SIMULTANEOUS EQUATIONS MODEL
A study of the simultaneous equations model was initiated by the researchers of the Cowles Commission at the University of Chicago in the 1940s. The model was extensively used by econometricians in the 1950s and 1960s. Although it was more frequently employed in macroeconomic analysis, we shall illustrate it by a supply and demand model. Consider
(13.3.1) yj = уіу2 + Х:Рі + uj and
(13.3.2) y2 = y2yi + X2p2 + u2,
where yi and y2 are T-dimensional vectors of dependent variables, Xj and
X2 are known nonstochastic matrices, and Uj and u2 are unobservable
random variables such that £u] = £u2 = 0, Uuj = ayl, Vu2 = cr2I, and £uju2 = a12I. We give these equations the following interpretation.
A buyer comes to the market with the schedule (13.3.1), which tells him what price (y0 he should offer for each amount (y2) of a good he is to buy at each time period t, corresponding to the <th element of the vector. A seller comes to the market with the schedule (13.3.2), which tells her how much (y2) she should sell at each value (yi) of the price offered at each t. Then, by some kind of market mechanism (for example, the help of an auctioneer, or trial and error), the values of yx and y2 that satisfy both equations simultaneously—namely, the equilibrium price and quantity—are determined.
Solving the above two equations for yj and y2, we obtain (provided that 7і7г * 1):
(13.3.3) у! = : _1yi72 (XjPj + Х2УІ02 + Uj + yiu2) and
(13.3.4) y2 = ! (x202 + +Щ + 72«i)-
We call (13.3.1) and (13.3.2) the structural equations and (13.3.3) and
(13.3.4) the reduced-form, equations. A structural equation describes the behavior of an economic unit such as a buyer or a seller, whereas a reduced-form equation represents a purely statistical relationship.
A salient feature of a structural equation is that the LS estimator is inconsistent because of the correlation between the dependent variable that appears on the right-hand side of a regression equation and the error term. For example, in (13.3.1) y2 is correlated with Uj because y2 depends on ub as we can see in (13.3.4).
Next, we consider the consistent estimation of the parameters of structural equations. Rewrite the reduced-form equations as
(13.3.5) yi = Хті! + Vj and
(13.3.6) y2 = Xtt2 + v2,
where X consists of the distinct columns of X! and X2 after elimination of any redundant vector and ті!, tt2, vb and v2 are appropriately defined. Note that тії and tt2 are functions of y’s and 0’s. Express that fact as
(13.3.7) (тть 7t2) = g(yь y2, 0i, 02).
Since a reduced-form equation constitutes a classical regression model, the LS estimator applied to (13.3.5) or (13.3.6) yields a consistent estimator of TTi or it2. If mapping g(-) is one-to-one, we can uniquely determine the estimates of y’s and P’s from the LS estimators of ті] and тг2, and the resulting estimators are expected to possess desirable properties. If mapping g(-) is many-to-one, however, any solution to the equation (ftb fr2) = g(yі, у2, Pi, P2), where ft! and fr2 are the LS estimators, is still consistent but in general not efficient. If, for example, we assume the joint normality of ut and u2, and hence of Vi and v2, we can derive the likelihood function from equations (13.3.5) and (13.3.6). Maximizing that function with respect to y’s, p’s, and ct’s yields a consistent and asymptotically efficient estimator, known as the full information maximum likelihood estimator.
A simple consistent estimator of y’s and P’s is provided by the instrumental variables method, discussed in Section 13.2. Consider the estimation of ух and px in (13.3.1). For this purpose, rewrite the equation as
(13.3.8) yx = Za + ub
where Z = (y2,X) and a = (yb Pi)’. Let S be a known nonstochastic matrix of the same size as Z such that plim T~lS’Z is nonsingular. Then the instrumental variables (IV) estimator of a is defined by
(13.3.9) cljv = (S’Z)-1S’yi.
Under general conditions it is consistent and asymptotically
(13.3.10) oijv ~ N[a, 0-1 (S’Z)_1S’S(Z’S)-1].
Let X be as defined after (13.3.6), and define the projection matrix P = X(X’X) 1X’. If we insert S = PZ on the right-hand side of (13.3.9), we obtain the two-stage least squares (2SLS) estimator
(13.3.11) a 25 = (Z’PZ)_1Z’Pyi.
This estimator was proposed by Theil (1953). It is consistent and asymptotically
(13.3.12) 012S ~ N[a, CT?(Z’PZ)-1].
It can be shown that
(13.3.13) plim T(Z’PZ)-1 < plim T(S’Z)_1S’S(Z’S)’1.
In other words, the two-stage least squares estimator is asymptotically more efficient than any instrumental variables estimator.
Nowadays the simultaneous equations model is not so frequently used as in the 1950s and 1960s. One reason is that a multivariate time series model has proved to be more useful for prediction than the simultaneous equations model, especially when data with time intervals finer than annual are used. Another reason is that a disequilibrium model is believed to be more realistic than an equilibrium model. Let us illustrate, again with the supply and demand model. Consider
(13.3.14) D, = 7lP( + xi(p! + ult and
(13.3.15) St = y2Pt + X2,32 +
where Dt is the quantity the buyer desires to buy at price Pt, and St is the quantity the seller desires to sell at price Pt. We do not observe Dt or St, but instead observe the actually traded amount Qt, which is defined by
(13.3.16) Q_t = min (Dt, St).
This is the disequilibrium model proposed by Fair and Jaffee (1972). The parameters of this model can be consistendy and efficiently estimated by the maximum likelihood estimator. There are two different likelihood functions, depending on whether the research knows which of the two variables Dt or St is smaller. The case when the researcher knows is called sample separation-, when the researcher does not know, we have the case of no sample separation. The computation of the maximum likelihood estimator in the second instance is cumbersome. Note that replacing
(13.3.16) with the equilibrium condition Dt = St leads to a simultaneous equations model similar to (13.3.1) and (13.3.2).
Although the simultaneous equations model is of limited use, estimators such as the instrumental variables and the two-stage least squares are valuable because they can be effectively used whenever a correlation between the regressors and the error term exists. We have already seen one such example in Section 13.2. Another example is the error-in-variables model. See Chapter 11, Exercise 5, for the simplest such model and Fuller (1987) for a discussion in depth.