To facilitate our discussion, we introduce the following notations and definitions.
736 PART FOUR: SIMULTANEOUS-EQUATION MODELS
The general M equations model in M endogenous, or jointly dependent, variables may be written as Eq. (19.1.1):
YM = M endogenous, or jointly dependent, variables XK = K predetermined variables (one of these X variables may take a value of unity to allow for the intercept term in each equation) uM = M stochastic disturbances , T = total number of observations P's = coefficients of the endogenous variables Y's = coefficients of the predetermined variables
In passing, note that not each and every variable need appear in each equation. As a matter of fact, we see in Section 19.2 that this must not be the case if an equation is to be identified.
As Eq. (19.1.1) shows, the variables entering a simultaneous-equation model are of two types: endogenous, that is, those (whose values are) determined within the model; and predetermined, that is, those (whose values are) determined outside the model. The endogenous variables are regarded as stochastic, whereas the predetermined variables are treated as nonstochastic.
The predetermined variables are divided into two categories: exogenous, current as well as lagged, and lagged endogenous. Thus, X1t is a current (present-time) exogenous variable, whereas X1(t— 1) is a lagged exogenous variable, with a lag of one time period. Y(t— 1) is a lagged endogenous variable with a lag of one time period, but since the value of Y1(t— 1) is known at the current time t, it is regarded as nonstochastic, hence, a predetermined variable.1 In short, current exogenous, lagged exogenous, and lagged endogenous where Y1, Y2,..., X1, ..., u1, u2,..., t = 1,2,...
'It is assumed implicitly here that the stochastic disturbances, the us, are serially uncorre-lated. If this is not the case, Yt—\ will be correlated with the current period disturbance term ut. Hence, we cannot treat it as predetermined.
CHAPTER NINETEEN: THE IDENTIFICATION PROBLEM 737
variables are deemed predetermined; their values are not determined by the model in the current time period.
It is up to the model builder to specify which variables are endogenous and which are predetermined. Although (noneconomic) variables, such as temperature and rainfall, are clearly exogenous or predetermined, the model builder must exercise great care in classifying economic variables as endogenous or predetermined: He or she must defend the classification on a priori or theoretical grounds. However, later in the chapter we provide a statistical test of exogeneity.
The equations appearing in (19.1.1) are known as the structural, or behavioral, equations because they may portray the structure (of an economic model) of an economy or the behavior of an economic agent (e.g., consumer or producer). The P's and ys are known as the structural parameters or coefficients.
From the structural equations one can solve for the M endogenous variables and derive the reduced-form equations and the associated reduced-form coefficients. A reduced-form equation is one that expresses an endogenous variable solely in terms of the predetermined variables and the stochastic disturbances. To illustrate, consider the Keynesian model of income determination encountered in Chapter 18:
Consumption function: Ct = P0 + p1Yt + ut 0 < p1 < 1 (18.2.3) Income identity: Yt = Ct + It (18.2.4)
In this model C (consumption) and Y (income) are the endogenous variables and I (investment expenditure) is treated as an exogenous variable. Both these equations are structural equations, (18.2.4) being an identity. As usual, the MPC Pi is assumed to lie between 0 and 1.
If (18.2.3) is substituted into (18.2.4), we obtain, after simple algebraic manipulation,
Equation (19.1.2) is a reduced-form equation;it expresses the endogenous variable Y solely as a function of the exogenous (or predetermined) variable I and the stochastic disturbance term u. n0 and n1 are the associated reduced-form coefficients. Notice that these reduced-form coefficients are nonlinear combinations of the structural coefficient(s).
738 PART FOUR: SIMULTANEOUS-EQUATION MODELS
Substituting the value of Y from (19.1.2) into C of (18.2.3), we obtain another reduced-form equation:
The reduced-form coefficients, such as n1 and n3, are also known as impact, or short-run, multipliers, because they measure the immediate impact on the endogenous variable of a unit change in the value of the exogenous variable.2 If in the preceding Keynesian model the investment expenditure is increased by, say, $1 and if the MPC is assumed to be 0.8, then from (19.1.3) we obtain n1 = 5. This result means that increasing the investment by $1 will immediately (i.e., in the current time period) lead to an increase in income of $5, that is, a fivefold increase. Similarly, under the assumed conditions, (19.1.5) shows that n3 = 4, meaning that $1 increase in investment expenditure will lead immediately to $4 increase in consumption expenditure.
In the context of econometric models, equations such as (18.2.4) or Qtd = Qts (quantity demanded equal to quantity supplied) are known as the equilibrium conditions. Identity (18.2.4) states that aggregate income Ymust be equal to aggregate consumption (i.e., consumption expenditure plus investment expenditure). When equilibrium is achieved, the endogenous variables assume their equilibrium values.3
Notice an interesting feature of the reduced-form equations. Since only the predetermined variables and stochastic disturbances appear on the right sides of these equations, and since the predetermined variables are assumed to be uncorrelated with the disturbance terms, the OLS method can be applied to estimate the coefficients of the reduced-form equations (the n's). From the estimated reduced-form coefficients one may estimate the structural coefficients (the ft's), as shown later. This procedure is known as indirect least squares (ILS), and the estimated structural coefficients are called ILS estimates.
We shall study the ILS method in greater detail in Chapter 20. In the meantime, note that since the reduced-form coefficients can be estimated by the
2In econometric models the exogenous variables play a crucial role. Very often, such variables are under the direct control of the government. Examples are the rate of personal and corporate taxes, subsidies, unemployment compensation, etc.
3For details, see Jan Kmenta, Elements of Econometrics, 2d ed., Macmillan, New York, 1986, pp. 723-731.
CHAPTER NINETEEN: THE IDENTIFICATION PROBLEM 739
OLS method, and since these coefficients are combinations of the structural coefficients, the possibility exists that the structural coefficients can be "retrieved" from the reduced-form coefficients, and it is in the estimation of the structural parameters that we may be ultimately interested. How does one retrieve the structural coefficients from the reduced-form coefficients? The answer is given in Section 19.2, an answer that brings out the crux of the identification problem.
By the identification problem we mean whether numerical estimates of the parameters of a structural equation can be obtained from the estimated reduced-form coefficients. If this can be done, we say that the particular equation is identified. If this cannot be done, then we say that the equation under consideration is unidentified, or underidentified.
An identified equation may be either exactly (or fully or just) identified or overidentified. It is said to be exactly identified if unique numerical values of the structural parameters can be obtained. It is said to be overidentified if more than one numerical value can be obtained for some of the parameters of the structural equations. The circumstances under which each of these cases occurs will be shown in the following discussion.
The identification problem arises because different sets of structural coefficients may be compatible with the same set of data. To put the matter differently, a given reduced-form equation may be compatible with different structural equations or different hypotheses (models), and it may be difficult to tell which particular hypothesis (model) we are investigating. In the remainder of this section we consider several examples to show the nature of the identification problem.
Consider once again the demand-and-supply model (18.2.1) and (18.2.2), together with the market-clearing, or equilibrium, condition that demand is equal to supply. By the equilibrium condition, we obtain
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