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| '''Simultaneous equation models''' are a form of [[statistical model]] in the form of a set of [[linear simultaneous equations]]. They are often used in [[econometrics]].
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| == Structural and reduced form ==
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| Suppose there are ''m'' regression equations of the form
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| : <math>
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| y_{it} = y_{-i,t}'\gamma_i + x_{it}'\;\!\beta_i + u_{it}, \quad i=1,\ldots,m,
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| </math>
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| 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
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| : <math> | |
| y_i = Y_{-i}\gamma_i + X_i\beta_i + u_i, \quad i=1,\ldots,m,
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| </math>
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| 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
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| : <math>
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| Y\Gamma = X\Beta + U.\,
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| </math>
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| 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.
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| Postmultiplying the structural equation by {{nowrap|Γ<sup> −1</sup>}}, the system can be written in the '''reduced form''' as
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| : <math>
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| Y = X\Beta\Gamma^{-1} + U\Gamma^{-1} = X\Pi + V.\,
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| </math>
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| 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.
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| === Assumptions ===
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| 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.
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| 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}}.
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| Lastly, the [[identification condition]]s requires that the number of unknowns in this system of equations should not exceed the number of equations. More specifically, the ''order condition'' requires that for each equation {{nowrap|''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'' of identifiability is that {{nowrap|rank(Π<sub>''i''0</sub>) {{=}} ''n<sub>i</sub>''}}, where Π<sub>''i''0</sub> is a {{nowrap|(''k − k<sub>i</sub>'')×''n<sub>i</sub>''}} 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.
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| == Estimation ==
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| === Two-stages least squares (2SLS) ===
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| The simplest and the most common<ref>{{harvtxt|Greene|2003|loc=p. 398}}</ref> estimation method for the simultaneous equations model is the so-called [[two-stage least squares]] method, developed independently by {{harvtxt|Theil|1953}} and {{harvtxt|Basmann|1957}}. 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>{{harvtxt|Greene|2003|loc=p. 399}}</ref>
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| : ''Step 1'': Regress ''Y<sub>−i</sub>'' on ''X'' and obtain the predicted values <math style="vertical-align:-.2em">\scriptstyle\hat{Y}_{\!-i}</math>;
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| : ''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>''.
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| If the ''i''<sup>th</sup> equation in the model is written as
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| : <math>
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| y_i = \begin{pmatrix}Y_{-i} & X_i\end{pmatrix}\begin{pmatrix}\gamma_i\\\beta_i\end{pmatrix} + u_i
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| \equiv Z_i \delta_i + u_i,
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| </math>
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| 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>{{harvtxt|Greene|2003|loc=p. 399}}</ref>
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| : <math>
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| \hat\delta_i = \big(\hat{Z}'_i\hat{Z}_i\big)^{-1}\hat{Z}'_i y_i
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| = \big( Z'_iPZ_i \big)^{-1} Z'_iPy_i,
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| </math>
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| where {{nowrap|''P'' {{=}} ''X'' (''X'' ′''X'')<sup>−1</sup>''X'' ′}} is the projection matrix onto the linear space spanned by the exogenous regressors ''X''. | |
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| === Indirect least squares ===
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| 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>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}}</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.
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| === Limited information maximum likelihood (LIML) ===
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| The “limited information” maximum likelihood method was suggested by {{harvtxt|Anderson|Rubin|1949}}. It is used when one is interested in estimating a single structural equation at a time (hence its name of limited information), say for variable i:
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| : <math> y_i = Y_{-i}\gamma_i +X_i\beta_i+ u_i \equiv Z_i \delta_i + u_i </math>
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| The structural equations for the remaining endogeneous variables Y<sub>−1</sub> are not specified, and they are given in their reduced form:
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| : <math> Y_{-i} = X \Pi + U_{-1} </math>
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| The explicit formula for this estimator is:<ref>{{harvtxt|Amemiya|1985|loc=p. 235}}</ref>
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| : <math>
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| \hat\delta_i = \Big(Z'_i(I-\lambda M)Z_i\Big)^{\!-1}Z'_i(I-\lambda M)y_i,
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| </math>
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| where {{nowrap|''M'' {{=}} ''I − X'' (''X'' ′''X'')<sup>−1</sup>''X'' ′}}, {{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>''′}}, and ''λ'' is the smallest characteristic root of the matrix
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| : <math>
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| \Big(\begin{bmatrix}y_i\\Y_{-i}\end{bmatrix} M_i \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big)
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| \Big(\begin{bmatrix}y_i\\Y_{-i}\end{bmatrix} M \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big)^{\!-1}
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| </math>
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| 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}}:
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| : <math>
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| \Big|\begin{bmatrix}y_i&Y_{-i}\end{bmatrix}' M_i \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} -\lambda
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| \begin{bmatrix}y_i&Y_{-i}\end{bmatrix}' M \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big|=0
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| </math>
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| ==== K class estimators ====
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| The LIML is a special case of the K-class estimators:<ref>{{harvtxt|Davidson|Mackinnon|1993|loc=p. 649}}</ref>
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| : <math> | |
| \hat\delta = \Big(Z'(I-\kappa M)Z\Big)^{\!-1}Z'(I-\kappa M)y,
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| </math>
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| with:
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| * <math> \delta = \begin{bmatrix} \beta_i & \gamma_i\end{bmatrix} </math>
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| * <math> Z = \begin{bmatrix} X_i & Y_{-i}\end{bmatrix} </math>
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| Several estimators belong to this class:
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| * κ=0: [[OLS]]
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| * κ=1: 2SLS. Note indeed that in this case, <math> I-\kappa M = I-M= P </math> the usual projection matrix of the 2SLS
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| * κ=λ: LIML
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| * κ=λ - α (n-K): {{harvtxt|Fuller|1977}} estimator. 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>{{harvtxt|Davidson|Mackinnon|1993|loc=p. 649}}</ref>
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| === Three-stage least squares (3SLS) ===
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| The three-stage least squares estimator was introduced by {{harvtxt|Zellner|Theil|1962}}. It combines [[2SLS|two-stage least squares]] (2SLS) with [[seemingly unrelated regressions]] (SUR).
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| == See also ==
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| * [[General linear model]]
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| * [[Seemingly unrelated regressions]]
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| * [[Indirect least squares]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| {{refbegin}}
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| * {{cite book
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| | last = Amemiya | first = Takeshi
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| | title = Advanced econometrics
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| | year = 1985
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| | publisher = Harvard University Press
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| | location = Cambridge, Massachusetts
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| | isbn = 0-674-00560-0
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| | ref = harv
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| }}
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| * {{cite journal
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| | last1 = Anderson | first1 = T.W.
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| | last2 = Rubin | first2 = H.
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| | title = Estimator of the parameters of a single equation in a complete system of stochastic equations
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| | year = 1949
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| | journal = Annals of Mathematical Statistics
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| | volume = 20 | issue = 1
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| | pages = 46–63
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| | jstor = 2236803
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| | ref = harv
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| }}
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| * {{cite journal
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| | last = Basmann | first = R.L.
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| | title = A generalized classical method of linear estimation of coefficients in a structural equation
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| | year = 1957
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| | journal = Econometrica
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| | volume = 25 | issue = 1
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| | pages = 77–83
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| | jstor = 1907743
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| | ref = harv
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| }}
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| * {{cite book
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| | last1 = Davidson | first1 = Russell
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| | last2 = MacKinnon | first2 = James G.
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| | title = Estimation and inference in econometrics
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| | year = 1993
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| | publisher = Oxford University Press
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| | isbn = 978-0-19-506011-9
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| | ref = harv
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| }}
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| * {{cite journal
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| | last = [[Wayne Fuller|Fuller]] | first = Wayne
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| | title = Some Properties of a Modification of the Limited Information Estimator
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| | year = 1977
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| | journal = Econometrica
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| | volume = 45 |issue=4
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| | pages = 939–953
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| | ref = harv
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| }}
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| * {{cite book
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| | last = Greene | first = William H.
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| | title = Econometric analysis
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| | publisher = Prentice Hall
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| | year = 2002 | edition = 5th
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| | isbn = 0-13-066189-9
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| | ref = harv
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| }}
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| * {{cite book
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| | last = [[Henri Theil|Theil]]| first = Henri
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| | title = Principles of Econometrics
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| | year = 1971
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| | publisher = John Wiley
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| | location = New York
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| | ref = harv
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| }}
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| * {{cite journal
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| | last1 = [[Arnold Zellner|Zellner]] | first1 = Arnold
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| | last2 = [[Henri Theil|Theil]] | first2 = Henri
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| | title = Three-stage least squares: simultaneous estimation of simultaneous equations
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| | year = 1962
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| | journal = Econometrica
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| | volume = 30 | issue = 1
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| | pages = 54–78
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| | jstor = 1911287
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| | ref = harv
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| }}
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| {{refend}}
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| == External links ==
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| *[http://economics.about.com/library/glossary/bldef-ils.htm About.com:economics] Online dictionary of economics, entry for ILS
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| [[Category:Multivariate statistics]]
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| [[Category:Econometrics]]
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