Understanding Indirect Least Squares (ILS)

Indirect Least Squares (ILS) is an econometric technique used primarily for estimating the parameters (coefficients) of structural equations within simultaneous equation models.

Structural vs. Reduced Form Equations

In econometrics, particularly when dealing with systems of equations where variables are determined jointly (simultaneity), we distinguish between structural and reduced form equations:

• Structural Equations: These equations represent the theoretical relationships hypothesized by economic theory. They specify how endogenous variables depend on other endogenous and exogenous variables. The coefficients in these equations have direct economic interpretations.
• Reduced Form Equations: These equations express each endogenous variable solely as a function of exogenous variables and error terms. They are derived by solving the system of structural equations. While useful for forecasting, their coefficients don't have the direct theoretical meaning of structural coefficients.

Application of Indirect Least Squares

The Indirect Least Squares (ILS) method works in two main stages:

1. Estimate Reduced Form Equations: First, the parameters of the reduced form equations are estimated, typically using Ordinary Least Squares (OLS) because each reduced form equation contains only exogenous variables (and potentially lagged endogenous variables treated as exogenous), thus avoiding simultaneity bias.
2. Derive Structural Coefficients: The estimated coefficients from the reduced form equations are then used to calculate estimates for the coefficients of the original structural equations. This indirect approach helps overcome identification problems that might arise if one tried to estimate the structural equations directly using methods like OLS.

Therefore, ILS is fundamentally a method for obtaining estimates of the coefficients that belong to the structural equation.

Why Other Options Are Less Suitable

• Over-identified equation: While ILS can be used when structural equations are over-identified, it's not the equation *type* that ILS directly estimates coefficients for; rather, it's a characteristic of the structural equation being estimated. ILS aims to estimate the structural coefficients themselves.
• Reduced form equation: Reduced form equations are usually estimated directly using OLS. ILS uses the estimates *from* the reduced form, but its target is the structural parameters.
• Under-identified equation: An under-identified equation cannot be estimated by any method, including ILS, because there isn't enough information. ILS is typically applicable to exactly identified or over-identified structural equations.