Identification

In the next table we mean that exogenous variables are available as instruments if they are not already in the equation in their own right. A structural equation is:

Underidentified if	# of exogenous variables available as instruments < # of endogenous variables requiring instruments
Just identified if	# of exogenous variables available as instruments = # of endogenous variables requiring instruments
Overidentified if	# of exogenous variables available as instruments > # of endogenous variables requiring instruments

Explain:

1) Equation (1) is just identified.

2) Equation (2) is underidentified but introducing the money supply as in equation (9.26) makes it identified.

3) In the system

(3)

(4) ( is the rate of growth of productivity)

the first equation is overidentified. What is bad about under- and overidenification?

1. Indirect least squares procedure

Step 1. If the system of structural equations is complete, derive the reduced form.

Step 2. Estimate the reduced form using OLS.

Step 3. Derive the original system coefficients from the estimates obtained, if the original equations are identified.

In fact the definition of under/just/overidentification is given in terms of the possibility of retrieving the coefficients of structural equations by way of ILS.

1. 2SLS (two stage least squares)

2SLS is applied to individual structural equations in case of overidentification. In (3)+(4) there are two instruments for . Instead of trying to choose between them, we can regress on and use the fitted value as an instrument. This way is better because:

(i) OLS yields estimates that are optimal according to three mutually equivalent criteria: (a) minimizing the sum of squares of the residuals, (b) maximizing , and maximizing the correlation between the actual and fitted values.

(ii) There is no need to choose between conflicting estimates.

2SLS in application to (3) consists of two steps:

Step 1. Regress on and save the fitted value.

Step 2. Regress on the fitted value of to estimate .

In case of overidentification, 2SLS is better than ILS because there are no conflicting estimates. In other cases they yield identical estimates.

1. Order condition for identification

This condition is necessary but not sufficient for identification.

Suppose there are endogenous variables in the simultaneous equations system, the system is complete and all equations are written so that on the left side there is only one (dependent) endogenous variable. Fix one such equation. The order condition for this equation states that the number of exogenous variables available is instruments (that is excluded from the given equation) is at least as large as the number of endogenous variables requiring instruments (that is included in the given equation).

Exercises 9.1, 9.2, 9.4, 9.9, 9.12

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