Recursive Model: Understanding Simultaneous Equations with Recursive Computation

A deep dive into Recursive Models, a specific version of simultaneous equations models characterized by a triangular coefficient matrix and no contemporaneous correlation of random errors across equations.

Overview

A Recursive Model is a type of Simultaneous Equations Model (SEM) commonly used in econometrics and statistics. It is characterized by a triangular coefficient matrix of current endogenous variables and no contemporaneous correlation of random errors across equations. This structure ensures that all parameters in each equation of the system can be calculated recursively, meaning each equation is individually identified.

Historical Context

The Recursive Model concept emerged from the need to analyze economic systems where multiple variables influence each other simultaneously. Traditionally, models like the classical linear regression were inadequate in capturing the complexities of simultaneous interactions between variables. Recursive models provided a structured way to handle these challenges by simplifying the system of equations involved.

Key Characteristics

1. Triangular Coefficient Matrix: In a Recursive Model, the matrix of coefficients for the current endogenous variables is triangular. This means that each variable only depends on itself and the previously listed endogenous variables.
2. No Contemporaneous Correlation: The error terms in the different equations are uncorrelated with each other at the same time period.
3. Recursive Calculation: Parameters are computed in a step-by-step manner, with each equation depending on the preceding ones.

Mathematical Formulation

Consider a system of equations:

$$ y_1 = a_{11} x_1 + \epsilon_1 \\ y_2 = a_{21} y_1 + a_{22} x_2 + \epsilon_2 \\ y_3 = a_{31} y_1 + a_{32} y_2 + a_{33} x_3 + \epsilon_3 $$

Here, the matrix of coefficients (A) of the endogenous variables (Y) is lower triangular:

$$ \begin{bmatrix} 1 & 0 & 0 \\ a_{21} & 1 & 0 \\ a_{31} & a_{32} & 1 \end{bmatrix} $$

And the system can be solved sequentially, making it easier to interpret and estimate.

Example

Consider a simple economic model where we have:

• \( y_1 \): Investment
• \( y_2 \): Savings
• \( y_3 \): Output

The recursive equations could be:

$$ y_1 = a_{11} x_1 + \epsilon_1 $$

$$ y_2 = a_{21} y_1 + a_{22} x_2 + \epsilon_2 $$

$$ y_3 = a_{31} y_1 + a_{32} y_2 + a_{33} x_3 + \epsilon_3 $$

These equations show how investment influences savings, which in turn affects the output.

Applicability

Recursive Models are widely used in macroeconomic modeling, policy analysis, and econometrics for:

1. Forecasting: They allow for predicting the values of endogenous variables based on exogenous inputs.
2. Policy Analysis: Assess the impact of policy changes by modifying exogenous variables.
3. Causal Inference: Identify causal relationships among variables due to the clear dependency structure.

Related Terms

• Simultaneous Equations Model (SEM): A general class of econometric models that includes multiple equations to model interdependencies between variables.
• Endogenous Variable: A variable whose value is determined within the model.
• Exogenous Variable: A variable whose value is determined outside the model.

FAQs

Why use a recursive model instead of a standard simultaneous equations model?

Recursive models simplify the estimation process and reduce the computational complexity by ensuring that equations can be solved step-by-step.

Can recursive models be used for non-linear systems?

Yes, while they are more commonly applied to linear systems, recursive approaches can also be adapted for non-linear relationships.

What are the limitations of recursive models?

Recursive models assume no contemporaneous correlation of errors, which may not hold true in all real-world scenarios.

References

1. Gujarati, D. N. (2003). Basic Econometrics. McGraw-Hill.
2. Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press.

Summary

Recursive Models provide a structured, simplified approach to dealing with simultaneous equations in economics and statistics. By leveraging a triangular coefficient matrix and uncorrelated error terms, they facilitate straightforward, recursive computation of parameters. This makes them particularly useful in economic forecasting, policy analysis, and causal inference.

This article explored the formulation, key characteristics, historical context, and practical applications of Recursive Models, solidifying their importance in the realm of econometrics.