CHAPTER 19: THE IDENTIFICATION PROBLEM

ü  Notations and Definitions

2 Types of Simultaneous-Equation Model

Endogenous – those determined within the model

Predetermined – those determined outside the model

2 Categories of Predetermined Variables

Exogenous

Lagged Endogenous

A reduced-form equation is one that expresses an endogenous variable solely in terms of the predetermined variables and the stochastic disturbances.

ü  The Identification Problem

-          Whether numerical estimates of the parameters of a structural equation can be obtained from the estimated reduced-form coefficients.

-          May be either exactly identified or overidentified.

Exactly identified if unique numerical values of the structural parameters can be obtained.

Overidentified if more than one numerical value can be obtained for some of the parameters of the structural equations.

ü  Rules for Identification

The so-called order and rank conditions of identification lighten the task by providing a systematic routine.

ü  The Order Condition of Identifiability

Definition 19.1 In a model M simultaneous equations in order for an equation to be identified, it must exclude at least M – 1 variables appearing in the model. It excludes exactly M – 1 variables, the equation is just identified. If it excludes more than M – 1 variables, it is overidentified.

Definition 19.2 In a model of M simultaneous equations, in order for an equation to be identified, the number of predetermined variables excluded from the equation must not be less than the number of endogenous variables included in that equation less 1, that is,

K – k ≥ m – 1

If K – k = m – 1, the equation is just identified, but if K – k > m – 1, it is overidentified.

ü  The Rank Condition of Identifiability

In a model containing M endogenous variables, an equation is identified if and only if at least one nonzero determinant of order (M – 1)(M – 1) can be constructed from the coefficients of the variables excluded from that particular equation but included in the other equations of the model. The rank condition tells us if it is exactly identified or overidentified.

ü  Tests of Simultaneity

Hausman Specification Test.

Step 1: Regress P_t on I_t and R_t to obtain v_t.

Step 2: Regress Q_t on P_t and v_t and perform t test on the coefficient of v_t. If it is significant, do not reject the hypothesis of simultaneity; otherwise, reject it.