Simultaneous equations model

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 ${\displaystyle \scriptstyle {\frac {1}{T}}X'\!X}$ 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.

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]

${\displaystyle {\begin{aligned}y_{1}&=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{12}z_{2}+\delta _{13}z_{3}+u_{1}\\y_{2}&=\gamma _{21}y_{1}+\delta _{21}z_{1}+\delta _{22}z_{2}+u_{2}\end{aligned}}}$

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 δ₁₃≠0, which is assumed to be true for the rest of discussion.

Now we impose the cross equation restriction of δ₁₂=δ₂₂. Since the second equation is identified, we can treat δ₁₂ as known for the purpose of identification. Then, the first equation becomes:

${\displaystyle y_{1}-\delta _{12}z_{2}=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{13}z_{3}+u_{1}}$

Then, we can use (z₁, z₂, z₃) as instruments to estimate the coefficients in the above equation since there are one endogenous variable (y₂) and one excluded exogenous variable (z₂) on the right hand side. Therefore, cross equation restrictions in place of within-equation restrictions can achieve identification.

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) harvtxt error: no target: CITEREFTheil1953 (help) and Basmann (1957).[8][9] 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 ${\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}$;

Step 2: Estimate γ_i, β_i by the ordinary least squares regression of y_i on ${\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}$ and X_i.

If the i^th equation in the model is written as

${\displaystyle y_{i}={\begin{pmatrix}Y_{-i}&X_{i}\end{pmatrix}}{\begin{pmatrix}\gamma _{i}\\\beta _{i}\end{pmatrix}}+u_{i}\equiv Z_{i}\delta _{i}+u_{i},}$

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]

${\displaystyle {\hat {\delta }}_{i}={\big (}{\hat {Z}}'_{i}{\hat {Z}}_{i}{\big )}^{-1}{\hat {Z}}'_{i}y_{i}={\big (}Z'_{i}PZ_{i}{\big )}^{-1}Z'_{i}Py_{i},}$

where P = X (X ′X)⁻¹X ′ is the projection matrix onto the linear space spanned by the exogenous regressors X.

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.[10][11] 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.

The “limited information” maximum likelihood method was suggested M. A. Girshick in 1947,[12] and formalized by T. W. Anderson and H. Rubin in 1949.[13] 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:

${\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i}\equiv Z_{i}\delta _{i}+u_{i}}$

The structural equations for the remaining endogenous variables Y_−i are not specified, and they are given in their reduced form:

${\displaystyle Y_{-i}=X\Pi +U_{-i}}$

Notation in this context is different than for the simple IV case. One has:

• ${\displaystyle Y_{-i}}$: The endogenous variable(s).
• ${\displaystyle X_{-i}}$: The exogenous variable(s)
• ${\displaystyle X}$: The instrument(s) (often denoted ${\displaystyle Z}$)

The explicit formula for the LIML is:[14]

${\displaystyle {\hat {\delta }}_{i}={\Big (}Z'_{i}(I-\lambda M)Z_{i}{\Big )}^{\!-1}Z'_{i}(I-\lambda M)y_{i},}$

where M = I − X (X ′X)⁻¹X ′, and λ is the smallest characteristic root of the matrix:

${\displaystyle {\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}{\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}^{\!-1}}$

where, in a similar way, M_i = I − X_i (X_i′X_i)⁻¹X_i′.

In other words, λ is the smallest solution of the generalized eigenvalue problem, see Theil (1971, p. 503):

${\displaystyle {\Big |}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}-\lambda {\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big |}=0}$

The LIML is a special case of the K-class estimators:[15]

${\displaystyle {\hat {\delta }}={\Big (}Z'(I-\kappa M)Z{\Big )}^{\!-1}Z'(I-\kappa M)y,}$

with:

• ${\displaystyle \delta ={\begin{bmatrix}\beta _{i}&\gamma _{i}\end{bmatrix}}}$
• ${\displaystyle Z={\begin{bmatrix}X_{i}&Y_{-i}\end{bmatrix}}}$

Several estimators belong to this class:

• κ=0: OLS
• κ=1: 2SLS. Note indeed that in this case, ${\displaystyle I-\kappa M=I-M=P}$ the usual projection matrix of the 2SLS
• κ=λ: LIML
• κ=λ - α / (n-K): Fuller (1977) estimator.[16] 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.[15]

The three-stage least squares estimator was introduced by Zellner & Theil (1962).[17][18] It can be seen as a special case of multi-equation GMM where the set of instrumental variables is common to all equations.[19] 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.