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á; ~~.~:~ Discussionw ~o. ~,.~,.~84141989

iomic Research a er., i~imuu uiiu~ i ii n~ i~nqi iii umi uup

No. 8944IDENTIFICATION AND ESTIMATION

OF DICHOTOMOUS LATENT VARIABLESMODELS USING PANEL DATA

by Cheng Hsiao

October, 1989

Identification and Eetimation

of Dicliotomous Latent Variables

Models Using Panel Data'

Cheng Hsiao

University of Southern CaliforniaLos Angeles, California 90089

U.S.A.

Revised September 1989

Abstract

Identification conditions for binary choice errors-in-variables models are explored.Conditions for the consistency and asymptotic normality of the maximum likelihood esti-mators of binary choice models with unbounded explanatoty variables are given. Two orthree step estimators to simplify computation are also suggested. Their consistency areproved and asymptotic variance-covariance matrices are derived. Conditions for the twoor three step estimator to achieve asymptotic ef6ciency are also given.

JEL Code 212

i

1. Introduction

When variables enter into an equation nonlinearly, and these variables are subject to

measurement errors, complicated identification and estimation issues arise. Realizing that

the usual large sample theory scems to fail to provide useful results for nonlinear errors-

in-variabies mo3els and consistent estimatars in thc usual scnsc arc not rcadily available,

Y. Amemiya ( 1985), Y. Amemiya and Fuller ( 1985, 88), Stefanski and Carroll ( 1985),

Wolter and Fuller (1982a,b), etc. have developed alternative asymptotic theories in terms

of the index n- a„6„ with {an},~ r and {6n},~ 1 being sequences of positive real numbers

representing the magnitudes of reciprocals of error variances and the number of data

points, respectively, ïor n- 1,2,.... Consistency and asymptotic normality of nonlineaz

least squares and instrumental variable estimators are demonstrated when n y oo and

an 1- O(n- ~ ) Or a„~- O(R- ~).

While this alternative approach is ingenious and yields useful approximations to the

properties of estimators when error variances are small and when sample size is large, not

all economic data possess the property of shrinking error variances when the number of ob-

servations increases. Unfortunately, without imposing more structural information, there

appears no alternative to the assumption oí shrinking error variances for deriving consis-

tent estimators for general nonlinear errors-in-variables models. However, in many samples

individual observations may be viewed as random draws from a common population. In

this paper we wish to explore the type of data which would allow us to identify a nonlinear

errors-in-variables model under tliis assumption and provide consistent estimators for the

unknown structural parameters when measurement error variances stay constant. We shall

focus our issues on the binary choice models.

We set up the model in section 2. Issues of identification are explored in section 3.

Conditions for the consistency and asymptotic normality ofmaximum likelihood estimators

are established in section 4. A computationally simpler two-step conditional maximum

likelihood estimator conditioning on a subset of the estimated parameters is suggested

2

and its asymptotic variance-covariance matrix is derived in section 5. Simple consistent

estimators for probit model and logit model are suggested in section 6. Conclusions are in

section 7.

2. Thc Model

Let (y;,x~.) be (K -~ 1} dimensional independently distributed random variables with

finite second order moments. Let the expected value of y conditional on x be

E(yt ~ ?;) - 9(?t, B o). (2.1)

where Bo is a p x 1 vector of unknown parameters, assumed to lie in the interior of a convex

compact set O C Ro, with Ro denoting a p-dimensional Euclidean space. Model (2.1) is

nonlinear if g(x~; Bo) is nonlinear in eithcr z or Bo.

The class of nonlinear models we will concentrate on is that of qualitative choice mod-

els, originally developed by psychologists and later adapted and extended by economists

for describing consumers' choices (e.g. see Amemiya (1981), McFadden (1976, 81, 84}, and

Train (1986) for general reviews). This class of models assumes that y takes a binary out-

come and relates the choice probabilities to observed attributes of the alternatives (such as

the price or cost associated with each alternative) and of the individuals (such as income)

in the form

Prob(y - 1 ~?) - E(y ~?) - 9(~0~3). (2.2)

When g(Bo~x) takes the form of the integrated standard normal, ~(Bo~x}, we have the probit

model. When g(Bo~x) takes the form of the logistic distribution, exp(Bo~x}~[1 t exp(Oo~2)~,

we have the logit model.

Suppose x are unobservable.~ Instead, we observe

z - Az f c, (2.3}

where E are assumed to be independent of ? and u and are assumed to be independently

normally distributed with mean zero and variance-covariance matrix f2. When A- IK, and

3

c; is e vector of K errors, (2.3) corresponds to the standard measurement errors aituation.

When A is an M x K matrix with rank K(M 1 K) and c; is an M x 1 vector of errors,

(2.4) corresponds to the factor analysis model with z; being the indicator (or manifest)

variables of the X latent variables x; (e.g. Anderson (i985), Anderson and Rubin (195s),

Lawley and Maxwell (1971, 73)).

Let the conditional distribution oí x given z be j(z ~ z; 60), where óo is an s x 1

vector of unknown parameters. For instance, under the assumption that x is also normally

distributed with mean p and variance-covariance matrix EZ, the conditional distribution

of x given z also has a multivariate normal distribution. This distribution is characterized

by the (conditional) mean

E(? I?) - r~ f Ern(n f nE~n~)-'(? - n,~), (a.4)

and (conditional) variance-covariance matrix,

var(x ~?) - EZ - E2n~(n t nE~n~)-'nEz - A, ( a.5)

The joint density of ( y, z) can be written in any one of the following equivalent forms:

j(y,?) - !(y ~ ?)j(?)

- J j(y,x,z)d?

- Jf f(y ~?,?)j(? ~?)d? ' j(?)(2.s)

- f j(y ~?)j(? ~?)d? ' j(i),

where the last equality follows from the assumption that under ( 2.1) and ( 2.3), we have

j(y I ?,?) - j(y I ?). (2.7)

Thus, for the model (2.1), we have

Prob(y - 1 ~ z) - 1 g(Bo'x) - j(x ~ z; 6o)dt. (2.8)

4

Under the assumption of (2.3), we have

x - (A'A)-'A'(z - c). (2.9)

Substituting ( 2.9) into ( 2.8), the specific type of model we are considering has the form

Prob ( y - 1 j z) - 1 q~BoJ ~(A`A)-' ( z - E)~ JÍ(c j z)ácJ (2.10)

- G(borz)

where G is some transformation of 6o'z and bo is a nonlinear continuous function of 6o and

60 , 60 - h(Bo , 60). When x andE are normally distributed, from ( 2.8) we know that the

probability of y- 1, conditional on z, for the probit model, is (e.g. Lien (1986))~

eo~~l~ f EzA(fI -~ AEtA')-1(z - Ap))Prob(y - 1 j ?) - G(óo'?) - ~ ~ (1 f Bo'ABo) ~(2.11)

and, for the logit model, is

Prob(y - 1 j z) - G(z)

- f~ 9{(Bo'ABo) ~ v f B'~p t EZA(f2 -~ AEIA')-' (z - A~)~} ' m(v)dv~(2.12)

whcre g(a) - exp(a)jl t exp(a)~-', and ~(.) is thc standard normal density function.

3. Identification

A structure S is a complete specification of the probability distribution function of the

random variable y, F(y). The set of all a priori possible structure J is called a model. The

identification problem consists in making judgements about structures, given the model J

and the observations y(e.g. Hsiao (1983)). In most applications, conditional on the m x 1

parameter vector 70, y is assumed to be generated by a known probability distríbution

function F(y ~ ry), but ryo is unknown. Thus, the problem of distinguishing structures is

reduced to the problem of distinguishing between parametcr points. In this framework,

we have

DEF. 3.1: ForryocN, when N is a convex compact subset of Rm, the structure F(y j ryo)

is said to be identífied if there is no other ry'cN such that F(y ~ ryo) - F(y j ryl) for all y.

Given Definition 3.1, the general condition for the ]ocal identification of a structure

as worked out by Rothenberg (1971), etc. is that:

5

THEOREM 3. 1: Let ryocN be a regular point of the information matrix, i.e. the infor-

mation matrix has constant rank for ry in an open neighborhood of ryo. Then ryo is locally

identifiable if and only if the information matrix evaluated at ryo is nonsingular.

From Theorem 3.1 we can conciude thát

THEOREM 3.2: The binary choice model of (2.2)-(2.3) is (locally) identified only ifbo can

be identified from the marginal distribution of z. When 9o is identióable given y and s,

this condiiton is also sufTcient provided Ezz' - M' being an M x M nonsingular matrix

with M ~ K.

For proof, see the appendix.

Theorem 3.2 states that in order to identify Bo from (y, z) it is essential that 6o can

be identifiable from the sample information of z. This immediately imposes restrictions

on thc type of data for which a nonlincar errors-in-variables modcl is identi6able. For in-

stance, under the assumption that s and c in (2.3) are mutually independent multivariate

normally distributed random variables, the conditional distribution of x given z is char-

acterized by the conditional mean (2.4) and conditional variance-covariance matrix (2.5).

In other words, in order to identify bo, we have to know A,p and the measurement error

covariance matrix f2. The mere existence of instruments w which are correlated with x and

uncorrelated with E is not sufficient to ensure the identifiability of óo, hence Bo. Stronger

conditions on the probabilíty distribution of z are needed in ordcr to identify 90.

There are many different ways one can identify A, fl and EI from observed z. For

instance, there could be consumer's responses (the indicator variables z) to a series of

attitudinal questions (e.g. Train, McFadden and Goett (1987)). In such a factor analysis

framework, one set of conditions for the identification of a factor analysis model (2.3) is

that (i) !2 is diagonal, (ii) each column of A has at least K- 1 specified 0's in a certain

column and the matrix composed of the rows of A corresponding to the 0's in a certain

column has rank K- 1, (iii) a norrnalization rule such as an element in each column of

A is 1, (iv) the number of components in the variance-covariance matrix of z, E~, and

the number of conditions ( sum of ( ii) and (iii)), zM(M t 1) f K~ exceeds the number

of parameters in A, EZ, and fl, i.e., z ~(M - K)~ - M- K~ ~ 0. (Anderson and Rubin

(195G)).

1n the standard measurement error framework, we have A - IX. One way to identify

fl, hence E2i from z, is Lo obtain replicated observations for xi,

zil - x; ~- ci~, t - 1,...,n;, (3.1)

where n; 1 2. Under the assumption that xi and E it are multivariate normal, the joint

densityof (zil,...,z;n ) ls

I(?iU...e?in:)-J(?ilr...,Zin: ~?i)Jl?i)~ t..,-nx 1 n' l

-. (27f)-i I ~ I-- , exp -2 ~(zil - Zi)~fZ-1(zil - Tti) 1i-1

-(2ir)- ~ ~ E: t n f2 ~-~ exP {-2(zi - Fi)~(E: ~- rz~)-1(zi -~)~~ l ,(3.2)

n;whete ii - R~ ~ zit. Taking the second partial derivatives of the logarithm of II~lJ(zi)

e-1with respect to ~ and f2, the limiting matrix has the form3

E(E~ t n f2) D

O ~-~i1 ~ f2sIt is clear from (3.3) Lhat a necessary and sufficient condition for (3.3) to be nonsingular

is that E(n; - 1) ~ 0. This is so if and only if there are repeated observations. That is ~

and SI can be identified if we have replicated observations.

4. Maximum Likclihood Estimation

ln this section we will focus on the issues of estimation when the identification of

(2.2)-('l.3) is achieved through replícated observations (3.1). Conventional proofs of the

consistency and asymptotic normality of the maximum likelihood estimator (MLF) for

the binary choice models typically assume that the explanatory variables are bounded

(e.g. Amemiya (1985), Giourieroux and )`lonfort (1981), McFadden (1974)). But in our

formulation, in particular, when z and c are assumed to be normally distributed, the s

are clearly unbounded. Therefore, we shall give a set of regularity conditions which will

ensure the desirable properties of the MLE. However, these conditions are not necessarily

the weekest They are chosen, for simplicity and ease in verification.

Consider the likelihood function

f(yil,...,yin:r?il,...,?in~) - flyil,...,yin. I ?il,...~Zin;)f(?i1,...~Z1n;)

r (4.1)-[J f(yile...,y,n~ ~2i)f(xi~?il,... Zin,)d?i]f(?ile... Zin, I?i)f(?ti)

where zi - n ~t' r zit. Taking the logarithm of ll~ r f(yi, zi), we have

n

Ln. - log L - ~ log H(yi ~ zi) f log LZ, (4.2)i- r

nwhere n' - ~ ni,

i-r

N(.Tli I?i) - I nt-lflyit I xi)f(?i I?il,...Z in.)dyi

artd

n n K n llogLz -~log!(?ir,... yin, ~?i) f~IoBÍ(?i) -- 2 ~(n; - 1) J log2n

i-1 i-1 i-1

n l n n;- 2 ~(n; - 1) J log ~ f2 ~ -2 ~ ~(?ie - ?i)rn-r(?ie - ?i)

i-t i-1t-i

- nh log2rr - n log ~ E: t 1 ft ~-~ ~(ii -!f)r(E~ f ~fl)-r(zi - i~)-2 2 n; Z n;i -: 1

Using a similar argumcnt as that of Couricroux and Munfort (1981), Iloadloy (197])

and y~'hite (19R0), we can show that

THEOREM 4. 1: Let ryn, be the estimator that Ln- (ryn) - max7eN Ln. (y). Assume that

(A1): log f]lé-1l(yit ~?i;B)f(?i ~?ii,...,?in,,6)d?i - 1ogFl(y; ~?,;ry) is a measure-

ablc function on Dy x Dr, whcre Dy and D~ denote the support of y and z,

respectively.

8

(A2): log H(y~ ~ z;; ry) is a continuous function of ry on N, uniformly in i, a.s. (P~; where

P denotes the probability law governing the data generating process of y and z.

(A3): T'hcre exists mea~urable function m(y~,z;) such that ~ IogH(y~ ~ z;;ry) ~G m(y~,

a,) for all ry! N anrl ínT all i; F, I m(y~;z~l I~t~G ~ G oo for some v~ 0, and

OG~ Goo.

(A4): The model is identifiable. In other words, there exists no such that

inf I L(ryo) - max L(ry)J ~ 0.n'1no l rycN

where N denotes the complement of a neighborhood of ryo in N, and

1 nL- n, ~ E[loBÍ(y;, z~ ~ 7)] .

t-i

hold, then "ryn:e~ry~.

B' L.,' (7)Let rR,;n (ry) and r,,,,X (ry) be the smallest and largest eigenvalues of aryary, - In- (ry),

the following conditions will imply A9:

(1i5): 7min('Yo) -~ oo as rL' -~ o0

fo,.. (ry)(AO): for ry in the neighborhood of ryo, there exists a e such that ~~ ~,yj G e, V n'.

THEOxEM 4.2.: Undcr the assumptions of Theorem 9.1 and

(A7): á log H(y. ~ z;; ry) and ~ log H(y. ~ z;; ry) exist and are continuous functionry i ry 7' i

of ry for rycN, uniformly in i, a.s. ~P~, and is a measurablc function of y and z.

(A8): E(~ logH(y; ~ 3;;1) ~ ry~ - 0

9

(A9): E~e logN(y, ~?.~7)~j, logH(y~ I?:~7) I 71 - -E(eé~ logH(y; I?.~7) I 7] -lY~s, and Iv1: is positive definite.

(A10): For some v~ 0,~~ ~ E ~ a'~ logH(y~ ~ z;;ry) ~~t~ ~ n']z}Yl~~ -. 0 for all a in

Rp.

(All): For some v~ O,Esup.t,N ~~ IogN(y~ ~ z;;ry) ~~t~C oo.

Thend

n~' (7n - 7~) ~ N(0, r-~),

where

( 821og L 1t--E` J .~7 7 1'- lo

As pointed out by Andrews (1987) and Pótscher and Prucha (1986), assumption A.2

(of theorem 9.1) is a fairly restrictive one. It often implies that z has to be bounded (e.g.

Amemiya (1985, p. 270)). But under our formulation z clearly is not bounded. However,

in the case that y;t can only take value of zero and 1, H(y~ ~ z;) takes the form

f](y~ ~ z„ 6) - J p(x; g)~~-~ v,. (1 - P(x; B)~":-~;`, a:~ f(x ~ z;; 6)dx

where P(z; B) - Prob(y - 1 ~ z; D). We only need relativcly míld assumption to ensure

the consistency and asymptotic normality of the maximum likelihood estimator without

having to impose the assumption that z belongs to a bounded set.

1 (1

Lemma 4.1 In addition to assumptions A.1 and A.4 (or A1, AS and A.6), we assume that

(A12): P(x; B) is a continuous function of x and B on D~ x O, where DZ denotes the

support of x and O is a p-dimensional convex compact set.

(A13): iili:iE E~i5t5 a.-~. ~..., ,...~!; th~t fnr oarti p in Fi F ~ g~ 2'B ~~ p and E I- i8P~ IG ~.

(A14): For each B in O, there exists an i such that 0 G P(i; B) G 1.

(A15): f(x I z) ~ 0 and is continuous for x E D~.

!(?~ry)(A18):~ For some v ~ 0, there exists a 0 G oo such that for ~~ z ~~ ~ c~ 0 supry~~, y y z.ry G

~~~?~~-~lt~).

hold. Then the MLE of ry is consistent and asymptotically normally distributed.

Proof: i7nder A12, P(s;0) is continous in x. Under A14, we know that there exists

an ~ in D~ and a ~ 0 such that P(i;B) ~ a 1 0. Furthermore, A.15 ensures that f(i ~

z; b) 1 0. By the continuity argument, we have H(y; ~ z~; ry) ~ 0 and H(yi ~ z;; ry) is

bounded because ~ P(x; B) ~G 1. Therefore, log H(y~ ~ z;; ry) is bounded and there exists a

positive v G 1 such that EsupryEN ~ log H(y; ~ z;;ry) ~1t~G oo.

Furthermore, for z in DZ under A12-A15,

~ IogH(y; ~ z;;ry)

1 f(Eey;~) P(?;B)E~v;~-i~l - P(x;B)~".-E~Y,~ aI~(2~B) f(x ~ z;b)dx- H(y~ ~ ?:;7) aryEY:e n~-EaY~e-1 ÓP(2;B)

- (n; - Ety;t - 1) P(x;B) ~1 - P(x;~)~ ary f(x ~ z;á)dx

{. r p(2, p)E~v,~ ~1 - P(x;B)~n:-E~v,~ ~ Í( ? ~ ?;6)d?JJ Y

(9.6)

11

is bounded because there exists an a~ 0(a may depend on z, and ry) such that H(y~ ~

z;;ry) ) a and EaPery`Bl and J ef~Bl~dz are bounded by A.13 and A.15. Furthermore,

the independence assumption of z; and A16 ensures that

lim I [)n~~ su a loB H y; ~?.;1') C o0n-~oo n u ry~N II ary - ~I

s-~ .

Therefore, by a theorem of Andrews ( 1987), R, log L converges to E log L. Hence the MLE

is consistent.

Similarly, we can show that A12-AI6 are sufficient for the conditions of theorem 4.2

to hold. Hence "ryMtE ~s asymptotically normally distributed.

In the case that P(x; B) is an integrated normal ( probit model) or logistic distribution

(logit model), P(x; B) is continous on DZ x O with bounded derivatives. Furthermore,

since BcO, a compact set, there exists an i in D~ such that 0 C P(i;B) G 1. Moreover, for

the probit model we have

aP(x; B) - ,àe - m(x e)x.

For the logit model, we have

aP(x; B) eB~z 1x ~

aB - (1 -~ ee ?) (1 ~- ee 2)

- P(s; B) ~1 - P(x; B)] x.

Therefore A.12-A13 hold.

When x and c are normally distributed, A.15 holds. So is A.16 because H(y ~ z; ry)

is bounded away (roru zero. Furthermore, using an argument similar to the proof of

Corollary 3 of Fahrmeir and Kaufman (1985) we can show that the following sampling

scheme is sufficient to ensure A.S and A.6:

(i) Let the number of data points (n) tend to infinity. The number of replications at

each data point (n,) does not necessarily have to go to infinity as long as n tends to a

nonzero constant, where n- the number of {i ~ n; ~ 1}.

12

Then, the MLE of B and á are consistent and asymptotically normally distributed.

On the other hand, the following sampling scheme:

(ii) The number of data points (n) is finite and the number of replications (n;) tendsn

to infinity provided ~~;f~ has full rank,i-1

is not sufficient to ensure A5 and A6. However, using an argument similar to that of Y.

Amemiya and Fuller (1988) we can show that the MLE of 9 is still consistent.

Hence, if one is only interested in the estimation of B for probit and logit models

under the normality assumption of x and c, the MLE of B is consistent and asymptotically

normally distributed when the number of data points remains finite but the number of

replication increases or when the number of replications remains finite as long as the

number of data points increases or when both the replications and data points increase.

Of course, under the latter sampling schemes other parameters can also be consistently

estimated.

5. A Conditional Maximum Like]ihood Estimation Procedure

The simultaneous estimation of (B',6') - ry' can be very complicated. However, as

discussed in section 3, under the assumption that B is identifiable had x been observable, ry

is identifiable from the probability distribution of (y, z) if and only if 6 is identifiable from

the marginal distribution of z alone. This suggests that we may first estimate 6 from the

marginal distribution of z alone, substitute the estimated 6 into the joint likelihood of y and

t, j(y, z ~ 9, 6- 6n) assuming that 6- 6n, then maximize the likelihood function with

respect to B alone. For instance, the conditional dístribution, j(x ~ z; 6), is characterized

by A,p, fl and EI((2.9) and (2.5)). In the standard measurement errors framework A- Ik,

if the identification of ~, f2, hence ï~, from z is achieved through rcplicated observations

(3.1), we can maximize the conditional likelihood IIi 1 j(z;l,-.-,z;n ~ i;) with respect to

f1 (Andersen (1970)) and obtain

1

' S.1f~ - ~(ni - 1)~ ~(?;t - z i)(?it - ?i) e ( )

i-1 i-1t-1

13

where1 n:

~i - n. ~Zit. t-i

WP obtain an esimate of p by maximizing the marginal likelihood of II~ 1 f(?i),

i l

~- `~~n~~- `,~t~z;tI.

Once f2 is obtained, we can estimate EZ by 1

EZ - ËZ - tl, (5.9)

where

~Z - ~t~ ni~ -1 ~`~ t~(?it - ~)(?it - Z)'~ .

The estimates of i, fl and E~ converges to ts, ít and ~Z at the rate 0(re-1. ). Thus, we

can substitute ó„ for 6o in (9.2) and estimate B conditioning on 60 - 6n. The resulting

conditional maximum likelihood estimator is consistent due to the following theorem:

THEOREM 5.1: Let (y;,z~) be independent random variables ftom a disttibution de-

pending on (Bo',óo') - ryo'cN, where N is a convex compact subset of m-dimensional

Euclidean space. Let b„ - ~i(z~,...,zn) 6e such that 6„ converges to óo in probability.

Let ~(y„ z,; ry) be a dilTcrrntiable function of ry for rycN and E ~~(y, z; ry) ~e oo. Suppose

I d7 IC L(y,z) for all rycN and E ~ L(y,z) ~c oo. Then

1 n 1 nn ~~Íy:, ?i~~,6n) ~ ~ ~b(yi,?si8,óo)

i-t ,-i

in probability.

14

The asymptotic variance-covariance matrix of the two-step estimator follows from

THEOxEM 5.2: Let the model (2.1) and (3.1) hold. Assume that the partial derivatives of

log L- L., (7) up to the third order exist on rycN and are bounded by integrable functions.

Let 6„ - r~(zl,... , zn) be a consistent estimator of 60. Let 8.,, be a root of the equation

8B ~"(B, 6„) - O

Then f( B„ - Bo) will have asymptotic variance-covariance matrix

LBBr{E(naLn(aB~60) aLnóBo~óo))}~eóVar(~6n)Ló8~ (5.3)t ~ L68 f LB6 ~ }LBB

where ,Cgó - E(neLeláo'ó~ ) and ~- E(~eC"~~ . f(b„ - 60)]. Moreover, if

the conditions that ensure f("ry - ryo) being asymptotically normally distributed holds,

~(B„ - Bo) is also asymptotically normally distributed.

For proof of theorems 5.1 and 5.2, see Gourieroux and Monfort ( 1987) or Hsiao ( 1989).

As shown in ( 5.3), the asymptotic variance-covariance matrix of a two-step estimator

of Bo,B,,, is, in general, larger than the MLE of B,~,rLE. The conditional MLE will have

the same asymptotic efficiency as the MLE if and only if either of the following conditions

hold:

Lemma 5.1. The MLE and the two-step estimator of Bo will have the same asymptotic

variancc-covariancc matrix if and only if eithcr

( 1 8~ IogC~(Bo~o))

i. E „ ~~r~ - ~

or

ii. The asymptotic variance-covariance matrix of b„ is the same as the asymptotic

variance-covariance matrix of the MLE of óo.

Proof: When (i) holds, then (5.4) becomes E(- n a~)éó , which is the asymptotic

variance-cocariance matrix of the MI,E of Bo -under the assumption that

E(a~ iog c~B",óo)) - 0.~~

is

When (ii) holds, we have E- 0. To show this, we follow a similar argument as that

of Rao (1973) by defining a new estimator

S„ t a~~ ~ ~ a~ ( eo, 60)1~n aB J'

which is consistent and has the asymptotic variance covariance matrix of the form

0 0var(fón) - Var (fón) f zaE~E t a~E~var(~a~(áe~ó ))~. (5.5)

Let c be an arbitrary s x 1 vector of constants, then

c' [ Var(f 6„) - Var(f b„)]c- -a [2c'E'Ec t~cE' Var(~ aC(ae' 60)

)Ec, (5.6)

It is obvious that for a e(0, -2(c'E' Var( ~ eC(Bo,6o) )Ec)-rc'E'Ec), ( 5.6) is non-negative.~~This is a direct contradiction to the staternent that Var(Jn6„) achieves the Cramer-Rao

lower bound unless E- 0.

When E- 0, the asymptotic variance-covariance matrix of the two-step estimator of

Bo, (5.3) becomes

- LBa' t CBS'Cea ~- Cb6 t LóBLgg1LBÓ] - tCseCéB'

- -(CBB - LB6L6b1L6B)-t.

which is identical to the asymptotic variance-covariance matrix of the MLE of Bo, where

Ceb -Ea~i~st"(0",b")~

Although the conditional maximum likelihood estimator may be less efficient than the

maximum likelihood estimator, it does simplify the computation substantially. Moreovcr,

if the surface of L„(ry) is íll-behaved, an iterative procedure to solve for B and 6 will have

difficulty converging. On the other hand, there could be cases where conditional on ó- 6,,,

the surface of C„(8,6„) is well-behaved in OcO, hence making the iterative procedure to

solve for aC" (-~~- 0 more easily convergeable to 8,,.

6. A Simple Consistent Estimator for the Probit and Logit Models

16

In section 5 we suggested a conditional MLE of 9 to simplify the computation. How-

ever, its implication remains complicated. For instance, consider the case of probit model

where g(B'z) -~(B'x). Conditioning on x, the likelihood function of ( y;l,-~ ~,y;,,;) has a

univariate probit form

Hi-i~(B~yr)v:. [1 - ~(B'xt)[~-vu

Conditioning on (z~l,...,z;n ), fl and E~, Lhe likelihood function of (y;l,.-.,y;,,.) be-

comes a multivariate probit model involving n;-dimensional integration. Reformulating

the likelihood of (y;l ,-.-, y;,,. ) in the form

I(Yi1,...,yin: I~i) -~ Hf-1~lCi f 8'[J - Í~(niEx f~)-~j~ti f B'LilJJJ(2y`~ - 1)f(v;)dv„

where

Í(Y:) ~-' N[~,aurn; t e ;e; . B'A;B],

A; - n` [ft - f2(n;Ez f tt)-lflj,

~, - B'(n(n t n;Ez)-'1~

and e; is an n; x 1 vector of ones, I„~ is an n;-rowed identity matrix, we can zeduce the

n;-dimensional integration into a K-dimensional integration, where K is the number ofexplanatory variables. However, if K is large, the K-dimensional integration will still bequite complicated.5 To obtain simple consistent estimate of B, we note that conditional on

z;, we have

Prob(ytt - 1 I?i) -~ f c-F B'(I - f1E~ i)(ii - N)1L (1 t B'A;B) ~ J

where

c - B'p,

E. - H t n;Ez.

17

Suppose n; - n~ - T, `di, j, then we have

Er-nfTEz-E~-Ei,

and

A; - A; - A' .

If we ignore the correlation between y;i and y;e, we can write down the pseudo likelihood

function of y, i- 1, .--, n, asfi

n T

Qn -~{~~yie 1og 4i(a' f B''z;) f(1 - y[e) log;-i t-i (6.4)

~1 - ~(a' -t- B''~t)~},

and use the standard Probit computer package to estimate a' and B'. Once an estimated

B' is obtained, we can solve for B by letting

B - (1 - B''C-'A'C-'B') ~C-'B', (6.5)

where

C - [I - it(f1 t TEx)-'~.

Provided that the estimated f2 and Et are n~- consistent, one can show that the two steppseudo maximum likelihood estimator of B is consistent.

Similarly, we can obtain simplified estimators of B for logit models. As shown by Efron

(1975), Maddala (1983), and McFadden (1976), etc. if the conditional distribution of x

given y is multivariate normal with mean ~ev and common variance-covariance matrix D.

Then through the use of Bayes' formula, we can express the coeflicients of the logit model

Prob(y - 1 ~ x) - exp(Do f B~T)I(1 f exp(Bo f B~x)~, (6.6)

in terms of ~ , D, and the marginal distribution of y,rry:-y

eo - 2 (~o - !~I)'D-' (!~p -f {tl) - en(~o~~i), (s.7))

18

B~ - D-~ (~~ - l~o). (6.8)

If there are repeated observations as in (3.1), we can obtain simple consistent estimators

of ~ay, ~y, and D by

~o - 7. ~ L ?it, {~~ - T ~ Lz;t, .o i e ~ e

i,eclo ....,,

T, T~~o - T. ~ ~i - T.

1 (D - 7.{ ~ (zie - ~o)(zit - il), } ~ 1?it - Í4 t)(?it - ~t)t} - ~~

i,tcl„ i,tc7,

where T' - E:ini,Ti - EiEtyit, To - T' - Ti,lo -{(:~t) ~ Yie - 0},li -{(t,t) ~

Yit - 1), and it is estimated by (5.1). Thus, we can substitute (6.9) into (6.7) and (6.8) to

obtain consistent estimators of B.

The simple consistent estimator can be used for its own right. Or it can be used

as an initial estimator to start the iterative process of obtaining maximum likelihood or

conditional maximum likelihood estimator.

7. Conclusione:

In this paper we have explored conditions for identifying binary errors-in-variables

models. It is shown that when measurement error variances do not decrease with the

sample size, contrary to the linear errors-in-variables models, it is almost impossible to

get a model identificd unless there are replicated observations. We have also explored

conditions for the maximum likelihood estimators to be consistent and asymptotically

normally distributed. Some two or three step estimators which substantially simplifies the

computation are also suggested and their loss of efficiency is also explored.

The discussion in the paper is based on the assumption that all components of x

are observed with errors. Similar conclusions can easily be drawn when only part of the

components oí z are observed with errors.

19

APPENDIX

To prove Theorem 3.2, we note that the joint likelihood function of f(y, t; 6a, 60) can

be wriiten as the product of the conditional likelihood, f(y ~ z;Bo,áo) and the marginal

likelihood f(z; Do), then we have

log f(y,z;Bo,óo) - 108 Í(y~z;Bo,bo) ~-log Í(?;6o)(A.1)

- log f Í(y ~ i; Bo)f (x ~ z; bo)dz f l06 Í(z; 60).

Hence, the information matrix of (Bo, óo) is of the form

I - Iy~z(eo,óo) f I:(óa), (A.2)

where Iy~z(Bo,6o) and IZ(6o) are the information matrices of log f(y ~ z;Bo,6o) and

log j(z; óo), respectively. By (2.]0), we have

Ly~z - log Í(Y I ?; Bo, 60) -~{yslog G(bo~?i)~

t (1 - yi)log~l - G(bo~?i)~ )

Using the chain rule of difíentiation, the score vector of (A.3) is

óLy~z - B~ ~ 3It - 1- y~ 1 G'z8B G; 1- G;J'"

(A.3)

(A.4)

BLy~z - C ( y: - 1- y. G~z A.58b ~ `G; 1 - G; ~ " ( ):

where B - ab' G, ab' and G; - G(b'zi). Therefore, the inïormation matrix of (A.3)~ - á~'

is equal to

20

( `- ) (~h;zz)(B C)IYIz --EsEv aa7~ aa,~;j - E~ B .-:~ ~

:BMB' BMC' ~ (A.6)

~C;Í1B' CMr'

where h; --~G;(1 - G;)]-~(G()Z and M - E(~h;z;z;). The,information matrix of:

log j(z; 60) has the form

(A.7)

because j(z; 60) is assumed not a function of Bo.

If we premultiply the first p rows of (A.6) by - CB- and add them to the last s rows

we have a matrix of the torm

0 0 (A.s)

where B- is the left inverse of B. Therefore, for I to have full rank, it is necessary that D

is of full rank s. Furthermore, if B111B' is of full rank p, then I is of full rank m- p-~ s.

The rank condition of BMB' is assured if M' is a full rank matrix and B is of rank p.

The rank condition of B is asured by the assumption that Bo is identifiable from (y, x). The

identification of Bo also implies that the information matrix of (y, x) is nonsingular. The

nonsingularity of the information matrix of (y, x) togethet with the relation (2.3) implies

that M will be nonsingular if Ezz' is nonsingular.

BMB' BMC',

21

FOOTNOTES

' This work was partially supported by National Science Foundation grant SES88-

21205. An earlier version of this paper was presented at Deuxiemes Journees D'etude Sur

L'tutilisation Des Donnees De Panel held at Universite Parïs Vai De Marne on June G

and 7, 1988. The revision was carried out while the author was visiting the Center for

Economic Research, Tilburg University. The author wishes to thank three referees for very

helpful comments. He also wishes to thank H. Cheng, F. Kozin, W. Shafer and coníerence

participants for helpful comments and discussions.

1. For examples of models when y are measured with errors, see Stapelton and Young

(1989).

2. An alternative derivation of (2.11) is to rewrite the probit model in terms of the

conventiona] latent variable formulation of

y: - Bo~?~ f v~ , v; ~- N(0,1),

and- 1 1 if y, 1 0, l

y' - l 0 otherwise J

Substituting x; by E(zi ~ z~) -~ q; where r~; ~ N(0, A) we have y~ - Bo'E(xi ~ z;) -~ vi f

Bo~n~ - Qo~E(x; ~?~) t v; , vi ~ N(o, l-~ Bo'ABo)

3. The symmetry condition in f1 has been ignored.

4. Alternatively, one may replace A.16 by some smoothness condition (e.g. IV1cFadden

(1984, p. 1907)).

5. Hajivassiliou (1989) has provided an efficient smooth unibased simulator for the

score corresponding to (6.2).

6. This formulation can be viewed as a special case of those suggested by Gourieroux

et.al. (1985).

22

References

Amemiya, T. (1981) "Qualitative Response Models: A Survey", Journat oj EconomicLiterature, 19, 1983-1536.

(1985), Advanecd Eeonometrics, CaTibridge: Harvard University Press.

Amemiya, Y. (1985~ "Instrumental Variable Estimator for the Nonlinear Errors-in-Variables Model , Journal oj Econometrics 28, 273-290.

Amemiya, Y. and W.A. Fuller (1985) "Estimation for the Nonlinear Functional Relation-ship", Proceedings oj the 1989 Business Economies Section, ASA, 45-56.

and ~1988) "Estimation for the Nonlinear Functional Rela-tionship", The Annals oj Stat~siics, 16, 147-160.

Andersen, E.B. (1970), "Asymptotic Properties of Conditional Maximum Likelihood Esti-mators" Journal of the Roya! Statistieal Society, B, 32, 283-301.

Anderson, T.W. (1985), An Introduction to Multivariate Analysis, 2nd ed. Wiley, NewYork.

Anderson, T.W. and II. Rubin (1956) "Statistical Inference in Factor Analysis", in Proeced-ings oj the Third Berkeley Symposium on Mathematiea! Statisties and Proóaóilities,vol. V, Berkeley and Los Angeles: University of California Press, 111-156.

Andrews, D.W.K. (1987), "Consistency in Nonlinear Econometric Models: A GenericUniform Law o{~ Large Numbers", Econometrica, 55, 1465-1471.

Efron, B. (1975) "The Efficiency of Logistic Regression Compared to Normal DiscriminantAnalysis", Journal oj the American Statistieal Association 70, 892-898.

Fahrmeir, L. and H. Kaufmann (1985), "Consistency and Asymptotic Normality of theMaximum Likelihood Estimator in Generalized Linear Modelsn, The Annuals oj Statis-tics 13, 342-368.

Gourieroux, G. and A. Monfort (1981), "Asymptotic Properties of the Maximum Likeli-hood Estimator in Dichotomous Logit Models", Journal oJ Econometrics 17, 83-97.

SAE.(1987), "Statistique des Modéles Econométriques~, Cours polycopié EN-

Gourieroux, G., A. Monfort and A. Trognon (1985), "A Gencral Approach to Serial Cor-relation", Econometric Theory, 1, 315-390.

IIajivassiliou, V.A. (1989), "Measurement Errors in Switching Regression Models: WithApplications to Price-Fixing Behaviour", Mimeo.

23

Hsiao, C. (1983), "Identification", in Handbook of Eeonometrica, vol. I. ed. by Z. Grilichesand M. Intriligator, North-Holland: Amsterdam.

(1989), "Consistent Estimatiw~ for Some Nonlinear Errors-in-VariablesModels", Journal of Econometrics, 41, 159-185.

Lawley, D.N. and A.E. Isiaxwel! (1977 ); Faetor Analysis as a Statisticd Method, London:Butterworth.

(1973), "Regression and Factor Analysis", Biometrika 60, 331-338.

Lien, D.H. (1986), "Moments of Ordered Bivariate Log-Normal Distributions", EconomicLetters 20, 45-47.

Maddala, G.S. (1983), Limited Dependent and Qualitative Variables in Economttries, Cam-bridge University Press: Cambridge.

McFadden, D.J. (1974), "Conditional Logit Analysis of Qualitative Choice Behavior", inFrontiers in Econometrics, ed. by P. Zarembka, New York: Academic Press.

McFadden, D.J. (1976), "A Comment on Discriminant Analysis versus Logit Analysis",Annals oJ Economie and Social Measuremtnt, 5, 511-523.

(1981) "Econometric Models of Probabilistic Choice", in Struetura! Anal-ysis of Discrete Data with Eeonometrie Applieations, ed. by C.F. Manski and D.McFadden, Cambridge: MIT Press.

1984), "Qualitative Response Models", in Handbook oJ Eeonometrics, Vol.II, edited y Z. Giriliches and M. Intriligator, Amersterdam: North-Holland, 1395-1957.

Pótscher, B.M. and I.R. Prucha (1986), "Consistency in Nonlinear Econometrics: AGeneric Uniform Law of Large Numbers and Some Comments on Recent Results",Department of Economics Working Paper ~86-9, University of Maryland.

Rao, C.R. (1973), Linear Statistiea! Inferenee and Its Applications, 2nd ed., Wiley: NewYork.

Rothenberg, T.J. (1971), "Identification in Parametric IVlodels", Econometrica 39, 577-592.

Stapelton, D.C. and D.J. Yovng ( 1989~, "Censored Normal Regression with MeasurementError on the Dependent Variable , Econometriea 52, 737-760.

Stefanski, L.A. and R.J. Carroll ( 1985), "Covariate Measurement Error in Logistic Regres-sion", The Annals of ,Statistics 13, 133Fi-1351.

Train, K., D. McFadden and A. Goett ( 1987), "The Incorporation of ALLitudes in Econo-metric Models of Consumer Choice", Revitw of Economics and Statistics 69, 383-391.

Train, K. (1985) Qualitative Choice Analysis, Cambridge: MIT Press.

24

White, H. (1980), "A Heteroskedasticity - Consistent Covariance Matrix Estimator and aDirect Test for Hetetoskedasticity", Economttrica 98, 817-838.

Wolter, K.M. and W.A. Fuller ( 1982a), "Estimation of Nonlinear Errors-in-Variables Mod-els", Annua(s oJ Statistics 10, 539-548.

(1982b) "Estimation of the Quadratic Errors-in-Variables Model",Biometrika 69, 175-182.

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