Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 16Simultaneous equations

1 INTRODUCTION

In Chapter 13, we considered the simple linear regression model

(1)

where y_t and u_t are scalar random variables and x_t and β₀ are k × 1 vectors. In Section 15.8, we generalized (1) to the multivariate linear regression model

(2)

where y_t and u_t are random m × 1 vectors, x_t is a k × 1 vector, and B₀ a k × m matrix.

In this chapter, we consider a further generalization, where the model is specified by

(3)

This model is known as the simultaneous equations model.

2 THE SIMULTANEOUS EQUATIONS MODEL

Thus, let economic theory specify a set of economic relations of the form

(4)

where y_t is an m × 1 vector of observed endogenous variables, x_t is a k × 1 vector of observed exogenous (nonstochastic) variables, and u_0t is an m × 1 vector of unobserved random disturbances. The m × m matrix Γ₀ and the k × m matrix B₀ are unknown parameter matrices. We shall make the following assumption.

Assumption 16.1 (normality)

The vectors {u_0t, t = 1, …, n} are independent and identically distributed as N(0, Σ₀) with Σ₀ a positive definite m × m matrix of unknown ...