Econometrics pp 268-306

# Simultaneous Equations Model

## Abstract

Economists formulate models for consumption, production, investment, money demand and money supply, labor demand and labor supply to attempt to explain the workings of the economy. These behavioral equations are estimated equation by equation or jointly as a system of equations. These are known as simultaneous equations models. Much of todays econometrics have been influenced and shaped by a group of economists and econometricians known as the Cowles Commission who worked together at the University of Chicago in the late 1940’s, see Chapter 1. Simultaneous equations models had their genesis in economics during that period. Haavelmo’s (1944) work emphasized the use of the probability approach to formulating econometric models. Koopmans and Marschak (1950) and Koopmans and Hood (1953) in two influential Cowles Commission monographs provided the appropriate statistical procedures for handling simultaneous equations models. In this chapter, we first give simple examples of simultaneous equations models and show why the least squares estimator is no longer appropriate. Next, we discuss the important problem of identification and give a simple condition that helps check whether a specific equation is identified. Sections 11.2 and 11.3 give the estimation of a single and a system of equations using instrumental variable procedures while, section 11.4 gives a necessary and sufficient condition for identification. Section 11.5 gives a test of over-identification restrictions whereas, section 11.6 gives a Hausman specification test. Section 11.7 concludes with an empirical example.

### Keywords

Covariance Income Oates

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### References

1. Anderson, T.W. and H. Rubin (1950), “The Asymptotic Properties of Estimates of the Parameters of a Single Equation in a Complete System of Stochastic Equations,” Annals of Mathematical Statistics, 21: 570–582.
2. Baltagi, B.H. (1989), “A Hausman Specification Test in a Simultaneous Equations Model,” Econometric Theory, Solution 88.3.5, 5: 453–467.Google Scholar
3. Basmann, R.L. (1957), “A Generalized Classical Method of Linear Estimation of Coefficients in a Structural Equation,” Econometrica, 25: 77–83.
4. Basmann, R.L. (1960), “On Finite Sample Distributions of Generalized Classical Linear Identifiability Tests Statistics,” Journal of the American Statistical Association, 55: 650–659.
5. Durbin, J. (1954), “Errors in Variables,” Review of the International Statistical Institute, 22: 23–32.
6. Farebrother, R.W. (1985), “The Exact Bias of Wald’s Estimator,” Econometric Theory, Problem 85.3.1, 1: 419.Google Scholar
7. Farebrother, R.W. (1991), “Comparison of t-Ratios,” Econometric Theory, Solution 90.1.4, 7: 145–146.Google Scholar
8. Fisher, F.M. (1966), The Identification Problem in Econometrics ( McGraw-Hill: New York).Google Scholar
9. Haavelmo, T. (1944), “The Probability Approach in Econometrics,” Supplement to Econometrica, 12.Google Scholar
10. Hall, A. (1993), “Some Aspects of Generalized Method of Moments Estimation,” Chapter 15, Handbook of Statistics, Volume 11, North Holland, Amsterdam.Google Scholar
11. Hansen, L. (1982), “Large Sample Properties of Generalized Method of Moments Estimators,” Econometrica, 50: 646–660.Google Scholar
12. Hausman, J.A. (1978), “Specification Tests in Econometrics,” Econometrica, 46: 1251–1272.
13. Hausman, J.A. (1983), “Specification and Estimation of Simultaneous Equation Models,” Chapter 7, Handbook of Econometrics, Vol. I, eds. Z Griliches and M.D. Intriligator, North Holland, Amsterdam.Google Scholar
14. Holly, A. (1987), “Identification and Estimation of a Simple Two-Equation Model,” Econometric Theory, Problem 87.3.3, 3: 463–466.Google Scholar
15. Holly, A. (1988), “A Hausman Specification Test in a Simultaneous Equations Model,” Econometric Theory, Problem 88.3.5, 4: 537–538.Google Scholar
16. Holly, A. (1990), “Comparison of t-ratios,” Econometric Theory, Problem 90.1.4, 6: 114.Google Scholar
17. Kapteyn, A. and D.G. Fiebig (1981), “When are Two-Stage and Three-Stage Least Squares Estimators Identical?,” Economics Letters, 8: 53–57.
18. Koopmans, T.C. and J. Marschak (1950), Statistical Inference in Dynamic Economic Models ( John Wiley and Sons: New York).Google Scholar
19. Koopmans, T.C. and W.C. Hood (1953), Studies in Econometric Method ( John Wiley and Sons: New York).Google Scholar
20. Laffer, A.B., (1970), “Trade Credit and the Money Market,” Journal of Political Economy, 78: 239–267.
21. Lott, W.F. and S.C. Ray (1992), Applied Econometrics: Problems with Data Sets ( The Dryden Press: New York).Google Scholar
22. Mariano, R.S. (1982), “Analytical Small-Sample Distribution Theory in Econometrics: The Simultaneous Equations Case,” International Economic Review, 23: 503–534.
23. Nelson, C.R. and R. Startz (1990), “The Distribution of the Instrumental Variables Estimator and its t-Ratio when the Instrument is a Poor One,” Journal of Business, 63: S 125–140.
24. Sapra, S.K. (1997), “Equivariance of an Instrumental Variable (IV) Estimator in the Linear Regression Model,” Econometric Theory, Problem 97.2.5, 13: 464.Google Scholar
25. Singh, N. and A N. Bhat (1988), “Identification and Estimation of a Simple Two-Equation Model,” Econometric Theory, Solution 87.3.3, 4: 542–545.Google Scholar
26. Theil, H. (1953), “Repeated Least Squares Applied to Complete Equation Systems,” The Hague, Central Planning Bureau (Mimeo).Google Scholar
27. Theil, H. (1971), Principles of Econometrics ( Wiley: New York).Google Scholar
28. Wooldridge, J.M. (1990), “A Note on the Lagrange Multiplier and F Statistics for Two Stage Least Squares Regression,” Economics Letters, 34: 151–155.
29. Wald, A. (1940), “Fitting of Straight Lines if Both Variables are Subject to Error,” Annals of Mathematical Statistics, 11: 284–300.
30. Wu, D.M. (1973), “Alternative Tests of Independence Between Stochastic Regressors and Disturbances,” Econometrica, 41: 733–740.
31. Zellnner, A. and Theil, H. (1962), “Three-Stage Least Squares: Simultaneous Estimation of Simultaneous Equations,” Econometrica, 30: 54–78.