Bekker, P.A.

English

Research Papers in Economics

CBM
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7626 1
1985 ~
169
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faculteit der economische wetenschappen
RESEARCH MEMORANDUM
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TILBURG UNIVERSITY
DEPARTMENT OF ECONOMICS
Postbus 90153 - 5000 LE Tilburg
Netherlands
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169
A NOTE ON THE IDENTIFICATION OF
RESTRICTED FACTOR LOADING MATRICES
by Paul A. Bekker
1985
The author would like to thank Arie Kapteyn for his comments and sugges-
tions. Financial support by the Netherlands Organization for the Advan-
cement of Pure Research (ZWO) is gratefully acknowledged.
Tilburg University, Department of Econometrics, P.O. Box 90153, 5000 LE
Tilburg, The Netherlands.Abstract
It is shown that problems of rotational equivalence of restricted factor
loading matrices in orthogonal factor analysis are equivalent to pro-
blems of identification in simultaneous equations systems with covarian-
ce restrictions. A necessary and sufficient condition for local unique-
ness is given and a counterexample is provided to a theorem by J. Algina
concerning necessary and sufficient conditions for global uniqueness.
Key Words: identification, orthogonal rotation, confirmatory factor ana-
lysisi
1. Introduction
Consider the orthogonal factor analysis model E- AA' t~,, where
E is a positive semi-definite pXp-matrix which can be consistently estí-
mated, ~ is a pXp-diagonal matrix of unique variances, A is a pmn-factor
loading matrix of full column rank (p ~ m). Identifíability of the ma-
trices A and ~ amounts to the question whether these matrices can be
solved uniquely from these equations. In what follows, it ís assumed
that the matrix ~ is identified. Thus iden[ifiability of A stems from
the pruperties of the equations C- E-~- AA'. In general, these equa-
tions do not yield a unique solution for A. Identifiability may be aided
by invoking prior information represented by linear restrictions on the
columns of A-(al,...,am). The identifying equations for A are thus
given by
(I) c - An',
(Z) R~a~ - r~ ~ J - 1.-.-~m~
where R~ is a k~xp-matrix and r~ a k~-vector, representíng k~ linear
restrictions on a,.
J
In his article on confirmatory factor analysis JSreskog (1969)
attempted to give conditions for uníqueness of A based on the specifica-
tion of m(m-1)~2 elements. Dunn ( 1973) and Jennrich ( 1978) gave counter-
examples showing that the conditions given by JSreskog were no[ suffi-
cient. Dunn also gave sufficient conditions for identífication (up to
column sign changes) based on the specification of m(m-1)~2 zeros. Jenn-
rich showed that when the m(m-1)~2 specified values are not necessarily
zero, the number of solutions for A equals 2m. Geweke and Singleton
(1981) gave the same conditions as Dunn had given and noticed that these
conditions are analogous to sufficient conditions for the identifiabí-
lity of simultaneous equations systems with uncorrelated disturbances.
Recently, Hausman and Taylor ( 1983), Hausman, Newey and Taylor (1983)
and Bekker and Pollock ( 1984) treated the problem of identification ín
simultaneous equations systems with zero covariance restrictíons on the
disturbances. It may therefore be helpful to further develop the analogy2
nuticed by Geweke and Singleton (1981). Another purpose of this note ís
to Jemonstrate that Algina's (1980) necessary and sufficient conditions
for identiEication, in his theorem 3, are, in fact, not necessary and
consequently, that the necessity part of his proof is wrong.3
'L. An AnalogY
The equations in (1) can be seen as a special case of the more
general equations C- A~A', where ~ is a positive defínite m~nn-matrix.
Let n be partitioned as A' -(Ai,n2), with a curresponding partitioníng
of C. It is assumed, without loss of generality, that A1 and C11 are
nonsingular mxm-matrices. Furthermore, let A be an arbitrary positive
definite (p-m)x(p-m)-matrix. Then
Lc21 c22}A ~
nl o
AZ I
and H is positive definite. Clearly then
(4) H 1
nl o
nZ I
m o
0 A
~-1 0
0 A 1
rn' n' ~
1 2
0 I
The classical simultaneous equatíons system of econometrics is
giveii by y'B t x'r - E', where y' is a m-vector of endogenous variables,
x' is a(p-m)-vector of exogenous variables, B is a nonsingular mXm-ma-
trix, r is a(p-m)xm-matrix, and the m disturbances in e' are uncorrela-
ted with the exogenous variables in x'. Denoting covariance matrices
by E, with subscripts refering to the variables, ít follows that
Ír' B'~
(5)
0 I
yy yx rE E -
E E
xy xx
B o
r I
EcE o
0 E
xx
The analogy with (4) is obvious. The structural coefficients in B and r
currespond to A1 and nZ respectively and the covariances of the distur-
bances in EEe correspond to ~-1.
When ~ is unrestricted as in oblique factor analysis, then also
m-1 is unrestricted and necessary and sufficient conditions for global
identiEication, with a unique solution for A, may be adapted from simi-4
lar identification conditions for simultaneous equations sys[ems. These
conditions were given by many authors, among others by Koopmans and
Reiers~l (1950), Anderson and Rubin (1956) and, more recently by Algina
(1980) and Geweke and Singleton (1981).
When ~ is restricted to be block diagonal, then also ~ 1 is
block diagonal and conditions for identification of A are similar to
conditions for identification in simultaneous equations systems
where Ece is block diagonal. Conditions for identification in these
cases were given by F.M. Fisher (1966). For example, Dunn's condition
can be seen as a special case of Fisher's conditions. Conditions for the
identiFication uf single rows of (K',t'), or A', were gíven by Hausman
and Taylor (1983) and Bekker and Pollock (1984). A necessary and suffí-
cient condition for local ídentification, based on the Jacobian matrix
of the equations in (5) is given by Bekker and Pollock (1984) and Haus-
man, Newey and Taylor (1983). An analogous condition for orthogonal fac-
tor analysis is the following. Let K be a permutation matrix, i.e. K is
a partitioned matrix of order m2xm2 whose (j,í)-th block is an mxm-
matrix which has a unit in the (i,j)-th position and zeros elsewhere:
K- E E(e.e:t~e.e'.). Let ttie restrictions (2) be written as R vec(A) - r,
J 1 1 J i j
where R is a E k,xmp-matrix and r is a E k, vector. It is assumed that
j ~ ~ ~
A is a regular point of the matrix:
I 2~-K
m
(6) F(A) -
i.e. the rank of F(A) is constant in an open neighbourhood of A. The
condition is the following.
Proposition: A necessary and sufficient conditions for a solution A of
the equations (1) and (2) to be locally unique is that rank
{F(A)} - m2.
The proof is omitted.
Algina (1980) has given a much simpler necessary and sufficient
condition Eor global identification of A. However, by using an example,similar to an example gíven by Bekker and Pollock (1984), I will show
ttlat A1Kina's condition is not necessary.6
3. .A counterexample
Algina's theorem 3 is as follows: Let Ej denote the first (j-1) columns
of the mxm identity matrix, then, under the restrictions given by (2),
where rj ~ 0, a necessary and sufficient condition for identification
of A is that rank(R1A) - m and rank(A'R'.,E,) - m for j- 2,...,m.
J J
In order to give a counterexample to this theorem, let A be par-
titioned as above, and let A1 be a 3X3 nonsingular matrix which is re-
stricted as
r
~11 ~12 0
(~) A1 - I 0 ~22 ~23
I; ~31 0 ~33
~
Civen these restrictions, one of the equations in Ci1 - AlA1 is
(8) (C11)31 - A31A11'
Considering the equations A1C11A1 - I, it follows that
(9)
(C11)I1~11 } 2(C11)13A11~31 t (C11)33~31 - 1'
Any two solutions Ail) and Ai2j, say, must thus satisfy
(10) a(1)a(1) - a(2)a(2) - (C ) . 31 11 31 11 11 31
and
(11) (C-1) ~(1)2 } (C 1) ~(1)2 - (C-1) ~(2)2 } (C-1) ~(2)2 .
11 11 11 11 33 31 11 11 11 11 33 31
2 2
h1u1[iplying (11) by
~11) All)
and using (10) yields
(L2) (C-1)
~(1)2~(2)2(a(1)2-a(2)2) - (C 1) (C )2 (a(1)2-a(2)2).
11 11 11 11 11 11 11 33 11 31 11 117
2 2
Therefore if a(1) ~ a(2) it must hold true that
' L1 11 '
-1 2
(13)
~(1)2~(2)2 - (C11)33(C11)31 .
11 11 (C-1)
11 11
This equation shows that if a(1), or -a(1), is a solution for a, then
there can be at most two other solutiuns aii) and -aii) which ldiffer
only by sign. Analogous expressions hold for the other parameters. Fur-
ttiermure, equation (10) shows that, within a single column of A1, the
value of one parameter, all say, fixes the value of the other free para-
meCer a31. ThereEore, up to column sign changes, there are no more than
two distinct solutions for A. As A- C A'-1
1 2 21 1~
each solution for A1
corresponds to a solution for A2. Consequently, if A1 is restricted ac-
cording to (7), there are, up to column sign changes, no more than two
distinct solutions for A, i.e. A is locally identified.
A nummerical example will show that the two distinct solutions
may indeed exist. Let C11 be the following matrix:
81 54 45 ~
(14)
C11 -
1
100
54 86 5
45 5 50
then two solutions for A1 are given by:
(15)
and
A(1) - 1
1 10
A(2) - 1 1 io
3.(5)~ 6 0
0 9 (5)~
3.(5)1` 0 (5)~
I9.(19)~ 27 } 0
(19)
0 2.(19)~ (10)~
(95)~ 0 }(10)~
2 L -8
Of course, within each colwnn signs may be changed.
Dunn's specification of 2eros leads to local identification of A
where A is uníque up to column sign changes. The example shows that spe-
cification of zeros may also lead to local identification of A where A
is not unique up to column sign changes. Further specification of a
value, e.g. all - 3.(5)~` leads to uniqueness of A up to sign changes of
the second and third columns. If also the values a22 - 9 and a33 -(5)~
are specified, then A is globally unique. This last specification may
serve as a counterexample to Algina's theorem 3. The restrictions on the
columns, R.a. - r., are such that r. ~ 0, so that, accordíng to the
J ] J ~
theorem, global identification of A implies the existence of at least
one culumn, ak say, such that rank(RkA) - 3. That is to say, at least
one column must have three specified values. In the example, specifica-
tion of only two values in each column leads to global identification.
Therefore Algina's conditions are not necessary for identification and
consequently, the necessity part of his proof is wrong.9
4. Iliscussiou
The sufficient conditions of Dunn (1973), Jennrich (1978), and Algina
(1980) for identificatíon of A are all based on the same principle. The
columns of A are identified sequentially. In the first step a column is
identified by means of linear restrictions Rlal - rl in (2) and
-1
(O,I)H-lal - 0 in (4) and a normalization aiH al - 1 in (4). In the
second step another column is ídentified using similar restrictions and
an additional bilinear restriction aiH laZ - 0 in (4), which becomes
linear if al is previously identified. If al and aZ are previously iden-
tified, then the third column has two additional linear restric-
tions (al,aZ)'H la3 - 0, etc..
-1
In this procedure the bilinear restrictions A'H A- I in (4)
behave as if they were linear restríctions on separate coliunns of A. As
all restrictions are linear, apart from the normalizations, the conse-
quent conditions for uniqueness of A are relatively simple. However,
these conditions, including Algina's condition, are not necessary. The
example presented here shows that A may be locally identified while no
column of A is previously identified by means of linear restrictions and
a normalization. In such cases the restrictions in A'H-lA - I truly be-
iiave as bilinear restrictions which cannot be reduced to linear restric-
tions, and the columns of A are identified simultaneously instead of
sequentially. A necessary and sufficient condition for identification
must therefore be a system-wide condition instead of a sequence of rela-
tively simple conditions for separate columns.
Ttie necessary and sufficient condition for local identífication
of A that is presented here is system-wide. However, the evaluation of
the rank of the matrix F(A) may be very complícated. Therefore, in my
opiuion, the problem of identification of A, or the equivalent problem
of identifiability of simultaneous equations systems with covariance
restríctions, cannot be considered as completely solved. Moreover, an
additional problem is encountered in confirmatory factor analysis.
Usually it is assumed, just as in this note, that the matrix of unique
variances ~y is identified. Although it is not hard to give necessary and
sufficient conditions for the identification of y~ ín case there are no
prior restrictions on elements of A(see for instance Shapiro (1983)),10
restrictions on A may affect the identification of ~. Therefore the
overall identification problem in confirmatory factor analysis is far
from being solved.11
Ketercnces
Algina, J., 1980, A note on identification in the oblique and orthogonal
factor analysis models, Psychometrika 45, 393-396.
Anderson, T.W. and H. Rubin, 1956, Statistical inference in factor ana-
lysis, Proceedings of the Third Berkeley Symposium on Mathematical
Statistics and Probability 5, 111-150, (Jerzy Neyman, ed.), Universi-
ty uf California Press.
I3ekker, P.A. and D.S.G. Pollock, 1984, IdentiEication of linear sto-
cha5tic models with covaríance restrictions, Research Memorandum L59,
Tilburg University.
Dunn, J.l;., 1973, A note on a sufficiency condition for uniqueness of a
restricted factor matrix, Psychometrika 38, 141-143.
Fisher, F.M., 1966, The identification problem in econometrics, McGraw-
Hill, New Yurk.
Ueweke, .I.I~'. and K..I. Singleton, 1981, Maximum iikelíhood confirmatory
factor analysis of economic time series, Internatiunal Economic Re-
view 22, 37-54.
Hausman, J.A. and W.E. Taylor, 1983, Identification in linear simulta-
neous equations models with covariance restrictions: an instrumental
variables interpretation, Econometrica 51, 1527-1549.
Hausman, J.A., W. Newey and W. Taylor, 1983, Efficient estimatíon and
identification of simultaneous equation models with covariance re-
strictions, MIT Working Paper 331.
Jennrich, R.I., 1978, Rotational equivalence of factor loading matrices
with specified values, Psychometrika 43, 421-426.
JSreskog, K.G., 1969, A general approach to confírmatory factor analy-
sis, Psychometrika 34, 183-202.
Koopmans, T.C. and 0. Reiersbl, 1950, The identification of structural
cliaracteristics, Annals of Math. Stat. 21, 165-181.
Reiers~l, 0., 1950, On the identifiability of parameters in Thurstone's
multiple factor analysis, Psychometrika 15, 121-149.
Shapiro, A., 1983, On local identifiability in structural models, Re-
search Report, University of South Africa, Dept. of Math. and Appl.
Math..1
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A note on the identification of restricted factor loading matrices

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