Corollary 1. In the context of Theorem 4, the system as a whole is identi- fied by the exclusion and normalization conditions L*o'vec(A*) = vec(H) ,
3.2.6 Covariance and Cross Equation Restrictions
In previous sections, we considered the identification problem in the clas- sical context. The major concern, therein, was to obtain conditions under which identification of the structural parameters of a given equation, may be established by imposing restrictions on that equation alone, leaving the covariance structure of the error process completely unconstrained. It is interesting to note that many situations arise in which it is natural to im- pose covariance restrictions; this is the case, for example, in panel data based models, where it is not unreasonable to argue that the error process consists of two components, one relating to the particular individual in the sample, and other pertaining to the time of the observation. In this frame- work, it is typically asserted that the two components are independent.
This amounts to a set of covariance restrictions.
In a somewhat different context, when dealing with "rational expecta- tions" models, we encounter cross equation restrictions. This arises be- cause, in such models, "expected" quantities are represented as the condi- tional expectation of the relevant variables, as obtained from the (solution of the) model. Consequently, the representation of an "expected" quantity may well (and typically does) involve parameters from other equations, thus creating cross equation restrictions.
Let us examine first how covariance restrictions may aid in the identifi- cation process. Consider again the simple supply-demand model
Making use of the third equation above, and properly normalizing the sys- tem, we obtain
a l l *
Pt = -b + bqt - bU t2 .
As is evident from the preceding discussion, neither of the two equations above are identified.Ifit is given that a = 0 , the second (price) equation is identified but the first (quantity) is not. This is so because the quantity equation cannot be distinguished from a linear combination of the quantity and price equations since, in the standard context, there are no covariance restrictions. Even though, typically, we leave the covariance matrix of the structural errors unrestricted, here we are interested in the question: would restrictions on the latter bring about identification of the first equation?
To provide a framework for this discussion, we observe that, in nearly all estimation procedures explored this far, we have made use only of first and second moments. Since we have two variables, we can compute from
our sample five moments, viz. the two means p, ij, and the three second moments Sqq, spp, and Spq, i.e. the sample variance of quantity and price, respectively, and their sample covariance. The model, on the other hand, contains six parameters, viz. b, 0:,(3,O"n,0"22, and 0"12, in the obvious notation. Thus, if it is true that 0"12 = 0 there is at least the possibility that the model will become completely identified. To this end, we employ the method of moments, in which we use the structure above to write the sample moments of the observables as a function of the uknown parameters.
Before we do so, we write the model in the more convenient form,
where Un = U;l' b' = (lib), and Ut2 = -b'U;2' Although it is some- what inaccurate, we shall continue to denote the variances of Uti, by au , respectively, i = 1,2.It is still true that Cov(un,Ut2) =O.
Omitting terms whose probability limit isnull as, for example, 'Ill , U2, the first and second moments of the observables are given by
ij 0: +(3p, P b'ij (3.26)
Sqq (3spq +Sq1, Sqp (3.27)
Spp = b'spq+Sp2, Spq = b'sqq +Sq2, (3.28) where Sqi is the sample covariance between q and Uti and Spi is the sam- ple covariance between P and Uti, respectively, i = 1,2. From the second equation of Eq. (3.26) we find l/ = (p/iJ) . After further manipulation we obtain,
/J = Sqp - (p/iJ)Sqq ,
spp - (plq)spq a= ij - /Jp, b= -.A ij
p (3.29)
It is possible to obtain estimators of the variances by substituting the es- timators above in Eqs. (3.27) and (3.29); the estimators of the variances, thus obtained, could not be guaranteed to be positive, however. To ensure that, it is preferable to estimate the variances in the standard fashion,
T
A 1~ A A 2
O"n = T ~(qt - 0: - (3Pt) , t=l
T
A 1~( A')2
0"22 = T Z:: Pt - b qt . t=l
(3.30) Since the sample moments may be easily shown to converge, in probabil- ity, to the corresponding parameters, the parameter estimators obtained above will also converge, in probability, to the corresponding parameters.
Hence, we have shown how covariance restrictions may aid in identifying the parameters of a model.
Recursive Systems
A case where covariance restrictions play an important role in the identifi- cation and estimation of structural parameters, is the recursive model. We have
Definition 9. Consider the standard GLSEM YB* =XC+U.
Ifthe matrix B* is upper triangular, i.e. bij = 0 for i >j ,and moreover, if ~ is diagonal, the GLSEM is said to be simply recursive.
Let B* be a block matrix,i.e,
[Bi>
Br2
B;n1
B* B 22 B 2n
B* = ~1 . , (3.31)
B~l B~2 B~n
where each block, Blj ,is an s; xSj matrix. Partition ~ conformably, i.e.
let
[~U~21 ~12~22 ~'n~2n1
~= . . , (3.32)
~nl ~n2 ~nn
where ~ij is a matrix of dimension s; x Sj . If Blj = 0, for i > j, and
~ij = 0, for i i- j ,i.e, if B* is upper block triangular and ~ is block diagonal, the GLSEM is said to be a block recursive system.
Consider, now, the second equation of a simply recursive system, impose the normalization convention and suppose no other prior restrictions are available. The second equation reads
Yt2 = f312Ytl +Xtã Cã2+Unã (3.33)
Ifwe apply to this equation, mechanically, the criteria of Theorem 2, par- ticularly the necessary (order) condition of Remark 4, we shall find that
m2 = 1, G2 = G. Since, evidently, m2 +G2 > G, the necessary condi- tion for identification fails, and hence, we should be obliged to conclude that this equation is not identifiable. This, however, would be a gross error and highlights the significance of Remark 2, of Chapter 1. It was stated, therein, that the division between current endogenous and predetermined variables is an artificial one, from the point of view of econometric theory.
What is important, in an estimation context, is whether an "explanatory"
variable is or is not correlated with (or independent of) the structural error of the equation in which it occurs. To explore this insight let us consider
the correlation structure between the vector Yt. , and the structural error,
Ut . .From the reduced form we find
D = B*-I. (3.34)
Consequently,
(3.35) Since, in the case of a simply recursive system, D is upper triangular and I; is diagonal, it follows that
O"l1dl1
0"11d12
D'I; = 0"11d13
o o
o (3.36)
Thus, Ytl is not correlated with (or is independent of) Uti, for i > 1 ; in general, Eq. (3.36) shows that Yti is not correlated with (or is independent of) Utj, for j > i. It follows, therefore, that, in the it h equation of the system, the Ytj, for j < i ,can be treated as "predetermined" variables.
Consequently, even though no (zero) restrictions are placed on the elements of the matrix C , all parameters of the it h equation are iden- tified. On the other hand, if we apply, mechanically, the results of Theorem 2, we should find that, since G, = G, the necessary (order) condition for identification is violated. The point to be noted, however, is that for this equation the number of "predetermined" variables is (at most) G+i - 1 , and the number of parameters to be estimated is also (at most) G+i - 1 . Thus, this equation is certainly identified.
Similarly, in the case of block recursive systems, we have D~II;l1
DpI;l1
D13I; 11 (3.37)
To examine identification issues, more precisely in this context, partition conformably the vectors, Yt. and Ut. ,
( 1 2 n)
Ytã = Ytã, Ytã, ... ,Ytã , ( 1 2 n)
Utã = Ut., Ut.,ããã, Ut. ,
where yt and ut are Si-element row (sub )vectors. The it h subsystem implied by Eq. (3.31) may be written as
i - I
i B* - "jB* + C i
Ytã ii - - 6 Ytã ji Xtã (i) +Utã, j=1
where C = (C(1), C(2)"'" C(nằ), and each C(i) is a G xSi matrix corre- sponding to the it h subsystem.
The implication of Eq. (3.32) is that, in the subsystem above, the "de- pendent (explanatory) variables" y{, j < i, are uncorrelated with (or independent of) ut and hence, in the it h subsystem, may be treated as predetermined variables. Consequently, in deciding on the identifiability of parameters in this (it h ) subsystem, only the subset of y's having coeffi- cients in the matrix B1i are the "current endogenous" variables.
In both, the simply recursive and block recursive cases, it is rather simple to see how restrictions on the covariance matrix of the system contribute to the identification of parameters. They do so by creating what may be called equation specific predetermined variables; precisely, they do so by drawing a sharp distinction between variables that are jointly dependent in an economic sense, i.e, in the sense they are simultaneously determined by the operation of the economic system, and variables that are correlated with (or jointly dependent on) the structural errors. This is close to the concept of weak exogeneity which we shall briefly examine in Chapter 6.
Panel Data Based Models
Panel data sets consist of observations over a number of individuals, say N ,over several time periods, say T. The typical simultaneous equations model may be of the form
Yti.B* = Xti.C+Utiã, i =1,2, ... , N, t = 1,2, ... ,T. (3.38) In this particular example the (structural) matrices B* and C do not vary over individuals or time. Some partial variations may be allowed in the form of fixed effects. In this discussion, we ignore this aspect and concentrate on the error process specification.
What is indicated by Eq. (3.38) is that (over time) we have T observa- tions on the same number of individuals, N . The vectors Uti. and Yti. each contain m elements, and their /h element pertains to the /h equation;
each equation describes certain aspects of the behavior of the individual in question, at time t. Ifthe specification of Uti. is left completely gen- eral then for, the entire set of observations, we should be dealing with a covariance matrix of order mTN ,containing an impossibly large number of parameters. This alone would have the effect of preventing the identi- fication of parameters in the model. Ifwe look at the typical element of the error vector, i.e. Utij we note that it has three points of reference.
It is the disturbance adhering to the jth equation of the it h individ- ual at time t. It is typically assumed, in such models, that the three effects act independently and linearly. Thus, one tends to write Utij =
gj(1]i+Et)+'l/.Jtij, g = (gl,g2,oO.,gm)', where {1]i: i = 1,2,3,oO.,N}
is a sequence of i.i.d. random variables, with mean zero and variance O'~, {Et :t = 1,2,3, ... ,T} is, similarly, a zero mean sequence of i.i.d. random
variables with variance cy; and {'Ij;ti. :t =1,2,3, ...,T, i = 1,2, ...N} is, also, a sequence of zero mean i.i.d. random vectors with covariance matrix
~. Moreover, the three components are mutually independent. Ifwe write down the observations on this model, lexicographically, ordering first the equations from one to m, in a row, then for fixed time ordering individ- uals from one to N and finally, ordering "time" from one to T, we shall obtain the matrix of error terms, say U, as
(3.39) where
U(t) = (Uti.)' i = 1,2, ... ,N, (3.40)
Uti. being the it h row of U(t) . From the preceding discussion it follows immediately that
E (U~i.Utiã ) E(U~i.Uts.) E(U~i.Ut'i.) E(U~i.Ut's.)
(cy~+cy;)gg'+~, for all i and T cy; gg', for all t ands of.i
cy~gg', for all i and t of.t'
0, for t of. t' and s of.i. (3.41) In fact, examining the results above, we note that the equation-relevant parameters, gj ,appear always in conjunction with cy; or CY; ,or both.
For this reason, these parameters cannot be separately identified, which compels us to normalize one of them, for example to take, say CY; = 1 .
The reader desiring more detailed exposition of such matters is referred to Hsiao (1983), (1985).