Editing Simultaneous equations model (section)

=== 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" />