Indirect Least Squares Definition & Examples

Published Apr 29, 2024

Definition of Indirect Least Squares

Indirect Least Squares (ILS) is a method used in econometrics and statistics for estimating the parameters of simultaneous equations models. These models are common in economics where multiple equations describe the relationships between variables, and each equation may contain endogenous variables—variables that are determined within the system of equations. The ILS method involves estimating the structural parameters indirectly by first estimating the reduced form of the model and then solving these estimates to obtain the structural parameters.

Example

Consider a simple economic model with two equations representing the demand and supply for a good:

Demand: \(Q_d = a – bP + e_d\)
Supply: \(Q_s = c + dP + e_s\)

Here, \(Q_d\) and \(Q_s\) are the quantity demanded and quantity supplied, \(P\) is the price, \(a\), \(b\), \(c\), and \(d\) are parameters to be estimated, and \(e_d\) and \(e_s\) are error terms. In an ILS approach, you would first estimate the reduced form of this system, which directly relates the independent variables to the price and quantity (the endogenous variables) without specifying a functional form for the supply and demand relationship. After obtaining the reduced form estimates, you can then solve for the structural parameters (\(a\), \(b\), \(c\), and \(d\)).

Why Indirect Least Squares Matters

Indirect Least Squares is particularly useful in econometrics for estimating simultaneous equation models where classical methods like Ordinary Least Squares (OLS) fail due to endogeneity—when predictors are correlated with the error term. Endogeneity in the model can lead to biased and inconsistent estimates. By using ILS, economists can circumvent some of these issues, obtaining consistent estimates of the model’s parameters. This method is essential for accurately understanding economic phenomena and providing reliable empirical evidence to inform economic policy and decision-making.

Frequently Asked Questions (FAQ)

How does Indirect Least Squares differ from Ordinary Least Squares?

Indirect Least Squares differs from Ordinary Least Squares in the context and application. OLS is used primarily for single-equation models and may produce biased estimates in simultaneous equations due to endogeneity. ILS, on the other hand, is specifically designed for simultaneous equation models, estimating parameters indirectly to address the issue of endogenous variables.

What are the limitations of Indirect Least Squares?

The main limitation of ILS is its reliance on the correctness of the model’s specifications and the assumption that the model is exactly identified or over-identified. If the model is under-identified (there are more parameters to estimate than there are equations), ILS cannot be used. Additionally, ILS assumes that the errors in the equations are not correlated; violations of this assumption can lead to biased estimates.

Can Indirect Least Squares be used for any type of simultaneous equations model?

ILS is suitable for exactly identified or over-identified models where the number of instrumental variables (exogenous variables) is equal to or greater than the number of endogenous variables needing estimation. It is not suitable for under-identified models where the reverse is true. Therefore, its applicability depends on the structure of the model and the availability of appropriate instruments.

What are the practical applications of Indirect Least Squares?

Practical applications of ILS include estimating the parameters of economic models that describe the behavior of markets (e.g., supply and demand interactions), fiscal policy impacts, and other macroeconomic relationships. It allows economists to dissect complex relationships where variables influence one another simultaneously, such as in models of monetary policy, where interest rates, inflation, and output growth may all be endogenously determined.

Indirect Least Squares plays a crucial role in the econometric analysis of simultaneous equations, enabling economists to derive consistent and unbiased estimates of model parameters. This method enhances the accuracy and reliability of economic research, contributing to better policy formulation and economic decision-making.