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Twp?"

 

 

This is to certify that the

dissertation entitled

SOME APPLICATIONS OF THE LAGRANGE MULTIPLIER
TEST IN ECONOMETRICS

presented by

Tsai-Fen Lin

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Economics

"P53“ 319%

Major professor

Date August 6, 1982

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

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SOME APPLICATIONS OF THE LAGRANGE HULTIPLIER TEST

IN ECONOHETRICS

BY

Tsai-Fen Lin

A DISSERTATION

Submitted to
Michigan State University
in partial Fulfillment of the requirements
For the degree OF

DOCTOR OF PHILOSOPHY

Department 0? Economics

1982

ABSTRACT

sour: APPLICATIONS or THE IAGRANGE MULTIPLIER TEST 1N ECONOME'IRICS
By

Tsai-Fen Lin

There are three kinds of tests for model specification - the Wald test. the
likelihood ratio test and the Lagrange multiplier test. They have the same
asymptotic power. Therefore. the choice among them depends on computational
convenience. Since the Lagrange multiplier test is based on the restricted esti-
mates. we choose the Lagrange multiplier test when estimation is easier in the

restricted model than in the unrestricted model.

Since the Lagrange multiplier test is not well known. and the derivation for
the test statistic is complicated. in this thesis. I develop the Lagrange multiplier
test statistic for some commonly used econometric models so that they can be
used readily by applied economists. These models include distributed lag
models. qualitative and limited dependent variable models. and stochastic pro-
duction and cost frontiers. In the distributed lag models. the Lagrange multi-
plier test statistic is shown to be asymptotically equivalent to the 1“ statistic in
testing the coeflicients of the lagged explanatory variables when they are added
to the restricted model. In Heckman’s sample selection bias model. the
Lagrange multiplier test statistic is asymptotically equal to the. square of the t
test statistic in testing the coeflicient of the correction term for the sample
selection bias when this correction term is added to the’restricted model. In
Poirier's partial observability model. the Lagrange multiplier test statistic is
equivalent to the explained sum of squares in a regression of residuals on a set

of regressors. In the stochastic production and cost frontiers models. the

'2' Tsai-Fen Lin
Lagrange multiplier test fails in some cases. and alternative tests are suggested.

In summary. the Lagrange multiplier test. except in a few cases. can be
used to test the adequency of the simple models. Since the simple model usually
involves a simple estimation method or less computational cost than the more

complicated alternative. the Lagrange multiplier test can be useful.

ACKN OWIEDGMENTS

I am deeply indebted to my advisor. Professor Peter Schmidt. for sparking
my interest in these topics and for all his guidance and help. I also thank other
members of my dissertation committee. William Quinn. John Goddeeris. and Paul

Menchik.

ii.

 

“‘-

,..

 

 

Chapter

I

II

III

TABLE OF CONTENTS

Introduction

1.1

Introduction

1.2 The LM Test

Distributed Lag Models

2.1

2.

PJ

2.

bid

2.

Introduction
The Geometric Lag Model

The Rational Lag Model

Conclusions

Qualitative and Limited Dependent Variable

Models
3.1 Introduction
3.2 Test of the Tobit Specification

against Cragg’s Extension 0? the Tobit Model

3.2.1 The Tobit Model and Cragg’s Extension

3.2.2 The LM Test

Test a? Sample Selection Bias

3.3.1 Heckman’s Sample Selection Bias Model
and His k lest

3.3.2 The LM Test OP Sample Selection Bias

Test 0? Independence in Poirier’s Partial

Observabilitg Probit Model

iii

page

he
[U m 0‘

[U
M

34

36

IV

3.5

3.4.1 Poirier’s Partial Observabilitg
Probit Model

3.4.2 The LM Test

Conclusions

Stochastic Production/Cost Frontiers

4.1

4.2

4.6

Introduction

Test 09 Zero Mode For the Technical
Inefficiencg in Stevenson’s Extension 0?

the ALS Model

4.2.1 Stevenson’s Extension 09 the ALS Model
4.2.2 The LM Test

Test 0? Allocative InePPiciencg in SL I Model
4.3.1 SL I Model

4.3.2 The LM Test

4.3.3 An Alternative Test

Test or Systematic Allocative IneFFiciencg

in SL II Model

4.4.1 SL II Model

4.4.2 The LM Test

Test of Independence between Technical
Inefficiencg and Allocative IneFFiciencg

in SL III Model

4.5.1 SL III Model

4.5.2 The LM Test

4.5.3 Alternative Tests

Conclusions

iv

41

47

4e

51

63

63

68

69

73

V Summarg and Conclusions

Notes

Appendix
Appendix
Appendix

Appendix

mommy

Appendix

Bibliography

74

B4
85

9O

97

CHAPTER I
INTRODUCTION

1. 1 Introduction

In statistics and econometrics. there are three basic principles for the con-
struction of test statistics for model specification. They are the Wald test
(Wald(1943)). the likelihood ratio (LR) test and the Lagrange Multiplier (LM) test.
Suppose there are two possible model specifications. one of which is a special
case of the other one under some restrictions. Let's call the special case the
restricted model. and the generalized case the unrestricted model. The Wald
test is based on the estimates from the unrestricted model. while the LM test is
based on the estimates from the restricted model. and the LR test is based on
both sets of estimates. These three principles yield tests which are equivalent in
large samples when the restrictions are true (see Silvey(1959)). Their small sam-
ple properties are unknown. except in special cases. Therefore the choice among
them will often be based on computational convenience. The LM test is very use-
ful in cases in which the restricted model is easier to estimate than the unres-
tricted model. This will often be the case when one is testing the adequacy of a
particular model. Then the null hypothesis is that a relatively simple model is
adequate. while the alternative is that a more complicated model is necessary.

The LM test permits a test of this hypothesis without having to estimate the

more complicated model.

Although the LM test was suggested by Aitchison and Silvey in 1958. it did
not receive much attention from econometricians until recent years. Therefore.

not many economists are aware of the LM test and its computational advantages

-2-

in many cases. It is the responsibility of the econometricians to introduce the
LM test to applied economists by developing LM test statistics for common
models in econometrics. The LM test has been applied successfully in testing for
a liquidity trap. autocorrelation. the error components model. seemingly unre-
lated equation systems and various non-nested hypotheses. (See Breusch and
Pagan(1980) for a survey and references.) In this thesis. I report the successful
application of the LM test to distributed lag models in chapter 2. to some quali-
tative and limited dependent variable models in chapter 3. and to stochastic

production/cost frontiers in chapter 4.

1.2 The L]! Test

Let 19 = (13,. ' ' - .d,)' be a set of parameters. L03) be the log-likelihood fuc-
tion. h(1$) = [111(6), - - - .h,(13)]' = 0 be a set of r restrictions. A = [A1. - - - .N]' be
a set of Lagrange Multipliers. 1(13) be the information matrix and n be sample
size. Define the Lagrangian function for the maximization of the likelihood sub-
ject to the restrictions as

L30“) = Lw) + Nh(1$)
A constrained maximum of L03) is obtained at a stationary point of LR (13). By

difierentiating LR(13.>\) with respect to u and A . we have the first order condi-
tions:
D03) + H; A = 0
h(13) = o (1.1)

’ 6h 13
where 0(1)) is the sxl vector. {65:1 . and H1, is the er matrix . [ 6’1: ) . By
. . . i ' i

solving eq.(1.1). we obtain the restricted MLE 3' and 7". When the restrictions are
in fact true (h (13) = 0). the restricted estimates 1? will tend to be near the unres-
tricted estimates. and 0(5) and X will tend to be near zero. It seems reasonable
to decide that h('6) = O is true if 3: is in some sense near enough zero. Aitchison
and Silvey (1958) proved that under the null hypothesis that h(13) = O . V723: is

asymptotically distributed as normal with mean zero and covariance matrix
~ --1 ~ ~

[h'g'n[1(13)]‘1 H3] where H; and [(13) are H, and I (13) evaluated at 1? respec-

tively. They suggested a test statistic which is based on the estimated Lagrange

Multipliers (All) and called this the Lagrange Multiplier test statistic:

LM test statistic = X'H;'[I(5)]-1H3X (1.2)
This statistic asymptotically follows a chi-square distribution with r degrees of

freedom when Ms) = 0 is true. The region of acceptance of the null hypothesis

-4-

h.(13) = O is X'H3'[I(5)]“H3Xs K. where K is determined by
Prob ( x3 5 K) = 1- significance level. .

Note that from eq.(1.1). 1135: = — 0(5). so eq.(1.2) can be rewritten as
LM test statistic = [0(5)]'[1(5)]-1[D(5)] (1.3)
The right hand side of eq.(1.3) is just Rao's score statistic (Rao(1947)). Hence

the LM test statistic is the same as Rao's score statistic. Since eq.( 1.3) is easier

to use. in the following chapters eq.(1.3) instead of eq.(1.2) will be used.

When 13 is partitioned into 2 subsets. 131 and 192. and the restriction under
test is that one of the subsets of parameters equals particular values. i.e.

Hon). = 73m- then we can establish a simpler form of the LM test statistic. From

eq.(1.1). 0(3) = —H3’X’. therefore.

 

 

 

 

[61. 912522., . . “11") W
6.61 - 6'61 6131 ,
6L T - 6111(13) dh.,.('0) ‘
6192 l at"; ‘ ' ' as”; ”A,
7' 6h 13 ~
2 61.1; )l‘j
= _ {=1 1 (1 4)
o a
6h 13 6h ~
where 61’ and —%—)—ar and —-L(—)- evaluated at 13 respectively.
GT; 5761; 61!;

BL

T: 0 because hj (13) is not a function of 132. Partitioning 1 (1’5) conformably.
z

eq.(1.3) becomes

6L ' -1! 6L

LM test statistic: [631211 I121 [501
21 22

u ' ~ ~ ~ - ~ -1 a
= [6 1 {[11 “112122 1121] [a—é‘] (1-5)
If I ('5) is block diagonal. the LM test statistic can be further simplified as

. . 6L _. M
= .6
LM test statistic [6131 In [6191 (1 )

-5-

When I (13) is difiicult to calculate. we can use the negative of the Hessian (

matrix of second derivatives ) or its limiting form to construct the LM statistic

6% is 621.
6666' acac'

 

~ I 2
because in many cases. plim {[I(19)]“l— 4115—1 -.- [3, where

was .

W

 

evaluated at 5'. Also note that whenever the usual regularity conditions hold.

[(19) can be obtained from the first partial derivatives of the log-likelihood func-

6'

I 2 l f H 1'
tion. ‘That is, [(19) = Ell- Eta-9%“ = E{{%€'-}l§-Lll‘ Besides these. there is an

1
indirect approach using the scoring algorithm ( Newton-Raphson algorithm ) to
compute the LM statistic indirectly (Breusch and Pagan (1980)).

If I (13’) is not of full rank. say. rank [[6313] = s - t < s. then [(3) is singular

and is therefore not invertible. Silvey (1959) assumes there exists a sxt subma-
trix H1 of H, such that 3:4(5) + H 1H 1' is positive definite. Then. he proposed a

modified LM test statistic
LM' - 1 'D(§)l'[l-I(5) + H H 'l-‘ID {i ] i (1 7)
- m n 1 1 l ( ) -

which asymptotically follows a chi-square distribution with (s - t) degrees of free-

dom. This case arises in one of our analyses of chapter II.

-3-

CHAPTER II
DISTRIBUTED LAG MODEIS

2.1 Introduction

A distributed lag model describes how the lagged independent variable
aflects the dependent variable over time. The length of the lag may sometimes
be known a priori. but usually it is unknown and in many cases it is assumed to

be infinite. Thus we consider a distributed lag model of the general form

y; = 23051th + 8:
where y, is the dependent variable. 2,4 is the lagged independent variable. .6. is
the distributed lag weight. a; is a disturbance term. Infinite lag distributions
involve an infinite number of unknown parameters. and thus it is impossible to
estimate all these parameters. To make estimation possible. it is necessary to
make some reasonable assumption about the pattern of the distributed lag
weights. The earliest distributed lag model is the geometric lag model proposed
by Koyck (1954). He assumes that the lag weights decline geometrically. i.e.

a. = W. for i = 0.1.2....
where O s A < 1. Since the lag weights of the geometric lag model decline mono-
tonically. and this may not always be reasonable. various alternative models
have been proposed. For example. the Pascal lag model proposed by Solow

(1960) permits a hump in the lag weight distribution curve. In 1966. Jorgenson

proposed a more general rational lag model

_ .4ng
y; - 30:) 3: +114

-7-

where A(L) and B (L) are polynomials in the lag operator of order p and V.
respectively. He also proved that any arbitary lag model can be approximated
to any desired degree of accuracy by a rational lag model with sufficiently high
values of p and v. If we take A(L) = 6(1 — A) and B(L) = 1 - M. the rational lag
model is Koyck’s geometric lag model. If we take A(L) = 6(1 — A)’ and
B(L) = (1 - M)’. the result is the Pascal lag model.

In this chapter. two distributed lag models are discussed. The geometric

lag model is discussed in section 2.2. and rational lag model is discussed in sec-

tion 2.3.

2.2 The Geometric Lag Model

Following Klein (1958). the geometric lag model can be expressed as

3!: = B‘onlzt-i + 8:

= We +noA‘ + 81 . t = 1. ° - ~ .T. (2.2.1)
6-1 a

where 05>. < 1. w. = 21%.... no= 52162.... a. iid N ( 0. 0'2). 11x: 0. or 1:
i=0 i=0

we know the value of A. we could estimate eq.(2.2.1) by OLS. Usually.we don't
know the value of A and we use a search procedure to estimate eq.(2.2.1). Since
the search procedure is not simple. it may be useful to test whether A = 0 before
we start the search procedure. The restriction A = O is easy to impose and the
restricted model can be estimated by OLS of y, on 2:, only. Therefore. the LM

test is very suitable in this case.

It is well known that the parameter no can not be estimated consistently.
and that indeed the information matrix is singular asymptotically when no is
included in the list of parameters to be estimated. See Appendix A for details.
However. Schmidt and Guilkey (1976) showed that it makes no difference
asymptotically whether one drops or estimates the truncation remainder term
in the maximum likelihood estimation of distributed lag models. Maximum likeli-

hood estimation of eq.(2.2. 1) amounts to minimizing the sum of squares

i (y; " £11), "00”)2
£81
with respect to A. .3. and no. Since noA‘ disappers asymptotically. this is
' 1'
equivalent to minimizing 2 (y; - 6111‘)”; that is. to setting no = 0. and applying
t=i

OLS to the model

yt=fiwt+8¢ . t=l."'.T. (2.2.2)

2—1
where w, = 2A‘z,-i. Us A < 1. a; iid N (0. 02). Also. the estimated variances
i=0

of A and 5 resulting from estimation of eq.(2.2.2) are asymptotically the same as
ones from eq.(2.2.1). This is so because after deleting the row and column
corresponding to no. the resulting submatrix of the inverse of the information
matrix corresponding to eq.(2.2.1) is asymptotically the same as the inverse of
the information matrix corresponding to eq.(2.2.2). Therefore. we can construct

our LM test statistic based on eq.(2.2.2) instead of eq.(2.2.1).

The log-likelihood function for eq.(2.2.2) is

1'

L = constant - g-logo‘e — 31:5; (3;. - fiwdz
:1
The first partial derivatives are

61. _ 1 T - d1”: -“‘- 11—:
6A - 02 Effie; where R; - dA -Ԥ11A 2.4
OZ. "
—= — w a
66 2 {:1 t ‘
6L 1 T

The elements of the information-matrix are

(
62L 1
1M = — E fl. = fig (131%)2

 

. 881
_ 62L _ 1 T 2
I” ' 'Elaaz ' 35.2.3.”

 

_ lazL _
IP°"'E__—']a,saa2 -0

The restricted model is

-10-

y¢=ztfi+8g I t=l."’,T.

"_t=l

Let '3': (75.6.1?) where A: 0. §= OLS estimate. E} = y, - 33:; and 02 T

 

T ~ 1
5?: ‘2131-1 2 = ‘gg—Xt-let
~ "" 1' ~2
1AA = $213124 - ~2 Xt-ixc-i

1M - ~2
i=1
1;” = .}_27 3‘2 — 1_](")(‘
a2 .3, 52
where
Xt-l = (30-31- ' ' ' 31-1).
X: =(31- " ' 137').
8‘ = (31' ' ' ' 037').

Let 191 = A. 132 = (3.02). Then we can use eq.(1.5).

-1
1

 

 

1' f“: ' ~_ [~
. . ._ 6L) N _ 11p [”1 ~O [An 6L
LM test statistic — [57] In [0 ll 0 1:22,: I 0 6A

' -1
6L ~ ~ -~ _ ~ 6L
1.1111 1.1

= eixt -1(X¢'-1M1X¢ -1)‘1X¢'-1 e:

‘52

{(Xt.-1Ml){t-1)-1(Xt.-1Ml 1,012

32(X£-1M1X‘_1)-1 (Note 1) (2.2.4)

where M; =1 -X¢(X[X¢)‘1X¢' and Y: =[y1_ .yy-I. The last equality holds

because

Mi}? = Y: -%(X¢')Q)"X[Y: = Y: -X¢5 = e:-

-11-

Note that this LM test statistic can be expressed as the square of the t
statistic'for the coeflicient of X,_1 in regression of Y; on (X,,X¢-1). This point can

be clarified from the following discussion. Consider the regression

 

11 = m +XHc + u, , u, m N (0.0”) (2.2.5)
A O O
OLS estimate for c is c = (X¢_1M1X¢-1)’1(X,-1M1Y,) and OLS estimate for 02 is
7' A
A $1112 A
02 = “IT where u; is the OLS residual. The OLS estimate for 05 is equal to

32(Xt'-1M,X;_1)‘1. The t test statistic for c = 0 is

I‘
_ c — O
t — .T—
a?
_ (Xt'-1M1X:-1)—1(Xt'-1M1 Yt)
[32(X:'-1M1X:—1)"]”
[(Xg-lMiXt-O-th—IMIYt)]2
E\’2(Xt'-11‘11Xt-i)nl

 

Therefore. t2 = which diflers from eq.(2.2.4) only in

one term. namely the estimate for 02. The test of c = O in eq.(2.2.5) is asymptoti-
cally equivalent to the test of A = 0 in eq.(2.2.2). since when c = 0 is true . 32 is
near 32.

This is an interesting result. We can test the existence of a lag (A = 0) in the
geometric lag model by testing the significance of the single lagged term. X¢-1.
in the OLS regression of y; on (X, .Xg-1). This provides an asymptotically optimal

test. despite the fact that the geometric lag is a lag of infinite order.

-12-

2.3 The Rational Lag Model

The rational lag model is a rather general distributed lag model. It can be

expressed as follows:

HUI) (2.3.1)

where L is the lag operator defined as

y‘=fl£)_.zt+u‘. .t=1.....T.

£1th = 2‘4, , k = 0.1. ' ° ' . L0 = I . 13‘ = 2t

v o
and A(L) = f aiL‘.~B(L) = 2 (7,-1.1. b0 = 1.1.1. < v. The independent variable 2, is
i=0 j=0

assumed to be nonstochastic. or if stochastic. uncorrelated with the random
term at. We also assume u, is independently. identically distributed as N (0.02).
Dhrymes. Klein and Steiglitz (1970) suggested that this model can be estimated
by maximum likelihood methods through a search procedure ( search :1... given
b, ). or through an iterative procedure for all of the parameter estimates simul-

taneously. Then. using the estimates for ti. and bj. one can estimate 02 easily

from the first order conditions.

Since the estimation of a..- and b; is not easy. we have two alternative model
specifications which can be estimated by OLS. The test of B(L) = 1 is given in

section 2.3.1. and the test of A(L) = a0 and B(L) = 1 is given in section 2.3.2.

2.3.1 Test of B(L) = 1

The restriction B(L) = 1 can be written as

221:0 . j=1. .11. (2.3.2)

-13..

Under the restrictions. eq.(2.3.1) becomes

31. = A(L)z: + u: = iniL‘z. + u. . t = 1. -~ .T. (2.3.3)
i=0

Despite the fact that there might exist high multicollinearity among x's. we still

can use the OLS method to estimate this restricted model. and indeed OLS pro-

 

vides MLE’S subject to the restriction. Let 131 = (b 1. - - ' .b..)'.

132 = (0.0.01. ' - ' .awaz)’ and 13’: (31. - - - .gv.;o.;1. - . - Epic-2y where
7' .2
211-22

b1: =b.,=o. a; = OLS estimates fora. .7: =0. -~ .u. and 3?: ”‘7,

where 17.} = y; - X(L)x. and Z(L) = f: ELL". We can use eq.(1.5) to construct the
i=0

LM test statistic.

The log-likelihood function is

L = - 330114210

             

-A(L)z,)2
The first partial derivative with respect to 132 evaluated at 1? is zero (see

eq.(1.4)). The first partial derivative with respect to 13, evaluated at 5' is

 

 

61.13;; , aL
l6b1'6b,
1” T.... '
= "— 217 u, A(L)2¢- _1. .ZugA(L)z;_..
-02 =1 t=l
= .. .145
52
where .
34(1031—1 - ' - A(L)31-v‘
X =
.A(L)zT-l ' - - A(L)zT-v.
and
U= [171. {Erl'

-14..

The information matrix evaluated at 1; is

 

 

 

 

.. _ P131 52
[(19)- Vzi 122
r11.}. 12.. o
= [:5 f“ 0
'0 0 1°20;
where
1:101 Eloy]
1» =
[121»! ' ' ' Kuhn
[r .2 A. 7' ~ ~ l
2[A(L)x._11[A(L)z.-.] . . . g21[A(L)a.-...][A<L)z....]
{=1 3
= 72-
7' ~ . ~ . . . 7' ~ . ~
L2 [A(L)zt—v][A(L)zt—l] - - - 2[A(L)zt-v][A(L)zt-v]
=1 ‘31 i
1
= rXX
02
151“!) 2:10“
Ibo =
Elfin Kipp
1' ~ 7' ~ ‘
L2 [A(L)z:—1]x: - - - ‘21[A(L)zt—l]x¢-p
=1 =
_ .1.
7' ~ 7' ~
“21[A(L)2‘_v]2¢ ' - ' £2 [A(L)zt-v]zt-p
= :1 ‘

 

 

 

 

 

-15-

II

I
IH
N
5

g“?
u

 

 

 

 

With

2121-1 - - - zl-n

 

 

537' 27-1 - - - zT-pl
and

~ _ T
[gens-‘27?

From eq.(1-5).

. ~ 1 - 0 IV"

. ' - 6L ~ r a lab 6L
Mt t tmn - - “‘

L es s a C [6191 Ibb [166 O 0 42,210] [6131

~16-

l— 2i—i- 22i- i‘i-

~ -1 ~
UX M XU
= [X a: (2.3.4)

 

 

where M3 = I - X.(X.'X.)'1X.'. Note that in this case. the LM test is essentially

the F test of significance when X is added to the regression. i.e.

Y = Xoa + Xc + a (2.3.5)
where e is distributed as N (0.021). The OLS estimate for c is

A

8 = (XMsxrlXMaY. and 32 = a; where S is OLS residual. The F test statistic

 

for the null hypothesis 0 = O is

_ 3.(X’M1X)8/ V
" ’é'é/[T - (p.+v+1)]
The restriction of B(L) = 1 in eq.(2.3.1) is equivalent to the restriction of c = 0

 

(2.3.6)

in eq.(2.3.5). When the restriction is true. 32 should be close to 32. Therefore.

eq.(2.3.4) can be rewritten as

_ YM3X(X'M3X)“X'M3Y _

LM test statistic - ~2 because MSY = i7

 

 

 

a
_ 3'(XM3X)3
_ 52
A. h
C(XM c
. LM/v
From eq.(2.3.6) and eq.(2.3.7). Fm In a large sample.

 

T/ [T - (u+v+1)]'

T

T _ (I1 +1, + 1) 4 1 and the F test is asymptotically equivalent to the LM test.

This gives the justification of doing the F test in this case.

2.3.2 Test ofA(L) = a0 and B(L) = 1

- 17 -
This section might seem to be a special case of last section. In fact. in this
case a singularity problem arises and needs special discussion.

The restrictions A(L) = a0 and B(L) = 1 can be expressed as

a1=a2=~~=a,.=b.=b2=-~=b.,=o (2.3.3)

Under the restrictions. eq.(2.3.1.) becomes

yt=aoxt+ut . t=1. .T. (2.3.9)
We can apply the OLS method to eq.(2.3.9) and obtain the restricted estimates

= [0-1. ' ' ' .a#,'gl. ' ' - '8'...~o.'5’2]'

where E.=~-=a',.=8'.=-~= ”=0 . EB=OLSestimateofam and

[‘43

{1‘2
1
T I

2 = with 17, = y; - 503,. The information matrix evaluated at 13 is singu-

 

'5
lar because

[Coat [Cob [3'30
~ 1.1.» 1» Iago
I 13 := ‘7 ‘7 '*'

( ) [coco [baa [coco
[0 O 0 10303

cCDOC)

 

 

 

 

 

 

 

where

[131:31 . - - 1016,,

~ _ __ 1

In“. - . . . . . - fiXoXo
{aflal - - - [afiap‘
[C101 ' [I‘DJ

~ . . . . . an ' . ‘

[0.5 = . . . . . = - 52 ‘XoXu
Haul?! Iafibu‘
~ ~ 1

~ _ _ 55 .

[ob - . . . . . - 1.32—‘XooXoo
{oval - - - fave.)

 

 

-13-

 

 

 

 

 

 

 

 

[“90
j" ' _ 1
a'co . - Egg—XOX
.I“#“°l
”51%
rv . a’o .
[”0 = "" g'é—XOOX
(1......
’V _ 1
where
I21
X: I
2'7 .
'1’1-1 zl-fifi
X0 =
27-1 . . . 27-“!
and
{21-1 . . . 21..“ . . . 21-9!
X0. =
FT—l - o o 27'.“ u s . 27-”.

 

 

Note that . in [(5). the (p. + k)th column is just ( - 3’0) times kth column. where
Ic = 1. - - - 41.. Hence [(5) is singular with a rank equal to (11 +2). In order to util-
ize eq.(1.5). we have to find some good reason to reduce the size of {(13) so that

the resultant information matrix is of full rank. Since

6L 1 T 2
T: :— 2 . 1. = l. .
601 02 ‘gluf Id! ’1'
and
BL iTo 7'...
= - — u I = I I
6wa 32 .§. ‘2” J 1 V

we have

-19-

 

 

 

 

 

2L: _ g 3.1:. - = 1
ab“. ° ac’i'j ' '7 ' ""
That is gill-:1 1’. = 1. y. do not contain any information not contained in 361%—
1
j = 1. ... .1). Thus we can drop 567L_ 1'. = 1. . H ..u. from the vector of the first
partials and drop the rows and columns corresponding to a... 1'. = 1. . ' - .p. in the
information matrix without sacrificing any information.
Define
[~] '61. oz. 61. 6LT
D130="°'—."'.~.~u~
l H (ab. 6b., Bag 602]
= L- g—E‘E,Xoo. 0 . 0
rflb KOO O
[(13)' = [ciao [coco 0
_ 0 0 10202]
The resulting LM statistic is
LM JD 5 "1 5 '1' ~
o-l ( )ol()o lD(13)o
~ -1 ~
a "'3 ~ ~ ~_ ~ a0 , ~
:: [— #UXu]{Ibb - IDCOIOOLOIGOb} [- fa'z—XOOU]
Iv. ' -l .
UXOO [XOOM‘XOO] X003
= ”2 (2.3.10)
0'

where M. I -X(XX)“1X. By the same reasoning as in section 2.3.1. MI. is

asymptotically equivalent to the F statistic in testing the coefficients of the

lagged x’s in X. when they are added to eq.(2.3.9)

This way of dealing the singularity problem may seem too simple and
without theoretical support. But eq.(2.3.10) turns out to be exactly the same as

Silvey’s modified LM test statistic eq.( 1.7). To see this. note that in this case.

 

 

'1,.o
01..
H6=oo
(00

 

-20-

 

 

 

 

 

 

 

 

 

 

II“
H1 = 8
0
Therefore
I!“ o o o
. _ 0 0 O O
HIHI " o o o o
l0 0 O 0
and
I1 ~ 1 v 1 ~ -1
fiam.+1p ‘YTJaJ Wang 0
1 ~ ~ ~
[1 ~ "1 find: ‘17?!» 'lf‘feao 0
fi(6)+H1Hi] = 1~ 1’7 1...
Flown ‘7710130 F1“o“o 0
1 ~
[ 0 0 0 70202
[A F K o
_ B G M O
' C J N ~0
0 O 0 Nazca“
where
1 ~ 1 ~ 1 ~ “1
.4 F K T ...,+1,. 2’” F’Mo
1 ’Y 1 ~ 1 ~
‘5 G M = #005 gr!» 7:!”0
C J N 1 ~ 1 ~ 1 ~
#0060 .1715”! '7'?!an
From eq.(1.7).
laL " am
660 60.0
251.4 F K 0 6L
LM.= L ab 3 G M 0 5?
T D C J N 0 0
- 0 - o o 0 TI ‘1 °

-21-

     

 

Jae-W

       

 

212. [212

 

 

 

 

36m]
(2.3.11)
where
M. _a__L 6L 1 .~
68.: 677' '65,. =52X‘U
61. IM 6L 30 .~
61: =lab. 'Bbv ’ 52X”U

and A. B. F. G. can be found by some manipulations involving the inverse of a

partitioned matrix ( see Appendix B ) :

 

A=1,.
I,
5:7}7-l10“
a
F=ll1w01
Go

{.0
oo

 

 

}

i7“ [,r.Ax;-on..3x;—&”ox.rx.'.+?igx.. 026.] 3'
r34
fl 0 o o N I _l 0 0 ~
U o o-XoXo-XoXo‘I' T02Xu hacMgXu] Xu'I'XoXo] U

T a‘

N , “'1
G = g—[Tazprumxn] +
co

Therefore eq.(2.3. 1 1) becomes

LM'=

 

~ ’ -1 . ~
U'X» ..M.X..] x..U

which is exactly the same as eq.(2.3.10).

 

2.4

-32-

2.4 Conclusion

The LM test statistic for the geometric lag model (eq.(2.2.3)) and for the
rational lag model (eq.(2.3.4) and eq.(2.3.10)) are similar because the geometric
lag model is a special case of the rational lag model. In both models. the LM
statistic can be constructed by adding some lagged values of the explanatory
variable to the restricted model. and testing their significance using the usual F
test (Note 2). This is a favorable result. since in practice many researchers may
prefer to estimate under the restrictions (using the OLS method). and to con-
sider more complicated estimation methods only if the LM test provides

significant evidence of the existence of a more complicated lag pattern.

Although we have considered a model with only a single explanatory vari-
able. the results do not depend on this assumption (Note 1). They would still hold

if there were additional regressors (not subject to the distributed lag).

3.]

de

-23-

CHAPTERIII

QUALITATIVE AND LIMITED DEPENDENT VARIABLE MODEIS

3. 1 Introduction

In many cases. economic studies have to deal with situations in which the
dependent variable is dichotomous ; that is. it is observed by its sign only. The
usual least squares method will cause many problems (see Judge. Grifliths. Hill
and Lee (1980) p.586). The most serious problem is that the predicted value of
the dependent variable will not be in the unit interval. One of the solutions to
this problem is the "probit model" (see Finney (1971)) which uses the cumula-

tive normal distribution function to transform the dependent variable into a pro-

bability. This model takes the form

y.'=z..3 + e.. a. ma. N(O. a2).
and

1 if 21;) O
y‘ = 0 flyis 0
Then

P705- (y¢=1) = Prob. (W30) = PW"- (ztfi + 8: > 0) = Prob. (8: < 2:5) = 4:5

where (PM is cumulative distribution function of N (0.1). Therefore.

a:
prob.(y. =0) = 1 -proe.(y. = 1) = 1 -<i>[ f.
The probit estimate for gcan be obtained by maximizing the following likeli-

hood function:

xtfl I 53:31
L = y1._=I1¢[7_Lg-oll - (I>[ a .

 

H

 

16.631

1110

1101

W t

-24-

Since we cannot identify [3. a separately. we choose the normalization 0:1 to
identify 6. The classical example of the probit analysis in economics is the study

of the consumer’s decision of buying a durable good.

In this chapter. we consider three models which are extensions of the probit
model. Section 3.2 considers the Tobit model. and Cragg's extension of it. in
which the dependent variable is observable in a limited range. Section 3.3 con-
siders Heckman's sample selection bias model which consists of two equations.
one of which is a probit equation representing the rule for sample selection. Sec-
tion 3.4 considers Poirier‘s partial observability probit model. which consists of

two probit equations with a condition of partial observability.

-25-

3.2 Test of the Tobit Specification against Cragg’s Extension of the Tobit Model

3.2.1 The Tobit Model and Cragg’s Extension

Tobin(1958) considered a case in which the dependent variable is observ-
able in a limited range and the analyst is not only interested in the probability of
limit and non-limit responses. but also in the value of non-limit responses. Probit

analysis is not suitable for this purpose. He proposed the following model. called

the Tobit model:

yg.= $35 + 8; . 8; 12d N (0. 02) .
g. If y‘.> 0
t _= 1.....T

where y,’ is unobservable and y; is observable. y. has a lower limit which is zero.
That is. there is an event which at each observation may or may not occur. If it
does occur. associated with it will be a continuous positive random variable. If it
does not occur. this variable has a zero value. An example is an individual’s deci-
sion whether or not to buy a new car. and the amount he spends if he does buy

one.

According to eq.(3.2.1). for y, > D. the probability density function (p.d.f.)

for y; is

f (21.) = 72%?pr — $21!. —z.a)2} (3.2.2)

and for y; = 0. the probability of observing y, = O is

Prob. (y; = 0) = Prob. (yfs O)
= Prob. (3.3 + a; s O)
= Prob. (a, s -z;fi)
-z‘p
= f —1——exp - -1-—52 ds
V2770 2a2

Wh!

ob

rh’

-25..

- a (3.2.3)

 

where <I>(-) is the c.d.f. of the standard normal distribution. The probability of

observing y, = 0 is represented by the shaded area in Fig.3.2.1.

122*)

 

 

 

5/:

 

Fig.3.2. 1

Note that there is one and only one 6 to determine both the probability of
y; = 0 and the shape of the probability distribution for y; > 0. That is. in the
example of purchases of a durable good. the decisions on whether to acquire and
on how much to spend if acquisition occurs are basically the same in this model.
in the sense that the same variables and parameters occur in eq.(3.2.2) and
eq.(3.2.3). Cragg(1971) argues that "In some situations the decison to acquire
and the amount of the acquisition may not be so intimately related. In particu-
lar. even when the probability of a non-zero value is less than one half. one might
not feel that values close to zero are more probable than ones near some larger
value. given that a positive value will occur." In the case of buying a new car. this
argument is certainly true. The probability of buying a new car for an individual
in a particular year is probably less than one half. From Fig.3.2.1. the Tobit
model implies that. if a new car is purchased. smaller expenditures(e.g. 5 dol-

lars) are more likely than larger expenditures(e.g. 5000 dollars). This foolish

imp

min

92>

ter:

the

Tot

F1!"

210

-27-

implication is due to the fact that there is only one set of parameters to deter-
mine the probability of y, = 0 and the shape of the probability distribution for

y¢>0.

Cragg(1971) proposed a more general model which uses two sets of parame-
ters. One set determines the probability of y, = 0. and the other set determines
the shape of the probability distribution for y, > 0. Cragg’s extension of the
Tobit model can be written as a two-stage decision process.

First-stage -- decision on whether to acquire
The probability of not buying a durable good is

Prob. (y¢=0) = Prob. (x,fil+u1<0) = <I>(-x,fil) (3.2.4)
and the probability of buying a durable good is

Prob. (y¢>0) = 1 - Prob. (yt=0) = 1 - <I>(-z“81) = @(z‘fil) (3.2.5)
where a, is normalized as 1 because we can not identify 61 and 01
separately in a probit model.

Second-stage -- decision on how much to acquire if acquisition occurs
The probability density function for y,. given acquisition occurs. is

f (y; I y; > 0) = N (2; fig. 0%) truncated at zero
1 [_ (y; '1': 592i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ Mag , 203
J- 1 x _(yt-2882)2‘d -
0 flag p 20% y;
1 L (ye-31132)“
_ mag“? 20.2 1 (326)
— q, 3:52I ' .
L 02 1
since
(2111- 232232)] __ 3:52
IMO—i? xPl 1113;. -Q[ 02

The unrestricted estimates for 51. 62 and 02 can be obtained by maximizing

the 1

WI}:

-23-
the following likelihood function:

1.12.6222) = H {Prob (y.=0)] H [Prob (yt>0)'f (y: I yam]
y‘=0 yt>0

T 1' t t
= E{Hob.(y.=0)] ‘ [Prob.(y.>0)-f(y. Iy.>0)]‘

 

l- 2: I _ ‘9
:fi{¢('x‘fil)] d‘ 43(151) 1 eXpl— (y: 1132?] (3.2.7)

¢[2‘ 62 mag 2022 J

02
where d,=1 if yg>0. d¢=0 if y¢=0. Equivalently. one can maximize the log-

likelihood function

[4131.52.09 =1I1L.(51.I32.0'2)

 

=‘)_51{<1-d.)1n[¢<-z.2>1 (32-8)
+ d. {1n<1>(z.5.)—1n<1>[—%z‘82 —ln(\/2'1?a'2)- (.y‘ ”‘25 2021
02 202 I

3.2.2 The LM Test

In order to derive the LM test statistic. it is convenient to reparametize in a

way similar to that suggested by Olsen(1978). That is. letting

- 5;
_ 02
6= 51 - 5 (3.2.9)
h. = -1—
02

eq.(3.2.8) becomes

“64”!) = §{(1’dt)lnq’[-zt(£+fl)] (3-2-10)

8:1
I
+ d. 11:122. (£+B)]-1n<1>(2=¢ B)-;—ln(21r)+1nh -;}.—<hy. 2.221}
Note that when £=0. eq.(3.2.10) reduces to

r I
L. =‘21{<1-d.)1n2<—x.m+d. -;—1n<21r)+1nh—-;—<hy.—z.a)2

 

] (3.2.11)

-39..

which is the log-likelihood function for the Tobit model. The restricted MLE is

5: (5.6.h) where E: 0 and E. 72. are obtained by maximizing eq.(3.2.11). i.e. F

and I: satisfy the following first order condition:

6L]. T N o N ~ 0
3736—: ‘gl-(l‘dt)'m(-ztfi)z¢+dt(hyt ”215)”: = 0 (3'2'12)

where m(-) =

{3% with rp(-) being the p.d.f. of N(0.1).

From eq.(3.2.10). the first partial derivatives of the unrestricted likelihood

are

'21:: 21-122...1-2.2.2.”.222212}
%é— = ,é1{—(1-dt)’m [‘zt (54-3)]‘239'4’4 {m [3! (f+fi)]—m [z‘fi]+hyt 41251271.}

and the second partial derivatives are

62—L- i '2' <t+a)-m[-z (6+6)]-m2I-x (221]
3555. - t if": t ‘

{=1

+dt{’zt(£+fi)'m[’zt(£+fi)]+m2[-31(6+fi)]"zt(€+fi)'m[xt(£+fi)]

 

 

 

- 77121222211}
6:261); = $111.31 {WEI-2: (f‘tfifl'zt (5H3) —m2[—z‘ ($473”)
"1: {m [‘53: (5+5)]'3t (5+5)'m2["zt (“fin-"l [3: (5+5Hzt (5+5)
-m2[=¢(£+fi)]}}
__ 62L
’ age;
62L _
agah " 0
62L _ 62L

 

T
6666' " atat' —.2212“[’ztfl'm(2:6)-m2(x¢fi)+1]

-30..

62_._L__ 7'
aTah =81 dtztyt

2
'g—“hLz '-' “X: dt(h—2-+ 11:2)

The elements of the information matrix (see Appendix C) are

I 62L
[it = ”Eljggl

= 222. m [-2.122] mixtiwn
= fee
I... = —E[%—]= o
= 42:22:: 212. (6+fill'I-ztfi'm (sum-mam 2+1]

-_- _ ‘21 ztél: 2‘ (€+fl)] —'[Z¢ B+m (ztfi)]

62L
M=’flifi

= 252122212222";(2.21212 (25)]

 

If we let 131:6. 132=(fi.h). then

2%: g—g’e evaluated at 13
1
=H-ZZI[ -(1-d:) m I-ztfi)2£+dt m(2¢§)zt'] (3.2.13)
=jgld,[(hy,-z,fi)-m(z,§)]z¢' by eq.(3.2.11).
Also
Ifli
a—aé—= 2‘: evaluated at 5
2 _.
6h.

 

-31..

r
=13

 

by eq. (1.4)

The information matrix evaluated at 1; is

[~ ~

Ice {55 {5h

“13) = In: {53 Ian
{as [up [an

'XAX XAX o
= {our X'(A+B)X (620'
o

 

 

 

 

 

 

CX D
where
'311 - - - 311:] F311
X :: :-
.zn - . - 372‘ 531‘
’a, o o o'
0 02 t) . . 0
0 0 as . . 0 ~ ~
A = where a, = m(-z¢fl)-m(zgfi). t=1....T.
IO (3 O . . QT;
r1:, 0 o o'
0 b2 0 . . O
0 O 63 . . 0 ' ~ ~ ~ ~
3 = . , . . . . where 5: '-' ‘I’(Ztl3)[1"z¢fi'm(3:5)‘m2(3¢5)]
[0 0 O . . bTJ

 

 

c = -';-§—{¢<21§)[z1§+m<z1&)1. . . . -‘P(ZT§)[ZT§+m(-rrm]]
n = )5 file-{mmfirmfi-m(am-m3)

2:1
Let the inverse of [(13) be

. ,2; Ra Ra
[mm-1*= 136 g" If"
[hs [hp Ihh

Then. from eq.(1.3) and eq.(3.2.14). (Note 3)

-32-

6LT ,_, ~ 'aLl
To! [es [en fin T5131

LM statistic

fit if: [an
fie FM 1» l

 

 

 

 

O O
0 0

L

M '76 61.
[6‘57]! [WI] (3.2.15)

We now look for an explicit formula for I“. Let us rewrite I (13) as

 

 

.. [P 1?
103): IR" Q
where
_ 'XAX XAX
P ' WAX X(A+B)
l 0 l
R = .
L(CX)
Q = 0
Therefore.
[Fe 133 .
[five 1735 = U” ~RQ“R>"
_ M N"
- N' S
where
M = N = X'AX

S = X[(A+B)-CD"C]X
Note that M’IN = 1. Thus.

~

I“ = M"1-l-M"1N(.51-N'1ll"1N)"1N'1ll'1
-1
= (X'AX)‘1+{X[(A+B)-C'D"CPI-XXX}

, -l
= (XAX)‘1+{XBX @0300} (3.2.16)

Substituting eq.(3.2.13) and eq.(3.2.16) back into eq.(3.2.15). we have the LM test

 

statistic.

-33-

Although I “ has no obvious interpretation. it is easily calculated from the

Tobit estimates. On the other hand. aLEI—is both easily calculated and also easiy
l

interpreted as a vector of cross products between the explanatory variables and
the Tobit "residuals" for the non-limit observations. That is so in the sense that

E[(hye-z:§) I 1100] = 771.0513) by 99-(C'1)
and thus the term in brackets in eq.(3.2.13) can be regarded as the Tobit "resi-

dual."

-34-

3.3 Test of Sample Selection Bias

3.3.1 Heckman's Sample Selection Bias Model and His A Test

In some cases. the dependent variable is unobservable while the
corresponding independent variables are still available. That is. we have an

incomple sample(or censored sample). Heckman(1976.1979) proposed a two-

equation model to deal with this situation:

ya = 311131 + “11 (3-3-1)
3121 = 32152 + “21' (3-3-2)
with
[an
I“ iid N(O. 2). i=1....,n.
2i
where
_ '01" pm
2 ' a, 1

since eq.(3.3.2) is a probit equation. We observe the sign of ya and we observe

y u it and only if ya > 0. That is. ya > O is the sample selection rule and we have

a nonrandomly selected sample.

This model can be estimated by the maximum likelihood method. But if we
use least squares for eq.(3.3.1) and probit analysis for eq.(3.3.2) instead . the

resulting estimates of 61 will be biased. This is so because

E(y“ l swsample selection rule) E(yu | :3“. ya > O)

31151 + 30111 I 3121 > 0)
. ' = 31151 + P01)”:
(See Johnson and Kotz (1970).p.81). where the inverse of the Mill’s ratio is

_ ¢(‘32~;§)
M - 14130-22132)

with ¢(') and <I>(-) being the p.d.f. and c.d.f. of a standard normal distribution

- 35 -
respectively.

Thus. the expectation of y“ for the nonrandom sample is not equal to the
expectation of yu- for the complete (random) sample unless Emu | ya.- > 0)
equals zero; that is. unless p=0. Therefore. by using eq.(3.3.1).we have the
equivalent of an omitted variable problem which will result in a bias in the esti-
mate. This bias is called "sample selection bias." This bias will be eliminated if
the conditional mean of u“ is included as a regressor. However. since 62 is a
parameter to be estimated. the M’s are unknown. Heckman proposed a simple
two-stage estimation procedure to estimate the parameters.

Step 1 -- Probit Analysis

Let
1ify2i>0
di: 0 lfyZiSO
n
2de=ni

1
Gt = P705- (yeis 0 I 1'21) = ‘I’("32i52)
The probit estimate 52 is obtained by maximizing the following likelihood func-

1.

tion:

it _
L. = II(1-G.-)““G.-‘ “

1:1

Step 2 -- Least Squares

Let

a" = 54-22131!)
1414-32132)

then apply OLS method to the following equation

1;“ = mum + Xe +verro'r. i=.....n (3.3.3)
where c = pol. The OLS estimates are

>

51 = (XiX1)“Xi Y1 " (XiXxl'lXiXd'Mlifti'MzY1
8 = (1..“ 1X)-1XOM1Y1

-35-

where X1 is a nlxlc matrix which consists of mu. Y1 is a nlxl vector which con-
sists of y“. A is a nlxl vector which consists of A; corresponding to observed

y“. M1 = I - X1(X'1X1)"Xi. Note that g, is a consistent estimate of 51.

If p=0. E(yu I 2:“. ya > 0) = 211,31. Then. there is no sample selection bias
even if we apply OLS to eq.(3.3.1). Therefore. the test of sample selection bias is
equivalent to the test of p = 0. Since c =p01 in eq.(3.3.3). the test of p = O is
equivalent to the test of c = O. Heckman uses the standard t test to test the
hypothesis c = 0.(We will refer to it as the "A test") The t statistic is

. ~ _1~.
= W)... $511111 (3'34)

where 3,2 is the usual variance estimate (SSE divided by 11.1. or degrees of free-

 

dom) from OLS to eq.(3.3.3). This model and the A test have been widely used.
especially in labor economics. Many applications have reported an insignificant
value for A test statistic. One possible conjecture is that the A test is not a very
powerful test of sample selection bias. However. this turns out to be a false con-
jecture. The A test is asymptotically equivalent to the LM test. as is shown in the

next section. and thus has good asymptotic power properties.

3.3.2 The LM Test of Sample Selection Bias

Let

Ft = Prob. (3111- “.921 > 0 l zit: 221')

f Mam-2.1191. “21)‘11‘21 (53-3-5)

“3213:

where h(-.') is the p.d.f. of N(O. E ). Then the log-likelihood function for eq.(3.3.1)
and eq.(3.3.2) is

L = iid-an. + (1-d.)1nc.] (3.3.6)
i=1

-37-

The restricted model is the one in which p = 0 is imposed. When p = 0. eq.(3.3.5)
becomes
Ft = f hi(yii-2iifii)'¢(u21)du21
”up:

= hi(yii‘xiifii)'[1 ’ ‘I’('22152)] (33.7)

where MO is p.d.f. of N(O. 0?). Hence. whenp = O. eq.(3.3.6) becomes

L. = i:[d-ilnh-K‘tl11"“: 11131) + diln[1—Gi] 4' (l-di)1nGi (3-3-3)
The restricted MLE 5 = (551.523,) is obtained by maximizing L' w.r.t. £1. .32 and
0,; i.e.. S: O. 51 = OLS estimate from eq.(3.3.1). 52 = probit MLE of eq.(3.3.2).
i": -—eiel where 21 = Y1 -X1§1. That is. when p = 0. we can estimate 51. of

~ 1
a
in.l

from eq.(3.3.1) by OLS and estimate {32 from eq.(3.3.2) by probit analysis.

Letting 131, = p. 192 = (61.32.01). we can use eq.(1.5) to construct the LM statis-
tic. From eq.(3.3.6). the first partial derivatives evaluated at 13‘ are (see Appendix

D)

.25
6fl
an

55;-

E.
552
BL

.W

Note thatA is a nlxl vector. not a nxl vector. The information matrix evaluated

>43
to

32

by eq. (1.4) (3.3.9)

mo:
LN

 

 

COD

'7

 

 

at :5 (see Appendix D) is

,N ~ ~

[pp [p31 195; [9011

[5151 [5152 [5101
[3252 152‘:
[‘10:]

 

 

-33-

F

 

 

n M.»
n ~~ 2(1-GilAizu
§_‘,(1--c;.)>..a "1 ... o 0
=1 01
n ~ '
;(1-Gi)zlizli
=1
. .... 0 0
a?
= n ' MN (3.3.10)
‘ZzaZZiAim-i 0
=1

A
H
l
.53
V

 

 

 

where

51 = @('221'.§2)

~ = ¢(-32i§2)
m... ¢(”32i52).

N 1:
Since pl'im(1 - G.) = 1 - Gi. E(d,;) = 1 - Gi.and palm-1172 [(1. - (l-G¢)]-H.- = 0
i=1 -
with H.- being a nonstochastic variable. we can replace (1-5.) by d. in [(13)

without affecting plimi—fla). Thus. eq.(3.3.10) becomes

 

 

 

 

'~.~ AX 1
AA 1—1- 0 o
01
xx x'x
.3 3.21 o 0
[was .. . ..- (3.3.11)
0 0 Pagixakm.‘ 0
=1
2n
0 o o ~2‘
l “I

-39-

From eq.(3.3.9). eq.(3.3.11) and eq.(1.5). we have

F

 

 

 

 

 

( . -1
X X .e
[ $21 0 0 lXiAl
1 All
LM statistic N[ .51 (AA-1&- 0 Cl 0 [ixz'izzix‘imi] 0 0
01 01 i=1 l 0
0 0 512
' ate:
51
_ Xe! . ~. ~ ’1 X 1
- [‘5‘] [W] [..
= [(A'figil'1A'e 112 (3 3 12)

 

3f(A'M1A)“

where

M1 = 1" X1(XlX1)-1Xl-

Comparing eq.(3.3.12) and eq.(3.3.4). we see that the LM test statistic is
almost the square of the t test statistic used to test the coefficient of {when it
is added to eq.(3.3.1). The only difl'erence is the diflerence between 312 and 512.

which is asymptotically negligible when p = 0. In other words. Heckman’s A test

 

 

 

-40-

is almost the LM test; the simple A test therefore has desirable large sample

properties (Note 4).

-41..

3.4 Test of Independence in Poirier’s Partial Observability Probit Model

3.4.1 Poirier's Partial Observability Probit Model

In a recent paper Poirier(1980) has proposed the partial observability pro-

bit model:

yfi = 3151 4’ ‘Uii
3152 = $1132 + ”1:2
1 if yfl > O
y“ = {o iiyg. s o
{1 ifyi'e > 0
3112 =

21 = yilyiz
1'. = 1....n.

‘ 1
[:11] 1111:! N (0.2) where 2 = [a q]
12

Here yfi. yi'g. y“ and 31.2 are unobservable. We observe only 2‘ and 21. We

observe z. = 1 if and only if y“ = yiz = 1. and z. = 0 if y“ = 0 or 31.2 = 0 or both.
Some examples of this model are

1) Retention of trainees (see Gunderson (1974))

2) Two-member committee voting anonymously under a unanimity rule (see

Poirier (1980)).

3) Colletive bargaining between cities and municipal employees' unions in Michi-

gan; binding arbitration is imposed if either side asks for it (see Connally

(1982)).

If y“ and gig were individually observed. we would simply have a system of
two probit equations. Instead we observe only the product of y“ and yig. and

estimation is correspondingly more diflicult. If we define

p¢=Prob. (21: 1):}3’05- (911:1 and yi2=1)=F(zifli-3152§P) (3-4'2)

- 43 -
1-p.=Prob. (z.=0)=Prob. (y..=o or y.z=0)=1-F(z.p..z.-az:p) (3.4.3)
where F is the bivariate standard normal cumulative distribution function. then.

the log-likelihood function for this model is

L(P.51.52) = gem .+ (lvziflnflvpi) (3.4.4)
It can be maximized numerically with respect to the parameters p. 31. fig. The
main numerical difficulty involved is the accurate evaluation of the bivariate
normal c.d.f. for arbitary p. Furthermore. there is some (limited) experience
with the model which indicates that p is rather hard to estimate. These prob-
lems would be avoided if the restriction p = 0 were imposed. Then the bivariate

standard normal c.d.f. factors into the product of two univariate standard nor-

mal c.d.f.'s:

F(zi51-315230) = ‘I’Wtfiifi’uififl (3-4-5)
Since univariate normal c.d.f.’s are fairly easy to evaluate. and since the param-
eter p need no longer be estimated. the cost savings from the restriction p = 0
can be substantial. Given that p = 0 is a potentially valuable restriction. and the
estimation in the restricted model is easier. the LM test can be used to test the

hypothesis p = 0.

3.4.2 The LM Test

In order to construct the LM test statistic. we need the first partial deriva-
tives and the information matrix. both evaluated at the restricted MLE.
1; = (15.51.32) where 5': 0. El and 32 are obtained by maximizing eq.(3.4.4) with
p = 0 being imposed. From eq.(3.4.4). the first partial derivatives of the unres-

tricted likelihood are

_6_L__§’j zi-Pi apt
6p .«-.pi(1-P.-)6p

-43-

an _ 3 21-101 61m
671 {=1pi(1-Pi) 551
6_L_ _ " zi_Pi aPi
a—_52 i=1P1'(1-Pi) 552'
The information matrix is

1(3) = c’c
where C is the nx(2k + 1) matrix with with row equalling

-l-
at = [Jodi-12.)] 2[f(at-bi3p)'¢(a-i)@(Ai)zin¢<bi)¢(Bi)zi]
with ¢(-) and f (-.';-) being the univariate and bivariate standard normal densi-

ties respectively. and where

-L
at = zifii . A1 = (1"P2) 2011-901)

-l.
bi. = zifiz . 31' = (1‘P2) Hat-phi)

The first partial derivatives evaluated at 1; are:

5.- : @(zifi)'¢(z¢§2) by eq. (3.4.2) and eq. (3.4.5)
= ¢(a’i)'@(bi) With a; = 3151 - 8; = 3:52.
a
d1 = fvil'¢(vii)dvil

= NEAEWH I ”1'1 < 3:)

@(fi)i—:E2; (see Johnson and Kotz (1970).p.83)

= - N52)
37
d2 = f‘Uiz'M‘UieMUie
= — ¢(b¢) by the same reasoning.
fl 2 —p:
%—= '2 ‘ " ———v<a’.) Wax. -- o by eq (1 4)
11Pi(1'P~i)

-44.-

6L_n zi-Pi ,.~...:
53?’ ‘§1W<e)e<e)e 0 by eq.(1-4)

In matrix form.

‘2 zm—Mwa'i) 1

{=1Pi(1"Pi)
6L- " 333 1. ~..
6—3-- JEWW W x.
n 2t "Pt
.¥.T——¢.<1-p) (mains.-

 

 

J

Elfip—lwawfl

0 (3.4.6)
o

 

The information matrix evaluated at 15’ is

1(5) = 65’ (3.4.7)

'eta’.)¢<8‘.) ¢(ai)¢(b .) ¢<€.>e(8’.)zl
Vplil-Plj V$1(1_p15‘31 VP1(1-Pl)

 

c = . Z . (3.4.3)

mm.) menial)“ twat);
flesh-pal Vpnll-Pnl Vfinll-p'nl‘"

From eq.(3.4.6) - eq.(3.4.8) and eq.(1.3). we have
6L2~~
LM statistic = [5:7] (C'C)fi‘
2

=LZ———v "P‘——-yv meta?) (55).?

 

 

 

=1pi(1-pi
where (5'5) {'1‘ is the upper left corner element of (275)“.

This expression can be further simplified as following :

Let

-45..

 

 

 

 

ae'Zrfiz... sea 1
[VINO-’1’!) VPnzl—Pnjj
then
§~ zg-P‘ -—--¢(<i.)90(5)W
1': 1Pi(1-Pi) i
E": i ‘1'"- . at ..
Q Eli-fl (1.?) ‘——-v(<31)<1>( or: (349)
21' __-___¢Pi
b i
Eiglpiu— P11) (a'i)¢( fit): I
- 22.
63 by eq.(3.4.6)
Therefore.

LM statistic = [-—]). {1f:)]- :[:L_]

— ) by eq...(347)andeq.(3..49)

._. 55°5- (3.4.10)

where B is the OLS estimate for the coefficient of E in the following regression:

= 53 + 8
Define
2:62
e=2-2

Eq.(3.4.10) becomes

LM statistic = ('Q' + e')Z‘k§
= 053 by e'E = O
= é'é

which equals explained sum of squares in a regression of 6 on E.

-43-

Note that Q can be interpreted as a vector of standardized residuals from
the restricted model. This is so since
E<Z¢) = 1PTOb. (2":1) = Pi
and
f 2
Var; (2‘) = E12,; -E(z..)]
= (l-p¢)2'Prob. (21:1) 4» (0-p¢)2-Prob. (2‘ =0)
= (l-Pi)2'Pi + P12'(1-Pi)

= Pi(l-Pi)-

-47-

3.5 Conclusions

In section 3.2 and section 3.4-. the LM test statistics are not very simple.
However. they are performed from the results of estimating the simpler. res-
tricted models. and they have the reasonable property that they are based on
the "residuals” from the estimated restricted model. This avoids estimation of

the more complicated alternative models. at least in cases in which the res-

tricted models are not rejected.

In section 3.3. the LM test statistic is almost equal to the square of the t
statistic for the "A test". Therefore. we can use the Heckman’s two-stage pro-

cedure to construct the LM statistic without estimating the whole system jointly.

-43-

CHAPTERIV

SI'OCHAS’I'IC PRODUCTION! COST FRONTIERS

4. 1 Introduction

The theoretical definition of a production function expresses the maximum
amount of output obtainable from given input bundles with fixed technology. On
the other hand. the traditional econometric methods (least squares of one kind
or another) for estimating a production function allow points above the fitted
line. Therefore. the resulting fitted function just represents the "average" rela-
tionship between inputs and output. and does not necessarily reflect the ”fron-

tier" relationship (the production function).

In 1957. Farrell first explored the possibility of estimating the frontier pro-
duction function in order to bridge the gap between theory and empirical work.
Later. other work was done by the use of mathematical programming techniques
under. a deterministic frontier assumption (see Forsund. Lovell and Schmidt
(1980) for references). However. mathematical programming techniques do not
lead to estimates with known statistical properties. since no statistical assump-
tions are made in those models. Schmidt (1976) explicitly added a one-sided
disturbance to the traditional production function. which yields the model

y: =f(zt:p)+8h t = 1.~--.T.
where y, is the observed output. f (21:53) is the maximum output obtainable from
inputs 2;. p is an unknown parameter vector to be estimated. and the distur-
bance term a; is non-positive. Although. given a distribution assumption for the
disturbance term. the model can be‘estimated by maximum-likelihood tech-

niques. the asymptotic distribution of the parameter estimates is not known

-49-

since the usual "regularity conditions" for the application of maximum likeli-
hood are violated (since y‘sf (2:56). the range of y depends on the parameters
to be estimated). In order to avoid this difficulty. Aigner. Lovell and Schmidt
(ALS)(1977) proposed a stochastic production frontier. f(z,:fi) + 11‘. with 11,
being a symmetric random disturbance -- v, is assumed to be 1°..i.d as N (0. 03).
Thus. the frontier itself can vary randomly across firms. or over time for the
same firm due to favorable or unfavorable external events which are beyond the
control of the firm. Errors of observation and measurement on output consti-
tute another source of variation in the frontier. Also. the ALS model allows the
firms to be technically inefficient relative to their own frontier rather than to
some sample norm. In summary. the ALS model is as follows (eq.(4.1.1) to
eq.(4.1.4))

yt=f(zt:fi)+‘vt ”uh t=1."'.T (4.1.1)
where u: is a non-negative disturbance term representing the deviation from the
stochastic frontier as a result of technical inefficiency. The following assump-

tions are made:

11., 1'3 1°..72.d. as N(0.a'5) truncated at zero. (4.1.2)
11., and v, are independent. (4.1.3)
it. is i.i.a. as N(0. 0'3). ‘ (4.1.4)

In 1980. Stevenson extended the ALS model by allowing a nonzero mode for
technical inefficiency. A test of zero mode is considered in Section 4.2. Besides
this. Schmidt andLovell (1979. 1980) extended the ALS model in another direc-
tion. That is. in SL models. not only technical inefficiency but also allocative
inefficiency are considered. The simplest SL model (515 I) which allows techni-
cal and allocative inefiiciency is in Section 4.3. A more general model (SL 11)
which generalizes SL I model to allow systematic allocative inefiiciency is in Sec-

tion 4.4. Lastly. a model (SL III) which generalizes SL 11 model and allows

-50-

correlation between these two inefficiencies is considered in Section 4.5.

-51-

4.2 Test of Zero Mode for the Technical Inefliciency in Stevenson's Extension
of the AIS Model

4.2.1 Stevenson's Extension of the ALS Model

Stevenson (1980) pointed out that the ALS specification about the level of
inefliciency (eq.(4.1.2)) implies that the likelihood of inefficient behavior mono-

tonically decreases for increasing levels of inefficiency. This point can be seen

clearly in Fig.4.2.1:

/C(u£)

 

 

o ”t
Fig. 4.2.1

where u; is the level of inefiiciency and k (114) is the (half-normal) density func-
tion of m. According to eq.(4.1.2). the mode is at zero. and the normal distribu-
tion is truncated at 0. therefore. lc (at) is a monotonically decreasing function of
u... Stevenson argues that some characteristics are not likely distributed with
such a monotonically declining density function over the population. The possi-
bility of a non-zero mode for the density function of u; would seem a more rea-
sonable assumption. He. thus. generalizes the ALS model by permitting a non-
zero mode for the density function of u,:
11.; ~ N(p. 05) truncated at zero.
Note that ALS model is a special case of a zero mode (p. = 0) in Stevenson's

model. Since the restriction of ,u. = 0 can be easily imposed in Stevenson's

-52-

model and estimation for the ALS model is easier. the LM test is a suitable one.

4.2.2 The LM Test

Before we derive the LM test statistic. we have to derive the likelihood func-

tion for Stevenson’s model. With a linear model. (e.g. Cobb-Douglas) . eq.(4.1.1)

becomes

y; = 2:5 + U; -1“, t=1. 4' ' ' .T.
where v; is i.i.d. as N (0.05). or explicitly

()- 1 I ”‘2 f u
g1); -—Wexpl 203 ora “U:

and u; is i.i.d. as N (p. 03) truncated at zero. or explicitly.
r
1 , 2pr (U: -u)2
r [‘ ‘——‘]2
Uu for 11.: >0
l0

otherwise
with <I>(-) being the c.d.f. of N(0.1). Therefore. the joint density function for

 

 

 

k(u:)=

w, = v, —ut is
_" 1 , _1 us-ulz “31+“! 2
Mm.) ‘fl exp 2 a + a an. (4.2.1)
o li-‘I>[- ‘E—HZTWuO'u u ‘ "
au
1 104-1» I 1 I ’1
= 3440—] 1-4[;{-£§-+ w.x]].l1-2[-§:—H

where a = (a: + 03)”. A = 53-. (at) = standard normal p.d.f. Eq.(4.2.1) util-
v

 

 

 

 

izes the following formula for integration:

- _ 2 = 1_ .1? b2-4ac . b
{eXp[ (au +bu+c)]du 2 erplL—E—Ll erfc[zTE-]

where erfc (p) = é—fexm-uzmu
p

-53-

The log-likelihood function for Stevenson's model is

‘ 7'
Mama) 1311mm.)
7'
= —211n(2n) - {we - #13114. «2.2mm

+t§lln[1-¢(a,)]- Tln[1-<I>(b)]

l
where at = fi-lf-i- (y,-x,fi)A]
b = —%—(A‘2+1)*

The first partial derivatives of the likelihood function are

4% 317‘: [(yt-ztfi)+#] + figmm.) — gramme)

6L __,1__"[ _ - >\_" . '
'7- agtll}; (y, 3tfi)+fl]zt 4' a‘§‘m(¢¢)3t

r
%£-= -%—§lfi%'+ (yt-ztfi)l'm(at)} + 7%{A‘24-1Y’9'm9)

36;Ié-=- 2—1524'2 —2:3[(y: "M5)"’#]2+ i251: m(at)- -9Mb)
where m(-) = Z:-

1-‘1’(')

The second partial derivatives are

3%” 2+ +02A222<4)-—-<x2+1)z<b)

where z(s)- - s-m.(s) -mz(s)

 

62L thll+2(a:)]
aflafi =012t= -1

-54..

62L ____12 _ _ .
apex Poet-177L010 :2?“ ill:- 2+(yt 11542010

+ -—-(A'2+1)‘*m(b) + —-9‘3—z(b)

 

 

 

 

2 7' 7'
6 L2 = fi-‘EKM’Ztm +-,u.] 2):, 2 §1m(at)+ 2:0 3 2101'”

+—a-?(A-'2+1)%-m(b) - -—()\’2+1)”b z(b)

= 3- Is'ztl-sz'zmtfl

 

0)
“lb
0.)
"‘70.
Q

M
II
p

—_861:8L)\ '= Ely—‘21 t{m(0t) " 3%“? (yg-2¢B))\]‘2(Gt)}

6“ = - -1— [(y -2 B)+ul z —§z'0(a)
65602 a . ‘ ‘ ‘ 2083:1‘ ‘

where Q(s)- =-s z(s)-m(s)

 

62L 2
55:-0 144-32.... )‘3 (at)‘1-{£‘+(yt'ztl3)l 'z(‘1¢)}

at=l

+ #A‘% 1)'”{[-3+A'2(A‘2+ 1)"]'m (5 Fit-{(444- 1)'*z (b )}

52L _ 1 _ 1 7' [E— - .
axaaz - 2031:1176 + (y: thfi)lm(at) afi—Zgia‘lhz + (y; Zgfl)] z(a.¢)

+ z—gidA-h 1)-%- 0(a)

821. = T
802 20‘

+ —T;-b -m(b)'[3—b2+b m(b)]
4a

' fiEKye-zimwle + ital-Barmmdwtzzmd]

-55-

Since it is not possible to calculate analytically E[z(a,)], E[ag-z(a,)].
E[a,2-z(a,)]. E[Q(a‘)]..... we can use the negative of the Hessian instead of the
informaion matrix to construct the LM statistic. Let 3 = (EEK?) where
1:: 0. E X. 32 are the MLE estimates from ALS model. The first partial deriva-
tives evaluated at 5 are

3!
.6_L_— __1_T ._ ~ _1_. ~ _. T «1'2 ,5. 2
6/7 ‘ 52t§1(y‘ 1&5) + EX‘ém (at) 3'0 +1) [a (4-2-2)

where a.’ = i—(y2-z.§)A

—--=0 by eq.(1.4)

The elements of negative Hessian evaluated at i; can be used as substitutes for

the elements of I (:3). They are :

 

~ T 1 7.. 2T~-
H =:—-~~ 2- ~ had-1
2 .2 2.2.2.: ——<. .2 >

when Et = armfi't) - 7112(3)

H --—1-r 1 "
p.3- 52Ԥ121(+zt)

H 2.2 Lima). _1._§:<y -. a); - 242-..... 2 x
A 012331 ‘ 32A,.“ ‘ t t 3K3 77

~ 7' ~ 1' ~ ~ 5!
H z = "LE (y. '115) " 337‘ng - —Z-(A'2+l)”[1—2r-]

M 34:31 208

where 52 = 322'. -m(&)

’V 1 T . ~2~
H35 = 3'2—‘22‘Z‘(1-A 2‘)
=1

-55-

r'v 1 T , ~
Hm = 323212: Qt

I? 1 {x E) ' X f: '5
#02 04:81 ‘ ‘ t 203‘“ 3 ‘

N 1 T «.2...
H». = “272 (tilt-33:5) 2:
0' t=1

~

1 T A. ~
Hm? = —2~a 2 (yt‘ztfi)Qt
a t=1

 

 

 

 

 

 

 

 

N T 1 T "" 2 l T ~ ~ ~2 2...
H =-~+.—- -x -.. —3-m + -2
02,2 204 0'3 £21051: :5) 404;:1': at (at) at t]
The negative of Hessian evaluated at 5 is
2» He: ”in 133.2
19(13)— H,“ H”.
H0302.
Let the inverse of H (1;) be
line ,3... HM see?
.. g... 222 2222
[H(13)]_1 = ~M “A02
H H
l . . . £33202
then.
lar."... ~ ~ .. aflai'l
afi Hm HM HM Hut: 5;:-
o . if” 2?” A7502 0
_ 0 ' . . . $1,222 .0 ‘

 

 

 

 

 

-57-

M N”, u
[ail H 6;:

 

where %is given in eq.(4.2.2) and H“ is the left corner of the inverse of H(5).

-53-

4.3 Test of Allocative Inefficiency in SL I Model

4.3.1 SL I Model

Although the ALS model and Stevenson's extension can deal with technical
inefiiciency. there is another kind of inefiiciency which is not considered in the
above models. A production process can be ineflicient in two ways:

1) Technical inefiiciency
It fails to produce maximum output from a given input bundle. This
results in an equiproportionate overutilization of all inputs.

2) Allocative inefiiciency
The marginal revenue product of an input is not equal to the marginal cost
of that input. This results in utilization of inputs in the wrong proportions.
given‘input prices.

Since ALS model and Stevenson’s extension do not relate to input prices. they

cannot detect allocative inefliciency.

Schmidt and Lovell (SL)(1979) extend the ALS model to permit allocative
inefliciency. They assume that the firm seeks to minimize the cost of producing
its desired rate of output. subject to a stochastic production frontier constraint.
If the firm is technically inefficient. it operates beneath its stochastic produc-
tion frontier..and if the firm is allocatively ineflicient it operates 03 its least cost
expansion path. The least cost expansion path can be derived as follows:

Let 2‘. i=1. - - - .n be the n inputs. pi. i=1.- - - .n be the prices of the inputs. y;
be the flrm's output. We assume the firm's production technology is character-
ized by a Cobb-Douglas production function of the form:

n
y, = e‘ Hzia‘ew‘ (4.3.1)
i=1

-59-

where w, = v, — u, is a random disturbance. v, is a random disturbance due to
external shocks. u, is a random disturbance due to technical inefficiency. A is a
constant term. The minimum cost input combination can be obtained by minim-
izing the cost function

13
Ct = Spa-Tu
i=1

subject to eq.(4.3.1). From the first order condition. we have

.zL: gag-L. : . '1], (4.3.2)
it: pitai

This can be rewritten as
11'1th " 1112“ = Bu. i=2. ' ' ' .71. (4.33)

a
where 8.; = 1n p“ 1
pitai

Eq.(4.3.3) represents the least cost expansion path The deviation from the
least cost expansion path can be expressed as a disturbance term. 81. added to

the right hand side of eq.(4.3.3). That is. 8“ measures the percent by which the

21‘ O O I I I O .
chosen —2 ratio dev1ates from its cost minimizmg value.
a

In summary. the SL 1 model

it
lnyg = A 4' 2mm“ 4" wt. t=1. ' ' ' .T (Note 5) (4.3.4)

i=1

where w, = “u. -'u.,. v, is i.i.d. as N(O. 03). u. is i.i.d. as N(O. 05) truncated at

zero and
1M“ —1n2a = Ba + Ca. i=2. ' ‘ ' .17. (4.2.35)
where
a
Pitai
8: = (82:.' ' ' -5nt). 73-71(1- N(0. Zea):
r022 ' ' UZn‘
2.. = . . . . .
F132 ° ' 01ml

 

 

- 60 -
a, is independent of v; and 71¢.

Note that. when 2“ = 0. this SL I model reduces to the ALS model in which
there is no allocative inefficiency. That is. we can test the null hypothesis of
exact cost minimization by testing whether 2“ equals 0. However. the LM test

fails as shown in Section 4.3.2. A simple test is suggested in Section 4.3.3.

4.3.2 The LM Test

From eq. (4.3.4). the p.d.f. for w, is

 

hm.) = 341% {1-<I:(a.)] (4.3.6)
where
10M
:1, = a

a = (05 + 03)”

an
A = '—
all

From eq.(4.3.5). the p.d.f. for a; is

n-l

- l
ME.) = (an) 2 IE..I'*eXPl-%e£2&‘etl (4.3.7)

Since a, is independent of w; . the joint density function is

f (wt-5:) = h(wt)"'l’(€:) (4-3-3)
2 I 2
"" 1 - 1 - - w
= 2(2") n'a-lzecl ”[1-‘p(at)]exPl'z‘{3t2ulet+31%“
The Jacobian of the transformation. from (w,,s£)' to [1722“. - - ulnzmI is

n
r = 2a... Therefore. eq.(4.3.‘7) becomes
i=1

_.n_ I 2
901121;. ' ' ' .lnxn:) = 2(2") 2 Z—IE...|‘%[1-<Ia(a.)]expl-%-[s£2!&‘s:+302-“

-51-

where

n
wt = lny, "A :2qu
=1

8t=(8231"'-8M)'
8a =1nzlg -mig ’33

B“ = 1n[Pit “1
pitai I
Hence the log-likelihood function for SL I model (eq.(4.3.4) and (43.5)) is

L(1$)= tiling (lnzu. - - - .lnzm) (4.3.9)

7'
= Tln2 - 24112:“) + Tlnr — Tlna - gum“) + 21n[1-<P(a.¢)]
t=1

where 13 = (A:.:.2,,.a1. - ' - .an).

In order to construct the LM test. we need the first partial derivatives with
respect to elements of 2“. evaluated at the restricted MLE
5: (Z.X.;.0.El. - - - .3“). where 23.5.31. - - - .31; are the MLEs subject to the

restriction. The first partial derivative with respect to 2“ is

02-1
alnL T -1 _ _;_ 6836;: 8: (4.3.10)
t=l

The first term in the right hand side of eq. (4.3.10) does not exist when %—?L1 is
(:6
evaluated at 5 since 2;.) does not exist when 2“ = 0. Therefore. the LM test can

not be used in this model to test Ho: 2:, = 0.

-62-

4.3.3 AnAlternative Test

Since the LM test fails in SL Imodel. an alternative test is suggested in this

section. Recall that exact cost minimization is attained when eq.(4.3.2) holds.

i.e.. when the following equation holds:

Pa 21;: _ ai

Pltzlt a1

 

a:
A simple test of exact costminimization is to see if If)“ 2“ are the same for all
1: 1:

on
observations: they should be exactly equal to i—for each observation. There-
1

fore this test has a power which is equal to 1 (Le. under Ho. Prob.( type I error )

= 0. under HA. Prob.( type 11 error ) = 0 ). Since

 

 

 

Pizza
Pizza = C: = Lactor share for input i for observation t
pus“ purl, factor share for input 1 for observation t
C:

= ratio of factor shares for observation t

this test is based on the ratios of factor shares.

-53..

4.4 Test of Systematic Allocative Inefl’iciency in SI. 11 Model

4.4.1 SL 11 Model

The SL H model is composed of eq.(4.3.4) and

1X12" -1Ma = Bu ‘1" 8“. i=2. ' ' ‘ .71. (4.4.1)
where
Bit = 114%.}
Pitch;
'022 (TenW
8; = (82:. ’ ' ' .8“). i.i.d. N(£.Ecc). With 2“ =
(01:2 ' anal

 

 

a; is independent of v; andug.

Note that E(e,) = 5. If E = 0. this reduces to SL I model and implies that
there is no systematic tendency to over- or under-utilize any input relative to
any other input. There is a well-known argument (the Averch-Johnson
hypothesis) that. suggests that firms which are subject to rate of return regula-
tion will tend to use higher ratios of capital to other inputs than cost minimiza-
tion would dictate. Schmidt and Lovell (1979) supported this hypothesis by
using data on steam-electric generating plants. In their paper. they used the
likelihood ratio test for the null hypothesis Ho: 5 = 0. Since the SL 11 model is
easier to estimate when 5 = O is imposed. we can use the LM test to test this null

hypothesis.

4.4.2 The LM Test

From eq. (4.4.1). the p.d.f. for 8; is

-64-

n-l 1

-— —— I
V48!) = (277) 2 lzccl .zexP -%(81—€)'2Ec‘(8¢-€)] (4-4'2)

Since a; is independent of w, from eq.(4.3.6) and eq.(4.4.2), the joint density

 

function is
f(‘wt.8¢) = h(w:)‘19(8c) (4-4-3)
‘2' 1 1 . 1 we 2
= 2(21r) 2';- lzul'”°[1-‘P(a:)]exp[- 2—(81-5) 2.1‘(e¢-£)-2‘[T]
The Jacobian of the transformation from (w..s.)‘ to [lnxu.--°.lnzng]' is

B
r = 2 a... Therefore eq.(4.4.3) becomes
i=1

-1.
2

g<1nz...~-.1nz...>=<2n> -§-Iz..I—*-2[1-¢<a.)1

2

n
11);: lnyg “A -{2ailnzu.
=1

whe re

8: = (82:. ' ' ' £110.:
Ea =1nzu -1111“ "Bu.

Ba = [Pam .
pitai]

Hence the log-likelihood function for SL 11 model (eq. (4.3.4) and eq.(4.4.1)) is

L(19)= ‘Erllng (1m... - - - inc...) (4.4.4)

7'
= --T—;'-ln21r + Tlnr — Tlna - glnlzul + Elfin-Mad]
881

12
T1112 2g“: 92““: 6) 2m 0

where 19 = (£.A.A.a.2wa1. ' ' - .an).

 

 

l

~~~

The restricted MLE is 3=(5.A.A.E.E...E..---.'&’,.) where 2:0 and

331.3%”.31. - - . .211. are the MLE's from SL 1 model when £= 0 is imposed.

-65..

Then. from eq.(E-l) in Appendix E. and eq.(1.4). the first partial derivatives

evaluated at 5 are

. [N
[6L 25:1[27' gt]

66 _ ¢=l

0 - 0 (4.4.5)

where Egg. 2'. are 25:1. 1:; evaluated at 5 respectively. Also. from Appendix E. the

0(5) =

information matrix evaluated at 19 is

Tee 2 0 1:;
~ _ 0 122 ~ 0 [24
[(13) — 0 0 18.33.. 0 (4.4.6)

.1 éa I24 0 Ian.

 

 

 

 

 

 

where
'1'” - - 7 l
5252 teen
[55 = . . . . = TEc-cl
{i.e.. ' ' 14.5..
F” ~ ‘
1‘2“: ' ' 1‘2“»
[Ga =
L‘fla1 ' ° 1‘75““
With
A; “-1 n ~u
1‘1“! = Tal ‘gza l l = 21 . ' ' .71..
7....“ = -T&,:IE'“. 1.. h = e. - ~ . .n.
~ '1... {._4. .9.
122 = ' In {15a

[00
(see eq.(E-2)‘ to (13-9)”)

1491. "1Aa,.
124: [M1' "11.11,.

 

 

1;“: . . . I...

b

-35-

(see eq.(E-6)’. (E-lO)’. (E-IB)’. and (E-15)')

1w. - - - 14.4.1

22:23:
1a

M028 ' ' ' [Winona

(see eq.(E-l 1)’ to (E-13)')

 

 

~

halal ' ' ° 1‘11“”. 1

 

 

{anal ° ° - land.
(see eq.(E-14)’)

From eqs.(4.4.5). (4.4.6). and (1.5).

r W-l

. if” 0 j;

LMstatistic - 6L .7 -[o o F] o .r~ 4 0 5L
- a; “ ' ' 6a ~ tutu ~ ’

3 . as
1;. o I... 16¢

1 .

' -1
6L ~ ~ ~ I". - ~ - at.
= {3?}{1ee " I £a[I can-12+, 221! 24] 115a} [3?

 

 

Note that the average of the a. .‘E. is distributed as normal with mean 6 and

varianve-covariance matrix -;.-En. Therefore. T5251: follows a x2 distribution

with n-l degrees of freedom. Since E is near 0 if E(a)=£=0. then we could
accept Ho: E(s)=£=0 if T525125 is near zero. However. we do not observe E and

)3“. We could try to use their estimated values. say 3 and in. to construct the

'

6’

test statistic Tg'fgs. However. this does not work: this test statistic does not

have the same asymptotic distribution (XE-1) as the test based on 1‘5 and 2". The

-37-

reason is that the difierence between E and '5 does not go to zero any faster than
if does. Explicitly.

g; = '5; + (1116‘ " Ina.) "' (1111;; -' lnal)
so that plim VT ('5 - E)¢0. A test can be based on the value of '3. and indeed it is
clear from (4.4.5) that the LM test itself is based on E’. but the correct covari-

ance matrix is more complicated than just 2;}.

-68-

4.5 Test of Independence between Technical Inefliciency and Allocative
Inefliciency in SL III Model

4.5.1 SL 111 Model

The Si. III model allows correlation between technical inefficiency and allo-
cative inefficiency. i.e. correlation between 11.. and the absolute value of 8,. This

can be formulated as follows:

n
my; = A + Bailnxu + 1.0,. t=1. ' ' ' .T. (Note 5)
i=1
1M1; -1nza = Bu + Ea, i=2. ‘ ‘ ' .n
WNW} 2]
8: f '
where

wig”! ‘1‘:

B“ ___ in Paul
Pita:

8: =(82:.' "£710.
£= (£2. ' ' ' .tnl
11-: = lufl
[03 211:]
2 " >3... 2..
Eu: = (0142- ' ' ' noun)
F022 . . . 0’23.
' 286 =
Lang 0 . 0 am‘

 

 

v; is independent of v." and 8;.
v; i.id. N(O. 03)
Note that cov (u, .83) = 0 but that

-69-

0°”(uts lea l) = (ZUuVO‘u/"MV 1-pf+p.arcsin(pi)-1]
> o. 11p.- .. o
where p. is the correlation of u; and a... -1<p.-,<1. (See Schmidt and Lovell

(1980) p.87.) That is. u, is positively correlated with [NI as long as p. 7! O. or
equivalently. as long as a... .4 0. Thus. the SL 111 model allows a non-zero correla-
tion between technical and allocative inefficiency. When 23..., = 0. the SL 111

model reduces to the SL 11 model.

Since the SI. II model is easier to estimate than the Sf. 11] model. it seems
that we might use the LM test to test the null hypothesis Ho: 2.“ = 0. But as we

will see in Section 4.5.2. the LM test fails again. An alternative test is suggested

in section 4.5.3.

4.5.2 The LM Test

The log-likelihood function for SL 111 model is

L = TIDT + ilD‘f 1‘ + fag) (4.51)
t=1
where
7' = n a.
i=1
fjt = [1“I’(Cj¢)l(277)-§-l G1 |‘”exp(D;.)

_ w.+(-1)’“052“‘(s.-£) jg]:
C}; - av \/ lGIl

 

1 . -1 wt
Djt = "'z‘l‘wt- (Et‘€)lGj “.1
I 05"“? (-l)j+12uc
G} = l(-1)j+lziu 2::
GN' (_1)j+lGNE'
0:" = (_1)5+1Gsv 653
3' =1.2

 

 

- 7o -
and

2‘“ = -' '—1_zu:2:-cl
1?}:
RE = 05 " Baez‘s-£121.":

N! ___ .1_.
G Ra
05"“ = 2.7.1 + 2.:‘Ei.fi—E...E:.‘
0
RC = (05+03) "' Banzai-12in:

n
lzl = aglzul + Bani cofa0t07(aui)
i=2

1%
lGll = (05+05N23cl + 2201“ conCt°r(aui)
%3
Since the null hypothesis is 2..., = 0. we need the first partial derivative of the
log-likelihood function with respect to )3.“ evaluated at the restricted MLE
3 = (i....Z.E..33.3’3.§...E). where if.“ = D. and KENSEEEECUE are the MLS‘s from

the SL 11 model. From eq.(4.5.1).

 

 

 

 

 

 

l 61. '
ang
at. =
62.“;
5L
(dam.
where
.64.: f __1__{af .. . ”ed (4.5.2)
30:41 g=1f1:+f2¢ 60:“ 50ml
B—I’L’l- (c )ac. (2 )Jé'lo I'*e (n) (453)
Baud-llai‘aaui 7’ 1 XP.;: ..
n[ 3
"' 1 "'5lGil 6D;
+[1-‘P(°j:)](2fl') zl-E'IGII 2 60...- eXP(Djt)+fj¢5';::'z

-71-

6c- u:

_L‘_= _ 1+1__.(62 _ "El:

601“ ( 1) 60‘“ 8‘ £)\/ IG

BIZ] aIGII

160 -m-—
u'l.

 

 

 

 

 

+ .....(-1):+21..3 News) \/TGT. “’3 ac...
lzl lGlla

501': _ 1 althN'wz+(-1)j+1(8¢’$)'Gsvw2+(‘1)j+lw¢GNE(3¢"€)]
60m - —2 6015‘

_ 1 6[(s.-£)'GSE(8.-€)]

2 60"“
66:21 = 6615;" = cofactar(a.d)
ui wt

60”, = 1 6(2uc2c-elzve)
601;; RE 60'“;

a[2l:-€12‘;5 6214:2811]
6053 _ 60...
60m - R02

- - 35’s
- Ecclzuczuczul 50m

 

G

~ 6 GI
Note that if Eu; = 0 then Cu = 02‘. Du = Dag. f“ = f2; and —‘L—‘l—= I ll —= 0

since cofactor (a...) has 2.“ as its first column. Thus. when in = 0. from
eq.(4.5.2).

afu . afzt I501: 6‘32:

60... T 50.“ = ‘¢(°1:)(21r) 2 IGII'xeXMDiUW do... {“30 (4.5.4)

[an an
It | 2:

+ fit'
60.“ 60...- swan

= 0,
since

I501: 1 502: I
an... ' ac... I3...”

 

_ 62'“ _ -
‘I‘1+l)'_(aa... 2. 9V?-

-73-

 

 

and
[60,. . 602.] _ I_6[w.GN'w.+(8:-£)'GSE(8:-£)]I __ o
bamfi do... Jinan I 60'.“ Ifwao " .
N! 53
(because 6G .. = 60 ... = 0)
60m; 2m=° 60.4 xut=o

 

 

Substituting eq.(4.5.4) into eq.(4.5.2). we have

.61. i T 1 (61.. are.)
" = I ~ = 0 4.5.5
[aaui zu¢=o ‘gl 2! It [Bout aaui Jzuc=° ( )

Eq.(4.5.5) holds not just at the restricted MLE 13. but everywhere when

 

E.“ = 0 is imposed. Therefore. the LM test fails in this case.

4.5.3 Alternative Tests

Since v, is independent of ca anduf. we can test the correlation between 11.,
and Is, I by seeing whether w.(av, an.) correlates with I81: I. If we observed w,
and 8‘, we could calculate the correlation of w, with Ital . i=2. - - ' . n and do
(n-1) univariate t tests. or regress w, on a constant term plus leg, I. - - - . Is,“ I

and do an F test. Since w. and Isa I are non-normal. these tests would hold only

asymptotically.

Because 11:, and I a“ I are not observed. they are replaced by the estimates
1'13, and I?“ |. The above tests still hold asymptotically since the estimation

error part of 17). and 311: goes to zero as T-m while the w, and a“ part does not.

-73-

4.6 Conclusions

There are several extensions of the basic ALS model. Stevenson's non-zero
mode for technical inefiiciency is discussed in Section 4.2. The SL I model
which allows allocative inefficiency in addition to technical inefficiency is dis-
cussed in Section 4.3. The 31. 11 model which allows systematic allocative
inefliciency is discussed in Section 4.4. The SL 111 model which allows correla-

tion between technical and allocative inefficiency is discussed in Section 4.5.

The SL 111 model reduces to the SL 11 model when 2... = 0; the SL 11 model
reduces to the $1. I model when S = O; and the SL I model reduces to the ALS
model when 2,; = 0. Also. Stevenson’s model reduces to the ALS model when
,4. = 0. Since the estimation for the more general models is more complicated. it
is reasonable to test the simpler models based on the estimates which impose
these restrictions. However. the LM test fails in testing 2“ in Section 4.3. A
simple test based on the ratio of factor shares is suggested. The LM test fails
again in testing 2.“ = 0 in Section 4.5. Tests based on the correlation between

w, and ls. l are suggested.

-74.-

CHAPTERV

SUMMARY AND CONCLUSIONS

There are three kinds of tests for model specification -- the Wald test. the
likelihood ratio test and the Lagrange multiplier test. They have the same
asymptotic power. Therefore. the choice among them depends on computa-
tional convenience. Since the LM test is based on the restricted estimates. we
choose the LM test when estimation is easier in the restricted model than in the

unrestricted model.

In Chapter 2. the LM test is applied to distributed lag models to test
different alternative specifications. The test of the geometric lag specification
against the alternative of no lag is discussed in Section 2.2. The corresponding
LM test statistic is equivalent to the square of the t statistic for the coefiicient
of the lagged independent variable when it is added to the restricted model. In
Section 2.3. the rational lag specification is tested against [two simpler alterna-
tive specifications. The results are similar - the LM test is essentially the F test
of significance when lagged independent variables are added to the restricted
model. For example. comparing eq.(2.2.4) and eq.(2.3.10). the LM statistics are
similar (note that M, in eq.(2.2.4) is the same as M; in eq.(2.3.10)) because the
geometric lag model is a special case of the rational lag model and both models
have the same alternative specification (no lag). Note that the LM test statistics
' in Chapter 2 can be constructed from OLS residuals without running an MLE

procedure. since the estimation of the restricted model requires only OLS.

In Chapter 3. the LM test is applied to qualitative and limited dependent
variable models. Since the estimation of the restricted models can not utilize

the OLS method. the LM statistic can not be expected to be constructed from

-75..

OLS results. The simplest example is in testing for sample selection bias in
Heckman's sample selection bias model (Section 3.3). The restricted model can
be. estimated first by probit analysis. and then by OLS. Therefore. the
corresponding LM test statistic can be constructed from the result of this two
stage estimation procedure. Actually. this LM test statistic is equivalent to the
square of the t statistic for the coefiicient of the probit estimate for the inverse
of the Mill’s ratio when it is added to the regression. This means the simple "A
test" proposed by Heckman is asymptotically the same as the LM test. There-
fore. the LM test gives the justification for using the A test in large samples. The
LM tests in the other two examples (Section 3.2 and Section 3.4) do not have
such clear interpretations. In the test of the Tobit specification against Cragg's
generalization of the Tobit model (Section 3.2). the LM statistic itself is not very
complicated. but not much can be said about it except that it is indeed based on
the Tobit residuals. In the test of independence in Poirier’s partial observability
model (Section 3.4). the LM statistic is equivalent to the explained sum of
squares in a regression of residuals on a set of regressors. The regressors are
related to the terms in the information matrix. but otherwise have no clear

interpretation.

In Chapter 4. the LM test is applied to stochastic production and cost fron-
tiers. A basic model proposed by Aigner. Lovell and Schmidt (ALS model) is
presented in Section 4.1. There are several extensions of the ALS model in the
literature. Stevenson considers a non-zero mode of technical inefficiency. while
Schmidt and Lovell consider the possibility of allocative inefiiciency (SL I
model). of systematic allocative inefiiciency (SL 11 model). and of correlation
between technical and allocative inefiiciency (SL III model). Stevenson’s model

reduces to the ALS model when the mode of technical inefficiency is zero. and

-76-

the LM test of zero mode is presented in Section 4.2. The 51. I model reduces to
the ALS model when the variance-covariance matrix of the allocative
inefficiency errors is zero. In Section 4.3. the SL 1 model is presented and the
LM test is shown to fail in testing the zero variance-covariance matrix of the
allocative inefiiciency errors. A simple test based on the ratios of factor shares
is suggested in Section 4.3.3. The SL II model reduces to the SL I model when
the mean of allocative inefficiency is zero. The LM test of zero means is
presented in Section 4.4. The SL 111 model reduces to the SL II model when
technical and allocative inefficiency are independent. In Section 4.5. the LM
test is shown to fail in testing the independence of technical and allocative

inefficiency. Alternative tests based on the restricted estimates are suggested

in Section 4.5.3.

In summary. the LM test. except in a few cases. can be used to test the ade-
quency of the simple models which we discuss. Since the simpler model usually
involves a simpler estimation method or less computational cost than the more

complicated alternative. the LM test can be useful.

-77-

Note 1:

If the geometric lag model is

{—1
y, = 61», +72, + 8;. where w, = Skirt-.. t=1. - ' ' .T
i=0

then the LM test statistic equals
. _ . 2
{04441240 ”(t-15127:]
32(Xt.-1M2Xt-l)
which is the square of the t statistic for the coefficient of X.-. in the regression

0f Y: 011 (X:- zt- Xt-1)- Here M2 = I " (Xt. ZJIUQ- ztllxtu 21)]-109: 21). and

7'
2'5?
¢=l

2 -.- T with 3, = y, -EX, -;z,. and a; are the OLS estimates.

 

0'

Note 2:

Godfrey (1978) applies the LM test on an autoregressive model (.41? model) and
has a similar result. After all. both the AR model and the rational lag model are
special cases of the ARMAX model (see Nicholls. Pagan. and Terrell (1975)).

Note 3:

We can use eq.(1.5) directly. but the resulting LM statistic looks messy.

Note 4:

The same conclusion had been derived independently by Angelo Melino of NBER.

-73-

His paper will appear in Review of Economic Studies.

Note 5:

y; is exogenous. while za's are endogenous.

APPENDICES

-79-

APPENDIX A
INFORMATION MATRIX FOR THE GEOMETRIC MODEL

The log-likelihood function for the geometric lag model is
L = constant - g—logaz- —022(y - 5w, - no”)?
i=1
The first partial derivatives are

_O_L__ 1— {-1
6A - azgffim 4' “704 )8:

 

6L 1 T .
._ we
67' =a2.§. ‘
6L 1 T . 1
—= — A s
6770 02:?1 ‘
6L T 1 T 2
= ———+ s
602 202 204.31 ‘ 1
wt 0
where R, = TA =2iA“ 12,...
i=1

The second partial derivatives are

 

a_f_L 1 .
= — R + t M” 2 + a -som thin

5T2 ~02§1{(fi t 770 ) c 9 9}
a__zL _ 1 7
6T2 — 02 MW
621. _ _1 T A2.

2 " 2
5’00 -0 1:1

 

1 T t-1
6A66 021:1[w‘wm 770A ) R, at}

 

 

62L 1 1- -1 t t- -1
= -— -t

 

= ‘— :(35’: + “704‘ ”8:

-30-

 

 

 

The elements of information matrix are

 

52L 1 :-1 2
I = -E = — + t
M [673'] ag‘éle: 7704 )
62L 1 T
I E _—
25 [6132 02 ‘2‘”:
I ___ E'azL =__ 1 12‘
"0’70 (6770 021:1
, = _El 62L = -_L... .T.: .2;
0203 b6(a.2)2 2 0.4 204

 

 

 

11mg -EI6>6\::O = 'JT‘ZQWRt 'I' ”ION-1)”
- 4.22:2 -

[W " ..E'angh '37:“:

1.3.2 " -EIa;:I;2 - 0

 

-31-

The information matrix is

1

U... I... 1....) o
In}. I” 15710 0
Incl Inofl loan.) 0
I O O 0 ”102,2‘

[(13) =

 

 

'Under reasonable assumptions about the explanatory variables. lyrfnm. 17,-!an

and IFIMO all converge in probability to zero. Thus the information matrix is

singular asymptotically. if ’00 is estimated as a parameter.

 

 

-82-

APPENDIX B

INVERSE OF A PARTITIONED MATRIX: "OR 'I'HE RAT'ONAL LAG MODE L

 

 

 

 

 

 

 

 

 

C J N RI Q‘
where
[1 «I 1 ~ .1 1 1 Go I
P Tina”... 71.1.1. _ $32—$95” T~2X'X“
1 = 1 ’V 1 'V - 10 6'0 .150 .
._., fi ___ no; a 001 no
Too. T» I T‘b’ZXY TEZXY‘
1 ~ 1 1 I
R _ FIMO _ TEZX'X
1- 1 ~ - _1 E730
7.71“," L -T,, —-—2-X00X
_ 1 v _ 1 1
Q1- -7w—Ill°a°- Ta ___-XX
Step 2
IA F 1 _
B G- [Pl-R1 Q1 R1]
II + 1 X'MJ 3° X'M4X -1
_ # T~2 O O T62 O O.
- E J¥.M;X'o 53 Y'Ah)?
.T002 .0 T521 O. .04
[P2 R21"
" 32 02

Hence.

= (P2 " 32425135)“

-83..

-1

_ 1 ' 1 ' ' t - ' t
_ I“, + WYOIJ4XO — .T—aTz—XoIJoXoo (XOOJ'J41Y00) Una-141$

:1“

Sinc e

X;M4Xo " X;IJ4X00(X;0!J4Xoo)—1X;01M4Xo= X;[I " .Xu(x;a.Xoo)_1Xu]Xo
= 0
where X. = [J4X.. X» = mx... The second equality holds because X. is contained

111x...

3: 'QEIRé(P2 - ReQz-lfiérl
= -Qz'"f?éA

= 8L(XL.M4X..)*1XL.M,X.
0

_ 1 [’1‘]
' 2710
F: ‘(Pz " RzQilRé)-132051

= “M2051
= é—[Iw 0]

‘10

G: (25‘ + 0511?;(1’2 — Rzai‘Ré)“RgQi’

52 , ‘1 . . . .
[#Xufl‘x”] + (3:5 "1(X..M4X..)“(Xufly‘fi)(X.f/I4X..)(Xufithrl

1 ~2 . -1 [’1‘
= 302 To (X00514Xoo) + [0

 

[Ip' 0]]

-84..

APPENDIX C
SOME EXPECTATIONS FOR CRAGG'S EICI'EI‘ISION OF THE TOBI'I' HODEL

E(d,)= l-Prob(d‘=1)+ O-Prob(d, =0)
= Prob(y,>0)
= M21“ + 5)]

E(dgy,)= ECU: lyt>0)-Prob (yt>0)

 

= [37:52 + m

 

 

 

 

2;;5—2‘1'021'Wxtfii)
= 112113 + Mam-@1241 + 6)] by eq. (3.2.9)
Ed( 12:31:) E(y;2|y¢>0)- -Prob (yt>0)
= WWW: lyt>0) + [E(yt lyt>0)]2
= [1 — $77?- Eflm " 223:2 ‘72 +{\31162)2 +2(3152)02' m[‘—}+U§ m 212%}
'-' 022 + (1': 592 + 02 ($132) "LI-£1
= EEU + (2:5)2 + ztfi-m(=:fi)]"1’[x:(£ + 5)] by eq. (3.2-9)
"' 3:132
02

|y>0

 

E(hy¢ ‘3tfilyt>0)= Er!
_ [3:32
—m ,
02
since

E(ytlyt>0) =3152+m a 02.

 

3:52 .
2

-35-

APPENDIX D
INFORLIATION HA’I‘RIX FOR THE SAMPLE SELECTION BIAS MODEL

The log-likelihood function is

L =-id‘1nFi + (1 - 40an

i=1

The first partial derivatives are

11;: " I e _ p (My‘ZuHJZ + (la-pa) (3111“31151)A¢ _ _p Bi]
6p {.1 1-pz (1-p2)2 012 (1—p2)2 01 Ft (1-P2)2 Ft]

Egg-___ id-‘Hyu‘zufiflzit __ Jo 3'11 AL]
361 m «fa—p2) l-pz 0: A]

I’ o
21.- .. - m _ .. -
652 - ig‘ldiza F1 (1 dl)z2i7nl

 

 

 

 

6L = i“: . __1__+ (yu’ziifii)2 _ J (ya-31151) At]
501 m 0: (l-pzki‘ 1-92 612 Ft}
where

ht.= h(yu '31131. -22152).

A¢= f ua'hWit'zufliuuaddua:

“32152

Bi.

f ugt'th-zufii- “21)‘11‘21:
43152

F1: f h(yu"$ufli-ua)duziu

"2132
= M'Zztflz)
m.‘ “"32152).

Whenp = O.

 

-55-

f ua'¢(u2¢)duzi

 

%:= «an: ' "-' 3(u21lu21>-321132) = M (D'l)
fa @(uztIdU-zi
“21 2

Hence the first partial derivatives evaluated at 13 are

~ ~

 

 

6L " 3111-31151 X91 ~ ¢(‘22i52)
—:..-= ‘ ... I‘ = —-— where . = ... ,
6p Ed‘ 01 w 01 AI 1-‘I’(-32u32)

6L_6L_BL

- =0 (29.1.4
7-51 35-; 601 y 9()

The second partial derivatives are

 

 

2 2
'3‘}: (11:p2)2_—L_;¥ldl {(112312);«Izzddyu-zl‘mf". 28(3;2[;:) a? ‘§1d1(yu-2u51)—

04-102)2 _ A12 _p_=__n [_B__¢:_ D:
+(1-p2)‘ a? 121“?!“ 31251;:[8'F-B F} T75]- (1_p2)4i21 F12 F1

-ML“ 31 _ 2pL1+pz) 1 _ _Buh]
(1-p2)3 “I F" (1_ pg)‘ 0 Eddy“ 21151)[? F3; F22 1

 

 

 

 

I 2
2 P(1+p2) (ya-21150131 __A; H
fl: n szii(yli‘zlifi) _zl‘[(1+p)%- + (1‘102) 01 [F1 F12 2
apafil i=1 (1-'/Jz)2012 (1-p2)201

 

 

 

. I p2 Pi- A‘BiI
zitl1_p2 LF‘ R2 4
(1'P2)201 I

 

 

 

2 h." A;
626+2' = ”((11:52?2 1 it @321“! H. "'3 1‘51) —'[zzifiz+ 2.1:]

- —&—§ld1321(1_p2)2 F—:{(32152)2" %]

-37-

 

 

 

62L 2p21+ 1 n A:
apaal = (l-p a8 1‘2‘1‘(y“-z“fil)2- ((1 $32320 “Edam-21:51)}:-
62L - n -21‘3“ f p2 zitzu Af-BiF‘j

 

amen; 1?, 0?(1-p2)_(1-p2)2 of F3 1

66165;. m 1-p a: I F‘ WM]

 

62 L “Edi p 311221 [-22:52th _ A: ’11.]

 

62L = i {-2(yli-zlifil)zii + J 311 [141+ p (ya-21151) IB‘ _A‘ZI]
63160, {=1 (1-p2)a? 1—p2 of [Ft 1—p2 0; [Ft F? J

62L 2
0132862: l-pz an?“ 2‘ 2““

‘2dt32132t i;.2—+ 2(1'dn3322321(32ifiz)mt " 2(1‘dtk2t32tm12
(all i (81 {-1

                           

 

2 0
666261271 = T310210 24:321(y1t‘31¢51)%‘[12152 4' 2‘7}

 

 

62L J2
6012 = Egldt- (1_ :2)a4“fi12-'d‘(yli-zlifll) (l- pz)2 a4“1fi1d((y1“2u51)2 [-A-:-— %
Tag—E'ad‘ (91¢ '31¢51)2—

where

f ugt'thu'zufli- “21)‘11‘21
’32:”:
f ua-Myu-zuanumdua
'32":

Whenp = O.

-33-

 

 

 

 

A.‘ _ _
F:- M . (D-l)
ht. __ ‘P('32¢52) _ _
n " 1 - u-zaaz) ‘ "‘ (D 2)
E f “221:9”(1‘20‘11‘2: -
F‘ = 42‘? = E(u§1lu2¢>‘zafiz) (D-B)
i
f ¢(u2i)du2¢
”at”:

‘3 [EWa lu2i>-32152)]2 + V°T(u2i |u2i>‘32-;52)

=M2+ (1 + ZiM “A3

= 1 + zix‘
where z‘ = 21:362.

The elements of the information matrix evaluated at '3‘ are

a; = -E[g:lz‘ evaluated at 1’;
= -§<1-aa + :1?" sax-5;) - -..i,,-"'a?<1-6;>[<1+am-Xfl+ Sin-annex)
{=1 01 (:1 01 {=1 ‘81
= 1:51-53)th unth 2‘ -' “3213.2

since eq.(D-l). (D-3). and

E(d¢) = l'Prob(d.‘=1)+ O-Prob(d¢=0) = 1 - @(-zafiz) = 1 - Q (D-4)
and

Midas?» = imuamawmuma) 09-5)

{:1 i=1

-39-

= 1:1(1+P221M)012(1'Gi)-

1,51.—‘§1(1-a)—r“*‘ by eq.(D-1)andeq.(D-4).

~

IP52 = 0. since

E(d¢uu) = Emu ldt=1)'P"°b 011:1) (D-6)
= 012N‘P70b (di=1)
=POIN'Pr05 (di=1)

1m,1 = o by eq. (12-6).

 

i (1-§)ziizu

131,1 = 312 by eq. (D-4).

{81
[p152 = 0

1;"! = 0 by eq. (D -6)

gape:21¢(-32i52)321132t(32132) 4' f: %é‘za
[280(- 21:52)]2

 

' leétzzi(zzifiz)¢(-22¢fiz) 4' g1 M42132) zaza

= ‘fil‘i‘mzé‘za, .by eq. (D-Z) and eq. (D-4)
54-32132)

where ~ = —=—
mi @(‘32150

1,3,1-0

A. n N

T1;- <1-e>+§—:af<1-§)
011:1 01
2 2 ~

= .72.. (1— ) by eq. (12-4) and 99-(9‘5)

 

-5“)-

APPENDDK E
INFDRMATION MATRIX FOR SL 11 MODEL

From eq.(4.4.4). the first partial derivatives are

5L Tn
_ =22(8a_€i)0a.l=2.'n,n

=11 =2

Q)@
a

 

32—- §5m(at)_ Trn(b)z>‘_2++1)%_2[g;_]
£81 :31 0

 

6L _ 1”
6A - a’Eganz(a¢)1D‘]
6L .. I. _1_’ _
Ba - a + azmwtxmmm 03:33;
6L ._ TBlnlEccl+ 1 n n «U; j
60M: - 2 60M +=2‘ 7.13;;ng (3X81! Efla a”
where
amlzccl __ 0M ifh=k
60M, - 20"" ifhatlc
h=2.-ooln' k=2....n
6L T+
a1 1' 0““ Tlt=1j=2i=z
1 ‘7'
+ T wilful:

(E'l)

 

-91-

BL T

A T 1 7' n , 1 7'
—=—+— m 'lnx —— 2 --a"‘+— wlnx
6a,, 7. “a (at) m “Mag; it £1.) 02‘; t n:
h = 2. n.

where a“ is the (i.l)th element 0:251. m(-) = l%'

The seCond partial derivatives are

62L

 

 

 

 

—=-To"“. l=2.~-.n, m=2.°--.n
aézaém
62L
= 0 I: 2.- .n
afial‘
62L - - . o .
6£¢6A-0' l-2. .n
621. _ _ ...
6&6A-0' l—2, .n
62L - - o I o
6&60’ -0. l-2. .n
_6_2L__= -ZT fi(ea-£i)a"‘a“. 1.11.]: = 2, - ~ - .71.
65:50».- t=u=2
62L T " a
= -— a . I: 2. .n
afiaai (Mtge
62L = T l: 2 . n

 

 

aAaa

62L

6/1601,-

62L

aAaoq

62L
axaa

621.

KT. 27
= 33?EJQUH)":;F23u“

-92-

T A
= 272{ (ch) - pvrzwa}

=1

t=l

==O

A2 7' 1 .
= Fawn,“ mm.
3 =1

1 r
= -?Ew‘Q(a,)

‘81

6A6 a.“

62L
mam

62L =
602

62L

6060”

_ 1 T A
.. ——-2 {3'10¢Z(01)Inzfl - m(mflnza}

a,

T

7+ _2 (hi-701:) - 25?“) - 4 2w!
02t=1 0' 0' i=1

0'

==0

(13'?)

(3'3)

(3'4)

(E'5)

(PS-6)

(E‘7)

(E-B)

(13-9)

-93-

2
%= —2Aa(ag)1nzu02. - —22w:1nzu (B-w)

£81

62L 7' n n _ 7"
—.= 2 2 2 (cu —$i)(ej,-£j)a"‘a""a"j —- E-(dmfi’. h=2. - - ' .n (E-ll)

aohhaahh (=1j:2i=2

2 T n n
__6L__= -£o’“a"”‘ + 1—2 2 E (ea-£¢)(s,-g-$j)(a“a”"‘a’"+afl'amam’) (E-12)
aamaalm 2 2t=lj=2i=2
h.l,m=2,-'-.n
62L 1 T n n
*= -Ta"‘o”"‘ + —2 2 2(Eu-6t)(813 -$j)(a“am"a"’+o auam’)
hath. h,lc.l.m=2. .n

12—: "LET: fitézkea ‘50 4' (ejg-£5)]a"‘o*j

 

 

Bauaal 20: “Fe
h. k = . .n
62L 1 1' " n
—-——=--— e-- omakj-i- e-- afl‘a”
601125“: 2a; t§1[j§2( 1‘ £1) E); g 6‘) ]
hi kcl = 2. ' ' ' .n
62 2 1 1'
Bali“; = 'Lz+ %§Z(W)’(‘mu)z’ 32-210mm? (El-13)
r
12 2 i: Z°g(2+5a+€jc‘€i‘€j)
—2a1 t: lj=2i=2
62L _ T X2 7' 1 -
_aalaah — r2 + 02‘=12(a:)(1nzu)(lnzm) a2,é(m“)(lm“) (E 14)
n
+ am' h = 2' . o . 'n
“1“); gauge
62L _= _L... firz( )(lnx )(lnx )_ Li (Zn: “In: ) (El-15)
anhaak 1'2 02::1 a: M ’“ (,2:=1 M kt

 

T

- ma”. h. k 2. '.n
where
2(8) = S'm(8) - m2(8)
0(8) = -m(S) + 82mm -S'm2(8)

P(s) = -2m(s) + szom(s) -s-m2(s)

The elements of the information matrix are

IGsz = To”
[6:4 = 0
[613:0
1“,: O

[Glut = _ 0'
T m
[61% = -

62

I,“ N— 6751A— see eq. (E -2)

(3‘2?

 

-95-

2

 

 

as... .-
IAA BABA see eq. (E 3)
[4 R3- 62L see eq (E-4)
° aAaa '
1A,” — o

 

 

 

L
as— -
[M GAGA see eq. (E 6)
[A - 62L see eq(E-7)
a 6A6
[A0331 = 0
[A N- 62L see eq. (E-B)
a‘ akaa;
1 RI- 621’ see 2:] (5—9)
W 6060 '
1,,“ = o

621.
a-
Ica‘ 60' 6 at

 

see eq. (E-19)

: 2£(ahh)2

[UMUM

(E-3)'

(E'4)'

(E-5)'

(E'6Y

(E'7)'

(E'3)'

(E'9)'

(E-lO)’

(EH 1)’

 

-95-

Iowm = E—a’i‘amh (E-12)’

[ahkatm - Toma” (E-13)’

I‘M“; = 0. l: 1. .n

[ahakz__§_2L__. h. Ic = 1. ' - ° .17.. see eq. (E-13) - (E-15) (E-l4)’
Bahaab

Some elements of the information matrix are difiicult to find and are approxi-
mated by the negative of the second partial derivatives since this will not aflect

the probability limit of the resulting "information matrix."

 

BIBLIOGRAPHY

 

-97-

BIBLIOGRAPHY

Aigner. D. J.. C. A. K. Lovell and P. Schmidt (1977). "Formulation and Estimation
of Stochastic Frontier Production Function Models." Journal of Econometrics.
6. 21-37.

Aitchison. J. and S. D. Silvey (1958). "Maximum-likelihood Estimation of Parame-
ters Subject to Restraints." Annals of Mathematical Statistics. 29. 813-828.

------ (1960). "Maximum-likelihood Estimation Procedures and Associated

Tests of Significance.” Journal of the Royal Statistical Society. Series B. 22.
154-171.

Breusch. T. S. and A. R. Pagan (1980). "The Lagrange Multiplier Test and Its Ap-
plication to Model Specification in Econometrics."
Review of Economic Studies. XLVII. 239-253.

Connally. M. (1982). "The Impact of Final Ofier Arbitration on the Bargaining Pro-
cess and Wage Outcomes." unpublished Ph.D. thesis. Michigan State University.

Cragg. J. G. (1971). "Some Statistical Models for Limited Dependent Variables
with Application to the Demand for Double Goods." Econometrica. 39. 329-844.

Dhrymes. P. J.. L. R. Klein and K. Steiglitz (1970). "Estimation of Distributed
Lags.” International Economic Review. 11. 235-250.

Farrell. M. J. (1957). "The Measurement of Productive Efficiency."
Journal of the Royal Statistical Society (A. Goneral). 120. 253-281.

Finney. D. J. (1971). Probit Analysis; Third edition. Cambridge University Press.

Forsund. F. R.. C. A. K. Lovell and P. Schmidt (1980). "A Survey of Frontier Pro-
duction Functions and of Their Relationship to Efliciency Measuremen ."
Journal of Econometrics. 13. 5-25.

Godfrey. L. G. (1973). "Testing Against General Autoregressive and Moving Aver-

age Error Models When the Regressors Include Lagged Dependent Variables."
Econometrica. 46. 1293-1301.

Gunderson. M. (1974). "Retention of Trainees : A Study with Dichotomous Depen-

 

-93-

dent Variables." Journal of Econometrics. 2. 79-93.

Heckman. J. J. (1976). "The Common. Structure of Statistical Models of Trunca-
. tion. Sample Selection and Limited Dependent Variables and a Simple Estimator
for Such Models." Annals of Economic and Social Measurement. 5. 475-492.

 

(1979). "Sample Selection Bias as a Specification Error.” Econometrica.
47. 153-161.

Johnson. N. L. and S. Kotz (1970). Continuous Univariate Distribution —I. Boston
: Houghton Miflin Company.

Jorgenson. D. W. (1966). "Rational Distributed Lag Functions." Econometrica. 34.
135-149.

Judge. G. G.. W. E. Griffiths. R. C. Hill and T. C. Lee (1980).
The Theory and Practice of Econometrics. New York : Wiley.

Klein. L. R. (1956). "The Estimation of Distributed Lags." Econometrica. 26. 553-
565.

Koyck. L. M. (1954). Distributed Lags and Investment Analysis. Amsterdam :
North-Holland.

Melino. A. (1962). "Testing for Sample Selection Bias."
Review of Economic Studies. forthcoming.

Nicholls. D. F.. A. R. Pagan and R. D. Terrell (1975). "The Estimation and Use of
Models with Moving Average Disturbance Terms : A Survey."
International Economic Review. 16. 113-133.

Olsen. R. J. (1978). "Note on the Uniqueness of th Maximum Likelihood Estimator
for the Tobit Model." Econometrica. 46. 1121-1215.

Poirier. D. J. (1980). "Partial Observability in Bivariate Probit Models."
Journal of Econometrics. 12. 209-217.

Rao. C. R. (1947). "Large Sample Tests of Statistical Hypothesis Concerning
Several Parameters with Applications to Problems of Estimation."
Proceedings of the Cambridge Philosophical Society. 44. 50-57.

 

-99-

Schmidt. P. (1976). "On the Statistical Estimation of Parametric Frontier Pro-
duction Functions." Review of Economics and Statistics. 58. 238-239.

----- and D. K. Guilkey (1976). "The Effects of Various Treatments of Truncation
Remainders on Tests of Hypotheses in Distributed Lag Models.”
Journal of Econometrics. 4. 211-230.

----- and C. A. K. Lovell (1979). ”Estimating Technical and Allocative Inefiiciency
Relative to Stochastic Production and Cost Frontiers.”
Journal of Econometrics. 9. 343-366.

----- and C. A. K. Lovell (1980). "Estimating Stochastic Production and Cost
Frontiers when Technical and Allocative lnefiiciency Are Correlated."
Journal of Econometrics. 13. 83-100.

Silvey. S. D. (1959). "The Lagrangian Multiplier Test."
Annals of Mathematical Statistics. 30. 389-407.

Solow. R. M. (1960). "On a Family of Lag Distributions." Econometrica. 28. 393-
406.

Stevenson. R. E. (1980). "likelihood Functions for Generalized Stochastic Fron-
tier Estimation.” Journal of Econometrics. 13. 57-66.

Tobin. J. (1958). "Estimation of Relationships for Limited Dependent Variables."
Econometrica. 26. 24-36.

Wald. A. (1943). "Tests of Statistical Hypotheses Concerning Several Parameters
when the Number of Observations is Large." Trans. Am. Math. Soc. . 54. 426-482.

0 ._L