Indirect Least Squares Method (ILS)

Suppose we wish to estimate a structural equation containing say, three endogenous variables. The first step of the ILS technique is to estimate the reduced-form equations for these three endogenous variables. If the structural equations for these three endogenous variables. If the structural equation in question is just identified, there will be only one way of calculating the desired estimates of the structural equation parameters from the reduced-form parameter estimates. The structural parameters are expressed in terms of the reduced-form parameters, and the OLS estimates of the reduced-form parameters are plugged in these expressions to produce estimates of the structural parameters. Because these expressions are nonlinear, however, unbiased estimates of the reduced-form parameters produce Only Consistent estimates of the structural parameters, not unbiased estimates.

If an equation is over-identified, the extra identifying restrictions provide additional ways of calculating the structural parameters from the reduced-form parameters, all of which are supposed to lead to the same values of the structural parameters. But because the estimates of the reduced-form parameters do not embody these extra restrictions, these different ways of calculating the structural parameters creates different estimates of these parameters. (This is because unrestricted estimates rather than actual values of the parameters are being used for these calculations.) Because there is no way of determining which of these different estimates is the most appropriate, ILS is not used for over-identified equations. The other simultaneous equation estimating techniques have been designed to estimate structural parameters in the over-identified case; many of these can be shown to be equivalent in the over-identified case; many of these can be shown to be equivalent to ILS in the context of a just-identified equation, and to be weighted averages of the different estimates produced by ILS in the context of over-identified equations.

Here is a basic procedure to implement ILS:
1. Rearrange the structural form equations into reduced form, Estimate the reduced form equations;
2. Estimate the reduced form parameters;
3. Solve for the structural form parameters in terms of the reduced form parameters, and substitute in the estimates of the reduced form parameters to get estimates for the structural ones.
Note: If structural equation is exactly identified, there will be  a unique way to calculate the parameters. Estimates of reduced form parameters are unbiased, but estimates of the structural parameters will not be.  Both are consistent.