On the estimated variances of regression coefficients in misspecified error components models

Tilburg University

On the estimated variances of regression coefficients in misspecified error

components models

Deschamps, P.J.

Publication date:

1992

Document Version

Publisher's PDF, also known as Version of record

Link to publication in Tilburg University Research Portal

Citation for published version (APA):

Deschamps, P. J. (1992). On the estimated variances of regression coefficients in misspecified error

components models. (Reprint Series). CentER for Economic Research.

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners

and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.

• You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal

Take down policy

If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately

and investigate your claim.

CRM

-t'I III I I "J

-sszs

for

i 982 omic Research

76

~~~~

IIIIIIIIIIIPoIIIIIINIIIIUUIIIN~II,llllllll

On the Estimated Variances of

Regression Coefficients in

Misspecified Error Components

Models

by

Philippe J. Deschamps

Reprinted from Econometric Theory,

Vol. 7, No. 3, 1991

Reprint Series

Research Staff

Helmut Bester Eric van Damme

Board

Helmut Bester

Eric van Damme, director

Arie Kapteyn

Scientific Council

Eduard Bomhoff Willem Buiter Jacques Drèze

Theo van de Klundert Simon Kuipers Jean-Jacques Laffont Merton Miller Stephen Nickell Pieter Ruys

Jacques Sijben

Residential Fellows Svend Albaek Pramila Krishnan Jan Magnus Eduardo Siandra Dale Stahl II Hideo Suehiro Doctoral Students Roel Beetsma Hans Bloemen Sjaak Hurkens Frank de Jong Pieter Kop Jansen

Erasmus University Rotterdam Yale University

Université Catholique de Louvain Tilburg University

Groningen University

Université des Sciences Sociales de Toulouse University of Chicago

University of Oxford Tilburg University Tilburg University

E:uropean University Institute San Francisco State University Tilburg University

UCLA

University of Texas at Austin Kobe University

~

for

Economic Research

On the Estimated Variances of

Regression Coefficients in

Misspecified Error Components

Models

by

Philippe J. Deschamps

Reprinted from Econometric Theory,

Vol. 7, No. 3, 1991

Reprint Series

ON THE ESTIMATED VARIANCES

OF REGRESSION COEFFICIENTS

IN MISSPECIFIED ERROR

COMPONENTS MODELS

PHILIPPE .J. DESCHAMPS

University of Fribourg

In a regression model with an arbitrary number of error components, the

co-variance matrix of the disturbances has three equivalent representations as linear

combinations of matrices. Furthermore, this property is invariant with respect

to powers, matrix addition, and matrix multiplication. This result is applied

to the derivation and interpretation of the inconsistency of the estimated

co-efficient varances when the error components structure is improperly restricted.

This inconsistency is defined as the difference between the asymptotic variance

obtained when the restricted model is correctly specified, and tlie asymptotic

variance obtained when the restricted model is incorrectly specified; when some

error components are improperly omitted, and the remaining variance

compo-nents are consistently estimated, it is always negative. ln the case where the time

component is improperly omitted from the two-way model, we show that the

difference between the true and estimated coefficient variances is of order

greater than N-t in probability.

1. INTRODUCTION

Since the pioneering article by Balestra and Nerlove [3], many authors have

studied the estimation of linear regression models with error components.

The Balestra-Nerlove specification was limited to a two-way classification

of data, and otnitted time-specific effects. This restriction was later lifted by

Wallace and Hussain [10] and Nerlove [8], who considered the following

er-ror specification:

u;,-u;fw,fe;,

fori-l,...,n

and

t-1,...,T

(1)

where u;, w„ and e;, have zero expectations, constant variances, and are

pairwise and serially uncorrelated. Nerlove, in particular, found the spectral

form of the error covariance matrix V implied by equation (1), expressing

V as a linear combination of symmetric, idempotent, and pairwise

orthog-Pan of this research was conducted white the author held a Faculty Research Fellowship at CentER, Tilburg University. I wish to thank Pietro Balestra, Badi Baltagi, and anonymous referccs tor helpful comments. A prcwious version of this paper was presented at CORE, CentER, and at the European Mceting of the Econo-metric Society, hlunich, 19g9. Any remaining error is my own.

onal matrices that sum to an identity matrix. All the rational powers of V

can be similarly represented; this is particularly useful for interpreting and

computing the Aitken estimators of the regression coefficients. The spectral

form also leads to a particularly simple expression of the ANOVA table for

the variance components of u;, (for a classical presentation of this table, see,

e.g., Graybill [6, p. 349]).

More recently, Searle and Henderson [9] and Deschamps [4] have studied

the structure of the error covariance matrix in a model with a general, p-way

classification of data ( where there are 2p - 1 error components). As will be

shown, their results imply that V has three equivalent representations as

lin-ear combinations of matrices. The coefficients of the first one are the

vari-ance components; the coefficients of the second one (the spectral form) are

the eigenvalues of V; the coefficients of the third one are the elements of V.

Each linear combination is, in a sense, isomorphic to the two others.

Fur-thermore, the property is invariant with respect to powers, linear

combina-tions, and matrix multiplication. This means that if the nonsingular matrices

V,, VZ, V~, V, have the same form as Vand if a,b,a,R,y,b are scalars, then

V; (aVZ f bV~ ) V; also has the same form; it can be represented as three

linear combinations of the same matrices that appear in the three expressions

for V. Furthermore, the three sets of coefficients are easily derived.

Section 2 of this paper will present the assumptions of the model, and a

new notation. In Section 3, we present Theorem 1, which is a general result

on the misspecified error components model. Theorem 1 can be used to

derive and interpret the inconsistency (plim N(à,? - a?)) of the estimated

variances of the regression coefficients when some error components are

im-properly omitted (even though their variances are nonzero), and the

remain-ing variance components are consistently estimated. We will show that the

true coefficient variances are always underestimated in the misspecified

model, for any p-way classification and for any number of omitted

compo-nents. More generally, we find that the inconsistency can be expressed as

three linear combinations of quadratic forms, in the same way as there are

three representations of V. This turns out to be very helpful for the

interpre-tation of the inconsistency.

When a single error component is omitted in the misspecified model

and when the remaining variance components are consistently es[imated,

our results also imply that the inconsistency is proportional to the variance

of the omitted component. The coefficient of proportionality is a linear

com-bination of sums of squares of partial sums (as in, e.g., a, ~, (~; Z;,)2 f

aZ E; (E, Z;,) Z) ; it can be estimated as a by-product of generalized least

squares on the misspecified model. This greatly facilitates the sensitivity

anal-ysis of misspecification.

Theorem 1 also applies to other types of misspecification: the inconsistency

In Section 4, we illustrate the previous results by analyzing the

incon-sistency when the time-specific component w, is improperly omitted from

equation (1). This case is of particular interest since it corresponds to the

ear-liest (Balcstra-Nerlove) specification, and is therefore found in most

empir-ical implementations. In this instance the inconsistency has a particularly

simple, intuitively appealing form, whose properties are easily investigated.

lt will be shown that this inconsistency is unbounded, unless the matrix of

rcgressors satisfies very restrictive assumptions.

Lemma 1 of Appendix A presents the [hree isomorphic representations of

V, and can be viewed as a synthesis of Searle and Henderson [9] and

Deschamps [4J. This lemma will be used in Appendix B, where we prove

The-orem 1 of Section 3. Appendix C presents estimators of the variance

compo-nents which are consistent under misspecification, and coincide with the ones

proposed by Amemiya [ I] for a correctly specified three-comporient model.

2. A GENERAL ERROR COMPONENTS MODEL

Our regression model has 2~ error components, one of which is identically

zero. It can be written as

y-Xafu

(2)

where y is an N x 1 vector of observations on a dependent variable, X is an

N x k matrix of observations on nonstochastic regressors, a is a k x 1

vec-tor of regression coefficients, and u is an N x 1 vecvec-tor of compound

distur-bances with the following structure:

i~. .i

u- ~ ( S~ i a l O... O S~v a0 ) Va u-W...O

(3a)

with

E(~~) - D~r,.

(3b)

E( ~~. ~~ ) - e~ I4,.

(3c)

E(~.,tJ~) - 4u..x4~

(a ~ Q).

(3d)

ln equations (3a)-(3d) and in the rest of this paper, the Greek subscrip~s

a, (3, and y indicate binary numbers with p digits, for example,

a-a~ a, ... a,,, where a; is either zero or one. The letters n~,..., n„ denote

Specification (3a)-(3d) obviously implies the following form for the

covar-iance matrix of u(see Note 1):

~i ..i

V - E(i1U') - ~ Bo(S~i o~ (~ . . . [~ Sn~, `~~') (4)

a-00...0

where Sn, is a square matrix of ones of order n; and S~ - In,.

The model in Nerlove [8J is obtained as a special case of (4) by letting

p- 2, n~ - n, and nZ - T:

V- BOUSnT } B01 (Sn 0 IT) f e10(In ~ ST) f BII InT (5)

where B~ - 0, t3o, ~ aW, 6~0 ~ a~, and B~ ~~ a? (compare equation (1)). By

let-ting Bo, - 0 in (5), the model specializes further to Balestra and Nerlove [3].

The reader will have no difficulty in verifying that when p 3, if B~

-ecxii - duio - 9oi~ - eioi - 0 and if 9ioo ~ Qó. Bi~o á aW, Biii ~ a,2, equation

(4) specializes to the nested specification in Fuller and Battese [5], implied by

u;~~ - u; ~- w;~ t e;;,.

(6)

So the binary number a may serve to interpret the variance component 6~

in equation (4): it is simply the variance of that error term which has tlie

par-ticular index subset identified by the unitary digits in a.

Equation (4) is the first of our three isomorphic characterizations of V;

the two others are given in Appendix A.

3. THE CONSEQUENCES OF MISSPECIFICATION

We consider the case where V in equation (4) is estimated by:

~~ ..i

N

V', -

~

g~ 0 S~-~,

_{a n,}

~.-00...0 r-1

(7)

where 8' is a possibly inconsistent estimator of the 2~ x 1 vector ~. We

es-timate I3 by

à - (X'~:'X)-'X'~:'y.

(s)

The covariance matrix of (3 is estimated by:

C - (X'V;'X)-'

whereas the true covariance matrix, conditional on 6', is

V(QI9') - (X'V;'X)-'X'V;'VV; ~X(X'V;'X)-'.

We let B' - plim B' ( see Note 2), and assume that V. -~;~ B„' Qx N~ 5,;; "~ is

positive definite. We also assume that X' V; 'X~N tends to a finite, positive

dcfinite matrix M.. We further note that

X'V;'X

-'

NC

and

- ~X'V;'Xl-'~X'V;'VV;'X1

(X'lV;'X1-'

NV((3~~')

_N

_J

_N

J l

N

_J

We wish to evaluate~

plimN(C- V((3~B'))

- plimN((X'V;'X)-~ - (X'V;'X)-'X'V;'VV;'X(X'V;'X)-')

- IimN((X'V;'X)-' - (X'V;'X)-'X'V;'VV;'X(X'V;'X)-')

- IimN((X'V; ~X)-~(X'V; ~V,V; ~X-X'V;'VV;'X)(X'V; ~X)-')

- M;' lim ~X~NX'~M; ~

(9)

with X. - V; '`'`X and D- V; 12(V, - V)V; 12.

Equation ( 9) implies the following expression for the inconsistency of the

estimated variance of ~Q,.:

(Z'(P)DZ(P) 1

plim N(C,~,. - V(a,.~B')) - limt

N

1

(10)

where Z(P) is the Pth column of the matrix X.M; ~.

The following definition provides the key to the three equivalent

tations of V given in Appendix A, and to the three corresponding

represen-tations of (10) given by Theorem 1 of this section.

DEFINITION 1. Let, as before, S,,, 6e a matrix vf ates of order n; and

~ero powers denvle iden~ity ma~rices. We define

v

1

K-0

_,-,

₁

0

₁

_'

Ka-,OO ~~~

1~;

`0

1

(12)

v

L~ - Qx S,~,; ~~;

(13)

P ~ -A~

k~ - ~

` 1 S~~I

``n

,-i

~~;

P

N1a - OO ( 5,,, - l,,, ) ~ -a, .

- I Sn,`a,'

n;

il

(14)

linear combination of the elements of rt, with coefficients that depend on

B', but not on B.

THEOREM 1. Let ,t - plim 6' - B- B` - B, !et V. -~u B,;La, let

D-V; 1z( V. - V) D-V; 12, and let ~ be the diagona! ma[rrx with [he elements oJ

KoB' on the diagonal. Then D can be equiualently expressed as

D-

ii. .i

_~i

_~]ti

ii...i

_~

_{aa-r Ra}

ti-oo...o

p-oo...o

(16)

D-~t ..i

~i...i

~i

Tlti

F

b~,L3

ti-ai...o

t~-oo...o

ii...i

ii...i

~

nti

~

cati Ma

ti-oo...o

r~-oo...o

(17)

(18)

~vhere a,~,, bay, and c~, are the elements in row Q and column y of ~-'Ko,

K~ ' 0-' Ko, and KKó ' ~'' Ko, respectiuely.

~

The most interesting type of misspecification occurs when one, or several,

variance components B, are improperly excluded from equation (4). This

follows from letting By - 0, so that By - 0. [t is shown in Appendix C that

the remaining variance components can always be consistently estimated.

In this case the inconsistency (10) is always negative: indeed, it is shown in

Appendix A that the.matrices R~ in (16) are idempotent, pairwise

orthogonal, and add up to an identity matrix, so that the eigenvalues of D are ~p

-Eti ~,a~,. Since a~, ? 0 and since ,t, - 0 when component y is included,

whereas rt, --9, s 0 when component ti is excluded, we have that ~~ s 0

for all (3.

We may also note that the opposite type of misspecification whereby a

variance component By is improperly included is of no consequence

asymp-totically if plim By - B, - 0, so that rt, - 0 in Theorem 1.

Equation (17) implies that the quadratic form in (10) is a linear

combina-tion of sums of squares of partial sums of the elements of Z( P), as is

obvi-ous from the definition of Le in (13). This fact will be used in Section 4.

Similarly, equation (18) expresses the quadratic form as a linear

combina-tion of sums of cross-products of the elements of Z( P) (see the definicombina-tion of

M;, in (15)). Using the arguments in Appendix A, it is easy to show that the

coefficients ~~ rt~c,~, in equation (18) are in fact the elements of D.

4. AN ILLUSTRATION

at the end of this section. For now, we assume that 9W - B~, - 0, B;o - t7,~,

and B;, - 8,,.

As prcviously noted, the elements of KoB' in Theorem I are the

eigcnval-ues of V.. Using equation ( 12), they may be written as

~~~K, I

~nT n

T

11( 01

_{~Teio f eii I}

~c,

0

~t

0

1

0

B„

~,o

-

0

0

T 1

B,o

-

TB,a f B„

)`„

0

0

0

1

B„

9„

Furthermore, the coefficients b~, in equation (17) are elements of the

matrix:

i

i

Kó ~- Ko

1

- nT

.

ol

-1

1

~~

o

0

0

-T

0

~ó,~

0

n

-n

0

0

~;ó

0

0

0

nT

.

0

0

(19)

nT

n

T

1

0

n

0

1

0

0

T

1

~ 0

0

0

~;,~~ l 0

0

0

1~

nT~~~~

n(~~ -~ó,~)

T(~óó - ~ió)

~~ -Aó,~-~ió -~~„~

0

nT~ó,~

0

T(~á,~ - ~ii~)

0

0

nT~~ó

n(~ió -~ii~)

0

0

0

nT~;,'

(20)

Since the only inconsistency occurs in the time component, we have no,

--Bo, and n~ - n,o - n„ - 0, so that we only need those coefficients bQ, that

are located in the second column of the preceding matrix (which hás the

bi-nary index O1). Upon substituting (13), (19), and (20) in equation (17), we

obtain

D - -Bo, ~ 1 ~

1

- ~ ~S„T {- B~~ (S„ OO ~T)~

`T `T~,o f eii

Bii

- -Bo, (

-B,o

S~r f (Sn ~ Ir)~

B„ `re,~ f e„

so that the quadratic form in equation (10) is

(21)

r

Z'( e)DZ( Q) -- 8~~ ~~i Z ~( Q) - T6,o If B„ Zï ( e)

_~i

_,-i

(22)

Since B,o~(TB,o f B„ ) e 1~T, an upper bound for the expression in (22)

is given by

Z'ee)DZ(e) ~ -Ba~ É (z.,(P) - z..(e)lz

_{e„ ,-,}

_{T J}

and it is seen that for given 60, and 9,,, IimT,,, Z'(P)DZ(P) - -oo, unless

Bo, - 0 or unless Z,, ( Q) - a for all t. !n the latter case, equation (22) is

eas-ily shown to imply

Z'(l')DZ(P) -

- T~ZBoi

TB,o f 9„

whose limit equals -a Z Bo, ~6,o as T tends to infinity.

To summarize, the inconsistency of the Fth estimated variance in the

mis-specified model is unbounded as n-. oo and T--~ oo. This may be seen by

di-viding equation (22) by n T, and noting that ET , Z;( F)~nT does not

converge in general. The unboundedness occurs unless the "observations" in

the ('th column of X.M;' repeat their average pattern over time. In this

(very special) case where Z., ( P) - a for all t, we have

plim nT(Cr,. - V(Rr~e~)) - lim

Z'(PnDZ(Y) -

lim n

17,(-~~Bo' 1 - 0.

`

io

J

We will now show that the previous result remains essentially unchangcd

when 9;o and 9;, are the one-way estimators:

1

e~~ - n(T - 1) (u (j~

xO Ri)u)

.

1

u~(l~~ 0 Ro)u

u'(f., O Ri)uI

B~o- T

n

-

n(T- 1)

where Ro - T-'ST, R, -!T - Ro, and where u is a predictor of u. If the

true disturbances u;, are known and u u, it is easy to show that E(B;,)

-Bo, t B„ and E(8;o) - B,o; we will therefore assume that B;, - Uo, t B„ and

that B;o - B,o. As before, we have B~ - B~, - 0. The eigenvalues of V. are

again given by KoB', this time as

r~W~

Te,~ t e~, f e„

Bo, t e„

~~o } eo~ f B~~

T}ie vector ,! of variance component inconsistencies is this time given by:

r~~~,~ r o ~

rio l

-e~,

0

~nll~

l eoI )

Upon using again the methodology of Theorem 1, we see that D- D, f

Dz, with

DI --eol ( T( T6

_lo

~- B

_ol

f 8

II

- B

ol

f B ) S"r } 6

II ol

~- B

II

(S" x0 f r))

Dz-eo1(T(T9

_fB

_-fB

_{- B}

_fB

)(fnOsr)} 9-~B ~"r~.

10 01 II 01 II UI II

The matrix DI has the same form as D in (21), with 611 replaced by Bo, -~

911; hence the preceding analysis also applies to Z'(P)D,Z(P)~nT, which is

negative and generally unbounded. By analogy with (22), the second term

may be written as

z'(P)Dzz(P) -

~~ ~Z,(P) -

e 'o

~ZZ(P))

Bol ~- B11

;-1 r-1

Telo f eol f B11 ;-1

e~,

n T 2

~

BoI

_{~ ~` (Zu ( P) - Z~ TP) ~ ? 0}

₍₂₃₎

B01 f ell i-1 r-I `

and Z'( P)DZZ( P)InTconverges to a finite, nonnegative number if E; , Z;( P)~

nT and E; Z?( P)~nTZ are bounded. Furthermore the lower bound in (23)

vanishes if, and only if, Z;r(P) - c; for all !; in this case the limit of

Z'( P)DZZ( P)~nT is easily shown to vanish when B,o ~ 0.

Collecting resu(ts, we may say that the inconsistency remains negative and

generally unbounded, since the negative term dominates the positive one.

Furthermore, when B,o ~ 0, the inconsistency now vanishes if (and only if)

the elements of the relevant column of X.M; t exac!!y repeat their pattern

over time.

5. CONCLUDING REMARKS

two-way classification model, this inconsistency is in general unbounded, in the

sense that C,r - V(ár~9') is of order greater than N-' in probability, so

that plim NV((3,~ ~ B' ) does not exist. Since plim NV( (3r ~ 6' ) is the asymptotic

variance when the restricted model is incorrectly specified, this implies that

~(I3t - Qr) does not have a proper limiting distribution in such a case.

The arguments in Appendixes A and C indicate that the p-way model is

essentially a tensor generalization of the 2-way model, and is, as such, not

much more difficult to handle. It is therefore unfortunate, in view of the

po-tentially serious consequences of misspecification, that most contributions

to the litcrature on error components have been limited to the

Balestra-Nerlove specification.

NO TES

I. Note that equation ( 4) differs from Searle and Hendcrson (9, equation 2.2J in the index-ing of the variance components 8,,. A corresponden,ce between the two indexindex-ing schemes is ob-tained by writing the digits in a in reverse order, and taking the complement to unity of each digit: for instance, Bo„ in our notation corresponds to Br„t, in (9J. As we will see, our notation has definite advantages for interpreting the variance components, and for stating the isomor-phisms of Appendix A.

2. Unless otherwise indicated, all the limits in this paper are taken as n, -~ m for all i. 3. If V- à~ I with á~ the OLS estimator of the error variance, it may be shown ( Grcenwald,

(7J) that (X'X)-~X'(V, - V)X(X'X)-~ approximates the bias of the estimated covariance

ma-trix whcn N is large.

4. Note that c„p - 0 for (3 - 00.. .0.

REFERENCES

I. Amemiya, T. The estimation of the variances in a variance components moclel. Internutiunul

Ecunomic Review 12 (1971): I-13.

2. Balestra, P. Dest quadratic unbiased estimators of the variance-covariance matrix in normal regression. Journa!oj Economerrics 1(1973): 17-28.

3. Balestra, P. 8t M. Nerlove. Pooling cross section and time series data in the estimation of a dynamic modcl: the demand for natural gas. Econuinrtrica 34 (1966): 585-612. ~3. Dexhamps, P. A note on isomorphic characterizations of the dispersion matrix in error

components models. Linear Algebra and Its Applications I I 1(1988): 147- I50.

5. Fuller, W.A. 8c G.E. Battese. Transformations for estimation of linear models with nestcd-error structure. Journul ojlheAmerican S!alisticul Association 68 (1973): 626-632. 6. Graybill, F.A. An Inrroducrion lo Lineur Statisticul Models, vol. 1. New York:

MrGraw-Hill, 1961.

7. Greenwald, B.C. A general analysis of bias in the estimatcd standard errors of Ica.,t syuares coefficients. JuurnulojEconomerrics 22 (1983): 323-338.

8,, Nerlove, M. A note on error components models. Ecunometricu 39 (1971): 383-39G. 9. Searle, S.R. Bc H.V. Hendrrson. Dispersion matrices for variance components model~.

JuurnuJ oj rhe ,4merican Srulisticu! Associution 74 (1979): 465-470.

APPENDIX A. CHARACTERIZATIONS OF

THE ERROR COVARIANCE MATRIX

It is easily shown that the following lemma can be extended to all the rational

pow-ers of V, and also to linear combinations or products of matrices having the same

structure as V(an example of such an extension is in fact given by Theorem I).

LEMMA 1. Let u haue the specijication (3a)-(3d), and let V- E(uu'). Tlren

ii.

i

ii...i

ii

i

V-

~

B,. L~. -

~i

~,. R~, -

~i

p.. M,.

,.-W...U n-lM1...U u-W...U

where L,,, R,,, and M„ are giuen by DeJinition 1. The ~„ are the eigenualues oJ

V and

tlre p„ ure the elernents oJ V. Furthennore, iJ 1`, p ond B denote the 2N x I uecrors

~vith elentents J~,,, p,,, and 8,,, we haue ), - KaB and p- ti6, with K aird K~

given

by (11) and (l2).

Proof. Equation (4) may also be written as

V - (8, ~ IN)L

(24)

where

L-Lii...i

and where the matrices L„ are as defined in (13). Using 1` - K~B and p - KB, (24) is

equivalent to

V- (e" ~ IN)(Kó OO IN)((Kó)-' ~ jN)L

-(Í~" OO ~N)((KO)-~ OO ~N)L ~ (~" ~ ~N)R (ZS)

and

V - (e" ~ IN)(K" ~ IN)((K")-~ ~ ~N)L

- (p" OO 1N)((K")-~ ~ IN)L ~ (P" ~ ~N)M.

(26)

If we partition R and M in the same fashion as L,

R-0

MW...o

and

M -

.

i

it follows from (25) and (26) that

Mii..

ii.

i

Ra -

~

k~ou L~

(27)

and

ii...i

Mv - .~-, k~i~ L,. ( 28)

a-W. .0

where k~~ and ku„ are the elements in row a and column a of (K„)-' and (K') ',

respectively.

We now show that ( 27) is the expansion of (14). When p- 1, ~vc have

1

0

( Kó ) - ~ - no ~

-1 r1 ) '_v Lo - Sno ~ L i - ~~~,

so that (27) obviously implies (14). It is easy to see that the property is true for p

when-ever it is true for p- I. Indeed, (Kó)-' s(Kó(p))-' and L~ L(p) may be del'incd

recursively as

(Kó(P))-,

- ( ni ~(Kó(P - 1))-~

O

- `-ni ~(Kó(P - i))-~

(Kó(P - 1))-' )

rS~, ~ Loo...o(P - 1) I

L(P) -

5,,, ~ Lii...i(p - i)

I,,, OO Loo...o(P - 1)

1~n,0 L~~...~(P- 1) .

so that, upon letting (3 -((3, y, y2 ..~ yp-, ), (27) may be written if (3, - 0 as

R~(P) - F, (ni ~ká(P - i))(S~, ~ La(P - 1))

a - ni rS~, O~kó(P - 1)La(P - i) a

- ni ~Sn, ~ R,(P - i)

and if (3, - I as:

Ra(P) - F, (-ni ~ká(P - 1))(5,,, ~ L.,(P - )))

_a

a

- (~~, - ni ~5~,) ~ Ry(P - 1)

where we note that a and y now involve p- 1 binary digits rather than p. The proof

that (28) is the expansion of (IS) is exactly similar.

We must show that the matrices R~ in (14) are idempotent, pairwise orthogonal,

and add up to an identity matrix, thus proving our clairn that ~ is indeed the vector

f Fi (k ó(P - 1))(In, ~ La(P - 1))

_R

of eigenvalues. RqR~ - R~ is obvious since R~ is a Kronecker product of idempotent

matrices. RaRw O for a ~ Q follows from [he fact that n; ~S,,,(l,,; n; ~S„;)

-O; indeed, a~ Q implies the existence of an index i such that a; - 0 and (3; - l, or

a; - 1 and Q; - 0. So Ey Rp is symmetric and idempotent; in order to show that it

equals IN, we show that its trace is nonzero. This follows easily from (27), which

im-plies

ii. .i

ii...i

ii...i

ii...i

ii...i

lr ~

~

R~~ -

~

~

kaa tr(La) - N

~

~

kva - N

` d-00...0 (i-00...0 a-00...0 (i-00...o a-W...O

since E~ ~okw is the sum of the elements of (K~)-~, which is easily seen to equal

unity:

s"(Kó)~s

-` ;O (!

!)I `rO ` nn;!~

III `;O `l ~I -

1.

It was shown in Deschamps [4] that the elements of p are also the elements of V.

Since, from ( IS), the elements of M~ are 0 or I, this is also implied by E~ Mp - SN,

which is readily verified from (28).

APPENDIX B. PROOF OF THEOREM 1

It is an immediate consequence of Lemma 1 that V, V , and V- V can be expressed

as (different) linear combinations of the same matrices R~. The coefficients in the

linear combinations are the eigenvalues; in the case of V. - V, they form the vector

KeB' - K„B - Ko,t.

As is easily seen from the properties of the matrices Ra, both V; '~2 and D can

again be represented as linear combinations of the same matrices Ra. The

eigenval-ues of D are obtained by multiplying the eigenvaleigenval-ues of V. - V by those of V; t, so

that they form the vector 0-'Kon, with elements EY n,aqY. This proves that (16) is

indeed the spectral form of D.

ln order to obtain equation (17), we substitute (27) into (16):

~~. .i

ii...i

:i...i

u...i

ii...i

~i.

i

D- ~ ~t ~ ad, ~ kBa La - ~ nY ~i ~ kd~ aut L~

Y-00...0 Q-00...0 u-00...0 y-00...0 u-00...0 (i-00...0

which is equivalent to (17) since k,~a is the element in row a and column Q of K~'.

In order to obtain equation (18), we note that (25) and (26) imply R-((K~)-' Q

~h)L and L-(K' ~x IN)M, so that

R - ((Ku)-~K' Q Iti)M

and we may write, similarly to (27) and (28),

ii. .~

R~ -

~

kR~ Mo

(29)

where k~„ is the element in row Q and column a of (KKó ~)'. Upon substituting (29)

into (16), we obtain

ii

i

ii

i

ii

~

ii. .i

ii. .i

ii

i

~ -

~i

~,

~

aa,

~

ku., M~ -

~

~,

~

(

~

ku.. Qd, ) tif..

~-vv.. o

d-cxi

o

~-oo...o

ti-c~o . o

..-oo

o

u-ui

o

and (I8) follows upon noting that k~o is the element in row c~ and column (3 of KK„~.

APPENDIX C. ESTIMATION OF THE

VARIANCE COMPONENTS

This appendix may be viewed as a generalization of the first part of Amemiya [I] to

the p-way classification; however, the method of proof is partly new and relies on

Lemtna I. We first note that, since B- Kó '~, unbiased estimators of the variance

components are easy to obtain from unbiased estimators of the eigenvalues; however,

since B,p,.,o - 0 is known and since Kó' is upper triangular, we need only estirnate

the last 20 - I eigenvalues. For this reason we will always assurne in this appendix

that a ~ 00. . .0.

We will first present unbiased estimators ~„ of the eigenvalues when the true

dis-turbances are known. The 2~ - t last equations of B- Kó ~~ then provide unbiased

estimators of the variance components 9a.

We will show that ~a is consistent under the normality of u by proving that

V(B„) -~ 0; the usual continuity argument does not apply since Kó ~ is a function of

the numbers of observations n;. We will then show that predicting the truc

distur-bances by the analysis of covariance residuals ( a procedure suggested by Arnemiya)

leads to estimated variance components BQ with the same asymptotic distribution as

the BQ. It follows that Ba is a consistent, and asymptotically unbiased, estimator of Ba.

In the omitted component case of Section 3, we let By - 0, and 9Q - Bu for a~

y; misspecification does not affect the consistency of Bá since ~ is estimated from the

correctly specified model.

We first show that when the true disturbances are known, unbiased estimators

of the eigenvalues are given by Xa ~á'u'R4u, where, from (14), ~„ tr(Ru)

-j-j;' i(n; - I)"~. This follows from Balestra [2, Lemmá I]:

E(~.~) - 1~..~ tr(R..V) - ~„t tr~R.. ~i~aRd) - ~,.l~u~ tr(R.,) -

~..-u

Under the normality of u, we have, from Balestra [2, Lemma 6]:

V(.4'R„u) - 2tr(RQVRaV) - 2tr(~~RQ) - 2~a~a;

furthermore u'R„u and u'Ruu are independent for a~ Q. Upon Ictting Ó„ -~~i c„U~~i

with c,,,~ the element in row a and column (3 of Kó ~, we then havc

where 0 and 0~ are diagonal matrices with the ta~' and the cáW on the diagonal,

re-spectively. It may be checked that

1

p - v

~ (n; - I)

,-ó (n; - 1

O1

;-, `

0

I J

I

o

0

0 tr~

~~ - Z ~

z

N

;-,

0

n;

so that the matrix in the quadratic form (30) may be written

I

o

_n;

KóO~aKa -

z p

Q

X

N ~(n;-1) ;a,

I

;~

Oló~n;-1

Ol

IJ;L,

0

IJ

O r 0

0 la O! n,

I 1

;-, l0 n? J;:, l 0 I J

(31)

(32)

l

P!0

0 1"~ (n?(n; - 1)

n;(n; - 1)1'-"~

z

-

o

~

1`

zJ

l

J

N~( n; - 1) ; s t

0

n;

n; ( n; - I)

n;

;-i

- ó~0

0

~a~ ~ 1

n; '

_{~'-"~ arr}

P

(33)

;-~

0

(n; - I)-'

n; '

n~ '(n; - I)-,

a

It is easily seen from (33) that lim Pa - O, so that V(8„) -~ 0 in (30). Furthermore,

it can be checked that lim ~aPa - eaeá, where ea is the 2o x I unit vector with

ele-ment ~ equal to unity; equation (30) then implies lim ~aV(Ó„) - 28,;. This result, as

well as equations (30) and (33), is consistent with Amemiya [1, pp. 5 and 6].

We now show the asymptotic equivalence when u is replaced by u- My - Mu,

where

M - IN - ~ SN - ~IN - N SN~ (.Y(X'QX)-'.Y'Q)

with Q R,,,.., `QP i(I,,; n; 'S,,;) (see Amemiya [1]). Since RuSN SNR,.

-O, we have

M'R„~~f - R„ - Q'X(X'QX)-'X'R~,X(X'QX)-'X'Q.

Hence, upon letting

The estimators Ba and ~Q will be asymptotically equivalent if the last terms in (34)

converge in probability to zero. If (X'QX)~Ntends to a finite, positive definite matrix

and if (X'R~X)IN is bounded, this will be true when ~c,,,i~~ ~ tends to 0 for all

Q, by the same argument as in Amemiya [I, equation (16)J. But c„~ita,~ ~ is the elemcnt

in row ~ and column Q of Kó ~0, where ~ is defined in (31). Since, from (12) and

(3 I ).

I

i

p

I

I

n, - l

0

1

`'

I

-( n, - I)-'

0

0

r

K~- N~(n;-I) ,--Oi

0

n,

0

I~-NO `0

n;(n;-I)-~~~

No. 1 G. Marini end F. van der Plceg, Monetary and fiscal polícy in an optimising model with capital accumulation and finite lives,

The Economic Journal, vol. 98, no. 392, 1988, pp. 772 - 786.

No. 2 F. van der Plceg, International polícy coordination in interdependent monetary economiea, Journal of International Economics, vol. 25,

1988. PP. 1 - 23.

No. 3 A.P. Barten, The history of Dutch macrceconomic modelling

(1936-1986), in W. Driehuis, M.M.G. Fase and H. den Hartog (eds.), Challenges for Macroeconomic Modellíng, Contributions to Economic Malysis 178, Amsterdam: North-Hollend, 1988. PP. 39 - 88. No. 4 F. van der Plceg, Disposable income, unemployment, inflation and

state spending in s dynamic political-economic model, Public Choíce, vo1. 60, 1989. Pp. 211 - 239.

No. 5 Th. ten Raa end F. van der Ploeg, A statistical approach to the problem of negatives in input-output analysis, Economic Modelling, vol. 6, no. 1, 1989. PP. 2- 19.

No. 6 E. van Damme, Renegotiation-proof equilibria in repeated prisoners' dilemma, Journal of Economic Theory, vol. 47, no. 1, 1989.

pp. 206 - 217.

No. 7 C. Mulder and F. van der Plceg, Trade unions, investment and employment in a amall open economy: a Dutch perspective, in J. Muysken and C. de Neubourg (eds.), Unemployment in Europe, London: The MacMillan Press Ltd, 1989. PP. 200 -

229-No. 8 Th. van de Klundert and F. ven der Plceg, Wage rigidity and capitel mobility in an optimizing model of a small open economy, De Economist

137. nr. 1, 1989, PP. 47 - 75.

No. 9 G. Dhaene and A.P. Barten, When it all began: the 1936 Tinbergen model revisited, Economic Modelling, vol. 6, no. 2, 1989.

PP. zo3 - 219.

No. 10 F. van der Plceg and A.J. de Zeeuw, Conflict over arms accumuletion

in market end command economies, in F. van der Ploeg and A.J. de Zeeuw (eds.). Dynamic Policy Cames ín Economics, Contributions to

Economic Malysis 181, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1989. PP. 91 - 119.

No. 11 J. Driffill, Macrceconomic policy games with incomplete information:

some extensions, i n F. van der Plceg and A.J. de Zeeuw ( eds.), Dynamic Policy Games in Economics, Contributions to Economic Analysis 181, Amsterdam: Elsevier Science Publishers B.V. (North-Holland),

1989. pP. 289 - 322.

No. 12 F. van der Plceg, Towards monetary integration in Europe, in P. De Crauwe e.a., De Europese Moneteire Integratle: vier visies,

Development, Berlin~Heidelberg: Springer-Verlag. 1989. pP. 272 - 305. No. 14 A. Hoque, J.R. Magnus and B. Pesaren, The exact multi-period

mean-square forecast error for the first-order autoregressive model,

Journal of Econometrics, vol. 39. no. 3. 1988. PP. 327 - 346.

No. 15 R. Alessie, A. Kapteyn and B. Melenberg, The effects of liquidity constraints on consumption: estimation from household panel data, Eurocean Economic Review 33. no. 2I3. 1989. PP- 547 - 555. No. 16 A. Holly end J.R. Magnus, A note on instrumental variables and

maximum likelihood estimation procedures, Annales d'Économie et de Statistigue, no. 10, April-June, 1988, pp. 121 - 138.

No. 17 P. ten Hacken, A. Kapteyn and I. Woittiez, Unemployment benefits and the labor market, a micro~macro approach, in B.A. Custafsson and N.

Mders Klevmarken ( eds.), The Politícal Economy of Social Security, Contributions to Economic Malysis 179, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1989. pP. 143 - 164.

No. 18 T. Wensbeek and A. Kapteyn, Estimation of the error-components model

with incomplete panels, Journal of Econometrics, vol. 41, no. 3,

1989. PP. 341 - 361.

No. 19 A. Kapteyn, P. Kooreman and R. Willemse, Some methodological issues in the implementation of subjective poverty definitions, The Journal of Human Resources, vol. 23, no. 2, 1988, pp. 222 - 242.

No. 20 Th. van de Klundert and F. van der Plceg, Fiscel policy and finite

lives in interdependent economies with real and nominal wage rigidity, Oxford Economic Papers, vol. 41, no. 3, 1989. PP 459

-489.

No. 21 J.R. Magnus and B. Pesaran, The exact multi-period mean-square forecast error for the first-order autoregressive model with an intercept, Journel of Econometrica, vol. 42, no. 2, 1989.

PP. 157 - 179.

No. 22 F. van der Plceg, Two essays on political economy: (i) The political economy of overvaluation, The Economic Journal, vol. 99. no. 397.

1989. PP. 850 - 855; (ii) Election outcomes and the stockmarket, European Journal of Political Economy, vol. 5, no. 1, 1989. PP. 21

-30.

No. 23 J.R. Magnus and A.D. Woodland, On the maximum likelihood estimation of multivariate regression models containing serially correlated error components, International Economic Review, vol. 29, no. 4, 1988. DP. 707 - 725.

No. 26 A.P. Barten and L.J. Bettendorf, Price formation of fish: M application of en i nverse demend system, European Economic Review,

vol. 33. no. 8. 1989. PP. 1509 - 1525.

No. 27 G. Noldeke end E. van Demme, Signelling in a dynamic labour market, Review of Economic Studies, vol. 57 (1), no. 189, 1990, pp. 1- 23 No. 28 P. Kop Jansen and Th. ten Raa, The choíce of model in the

construction of ínput-output coefficienta matrices, International Economic Revíew, vol. 31, no. 1, 1990, pp. 213 - 227.

No. 29 F. van der Plceg and A.J. de Zeeuw, Perfect equílibrium i n a model of competítive arms accumulation, Internationel Economic Review, vol.

31, no. 1, 1990, pp. 131 - 146.

No. 30 J.R. Magnus and A.D. Woodland, Separabílity and aggregation, Economica, vol. 57. no. 226, 1990, pp. 239 - 247.

No. 31 F. van der Ploeg, International interdependence end policy

coordination ín economiea wíth real and nominal wage rigidity, Creek Economic Review, vol. 10, no. 1, June 1988. PP. 1- 48.

No. 32 E. van Damme, Sígnaling and forward índuction in a market entry context, Operations Research ProceedinBS 1989. Berlin-Heidelberg: Springer-Verlag, 1990, pp. 45 - 59.

No. 33 A.P. Barten, Toward a levels version of the Rotterdam and related demand systems, Contributions to 0 erations Research and Economics, Cambridge: MIT Press, 1989, pp. 41 - 65.

No. 34 F. ven der Plceg, International coordination of monetary policies under alternative exchange-rate regímes, Advanced Lectures in Quantitative Economics, London-Orlando: Academic Press Ltd., 1990, PP. 91 - 121.

No. 35 Th. ven de Klundert, On socioeconomic causes of 'wait unemployment',

European Economic Review, vo1. 34, no. 5. 1990, pp. 1011 - 1022. No. 36 R.J.M. Alessie, A. Kapteyn, J.B. van Lochem and T.J. Wansbeek,

Individual effects in utility consistent models of demand, ín J. Hartog, G. Ridder and J. Theeuwes ( eds.), Panel Data and Labor Market Studies, Amsterdam: Elsevier Science Publishers B.V.

(North-Holland), 1990, pp. 253 - 278.

No. 37 F. van der Ploeg, Capitel accumulation, inflation and long-run

conflict i n international objectives, Oxford Economic Papers, vol. 42, no. 3, 1990, pp. 501 - 525.

sector model with a dual labour market, Metrceconomica, vol. 40, no.

3. 1989. pp. 235 - 256.

No. 40 Th. Nijman and M.F.J. Steel, Exclusion restrictions in instrumental variables equations, Econometric Reviews, vol. 9, no. i, 1990, pp. 37

- 55.

No. 41 A. van Scest, I. Woittiez and A. Kapteyn, Labor supply, income taxes. end hours restrictiona in the Netherlands, Journal of Human

Resources, vol. 25, no. 3, 1990, pp. 517 - 558.

No. 42 Th.C.M.J. van de Klundert and A.B.T.M. van Schaik, Unemployment persistence and loss of productive capacity: a Keynesien approach, Journal of Macrceconomics, vol. 12, no. 3, 1990, pp. 363 - 380. No. 43 Th. Nijman and M. Verbeek, Estimation of time-dependent parameters in

linear models using cross-sections, panels, or both, Journel of Econometrlcs, vol. 46, no. 3. 1990, pp. 333 - 346.

No. 44 E. van Damme, R. Selten and E. Winter, Alternating bid bargaining with e smallest money unit, Games and Economic Behavior, vol. 2, no. 2, 1990, pp. 188 - 201.

No. 45 C. Dang, The D1-triangulation of Tn for simplicial algorithms for computing solutiona of nonlinear equations, Mathematics of Operations Research, vol. 16, no. 1, 1991, pp. 148 - 161.

No. 46 Th. Nijman and F. Palm, Predictive accuracy gain from disaggregate sempling in ARIMA models, Journnl of Business d, Economic Statistics. vol. 8, no. 4, 1990, pp. 405 - 15.

No. 47 J.R. Magnus, On certain moments relating to ratios of quadratic forms in normal variables: further results, Sankhya: The Indian Journal of Statistics, vol. 52, series B, part. 1, 1990, pp. 1- 13.

No. 48 M.F.J. Steel, A Bayesian enalysía of simultaneous equation models by combining recursive analytical and numerícal approaches, Journal of Econometrics, vol. 48, no. 1~2, 1991, pp. 83 - 117.

No. 49 _{F. ven der Plceg and C. Withagen, Pollution control end the ramsey} problem, Environmental and Resource Economics, vol. 1, no. 2, 1991, pp. 215 - 236.

No. 50 F. van der Plceg, Money end capital in interdependent economies with overlappíng generations, Economica, vol. 58, no. 230, 1991,

pp. 233 - 256.

No. 51 _{A. Kapteyn and A. de Zeeuw, Changing íncentives for economic research}

in the Netherlands, European Economic Review, vo1. 35,.no. 2~3, 1991,

pp. 603 - 611.

Economic Journal, vol. 101, no. 406. 1991, pp. 404 - 419. No. 54 W. van Grcenendaal and A. de Zeeuw, Control, coordination and

conflict on international commodity markets, Economic Modelling, vol. 8, no. 1, 1991, pp. 90 - 101.

No. 55 F. van der Plceg end A.J. Markink, Dynamic policy in linear models with rational expectations of future events: A computer package, Computer Science in Economics end Management, vol. 4, no. 3, 1991, PP. 175 - 199.

No. 56 H.A. Keuzenkamp and F. ven der Plceg, Savings, investment, government finance, and the current account: The Dutch experience, in G.

Alogoskoufls, L. Papadeeos and R. Portes ( eds.), Externel Constraints

on Macrceconomíc Policy: The European Experience, Cambridge: Cembridge University Press, 1991. PP. 219 - 263.

No. 57 Th. Nijmen, M. Verbeek and A. van Scest, The efficiency of

rotating-panel designs in an analysis-of-varience model, Journal of

Econometrics, vol. 49, no. 3. 1991. pp. 373 - 399.

No. 58 M.F.J. Steel and J.-F. Richard, Bayesian multivariate exogeneity

analysis - an application to a UK money demand equatíon, Journal of Econometrics, vol. 49, no. 1~2, 1991, pp. 239 - 274.

No. 59 Th. Nijman and F. Palm, Generalized least squares estimatíon of linear modela containing rational future expectations, International Economic Review, vol. 32, no. 2, 1991, pp. 383 - 389.

No. 60 E. van Damme, Equilibrium selection i n 2 x 2 games, Revista Espanola de Economía, vol. 8, no. 1, 1991. Pp. 37 - 52.

No. 61 E. Bennett and E. van Damme, Demand commitment bargaining: the case

of apex gamea, in R. Selten (ed.), Game Equilibrium Models III

-Strategic Bargaining, Berlín: Springer-Verlag, 1991, pp. 118 - 140. No. 62 W. Gi1th and E. van Damme, Gorby games - a game theoretlc analysis of

disarmament cempaigns and the defense efficiency - hypothesís -, ín

R. Avenhaus, H. Karkar and M. Rudnianski (eds.), Defense Decision Making - Anelytícal Support and Crisis Management, Berlin: Springer-Verlag, 1991. PP. 215 - 240.

No. 63 A. Rcell, Dual-capacity trading and the quality of the market,

Journal of Financial Intermediation, vol. 1, no. 2, 1990, pp. 105 - 124.

No. 64 Y. Dai, G. van der Laan, A.J.J. Telmen and Y. Yamamoto, A simplicial algorithm for the nonlinear statíonary point problem on an unbounded polyhedron, Siam Journal of Optimization, vol. 1, no. 2. 1991, pp.

151 - 165.

No. 65 M. McAleer and C.R. McKenzie, Keynesian and new classical models of unemployment revisited, The Economic Journal, vol. 101, no. 406,

No. 67 J.R. Magnus end B. Pesaran, The bias of forecasts from a first-order autoregression, Econometric Theory, vol. 7, no. 2, 1991, pp. 222

-235.

No. 68 F. van der Plceg, Macrceconomic policy coordination issues during the various phases of economic end monetary integration in Europe, European Economy - The Economics of EMU, Commission of the European Communities, special edition no. 1, 1991, pp. 136 - 164.

No. 69 H. Keuzenkamp, A precursor to Muth: Tinbergen's 1932 model of ratíonal expectations, The Economic Journal, vol. 101, no. 408, 1991, pp. 1245 - 1253.

No. 70 L. Zou, The target-incentive system vs. the price-incentive system

under adverse selection end the ratchet effect, Journal of Public Economics, vol. 46, no. 1, 1991. PP. 51 - 89.

No. 71 E. Bomhoff, Between price reform and privatization: Eastern Europe in trensition, Finanzmarkt und Portfolio Management, vol. 5, no. 3,

1991. pP. 241 - 251.

No. 72 E. Bomhoff, Stability of velocity in the major industrial countries:

a Kalman filter approach, International Monetary Fund Staff Papers,

vol. 38, no. 3, 1991, Pp. 626 - 642.

No. 73

E. Bomhoff, Currency convertibility: when and how? A contribution to the Bulgarien debete, Kredit und Kapital, vol. 24, no. 3, 1991, pp.

412 - 431.

No. 74 H. Keuzenkamp and F. ven der Plceg, Perceived constraints for Dutch unemployment policy, in C. de Neubourg (ed.), The Art of Full Employment - Unemployment Policy in Open Economies, Contributions to Economic Analysis 203, Amsterdam: Elsevier Science Publishers B.V.

(North-Holland), 1991. PP. 7 - 37.

No. 75 H. Peters end E. ven Damme, Characterizing the Nash and Raíffa bargaining solutions by disagreement point axions, Mathematics of

Operations Research, vol. 16, no. 3, 1991, pp. 447 - 461.

No. 76 P.J. Deschamps, On the estímated variances of regression ccefficients in misspecified error components models, Econometric Theory, vol. 7,

Bibliotheek K. U. Brabant