Simultaneous Equations Models

A simultaneous equations model is one in which there areendogenous variables which are determined jointly.

e.g. the demand-supply model

1 2 3 1,

1 2 3 2,

t

dt t t

st t t t

d st t

q p y u

q p w u

q q

In this model q and p are endogenous variables (determinedwithin the model) while y and w are exogenous variables(determined outside the model). u1 and u2 are random errors.

The structural form of an econometric model is the formsuggested by economic theory.

For example, the structural form of the demand curve is:

1 2 3 1,t

dt t tq p y u

The structural parameters are the alpha parameters from thisequation.

Most often we will be interested in estimating the structuralparameters of the model.

The reduced form of a model is obtained by solving out thestructural form so that each equation contains only oneendogenous variable.

For example:

1, 2,31 1 2

2 2 2 2 2 2 2 2

t tt t t

u up y w

is the reduced form equation for price in the demand-supplymodel.

Note that in the reduced form model each endogenous variableis a function of all the exogenous variables and all the randomerrors.

Now suppose we wish to estimate the structural parametersof the demand curve:

1

1 2 3 1,

21,

2 2

1cov ,

t

dt t t

t t u

q p y u

p u

From the reduced form, p is correlated with u1 and thereforeOLS will produce inconsistent estimates.

We therefore need to develop alternative estimators if we are going to obtain consistent estimates.

Another example

The structural form of the Keynesian income-expendituremodel can be written:

t t t t

t t t

Y C I GC Y u

The reduced form equation for Y is:

11t t t tY I G u

It follows that:

21cov ,1t t uY u

and therefore OLS estimates of the consumption functionwill be inconsistent. In fact we can show that:

2

2

1ˆplim1

u

Y

Indirect Least Squares

Use the reduced form estimates to estimate the structural parameters.

For example, if we estimate the reduced form of the Keynesianincome-expenditure model we obtain:

The relationship between the structural and reduced form parameters is given by the expression:

1 ˆˆ11 1ˆ 1 1 0.3322ˆ 1.4975

Now compare this with the OLS estimates of the structuralform:

The OLS estimate of the structural form parameter is0.62 but this is inconsistent. The ILS estimator is 0.33 andthis is a consistent estimator.

Problem: The ILS estimator may not be unique.

We have based our calculation around the reduced formequation:

1, 1 1,

1

11

1ˆ 1ˆ

t t t t tY A v A v

but we have an alternative reduced form equation:

2, 2 2,

2

2

1ˆˆ

ˆ1

t t t t tC A v A v

1.042279ˆ 0.51031 1.042279

The ILS estimator is not unique because the system isoveridentified. There are more reduced form parametersthan structural parameters.