the included exogenous variables. Using cross-equation restrictions to achieve identification In simultaneous equations models, the most common method to achieve identification is by imposing within-equation parameter restrictions. 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 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 , which is assumed to be true for the rest of discussion. Now we impose the cross equation restriction of . Since the second equation is identified, we can treat as known for the purpose of identification. Then, the first equation becomes: Then, we can use as instruments to estimate the coefficients in the above equation since there are one endogenous variable () and one excluded exogenous variable () on the right hand side. Therefore, cross equation restrictions in place of within-equation restrictions can achieve identification. Estimation 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, developed independently by and . 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: Step 1: Regress Y−i on X and obtain the predicted values ; Step 2: Estimate γi, βi by the ordinary least squares regression of yi on and Xi. If the ith equation in the model is written as where Zi is a T×(ni + ki) matrix of both endogenous and exogenous regressors in the ith equation, and δi is an (ni + ki)-dimensional vector of regression coefficients, then the 2SLS estimator of δi will be given by where is the projection matrix onto the linear space spanned by the exogenous regressors X. Indirect least squares 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. 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) The “limited information”