Simultaneous Equation Models

Simultaneous Equation Models Sandy Marquart-Pyatt Utah State University This course considers systems of equations. In contrast to single equation models, simultaneous equation models include more than one dependent variable. These models can be grouped into two major types: 1) recursive models, and 2) nonrecursive models. Recursive models do not create any special problems, while nonrecursive models require special treatment. For each of these major types of models, we will discuss the specification, identification, estimation, and assessment of these systems of simultaneous equations. Nonrecursive models introduce the problem of identification, or, how to establish that the parameters of the model are estimable. Nonrecursive models also require alternative estimation techniques. As time permits, advanced topics will be covered such as limited dependent variables, measurement error, and handling longitudinal data. As background, students should have a good understanding of the classical linear regression model and matrix algebra. Most of the readings are drawn from four econometric texts: Greene, William H. 2005. Econometric Analysis (5th ed.). Upper Saddle River, NJ: Prentice Hall. Gujarati, Damodar. 2003. Basic Econometrics (4th ed.). New York: McGraw-Hill. Johnston, J. 1984. Econometric Methods (3rd ed.). New York: McGraw-Hill. (Note: pages for Johnston, J. and J. DiNardo s Econometric Methods 4 th ed. (in parentheses)). Kmenta, Jan. 1986. Elements of Econometrics (2nd ed.). New York: Macmillan All of the readings are available in the summer program library. Topics and I. Introduction to Simultaneous Equation Models a. A brief introduction to simultaneous equation models II: Review of the Classical Linear Regression model a. Review of matrix algebra Bollen, Kenneth A. 1989. Structural Equations with Latent Variables. New York: Wiley. Appendix A. Johnston: pp.89-100, 122-138 (ed. 4: 459-483)

II: Review of the Classical Linear Regression model (cont.) b. Classical linear regression model Gujarati: chp.9 Greene: chp.6 Johnston: chp.5 (ed 4: chap 3) Note: if you feel you need to review, read Gujarati chp.2 & 6 or Johnston chp.1-2 first. III: Overview of simultaneous equation models Recursive vs. nonrecursive models; path diagrams/equations/matrices; reduced vs. structural form; direct, indirect and total effects. Gujarati: chp.18 Bollen: pp.32-34; 36-39 Kmenta: 13.1 or Greene: 16.1-16.2 IV: Recursive models a. Specification Kenny, David. 1979. Correlation and Causality. Wiley. p. 13-21 (chap 2) Gujarati: p. 764 Johnston: pp.467-468 (ed 4: 305-309) Kmenta: pp.719-720 b. Identification Bollen: p. 88-98 Kenny: p. 34-41, 61-62 Greene: examples 16.11-16.12 c. Estimation readings: Gujarati: p. 681-682 Johnston: p. 468-469 (ed. 4: 314-318) Kmenta: p. 720

IV: Recursive models (cont.) d. Decomposition of effects Bollen: pp.36-39 Fox, John. 1980. Effect Analysis in Structural Equation Models. Sociological Methods and Research 9(1): 3-14 and 19-22. Sobel, Michael. 1988. "Direct and Indirect Effects in Structural Equation Models." pp.46-53 in J. Scott Long (ed.) Common Problems/Proper Solutions: Avoiding Error in Quantitiative Research. Newbury Park, CA: Sage. Defining mediation. Baron, Reuben M., and David A. Kenny. 1986. The Moderator-Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic, and Statistical Considerations. Journal of Personality and Social Psychology 51(6): 1173-1182. Collins, Linda, John W. Graham, and Brian P. Flaherty. 1998. An Alternative Framework for Defining Mediation. Multivariate Behavioral Research 33(2): 295-312. Optional MacKinnon, David P., Chondra M. Lockwood, Jeanne M. Hoffman, Stephen G. West, and Virgil Sheets. 2002. A comparison of methods to test mediation and other intervening variable effects. Psychological Methods 7:83-104. [a Monte Carlo simulation assessing the performance of different techniques for estimating mediation] Shrout, Patrick E. and Niall Bolger. 2002. "Mediation in experimental and nonexperimental studies: New procedures and recommendations." Psychological Methods 7:422-445. [propose a bootstrapping strategy] MacKinnon, David P., Amanda Fairchild and Adam Fritz. 2007. Mediation Analysis. Annual Review of Psychology 58:593-614. V: SUR (seemingly unrelated regressions) models Greene: pp.674-688 Kmenta: 12.3 Example: Sampson, Robert J. 1987. Urban Black Violence: The Effect of Male Joblessness and Family Disruption. American Journal of Sociology 93(2): 348-382.

VI: Nonrecursive simultaneous equation models a. Specification. Gujarati: 18.3-18.4 b. Identification Gujarati: chp.19.1-19.3 Rigdon, Edward E. 1995. "A Necessary and Sufficient Identification Rule for Structural Models Estimated in Practice." Multivariate Behavioral Research 30:359-383. c. Estimation Gujarati: 20.1 c1. Indirect least squares Gujarati: 20.3 Johnston: pp. 469-472 (ed 4: 314) c2. 2SLS Gujarati: 20.4, 20.5 Greene: 16.4, 16.5.2, and 16.5.2b Kmenta: pp.681-687 Examples: Bollen, Kenneth and Robert Jackman. 1985. Political Democracy and the Size Distribution of Income. American Sociological Review 50: 438-457. Brehm, John and Wendy Rahn. 1997. "Individual-level evidence for the causes and consequences of social capital." American Journal of Political Science 41:999-1023. Erikson, Robert S. and Thomas R. Palfrey. 1998. "Campaign Spending and Incumbency: An Alternative Simultaneous Equations Approach." The Journal of Politics 60:355-373. c3. 3SLS Johnston: pp.486-490 Kmenta: pp.695-701 c4. MLE Greene: 16.6.2 Example: Frone, Michael R., Marcia Russell, and M. Lynne Cooper. 1994. Relationship Between Job and Family Satisfaction: Causal or Noncausal Covariation? Journal of Management 20(3): 565-579.

VI: Nonrecursive simultaneous equation models (cont.) d. Comparison of Methods Greene: 16.7 Kmenta: pp.711-714 e. Decomposition of Effects Bollen: pp.376-389 Fox, J. 1980. "Effect Analysis in Structural Equation Models." Sociological Methods and Research 9:3-28. Bollen, Kenneth A. 1987. "Total, Direct, and Indirect Effects in Structural Equation Models." pp. 37-69 in C.C. Clogg, ed., Sociological Methodology 1987. Washington D.C.: American Sociological Association VII. Assessment of models a. Equation by equation a1. Endogeneity tests: Gujarati: 19.4-19.5 Greene: 16.8 Hausman, J. A. 1978. "Specification Tests in Econometrics." Econometrica 6:1251-1271 a2. Assessment of Instruments Bound, John, David A. Jaeger, and Regina M. Baker. 1995. Problems with Instrumental Variables Estimation When the Correlation Between the Instruments and the Endogenous Explanatory Variable is Weak. Journal of the American Statistical Association 90(430): 443-450. Bartels, Larry M. 1991. "Instrumental and 'Quasi-Instrumental' Variables." American Journal of Political Science 35:777-800. Optional reading: Basmann, R. L. 1960. On Finite Sample Distributions of Generalized Classical Linear Identifiability Test Statistics. Journal of the American Statistical Association 55(292):650-659. b. Global goodness of fit statistics for overidentified models Bollen: pp.263-289

Additional Topics: (covered as time permits) Modeling change Finkel, Steven E. 1995. Causal Analysis with Panel Data. Sage. Optional Kessler, Ronald C. and David F. Greenberg. 1981. Linear panel analysis: Models of quantitative change. New York: Academic. [Classic textbook on longitudinal models] Consequences of measurement error Bollen: chp.5, Greene: 9.5 Simultaneous equations with limited dependent variables Winship, Christopher and Robert D. Mare. 1983. "Structural Equations and Path Analysis for Discrete Data." American Journal of Sociology 89:54-110. Muthen, Bengt. 1984. A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika 49:115-132. Bollen: 433-446. Example: Bollen, Kenneth A., David K. Guilkey, and Thomas A. Mroz. 1995. "Binary Outcomes and Endogenous Explanatory Variables: Tests and Solutions with an Application to the Demand for Contraceptive Use in Tunisia." Demography 32:111-131. Optional readings: Maddala 5.1, 5.8, chapter 7 and chapter 8. MacKinnon, David P. and James H. Dwyer. 1993. "Estimating mediated effects in prevention studies." Evaluation Review 17:144-158. [Discusses estimating mediation with a dichotomous mediator] Englin, Jeffrey, Peter Boxall, and David Watson. 1998. "Modeling recreation demand in a poisson system of equations: An analysis of the impact of international exchange rates." American Journal of Agricultural Economics 80:255-263. [Simultaneous equations with count outcomes] Terza, Joseph V. 1995. "Estimating count data models with endogenous switching: Sample selection and endogenous treatment effects." Journal of Econometrics 84:129-154. [Simultaneous equations with count outcomes]

Power Issues in Simultaneous Equations Bielby, William T., and Ross L. Matsueda. 1991. Statistical Power in Nonrecursive Linear Models. Sociological Methodology 21:167-197. Lagged Endogenous Variables with autocorrelation Kmenta: 13.5 Fair, Ray C. 1970. The Estimation of Simultaneous Equation Models with Lagged Endogenous Variables and First Order Serially Correlated Errors. Econometrica 38(3): 507-516. Standard Errors of indirect effects Sobel, Michael E. 1982. Asymptotic Confidence Intervals for Indirect Effects in Structural Equation Models. Sociological Methodology 13:290-312. Sobel, Michael E. 1986. Some New Results on Indirect Effects and Their Standard Errors in Covariance Structure Models. Sociological Methodology 16:159-186. Sobel, Michael E. 1988. Direct and Indirect Effects in Linear Structural Equation Models. Pp. 53-64 in Common Problems/Proper Solutions: Avoiding Error in Quantitative Research, edited by J. Scott Long. Sage. Endogeneity tests for models with dichotomous dependent variables Rivers, D. and Q. Vuong. 1988. "Limited Information Estimators and Exogeneity Tests for Simultaneous Probit Models." Journal of Econometrics 39:347-66. Using simultaneous equations to handle spatial effects Land, Kenneth C., and Glenn Deane. 1992. On the Large-Sample Estimation of Regression Models with Spatial- Or Network-Effects Terms: A Two-Stage Least Squares Approach. Sociological Methodology 22:221-248. Autocorrelation or heteroskedasticity in simultaneous equations Kmenta: 13.5 Harvey, A. C., and G. D. A. Phillips. 1980. Testing for Serial Correlation in Simultaneous Equation Models. Econometrica 48(3): 747-760.

Application Papers: Recursive: Presentation on Monday, July 30. Wickrama, K.A.S., et al. 1997. Linking Occupational Conditions to Physical Health through Marital, Social, and Intrapersonal Processes Journal of Health and Social Behavior 38(4): 363-375. SUR: Presentation on Thursday, Aug 2. Sampson, Robert J. 1987. Urban Black Violence: The Effect of Male Joblessness and Family Disruption. American Journal of Sociology 93(2): 348-382. Nonrecursive: Presentation on Monday, Aug 6. Bollen, Kenneth A., and Robert W. Jackman. 1985. Political Democracy and the Size Distribution of Income. American Sociological Review 50: 438-457. Nonrecursive: Presentation on Thursday, Aug 9. Brehm, John and Wendy Rahn. 1997. "Individual-level evidence for the causes and consequences of social capital." American Journal of Political Science 41:999-1023. Categorical Variables: Presentation on Wednesday, Aug 15. Bollen, Kenneth A., David K. Guilkey, and Thomas A. Mroz. 1995. "Binary Outcomes and Endogenous Explanatory Variables: Tests and Solutions with an Application to the Demand for Contraceptive Use in Tunisia." Demography 32:111-131. Issues to consider related to the application papers 1. Provide a very brief description of what the paper is trying to do, highlighting the theoretical model and how it is implemented statistically. 2. Evaluate how well the first issue is accomplished. 3. What complexities are encountered, and how are these resolved? 4. In what ways do these intersect with issues we ve discussed in class? 5. Evaluate the paper overall. Discuss why you have given this assessment.