Editing Simultaneous equations model (section)

=== 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''.