Note that the entries above the main diagonal are zeros (why?).
766 PART FOUR: SIMULTANEOUS-EQUATION MODELS
where P = rate of change of price per unit of output W = rate of change of wages per employee R = rate of change of price of capital M = rate of change of import prices L = rate of change of labor productivity UN = unemployment rate, %6
The price equation postulates that the rate of change of price in the current period is a function of the rates of change in the prices of capital and of raw material, the rate of change in labor productivity, and the rate of change in wages in the previous period. The wage equation shows that the rate of change in wages in the current period is determined by the current period rate of change in price and the unemployment rate. It is clear that the causal chain runs from Wt-1 ^ Pt ^ Wt, and hence OLS may be applied to estimate the parameters of the two equations individually.
Although recursive models have proved to be useful, most simultaneous-equation models do not exhibit such a unilateral cause-and-effect relationship. Therefore, OLS, in general, is inappropriate to estimate a single equation in the context of a simultaneous-equation model.7
There are some who argue that, although OLS is generally inapplicable to simultaneous-equation models, one can use it, if only as a standard or norm of comparison. That is, one can estimate a structural equation by OLS, with the resulting properties of biasedness, inconsistency, etc. Then the same equation may be estimated by other methods especially designed to handle the simultaneity problem and the results of the two methods compared, at least qualitatively. In many applications the results of the inappropriately applied OLS may not differ very much from those obtained by more sophisticated methods, as we shall see later. In principle, one should not have much objection to the production of the results based on OLS so long as estimates based on alternative methods devised for simultaneous-equation models are also given. In fact, this approach might give us some idea about how badly OLS does in situations when it is applied inappro-priately.8
6Note: The dotted symbol means "time derivative." For example, P = dP/dt. For discrete time series, dP/dt is sometimes approximated by AP/At, where the symbol A is the first difference operator, which was originally introduced in Chap. 12.
7It is important to keep in mind that we are assuming that the disturbances across equations are contemporaneously uncorrelated. If this is not the case, we may have to resort to the Zellner SURE (seemingly unrelated regressions) estimation technique to estimate the parameters of the recursive system. See A. Zellner, "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias," Journal of the American Statistical Association, vol. 57, 1962, pp. 348-368.
8It may also be noted that in small samples the alternative estimators, like the OLS estimators, are also biased. But the OLS estimator has the "virtue" that it has minimum variance among these alternative estimators. But this is true of small samples only.
CHAPTER TWENTY: SIMULTANEOUS-EQUATION METHODS 767
20.3 ESTIMATION OF A JUST IDENTIFIED EQUATION: THE METHOD OF INDIRECT LEAST SQUARES (ILS)
For a just or exactly identified structural equation, the method of obtaining the estimates of the structural coefficients from the OLS estimates of the reduced-form coefficients is known as the method of indirect least squares (ILS), and the estimates thus obtained are known as the indirect least-squares estimates. ILS involves the following three steps:
Step 1. We first obtain the reduced-form equations. As noted in Chapter 19, these reduced-form equations are obtained from the structural equations in such a manner that the dependent variable in each equation is the only endogenous variable and is a function solely of the predetermined (exogenous or lagged endogenous) variables and the stochastic error term(s).
Step 2. We apply OLS to the reduced-form equations individually. This operation is permissible since the explanatory variables in these equations are predetermined and hence uncorrelated with the stochastic disturbances. The estimates thus obtained are consistent.9
Step 3. We obtain estimates of the original structural coefficients from the estimated reduced-form coefficients obtained in Step 2. As noted in Chapter 19, if an equation is exactly identified, there is a one-to-one correspondence between the structural and reduced-form coefficients; that is, one can derive unique estimates of the former from the latter.
As this three-step procedure indicates, the name ILS derives from the fact that structural coefficients (the object of primary enquiry in most cases) are obtained indirectly from the OLS estimates of the reduced-form coefficients.
An Illustrative Example
Consider the demand-and-supply model introduced in Section 19.2, which for convenience is given below with a slight change in notation:
where Q = quantity P = price
X = income or expenditure
Assume that X is exogenous. As noted previously, the supply function is exactly identified whereas the demand function is not identified.
9In addition to being consistent, the estimates "may be best unbiased and/or asymptotically efficient, depending respectively upon whether (i) the z's [ = X's] are exogenous and not merely predetermined [i.e., do not contain lagged values of endogenous variables] and/or (ii) the distribution of the disturbances is normal." (W. C. Hood and Tjalling C. Koopmans, Studies in Econometric Method, John Wiley & Sons, New York, 1953, p. 133.)
Demand function: Qt = a0 + a1 Pt + a2Xt + u1t Supply function: Qt = fa0 + fa Pt + u2t
768 PART FOUR: SIMULTANEOUS-EQUATION MODELS
The reduced-form equations corresponding to the preceding structural equations are
where the n's are the reduced-form coefficients and are (nonlinear) combinations of the structural coefficients, as shown in Eqs. (19.2.16) and (19.2.18), and where w and v are linear combinations of the structural disturbances u1 and u2.
Notice that each reduced-form equation contains only one endogenous variable, which is the dependent variable and which is a function solely of the exogenous variable X (income) and the stochastic disturbances. Hence, the parameters of the preceding reduced-form equations may be estimated by OLS. These estimates are n 1 = (20.3.5)
where the lowercase letters, as usual, denote deviations from sample means and where Q and P are the sample mean values of Q and P. As noted previously, the ni's are consistent estimators and under appropriate assumptions are also minimum variance unbiased or asymptotically efficient (see footnote 9).
Since our primary objective is to determine the structural coefficients, let us see if we can estimate them from the reduced-form coefficients. Now as shown in Section 19.2, the supply function is exactly identified. Therefore, its parameters can be estimated uniquely from the reduced-form coefficients as follows:
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