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Underestimation and overestimation of the Leontief inverse revisited
181
Economics Letters 18 (1985) 181-186 North-Holland
UNDERESTIMATION AND OVERESTIMATION OF THE LEONTIEF INVERSE REVISITED * Sajal LAHIRI Southern Methodist University, Dallas, TX 75275, USA University of Essex, Colchester CO4 3SQ, UK
Steve SATCHELL University of Essex, Colchester CO4 3SQ
UK
Received 23 October 1984
This note re-examines the relationship between the expected value of the Leontief inverse, (I - A)-’ and its true value when A is a stochastic input-output matrix. Two alternative stochastic specifications are considered and several new results are proved.
1. Introduction It has been shown by Quandt (1958) Simonovits (1975) and Lahiri (1983) that even when the estimates of the input-output (IO) coefficients are unbiased, the derived estimates of the elements of the Leontief inverse (LI) are not necessarily so. In particular, Simonovits (1975) has proved that if the elements of the IO matrix, A, are mutually independent, the derived estimate of the LI, (I - A)-‘, will be overestimated in the sense that E(Z - A)-’ 2 (I - E(A))-‘. Lahiri (1983) has shown that Simonovits’s results go through even when the elements of A are not independent but are biproportionally stochastic. In this note we shall consider two types of stochastic specifications for the elements of A. In the first case, the source of stochastic errors are the prices at which the IO coefficients are valued. In this case, we shall develop a necessary and sufficient condition for the overestimation (or underestimation) of the elements of the LI. We then obtain six corollaries which give us more specific results under special cases. In the second case, all the elements of A are affected by a single uniformly distributed random variable. Here, we obtain an exact expression for the bias of the LI, derive a condition under which the biases are non-negative and show that each and every bias monotonically decreases to zero as the variance of the random variable goes to zero.
2. The case with stochastic prices Here we shall assume that the IO coefficients measured at the prices of a certain base period are deterministic. However, the same coefficients measured at the prices of a different period - to be * This note forms part of a wider study on stochastic input-output analysis which is now being carried out. The present note is a revised version of a part of Lahiri and Satchell (1984) which was presented at the 1984 European meeting of the Econometric Society, held in Madrid. We are grateful to Professors R.E. Quandt and G. Pyatt for comments. 0165-1765/85/$3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)
182
S. Lahiri, S. Satchel
/ Estimation
of the L.eontief inverse
called the second period - are stochastic. That is to say, the prices of the second period are subj error. As the commodities in the IO table are typically aggregates of a large number of not neces homogeneous commodities, it is very difficult to obtain accurate estimates of the relative p Therefore, attempts to change the prices at which all the variables are measured would intro randomness in the IO coefficient and it is this type of randomness that we shall be dealing wi this section. Let apj be the IO coefficient measured at the base period prices and P, be the price of th commodity in the second period relative to its base period price. Then the revalued IO coeffi aij’s are given by aij = a,“iPi/Pj.
Now, 19’s are assumed to be stochastic and we write Pi = i$;,
where ci( > 0) is a random variable with E(ci) = 1 and pi is the true price. The true value of the LI is given by
where F is the diagonal matrix with pj in its (i, i)th cell and A0 = (aij). We shall also define th for the base period as Q” =
(q;.) = [Z-/lo]-',
and the practical estimator of the LI as
Q= (qij)=
[Z-A]-‘.
The following result provides a necessary and sufficient condition for the over- or underestimatio (i, j)th element of the LI: Theorem 1 E(qii)=qii
foralli
E(qjj) $ qij
ifandonly if
proof.
0 = [I -
qij = qp,Fyq.
andforizj,
E(ei/cj)
i, j=l,...,n,
$1.
~,~‘$-l]-l= F[z - ~o]-‘~-’ = g~Og-1.
That
is,
S. Lahiri, S. Satchel1 / Estimation
183
of the L.eontiej inverse
Similarly,
[using (2)1, [using (5)] .
(6) From the above equation the conclusion follows.
Q.E.D.
We can now obtain a number of corollaries and these are: Corollary 1.
A sufficient condition for the overestimation of the (i, j)th
COV((,, (l/c,))
2 0.
element of the LI is that
Proof
= E(E;,‘c,) - E(ri)E(l,A,),
cov
<
E(cl/cJ)
-
l/E(cJ)
[using Jensen’s inequality and since E( C, ) = 11,
= E( c,/E~) - 1, :. E( C/C/) - 1 > cov( c;, (1,‘~~)) 2 0
(from the hypothesis).
Therefore, using Theorem 1, it follows that E( qi,) > qiJ. Corollary 2.
Q.E.D.
When all the E;‘S defined in (2) are independent E(q,j) > q,j for i ZJ’.
Prooj
It is easy to check that when ci and cj are independent, so are cr and l/r,. Therefore, from whence using Jensen’s inequality one has E( qij) > (6) one gets E(qij) = qijE(ei)E(l/ej), Q.E.D. qijE(ci)/E(rJ) = qiJ [since E(ei) = i]. For the next four corollaries we shall assume the C;‘S to follow a multivariate lognormal distribution. For the sake of brevity, we shall state this assumption formally below and make no reference to it in the statements of the corollaries. Assumption
1. (zl,. . . , c,,) follow a multivariate var(log ei) = ui2 and cov(log ci, log zj) = ujj(i #j). Corollary 3.
ProoJ:
E( qij) $ qij if and only if a,’ $ oiJ.
We know that
E( ei) = e !%+4/2 = 1.
lognormal
distribution
with E(log ci) = pi,
S. Lahiri. S. Satchel1 / Estmution
184
ofthe Lmmtiefinoerse
Therefore, j.l, = - a,2/2.
(7)
Now,
[using (7)],
= p, - p, = a,‘/2 - 0,~/2
(8)
and
= a,2 + a,2 - 2 u, , var log: i /i
(9)
Therefore, =
e0,2,‘2-~n,~/2+(o,~+0,~~2”,,,/2=,a,~
From (lo), it is clear that E(e,/e,) Q.E.D. follows. Corollary u,, < 0. Proof:
4.
A sufficient
Proof 0<
If the (i, j)th
condition for the overestimation
of the (i. j)th
element
1, the result
of the LI is that
Q.E.D.
element of the LI is underestimated,
element
from Theorem
of the LI is underestimated,
then the (j, i)th element of the LI is
then from Corollary
3 it follows that
(11)
a,2< u 11’
Now, suppose 0<
> 1 if and only if a,’ > a,,. Therefore,
The proof follows from (6) and (10).
Corollary 5. If the (i, j)th overestimated.
(10)
[using (8) and (9)].
0, /
that the (j, i)th element
of the LI is also underestimated.
Therefore
once again one has
q2 -=zu,,.
(12)
From (11) and (12) one gets u12uJ2< ~,‘a,, < a,: or u,:/( u,~u,~) > 1. That is, the square of the correlation Q.E.D. coefficient between log e, and log E, is greater than unity. This is a contradiction. Corollary 6. In the case of homoskedastic E( 4, , ) 2 ij,, for all i and j. Proof. u2
Now suppose
errors - i.e. when u,’ = a,’ = u2 (sa_v) for all i and .j
E( q,,) < q,, for some i and j. From Corollary
3 one then has (13)
S. Lohiri, S. Satchel1 / Estimation of the Leontief inverse
185
Therefore,
This is a contradiction
and hence the proof.
Q.E.D.
3. Uniform error and monotonicity of the bias The qualification ‘uniform’ in the caption of this section is used to signify two things. First, we shall assume that the elements of A are uniformly affected by a single random variable and second, that the random variable itself will be assumed, for the second half of this section, to follow a uniform distribution. In symbols, our specifications of the errors are as follows: u~~=S,~+C~,U
or
A =A+uC,
(14)
where u is a random variable, c,~‘s are deterministic but not necessarily positive numbers, and C = (c,,). There are however restrictions on c,/‘s and these are stated below.
A= (a;,)
Assumption 2. C( Z - A)-’ 2 0, u is a symmetrically distributed random variable and, for every realization ti of U, k is less than the reciprocal of the dominant eigen value of C( Z - A))‘. Our first result is that all the elements Theorem 2. Proof
of the LI are overestimated.
E(Q)hzwhereQ=(qjj)=(Z-A)-’
and~=(~,,)=(Z-A)-‘.
Note that
Q= (Z-A)-‘= = (Z-q-‘(Z-
(Z-/&C)-’
[from (I4)1,
uC(Z-X)-‘)-I,
or
Q=(Z-q-‘+(I-A3-‘(uD+u*D*+...)
[fromAssumption2],
where D=C(Z-A3-‘. Now, since u is symmetrically
distributed,
E(Q)=~+?~[E(u~)D*+E(u~)D~+...], 2 3
[from Assumption
E( u’) = 0 for odd integers
r and therefore, 05)
21.
Q.E.D.
We now examine the properties of the biases of the elements of the LI as the variance of u tends to zero. Clearly, when the variance is zero, the a,,‘~ are no longer stochastic and the biases disappear.
S. Lahirr, S. Satchel1 / Estmatron
186
a/ the
Lmntrefrnoerse
We, however, ask the following question: Does each bias decrease below that the answer is yes when u is uniformly distributed.
monotonically
Theorem 3. Suppose that u is uniformb distributed over an interval E( q,,) - a,, decreases monotonically to zero as var( u) goes to zero. ProoJ
It is easy to check that E( ur) = a’/(1
E(Q)-z=zf
a2’D2’/(1+2r).
+ r). Therefore,
( -a.
to zero? We prove
a ). Then for ull i and j,
from (15) one gets
(16)
r=l From Theorem 2, we know that the right-hand side of (16) is non-negative. Now since e 2 0, D 2 0, it is evident from (16) that the right-hand side of it decreases monotonically to zero as 5’ goes to Q.E.D. zero, i.e., as var(u) goes to zero. It is to be noted that the right-hand side of (16) gives us an exact expression for the bias. We are currently working on the problem of estimating a more efficient estimate of the LI using this expression.
References Lahiri, S., 1983, A note on the underestimation and overestimation in stochastic input-output models, Economics Letters 13, 361-366. Lahiri, S. and S. Satchell, 1984, Properties of the expected value of the Leontief inverse under different stochastic specifications, Discussion paper 237 (Department of Economics, University of Essex, Colchester). Quandt, R.E., 1958, Probabilistic errors in the Leontief system, Naval Research Logistics Quarterly 5, 155-170. Simonovits, A., 1975, A note on the underestimation and overestimation of the Leontief inverse. Econometrica 43, 493-498.
Economics Letters 18 (1985) 181-186 North-Holland
UNDERESTIMATION AND OVERESTIMATION OF THE LEONTIEF INVERSE REVISITED * Sajal LAHIRI Southern Methodist University, Dallas, TX 75275, USA University of Essex, Colchester CO4 3SQ, UK
Steve SATCHELL University of Essex, Colchester CO4 3SQ
UK
Received 23 October 1984
This note re-examines the relationship between the expected value of the Leontief inverse, (I - A)-’ and its true value when A is a stochastic input-output matrix. Two alternative stochastic specifications are considered and several new results are proved.
1. Introduction It has been shown by Quandt (1958) Simonovits (1975) and Lahiri (1983) that even when the estimates of the input-output (IO) coefficients are unbiased, the derived estimates of the elements of the Leontief inverse (LI) are not necessarily so. In particular, Simonovits (1975) has proved that if the elements of the IO matrix, A, are mutually independent, the derived estimate of the LI, (I - A)-‘, will be overestimated in the sense that E(Z - A)-’ 2 (I - E(A))-‘. Lahiri (1983) has shown that Simonovits’s results go through even when the elements of A are not independent but are biproportionally stochastic. In this note we shall consider two types of stochastic specifications for the elements of A. In the first case, the source of stochastic errors are the prices at which the IO coefficients are valued. In this case, we shall develop a necessary and sufficient condition for the overestimation (or underestimation) of the elements of the LI. We then obtain six corollaries which give us more specific results under special cases. In the second case, all the elements of A are affected by a single uniformly distributed random variable. Here, we obtain an exact expression for the bias of the LI, derive a condition under which the biases are non-negative and show that each and every bias monotonically decreases to zero as the variance of the random variable goes to zero.
2. The case with stochastic prices Here we shall assume that the IO coefficients measured at the prices of a certain base period are deterministic. However, the same coefficients measured at the prices of a different period - to be * This note forms part of a wider study on stochastic input-output analysis which is now being carried out. The present note is a revised version of a part of Lahiri and Satchell (1984) which was presented at the 1984 European meeting of the Econometric Society, held in Madrid. We are grateful to Professors R.E. Quandt and G. Pyatt for comments. 0165-1765/85/$3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)
182
S. Lahiri, S. Satchel
/ Estimation
of the L.eontief inverse
called the second period - are stochastic. That is to say, the prices of the second period are subj error. As the commodities in the IO table are typically aggregates of a large number of not neces homogeneous commodities, it is very difficult to obtain accurate estimates of the relative p Therefore, attempts to change the prices at which all the variables are measured would intro randomness in the IO coefficient and it is this type of randomness that we shall be dealing wi this section. Let apj be the IO coefficient measured at the base period prices and P, be the price of th commodity in the second period relative to its base period price. Then the revalued IO coeffi aij’s are given by aij = a,“iPi/Pj.
Now, 19’s are assumed to be stochastic and we write Pi = i$;,
where ci( > 0) is a random variable with E(ci) = 1 and pi is the true price. The true value of the LI is given by
where F is the diagonal matrix with pj in its (i, i)th cell and A0 = (aij). We shall also define th for the base period as Q” =
(q;.) = [Z-/lo]-',
and the practical estimator of the LI as
Q= (qij)=
[Z-A]-‘.
The following result provides a necessary and sufficient condition for the over- or underestimatio (i, j)th element of the LI: Theorem 1 E(qii)=qii
foralli
E(qjj) $ qij
ifandonly if
proof.
0 = [I -
qij = qp,Fyq.
andforizj,
E(ei/cj)
i, j=l,...,n,
$1.
~,~‘$-l]-l= F[z - ~o]-‘~-’ = g~Og-1.
That
is,
S. Lahiri, S. Satchel1 / Estimation
183
of the L.eontiej inverse
Similarly,
[using (2)1, [using (5)] .
(6) From the above equation the conclusion follows.
Q.E.D.
We can now obtain a number of corollaries and these are: Corollary 1.
A sufficient condition for the overestimation of the (i, j)th
COV((,, (l/c,))
2 0.
element of the LI is that
Proof
= E(E;,‘c,) - E(ri)E(l,A,),
cov
<
E(cl/cJ)
-
l/E(cJ)
[using Jensen’s inequality and since E( C, ) = 11,
= E( c,/E~) - 1, :. E( C/C/) - 1 > cov( c;, (1,‘~~)) 2 0
(from the hypothesis).
Therefore, using Theorem 1, it follows that E( qi,) > qiJ. Corollary 2.
Q.E.D.
When all the E;‘S defined in (2) are independent E(q,j) > q,j for i ZJ’.
Prooj
It is easy to check that when ci and cj are independent, so are cr and l/r,. Therefore, from whence using Jensen’s inequality one has E( qij) > (6) one gets E(qij) = qijE(ei)E(l/ej), Q.E.D. qijE(ci)/E(rJ) = qiJ [since E(ei) = i]. For the next four corollaries we shall assume the C;‘S to follow a multivariate lognormal distribution. For the sake of brevity, we shall state this assumption formally below and make no reference to it in the statements of the corollaries. Assumption
1. (zl,. . . , c,,) follow a multivariate var(log ei) = ui2 and cov(log ci, log zj) = ujj(i #j). Corollary 3.
ProoJ:
E( qij) $ qij if and only if a,’ $ oiJ.
We know that
E( ei) = e !%+4/2 = 1.
lognormal
distribution
with E(log ci) = pi,
S. Lahiri. S. Satchel1 / Estmution
184
ofthe Lmmtiefinoerse
Therefore, j.l, = - a,2/2.
(7)
Now,
[using (7)],
= p, - p, = a,‘/2 - 0,~/2
(8)
and
= a,2 + a,2 - 2 u, , var log: i /i
(9)
Therefore, =
e0,2,‘2-~n,~/2+(o,~+0,~~2”,,,/2=,a,~
From (lo), it is clear that E(e,/e,) Q.E.D. follows. Corollary u,, < 0. Proof:
4.
A sufficient
Proof 0<
If the (i, j)th
condition for the overestimation
of the (i. j)th
element
1, the result
of the LI is that
Q.E.D.
element of the LI is underestimated,
element
from Theorem
of the LI is underestimated,
then the (j, i)th element of the LI is
then from Corollary
3 it follows that
(11)
a,2< u 11’
Now, suppose 0<
> 1 if and only if a,’ > a,,. Therefore,
The proof follows from (6) and (10).
Corollary 5. If the (i, j)th overestimated.
(10)
[using (8) and (9)].
0, /
that the (j, i)th element
of the LI is also underestimated.
Therefore
once again one has
q2 -=zu,,.
(12)
From (11) and (12) one gets u12uJ2< ~,‘a,, < a,: or u,:/( u,~u,~) > 1. That is, the square of the correlation Q.E.D. coefficient between log e, and log E, is greater than unity. This is a contradiction. Corollary 6. In the case of homoskedastic E( 4, , ) 2 ij,, for all i and j. Proof. u2
Now suppose
errors - i.e. when u,’ = a,’ = u2 (sa_v) for all i and .j
E( q,,) < q,, for some i and j. From Corollary
3 one then has (13)
S. Lohiri, S. Satchel1 / Estimation of the Leontief inverse
185
Therefore,
This is a contradiction
and hence the proof.
Q.E.D.
3. Uniform error and monotonicity of the bias The qualification ‘uniform’ in the caption of this section is used to signify two things. First, we shall assume that the elements of A are uniformly affected by a single random variable and second, that the random variable itself will be assumed, for the second half of this section, to follow a uniform distribution. In symbols, our specifications of the errors are as follows: u~~=S,~+C~,U
or
A =A+uC,
(14)
where u is a random variable, c,~‘s are deterministic but not necessarily positive numbers, and C = (c,,). There are however restrictions on c,/‘s and these are stated below.
A= (a;,)
Assumption 2. C( Z - A)-’ 2 0, u is a symmetrically distributed random variable and, for every realization ti of U, k is less than the reciprocal of the dominant eigen value of C( Z - A))‘. Our first result is that all the elements Theorem 2. Proof
of the LI are overestimated.
E(Q)hzwhereQ=(qjj)=(Z-A)-’
and~=(~,,)=(Z-A)-‘.
Note that
Q= (Z-A)-‘= = (Z-q-‘(Z-
(Z-/&C)-’
[from (I4)1,
uC(Z-X)-‘)-I,
or
Q=(Z-q-‘+(I-A3-‘(uD+u*D*+...)
[fromAssumption2],
where D=C(Z-A3-‘. Now, since u is symmetrically
distributed,
E(Q)=~+?~[E(u~)D*+E(u~)D~+...], 2 3
[from Assumption
E( u’) = 0 for odd integers
r and therefore, 05)
21.
Q.E.D.
We now examine the properties of the biases of the elements of the LI as the variance of u tends to zero. Clearly, when the variance is zero, the a,,‘~ are no longer stochastic and the biases disappear.
S. Lahirr, S. Satchel1 / Estmatron
186
a/ the
Lmntrefrnoerse
We, however, ask the following question: Does each bias decrease below that the answer is yes when u is uniformly distributed.
monotonically
Theorem 3. Suppose that u is uniformb distributed over an interval E( q,,) - a,, decreases monotonically to zero as var( u) goes to zero. ProoJ
It is easy to check that E( ur) = a’/(1
E(Q)-z=zf
a2’D2’/(1+2r).
+ r). Therefore,
( -a.
to zero? We prove
a ). Then for ull i and j,
from (15) one gets
(16)
r=l From Theorem 2, we know that the right-hand side of (16) is non-negative. Now since e 2 0, D 2 0, it is evident from (16) that the right-hand side of it decreases monotonically to zero as 5’ goes to Q.E.D. zero, i.e., as var(u) goes to zero. It is to be noted that the right-hand side of (16) gives us an exact expression for the bias. We are currently working on the problem of estimating a more efficient estimate of the LI using this expression.
References Lahiri, S., 1983, A note on the underestimation and overestimation in stochastic input-output models, Economics Letters 13, 361-366. Lahiri, S. and S. Satchell, 1984, Properties of the expected value of the Leontief inverse under different stochastic specifications, Discussion paper 237 (Department of Economics, University of Essex, Colchester). Quandt, R.E., 1958, Probabilistic errors in the Leontief system, Naval Research Logistics Quarterly 5, 155-170. Simonovits, A., 1975, A note on the underestimation and overestimation of the Leontief inverse. Econometrica 43, 493-498.