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atrix of endogenous regressors on the right hand side of the i th equation finally we can move all endogenous variables to the left hand side and write the m equations jointly in vector form as y γ x b u displaystyle y gamma x mathrm b u this representation is known as the structural form in this equation y y 1 y 2 y m is the t m matrix of dependent variables each of the matrices y i is in fact an n i 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 i or zeros depending on which columns of y were included in the matrix y i 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 i is a k i columned submatrix of x matrix β has size k m and each of its columns consists of the components of vectors β i and zeros depending on which of the regressors from x were included or excluded from x i finally u u 1 u 2 u m is a t m matrix of the error terms postmultiplying the structural equation by γ 1 the system can be written in the reduced form as y x b γ 1 u γ 1 x π v displaystyle y x mathrm b gamma 1 u gamma 1 x pi v 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 π displaystyle scriptstyle hat pi into the individual factors β and γ 1 is quite complicated and therefore the reduced form is more suitable for prediction but not inference assumptions edit firstly the rank of the matrix x of exogenous regressors must be equal to k both in finite samples and in the limit as t this later requirement means that in the limit the expression 1 t x x displaystyle scriptstyle frac 1 t x x 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 th row of matrix u is denoted by u t then the sequence of vectors u t should be iid with zero mean and some covariance matrix σ which is unknown in particular this implies that e u 0 and e u u t σ lastly assumptions are required for identification identification edit the identification conditions 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 k i n i 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 of π i 0 equals n i where π i 0 is a k k i n i matrix which is obtained from 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 edit in simultaneous equations models the most common method to achieve identification is by imposing within equation parameter restrictions 6 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 6 y 1 γ 12 y 2 δ 11 z 1 δ 12 z 2 δ 13 z 3 u 1 y 2 γ 21 y 1 δ 21 z 1 δ 22 z 2 u 2 displaystyle begin aligned 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 aligned where z s are uncorrelated with u s and y s are 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 δ 13 0 which is assumed to be true for the rest of discussion now we impose the cross equation restriction of δ 12 δ 22 since the second equation is identified we can treat δ 12 as known for the purpose of identification then the first equation becomes y 1 δ 12 z 2 γ 12 y 2 δ 11 z 1 δ 13 z 3 u 1 displaystyle y_ 1 delta _ 12 z_ 2 gamma _ 12 y_ 2 delta _ 11 z_ 1 delta _ 13 z_ 3 u_ 1 then we can use z 1 z 2 z 3 as instruments to estimate the coefficients in the above equation since there are one endogenous variable y 2 and one excluded exogenous variable z 2 on the right hand side therefore cross equation restrictions in place of within equation restrictions can achieve identification estimation edit two stage least squares 2sls edit the simplest and the most common estimation method for the simultaneous equations model is the so called two stage least squares method 7 developed independently by theil 1953 and basmann 1957 8 9 10 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 7 step 1 regress y i on x and obtain the predicted values y i displaystyle scriptstyle hat y _ i step 2 estimate γ i β i by the ordinary least squares regression of y i on y i displaystyle scriptstyle hat y _ i and x i if the i th equation in the model is written as y i y i x i γ i β i u i z i δ i u i displaystyle 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 where z i is a t n i k i matrix of both endogenous and exogenous regressors in the i th equation and δ i is an n i k i dimensional vector of regression coefficients then the 2sls estimator of δ i will be given by 7 δ i z i z i 1 z i y i z i p z i 1 z i p y i displaystyle hat delta _ i big hat z _ i hat z _ i big 1 hat z _ i y_ i big z _ i pz_ i big 1 z _ i py_ i where p x x x 1 x is the projection matrix onto the linear space spanned by the exogenous regressors x indirect least squares edit indirect least squares is an approach in econometrics where the coefficients in a simultaneous equations model are estimated from the reduced form model using ordinary least squares 11 12 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 edit the limited information maximum likelihood method was suggested by m a girshick in 1947 13 and formalized by t w anderson and h rubin in 1949 14 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 y i y i γ i x i β i u i z i δ i u i displaystyle y_ i y_ i gamma _ i x_ i beta _ i u_ i equiv z_ i delta _ i u_ i the structural equations for the remaining endogenous variables y i are not specified and they are given in their reduced form y i x π u i displaystyle y_ i x pi u_ i notation in this context is different than for the simple iv case one has y i displaystyle y_ i the endogenous variable s x i displaystyle x_ i the exogenous variable s x displaystyle x the instrument s often denoted z displaystyle z the explicit formula for the liml is 15 δ i z i i λ m z i 1 z i i λ m y i displaystyle hat delta _ i big z _ i i lambda m z_ i big 1 z _ i i lambda m y_ i where m i x x x 1 x and λ is the smallest characteristic root of the matrix y i y i m i y i y i y i y i m y i y i 1 displaystyle 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 where in a similar way m i i x i x i x i 1 x i in other words λ is the smallest solution of the generalized eigenvalue problem see theil 1971 p 503 y i y i m i y i y i λ y i y i m y i y i 0 displaystyle 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 k class estimators edit the liml is a special case of the k class estimators 16 δ z i κ m z 1 z i κ m y displaystyle hat delta big z i kappa m z big 1 z i kappa m y with δ β i γ i displaystyle delta begin bmatrix beta _ i gamma _ i end bmatrix z x i y i displaystyle z begin bmatrix x_ i y_ i end bmatrix several estimators belong to this class κ 0 ols κ 1 2sls note indeed that in this case i κ m i m p displaystyle i kappa m i m p the usual projection matrix of the 2sls κ λ liml κ λ α n k fuller 1977 estimator 17 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 16 three stage least squares 3sls edit the three stage least squares estimator was introduced by zellner theil 1962 18 19 it can be seen as a special case of multi equation gmm where the set of instrumental variables is common to all equations 20 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 two stage least squares 2sls with sur applications in social science edit 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 21 or public opinion and social policy in political science 22 23 road investment and travel demand in geography 24 and educational attainment and parenthood entry in sociology or demography 25 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 26 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 27 see also edit general linear model seemingly unrelated regressions reduced form parameter identification problem references edit 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 2 wooldridge j m econometric analysis of cross section and panel data mit press cambridge mass 1 2 3 greene william h 2002 econometric analysis 5th ed prentice hall pp 398 99 isbn 0 13 066189 9 theil h 1953 estimation and simultaneous correlation in complete equation systems memorandum central planning bureau reprinted in henri theil s contributions to economics and econometrics springer 1992 doi 10 1007 978 94 011 2546 8_6 basmann r l 1957 a generalized classical method of linear estimation of coefficients in a structural equation econometrica 25 1 77 83 doi 10 2307 1907743 jstor 1907743 theil henri 1971 principles of econometrics new york john wiley isbn 978 0 471 85845 4 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 vajda s valko p godfrey k r 1987 direct and indirect least squares methods in continuous time parameter estimation automatica 23 6 707 718 doi 10 1016 0005 1098 87 90027 6 first application by girshick m a haavelmo trygve 1947 statistical analysis of the demand for food examples of simultaneous estimation of structural equations econometrica 15 2 79 110 doi 10 2307 1907066 jstor 1907066 anderson t w rubin h 1949 estimator of the parameters of a single equation in a complete system of stochastic equations annals of mathematical statistics 20 1 46 63 doi 10 1214 aoms 1177730090 jstor 2236803 amemiya takeshi 1985 advanced econometrics cambridge massachusetts harvard university press p 235 isbn 0 674 00560 0 1 2 davidson russell mackinnon james g 1993 estimation and inference in econometrics oxford university press p 649 isbn 0 19 506011 3 fuller wayne 1977 some properties of a modification of the limited information estimator econometrica 45 4 939 953 doi 10 2307 1912683 jstor 1912683 zellner arnold theil henri 1962 three stage least squares simultaneous estimation of simultaneous equations econometrica 30 1 54 78 doi 10 2307 1911287 jstor 1911287 kmenta jan 1986 system methods of estimation elements of econometrics second ed new york macmillan pp 695 701 isbn 9780023650703 hayashi fumio 2000 multiple equation gmm econometrics princeton university press pp 276 279 isbn 1400823838 page benjamin i jones calvin c 1979 12 01 reciprocal effects of policy preferences party loyalties and the vote american political science review 73 4 1071 1089 doi 10 2307 1953990 issn 0003 0554 jstor 1953990 s2cid 144984371 wlezien christopher 1995 01 01 the public as thermostat dynamics of preferences for spending american journal of political science 39 4 981 1000 doi 10 2307 2111666 jstor 2111666 breznau nate 2016 07 01 positive returns and equilibrium simultaneous feedback between public opinion and social policy policy studies journal 45 4 583 612 doi 10 1111 psj 12171 issn 1541 0072 xie f levinson d 2010 05 01 how streetcars shaped suburbanization a granger causality analysis of land use and transit in the twin cities journal of economic geography 10 3 453 470 doi 10 1093 jeg lbp031 hdl 11299 179996 issn 1468 2702 marini margaret mooney 1984 01 01 women s educational attainment and the timing of entry into parenthood american sociological review 49 4 491 511 doi 10 2307 2095464 jstor 2095464 wong chi sum law kenneth s 1999 01 01 testing reciprocal relations by nonrecursive structuralequation models using cross sectional data organ...