THE RANK CONDITION FOR STRUCTURAL EQUATION IDENTIFICATION RE-VISITED: NOT QUITE SUFFICIENT AFTER ALL. Richard Ashley and Hui Boon Tan

Transcription

1 THE RANK CONDITION FOR STRUCTURAL EQUATION IDENTIFICATION RE-VISITED: NOT QUITE SUFFICIENT AFTER ALL Richard Ashley and Hui Boon Tan Virginia Polytechnic Institute and State University 1 Economics Department Working Paper #E94-16 (revised: August, 1995) JEL: C3 ABSTRACT The usual rank condition for the identification of a structural equation is shown to in fact be insufficient. A counter-example and corrected condition are given. Simulations are presented illustrating the potential consequentiality of this flaw in the rank condition. 1Economics Department (361), Virginia Polytechnic Institute and State University, Blacksburg, VA The authors may be reached at (54) or at ashleyr@vt.edu.

2 1. Introduction The identification status of an equation which is part of a simultaneous equation system what identification means, the necessary (order) condition, the necessary and sufficient (rank) condition these concepts are so fundamental that practically every graduate econometrics sequence has allocated time to cover them for the last twenty years. Nevertheless, we have uncovered a flaw in two of the three forms usually used to express the necessary and sufficient rank condition for the identification of a structural equation. Once the flaw is understood, it is easy to construct examples where these flawed forms of the rank condition are satisfied despite the fact that the structural equation is under-identified. Indeed, the structural equation coefficients in such examples are not even well-defined where identification fails; such a counter-example is given below. How can this be? The confusion is partly due to the fact that so many different authors have stated and/or derived (supposedly) necessary and sufficient conditions, each using his/her own notation. More importantly, however, this situation has occurred because the flaw appears at first glance to be inconsequential due to the fact that it affects the sufficiency of the rank condition only for certain values of the reduced form coefficients. Nevertheless, the flaw is consequential simulation experiments with simultaneous equation systems capable of exhibiting this defect in the rank condition produce 2SLS coefficient estimates whose sample 2 dispersion becomes quite large over a substantial interval of reduced form coefficient values. Thus, even a near- failure of the rank condition can be important in practice. 2. The Necessary and Sufficient Rank Condition Different authors have varied substantially in the manner in which they derive and/or express the rank condition, leading to the three main forms of the necessary and sufficient condition summarized in Table 1. All of these conditions are referred to as "the rank condition for structural equation identification" in the literature since they would all be equivalent if all of them were, in fact, necessary and sufficient for identification. finite. 2The sample interquartile range is used to measure this dispersion since the variance of these estimators is not 1

3 Table 1 Generic Forms of the Rank Condition a Rank Condition Dimension Form Matrix Rank s Always Valid? Authors Kmenta(1971, 1986), Dhrymes(1978), Wonnacott & b I G x K G - 1 NO Wonnacott(1979), Pindyck & Rubinfeld(1981), Chow(1983) c 1 II G -1 x K G - 1 YES Davidson & McKinnon(1993), Greene(1973), Theil(1971), Dhrymes(1974), Spanos(1986), Lardaro(1993) Fisher(1966), Farebrother(1977), Dutta(1975), d III [ ] G x R G - 1 NO Chow(1983), Johnston(1984), Judge, et al.(1985), Madalla(1992) athe notation for simultaneous equation systems varies across authors; the notation in this Table is that used by Kmenta(1986) and others. It is assumed that the G equations of the system have been re-numbered so that it is the first structural equation whose identification status is being examined. Based on the explicit assumption that this is the equation "determining" endogenous variable number one, it is assumed that theory has specified that G = G - G endogenous variables (numbered G + 1 G) do not appear in this structural equation and that K predetermined variables (numbered K + 1 K) do not appear in this equation. Thus, G - 1 endogenous variables and K predetermined variables do appear as explanatory variables in the first structural equation. b is a submatrix of the G K reduced form coefficient matrix,, containing G rows (one for each endogenous variable appearing in the first structural equation) and K columns (one for each predetermined variable not appearing in the first structural equation.). c 1 omits the first row of ; this row contains the coefficients in the first reduced form equation on the K predetermined variables omitted from the first structural equation. d is a (G+K) x R matrix such that [ ] 1 = expresses the R homogeneous linear restrictions on the coefficients in the first structural equation. {Here is the GxG matrix whose gth row contains the coefficients on all G endogenous variables as they appear in the gth structural equation, is the GxK matrix whose gth row contains the coefficients on all K predetermined variables as they appear in the gth structural equation, and [ ] 1 is the first row of the [ ] matrix.} For the case considered here (where the coefficients on G endogenous variables and on K predetermined variables are restricted to be zero) the first row of [ ] is an R dimensional zero vector and the remaining G - 1 rows are the [ ] = matrix defined by Kmenta(1986, p ). Kmenta shows that, if B exists, then r( ) = r( ) + G ; thus, this condition is essentially equivalent to condition I. 2

4 The three forms of the rank condition given in Table 1 are not all equivalent, however. Form II of the rank condition truly is necessary and sufficient for identification regardless of what specific values the coefficients in take on. In contrast, Form I is not. Both of these forms of the rank condition are derived in the Appendix. Most of this derivation is quite standard; it is given here so as to delineate the defect in Form I of the rank condition. Forms I and II are both common in the literature, but Form I (the defective one) is essentially 2 equivalent to Form III which is therefore equally defective. This is important because due to the fact that it is couched in terms of the structural rather than the reduced form coefficients Form III is quite commonly given and used. 2A proof of this equivalence (for the simple exclusion restrictions considered here) is sketched in footnote d of Table 1. The linear algebra omitted from this sketch provides no additional insight into the nature of the problem and can, in any case, be found in Kmenta(1986, pp ). 3

5 3. An Example Where the Usual Rank Condition Is Not Sufficient Suppose that, based on exclusion restrictions derived from a particular macroeconomic theory, the structural equation system is of form: 4 C t 14 Y t 12 r t C t 1 u 1t r t 24 Y t M t u 2t (1) I t 32 r t 31 u 3t Y t C t I t G t u 4t and that the reduced form equations for this system, based only on a presumption that the endogenous variables {C, r, I, Y } are jointly determined by a set of stochastic equations linear in the predetermined variables {C, M, t t t t t-1 t G } is: t C t 3..45C t 1 3.M t.5g t v 1t r t.1.3c t 1 23 M t 1.G t v 2t (2) I t.3.12c t 1.8M t.4g t v 3t Y t C t 1 2.2M t 1.1G t v 4t Then (3) and 4 So as to eliminate any possibility of confusion regarding the treatment of identities, the variance of u is not set 4t to zero. This error term could be intrepreted as the statistical discrepancy in the national income accounts. 4

6 Note that r( B ) = 2 = G -1 so that Form I of the rank condition is satisfied and that r B G 1 3 (4) (5) so that Form III of the rank condition is also satisfied. Nevertheless,,,, and are not identified when equals two since, in that case, the rank of (6) is less than G -1 = 2. Consequently 1 is not of full rank so it is not possible to solve equation 13 (see Appendix): t (7) for and. Thus, the structural equation for C is simply not identified when equals two, even though t 23 Forms I and III of the rank condition are satisfied. Fisher(1966, p. 45) recognizes this possibility, but dismisses it as inconsequential: It follows that such a determinant will be zero and the rank condition will fail only on a set of measure zero in the space of those elements. Loosely speaking, the probability is zero in this situation that the last M - 1 rows of A will just happen to be such as to make the rank fail. Such failure will occur only by accident. The situation is the same as that encountered when considering the chances that a number picked from the real line happens to be zero. It can happen, but one feels rather safe in neglecting such a possibility unless one knows in advance that it must happen. 5

7 Inter-Quartile Range And it would be inconsequential if all we cared about were identification per se. In practice, however, it also behooves us to care about the sampling properties of instrumental variables estimators of,,, and in finite samples. Since these sampling properties fall apart when identification fails, the situation is in fact more akin to considering the expected value of a dirac delta function times whatever number we pick from the real line the set of points where the rank condition fails is indeed of measure zero but that set can be consequential if the dispersion of the instrumental variables estimator becomes unboundedly large on that set. To illustrate this point, we simulated the reduced form system (equation 1), generating 1 independent samples {C, r, I,Y, t = } for each of 41 equi-spaced values of in the interval [, 4]. t t t t 23 Figure 1 plots the sample interquartile range of the 2SLS estimate of across these 1 samples against the 14 value of 5 23 used in generating the data. FIGURE 1 2SLS MPC ESTIMATE IQR The exogenous variables {G, M, t = } Reduced were generated Form Coefficient t t only once, with [G t, M t] an independent bivariate gaussian draw in each period with E{G t} = E{M t} =, var{g t} = 1., var{m t} = 1.49, and cov{g t, M t} = 1.. The {v it, t = , i = 1,2,3,4} were generated afresh for each of the 41, simulations, with v it NIID[,.1]. The sample interquartile range is used to measure the sample dispersion in the 2SLS estimator because it is known that the population variance of this estimator does not in general exist for finite samples. 6

8 4. Conclusions For arbitrarily large samples all that really matters is identification. For such samples any consistent estimator will precisely recover the structural parameters e.g.,,,, and in the example given above In that case it doesn't matter how close to singularity the crucial matrix { 1 } is. Nor does it matter how large the dispersion of the 2SLS estimator is if this dispersion is too large, one can always make the sample size a billion times larger. In real life, however, our samples are distinctly too small. This is reflected in estimated sampling variances which are distinctly non-negligible. In this setting we see from the simulations above that the flaw in Forms I and III of the rank condition can easily lead one to unknowingly work with a specification for which, in the population, the structural coefficients are actually unidentified or nearly unidentified. In such an instance the structural coefficients are meaningless (or nearly meaningless) and the resulting structural parameter estimates have a finite (albeit perhaps quite large) estimated dispersion only because of the limited amount of sample data available. 7

9 5. References Chow, G. C. (1983), Econometrics, McGraw-Hill, New York. Davidson, R. and MacKinnon, J. G. (1993), Estimation and Inference in Econometrics, Oxford University Press, New York. Dhrymes, P. J. (1974), Econometrics, Springer-Verlag, New York. Dhrymes, P. J. (1978), Introductory Econometrics, Springer-Verlag, New York. Dutta, M (1975), Econometric Methods, South-Western Publishers, Cincinnati. Farebrother R. W. (1971), "A Short Proof of the Basic Lemma of the Linear Identification Problem," International Economic Review 12(3), Fisher, F. M. (1966), The Identification Problem in Econometrics, McGraw-Hill, New York. Greene, W. H. (1993), Econometric Analysis, Macmillan, New York. Johnston, J. (1984), Econometric Methods, McGraw-Hill, New York. Judge, G. G., Griffiths, W. E., Hill, R. C., Lütkepohl, H., and Lee, T. (1985), The Theory and Practice of Econometrics, Wiley, New York. Kmenta, J. (1971, 1986), Elements of Econometrics, Macmillan, New York. Lardaro, L. (1993), Applied Econometrics, Harper-Collins, New York. Maddala, G. S. (1992), Introduction to Econometrics, Macmillan, New York. Pindych, R. S. and Rubinfeld, D. L. (1981), Econometric Models and Economic Forecasts, McGraw-Hill, New York. Spanos, A. (1981), Statistical Foundations of Econometric Modelling, Cambridge University Press, Cambridge. Theil, H. (1971), Principles of Econometrics, Wiley, New York. Wonnacott, T. H. and Wonnacott, R. J. (1989), Econometrics, Wiley, New York. 8

10 Appendix Derivation of Form I and Form II of the Rank Condition For the ith observation on a G dimensional simultaneous equation system with K predetermined variables, the G structural equations can be represented as: By i x i u i (8) where B is a GxG matrix of structural coefficients on the endogenous variables, is a GxK matrix of structural coefficients on the predetermined variables, y is a G dimensional column vector containing observation i on each i of the G endogenous variables, x is a K dimensional column vector containing observation i on each of the K i predetermined variables, and u is a G dimensional column vector containing observation i on the random error i term in each equation. Similarly, the G reduced form equations can be represented as: y i x i i (9) where is the GxK matrix of reduced form coefficients. Thus, the structural coefficients (B and ) are related to the reduced form coefficients ( ) via the matrix equation: B (1) Following Kmenta(1971, 1986), it is most convenient to number the G endogenous variables in such a way that the structural equation whose identification status is at issue is the equation determining the first endogenous variable (equation number one) and in such a way that all of the G endogenous variables which actually appear in the first structural equation (as either the dependent variable or as explanatory variables) have smaller indices than any of the G = G - G endogenous variables which do not appear in the first equation. It is also convenient to number the K predetermined variables in such a way that all of the K predetermined variables which actually appear as explanatory variables in the first structural equation have smaller indices than any of the K = K - K predetermined variables which do not appear in the first equation. In that case the G + K non-zero coefficients 9

11 in the first structural equation are related to the reduced form coefficients solely via the first row of equation 1, which can be written: t, t t, t (11) where is a G dimensional column vector containing the (non-zero) coefficients on the endogenous variables appearing in the first structural equation. The first component of is one since the first structural equation is, by construction, the equation for the first endogenous variable. The vector consists of G zeroes. Similarly, is a K dimensional column vector containing the (non-zero) coefficients on the predetermined vaiables appearing in the first structural equation and the vector consists of K zeroes. The matrix is partitioned conformably; that is, is G x K, is G x K, and so forth. Multiplying equation 11 out and transposing yields: t t (12) Looking at the first part of equation 12, can be easily obtained as a function of subcomponents of the matrix if and only if the second part of equation 12 can be solved for. Let the K dimensional column vector -b denote the first column of [ ] t ; this vector contains the coefficients in the first reduced form equation on the K predetermined variables excluded from the first structural equation. And let the 1 t t K x G - 1 dimensional matrix [ ] denote the remaining G - 1 columns of [ ] ; the jth column of this matrix contains the coefficients in the (j+1)st reduced form equation on the K predetermined variables excluded from the first structural equation. the second part of equation 12 becomes: t b, 1 t 1 (13) 1

12 where is a G - 1 dimensional column vector containing components 2 G (the unknown components) of so that 1 t b (14) Thus, the structural coefficients ( and ) are identified (expressible in terms of components of the reduced form coefficient matrix, ) if and only if the K x G -1 dimensional matrix { 1 t } is of full column rank. (The structural coefficients are over-identified if and only if K > G -1 so that K - (G - 1) of the K rows of equation 14 are redundant, leading to a multiplicity of ways to express as a function of components of ; otherwise 1 t 1 t -1 { } is square and of full rank, leading to a unique solution ({ } ) b for and hence to exact identification.} Thus, a correct necessary and sufficient condition for the identification of the structural coefficients ( 1 t and ) is that the G - 1 columns of the matrix { } or, equivalently, the last G - 1 rows of the G x K matrix, are linearly independent. {The jth of these G - 1 rows contains the K reduced form coefficients on the predetermined variables which were excluded from the structural equation in question, from the reduced form equation for the jth of the G - 1 endogenous variables which do appear as explanatory variables in the structural equation in question.} This is the form of the rank conditon denoted "Form II" in Table 1. In contrast, Form I of the rank condition is r G 1 (15) which is thus sufficient for identification if and only if the first row of the G x K matrix is not linearly independent of the remaining G - 1 rows of the matrix. If the first row of is linearly independent of the rest, then b is linearly independent of the columns of { 1 t } so there is no linear combination of the columns of 1 t { } which equals b. Hence, in this case there is no vector which solves equation 13, so it is not possible to express the unknown coefficients in this structural equation ( and ) in terms of the elements of the matrix of reduced form coefficients, this structural equation is under-identified even though the usual rank condition is satisfied. 11

13 The problem is that the reduced form equations are such that any (structural) equation for the first endogenous variable which includes only endogenous variables number 2 G as explanatory variables and excludes this particular set of K predetermined variables forces endogenous variable number 1 to enter this structural equation with coefficient zero which is, of course, inconsistent with this being the equation determining the first endogenous variable. And since the exclusion restrictions on this structural equation are (presumably) predicated on the assumption that this is the equation for endogenous variable one, we are not free to re-number the endogenous variables so that this equation is, in effect, the equation for some other endogenous variable. Obviously, the data (embodied in the consistently-estimatable reduced form equations) is telling us in the clearest possible terms that this structural equation is mis-specified. 12