equations such that the individual-specific unobserved effect can
be eliminated by the differencing transformation and the like and hence the lack of
knowledge of a,- no longer affects the estimation of parameters of interest ( Manski [
1987]). Recently, Honoré and Kyriazidou (2000) derive effective moment conditions
for the unobserved effects logit model with one-period lagged dependent variable
from an objective function that identify the parameters. The interesting advantages
of semiparametric approaches is to allow estimation of parameters without specifying
distributions for the unobserved effects, although the estimators may not possibly
converge at the rate \/N (Hahn [ 1997]). The nature of semiparametric approaches,
nevertheless, can not suggest the estimators of partial effects on mean responses.

The nonlinear model with unobserved effects might be estimated by a random
effect approach. Such an approach requires the specification of the statistical rela-
tionship between the observed cmtariates and unobserved permanent individual het-
erogeneity. Furthermore, it entail specify the distribution of initial condition if the
list of explanatory variables include the lagged dependent variables. The inherent
defect is the misspecification of these distributions. We are to use the parametric to
solve the dynamic logit model with unobserved effects by specifying the conditional
distribution for the unobserved heterogeneity and hence we also incur the question,
which misspecification of this distribution generally leads to inconsistent parameter
estimates. We, nevertheless, have set. up a simple conditional maximum likelihood

estimators and moreover, the quantities of interest in nonlinear case can be obtained

139

based on the assumptions employed, in particular, the partial effects on the mean re-

sponse, averaged across the population distribution of the unobserved heterogeneity.

4.2.2 Conditional Maximum Likelihood Estimator

First, we construct the conditional likelihood function for the conditional maximum
likelihood estimator in dynamic logit model with unobserved effects (4.1), given 6 = 0.
Generally, we assume that 5,, is symmetrically distributed about zero, which means

that 1- C(—z) = C(z) for all real numbers 2. We make the assumptions as follows:

 

Assumption 4.1 5,, is independent of 5,,,-“ . . . , 5,1, ym, and a,.

 

 

 

 

Assumption 4.2 a,|y,~0 ~ Normal(a0 + 01 y'io,o§).

 

According to Assumption 4.1, the conditional density function is as follows:

“yell/6,14, - - - 43110.04) :- Afl) 3113—1 + atly” (1* (\(6 913—1 + Gina—y“), (4-7)

where

. . _ exp(P yi,t—1 + ail
M10 62.14 + a.) _ 1+ exp(P 1113—1 + a1)

 

and hence the density function of T-period observations of cross-section i:

T
f(yiTa - - -1yi1ly101ai) = HAW yi,t-—l + 0,1)“ (1 — A(p y,,t_1+ a,))“‘y") (4,8)
1:1

To obtain a conditional log-likelihood function for the T-period observations of cross-
section i, we specify a distribution for the unobserved effects h.(a,-0]y,-o;o') and then
we obtain the likelihood function of T-periods of cross-section i conditioning on yio

by equation (4.8) and h(a,0]y,0; a) as follows:

5 (yiTa' ' 'eyiOiO) :

108 f [”1121 A(p yi,t—1 + a)!“ (1 — A(p 1123—1 + 0))(1—y")] h.(a,-]y,-O;a) da,
-00 (4.9)

140

where 6 = (p , a). We can solve the conditional maximum likelihood estimator by
maximizing the sum of equation (4.9) across i=l,. . ., N with respective to 6. We

write the maximizing problem as follows:

N
mgrx 22:; 6 (MT, . . . ,y,0; 6) (4.10)

In the chapter 2, we have discussed the consistency of conditional maximum like-
lihood function. Equation (4.9) satisfies the generic form (2.9) and thus satisfies
the inequality equation (2.15), Kullback-Leibler information inequality and equation
(2.16), this ensures that the true parameters 60 solve the relevant population maxi-
mization problem, but they still might not be the unique solutions. For identification,
we must assume that the inequality is strict. According to Assumption 4.1 and 4.2,

equation (4.10) can be rewritten as follows:

108 f” [1.132. Mp 61.1-1 + aly“ (1 - Mp 9.3—1 + 0))(1— ml °
-.,, (4.11)
( 1 leXP(:Ql(sz;)2)dQ

‘ 2
2600

 

7

where a,- = 00 + 011/20 + c.- and c, ~ Normal(0, 03,) from Assumption 4.2.

It is impossible to reach a formula of closed form for the estimator by directly
solving out the first conditions of sum of equation (4.10) across 2' from 1 to N. We
need to employ numerical methods. Under Assumption 4.1 and 4.2, the formula
for the evaluation of the necessary integral of (4.11) is the Hermite integral formula

2
foo e—Z g(z) d z = 2;, wjg(zj), Where K is the number of evaluation points, 10,- is

‘W
the weight given to the jth evaluation point, and g(zj) is g(z) evaluated at the jth
point of 7: (Butler and Moffitt [ 1982]). This formula is appropriate to our problem
because the normal density h in equation (4.11) contains a term can be expressed

~2 0 n o o o
as a form of e-" and the function of 9(2) IS, in our case, the densrty function of

T—period observations of cross-section 2'. Without finding a specific distribution for

141

h(ailyio), an integration by Gaussian method might be needed. For simplification
of numerical calculation, we always specify a distribution for h(ailyio) to fit for the
Hermite integral formula in the simulation.

The previous analysis is limited on the time—series observation of cross section,
{git}: if”, without the exogenous variables. In most of application, we add other
exogenous variables to study the response of the (explanatory variables to the future
and further the feedback from the unexpected movements in the outcome variable
to future values of the explanatory variables). Some experimental case where the
variable will be in the control of a researcher. Honoré and Kyriazidou ( 2000) give
a restriction on Ira to identify the parameters. For example, if :1?“ represents some
program participation, :13“ = Stat—1 means that the status of participation will not
change for successive periods.

We consider the model for union membership with unobserved heterogeneity.
The key explanatory variables, such as the school year or education diploma, grad-
uate or non-graduate, is more or less related to the union membership. A vari-
able such as a person’s age can be thought of as strictly exogenous variable if

we just study the male youth. Using the general framework of CMLE in Chap-

ter 2 under the strict exogeneity, we specify distributions for (yT,...,y1) given
(xT, . . .,131, y0)and a,- given ((177,...,:1:], yo) as D(yT,. .. ,ylle,...,:1:1,y0) and
H(a|:I:T, . . . ,$1,y0) in respective. The parameterized density function for the distri-
butions are f(yT,...,y1|:z:T,...,;1:1,y0,(5) and h.(a,~|a:7~,...,2:1,y0,6) in respective. In
practice, E(a,-|;r,~T, . . . $11,350) is assumed to be in a function of (51:17“, . . . , 313,1, 311-0). We
assume that E(a,~|:1:.-T, . . . ,:1:,;1,y,:0) = ao+alyio+agfifh where .713,- is a linear combination
of (11:11,. .. ,13“),.T 22,471.15“. We assume that 7'; is equal to 1 for t— —— 1,. .,T to

decrease the number of identification for parameters. We make assumptions about

142

a“ and a,- as follows:

 

2
Assumption 4.3] Eitlyi,t_1, . . . ,yio, (13,37, . . . ,$,1,a,- ~ Log'zlt(0, 1%).

 

 

 

Assumption 4.4] a,|:ciT,...,:c,-1,y,0 ~ Normal ((10 + 011 yio + x, 02, 03).

 

According to Assumption 4.3 and 4.4, f(yT,...,y1|:z:T,...,:cl,y0,6) is equal to
“3:1 A00 yi,t-—1 + flirt/3 + aily“ (1 " A00 Elm—1 + 33115 + (1,)“ — y“). Equation (49)
can be rewritten as follows:

e(yiTa'Hay‘iOaxiTa'"9$21;6) I

108 / [nirzi MP yut—l + 113:3 + aly“ (1 " MP yi,t-1 + 33116 + a)“ — yidl

h(alarn, . . - , Im, yiO; 0) da
(4.12)

Replacing the above equation into the equation (4.12) and then solve out the maxi—
mization of the objection function (4.12).

If X,- is not strictly exogenous, we can apply the suggestion of Wooldridge (2000a)
as follows. To parameterize g(:ct|Xt_1,zt,a; A0), where 2t is strictly exogenous and
build up the joint density of (Yt,Xt) given (ZT,Yt_1,Xt_1,a) and then apply the
same procedure as discussed previously to set up a log-likelihood function. We can

use the numerical method to solve out the CMLE.

4.3 Simulation Evidence

In order to investigate the performance of maximum-likelihood estimators given
the initial value, we conducted Monte Carlo studies. We divide this section by two
subsection: one is for the model without exogenous variable; the other is for the
model with strictly exogenous variable. We use the MLE software of Gauss to do

our simulation for the conditional maximum likelihood estimator. As the discussion

143

in previous chapter, the feasible computation of the Hermite integral depends on the
number of evaluation points at which the integrand must be evaluated for accurate
approximation. Several evaluations of the integral using seven periods of arbitrary
values of data and coefficients on two right-hand-side variables shows that the value
of K is chosen to be 21 is highly accurate. Although the value of K determines the
accuracy of the calculation of integral, we don’t discuss the relation of K and the
evaluation of integral as Butter and Moffitt ( 1982) did. K221 is highly accurate for
the evaluation of integral (4.12). We repeat the maximization of the model of interest

in the following for 500 hundreds. The notations for the simulation are as follows:

1. 6* means the conditional maximum likelihood estimators in each iteration.

2.9:1 5009+

500 i=1 1'

 

3. 6 means true value of parameter, where 6:(p, (10, 011,00): (p, 0.2, 0.4, v1.2)
in the model Without exogenous variables or 6:(p, ,6, cm, 01, a2, 0a):

(p, 0.15, 0.2, 0.4, 0.35, v1.2) in the model with strictly exogenous variables.

4.3.1 The Model. Without Exogenous variables

Let the true value of p be 0, 0.25, 0.5, 0.75, 0.9, and 0.95. We calculate the
frequency of rejecting the hypothesis of H0 : 6 = 60 to examine the performance of
the conditional maximum likelihood estimator. With the same procedure, we focus on
the estimator,p by calculating the frequency of rejecting the hypothesis of H0 : p 2 p0

under different true value of p, where p0 is 0, 0.25, 0.5, 0.75, 0.9, and 0.95. The results

144

are reported in Table 4.1 - 4.4. We begin with the true model as follows:

31:} 2 P0 "(Jim—1 + 01+ 5:1.
ya = llyf, > 0], (4.13)

where a, : 0.2 + 0.4 yio + C). The c,- comes from Normal(0,1.2). According to

e5
u
Assumption 4.1, the Inverse function of the logistic function, u— — m is Imm).

Therefore, we generate the 5a = haul—ft), where 11“ comes from the uniform
1

distribution of [0,1]. The conditional likelihood function (4.11) can be rearranged as

follows:
T _
00 , 1 2' _ 1 —l a 1'a 2
10g / H exp(y 1,0) .l/ ,t 1 + all . )e—g—(‘mL‘l da, (414)
-00 t=l 1 + exp(p Elm—1 + a.) \/27m?,
Let z, to be fifi9 and replace a, with 00 + 01 yio + floaz, into the function (4.14).
00

Therefore the conditional likelihood function of cross section i can be re—written as

 

follows:

00 _ 2
log / [ft TI:I explyit( pyi,—t 1+00+011 y70+\/—2_0az ZN] e—Z dz. (4.15)

‘001(1+epryi,—t1+00+alyi0+fi0az)

 

The integral of function (4.15) can be approximated by the Hermite integral formula:
f_oog(2)€ Z d 2 2' Syd-1159(2)). The likelihood functlon of (4.15) can be expressed

in the form of Hermite integral formula as follows:

00 2 K
log / g(z)e_z dz 2 longjg(zJ-), (4.16)

where
T exp [MAP Pyi t—l + 0'0 + 011 yiO + fiaazill

=f II

We maximize the sum of the likelihood function (4.16) away from the constant term

 

1+eXP(Pyi,—t1+Olo+011yi0+\/§Uazz‘)

across 2' from 1 to N to obtain the estimators as follows:

T exp [Ll/MP yi,t—1 + 010 + 01 MD + fiaazijll

N K
max 2 log 2 wj H
i=1 j=1

t=1 l-l- exp(p yi,t—1 ‘l‘ 0'0 + 0'1 yio + fiUaZij)

 

(4.17)

145

In the simulation, we set the number of evaluation point, K to be 21. We examine
the assumption about the conditional a): normality and non-normality. The result of
Table 4.1 and 4.5 is under the normality assumption of a,, while Table 4.3 and 4.7 is
under the non—normality. We assume the t-distribution with freedom 10 to explain
the non-normality assumption on a... Table 4.1 reports the CMLE estimates for the
data generated from the true model. We repeat 500 times for the same procedure of
maximizing the objective function (4.17) to obtain the CMLE estimates.

To examine the power test for the CMLE estimators, with 500 repetitions, we
calculate how many times the hypothesis of H0 : 6 : 60 will be rejected under a
certain power value, 0.1, 0.05, 0.10 respectively. For example, in the second column
of Table 4.1, the number of bracket is the average value of 500 estimates, p“; the
valuas of the second to fourth row represent the p-value 0.004, 0.046, and 0.09 under
the power 0.01, 0.05, 0.1 in respective when true value of p is zero. According to the
result of the simulation, the CMLE estimators perform well. Similar to the linear
case in Chapter 2, the value of p is likely to be rejected when the true value of p is
getting further away from zero.

The response probability of interest is mainly related to the p, so we construct
another Table 4.2— ( i ) - ( vi ) to examine the hypothesis H0 : p : p0, where p0 is 0,
0.25, 0.5, 0.75, 0.9, and 0.95 under the different true value of p, 0, 0.25, 0.5, 0.75, 0.9,
0.95. For example, when the true value of p = 0.25 and p020.6, the p-value is 0.2620
in Table 4.2— (iii) under the level of p-value, 0.01. Table 4.2- ( vi ), the true value of
p = 0.9 and the p = 0.55, the p-value is 0.24 under the same level of p-value. This
numerical evidence shows that the estimates away from the true value is more likely
to be rejected when the true value of p is getting closer to zero. When we decrease

the tolerance of confidence to 0.05 or 0.1, there is no crucial difference of p-value

146

whatever the true value of p is.

4.3.2 The Model With Exogenous Variables

We put an end to the simulation with including an exogenous variable 131-t. We
still hold the strict exogeneity assumption. From the likelihood function (4.12), we

construct the log liklihood function as follows:

 

oo _ 2
log/ [f TEGXPlyit( Pit/zy—t 1+ xit3+fla+fi0az le]e—z dz. (4.18)

‘00 1(1—exppyi,t—l+zit/3+Ha'l'f0'a

where pa = a0 + a1 yio + 5,02. We maximize the sum of the objective function

(4.18) away from the constant term across i=1.. . ., N as follows:

T exp [yit(p yi,—t l + mitt/5 + l-La + $0.021)”

maleog ij H1

i=1 —exp(p yl,t—1 + $268 + “a + faazij)

 

(4.19)

According to Assumption 4.3 and 4.4, we generate the 5,1 and c, as the former
model do. We report the results in Table 4.5 - 4.8. Because we assume that 51:“ is
strictly exogenous, it doesn’t matter form which logic distribution :rz-t generates. We
assume the :13“ is continuous variable coming from the standard normal distribution.
Table 4.6 shows that except the 6 the other estimates have the similar property of the
former model. In Table 4.6, 6 is not significantly different from zero. Under the power
001,005 and 0.10, the hypothesis of H0 : 6 = 0 can not be significantly rejected when
the true value of 6 is 0.15. We calculated its relevant p-values of the test are 0.006,
0.036 and 0.09 under the power 0.01, 0.05, and 0.1 in respective. It might be the fact
that we assume the conditional mean of unobserved effects a,- = 00 + al yio + (12 25,-,
where the Ti: 71-. Zia-117%- The 02 dominates the effect of :13“ and accounts for most

of its effect. This can be explained by the fact that the hatag is 0.523 while the true

147

value of (12 is 0.35. We calculate the p-values of H0 : 0’2 2 0 are 0.3640, 0.6180 and
0.7640 under the power 0.01, 0.05, 0.1, in respective. That is we must pay much

attention on the specification of h(a.,-|a:,:T, . . . ,x,1,y,~0).

4.4 Empirical Example

Statistical models developed for analyzing cross-sectional data essentially ignore
individual differences and treat the aggregate of the individual effect and the omitted-
variable effect as an incidental event. In this section we use the data from Vella and
Verbeek (1998) to study the status of labor union membership. Such a Panel data
make it possible, through the knowledge of the intertemporal dynamics of a worker
who joins the labor union, to separate a model of individual behavior from a model
of average behavior of a group of individuals. In particular, we might assume that
the heterogeneity across cross-sectional units is time-invariant, and these individual-
specific effects are captured by decomposing the error as a,- + 5,1. We always treat
a,- as random to prevent the problem of incidental parameters. And the application
of conditional maximum likelihood estimation into the model make it possible to do
without restrictions on unximzio.

The existence of such unobserved time-invariant components allows individuals
who are homogenous in terms of their observed characteristics to be heterogenous in
response probabilities, F (3m)- For example, heterogeneity implies that the sequential-
participation behavior of a worker, F(union¢1), Within a group of observationally
homogenous worker differs systematically from the average behavior of the group,
f F (union.,~t)d H (alunionio), H(a|1mion,~0) gives the population probability for a con-
ditional the initial status of labor-union membership.

We use the data from Vella and Verbeek (1998) to study the conditional maximum

148

likelihood estimator in estimating dynamic logit model using observations draw from
a time series of cross sections. These data are for young males taken from the National
Longitudinal Survey (Youth Sample) for the period 1980-87. We estimate a dynamic
model for labor union status. Each of the 545 men in the sample worked in every
year from 1980 through 1987. When a worker is a member of a labor union we set
the status variable is one; when a worker is not a member of a labor union the status
variable is zero. We examine the response probability of dependent variable, union
membership over time series of cross section. For example, how do union membership
of the past affect the probability of keeping the labor union membership at present,
the amount of state dependence. We express the corresponding latent variable model
as that union}, 2 p unionmq + a,- +c,- and we set up a logit model under Assumption
4.1 and 4.2 as follows:
P(un'ion,-t : llun'ioni,t_1, . . . ,unionio, ai)
= P(€,-t > —(p unionigpl + a,)|union,-,t_1, . . .,um'on,—o,a,-) (4.20)
= 1 - A(-—(p unionthl + a,))

: A(p unionmq + a,),

1 , union; > 0
where union“ :
0 , otherwise.

From equation (4.20), the partial effect of unionthl on the response probability
is (A(p + a2) — A(a,)). Therefore, the conditional partial effect of union,,t_1_ on the
response probability depends on the unxiomhl through the quantity g(p unionmn +
a,), meaning the difference of A(p uniomhl + a1) with respective to mation,-$4,
A(p + a.) — A(ai). In Table 4.9, through the CMLE the estimates of p, 00, and 01

are 1.4923, ~3.2775, and 2.669. These estimates are significantly different from zero.

The estimated mean of unobserved heterogeneity a,- is (—3.2775 + 2.669 - uniomo).

149

d,=E(a,-|union,0) is greater than -3.2775 and less than -0.6085.

That is , empirically, the individual unobserved heterogeneity of a worker tends to
decrease the probability of keeping union membership, so the quantifying unobserved
heterogeneity have negative effect on the response probability of union membership
when we quantify the unobserved l'ieterogeneity. The significant amount of unob-
served effects means that previous membership of labor union appears to be a deter-
minant of future membership mostly because it is a proxy for temporally persistent
unobservables that determines the choice. If there is no other exogenous explanatory
variable, the result shows that the effect of temporally persistent unobservables that
determines joining the labor union or not is significant.

In other words, a worker participate in the union not just because he used to be
a membership of the union; on the contrary, he might join the union in accordance
with his own preference or some thing like the unobserved individual specific per-
sistent heterogeneity. The result is consistent to the empirical result of Chapter 2,
which the union membership accounts for not much of the wage rate. The estimated
response of the current union membership into the probability of keeping union mem—
bership in the future is measure by (A(p — 0.6085) — A(—0.6085)) instead of p when
unionio = 1. The estimated state dependence for a person with average of a,- is mea-
sured by the value of (A(1.49+[1,a) - A(fl.a)), equal to 0.1793, where E(Aa,-) = ao+al yo,
and yo = 54% 2:?) 31,0. Replacing [1.0 with the lower and upper bounds: 62, -3.278 and
E, -0.609, the range of state dependence effects is the interval of [~0.6665,-0.1088].
The upper bound is A(1.49 +170) — ME) and the lower bound A(1.49 +112) — ME).

The average partial effect of response probability is of primary interest, we calcu-

late the average partial effect of model (4.20). Since the unobserved heterogeneity has

rarely, if ever, natural measurements, it is unclear what value we need to plug in for

150

a. A suggested solution into it is a0 = E ((1,) = (10 +01 yo. Under Assumption 4.2, the

distribution of a0 is Normal(o'0 + on yo, 03) and thus its relevant density function is

 

 

f(aol = «22—00 exp(—1/2(a0 _ ((183: (11 yo) )2). The estimated average partial effect.
of response probability with respective to mean of heterogeneity across i is calculated
by the following:

/ [Am + a1— A<a>1f<a)da. (4.21)

(1 ‘(510 + @1310)

 

‘ 1
where f a = —— exp(——1 2 .
( ) 2m? / ( 00

distribution is limN_.oo f(a) : f(a). We calculate the integral of (4.21) from -00 to

)2). It is obvious that the limiting

00.

4.5 Conclusion

In this chapter we examine logit model as a specific non-linear case of theoretical
framework of conditional maximum likelihood function in estimating non-linear, dy-
namic, unobserved effects panel data models with feedback (Wooldridge 2000). We
make use of a joint density conditional on the strictly exogenous variables and the
initial condition. Because we model the density of the unobserved effect conditional
on the initial condition and exogenous variables, this is different from the treatment
of initial condition as fixed. It is feasible to construct the conditional MLE and the
relevant asymptotic property hold as the cross section sample size increase.

The simulation shows that when the true value of p is closer to zeros, the estimates
away from the true value of parameter is more likely to be rejected and is less likely
to be rejected, vice versa when the true value is getting closer to 1. The empirical
evidence shows that the unobserved heterogeneity have the negative impact on the

willingness to have a union membership. We have considered the important problems

151

of estimating the partial effects at the average value of the unobserved heterogeneity
and the partial effects averaged across the distribution of the unobserved heterogene-
ity in the empirical.

Our approach is parametric method, but the auxiliary conditional density can be
arbitrarily modeled in a flexible way leading to a strait forward parameterization that
can be estimated using standard software. If we want to extend to consider the ex-
ogenous variable, we can use the suggestion of Wooldridge (2000) in which we might
carefully specify the conditional density function of h(a.,:|'w,:, yio), where 11),: including
exogenous variables and strictly exogenous variables, ac,- and 2,. Using the specific
case of nonlinear model, this chapter shows that the idea of specifying a conditional
distribution for the unobserved heterogeneity given the initial conditions should prove

useful for analyzing the dynamic logit model with unobserved effects.

152

6

= (p,0.2,0.4, v1.2)
60 = (p0,0.2,0.4, \/1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\p‘f 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.1200 0.0700 0.0360 0.0240 0.0040 0.0080
0.05 0.2720 0.2040 0.1340 0.0820 0.0500 0.0440 0.25
0.10 0.3980 0.2800 0.2140 0.1440 0.1040 0.0880
P\ pi. 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.0100 0.0200 0.0460 0.0740 0.1020 0.1760
0.05 0.0520 0.0800 0.1140 0.1960 0.2740 0.3820 0.25
0.10 0.1060 0.1260 0.2140 0.2840 0.4080 0.5320
p\694 0.60 0.65 0.70 0.75 0.80 0.85 p
0.01 0.2620 0.3680 0.4960 0.6540 0.7480 0.8320
0.05 0.5160 0.6720 0.7620 0.8400 0.9080 0.9420 0.25
0.10 0.6800 0.7640 0.8480 0.9100 0.9460 0.9760
P\ 60—. 0.90 0.95 1.00 p
0.01 0.8980 0.9380 0.9700
0.05 0.9760 0.9800 0.9980 0.25
0.10 0.9800 1.0000 1.0000

Repetitions=500, é : 35—0 2500 91 ,/1._2 2 1.0964

(ii)

153

i=1 J’

 

6 :(p,0.2,0.4,\/1.2)
60 =(p0,0.2,0.4,\/1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\p94 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.5860 0.4620 0.3700 0.2400 0.1640 0.1000
0.05 0.8140 0.7120 0.6020 0.4900 0.3780 0.2680 0.5
0.10 0.8800 0.8300 0.7380 0.6140 0.5160 0.3860

P\ (’94: 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.0540 0.0300 0.0240 0.0100 0.0080 0.0140
0.05 0.1760 0.1140 0.0660 0.0500 0.0480 0.0540 0.5
0.10 0.2760 0.1860 0.1320 0.0880 0.0880 0.1020

P\ ”of 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.0240 0.0400 0.0640 0.1080 0.1640 0.2420
0.05 0.0780 0.1240 0.1820 0.2840 0.3960 0.5060 0.5
0.10 0.1460 0.1940 0.2940 0.4060 0.5300 0.6360

P\ p94: 0.9 0.95 1 p
0.01 0.3820 0.4720 0.6120

0.05 0.6280 0.7540 0.8400 0.5
0.10 0.7640 0.8420 0.9000

Repetitions=500, 6 2: 57110 23:01 6;, m 2 1.0954

(iii)

154

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 =(p,0.2,0.4,\/13)

00 =(p0,0.2,0.4,\/1—.2)
P\"O—J: 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.9280 0.8820 0.8360 0.7620 0.6640 0.5700
0.05 0.9760 0.9620 0.9340 0.8960 0.8500 0.7800 0.75
0.10 0.9920 0.9780 0.9700 0.9420 0.9020 0.8540
P\ ”95‘ 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.4500 0.3500 0.2480 0.1640 0.0940 0.0440
0.05 0.6940 0.5840 0.4860 0.3860 0.2680 0.1760 0.75
0.10 0.7880 0.7100 0.6040 0.5000 0.3960 0.2860
P\pL: 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.0240 0.0140 0.0080 0.0100 0.0120 0.0240
0.05 0.1100 0.0600 0.0440 0.0420 0.0480 0.0780 0.75
0.10 0.1940 0.1360 0.0900 0.0820 0.1000 0.1440
P\ ”of 0.9 0.95 1 p
0.01 0.0300 0.0600 0.1080
0.05 0.1340 0.1820 0.2620 0.75
0.10 0.2060 0.2780 0.3800

Repetitions=500, 6: 500 30016;, \/—2 2 1.0954

 

(W)

155

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 =(p,0.2,0.4,\/1.—2)

60 =(p0,0.2,0.4,,/fi)
P\”°—» 0.00 0.05 0.10 0.15 0.20 0.25 p
0.01 0.9780 0.9600 0.9300 0.8940 0.8520 0.8040
0.05 0.9960 0.9940 0.9820 0.9660 0.9360 0.9040 0.9
0.10 0.9980 0.9960 0.9940 0.9880 0.9720 0.9400
P\ ”94: 0.30 0.35 0.40 0.45 0.50 0.55 p
0.01 0.7360 0.6540 0.5480 0.4340 0.3120 0.2400
0.05 0.8700 0.8220 0.7540 0.6900 0.5940 0.4700 0.9
0.10 0.9140 0.8780 0.8300 0.7600 0.7000 0.6160
P\p‘f 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.1680 0.0880 0.0540 0.0180 0.0160 0.0120
0.05 0.3500 0.2720 0.1800 0.1000 0.0620 0.0400 0.9
0.10 0.4900 0.3640 0.2820 0.1920 0.1220 0.1020
P\ p94: 0.9 0.95 1 p
0.01 0.0100 0.0140 0.0300
0.05 0.0540 0.0680 0.1040 0.9
0.10 0.0960 0.1180 0.1620

Repetitions=500,6— 57,0250“ 0* [221.0954

(V)

156

J'lj’

 

0 = (p, 02,04, «1.2)
60 = (p0,0.2,0.4,\/1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\pO—f 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.9700 0.9600 0.9500 0.9100 0.9000 0.8300
0.05 1.0000 1.0000 0.9800 0.9600 0.9600 0.9300 0.95
0.10 1.0000 1.0000 1.0000 0.9900 0.9700 0.9600
P\ p“ 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.7700 0.7100 0.6300 0.4800 0.3000 0.2000
0.05 0.9000 0.8500 0.7900 0.7300 0.6500 0.5300 0.95
0.10 0.9400 0.9000 0.8600 0.7900 0.7300 0.6700
P\p‘L. 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.1600 0.1000 0.0500 0.0100 0.0100 0.0100
0.05 0.3400 0.2400 0.1700 0.1000 0.0600 0.0200 0.95
0.10 0.5800 0.3800 0.2800 0.1800 0.1200 0.1000
P\ ’00—» 0.9 0.95 1 p
0.01 0.0100 0.0100 0.0200
0.05 0.0500 0.0600 0.0900 0.95
0.10 0.0700 0.0900 0.1100

Repetitions=500, 6 = 566 2:30:01 6;, \/1._2 2 1.0954

(vi)

157

 

0 = (p,0.2,0.4, 61.25)

90 = (p0, 0204, 61.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\p‘l» 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.1260 0.0800 0.0300 0.0140 0.0040 0.0000
0.05 0.3020 0.1900 0.1320 0.0860 0.0340 0.0280 0.25
0.10 0.3960 0.3100 0.1980 0.1420 0.1040 0.0720
P\pO—T 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.0000 0.0140 0.0380 0.0700 0.1140 0.2200
0.05 0.0400 0.0760 0.1320 0.2280 0.3360 0.4400 0.25
0.10 0.0900 0.1460 0.2420 0.3440 0.4460 0.5620
P\pO 0.60 0.65 0.70 0.75 0.80 0.85 p
0.01 0.3120 0.4260 0.5220 0.6460 0.7500 0.8340
0.05 0.5440 0.6580 0.7680 0.8400 0.9040 0.9480 0.25
0.10 0.6680 0.7760 0.8460 0.9060 0.9560 0.9760
P\p‘l»: 0.90 0.95 1.00 p
0.01 0.8960 0.9420 0.9700
0.05 0.9720 0.9920 1.0000 0.25
0.10 0.9920 1.0000 1.0000

Repetitions=500, 6 2 55—0 233:0, 6;, V175 2 1.1180

(ii)

158

 

0 = (p, 0.2, 0.4, 61.25)

90 = (p0,0.2,0.4, «1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\p°—+ 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.5560 0.4760 0.3820 0.2660 0.1720 0.1140
0.05 0.7920 0.6740 0.5740 0.4920 0.4020 0.2800 0.5
0.10 0.8680 0.8000 0.6840 0.5960 0.4980 0.4120
P\ p05 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.0580 0.0280 0.0140 0.0080 0.0040 0.0140
0.05 0.1900 0.1240 0.0740 0.0460 0.0400 0.0460 0.5
0.10 0.3000 0.2100 0.1460 0.1000 0.0700 0.1060
P\p‘f‘ 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.0200 0.0360 0.0680 0.1160 0.1860 0.2940
0.05 0.0820 0.1340 0.2040 0.3160 0.4260 0.5200 0.5
0.10 0.1520 0.2260 0.3300 0.4420 0.5320 0.6200
P\ p05 0.9 0.95 1 p
0.01 0.4060 0.5040 0.5940
0.05 0.6080 0.7240 0.8240 0.5
0.10 0.7440 0.8320 0.8980

Repetitions=500, 9 = $7, 23120;, 6135 2 1.1180

(111)

159

 

9 = (p, 02,04, 61.25)

90 = (p0,0.2,0.4, 61.25)

 

 

P\ ”‘14:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.9320 0.8820 0.8100 0.7340 0.6240 0.5380
0.05 0.9820 0.9660 0.9400 0.9000 0.8360 0.7500 0.75
0.10 0.9880 0.9820 0.9700 0.9440 0.9080 0.8480
P\ (’94: 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.4300 0.3240 0.2500 0.1400 0.0940 0.0560
0.05 0.6560 0.5700 0.4520 0.3620 0.2740 0.1620 0.75
0.10 0.7640 0.6700 0.5760 0.4680 0.3800 0.2760
P\ ”of 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.0260 0.1060 0.0040 0.0060 0.0100 0.0200
0.05 0.1060 0.0700 0.0440 0.0400 0.0400 0.0700 0.75
0.10 0.1840 0.1280 0.1040 0.0760 0.1040 0.1500
P\ ’00—): 0.9 0.95 1 p
0.01 0.0320 0.0600 0.1200
0.05 0.1340 0.2060 0.3040 0.75
0.10 0.2160 0.3200 0.4140

Repetitions=500, 6 = 5170 2500 6‘.‘ x/1—25 2 1.1180

iv

160

i=1 :1

 

0 = (p, 0.204, 61.25)

00 = (p0,0.2,0.4, «1.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\p°—»: 0.00 0.05 0.10 0.15 0.20 0.25 p
0.01 0.9860 0.9760 0.9600 0.9260 0.8720 0.8220
0.05 0.9940 0.9940 0.9880 0.9800 0.9640 0.9360 0.9
0.10 0.9980 0.9960 0.9940 0.9880 0.9820 0.9680
P\ p°—+ 0.30 0.35 0.40 0.45 0.50 0.55 p
0.01 0.7160 0.6040 0.5220 0.4140 0.3180 0.2240
0.05 0.8960 0.8360 0.7480 0.6420 0.5460 0.4480 0.9
0.10 0.9420 0.9120 0.8440 0.7700 0.6640 0.5620
P\ (’05 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.1600 0.0940 0.0460 0.0240 0.0140 0.0120
0.05 0.3520 0.2500 0.1740 0.1180 0.0720 0.0420 0.9
0.10 0.4700 0.3740 0.2680 0.1960 0.1400 0.0920
P\ ’00—): 0.9 0.95 1 p
0.01 0.0100 0.0120 0.0180
0.05 0.0340 0.0480 0.0740 0.9
0.10 0.0760 0.0940 0.1380

Repetitions=500, 6 = 5% Egg 6;, \/—175 2 1.1180

(V)

161

 

0 : (p,0.2,0.4, 61.25)

00 = 010,020.41, 61.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\p‘L»: 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.9920 0.9840 0.9700 0.9500 0.9300 0.8720
0.05 0.9960 0.9960 0.9940 0.9860 0.9740 0.9640 0.95
0.10 1.0000 0.9960 0.9960 0.9940 0.9880 0.9760
P\ ”2+: 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.8040 0.7120 0.5980 0.5120 0.4040 0.3020
0.05 0.9340 0.8960 0.8260 0.7460 0.6320 0.5340 0.95
0.10 0.9640 0.9420 0.9060 0.8400 0.7660 0.6540
P\ (’95 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.2200 0.1460 0.0920 0.0460 0.0200 0.0140
0.05 0.4400 0.3280 0.2440 0.1660 0.1080 0.0620 0.95
0.10 0.5540 0.4560 0.3580 0.2640 0.1860 0.1300
P\ ”of 0.9 0.95 1 p
0.01 0.0160 0.0100 0.0100
0.05 0.0340 0.0400 0.0520 0.95
0.10 0.0900 0.0700 0.0860

Repetitions=500, 6 = 54,5 2:500 01 m 2 1.1180

(vi)

162

i=1 1’

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 = (p, 0.15, 0.2, 0.4, 0.35, M)

90 2 (p0, 015,02, 0.4, 0.35, M)
P\p0—>= 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.5833 0.4467 0.3300 0.2167 0.1433 0.0767
0.05 0.8267 0.6967 0.6033 0.4867 0.3433 0.2367 0.5
0.10 0.8967 0.8333 0.7133 0.6167 0.4967 0.3633
P\ (’94— 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.0400 0.0233 0.0067 0.0067 0.0033 0.0100
0.05 0.1500 0.0867 0.0467 0.0367 0.0333 0.0367 0.5
0.10 0.2433 0.1567 0.1033 0.0733 0.0633 0.0767
P\ p94: 0.60 0.65 0.70 0.75 0.80 0.85 p
0.01 0.0167 0.0267 0.0467 0.0800 0.1567 0.2467
0.05 0.0567 0.1000 0.1700 0.2767 0.4033 0.5233 0.5
0.10 0.1033 0.1800 0.2900 0.4167 0.5467 0.6833
P\ ’00—): 0.90 0.95 1.00 p
0.01 0.3833 0.5100 0.6367
0.05 0.6600 0.7700 0.8533 0.5
0.10 0.7800 0.8600 0.9167

Repetitions=300, 9: 3—0—0 :33", 9;, ,/_2 2 1.0954

(ii)

163

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 = (p,0.15,0.2,0.4,0.35,\/1._2)

90 = (P0,0.15,0.2,0.4,0.35, J17)
P\p" 0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.9867 0.9767 0.9467 0.9167 0.8667 0.8233
0.05 0.9967 0.9933 0.9900 0.9800 0.9567 0.9267 0.9
0.10 0.9967 0.9967 0.9933 0.9900 0.9800 0.9700
P\ ”L 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.7533 0.6400 0.5267 0.4033 0.3067 0.2100
0.05 0.8733 0.8533 0.7733 0.6800 0.5600 0.4533 0.9
0.10 0.9333 0.8900 0.8600 0.7967 0.6967 0.5833
P\pEL. 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.1300 0.0767 0.0267 0.0133 0.0033 0.0067
0.05 0.3167 0.2333 0.1500 0.0967 0.0467 0.0300 0.9
0.10 0.4833 0.3467 0.2700 0.1767 0.1067 0.0700
P\ p94 0.9 0.95 1 p
0.01 0.0067 0.0067 0.0133
0.05 0.0233 0.0433 0.0833 0.9
0.10 0.0633 0.0967 0.1300

Repetitions=300, 6: 3——00 233016;, \/_—2_ 2 1.0954

(iii)

164

 

6

= (p, 0.15, 0.2, 0.4, 0.35, «1.25)

90 = (p0,0.15, 02,04,035, 61.25)

 

 

P\ (’95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.5533 0.4500 0.3533 0.2667 0.1767 0.1100
0.05 0.7533 0.6567 0.5700 0.4800 0.3733 0.2867 0.5
0.10 0.8467 0.7600 0.6600 0.5800 0.4967 0.3833
P\ (’05 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.0533 0.0267 0.0133 0.0067 0.0033 0.0033
0.05 0.1900 0.1167 0.0633 0.0300 0.0333 0.0533 0.5
0.10 0.2933 0.2000 0.1300 0.0933 0.0833 0.1167
P\ ”L: 0.60 0.65 0.70 0.75 0.80 0.85 p
0.01 0.0100 0.0367 0.0767 0.1367 0.2267 0.3233
0.05 0.0867 0.1500 0.2467 0.3367 0.4233 0.5333 0.5
0.10 0.1667 0.2500 0.3567 0.4433 0.5400 0.6600
P\ ”of 0.90 0.95 1.00 p
0.01 0.4067 0.5233 0.6133
0.05 0.6367 0.7100 0.8133 0.5
0.10 0.7233 0.8267 0.8993

Repetitions=300, 6 = 55-5 23:01 6;, x/T2—5 2 1.1180

(ii)

165

 

6

= (5,015,020.41, 0.35, 61.25)

90 = (p0,0.15,0.2,0.4,0.35, 61.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0.05 0.1 0.15 0.2 0.25 p
0.01 0.9933 0.9867 0.9767 0.9333 0.8733 0.7867
0.05 1.0000 1.0000 0.9967 0.9933 0.9800 0.9467 0.9
0.10 1.0000 1.0000 1.0000 0.9967 0.9933 0.9867
P\p°—> 0.3 0.35 0.4 0.45 0.5 0.55 p
0.01 0.7067 0.6100 0.5233 0.4033 0.2933 0.2133
0.05 0.8900 0.8233 0.7200 0.6500 0.5667 0.4467 0.9
0.10 0.9500 0.9067 0.8300 0.7333 0.6633 0.5733
P\pO—T 0.6 0.65 0.7 0.75 0.8 0.85 p
0.01 0.1433 0.0900 0.0600 0.0367 0.0200 0.0067
0.05 0.3500 0.2433 0.1600 0.1033 0.0733 0.0433 0.9
0.10 0.4633 0.3600 0.2567 0.1733 0.1167 0.0900
P\ p” 0.9 0.95 1 p
0.01 0.0067 0.0000 0.0067
0.05 0.0267 0.0267 0.0533 0.9
0.10 0.0667 0.0900 0.1467
Repetitions=300, 6 = $5 23:01 6;, m 2 1.1180

(iii)

166

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4.1: Model a ::H0 6— — 60, where p =0 ~ 0.95
= (p. 0.2, 0.4, v1.2)
90 = (,20,0.2,0.4, 61.2)
.. 6 do 021 6a
W) ( 1.2 x 10-3) (0.1998) (0.4053) ( 1.0933) (’0
0.01 0.0040 0.0080 0.0120 0.0040
0.05 0.0460 0.0260 0.0460 0.0540 0.00
0.10 0.0900 0.0600 0.1060 0.1020
P\6 (0.2453) (0.2036) (0.4049) ( 1.0916) p0
0.01 0.0080 0.0080 0.0080 0.0140
0.05 0.0440 0.0400 0.0560 0.0420 0.25
0.10 0.0880 0.0760 0.1120 0.0840
P\6 (0.4918) (0.2068) (0.4046) (1.0925) p0
0.01 0.0080 0.0120 0.0120 0.0100
0.05 0.0480 0.0400 0.0460 0.0520 0.5
0.10 0.0880 0.0820 0.1040 0.1060
P\6 (0.7442 ) (0.2073 ) (0.4067) ( 1.0937) p0
0.01 0.0100 0.0120 0.0140 0.0040
0.05 0.0420 0.0380 0.0440 0.0520 0.75
0.10 0.0820 0.0900 0.1220 0.1060
P\9 ( 0.8915) ( 0.2087) (0.4097) ( 1.0967) p0
0.01 0.0100 0.0100 0.0120 0.0020
0.05 0.0540 0.0480 0.0600 0.5500 0.9
0.10 0.0960 0.0980 0.1060 0.8500
P\6 (0.9413) (0.2093) (0.4096) ( 1.0986) p0
0.01 0.0100 0.0200 0.0100 0.0100
0.05 0.0600 0.0600 0.0800 0.0300 0.95
0.10 0.0900 0.0800 0.1700 0.0800
. . ‘ 500 ..
Repetltions=500, 6— 50—0 2']; 1 62" \/1.2 2 1.0954

167

 

Table 4.2: Model a: H0 : p = p0, where p =0 ~ 0.95

9 = (p, 0.2, 0.4, \/1.2)

90 = (p0, 0.2, 0.4, \/1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\p‘L. 0 0.05 0.1 0.15 0.2 0.25
0.01 0.0040 0.0100 0.0200 0.0380 0.0680 0.1160
0.05 0.0460 0.0460 0.0720 0.1180 0.1800 0.2760
0.10 0.0900 0.1020 0.1240 0.2040 0.2860 0.4200
P\ (’94 0.30 0.35 0.40 0.45 0.50 0.55
0.01 0.1740 0.2720 0.4040 0.4920 0.6630 0.7600
0.05 0.4120 0.5000 0.6540 0.7680 0.8560 0.9220
0.10 0.5060 0.6460 0.7760 0.8860 0.9260 0.9480
P\ ’00—. 0.60 0.65 0.70 0.75 0.80 0.85
0.01 0.8580 0.9160 0.9480 0.9740 0.9940 0.9980
0.05 0.9480 0.9740 0.9940 0.9880 1.0000 1.0000
0.10 0.9740 0.9940 1.0000 1.0000 1.0000 1.0000
P\ ’00—? 0.90 0.95 1.000

0.01 1.0000 1.0000 1.0000

0.05 1.0000 1.0000 1.0000

0.10 1.0000 1.0000 1.0000

Repetitions=500, 6 = 566 2:30:01 6;, \/1.2 2 1.0954

(1)

168

 

 

 

Table 4.3: Model b 2H0 : 6 = 60, where p =0 ~ 0.95

= (p,0.2, 0.4, 61.25)
00 2 (p0, (12,114, V 1.25)

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\é la —3 do 021 a}, P0
( —6.5 x 10 ) ( 0.2253 ) ( 0.3803 ) ( 1.0699)

0.01 0.0040 0.0060 0.0120 0.0200

0.05 0.0400 0.0440 0.0460 0.0760 0.00

0.10 0.0980 0.0960 0.0940 0.1540

P\6 (0.2392) (0.2253) (0.3878) (1.0733) p0

0.01 0.0000 0.0080 0.0120 0.0200

0.05 0.0280 0.0400 0.0480 0.0800 0.25

0.10 0.0720 0.0800 0.0780 0.1280

P\6 (0.4894) (0.2284) (0.3872) (1.0741) p0

0.01 0.0040 0.0080 0.0140 0.0180

0.05 0.0400 0.0340 0.0480 0.0740 0.5

0.10 0.0700 0.0880 0.0860 0.1400

P\6 (0.7357) (0.2326) (0.3874) (1.0778) p0

0.01 0.0060 0.0080 0.0100 0.0180

0.05 0.0400 0.0420 0.0520 0.0660 0.75

0.10 0.0760 0.0900 0.0940 0.1220

P\6 (0.8900) (0.2304) (0.3872) (1.0770) p0

0.01 0.0100 0.0100 0.0060 0.0200

0.05 0.0340 0.0420 0.0460 0.0780 0.9

0.10 0.0760 0.0920 0.0900 0.1200

P\6 (0.9379) (0.2322) (0.3905) (1.0794) p0

0.01 0.0100 0.0080 0.0040 0.0160

0.05 0.0400 0.0440 0.0480 0.0820 0.95

0.10 0.0700 0.0900 0.0940 0.1220

Repetitions=500, 6 = 35—0 2500 6"? \/1.25 2 1.1180

169

7:1 a?

 

Table 4.4: Model b: Ho : p = p0, where p =0 ~ 0.95

9 = (p, 0.2, 0.4, 61.25)
90 = (p0,0.2,0.4, 61.25)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\pQJ‘ 0 0.05 0.1 0.15 0.2 0.25
0.01 0.0040 0.0080 0.0160 0.0440 0.0820 0.1260
0.05 0.0400 0.0620 0.0880 0.1340 0.2200 0.3200
0.10 0.0980 0.1180 0.1520 0.2340 0.3220 0.4680
P\ ()9: 0.30 0.35 0.40 0.45 0.50 0.55
0.01 0.2140 0.3100 0.4480 0.5340 0.6380 0.7420
0.05 0.4640 0.5440 0.6520 0.7520 0.8460 0.9040
0.10 0.5520 0.6720 0.7640 0.8520 0.9100 0.9540
P\ ("LT 0.60 0.65 0.70 0.75 0.80 0.85
0.01 0.8360 0.8980 0.9520 0.9700 0.9920 0.9960
0.05 0.9540 0.9720 0.9920 0.9960 1.0000 1.0000
0.10 0.9720 0.9920 0.9960 1.0000 1.0000 1.0000
P\ (JO—f 0.90 0.95 1.000

0.01 1.0000 1.0000 1.0000

0.05 1.0000 1.0000 1.0000

0.10 1.0000 1.0000 1.0000

 

 

 

 

 

 

 

Repetitions=500, 9 = 5150 230:”, 9;, «1.25 2 1.1180
i

170

Table 4.5: Model c :Ho : 6 = 60, where p =0 ~ 0.9

9 = (p, 0.15, 0.2, 0.4, 0.35, \/1.2)
90 = (p0, 0.15, 0.2, 0.4, 0.35, «1.2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\‘9 P B 020 d1 692 5a P0
(-0.0108) (0.1449) (0.2085) (0.4093) (0.3741) (1.0886)

0.01 0.0000 0.0000 0.0100 0.0233 0.0100 0.0067

0.05 0.0267 0.0667 0.0367 0.0633 0.0600 0.0200 0.00

0.10 0.0900 0.0833 0.0867 0.1267 0.1200 0.0567

P\6 (0.4884) (0.1466) (0.2086) (0.4140) (0.3717) (1.0876) p0

0.01 0.0033 0.0100 0.0100 0.0133 0.0133 0.0033

0.05 0.0333 0.0300 0.0400 0.0700 0.0600 0.0333 0.5

0.10 0.0633 0.0833 0.0733 0.1300 0.1133 0.0600

P\6 (0.8901) (0.1471) (0.2129) (0.4141) (0.3670) (0.10910) p0

0.01 0.0067 0.0067 0.0100 0.0200 0.0167 0.0033

0.05 0.0233 0.0500 0.0433 0.0767 0.0433 0.0333 0.9

0.10 0.0633 0.0833 0.0900 0.1167 0.1000 0.0067

Repetitions=300, 6 = 5% 231016;, N 2 1.0954

0,- : Normal

171

 

Table 4.6: Model (:2 H0 : p 2 p0, where p =0 ~ 0.9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 = (p, 0.15, 0.2, 0.4, 0.35, «1.2)
60 = (p0,0.15,0.2,0.4,0.35,\/1.2)

P0

P\ —» 0 0.05 0.1 0.15 0.2 0.25
0.01 0.0000 0.0067 0.0100 0.0400 0.0567 0.1067
0.05 0.0267 0.0433 0.0633 0.1200 0.1900 0.2933
0.10 0.0900 0.0933 0.1400 0.1933 0.3033 0.4500
P\ p21 0.30 0.35 0.40 0.45 0.50 0.55
0.01 0.1900 0.2700 0.4167 0.5533 0.6767 0.7967
0.05 0.4333 0.5833 0.7000 0.8067 0.8600 0.9100
0.10 0.5933 0.7033 0.8100 0.8667 0.9100 0.9533
P\ ”L 0.60 0.65 0.70 0.75 0.80 0.85
0.01 0.0.8600 0.9000 0.9500 0.9767 0.9933 0.9967
0.05 0.9533 0.9767 0.9967 0.9967 1.0000 1.0000
0.10 0.9833 0.9967 1.0000 1.0000 1.0000 1.0000
P\ pi, 0.90 0.95 1.000

0.01 1.0000 1.0000 1.0000

0.05 1.0000 1.0000 1.0000

0.10 1.0000 1.0000 1.0000

Repetitions=300, 0 = 37116 23:01 0;, v1.2 2 1.0954

a,- : Normal

0)

172

 

 

 

Table 4.7: Model d 2H0 : 6 = 60, where ,0 =0 ~ 0.9

6 = (p, 0.15, 0.2, 0.4, 0.35, \/1.25)
00 = (p0,0.15,0.2,0.4,0.35, \/1.25)

 

 

 

P\6

A

8

 

 

 

 

 

 

 

 

 

 

 

 

 

P do 021 022 dc P0
(0.0075) (0.1454) (0.2225) (0.3845) (0.3777) (1.0678)
0.01 0.0000 0.0000 0.0033 0.0067 0.0067 0.0167
0.05 0.0267 0.0700 0.0433 0.0500 0.0400 0.0700 0.00
0.10 0.0767 0.1133 0.0867 0.0967 0.0967 0.1433
P\6 (0.4837) (0.1425) (0.2275) (0.3909) (0.3871) (1.0790) p0
0.01 0.0033 0.0533 0.0333 0.0500 0.0333 0.0533
0.05 0.0333 0.0533 0.0333 0.0500 0.0333 0.0533 0.5
0.10 0.0833 0.1233 0.0733 0.0933 0.0967 0.1067
P\6 (0.8909) (0.1441) (0.2307) (0.3825) (0.3779) (1.0789) p0
0.01 0.0067 0.0200 0.0033 0.0233 0.0033 0.0133
0.05 0.0267 0.0667 0.0267 0.0433 0.0367 0.0433 0.9
0.10 0.0067 0.1167 0.0767 0.1033 0.0700 0.0900
Repetitionsz300, 6 = 3170 23:01 6;, v1.25 9: 1.1180

a,- : N on-normal

173

 

Table 4.8: Model (1: H0 : p : p0, where p :0 ~ 0.9

6

= (p,0.15,0.2,0.4,0.35, «1.25)
00 : (p0,0.l5,0.2,0.4,0.35, 1/125)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P\pO—T 0 0.05 0.1 0.15 0.2 0.25
0.01 0.0000 0.0133 0.0167 0.0333 0.0667 0.1167
0.05 0.0267 0.0400 0.0767 0.1333 0.2100 0.3267
0.10 0.0767 0.0900 0.1400 0.2200 0.3400 0.4600
P\ (’05 0.30 0.35 0.40 0.45 0.50 0.55
0.01 0.2000 0.3033 0.4367 0.5233 0.6500 0.7600
0.05 0.4567 0.5400 0.6600 0.7667 0.8700 0.9100
0.10 0.5433 0.6667 0.7767 0.8733 0.9133 0.9600
P\ 60—»: 0.60 0.65 0.70 0.75 0.80 0.85
0.01 0.8600 0.9100 0.9467 0.9833 0.9933 0.9967
0.05 0.9467 0.9867 0.9933 0.9967 1.0000 1.0000
0.10 0.9867 0.9983 0.9967 1.0000 1.0000 1.0000
P\p‘f 0.90 0.95 1.000

0.01 1.0000 1.0000 1.0000

0.05 1.0000 1.0000 1.0000

0.10 1.0000 1.0000 1.0000

Repetitions=300, 6 = 3% 23:01 63-“, \/ 1.25 c: 1.180

ai : Non-normal

(i)

 

 

Table 4.9: Empirical Evidence for Labor Union Membership, Period21980 ~ 1987

 

 

 

coefficient. p“ do 021 60
CMLE 1.4923 -3.27750 2.6690 1.9997
t-statistics (9.498) (18.942) (8.993) (12.036)

 

 

 

 

 

 

 

174

Appendix A

Conditional mean and variance

The appendix A is to slove the E(y,-|y,—0)Tx1 and Var(y,~|y,-0)TxT of (2.19). The
regression equation (2. 1) can be rewritten in terms of yio and the errors as ya = pt 920+
—11—:_—%ta,- + 23:1,,0’” Ei,t—j+1- According to assumptions on a“ and a,- in Chapter 2,
we have E(az~|yio) = 0'0 + ail/1‘0 and hence E(yuly¢0) = Pt 920 + lI:_:Epi(01 + 10910) for
t=1,. . .,T. We can write the expectation of T observaitons on individual 2' conditional
on ym as follows:

E(yily10): (00 +(01+P)yz‘0 ll;_l%tC¥0 +(‘11;_'%t01+/0t)yio )Txl,
wheret=1,...,T. V
90/20) E Var(y,|y,~0)
:- E ((3/2' — E(yz~ly.~o))(y4 — E(yilyi0)),lyi0)
= E(y.y£ly.-o) — E(yilyiO)E(yilyiO),
0211 . . . 021T
ail/1'0) = 5 3 9
can . . . 0277
Where wtt = E(yz'21lyz'0) “' E(yuly10)2 and (Us: = E(yisyitlyi0) _ E(yislyiO)E(yitly70)' To

obtain all the elements of the covariance matrix, we need to solve the E (yfilyio) and

175

E(y,-s'y,t|y,:0) for t, s = 1,. . .,T. We write the results as follows:

1_ t 4) _ 2t
a)“ : (‘1—%)2 0; +(11—_'%T)052,t:1,...,T,

 

0),, = (11 _ppll '11:)03+p"‘3l(11——_%2;) 5,37éts t———1,. .,T.
Because
E6360) 2 E((Pt 3110 + 211E465}, + 23:1:pj_1€i,t—j+1)2|yio)
= p843 + <%;_%)2<E<a.ly.o>2 + E<c?ly.o>)+
E((Z;=1pj’15,,t_j+1)2|y,0)
: .0 2ty10+(lr_—%)2 ((00+011yz‘.)02 +05) )+—£T(E(5221lyio))

t

1— 1—
: ,or“’t3,/,20+(-I—'%)2((C10+O‘1’yz‘0)2 +0 9+ m0?)

1— p
E(yitly10)2 = (P 3110 + of:%( 00 + 01y10))

E(yz'2tlyi0) — E(yitly10)2

1_ t 1__ 2t
: PgtJl‘Qo + 01—} p)2((010 + 0'1910)2 + 03) + —'0—71 _ p 03—
t

P2t3/120 — (11::%)2((a0 + 019402)

= <%;_%’>202 + (Lg/£903.

By similar manipulation, we can express the E(y,~)y,~3|y,0) - E(y,~t|y,-0)E(y,-s|yio)

 

in terms of parameters p, 03 0E ,and time period t and s as (11:!) 1 — ’0 )0: +

2 p T-p
_3 1— S
P" '(1—_'%r)03-

If the model includes the exogenous variable 23,-) for t = 1, . . ., T, the conditional

mean will be

E(yily10,~$1) : ‘11—:%C10 + (flag + p)y,-O + 23:1P’_1$i,t—j+1,3 , where

Txl
t = 1, . . . ,T. It, however, can be easily shown that Var(y,t|y,-0, 110,-) of (2.32) in which

:13,- is assumed to be strictly exogenous, is the same as that of the basic model without

176

no other regressors beyond y,,t_1. The illustration is in the following.

E(yz‘2tly1'0~:ri)

Z E((Pt 3140 + 1: ptaz' + ZE=1_Pj"](1Ti,t—j+11’3 + 51‘,t—j+1))2lyiOa 513:)

= 623/30 +(‘11'::pp—t)2(E(ailyiOa$i)2 + E(C?lyio, 5174)) +
2(p‘y,0 + lf::%t(13((12'l’312'0, SENSE-=1 PF] E(5Ei,t—j+1)/’3)) +
51(2):) PF] (Ii,t—j+118 + 57,t—j+1))2ly20~ 332'.)

= thyEO + (1113;7th + (113/2'0 + 002713—21)2 + 0721)+ ll:_—’%2;(E(5221lyio, 5134)) +
2(ptyz'0 + lI;_'%t((0‘0 + 01940 + (12354)(Z;=1Pj—1E(~’172',t—j+1)/3)) +
(23:1/fl_1E($1,t—j+1)73)2

= pz‘yi) + $392040 + my... + a)? + 03) + 1,33%: +
2(Ptyz‘0 + 17%p;t((010 + 01920 + aZEiXZEEA pH E($¢"t—j+1),3)) +
(23:1pj—1E($i,t—j+l)fi)2'

E(yalyio, 372V

2 (Ptyio + $601026 + 0111/20 + 0252‘) + (23:1Pj_1E($i,t—j+1)3))2

= p248 + (‘11—:‘%t)2((00 + 01910 + mm
2(Pty10 + 1r—:%‘((ao + Oil/2:0 + QQEingzl [Oi—1 E($i,t-j+1)3)) +
(2321.07;1 E(5Fi,t—j+1.)3)2-
Therefore, E(y?tly,-0,2:,-) — E(y,t|y,0,:r,-)2 is equal to (£65202 + (flit-)0?

1 - p
BY similar manipulation, E(yityisly107Xi,max[t,s]) ' E(yitlyiOaXit)E(yislyiOaXis) is 938-

 

s t 2.9
ily proved to be equal to (11:pp 11:1?) )0: + p|‘*3|(11—:£p7)0€2.

For model (3.1) or (3.7), the conditional mean can be derived in the way similar

to the previous. Similarly, we assume that

Gill/2'0 N N ormal(a0 + 0130,03),

Emlyn (1,- ~ Normal(0, 052).

177

We can write the expectation of T observaitons on individual 2' conditional on yw
as follows:

1_5t
E(yilyi0):(a + a +6 , .. a +( a +6f) , ..) ,where
0 (1 )yo 11—_—5: 0 —5‘ 1 )yo TX]

t=1,...,T and 6,- E p+7(ao +0131“) +c,).
Q(Ll/2'0) E vaT(y-z'lyz'0)
= E ((91 - E(yily1‘0))(yi - E(yilyi0))’ly10)

= E(yiyilyi0) - E(yily10)E(yilyi0)l

(4)7“) . .. LUTT

where em = E(yi|yio) - 436.1100)? and wst = E(y.~syu|y.o) — E(yislyiO)E(yitlyiO)- The

elements of Q(y,0) are as follows:

 

1 .3 _ ¥ _8 _ .
cast: (1_6:1_5:)0'g+61t Wiggly—>052, SaétHS‘, t: ,...,T.
Because

_ 5? ‘._
43031.48) = Em: yio + lfiga + 23:54: ‘e.,._.-+1)2|y.o)
1 — 6.
Z 521,110+ +(T—__5:)2(E(ailyi0)2 + 5((1'?lyz'0))+
E((Zj_161j 15”— j+1)2ly:0)
6t2 62t

= 63‘9 930+(fig) ((00+alyio)+03)+—17(E( $096))
1- 5? 11—65t2

: 612ty220 +(1—:3:T)2((010 + 011910)2+ 0a)+ 3:42702 as

1— 6i
E(yitlyi0)2 = (5:910 + —_——5:(00 + 0413110))2

1 — 6-
62t9120++(T:3:)2((00 + 0‘1940)2)a

178

E(yi2tly2'0) " E(yulyz‘0)2

. 1—0. 1—62t
= 6,2‘y30 + (j _ .)2((010 + (1'1y70)2+ 0a)+ —7203 “

52ty 20 — (“—592 ((00 +Oly10)2)

 

—06’ 1—- 62‘
We can express the E(yztyisly70) ' E(yitly10) E(yisly70) in terms 0f parameters, ,0, 03,
__ s __ 23
0,2, and time period t and s as (11 _g'; i_ (203) + 6,“ S[(1—_—(:512r)0§.

I

If the model includes the exogenous variable (13,-, for t = 1, ..., T, we replace the

assumptions on a,- and 5,, with

.. . , , — 2
(It'll/10,551 N Normauao + 01310 + Xi (12:03),

Eitlyi, 0, ~ Normal(0, 052).

The conditional mean will be

E(yily10, 5132')

1—6‘ 1—

= —1—__—5:00 +( (4—5L01 + 6) )y,-0 + 2,- 153 1il?z',t—j+1,8 ’
Txl

where 6,- E p+'y(ao+o'1y,~0 +53, +q) and t = 1,...,T.

Var(y,)|y,0, 23,-) is different from that of the basic model without no other regressors
beyond y,,t_1. The illustration is in the following.
E(y?tlyi0, 332')
= E((5f 910 + 9,3357% + 23:16{_1(x1,t~j+16 + €4,t—j+1))2ly10,$1)
= 632730 + <1—:%§>2<E(a.1.40,62) + 12030.0. 6)) +

2(6fy,0 + E§<E(azlyioax6(ZJ-=163 1E(Ii,t—j+l)53)) +

E((Zj:162J 1(37 Fit— j+13 + 52':— j+1))2lyiOa 513;)

1—0 _ 6”
Z 63ty120 +(1—_—5:)2((O'0 ‘1' 01311-0 + 02:51)“ 02.)+ —12_(E (gill/1‘0, 5171)) +

179

1 — 6f _
2(55312'0 + 1—_3:((0‘0 + Oil/i0 + 02$'i)(Zj—161J 'E(:z:,-,t_j+1),z3)) +

(23-: 63‘E<rm+1>3>2

I 1— 6t _ 62t2
= 6f‘y30 + (fimao + my“) + xi>+ 610+ —so§+

2(6fym + i_}§§((00 + alyio + 02:13,)(Zj_16f 1E($i,t_j+1))8)) +

(z; ,5; ‘E(x,,,_,+l);3)2.
E(yitlyiOs 5131:)?
= (53110 + gig-R00 + alyiO + 0272') +(23:153—1E($i,t-1+1)fl3))2
—6,2‘y—,20+(11—j§’:7)2((oo + (113/,0 + 012.1“ )2)+

2(5f’yz’0 + 1%«0‘0 + (113/,0 + 0252-)(2321 56H1Ef$i,t—j+1).3)) +

(2; 163 ‘E(x,,,_,+1),3)2.

Therefore, E(y,2,|y,0,:r,) — E(y,t|yi0,x,~)2 is equal to (%—:—§§)20§ + (ma—)0;
By similar manipulation, E(y,ty,,|y,-0, Ximax[t,s]) - E(y,-t|y,-0, x,)E(y,-s|y,-o, X3) is easily
11335,: 3:303 +6? “(£5663-

_ 2'

Therefore, when the state dependence interacts with the unobserved effect, the

 

proved to be equal to (

autoregressive coefficient contains the unobserved effect and hence the conditional

mean and variance depend on ym and 35,-.

180

Appendix B

IV estimator for average
autoregressive coefficient across
population of unobserved

heterogeneity

In this appendix we prove that the IV estimator for dynamic model where the
state dependence depends on the unobserved effects is not consistent We define some
notations for polynomial in the process of proof for the simplicity of exposition as

follows:

1. Let b E Rn“ be a non-zero coefficient vector b :: (b0, b1, . . . ,bn). Denote Pn a

non-trivial polynomial of degree n :

Pn(;r,b) E b0 + 61:13 + 62 51:2 + - -- + bn :c" E erc].

2. For m E IR , [m] :2 {nl if n. _<_ m < 71+ 1 for some n E Z}.

181

According to the first difference equation of (3.1), we use the lagged Aythg as an

instrument for A y,,¢_1 and we can write the equation as follows:
E(Ay11y1,1_2) Z E(t91Ay1,1-1A3/1,1_2) + E(A€it AMT—2% (8'1)

where 19,- = p + '7 6,. We make the following assumptions:

 

 

 

Assumption B.1 : E(s,-t|y,-,t_1, . . .,yi0,a,) = O.

 

 

Assumption B2 : Var(5,tly,,t_1,...,y,«0) = 03.

 

 

Assumption B.3 : E(a,-|y,¢,t_1,...,y,-0) = E(a,). .

 

 

Assumption B.4 : Var(a,-|y,-,t_1,...,y,-0) = 0,2,.

 

 

 

 

 

 

Assumption B.5 : E(y,-0) 2 ago and Va’r(y,:0) = 050.

 

According to Assumption 8.1 and 8.3, equation (8.1) is equal to
E (Ay11y1,1—2) = 19 E(Ay1,1_1Ay1-,1_2) + 7 E(01Ay1,1_1y1,1_2), (B?)

where 19 = p + 711a. The 1V estiamtor is as follows:

21:1 21:1 Ayi,tAyi,t—2

21:1 23:1,} Ayijg—lAyu—z

= ,9 _,_ ’7 21:1 1:1 CiAyi,t—1Ayi,t—2
211::1 26:1 Ayi,t—1Ayi,t-2

2:1 1:1 AEitAyi,t—2

N T -
21:1 21:1 Ayi,t—1Ayi,t-2
where 19 = p + 71.10. Under Assumption 1 to 5, the probability limit of 191v is as

 

191v! :

+ (13.3)

 

 

follows:

. N T
131121 7v]? 21:1 21:1Ayi1tAyi1t-2

 

 

plim éjv =
N—ooo . N T
181.13.} 7%? 21:1 21:1 Ayi,t—1Ayi,t—2
- N T
7 R132: NIT 21:1 21:1 CiAyi,t—1Ayi,t—2
: 19 + ’ + (8.4)

. N T
13133 1111= 23:12.: Ayn—1661,:
. N T
181—1.1;} fi 21:1 Zt=lAEitAyi1t—2

. 1 N T '
13111;} W 21.—.1 21:1 Ayi.t—1Ayi,t—2

 

182

The proof of plim 317‘ 2:1 2;] ciAyi,¢_1Ayi,t_g 7f 0 is in the fowlling.
N—ooo

. N T T
B11111 VII Zi=1zt=1CiAyi,t—1Ayi,t-2 : Zt=3E(CiAyi,t-1Ayi,t—2)

'—"OO

T
: 21:3 E(Ciyi.t—1yi.t-2)_E(C‘iAyi2,t—3)

(A1) (A2)

(8.5)

According to Assumption 8.1 to BS, equation (8.5) is solved out as follows:

(A1) 2 23:3 (E ((21931th + Ci7’;l/22,t—2 + Ci (#a + Ci)yi,t—2 + Ci£i,t—1yi,t—2))

: 21:3 Efciflyiiza) + 2:3 Efcz'“/yz2,t—2) + 23:3 E(c,~ (#a + Ci)yi,t—2)
(B.6)

Putting the final expression of (B.6) into (A1) in (8.5), we can obtain the following

equation:

23:3 E(c,- Ayi,t~1 Ayn—2) : (19 — 1 — 7) 23:3 E (CiyEt—z) + 23:3 E (Q7yEt—2)

+ 23:3 E (Ci (#a + Ci)yi,t—2)
(37)

We can sermrately solve out the three terms of (8.7), 2:23 E(c,-yi2,t_2),

231:3 E ((iifil'i'yEt—Z) as well as 23:3 E (Ci (#0 + Gilt/m4)-

23:3 E [Gigi—2] : 23:3 E

 

_ 1 — 9?“? _ -_ 2
C,’ (19: 22/10 + mm + 23-22119: 15i,t—j-1) ]
= :23 020730 + 1.11§O>E(130>Pt_2<03, b) + 2:; 03131-2(03, d)

+ 2 2:3 pyoagpt_2(02, e).
(8.8)

By the manipulation similar to (B8), we can express the other terms, 23:3

E (cn'ygt_2) in terms of (t-2)-order polynomial of 03 and 2:3 E (Ci (110 + Ci)y,~,t_2) in

terms of (Lt—a—ljlorder polynomial of 03, and their relevant coefficients are function
2 2

2 - 1 N T 1 - ,
(mum/15,0, 00,05,0y0). Therefore, phi: W 21:1 21:1 ciAyi'tyl-¢_2 IS not zero when

T is fixed.

183

 

Similarly, we can prove that,

. y T
1 ~00

1 - N "
(2) Pllm w]? 21:1 2le Amt—191,14 ¢ 0'

~:oo

As we have shown, Assumption 8.1 to 8.5 can not lead to the fact that 31v is
consistent. Even more,if we make the extreme assumption that 0,, 5,1 and ym are
independent with each other, then 13111: fi 21:, 2;, c,Ay,-,t_1Ay,-,t_2 is not equal
to zero. It is obvious that such a strong assumption appears that many expectation
of terms in the right hand side of equation of (B7) vanished because any term in the
form of E [Ps((c:n'1€:tn’2y$'3))] is zero; there, nevertheless, exist such non-zero terms

E (Pm(c,-)cf ), where k is O or 1, since m > 2 and hence a standard IV method applied

to first-differenced equation can not consistently estimate the parameter of interest.

184

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185

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190