Simultaneous equations model

Simultaneous equations models are a type of statistical model in which the dependent variables are functions of other dependent variables, rather than just independent variables.^[1] This means some of the explanatory variables are jointly determined with the dependent variable, which in economics usually is the consequence of some underlying equilibrium mechanism. Take the typical supply and demand model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand and then set the price.^[2]

Simultaneity poses challenges for the estimation of the statistical parameters of interest, because the Gauss–Markov assumption of strict exogeneity of the regressors is violated. And while it would be natural to estimate all simultaneous equations at once, this often leads to a computationally costly non-linear optimization problem even for the simplest system of linear equations.^[3] This situation prompted the development, spearheaded by the Cowles Commission in the 1940s and 1950s,^[4] of various techniques that estimate each equation in the model seriatim, most notably limited information maximum likelihood and two-stage least squares.^[5]

^ Martin, Vance; Hurn, Stan; Harris, David (2013). Econometric Modelling with Time Series. Cambridge University Press. p. 159. ISBN 978-0-521-19660-4.
^ Maddala, G. S.; Lahiri, Kajal (2009). Introduction to Econometrics (Fourth ed.). Wiley. pp. 355–357. ISBN 978-0-470-01512-4.
^ Quandt, Richard E. (1983). "Computational Problems and Methods". In Griliches, Z.; Intriligator, M. D. (eds.). Handbook of Econometrics. Vol. I. North-Holland. pp. 699–764. ISBN 0-444-86185-8.
^ Christ, Carl F. (1994). "The Cowles Commission's Contributions to Econometrics at Chicago, 1939–1955". Journal of Economic Literature. 32 (1): 30–59. JSTOR 2728422.
^ Johnston, J. (1971). "Simultaneous-equation Methods: Estimation". Econometric Methods (Second ed.). New York: McGraw-Hill. pp. 376–423. ISBN 0-07-032679-7.

[1] Martin, Vance; Hurn, Stan; Harris, David (2013). Econometric Modelling with Time Series. Cambridge University Press. p. 159. ISBN 978-0-521-19660-4.

[2] Maddala, G. S.; Lahiri, Kajal (2009). Introduction to Econometrics (Fourth ed.). Wiley. pp. 355–357. ISBN 978-0-470-01512-4.

[3] Quandt, Richard E. (1983). "Computational Problems and Methods". In Griliches, Z.; Intriligator, M. D. (eds.). Handbook of Econometrics. Vol. I. North-Holland. pp. 699–764. ISBN 0-444-86185-8.

[4] Christ, Carl F. (1994). "The Cowles Commission's Contributions to Econometrics at Chicago, 1939–1955". Journal of Economic Literature. 32 (1): 30–59. JSTOR 2728422.

[5] Johnston, J. (1971). "Simultaneous-equation Methods: Estimation". Econometric Methods (Second ed.). New York: McGraw-Hill. pp. 376–423. ISBN 0-07-032679-7.

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