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Research Papers in Economics

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RESEARCH MEMORANDUM
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TILBURG UNIVERSITY
DEPARTMENT OF ECONOMICS
Postbus 90153 - 5000 LE Tilburg
Netherlands
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159
COMMENT ON: IDENTIFICATION IN THF.
LLNEAK F.RRORS IN VARIARLES MODEL
Ay Paul A. Bekkerl)
1) The author wishes to thank Arie Kapteyn and Tom Wansbeek for their
stimulating discussions on the subject. Financial support by the Nether-
lands Organization for the Advancement of Pure Research (ZWO) is grate-
fully acknowledged.1. Introduction
Kapteyn and Wansbeek [1] considered the following multiple linear re-
gression model with errors in variables:
(1.1) Yj - ~~B f Ej. (j z 1,...,n)
(1.2) xj - ~j f uj,
where ~j, xj, uj and S are k-vectors, yj, Ej are scalars. The ~j are
unobservable variables: instead the xj are observed. The measurement
errors u j are unobservable as well and ít is assumed that v j - N(O,S2)
and e j- N(~,a2) for all j. The v j and e j are mutually independent and
independent of ~j. The ~j are considered as random drawings from some,
as yet unspecified, multivariate distribution with zero mean.
For the case k-1 Reiersbl [2] has shown that normality of ~j is the only
distributional assumption which spoils identification. For the case k~
1 and the components of ~j are mutually independent, Wilassen [3] has
shown that none of the components of ~j should be normally distributed
to ~uarantee identifiabílity of S. Kapteyn and Wansbeek [1] did not
assume independency of the components of ~j and they stated the follo-
wing proposition: the parameter vector S is identified if and only if
there does not exist a lineair combination of ~j which is normally
dístributed. The necessity part in this proposition is incorrect, i.e.
it may well be that a normally distributed linear combination of ~j does
not spoil the identifiability of B. Here I present necessary and suf-
ficient conditions for identification of s.2
2. Statement of the Result and Proof
Proposition: Under the assumptions above, the parameter vector 6 is
identified if and only if there does not exist a nonsin-
gular matrix A-(a1,A2) such that l;~al is distributed no r-
mally and independently of ~~A2.
Proof: We first show that nonidentifiability of S implies the existence
of the matrix A. Let s be a scalar and t a k-vector. The characteristic
function ~y ~x ( s,t) of y) and x~ is
J J
(2.1) ~y ~x (s,t) - exp{-}(a2s2tt'S2t)} ~~(Bstt) ,
J ~
where m~(.) is the characteristic functíon of ~~. Assuming that B is not
fully identified amoun[s [o saying that there exíst parameter
2
sets {B,aZ,f2} and {p~,a~ ,52~}, with S~ B~, generating the same distri-
bution of y~, x~. Consequently ~ x(s,t) should be the same for both
sets of parameters: y~~ ~
2
(2.z) exp{-~(a2s2ft'f2t)} ~~(psft) 3 exp{-}(a~ sZft'S2~t)} m~(s~stt).
~ ~
1.et R- S sft, then ~~(Ssft) -~~(((3-B )sflC) L~ ~ (s,R). Thus
~~ca-s ),~
(2.2) caries over into
2
(2.3) ~ ~ ( s,R) - exp{-}Is2(a~ -a2) f
F'(s-s ),~
t (R-B~s)'(R~-S2)(k-S~s)]} ~~(k).
The characteristic function corresponding to the marginal distribution
~
of ~~(S-g ) is found by setting R- 0
J3
~
2 ~~ 2 ~, ~ ~
(2.4) ~ ~ ( s) - exp{-~s (a -a fs (S2 -S2)S )} ,
~' (s-a )
which is the characteristic function of a normally distributed variable.
In addition [o this result, which was obtained by Kapteyn and
Wansbeek [1], it wíll now be shown that nonidentifiability of S also
implies the existence of a matrix AZ such that ( a1,A2) is nonsingular
and ~'al i s distributed i ndependently of ~~AZ. The characteristic func-
J
tion corresponding to the marginal distributions of ~~ is found by set-
ting s-0 in (2.3):
(2.5) ~~(R) - exp{-~R.'(St~-St)k} ~~(R).
Thus, we may rewrite (2.3) as
(2.6) ~ ~ (s,R) - ~ ~ (s) ~~(k) exp{sB~'(St~-St)k}.
~'(B-S ).~ ~'(B-B )
~~ ~
Let B be a(kx(k-1))-matrix of full column rank such that B (S2 -S2)B -
0. F.quality (2.6) holds for all possible values of s and R. In particu-
lar (2.6) holds if we let R vary such that R ~ Bm, where m is a(k-1)-
vector:
(2.7) ~ ~ (s,Bm) - ~ ~ (s) ~~(Bm).
~'(6-B ),~ ~'(B-B )
or equivalently,
(2.8) ~ ~ (s.m) - ~ .~ (s) ~~,B(m).
~'(S-B ,B) ~'(B-B )
~r
Thus nonidentifiability of S implies the existence of a matrix (~-S ,B)
~
such that ~'.(B-B ) ís distributed normally and índependently of
J ,~ ~r
~'B. If rank (S-S ,B) - k then a matrix A is given by (s-S ,B). In the
~ ~ ~
trivial case where Rank (B-B ,B) - k-1, the variable ~~(B-S ) is distri-
buted independently of itself and must therefore be equal to zero iden-
tically (which is also considered as a normal distribution). In that
~
case any nonsingular matrix A whose first column equals ~-g wíll do.4
To prove the necessity part of the Propositíon we assume that
there exists a nonsingular matrix A-(a1,A2) such that ~~al is distri-
buted normally and independently of ~~A2. If we substitute t- AR -
a1R1 f A2R2 and B- AS - alsl } A2~2 (R1 and B1 are scalars, R2 and 62
are (k-1)-vectors) in (2.1), then the characteristic function of y~,x~
takes the following form:
(2.9) ~yJ~x,(s,AR) - exp{-}(o2s2tR.'A'S2AA.)} ~~(A(BsfR)).
The characteristic function ~~(A(BstR)) can be rewritten as follows:
(2.1(1) ~~(A(ssfA.)) - ~~,A(Bsi.k) - ~~,al(B1sfR1) ~~,A2(62sfX,2)
- expl-~(BlsfRl)2Var(~'al)} ~~rA (62sfk2).
2
Using (2.1~), (2.9) carries over into
(2.11) ~y ~x.(s1AR) - exp{-}(s,R')C(s,k')'} ~~,A (BZstk2) ,
where
i t 2
(2.12) C -
a2 0
0 A'S2A
t Var(~'al)
e B e e'
The ~k(ktl) t 2 nonzero elements of C are functions of }k(k~-1) f 3 para-
meters i n a2, St, 61 and Var(~'al), whereas the function ~~,A ( BZsi-RZ) is
2
not affected by these parameters. Clearly, different parameter values
~enerate the same distribution of y~,x~. The existence of a nonsingular
transformation such that ~~al is distributed normally and independently
of t;'A2 thus implies nonidentifiability of S. Q.E.D.
75
3. Discussion
Compared to Kapteyn and Wansbeek's proposition, the sufficiency part of
the proposition proved here is stronger. Nonidentifiability does not
only imply the existence of a normally distributed linear combina-
tion ~~al , but also the existence of AZ such that ~~al and ~~A2 are
mutually i ndependent. Consequently, the necessity part of their proposi-
tion must be wrong, because they fail to invoke the existence of a
matrix A2 such that ~~al and ~~A2 are mutually independent.
As an example, consider the model with two regressors ~~1 and
F~2, the first of which is normally distributed, ~~1 - N(O,a2), and the
second i s a function of the first ~~z s~~1 - a2. According to Kapteyn
and Wansbeek this model would not be identified since ~~1 is normally
distributed. However, this point of view would be too pessimistic.
Clearly there i s no nonsingular transformation ( al,a2) such that
(~~l,~~Z)al
is distributed independently of (~j1,~~2)a2 and so the model
is identified.References
[1] Kapteyn, A. and T..J. Wansbeek: "Identification in the Linear Errors
in Variables Model", Econometrica 51 (1983), 1847-49.
[2] Reíersbl, 0.: "Identifiability of a Linear Relation Between
Variables Which are Subject to Error", Econometrica 18 (1950),
375-389.
[3] Willassen, Y.: "Extension of Some Results by Reiersbl to Multi-
variate Models", Scandinavian Journal of Statistics 6(1979),
89-91.i
IN 1983 REF,DS VERSCHENF.N
126 H.H. Tigelaar
Identification of noisy linear systems with multíple arma inputs.
127 J.P.C. Kleijnen
Statistical Analysis of Steady-State Simulations: Survey of Recent
Progress.
128 A.J. de Zeeuw
Two notes on Nash and Information.
129 H.L. Theuns en A.M.L. Passier-Grootjans
Toeristische ontwikkeling - voorwaarden en systematiek; een selec-
tief literatuuroverzicht.
130 J. Plasmans en V. Somers
A Maximum Likelihood Estimation Method of a Three Market Disequili-
brium Model.
131 R. van Montfort, R. Schippers, R. Heuts
Johnson SU transformations for parameter estimation in arma-models
when data are non-gaussian.
132 J. Glombowski en M. Kruger
~n the R81e of Distribution in Different Theories of Cyclical
Growth.
133 J.W.A. Vingerhoets en li.J.A. Coppens
Internationale Grondstoffenovereenkomsten.
Effecten, kosten en oligopolisten.
134 W.J. Oomens
The economic interpretation of the advertising effect of Lydia
Pinkham.
135 J.P.C. Kleijnen
Regression analysis: assiunptions, alternatives, applications.
136 J.P.C. Kleijnen
~n the interpretation of variables.
137 G. van der Laan en A.J.J. Talman
Simplicial approximation of solutions to the nonlinear complemen-
tarity problem with lower and upper bounds.11
IN 1984 REEDS VERSCHENEN
138 C.J. Cuypers, J.P.C. Kleijnen en J.W.M. van Rooyen
Testing the Mean of an Asymetric Population:
Four Procedures Evaluated
139 T. Wansbeek en A. Kapteyn
Estimation in a linear model with serially correlated errors when
observations are missing
140 A. Kapteyn, S. van de Geer, H. van de Stadt, T. Wansbeek
Interdependen[ preferences: an econometric analysis
141 W..J.H. van Groenendaal
Discrete and continuous univariate modelling
142 J.P.C. Kleijnen, P. Cremers, F. van Belle
The power of weighted and ordinary least squares with estimated
unequal variances in experimental design
143 J.P.C. Kleijnen
Superefficient estimation of power functions in simulation
experiments
144 P.A. Bekker, D.S.G. Pollock
Identification of linear stochastic models with covariance
restrictions.
145 Max D. Merbis, Aart J. de 7.eeuw
From structural form to state-space form
146 T.M. Doup and A.J.J. Talman
A new variable dimension simplicial algorithm to find equilibria on
the product space of unit simplices.
147 G. van der Laan, A.J.J. Talman and L. Van der Heyden
Variable dimensíon algorithms for unproper labellings.
148 G.J.C.'rh. van Schijndel
Dynamic firm behaviour and financial leverage clienteles
149 M. Plattel, J. Peil
The ethicr political and theoretical reconstruction of contemporary
economic doctrínes
150 F.J.A.M. Hoes, C.W. Vroom
Japanese Business Policy: The Cash Flow Triangle
an exercise in sociological demystification
151 T.M. Doup, G. van der Laan and A.J.J. Talman
The (2TT}1-2)-ray algorithm: a new simplicial algorithm to compute
economic equílibriaiii
IN 1984 RF.EDS VERSCHENEN (vervolg)
l52 A.L. Hempenius, P.G.H. Mulder
Total Mortality Analysis of the Rotterdam Sample of the Kaunas-
Rotterdam Intervention Study (KRIS)
153 A. Kapteyn, P. Kooreman
A disaggregated analysis of the allocation of time within the
household.
154 T. Wansbeek, A. Kapteyn
Statistically and Computationally Efficien[ Estimation of the
Gravity Model.
155 P.F.P.M. Nederstigt
Over de kosten per zíekenhuisopname en levensduurmodellen
156 R.R. Meijboom
An input-output like corporate model including multiple
technologies and make-or-buy decisions
157 P. Kooreman, A. Kapteyn
F.stimation of Ra[ioned and Unrationed Household Labor Supply
Functions Using Flexible Functional Forms
158 R. Heuts, J. van Lieshout
An implementation of an inventory model with stochastic lead time~i~~~ui~~~~iáiu~~~ui
~ 7 000 O~ 059784 8

A new variable dimension simplicial algorithm to find equilibria on the product space of unit simplices.

de 7.eeuw From structural form to state-space form

der Heyden Variable dimensíon algorithms for unproper labellings.

Extension of Some Results by Reiersbl to Multivariate Models&quot;,

Groenendaal Discrete and continuous univariate modelling

Heuts Johnson SU transformations for parameter estimation in arma-models when data are non-gaussian.

Identification in the Linear Errors in Variables Model&quot;,

Kapteyn Estimation in a linear model with serially correlated errors when observations are missing

Kapteyn F.stimation of Ra[ioned and Unrationed Household Labor Supply Functions Using Flexible Functional Forms

Kapteyn Statistically and Computationally Efficien[ Estimation of the Gravity Model.

Kleijnen ~n the interpretation of variables.

Kooreman A disaggregated analysis of the allocation of time within the household.

Kruger ~n the R81e of Distribution in Different Theories of Cyclical Growth.

Lieshout An implementation of an inventory model with stochastic lead

Meijboom An input-output like corporate model including multiple technologies and make-or-buy decisions

Over de kosten per zíekenhuisopname en levensduurmodellen

Peil The ethicr political and theoretical reconstruction of contemporary economic doctrínes

Pollock Identification of linear stochastic models with covariance restrictions.

Regression analysis: assiunptions, alternatives, applications.

Schijndel Dynamic firm behaviour and financial leverage clienteles

Somers A Maximum Likelihood Estimation Method of a Three Market Disequilibrium Model.

Statistical Analysis of Steady-State Simulations: Survey of Recent Progress.

Superefficient estimation of power functions in simulation experiments

Talman Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds.11 IN

Testing the Mean of an Asymetric Population: Four Procedures Evaluated

The (2TT}1-2)-ray algorithm: a new simplicial algorithm to compute economic equílibriaiii IN

The economic interpretation of the advertising effect of Lydia Pinkham.

Theuns en A.M.L. Passier-Grootjans Toeristische ontwikkeling - voorwaarden en systematiek; een selectief literatuuroverzicht.

Tigelaar Identification of noisy linear systems with multíple arma inputs.

van Belle The power of weighted and ordinary least squares with estimated unequal variances in experimental design

Vingerhoets en li.J.A. Coppens Internationale Grondstoffenovereenkomsten. Effecten, kosten en oligopolisten.

Vroom Japanese Business Policy: The Cash Flow Triangle an exercise in sociological demystification

Wansbeek Interdependen[ preferences: an econometric analysis

Zeeuw Two notes on Nash and Information.

Comment on: Identification in the linear errors in variables model

Comment on: Identification in the linear errors in variables model

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