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12.4. Estimation methods

Once we have confirmed that our model is identified we can proceed with the estimation of the parameters of the structural coefficients. In this section we will present two methods of estimation that can be used to estimate coefficients of a simultaneous equation system.

12.4.1. Indirect Least Squares (ILS)

When all the equations are exactly identified one can use the method of Indirect Least Square to estimate the coefficients of the structural equations. It is done by the following three steps:

1) Form the reduced form equations

2) Estimate the coefficients of the reduced form using OLS

3) Use the estimated coefficients of the reduced form to derive the structural coefficients.

Example 12.4 (ILS)

Consider the following simple macro economic model:

This model has two endogenous variables (y and Ct) and one exogenous variable (it), and we would like to estimate the coefficients of the behavioral equation. Since one of the variables of the model is excluded from the consumption function it is identified according to the order condition. The two structural equations could be used to form the reduced form equations for consumption. If we do that we receive:

(13.3) and (13.4) show how the reduced form coefficients are related to the structural coefficients. By using the estimated values of the reduced form coefficients we can solve for the structural coefficients. We have:

(12.26) and (12.27) can now be used to solve for B0 and B1. Since (12.27) is an equation with only one unknown we solve for B1 first (remember that n1 is an estimate and therefore a number in this expression). Once we receive the value of B1 we can use it in (12.26) to solve for B0. Hence we receive:

In order to determine the standard errors for Б0 and Б1 we can use linear approximations to their expression based on the standard errors and covariance of the reduced form estimated coefficients. It can be shown that the corresponding variance for Б0 and Б1 is:

1 1 2 2

with a =- and Ъ =- and where cr0 is the variance of ж0, ax the variance for nx and cr12

the covariance between n0 and nx.

ILS will result in consistent estimates but will still be biased in small samples. When using larger systems with more variables and equations it is often burdensome to find the estimates, and in those cases the equations are often over identified, which means that ILS cannot be used. For that reason ILS is not used very often in practice. Instead a much more popular method called 2SLS is used.

12.4.2. Two Stage Least Squares (2SLS)

The procedure of 2SLS is a method that allows you to receive consistent estimates of the structural coefficient when the equations are exactly identified as well as over identified. However, the estimates will still be biased in small samples.

Consider the following model

This model has two endogenous variables and four exogenous variables. The first equation (12.28) contains four variables which means that from a total of six variables, two has been omitted. That means that it is over identified. The same can be said about the second equation (12.29), which means that the model is identified. Since both equations are over identified we cannot estimate the structural parameters using ILS, but instead we are forced to use 2SLS. We will now focus the discussion on the estimation of the first equation.

The basic steps of 2SLS applied for equation (12.28):

Step 1 Derive the reduced form equation for Y2 and estimate the predicted value of Y2 (Y2) on the reduced form using OLS.

Step 2 Replace Y2 in equation (12.28) with its predicted value from the reduced form and estimate the coefficient of the model using OLS.

If these two steps are applied we will receive consistent estimates of the parameters in (12.28). That is, since we replace the endogenous variable with its predicted value, it is no longer correlated with the residual term. Hence, the problem is solved. Remember that Y2 = Y2 + V2 which implies that the stochastic variable Y , consist of two parts, one that is a linear combination of the exogenous (predetermined) variables and one random part. A group of exogenous variables are by necessity uncorrelated with the random term.

Observe that X1 and X2 both appear in the specification of Y1 and Y2 , which means that there will be a correlation between the explanatory variables X1 and X2 and Y2 . This correlation will not be perfect unless X3 and X4 also is included in the structural model of the first equation and is therefore nothing to worry about. But if that happens, the equation would not pass the order condition for identification.

There is one additional complication to be aware of when working with 2SLS. When the predicted value is included in the specification, the variance of the error term will not be correct. To see this we will consider a simplified version of a model to make it clear where the problem appear. Consider the following equation:

In order to receive consistent estimates of b1 we replace Y2 with its predicted value and estimate the parameters of following regression model using OLS:

The estimator of the slope coefficient is therefore given by the following expression:

We have concluded that this form of the estimator is consistent but not unbiased. Since it is consistent we need to compare it with its asymptotic variance, that is, the formula of the variance when the number of observation is very large (has gone to infinity). It can be shown that the asymptotic variance of this sample estimator is given by the following expression:

This is good, because it is very similar to the variance given by the standard OLS. So what is the problem? The problem is related to the estimated variance of the error term. When running the regression using (12.31) our estimated residual would be given by:

Whereas the estimated residual should be given by the following expression

(12.34) is based on the observed variable Y2 multiplied with the sample estimator b given by (12.32), rather than the predicted version of the variable. Hence in order to receive consistent estimates of the standard errors, one has to use (12.34). When using commercial software with routines for 2SLS they automatically make the correction. But if we run 2SLS in two steps, as described above, we need to correct the standard errors, before we can perform any hypothesis testing.

In sum, the 2SLS has the following properties:

It generates biased but consistent estimates

The distribution of the estimators are normally distributed only in large samples The variance is biased but consistent when using (12.34)

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