Linear Simultaneous Equations

|| Unconstrained Klein's Model I Example || Constrained Klein's Model I Example ||

Discussion of linear simultaneous equations systems may be found in almost all econometric textbooks. For example, Chapter 18 in Estimation and Inference in Econometrics by Russell Davidson and James G. MacKinnon (ISBN 0-19506011-3, Oxford University Press, 1993), or chapter 19 in Econometric Analysis, 2nd Ed. by William H. Greene (ISBN 0-02-346391-0, Macmillan, 1993).


The linear simultaneous equation model can be represented by the matrix equation:

where Y is an TxK matrix of T observations on K endogenous variables, X is an TxL matrix of T observations on L exogenous variables, and is a KxK matrix of the coefficients among the endogenous variables, B is an LxK matrix of the coefficients of the endogenous variables on the exogenous, and Z is a TxK matrix of unobserved residuals.

Estimates for B and are found by minimizing the log-likelihood:

where

Not all of the elements of B and can be estimated. To "identify" the model it is necessary to fix some elements of B and to zero. It is also possible to fix elements of the residual covariance matrix to zero as well. The diagonal of is also fixed to one as a "normalization". See the references above for details about the identification of simultaneous models.

Unconstrained Klein's Model I Example

Klein's Model I is a well-known example that is presented in many econometrics textbooks, including the references above. From page 595, Greene (op.cit.) the equations for this model are:

where C is consumption, I investment, Wp private wages, X equilibrium demand, P private profits, K capital stock, and where G is government spending, T is indirect business taxes plus net exports, Wg is the government wage bill, A is time trend measured from 1931.

For this model, we get

and

and because the last three equations in the model are identities, the last three columns and rows in the residuals covariance matrix are fixed to zeros.

The program for estimating the coefficients in this model, klein.prg,is written in the GAUSS programming language and uses Aptech System's Maximum Likelihood (MAXLIK) applications module.

Estimates are produced for a 22 year time series of 9 variables of the U.S. economy from 1920 through 1941.

The results are presented in klein.out

Klein's Model I is a dynamic model because it contains autoregression coefficients, i.e., coefficients of endogenous variables on their values at a previous point in time. For such a model to be dynamically "stable" or stationary, the model estimates must satisfy a certain condition. Let be the submatrix of B associated with the relationships of the endogenous variables to the exogenous variables that are their lagged versions. Then stability requires that the eigenvalues of be less than one in absolute value.

For our estimates of Klein's Model I, we have

and

the absolute values of the eigenvalues of which are


  0.0000 
  0.0000 
  0.0000 
  0.0420 
  0.4911 
  0.4911 

This confirms the stability of the model implied by our estimates. ||Top||HomePage||
Constrained Klein's Model I Example

Greene's book (op.cit.) contains an additional 32 years of data for Klein's Model I. Estimating the model with this additional data presents some difficulties that require the special features of Aptech System's Constrained Maximum Likelihood (CML) applications module.

First, the data are highly correlated, causing difficulty for the estimation process, and second, the unconstrained estimation produces estimates that imply an unstable system.

To deal with the first problem, CML is used to bound the coefficient estimates. For bounds we will use the 99% confidence limits from the analysis of the 1931 to 1942 data. For the second problem we will use CML to constrain the eigenvalues of to be less than one in absolute value.

The program for estimating the coefficients in this model, cklein.prg,is written in the GAUSS programming language and uses Aptech System's Constrained Maximum Likelihood (CML) applications module.

Estimates are produced for the original 22 year time series of 9 variables of the U.S. economy from 1920 through 1941 plus in addition, the 32 year time series.

The results are presented in klein.out

The results of our stability test are:

  0.0000 
  0.0000 
  0.0000 
  0.0026 
  0.9627 
  0.9627 
The Lagrangean coefficients associated with the constraints on these eigenvalues are:

  0.0000 
  0.0000 
  0.0000 
  1.9556 
  0.0000 
  0.0000 
indicating that the model estimates are on the stability boundary.

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