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Testing separability of production using flexible functional form profit functions
Economics Letters North-Holland
265
26 (1988) 265-270
TESTING SEPARABILITY PROFIT FUNCTIONS
OF PRODUCTION
USING
FLEXIBLE
FUNCTIONAL
FORM
Rulon D. POPE Brigham
Young University, Prow,
UT 84602, USA
Arne HALLAM IowaState Uniuersity, Ames, IA 5001 I, USA Received Accepted
30 October 1987 22 December 1987
We show that the dual implications flexible form production systems. separability is maintained.
of separability depend crucially on whether profit is transformed non-linearly In any case where profit is non-linearly transformed (as in the translog),
in modern homothetic
1. Introduction Recently Lopez (1985) dichotomized flexible functional forms for profit functions based upon whether profit is linearly transformed (LFFF) or non-linearly transformed (NLFFF). In the former case, quasi-homothetic technology is implied with linear expansion paths. Further, input demands are strongly separable commodity-wise in input prices. In the NLFFF case, these restrictions are avoided. Examples of LFFF are the generalized Leontief and normalized quadratic. The most common NLFFF form is the translog where the transformation is the logarithm of profit. In this paper, we define the dual implications of separability for these flexible functional forms. It is shown that if one imposes separability in some partition on technology using the profit function, then the normalized profit function is also separable in the same partition in the NLFFF case. Similarly, imposing separability of profit using the production function implies separability of production in the NLFFF case. From a theorem of Lau’s (1978), this implies homothetic separability. Thus, one cannot independently test separability of technology using the profit function without imposing also separability of the profit function. This is a serious limitation of many commonly used functional forms. This restriction does not hold in the LFFF case and thus it is more ‘separability flexible’ than the NLFFF form.
2. Notation Smooth technology and profit functions (twice differentiable) are assumed. The efficient production set is described by y =f(x) where y is output and x is an n vector of input levels. Fixed inputs, if any, are notationally suppressed. Firms are assumed to be price taking and maximizing n * = py 0165-1765/88/$3.50
0 1988, Elsevier Science Publishers
B.V. (North-Holland)
R.D. Pope, A. Hahm
266
/ Testing separability of production
q’x where 7r* is profit, p is output price and q ’ is a row vector (n X 1) of input prices. Let R = a*/p be the normalized profit function with normalized input prices r’ = (4*/p,. . ,4,/p). First-order conditions are f, = r, where f, is the gradient of f(x). This yields the normalized profit function r(r) = max,[f(x) - r’x]. Some duality results of interest here are
7rr= -x,
fX=r,
(1)
where r,. is the gradient of profit and f,, and rrrr,,are the Hessians of the production function and normalized profit function, respectively [Lau (1978)]. ’ It is assumed throughout that rr and f, are non-zero.
3. Dual separability results Let N denote the set of indices { 1,. . . , n} and let ,N be the set (0, 1,. . . , n}. Let 52 be a mutually exclusive and exhaustive partitioning of the set N, 52 = {N,, . . , N,, . . . , N,}. A monotonic function g(z) is said to exhibit separability in the partition Q if for any s
a( t&/g, > =0
for all
i,j~N,;
k E N,“,
az!i where N,’ is the complement of N, in N, and where the subscripts on g denote derivatives, i.e., g, = ag/az-,. 2 Letting Hn be the bordered Hessian of g H,=
0
g:
gzz1 ’
[ EL
the separability condition in (3) can be equivalently written rank H,(N,,
Nf)
= 1,
(5)
where N, denotes the row indices in N, and N,” is the complement of N, in ,N and represents the column indices of H,. Thus, the determinants of all 2 X 2 matrices in (5) vanish. As noted in (3) one such minor yields
= 0.
It is helpful to call the definition in (3) applied to the production function [i.e., g = f (x)], direct separability of N, from its complement N,‘. Separability of the normalized profit function (e.g., g = a), hr, from its complement N,‘, is indirect separability. The former implies that marginal rates of technical substitution within a group are invariant to changes in input levels outside the group. ’ Monotonicity of 71 and f are assumed. * See, e.g., Lau (1978). Symmetric separability
is assumed
throughout.
267
R.D. Pope, A. Hailam / Testing separability of production
The latter [via (2)] implies that relative input demands within a group are invariant to changes in normalized input prices outside of the group. The following dual relationships between separability of f and 71 can be established using (3) [Pope and Barrett (1987)]. A fundamental stated as
Similarly,
dual separability
indirect
separability
Direct
result.
separability
of sector
of sector y, from N,’ is equivalently
y,
from
stated
IV,’ is equivalently
as
An examination of (6) indicates that direct separability requires rank f,-,‘(IV,, N,‘)< 1. Since by requires that rank T~,(N,, Nsc) < 1 (with the strict (2) -f;,’ = rr,, it follows that direct separability equality only under additivity). This rank condition will be referred to as the necessary condition(s) for direct separability. Since r,.,(N,, N,‘) is a submatrix of H,(N,, Nf), where H, is the bordered Hessian of 7~, it follows from (5) that rank rrJN,, N,“) Q 1 is necessary for both direct and indirect separability. Both (6) and (7) give the basis for testing separability of the function of interest using its dual. Since we concentrate here on the profit function, it will be exemplified. Direct separability of N, from N,” in (6) is given by the following conditions on the profit function:
(8)
=O
for all i, .j E N,; k E NT’, where v,, = ax,/&-, from (2). 3 In the next section, (8) will be.kxamined .in the context
of flexible
functional
forms.
4. Flexible functional forms for profit functions and separability Let the profit
function n
be written n
as n
e(dr)) = a0 + C a,h,(r,) + : C C j=l
(9)
Bjth,,(q)ht(rt),
J=ll=l
where G” is a monotonic transformation of 7r, B is symmetric (i.e., B,, = B,,) and where h,(q) is an arbitrary monotonic function of r/ [Lopez (1985)]. Following Lopez, when G” is linear, it will be denoted as LFFF, and when G” is non-linear it will be denoted by NLFFF. We note that indirect separability of 71 is unaffected by the transformation G” since G/C?, = (aC$/a~)~,,/(a(?/an)n = 71;/7~/. 3 The first column of the matrices in (8) can also be written (- ax,/ap, - Elx,/C?p) by the well-known [see (2) or Lau (1978)].
reciprocity
conditions
268
R.D. Pope, A. Hallam / Testrng separabdrty of production
Thus, separability of 7~and d are equivalent. It is convenient to let 6-i = G, where G is the inverse function of c”. Blackorby, Primont and Russell (1977) have shown that indirect separability of N, from N,’ in (9) is given by
5 (B,tB,k-B,tBjk)ht(rt)=O.
a,B,k-a,B,k+
I=1
If (10) is to hold for all h, and all 5, (10) implies that a, B,, - a, Blk = 0
for
all i, jE
N,;
B,,B,, - B,,B,, = 0
for
all k E N,‘,
(IIa) and
I= 1,. . ., n.
(lib)
We will refer to (lib) as necessary conditions. These correspond to the requirement that rank T~,( N,, Ns’) < 1. Comparing this condition with (5) and (6) it is clear that rank v~,.(N,, y,“) < 1 is necessary for both direct and indirect separability. Both conditions in (11) are necessary and sufficient for indirect separability of N, from NY’ and can be written as rank H,(N,, IVY@) = 1, where H, is the bordered Hessian of rr and N,” insures that the relevant portion of the first column of Ha is added to T~,.(N,, y,“). As Blackorby, Primont and Russell (1977) have shown, these conditions are very restrictive and apply to (9) regardless of whether it is LFFF or NLFFF. Direct separability Since most often empirical measurement uses the profit function, we now consider the implications of separability of technology on the structure of 7. Eq. (6) gives the general result which must be applied to the transformed eq. (9). The calculations for the necessary conditions (common to direct and indirect separability) for
1 0
alli,
k, k’ E Nf\
jEN,;
(12)
2
yields G’2[B,,B,n,-
B,,B,,,]
+ R,,G’G”[
B,,R,
- R,Bjk]
+ R,G’G”[B,,,R,
- R,B,,,]
= 0,
(13)
where R, = a, + I;=, B,,h,(r,), t = i, j, k, k’ and primes on G denote derivatives. Since, r,. is non-zero by assumption, it follows that R, is also non-zero since R, ah,/&, = TV,. There are two cases of interest. Case 1. B,,B,,(
G” = 0, the LFFF - B,,B,,!
=0
for
case. In such case, (13) yields the necessary conditions: all i, j E N,; k, k’ E Ns’.
These are a subset of the conditions in (llb).
(14)
R.D. Pope, A. Hallam / Testing separability of productton Case 2.
G” # 0, the NLFFF
Bj,R,-R,~Bik=(~iB/k-~llBik+
case. In such case, (13) implies
269
in addition
to (14):
5 (B,,Bj,-B,,B,k)hl(rO=O I=7
(15)
foralli,jEN,andallk(k’)EN,“. If (15) is to hold for all functions h, at any r, 2 0, then the necessary to (11) as indicated in Proposition 1.
Proposition separability
conditions
1. Under NLFFF, the necessary conditions, which are common in the partition Q, imply also indirect separability in 0.
become
identical
to both direct and indirect
Thus, imposing the necessary conditions also imposes the necessary and sufficient conditions in the NLFFF case. This result implies that in this case, one cannot use the normalized profit function to test for direct separability without imposing indirect separability. Thus, from a theorem in Lau (1978), homothetic separability, separability with homothetic aggregators, is implied. 4 Hence, the translog profit function cannot be used to test for direct separability without also imposing indirect separability. However, the structure imposed on r by the necessary conditions in the LFFF case is mild in comparison. Applying (6) to (9) for the LFFF case gives as the necessary and sufficient conditions for direct separability as
5
(16)
(B,,B,,-B,,B,,)~~~=O. I
I=1
If (16) is to apply for arbitrary B,,B,,-B,,B,,=O
for
h, and r,, then the necessary
all i, jEN,;
kEN,’
and
and sufficient
conditions
are
I=l,...,n,
(174
or. Bjk _=_
B,k
B/I assuming Bi, ’
non-additivity.
This is identical to the necessary conditions for indirect separability given in (llb). There is a special case of (16) that is far less restrictive. If (3hJar~)r~ = K, a constant, becomes
then (16)
n
BJk[FiBi/-B,kc
B,,=O
for
all i, jEN,;
I=1
4 See Theorem II-9 and 11,
p. 160-161 of Lau (1978).
kEN,“,
(17b)
270
R. D. Pope, A. Hallam / Testing separability
of production
or,
B/k
I=1
-=
assuming
non-additivity.
B,k
5 4,’ I=1 Therefore, when h,(r,) = In r,, then the necessary and sufficient conditions the arbitrary LFFF case. These results are summarized in Proposition 2.
are less stringent
than in
Proposition 2. Under LFFF, the necessary and sufficient conditions on T for direct separability in the partition s2 are given in (16). For arbitrary h,(r,), these conditions are given in (17a). In the special case where (ah,,Grl)r, is a constant (similar to the translog), the test for direct separability is given in (17b).
5. Concluding remarks Propositions 1 and 2 indicate some potential pitfalls in testing separability of production using the profit function. In particular, the NLFFF form seems especially poorly suited to this test if one wishes to test only separability without imposing homothetic separability on both normalized profit and production. Thus, if one rejects the null hypothesis, it may not be separability that is in question but the homotheticitiy condition. Further, if one wishes to establish the conditions under which simultaneous price and quantity indices exist (homothetic separability), merely verifying indirect separability in the NLFFF case is sufficient. In the LFFF case, as Lopez has shown, quasi-homotheticity of technology is imposed. This imposes commodity wise price additivity on input demands. However, normalized profit does not have these restrictions, and when indirect separability is imposed no additional homotheticity conditions are inadvertently imposed. Thus, when it comes to separability, the tradeoffs between LFFF and NLFFF may make LFFF more attractive in spite of the implied quasi-homotheticity. Finally, though the implications of separability of production on profit have been emphasized, it is clear from (6) and (7) that the arguments apply equally to restrictions on production from separability of profit.
References Blackorby,
C., D. Primont
Econometrics,
March,
Lau, L., 1978, Applications to theory Lopez,
of profit
and applications,
separability
restrictions
with flexible
functional
forms,
Journal
of
functions,
in: M. Fuss and D. McFadden,
among
flexible
eds., Production
economics:
A dual approach
Amsterdam). functional
form specifications
for profit
function,
International
Economic
26 593-601.
R. and W. Barrett,
71-77.
1977, On testing
Vol. 1 (North-Holland,
R., 1985, On discriminating
Review Pope,
and R. Russell, 195-209.
1987,
A result
on the rank of submatrices
and an economic
application,
Economics
Letters
23,
265
26 (1988) 265-270
TESTING SEPARABILITY PROFIT FUNCTIONS
OF PRODUCTION
USING
FLEXIBLE
FUNCTIONAL
FORM
Rulon D. POPE Brigham
Young University, Prow,
UT 84602, USA
Arne HALLAM IowaState Uniuersity, Ames, IA 5001 I, USA Received Accepted
30 October 1987 22 December 1987
We show that the dual implications flexible form production systems. separability is maintained.
of separability depend crucially on whether profit is transformed non-linearly In any case where profit is non-linearly transformed (as in the translog),
in modern homothetic
1. Introduction Recently Lopez (1985) dichotomized flexible functional forms for profit functions based upon whether profit is linearly transformed (LFFF) or non-linearly transformed (NLFFF). In the former case, quasi-homothetic technology is implied with linear expansion paths. Further, input demands are strongly separable commodity-wise in input prices. In the NLFFF case, these restrictions are avoided. Examples of LFFF are the generalized Leontief and normalized quadratic. The most common NLFFF form is the translog where the transformation is the logarithm of profit. In this paper, we define the dual implications of separability for these flexible functional forms. It is shown that if one imposes separability in some partition on technology using the profit function, then the normalized profit function is also separable in the same partition in the NLFFF case. Similarly, imposing separability of profit using the production function implies separability of production in the NLFFF case. From a theorem of Lau’s (1978), this implies homothetic separability. Thus, one cannot independently test separability of technology using the profit function without imposing also separability of the profit function. This is a serious limitation of many commonly used functional forms. This restriction does not hold in the LFFF case and thus it is more ‘separability flexible’ than the NLFFF form.
2. Notation Smooth technology and profit functions (twice differentiable) are assumed. The efficient production set is described by y =f(x) where y is output and x is an n vector of input levels. Fixed inputs, if any, are notationally suppressed. Firms are assumed to be price taking and maximizing n * = py 0165-1765/88/$3.50
0 1988, Elsevier Science Publishers
B.V. (North-Holland)
R.D. Pope, A. Hahm
266
/ Testing separability of production
q’x where 7r* is profit, p is output price and q ’ is a row vector (n X 1) of input prices. Let R = a*/p be the normalized profit function with normalized input prices r’ = (4*/p,. . ,4,/p). First-order conditions are f, = r, where f, is the gradient of f(x). This yields the normalized profit function r(r) = max,[f(x) - r’x]. Some duality results of interest here are
7rr= -x,
fX=r,
(1)
where r,. is the gradient of profit and f,, and rrrr,,are the Hessians of the production function and normalized profit function, respectively [Lau (1978)]. ’ It is assumed throughout that rr and f, are non-zero.
3. Dual separability results Let N denote the set of indices { 1,. . . , n} and let ,N be the set (0, 1,. . . , n}. Let 52 be a mutually exclusive and exhaustive partitioning of the set N, 52 = {N,, . . , N,, . . . , N,}. A monotonic function g(z) is said to exhibit separability in the partition Q if for any s
a( t&/g, > =0
for all
i,j~N,;
k E N,“,
az!i where N,’ is the complement of N, in N, and where the subscripts on g denote derivatives, i.e., g, = ag/az-,. 2 Letting Hn be the bordered Hessian of g H,=
0
g:
gzz1 ’
[ EL
the separability condition in (3) can be equivalently written rank H,(N,,
Nf)
= 1,
(5)
where N, denotes the row indices in N, and N,” is the complement of N, in ,N and represents the column indices of H,. Thus, the determinants of all 2 X 2 matrices in (5) vanish. As noted in (3) one such minor yields
= 0.
It is helpful to call the definition in (3) applied to the production function [i.e., g = f (x)], direct separability of N, from its complement N,‘. Separability of the normalized profit function (e.g., g = a), hr, from its complement N,‘, is indirect separability. The former implies that marginal rates of technical substitution within a group are invariant to changes in input levels outside the group. ’ Monotonicity of 71 and f are assumed. * See, e.g., Lau (1978). Symmetric separability
is assumed
throughout.
267
R.D. Pope, A. Hailam / Testing separability of production
The latter [via (2)] implies that relative input demands within a group are invariant to changes in normalized input prices outside of the group. The following dual relationships between separability of f and 71 can be established using (3) [Pope and Barrett (1987)]. A fundamental stated as
Similarly,
dual separability
indirect
separability
Direct
result.
separability
of sector
of sector y, from N,’ is equivalently
y,
from
stated
IV,’ is equivalently
as
An examination of (6) indicates that direct separability requires rank f,-,‘(IV,, N,‘)< 1. Since by requires that rank T~,(N,, Nsc) < 1 (with the strict (2) -f;,’ = rr,, it follows that direct separability equality only under additivity). This rank condition will be referred to as the necessary condition(s) for direct separability. Since r,.,(N,, N,‘) is a submatrix of H,(N,, Nf), where H, is the bordered Hessian of 7~, it follows from (5) that rank rrJN,, N,“) Q 1 is necessary for both direct and indirect separability. Both (6) and (7) give the basis for testing separability of the function of interest using its dual. Since we concentrate here on the profit function, it will be exemplified. Direct separability of N, from N,” in (6) is given by the following conditions on the profit function:
(8)
=O
for all i, .j E N,; k E NT’, where v,, = ax,/&-, from (2). 3 In the next section, (8) will be.kxamined .in the context
of flexible
functional
forms.
4. Flexible functional forms for profit functions and separability Let the profit
function n
be written n
as n
e(dr)) = a0 + C a,h,(r,) + : C C j=l
(9)
Bjth,,(q)ht(rt),
J=ll=l
where G” is a monotonic transformation of 7r, B is symmetric (i.e., B,, = B,,) and where h,(q) is an arbitrary monotonic function of r/ [Lopez (1985)]. Following Lopez, when G” is linear, it will be denoted as LFFF, and when G” is non-linear it will be denoted by NLFFF. We note that indirect separability of 71 is unaffected by the transformation G” since G/C?, = (aC$/a~)~,,/(a(?/an)n = 71;/7~/. 3 The first column of the matrices in (8) can also be written (- ax,/ap, - Elx,/C?p) by the well-known [see (2) or Lau (1978)].
reciprocity
conditions
268
R.D. Pope, A. Hallam / Testrng separabdrty of production
Thus, separability of 7~and d are equivalent. It is convenient to let 6-i = G, where G is the inverse function of c”. Blackorby, Primont and Russell (1977) have shown that indirect separability of N, from N,’ in (9) is given by
5 (B,tB,k-B,tBjk)ht(rt)=O.
a,B,k-a,B,k+
I=1
If (10) is to hold for all h, and all 5, (10) implies that a, B,, - a, Blk = 0
for
all i, jE
N,;
B,,B,, - B,,B,, = 0
for
all k E N,‘,
(IIa) and
I= 1,. . ., n.
(lib)
We will refer to (lib) as necessary conditions. These correspond to the requirement that rank T~,( N,, Ns’) < 1. Comparing this condition with (5) and (6) it is clear that rank v~,.(N,, y,“) < 1 is necessary for both direct and indirect separability. Both conditions in (11) are necessary and sufficient for indirect separability of N, from NY’ and can be written as rank H,(N,, IVY@) = 1, where H, is the bordered Hessian of rr and N,” insures that the relevant portion of the first column of Ha is added to T~,.(N,, y,“). As Blackorby, Primont and Russell (1977) have shown, these conditions are very restrictive and apply to (9) regardless of whether it is LFFF or NLFFF. Direct separability Since most often empirical measurement uses the profit function, we now consider the implications of separability of technology on the structure of 7. Eq. (6) gives the general result which must be applied to the transformed eq. (9). The calculations for the necessary conditions (common to direct and indirect separability) for
1 0
alli,
k, k’ E Nf\
jEN,;
(12)
2
yields G’2[B,,B,n,-
B,,B,,,]
+ R,,G’G”[
B,,R,
- R,Bjk]
+ R,G’G”[B,,,R,
- R,B,,,]
= 0,
(13)
where R, = a, + I;=, B,,h,(r,), t = i, j, k, k’ and primes on G denote derivatives. Since, r,. is non-zero by assumption, it follows that R, is also non-zero since R, ah,/&, = TV,. There are two cases of interest. Case 1. B,,B,,(
G” = 0, the LFFF - B,,B,,!
=0
for
case. In such case, (13) yields the necessary conditions: all i, j E N,; k, k’ E Ns’.
These are a subset of the conditions in (llb).
(14)
R.D. Pope, A. Hallam / Testing separability of productton Case 2.
G” # 0, the NLFFF
Bj,R,-R,~Bik=(~iB/k-~llBik+
case. In such case, (13) implies
269
in addition
to (14):
5 (B,,Bj,-B,,B,k)hl(rO=O I=7
(15)
foralli,jEN,andallk(k’)EN,“. If (15) is to hold for all functions h, at any r, 2 0, then the necessary to (11) as indicated in Proposition 1.
Proposition separability
conditions
1. Under NLFFF, the necessary conditions, which are common in the partition Q, imply also indirect separability in 0.
become
identical
to both direct and indirect
Thus, imposing the necessary conditions also imposes the necessary and sufficient conditions in the NLFFF case. This result implies that in this case, one cannot use the normalized profit function to test for direct separability without imposing indirect separability. Thus, from a theorem in Lau (1978), homothetic separability, separability with homothetic aggregators, is implied. 4 Hence, the translog profit function cannot be used to test for direct separability without also imposing indirect separability. However, the structure imposed on r by the necessary conditions in the LFFF case is mild in comparison. Applying (6) to (9) for the LFFF case gives as the necessary and sufficient conditions for direct separability as
5
(16)
(B,,B,,-B,,B,,)~~~=O. I
I=1
If (16) is to apply for arbitrary B,,B,,-B,,B,,=O
for
h, and r,, then the necessary
all i, jEN,;
kEN,’
and
and sufficient
conditions
are
I=l,...,n,
(174
or. Bjk _=_
B,k
B/I assuming Bi, ’
non-additivity.
This is identical to the necessary conditions for indirect separability given in (llb). There is a special case of (16) that is far less restrictive. If (3hJar~)r~ = K, a constant, becomes
then (16)
n
BJk[FiBi/-B,kc
B,,=O
for
all i, jEN,;
I=1
4 See Theorem II-9 and 11,
p. 160-161 of Lau (1978).
kEN,“,
(17b)
270
R. D. Pope, A. Hallam / Testing separability
of production
or,
B/k
I=1
-=
assuming
non-additivity.
B,k
5 4,’ I=1 Therefore, when h,(r,) = In r,, then the necessary and sufficient conditions the arbitrary LFFF case. These results are summarized in Proposition 2.
are less stringent
than in
Proposition 2. Under LFFF, the necessary and sufficient conditions on T for direct separability in the partition s2 are given in (16). For arbitrary h,(r,), these conditions are given in (17a). In the special case where (ah,,Grl)r, is a constant (similar to the translog), the test for direct separability is given in (17b).
5. Concluding remarks Propositions 1 and 2 indicate some potential pitfalls in testing separability of production using the profit function. In particular, the NLFFF form seems especially poorly suited to this test if one wishes to test only separability without imposing homothetic separability on both normalized profit and production. Thus, if one rejects the null hypothesis, it may not be separability that is in question but the homotheticitiy condition. Further, if one wishes to establish the conditions under which simultaneous price and quantity indices exist (homothetic separability), merely verifying indirect separability in the NLFFF case is sufficient. In the LFFF case, as Lopez has shown, quasi-homotheticity of technology is imposed. This imposes commodity wise price additivity on input demands. However, normalized profit does not have these restrictions, and when indirect separability is imposed no additional homotheticity conditions are inadvertently imposed. Thus, when it comes to separability, the tradeoffs between LFFF and NLFFF may make LFFF more attractive in spite of the implied quasi-homotheticity. Finally, though the implications of separability of production on profit have been emphasized, it is clear from (6) and (7) that the arguments apply equally to restrictions on production from separability of profit.
References Blackorby,
C., D. Primont
Econometrics,
March,
Lau, L., 1978, Applications to theory Lopez,
of profit
and applications,
separability
restrictions
with flexible
functional
forms,
Journal
of
functions,
in: M. Fuss and D. McFadden,
among
flexible
eds., Production
economics:
A dual approach
Amsterdam). functional
form specifications
for profit
function,
International
Economic
26 593-601.
R. and W. Barrett,
71-77.
1977, On testing
Vol. 1 (North-Holland,
R., 1985, On discriminating
Review Pope,
and R. Russell, 195-209.
1987,
A result
on the rank of submatrices
and an economic
application,
Economics
Letters
23,