It can be seen that the determinant of this matrix is zero:
Since the determinant is zero, the rank of the matrix (19.3.6), denoted by p (A), is less than 3. Therefore, Eq. (19.3.2) does not satisfy the rank condition and hence is not identified.
As noted, the rank condition is both a necessary and sufficient condition for identification. Therefore, although the order condition shows that Eq. (19.3.2) is identified, the rank condition shows that it is not. Apparently, the columns or rows of the matrix A given in (19.3.6) are not (linearly) independent, meaning that there is some relationship between the variables Y4, X2, and X3. As a result, we may not have enough information to estimate the parameters of equation (19.3.2); the reduced-form equations for the preceding model will show that it is not possible to obtain the structural coefficients of that equation from the reduced-form coefficients. The reader should verify that by the rank condition Eqs. (19.3.3) and (19.3.4) are also unidentified but Eq. (19.3.5) is identified.
As the preceding discussion shows, the rank condition tells us whether the equation under consideration is identified or not, whereas the order condition tells us if it is exactly identified or overidentified.
To apply the rank condition one may proceed as follows:
1. Write down the system in a tabular form, as shown in Table 19.1.
2. Strike out the coefficients of the row in which the equation under consideration appears.
3. Also strike out the columns corresponding to those coefficients in 2 which are nonzero.
4. The entries left in the table will then give only the coefficients of the variables included in the system but not in the equation under consideration. From these entries form all possible matrices, like A, of order M — 1 and obtain the corresponding determinants. If at least one nonvanishing or nonzero determinant can be found, the equation in question is (just or over) identified. The rank of the matrix, say, A, in this case is exactly equal to M — 1. If all the possible (M — 1)(M — 1) determinants are zero, the rank of the matrix A is less than M — 1 and the equation under investigation is not identified.
Gujarati: Basic Econometrics, Fourth Edition
IV. Simultaneous-Equation I 19. The Identification
© The McGraw-Hill Companies, 2004
CHAPTER NINETEEN: THE IDENTIFICATION PROBLEM 753
Our discussion of the order and rank conditions of identification leads to the following general principles of identifiability of a structural equation in a system of M simultaneous equations:
1. If K — k > m — 1 and the rank of the A matrix is M — 1, the equation is overidentified.
2. If K — k = m — 1 and the rank of the matrix A is M — 1, the equation is exactly identified.
3. If K — k > m — 1 and the rank of the matrix A is less than M — 1, the equation is under-
4. If K — k < m — 1, the structural equation is unidentified. The rank of the A matrix in this case is bound to be less than M — 1. (Why?)
Henceforth, when we talk about identification we mean exact identification, or overidentification. There is no point in considering unidentified, or underidentified, equations because no matter how extensive the data, the structural parameters cannot be estimated. However, as shown in Chapter 20, parameters of overidentified as well as just identified equations can be estimated.
Which condition should one use in practice: Order or rank? For large simultaneous-equation models, applying the rank condition is a formidable task. Therefore, as Harvey notes,
Fortunately, the order condition is usually sufficient to ensure identifiability, and although it is important to be aware of the rank condition, a failure to verify it will rarely result in disaster.8
If there is no simultaneous equation, or simultaneity problem, the OLS estimators produce consistent and efficient estimators. On the other hand, if there is simultaneity, OLS estimators are not even consistent. In the presence of simultaneity, as we will show in Chapter 20, the methods of two-stage least squares (2SLS) and instrumental variables will give estimators that are consistent and efficient. Oddly, if we apply these alternative methods when there is in fact no simultaneity, these methods yield estimators that are consistent but not efficient (i.e., with smaller variance). All this discussion suggests that we should check for the simultaneity problem before we discard OLS in favor of the alternatives.
As we showed earlier, the simultaneity problem arises because some of the regressors are endogenous and are therefore likely to be correlated with
8Andrew Harvey, The Econometric Analysis of Time Series, 2d ed., The MIT Press, Cambridge, Mass., 1990, p. 328. *Optional.
9The following discussion draws from Robert S. Pindyck and Daniel L. Rubinfeld, Econometric Models and Economic Forecasts, 3d ed., McGraw-Hill, New York, 1991, pp. 303-305.
19.4 A TEST OF SIMULTANEITY9
754 PART FOUR: SIMULTANEOUS-EQUATION MODELS
the disturbance, or error, term. Therefore, a test of simultaneity is essentially a test of whether (an endogenous) regressor is correlated with the error term. If it is, the simultaneity problem exists, in which case alternatives to OLS must be found; if it is not, we can use OLS. To find out which is the case in a concrete situation, we can use Hausman's specification error test.
A version of the Hausman specification error test that can be used for testing the simultaneity problem can be explained as follows10: To fix ideas, consider the following two-equation model:
Demand function: Qt = a0 + a1 Pt + a2 It + a3 Rt + u1t (19.4.1) Supply function: Qt = p0 + p1 Pt + u2t (19.4.2)
where P = price
Q = quantity I = income R = wealth u's = error terms
Assume that I and R are exogenous. Of course, P and Q are endogenous.
Now consider the supply function (19.4.2). If there is no simultaneity problem (i.e., P and Q are mutually independent), Pt and u2t should be un-correlated (why?). On the other hand, if there is simultaneity, Pt and u2t will be correlated. To find out which is the case, the Hausman test proceeds as follows:
First, from (19.4.1) and (19.4.2) we obtain the following reduced-form equations:
where v and w are the reduced-form error terms. Estimating (19.4.3) by OLS we obtain
10J. A. Hausman, "Specification Tests in Econometrics," Econometrica, vol. 46, November 1976, pp. 1251-1271. See also A. Nakamura and M. Nakamura, "On the Relationship among Several Specification Error Tests Presented by Durbin, Wu, and Hausman," Econometrica, vol. 49, November 1981, pp. 1583-1588.
CHAPTER NINETEEN: THE IDENTIFICATION PROBLEM 755
where Pt are estimated Pt and vt are the estimated residuals. Substituting (19.4.6) into (19.4.2), we get
Note: The coefficients of Pt and vt are the same.
Now, under the null hypothesis that there is no simultaneity, the correlation between vt and u2t should be zero, asymptotically. Thus, if we run the regression (19.4.7) and find that the coefficient of vt in (19.4.7) is statistically zero, we can conclude that there is no simultaneity problem. Of course, this conclusion will be reversed if we find this coefficient to be statistically significant.
Essentially, then, the Hausman test involves the following steps:
Step 2. Regress Qt on Pt and vt and perform a t test on the coefficient of v t. If it is significant, do not reject the hypothesis of simultaneity; otherwise, reject it.11 For efficient estimation, however, Pindyck and Rubinfeld suggest regressing Qt on Pt and vt .12
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