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Simultaneous equations model
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{{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> == Structural and reduced form == Suppose there are ''m'' regression equations of the form : <math> y_{it} = y_{-i,t}'\gamma_i + x_{it}'\;\!\beta_i + u_{it}, \quad i=1,\ldots,m, </math> where ''i'' is the equation number, and {{nowrap|''t'' {{=}} 1, ..., ''T''}} is the observation index. In these equations ''x<sub>it</sub>'' is the ''k<sub>i</sub>×''1 vector of exogenous variables, ''y<sub>it</sub>'' is the dependent variable, ''y<sub>−i,t</sub>'' is the ''n<sub>i</sub>×''1 vector of all other endogenous variables which enter the ''i''<sup>th</sup> equation on the right-hand side, and ''u<sub>it</sub>'' are the error terms. The “−''i''” notation indicates that the vector ''y<sub>−i,t</sub>'' may contain any of the ''y''’s except for ''y<sub>it</sub>'' (since it is already present on the left-hand side). The regression coefficients ''β<sub>i</sub>'' and ''γ<sub>i</sub>'' are of dimensions ''k<sub>i</sub>×''1 and ''n<sub>i</sub>×''1 correspondingly. Vertically stacking the ''T'' observations corresponding to the ''i''<sup>th</sup> equation, we can write each equation in vector form as : <math> y_i = Y_{-i}\gamma_i + X_i\beta_i + u_i, \quad i=1,\ldots,m, </math> where ''y<sub>i</sub>'' and ''u<sub>i</sub>'' are ''T×''1 vectors, ''X<sub>i</sub>'' is a ''T×k<sub>i</sub>'' matrix of exogenous regressors, and ''Y<sub>−i</sub>'' is a ''T×n<sub>i</sub>'' matrix of endogenous regressors on the right-hand side of the ''i''<sup>th</sup> equation. Finally, we can move all endogenous variables to the left-hand side and write the ''m'' equations jointly in vector form as : <math> Y\Gamma = X\Beta + U.\, </math> This representation is known as the '''structural form'''. In this equation {{nowrap|''Y'' {{=}} [''y''<sub>1</sub> ''y''<sub>2</sub> ... ''y<sub>m</sub>'']}} is the ''T×m'' matrix of dependent variables. Each of the matrices ''Y<sub>−i</sub>'' is in fact an ''n<sub>i</sub>''-columned submatrix of this ''Y''. The ''m×m'' matrix Γ, which describes the relation between the dependent variables, has a complicated structure. It has ones on the diagonal, and all other elements of each column ''i'' are either the components of the vector ''−γ<sub>i</sub>'' or zeros, depending on which columns of ''Y'' were included in the matrix ''Y<sub>−i</sub>''. The ''T×k'' matrix ''X'' contains all exogenous regressors from all equations, but without repetitions (that is, matrix ''X'' should be of full rank). Thus, each ''X<sub>i</sub>'' is a ''k<sub>i</sub>''-columned submatrix of ''X''. Matrix Β has size ''k×m'', and each of its columns consists of the components of vectors ''β<sub>i</sub>'' and zeros, depending on which of the regressors from ''X'' were included or excluded from ''X<sub>i</sub>''. Finally, {{nowrap|''U'' {{=}} [''u''<sub>1</sub> ''u''<sub>2</sub> ... ''u<sub>m</sub>'']}} is a ''T×m'' matrix of the error terms. Postmultiplying the structural equation by {{nowrap|Γ<sup> −1</sup>}}, the system can be written in the '''[[reduced form]]''' as : <math> Y = X\Beta\Gamma^{-1} + U\Gamma^{-1} = X\Pi + V.\, </math> This is already a simple [[general linear model]], and it can be estimated for example by [[ordinary least squares]]. Unfortunately, the task of decomposing the estimated matrix <math style="vertical-align:0">\scriptstyle\hat\Pi</math> into the individual factors Β and {{nowrap|Γ<sup> −1</sup>}} is quite complicated, and therefore the reduced form is more suitable for prediction but not inference. === Assumptions === Firstly, the rank of the matrix ''X'' of exogenous regressors must be equal to ''k'', both in finite samples and in the limit as {{nowrap|''T'' → ∞}} (this later requirement means that in the limit the expression <math style="vertical-align:-.4em">\scriptstyle \frac1TX'\!X</math> should converge to a nondegenerate ''k×k'' matrix). Matrix Γ is also assumed to be non-degenerate. Secondly, error terms are assumed to be serially [[independent and identically distributed]]. That is, if the ''t''<sup>th</sup> row of matrix ''U'' is denoted by ''u''<sub>(''t'')</sub>, then the sequence of vectors {''u''<sub>(''t'')</sub>} should be iid, with zero mean and some covariance matrix Σ (which is unknown). In particular, this implies that {{nowrap|E[''U''] {{=}} 0}}, and {{nowrap|E[''U′U''] {{=}} ''T'' Σ}}. Lastly, assumptions are required for identification. ==Identification== The [[identification condition]]s require that the [[system of linear equations]] be solvable for the unknown parameters. More specifically, the ''order condition'', a necessary condition for identification, is that for each equation {{math|''k<sub>i</sub> + n<sub>i</sub> ≤ k''}}, which can be phrased as “the number of excluded exogenous variables is greater or equal to the number of included endogenous variables”. The ''rank condition'', a stronger condition which is necessary and sufficient, is that the [[rank (linear algebra)|rank]] of {{math|Π<sub>''i''0</sub>}} equals {{math|''n<sub>i</sub>''}}, where {{math|Π<sub>''i''0</sub>}} is a {{math|(''k − k<sub>i</sub>'')×''n<sub>i</sub>''}} matrix which is obtained from {{math|''Π''}} by crossing out those columns which correspond to the excluded endogenous variables, and those rows which correspond to the included exogenous variables. ===Using cross-equation restrictions to achieve identification=== In simultaneous equations models, the most common method to achieve [[Parameter identification problem|identification]] is by imposing within-equation parameter restrictions.<ref name= "Woolridge">Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.</ref> Yet, identification is also possible using cross equation restrictions. To illustrate how cross equation restrictions can be used for identification, consider the following example from Wooldridge<ref name= "Woolridge" /> :<math>\begin{align} y_1 &= \gamma_{12} y_2 + \delta_{11} z_1 + \delta_{12} z_2 + \delta_{13} z_3 + u_1 \\ y_2 &= \gamma_{21} y_1 + \delta_{21} z_1 + \delta_{22} z_2 + u_2 \end{align}</math> where z's are uncorrelated with u's and y's are [[endogenous variable|endogenous]] variables. Without further restrictions, the first equation is not identified because there is no excluded exogenous variable. The second equation is just identified if {{math|''δ''<sub>13</sub>≠0}}, which is assumed to be true for the rest of discussion. Now we impose the cross equation restriction of {{math|''δ''<sub>12</sub>{{=}}''δ''<sub>22</sub>}}. Since the second equation is identified, we can treat {{math|''δ''<sub>12</sub>}} as known for the purpose of identification. Then, the first equation becomes: :<math>y_1 - \delta_{12} z_2 = \gamma_{12} y_2 + \delta_{11} z_1 + \delta_{13} z_3 + u_1</math> Then, we can use {{math|(''z''<sub>1</sub>, ''z''<sub>2</sub>, ''z''<sub>3</sub>)}} as [[instrumental variable|instruments]] to estimate the coefficients in the above equation since there are one endogenous variable ({{math|''y''<sub>2</sub>}}) and one excluded exogenous variable ({{math|''z''<sub>2</sub>}}) on the right hand side. Therefore, cross equation restrictions in place of within-equation restrictions can achieve identification. == Estimation == === Two-stage least squares (2SLS) === The simplest and the most common estimation method for the simultaneous equations model is the so-called [[two-stage least squares]] method,<ref name="Greene 2003 loc=p. 399">{{cite book | last = Greene | first = William H. | title = Econometric analysis | publisher = Prentice Hall | year = 2002 | edition = 5th | isbn = 0-13-066189-9 | pages = 398–99 }}</ref> developed independently by {{harvtxt|Theil|1953}} and {{harvtxt|Basmann|1957}}.<ref>{{cite report|first=H.|last=Theil|title=Estimation and Simultaneous Correlation in Complete Equation Systems|type=Memorandum|publisher=Central Planning Bureau|year=1953}} Reprinted in ''Henri Theil’s Contributions to Economics and Econometrics'' (Springer, 1992), {{doi|10.1007/978-94-011-2546-8_6}}.</ref><ref>{{cite journal | last = Basmann | first = R. L. |author-link=Robert Basmann | title = A generalized classical method of linear estimation of coefficients in a structural equation | year = 1957 | journal = [[Econometrica]] | volume = 25 | issue = 1 | pages = 77–83 | jstor = 1907743 | doi=10.2307/1907743 }}</ref><ref>{{cite book | last = Theil | first = Henri |author-link=Henri Theil | title = Principles of Econometrics | url = https://archive.org/details/principlesofecon0000thei | url-access = registration | year = 1971 | publisher = John Wiley | location = New York | isbn = 978-0-471-85845-4 }}</ref> It is an equation-by-equation technique, where the endogenous regressors on the right-hand side of each equation are being instrumented with the regressors ''X'' from all other equations. The method is called “two-stage” because it conducts estimation in two steps:<ref name="Greene 2003 loc=p. 399" /> : ''Step 1'': Regress ''Y<sub>−i</sub>'' on ''X'' and obtain the predicted values <math style="vertical-align:-.2em">\scriptstyle\hat{Y}_{\!-i}</math>; : ''Step 2'': Estimate ''γ<sub>i</sub>'', ''β<sub>i</sub>'' by the [[ordinary least squares]] regression of ''y<sub>i</sub>'' on <math style="vertical-align:-.2em">\scriptstyle\hat{Y}_{\!-i}</math> and ''X<sub>i</sub>''. If the ''i''<sup>th</sup> equation in the model is written as : <math> y_i = \begin{pmatrix}Y_{-i} & X_i\end{pmatrix}\begin{pmatrix}\gamma_i\\\beta_i\end{pmatrix} + u_i \equiv Z_i \delta_i + u_i, </math> where ''Z<sub>i</sub>'' is a ''T×''(''n<sub>i</sub> + k<sub>i</sub>'') matrix of both endogenous and exogenous regressors in the ''i''<sup>th</sup> equation, and ''δ<sub>i</sub>'' is an (''n<sub>i</sub> + k<sub>i</sub>'')-dimensional vector of regression coefficients, then the 2SLS estimator of ''δ<sub>i</sub>'' will be given by<ref name="Greene 2003 loc=p. 399"/> : <math> \hat\delta_i = \big(\hat{Z}'_i\hat{Z}_i\big)^{-1}\hat{Z}'_i y_i = \big( Z'_iPZ_i \big)^{-1} Z'_iPy_i, </math> where {{nowrap|''P'' {{=}} ''X'' (''X'' ′''X'')<sup>−1</sup>''X'' ′}} is the projection matrix onto the linear space spanned by the exogenous regressors ''X''. === Indirect least squares === Indirect least squares is an approach in [[econometrics]] where the [[coefficient]]s in a simultaneous equations model are estimated from the [[reduced form]] model using [[ordinary least squares]].<ref>Park, S-B. (1974) "On Indirect Least Squares Estimation of a Simultaneous Equation System", ''The Canadian Journal of Statistics / La Revue Canadienne de Statistique'', 2 (1), 75–82 {{JSTOR|3314964}}</ref><ref>{{cite journal | last1 = Vajda | first1 = S. | last2 = Valko | first2 = P. | last3 = Godfrey | first3 = K.R. | year = 1987 | title = Direct and indirect least squares methods in continuous-time parameter estimation | journal = Automatica | volume = 23 | issue = 6| pages = 707–718 | doi = 10.1016/0005-1098(87)90027-6 }}</ref> For this, the structural system of equations is transformed into the reduced form first. Once the coefficients are estimated the model is put back into the structural form. === Limited information maximum likelihood (LIML) === The “limited information” maximum likelihood method was suggested by [[Meyer Abraham Girshick|M. A. Girshick]] in 1947,<ref>First application by {{cite journal |first1=M. A. |last1=Girshick |first2=Trygve |last2=Haavelmo |title=Statistical Analysis of the Demand for Food: Examples of Simultaneous Estimation of Structural Equations |journal=[[Econometrica]] |volume=15 |issue=2 |year=1947 |pages=79–110 |doi= 10.2307/1907066|jstor=1907066 }}</ref> and formalized by [[Theodore Wilbur Anderson|T. W. Anderson]] and [[Herman Rubin|H. Rubin]] in 1949.<ref>{{cite journal | last1 = Anderson | first1 = T.W. | last2 = Rubin | first2 = H. | title = Estimator of the parameters of a single equation in a complete system of stochastic equations | year = 1949 | journal = [[Annals of Mathematical Statistics]] | volume = 20 | issue = 1 | pages = 46–63 | jstor = 2236803 | doi=10.1214/aoms/1177730090 | doi-access = free }}</ref> It is used when one is interested in estimating a single structural equation at a time (hence its name of limited information), say for observation i: : <math> y_i = Y_{-i}\gamma_i +X_i\beta_i+ u_i \equiv Z_i \delta_i + u_i </math> The structural equations for the remaining endogenous variables Y<sub>−i</sub> are not specified, and they are given in their reduced form: : <math> Y_{-i} = X \Pi + U_{-i} </math> Notation in this context is different than for the simple [[Instrumental variable|IV]] case. One has: * <math>Y_{-i}</math>: The endogenous variable(s). * <math>X_{-i}</math>: The exogenous variable(s) * <math>X</math>: The instrument(s) (often denoted <math>Z</math>) The explicit formula for the LIML is:<ref>{{cite book | last = Amemiya | first = Takeshi | title = Advanced Econometrics | year = 1985 | publisher = Harvard University Press | location = Cambridge, Massachusetts | isbn = 0-674-00560-0 | page = [https://archive.org/details/advancedeconomet00amem/page/235 235] | url-access = registration | url = https://archive.org/details/advancedeconomet00amem }}</ref> : <math> \hat\delta_i = \Big(Z'_i(I-\lambda M)Z_i\Big)^{\!-1}Z'_i(I-\lambda M)y_i, </math> where {{nowrap|''M'' {{=}} ''I − X'' (''X'' ′''X'')<sup>−1</sup>''X'' ′}}, and ''λ'' is the smallest characteristic root of the matrix: : <math> \Big(\begin{bmatrix}y_i\\Y_{-i}\end{bmatrix} M_i \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big) \Big(\begin{bmatrix}y_i\\Y_{-i}\end{bmatrix} M \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big)^{\!-1} </math> where, in a similar way, {{nowrap|''M<sub>i</sub>'' {{=}} ''I − X<sub>i</sub>'' (''X<sub>i</sub>''′''X<sub>i</sub>'')<sup>−1</sup>''X<sub>i</sub>''′}}. In other words, ''λ'' is the smallest solution of the [[Generalized eigenvalue problem#Generalized eigenvalue problem|generalized eigenvalue problem]], see {{harvtxt|Theil|1971|loc=p. 503}}: : <math> \Big|\begin{bmatrix}y_i&Y_{-i}\end{bmatrix}' M_i \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} -\lambda \begin{bmatrix}y_i&Y_{-i}\end{bmatrix}' M \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big|=0 </math> ==== K class estimators ==== The LIML is a special case of the K-class estimators:<ref name="DavidsonMacKinnon649">{{cite book | last1 = Davidson | first1 = Russell | last2 = MacKinnon | first2 = James G. | title = Estimation and inference in econometrics | year = 1993 | publisher = Oxford University Press | isbn = 0-19-506011-3 | page=649 }}</ref> : <math> \hat\delta = \Big(Z'(I-\kappa M)Z\Big)^{\!-1}Z'(I-\kappa M)y, </math> with: * <math> \delta = \begin{bmatrix} \beta_i & \gamma_i\end{bmatrix} </math> * <math> Z = \begin{bmatrix} X_i & Y_{-i}\end{bmatrix} </math> Several estimators belong to this class: * κ=0: [[Ordinary least squares|OLS]] * κ=1: 2SLS. Note indeed that in this case, <math> I-\kappa M = I-M= P </math> the usual projection matrix of the 2SLS * κ=λ: LIML * κ=λ - α / (n-K): {{harvtxt|Fuller|1977}} estimator.<ref>{{cite journal | last = Fuller | first = Wayne |author-link=Wayne Fuller | title = Some Properties of a Modification of the Limited Information Estimator | year = 1977 | journal = Econometrica | volume = 45 |issue=4 | pages = 939–953 | doi=10.2307/1912683 | jstor = 1912683 }}</ref> Here K represents the number of instruments, n the sample size, and α a positive constant to specify. A value of α=1 will yield an estimator that is approximately unbiased.<ref name="DavidsonMacKinnon649" /> === Three-stage least squares (3SLS) === The three-stage least squares estimator was introduced by {{harvtxt|Zellner|Theil|1962}}.<ref>{{cite journal | last1 = Zellner | first1 = Arnold |author-link1=Arnold Zellner | last2 = Theil | first2 = Henri |author-link2=Henri Theil | title = Three-stage least squares: simultaneous estimation of simultaneous equations | year = 1962 | journal = Econometrica | volume = 30 | issue = 1 | pages = 54–78 | jstor = 1911287 | doi=10.2307/1911287 }}</ref><ref>{{cite book |first=Jan |last=Kmenta |chapter=System Methods of Estimation |title=Elements of Econometrics |location=New York |publisher=Macmillan |edition=Second |year=1986 |pages=695–701 |isbn=9780023650703 |chapter-url=https://books.google.com/books?id=Bxq7AAAAIAAJ&pg=PA695 }}</ref> It can be seen as a special case of multi-equation [[Generalized method of moments|GMM]] where the set of [[instrumental variable]]s is common to all equations.<ref>{{cite book |first=Fumio |last=Hayashi |chapter=Multiple-Equation GMM |title=Econometrics |publisher=Princeton University Press |year=2000 |pages=276–279 |isbn=1400823838 |chapter-url=https://books.google.com/books?id=QyIW8WUIyzcC&pg=PA276 }}</ref> If all regressors are in fact predetermined, then 3SLS reduces to [[seemingly unrelated regressions]] (SUR). Thus it may also be seen as a combination of [[2SLS|two-stage least squares]] (2SLS) with SUR. == Applications in social science == Across fields and disciplines simultaneous equation models are applied to various observational phenomena. These equations are applied when phenomena are assumed to be reciprocally causal. The classic example is supply and demand in [[economics]]. In other disciplines there are examples such as candidate evaluations and party identification<ref>{{Cite journal|last1=Page|first1=Benjamin I.|last2=Jones|first2=Calvin C.|date=1979-12-01|title=Reciprocal Effects of Policy Preferences, Party Loyalties and the Vote|journal=American Political Science Review|volume=73|issue=4|pages=1071–1089|doi=10.2307/1953990|issn=0003-0554|jstor=1953990|s2cid=144984371 }}</ref> or public opinion and social policy in [[political science]];<ref>{{Cite journal|last=Wlezien|first=Christopher|date=1995-01-01|title=The Public as Thermostat: Dynamics of Preferences for Spending|jstor=2111666|journal=American Journal of Political Science|volume=39|issue=4|pages=981–1000|doi=10.2307/2111666}}</ref><ref>{{Cite journal|last=Breznau|first=Nate|date=2016-07-01|title=Positive Returns and Equilibrium: Simultaneous Feedback Between Public Opinion and Social Policy|journal=Policy Studies Journal|volume=45|issue=4|language=en|pages=583–612|doi=10.1111/psj.12171|issn=1541-0072|url=http://osf.io/wt376/}}</ref> road investment and travel demand in geography;<ref>{{Cite journal|last1=Xie|first1=F.|last2=Levinson|first2=D.|date=2010-05-01|title=How streetcars shaped suburbanization: a Granger causality analysis of land use and transit in the Twin Cities|journal=Journal of Economic Geography|volume=10|issue=3|pages=453–470|doi=10.1093/jeg/lbp031|issn=1468-2702|hdl=11299/179996|hdl-access=free}}</ref> and educational attainment and parenthood entry in [[sociology]] or [[demography]].<ref>{{Cite journal|last=Marini|first=Margaret Mooney|date=1984-01-01|title=Women's Educational Attainment and the Timing of Entry into Parenthood|jstor=2095464|journal=American Sociological Review|volume=49|issue=4|pages=491–511|doi=10.2307/2095464}}</ref> The simultaneous equation model requires a theory of reciprocal causality that includes special features if the causal effects are to be estimated as simultaneous feedback as opposed to one-sided 'blocks' of an equation where a researcher is interested in the causal effect of X on Y while holding the causal effect of Y on X constant, or when the researcher knows the exact amount of time it takes for each causal effect to take place, i.e., the length of the causal lags. Instead of lagged effects, simultaneous feedback means estimating the simultaneous and perpetual impact of X and Y on each other. This requires a theory that causal effects are simultaneous in time, or so complex that they appear to behave simultaneously; a common example are the moods of roommates.<ref>{{Cite journal|last1=Wong|first1=Chi-Sum|last2=Law|first2=Kenneth S.|date=1999-01-01|title=Testing Reciprocal Relations by Nonrecursive Structuralequation Models Using Cross-Sectional Data|journal=Organizational Research Methods|language=en|volume=2|issue=1|pages=69–87|doi=10.1177/109442819921005|s2cid=122284566 |issn=1094-4281}}</ref> To estimate simultaneous feedback models a theory of equilibrium is also necessary – that X and Y are in relatively steady states or are part of a system (society, market, classroom) that is in a relatively stable state.<ref>2013. “Reverse Arrow Dynamics: Feedback Loops and Formative Measurement.” In ''Structural Equation Modeling: A Second Course'', edited by [[Gregory R. Hancock]] and Ralph O. Mueller, 2nd ed., 41–79. Charlotte, NC: Information Age Publishing</ref> == See also == * [[General linear model]] * [[Seemingly unrelated regressions]] * [[Reduced form]] * [[Parameter identification problem]] == References == {{reflist}} == Further reading == * {{cite book |first1=Dimitrios |last1=Asteriou |first2=Stephen G. |last2=Hall |title=Applied Econometrics |edition=Second |location=Basingstoke |publisher=Palgrave Macmillan |year=2011 |isbn=978-0-230-27182-1 |url={{Google books |plainurl=yes |id=6qYcBQAAQBAJ |page=395 }} |pages=395 }} * {{cite book |first=Gregory C. |last=Chow |author-link=Gregory Chow |title=Econometrics |location=New York |publisher=McGraw-Hill |year=1983 |isbn=0-07-010847-1 |pages=[https://archive.org/details/econometrics0000chow/page/117 117–121] |url-access=registration |url=https://archive.org/details/econometrics0000chow/page/117 }} * {{cite book |first1=Thomas B. |last1=Fomby |first2=R. Carter |last2=Hill |first3=Stanley R. |last3=Johnson |chapter=Simultaneous Equations Models |title=Advanced Econometric Methods |location=New York |publisher=Springer |year=1984 |isbn=0-387-90908-7 |pages=437–552 }} * {{cite book |last1=Maddala |first1=G. S. |author-link=G. S. Maddala |last2=Lahiri |first2=Kajal |title=Introduction to Econometrics |chapter=Simultaneous Equations Models |pages=355–400 |location=New York |publisher=Wiley |year=2009 |edition=Fourth |isbn=978-0-470-01512-4 }} * {{cite book |first=Paul A. |last=Ruud |chapter=Simultaneous Equations |title=An Introduction to Classical Econometric Theory |publisher=Oxford University Press |year=2000 |isbn=0-19-511164-8 |pages=697–746 }} * {{cite book |first=Denis |last=Sargan |author-link=Denis Sargan |title=Lectures on Advanced Econometric Theory |location=Oxford |publisher=Basil Blackwell |year=1988 |isbn=0-631-14956-2 |pages=68–89 }} * {{cite book |first=Jeffrey M. |last=Wooldridge |author-link=Jeffrey Wooldridge |chapter=Simultaneous Equations Models |title=Introductory Econometrics |publisher=South-Western |edition=Fifth |year=2013 |isbn=978-1-111-53104-1 |pages=554–582 }} == External links == *{{YouTube|id=D5lt9bhOshc&list=PLD15D38DC7AA3B737&index=15|title=Lecture on the Identification Problem in 2SLS, and Estimation}} by [[Mark Thoma]] [[Category:Simultaneous equation methods (econometrics)]] [[Category:Regression models]] [[Category:Mathematical and quantitative methods (economics)]]
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