As we discussed at length in the previous section, the GLSEM is motivated by the general equilibrium system of economic theory, and is represented by the system of equations
Yt.B* = Xt.C +Ut., t =1,2,3, ... ,T, (1.2) where Yt. is 1x m, Xt. is 1xG and denote, respectively, the row vectors containing the current endogenous (or jointly dependent) and the predetermined variables of the model.
The equations comprising the system in Eq. (1.2) may be either behav- ioral (stochastic) equations or identities (nonstochastic). In the formal discussion of the GLSEM it is convenient to think of identities as hav- ing been substituted out so that Eq. (1.2) contains only behavioral equations; when this is so the vector of error terms Ut. will not contain elements which are identically zero (corresponding to the identities) and we may assert that {u~. :t = 1,2,3, ...T} is a sequence of independent identically distributed (i.i.d.) random vectors with
E(u~.)=0, Cov(u~.) =~ >0, (1.3) the covariance matrix ~ being positive definite (nonsingular). There is also another bit of complexity occasioned by the identification problem.
Ifno restrictions are placed on the matrices B*(m x m) and C(Gx m), which contain the unknown parameters of the problem, then multiplying the system in Eq. (1.2), on the right, by the arbitrary nonsingular matrix H, we have
Yt.B* H = Xt.CH+Ut.H. (1.4)
In Eq. (1.4) the parameter matrices B* H, CH are similarly unrestricted, and the error vector,
H'U~. :t = 1,2,3, ...T, is one of i.i.d. random variables with
E(H'U~.)=0, COV(H'U~.) = H/~H>0. (1.5) Ifa set of observations (Yt-, xd, t = 1,2,3, ... T, is compatible with the model of Eqs. (1.2) and (1.3) it is also compatible with the model in Eqs. (1.4) and (1.5), so that these two versions of the GLSEM are observa- tionally equivalent. Thus, if literally everything depends on everything else, there is no assurance that, if we use the data to make inferences re- garding (estimate) the parameters of Eq. (1.2), we shall, in fact, obtain what we asked for. Thus, in approaching a problem for empirical analysis the economist cannot begin from a state of complete ignorance. Ifhe does, of course, there is no reason why his intervention is required! Nor is there
any reason why anyone should be interested in what he has to say. He can be dispensed with, without cost.
It is only by asserting some restrictions on the relationships in Eq. (1.2) that the problem of inference can be solved. At the same time, however, the economist not only expresses a view as to the manner in which the economic phenomenon under investigation operates, but is also making a, potentially, falsifiable statement about the real world. It is precisely these aspects that make economic analysis interesting, for if everything can be rationalized and every proposition can be accepted sui generis, then we have neither understood anything nor do we have any assurances that today's "explanations" will be valid tomorrow.
It is worth noting that a controversy of precisely this type was widely discussed in the early 1960s with the contention, primarily by Liu (1960), that, indeed, we have no basis for any a priori restrictions and hence the GLSEM is, in principle, not estimable! This view was convincingly refuted, at the time, by F.M. Fisher (1961), who argued that even though we might admit philosophically that no element of B* and C is zero, yet it is patently the case that many elements of B* and C would be very small.
Ignoring very small elements is rather innocuous, and the impairment of properties suffered as a consequence, are correspondingly small as well.
Since in every discipline theoretical structures are only an idealization of reality, the argument advanced by Liu is no reason why the GLSEM is not to be deemed a useful tool of analysis. A potentially more serious problem is the case of errors (of observation) in the exogenous variables of the GLSEM, to be defined below.
Incidentally the same argument would proscribe the use of the GLM as well, simply by arguing that "everything" ought to be considered as an explanatory variable in any GLM formulation. Since we will never have enough data to estimate the parameters of such a model this approach amounts to stating that nothing is knowable empirically.
We shall not discuss these ideas here, since they belong, more properly, to the realm of metaeconometrics.
The ultimate justification of any scientific procedure, is the results it yields in terms of advancing our understanding of the phenomenon under investigation, enabling us to predict and/or control its evolution. Predic- tion and control in economics are far more complicated than in the physi- cal sciences, since our analysis is conditional on the exogenous variables whose study is not always within the universe of discourse of economics;
control is also hampered by the fact that the monetary and fiscal author- ities are not always able to exert complete control over the variables that serve as their control instruments and moreover their objectives may be continuously shaped (feedback) by the behavior of the system (endogenous variables) they seek to control. On the other side, the behavior of economic agents may change discontinuously in response to certain types of policy measures, as well as changes in the way in which they perceive themselves.
Many of these arguments may well have merit, but in the following text we shall not discuss the extent of their validity or merit; our purpose in this volume is confined to the development and exposition of the theory of estimation for the standard GLSEM and certain extensions of it.
1.4.1 Assumptions and Conventions
Whether the basic issues raised above have been dealt with satisfactorily or not, we shall begin the formal discussion of the GLSEM on the assertion that we are able to impose sufficient a priori restrictions on B* and C so as to make the model in Eq. (1.2) distinguishable from that in Eq. (1.4), i.e., to render the equations of Eq. (1.2) identifiable. A formal discussion of the identification problem is postponed to Chapter 3, at which time we shall deal with it extensively.
We begin by noting that the vector of predetermined variables is given by
Xtã = (Yt-h Yt-2ã, ... , Yt-kã, pd (1.6)
where Pt. is an s -element row vector of exogenous variables.
The basic set of assumptions under which we shall operate in much of our discussion is:
(A. 1) The matrix of exogenous variables
P =(pd t= 1,2,3, ...T, T > s, is of full rank and4
plim ~plP = Mp p
T->ooT
exists and is nonsingular (positive definite).
(A.la) It is also asserted, or is derived as a consequence of A.l and the stability of the model, that the matrix of predetermined variables, X = (xd, t = 1,2,3, ...,T, T> G, is of full rank and
plim ~X'X = Mx x
T->oo T
exists and is nonsingular (positive definite).
(A.2) The matrix B* is nonsingular.
(A.3) Some elements of B* and C are knowna priori to be zero (exclusion restrictions), so that the equations of the system are identified.
4The notation plimT~oo ~P' P =Mp p is meant to be understood as follows:
(a) as an ordinary limit if the exogenous variables are taken as nonstochastic, or (b) as a probability limit if the exogenous variables are asserted to be stochastic.
(A.4) Ifthe system is dynamic, i.e., it contains lagged endogenous variables, it is stable; this means that the roots of its characteristic equation (of the associated homogeneous vector difference equation) are less than unity in absolute value.
(A.5) The structural errors, {Ut. :t = 0, ±1, ±2, ±3, ... }, are a sequence of i.i.d. random vectors with E(u~.)= 0,Cov(u~.) = ~, where ~ is positive definite (notation: ~ >0).5 Moreover, they are indepen- dent of the exogenous variables, i.e., the elements of P.
Definition 1 (Structural Form). The representation of an economic system by the equation set in Eq. (1.2) is said to be the structural form of the system, and the equations in Eq. (1.2) are said to be the structural equations of the system.
Definition 2 (Reduced Form). The transformation of the set of equations in Eq. (1.2), as in
Yt.= Xt.II+vt-, II=CD, Vt.= Ut.D, D = B*-l, t =1,2, ... ,T, (1.7) is said to be the reduced form of the system, and the equations in Eq.
(1.7) are said to be the reduced form equations of the system.
Remark 1. In the context of a proper theoretical framework one may de- rive "the rules of behavior" of economic agents. When these are exhibited in a system of equations such as Eq. (1.2), we assert that this represents the structural form of the system. What makes the form "structural" is our assertion (presumably in close correspondence with the modus operandi of the real world phenomenon we study) that economic agents, by their collec- tive action, and given the information in the predetermined variables Xt. , assign values to the endogenous variables Yt.; and, further, that within an additive stochastic component, these values satisfy the equations of Eq. (1.2). Thus, the structural representation shows explicitly the linkages amongst the endogenous variables, as well as their (collective) direct de- pendence on the predetermined variables. By contrast, the reduced form shows only how economic agents by their collective actions assign values to Yt given the information in Xt . . The differences between the reduced and structural form is that the latter shows explicitly the direct effect of a (generally small) set of variables (both endogenous and predetermined) on a given endogenous variable, while the former shows both the direct and indirect effects (generally through other endogenous variables) of all predetermined variables on a given endogenous variable.
5In effect this means that any identities that the model may have contained have been removed by substitution.
Generally, the structural form is more revealing of the manner in which an economic system is operating. The reduced form is less revealing. Indeed, a reduced form as in Eq. (1.7), estimated without reference as to its origin, i.e., without taking into account that II = CD and that C, D-1 are restricted by (A.3), may be compatible with infinitely many structural forms so long as they encompassed Yt. and involved as predetermined variables only Xt ..
For completeness we offer the definition below, although this matter was covered in an earlier section.
Definition 3 (Classification of Variables). The elements of the m -element row vector Yt. are said to be the endogenous or jointly dependent variables of the model. Sometimes they are also called (somewhat redun- dantly) current endogenous variables. The elements of the G -element row vector Xt. are said to be the predetermined variables of the sys- tem. As the notation in Eq. (1.6) suggests, the predetermined variables are either lagged endogenousor exogenous. The basic characteristic of exogenous variables is that they are independent of the structural error sequence
{Ut. :t = 0,±1, ±2, ±3, ... }.