Editing Simultaneous equations model (section)

{{Short description|Type of statistical model}}
'''Simultaneous equations models''' are a type of [[statistical model]] in which the [[Dependent and independent variables|dependent variable]]s are functions of other dependent variables, rather than just independent variables.<ref>{{cite book |first1=Vance |last1=Martin |first2=Stan |last2=Hurn |first3=David |last3=Harris |title=Econometric Modelling with Time Series |publisher=Cambridge University Press |year=2013 |isbn=978-0-521-19660-4 |page=159 }}</ref> This means some of the explanatory variables are [[Endogeneity (econometrics)|jointly determined]] with the dependent variable, which in [[economics]] usually is the consequence of some underlying [[Economic equilibrium|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.<ref>{{cite book |first1=G. S. |last1=Maddala |first2=Kajal |last2=Lahiri |title=Introduction to Econometrics |publisher=Wiley |edition=Fourth |year=2009 |isbn=978-0-470-01512-4 |pages=355–357 }}</ref>

Simultaneity poses challenges for the [[Point estimation|estimation]] of the statistical parameters of interest, because the [[Gauss–Markov theorem|Gauss–Markov assumption]] of [[Gauss–Markov theorem#Strict exogeneity|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 [[Computational complexity|computationally costly]] non-linear optimization problem even for the simplest [[system of linear equations]].<ref>{{cite book |first=Richard E. |last=Quandt |chapter=Computational Problems and Methods |title=Handbook of Econometrics |volume=I |editor-first=Z. |editor-last=Griliches |editor2-first=M. D. |editor2-last=Intriligator |publisher=North-Holland |year=1983 |pages=699–764 |isbn=0-444-86185-8 }}</ref> This situation prompted the development, spearheaded by the [[Cowles Commission]] in the 1940s and 1950s,<ref>{{cite journal |first=Carl F. |last=Christ |title=The Cowles Commission's Contributions to Econometrics at Chicago, 1939–1955 |journal=[[Journal of Economic Literature]] |volume=32 |issue=1 |year=1994 |pages=30–59 |jstor=2728422 }}</ref> of various techniques that estimate each equation in the model seriatim, most notably [[limited information maximum likelihood]] and [[two-stage least squares]].<ref>{{cite book |first=J. |last=Johnston |author-link=John Johnston (econometrician) |chapter=Simultaneous-equation Methods: Estimation |title=Econometric Methods |location=New York |publisher=McGraw-Hill |edition=Second |year=1971 |pages=376–423 |isbn=0-07-032679-7 }}</ref>