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Online Econometrics Textbook - Regression Extensions - Simultaneous linear equations

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In simultaneous equation systems there are always feedback structures which make it (almost) impossible to use the easy methods we've described before.

As a matter of fact we have to make a firm distinction between two different kinds of variables: the jointly dependent variables (or endogenous variables), and the predetermined variables (or exogenous variables).

The jointly dependent variables may (but don't have to) be used as dependent and explanatory variables at the same time (in different equations). The predetermined variables however are all of those which are not explicitly explained by other variables in any equation. This means that a "lagged" version of a dependent variable is in fact considered to be exogenous or rather predetermined.

The variance-covariance matrix of the residuals is assumed to be of the form

Online Econometrics Textbook - Regression Extensions - Simultaneous linear equations


The model of the structural equations can be written as



which leads to


and thus to


in order to get the reduced form equations.

Remark that


which are called the reduced form parameters and the reduced form disturbance respectively.

In fact, by transforming a structural form into a reduced form, what one does is to express all jointly dependent variables as a function of all predetermined variables.

The asymptotic assumptions are expressed explicitly as


Since it is our objective to estimate the structural parameters of (III.III-2) we could start naively with OLS estimations to each equation separately (after having normalized the equations)


Except in rare cases this procedure will not yield unbiased parameter estimations since


Above that, the estimated value of the parameters does not converge in probability


The results in (III.III-9) and (III.III-10) are known as the least squares bias. It is therefore quite obvious that we have to find other ways of estimating the structural parameters, without bias.

We know that estimating the reduced form with OLS would be at least consistent since the assumptions of (III.III-7) can be used to prove that


The OLS estimator will also be unbiased (for estimating the reduced form parameters) if X doesn't contain lagged jointly dependent variables.

Suppose we use (III.III-11) in the context of SUR then we know that this is equivalent to GLS since all the exogenous variables are identical in all equations.

After having used OLS with SUR it is sometimes possible to compute the structural form parameters knowing that


This procedure is known as the Indirect Least Squares method (ILS).

When is it possible to find a unique solution with ILS? Answer: only when the equations are exactly identified. If an equation is under identified it is never possible to compute the structural parameters. If however an equation is over identified there are more than one unique solution to (III.III-12), and consequently some special techniques should be used to solve this problem.

Let us have a look at the i-th equation of a structural form




we know that


A necessary and sufficient condition for exact identification is


which is called the rank condition. Here ILS is consistent and efficient.

A condition for over identification is


where ILS is consistent but not efficient.

A condition for under identification is


where estimation with ILS is not possible.

A necessary condition for identification is therefore


Now we consider the problem of estimating the structural parameters of an over identified system by use of


the ILS approach


the GLS approach

The ILS approach

A single-equation structural form




can be written as


Since the reduced form parameters may be consistently estimated by


and since


it follows that


and by premultiplying by X'X we get


and thus



The GLS approach

When the model


is premultiplied by X' we obtain


It can easily be shown, using (III.III-30), that the GLS estimator is




To prove consistency, assume


which can be used to show that


The consistent Two Stage Least Squares estimator (2SLS) consists of the following two stages

first stage


second stage


Asymptotically efficient estimators

All the previous estimators are not efficient (in general) since they don't use the information to the full extent. Therefore we describe two Full Information methods: the Tree Stage Least Squares (3SLS), and the Full Information Maximum Likelihood Estimation (FIMLE).

The 3SLS estimator uses the following model


or simply


If X'X/T converges to a nonstochastic limit then


and accordingly the 3SLS estimator can be deduced as


If all M equations are identified, the 3SLS estimator is consistent. Above this, the 3SLS method is more efficient than the 2SLS method.

The FIMLE method can be used under the assumption that the distribution of the error terms is known (e.g. normal).

Assume that the errors are jointly normally distributed with a zero mean and covariance matrix


then the FIMLE method can be applied by means of for instance a numerical iterative algorithm. Some examples of algorithms are: the Grid search method, the Newton-Raphson method, the Gauss-Newton method, the Steepest Descend method, the Marquardt algorithm etc...

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