From the previous discussion we learned that when an equation belongs to a system of equation, estimating them separately using OLS would lead to biased and inconsistent estimates. Hence, in order to be able to estimate the parameters of the equation system it is important to consider the whole system since they interact with each other. Before going into issues of estimation we need to define some more concepts. Consider the following system of equations:

It is a macro economic model that extends the example from the previous section and is based on three equations. It is an income determination model, with two behavioral equations; one for consumption expenditure Ct, and one for net-investments It. The consumption function is a function of income Yt and the investment function is a function of interest rate Rt. The income equation that specifies the equilibrium condition is a function of consumption, investment and government spending Gt. This model has three endogenous variables, Ct, It, and Yt, and two exogenous variables Rt and Gt that are pre determined.

The system of equations given by (12.8)-(12.10) describes the structure of the economy that we would like to investigate. For that reason these equations are called structural equations. The coefficients of the structural equations represent the direct effect of a change in one of the explanatory variables. If we take (12.9) as an example, B1 represents the marginal propensity to invest as a result from a change in the interest rate. This represents the direct effect of a change in interest rate on the net-investment.

Assume that we increase the interest rate. That will have a direct effect on the investments in this model, which in a second step via the equilibrium condition will have an effect on the income. The income in its term will affect the consumption level, and since income is endogenous it will have an effect the error term U1 since they are correlated. The initial change in the interest rate, will in this way, affect the components in the system until the effect reaches its equilibrium level.

We can therefore talk about two types of effect; the short run effect and the long run effect. The long run effect can be received from the long run relationship that can be determined by solving the structural equation system with respect to the endogenous variables. To solve the system for Yt we simply substitute (12.8) and (12.9) into (12.10) and solve for Yt. If we do that we receive:

If we do the similar thing with respect to the other two endogenous variables we would receive the following expressions:

By solving the structural system of equations with respect to the endogenous variables we have determined the reduced form equations for income, consumption and investment. The coefficients of the reduced form equations represent the full effect when the system is in equilibrium. The full effect of a change in interest rate on income is represented by B1/(1-A1). It is also called the interest rate multiplier on income. There is a corresponding multiplier related to consumption and investments that can be found in their reduced form equations. Observe that the reduced form equation for investments only is a function of interest rate. Government spending does not have any affect on the investment, even though it has an effect on consumption and income.

The nice thing with the reduced form equations is that they may be estimated separately using OLS. That is, the coefficients in the reduced form equations can be consistently estimated using OLS. Since the structural parameters are part of the reduced form coefficients it is sometimes possible to indirectly find the structural coefficient using the estimated values of the reduced form coefficients. For that to be possible, a certain requirement needs to be fulfilled: the structural coefficients must be exactly identified.

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