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A new procedure for the maximum-likelihood estimation of dynamic econometric models with errors in both endogenous and exogenous variables is presented in this monograph. A complete analytical development of the expressions used in problems of estimation and verification of models in state-space form is presented. The results are useful in relation not only to the problem of errors in variables but also to any other possible econometric application of state-space formulations.



Chapter 1. Introduction

As is well-known, econometric estimation procedures have not placed much emphasis on the problem of errors in variables. What is more, the fact that the regression model with measurement errors in the independent variables is not identified may lead to the erroneous belief that the treatment of this problem in more complex situations is even less open to study.
Jaime Terceiro Lomba

Chapter 2. Formulation of Econometric Models in State-Space

In this chapter we shall analyze the relationship between the structural, reduced, and state-space forms of an econometric model. We do not use the state-space formulations usually employed in the econometrics literature, but employ one that is based on the minimum dimension for the vector of state variables and which explicitly takes into account the improper nature of econometric models.
Jaime Terceiro Lomba

Chapter 3. Formulation of Econometric Models with Measurement Errors

In the last chapter we obtained a state-space parametrization for the dynamic econometric model given by (2.3).
Jaime Terceiro Lomba

Chapter 4. Estimation of Econometric Models with Measurement Errors

Having formulated the econometric model with errors in the variables in state-space, according to the following:
$$ {{x}_{t+1}}=\Phi {{x}_{t}}+\Gamma {{u}_{t}}+E{{w}_{t}} $$
$$ {{z}_{t}}=H{{x}_{t}}+D{{u}_{t}}+C{{v}_{t}} $$
$$ E\left[ {{w}_{t}} \right]=0,\,E\left[ {{v}_{t}} \right]=0 $$
$$E\left[ {\left[ {\begin{array}{*{20}c} {w_{t_1 } } \\ {v_{t_1 } } \\ \end{array} } \right]\left[ {w'_{t_2 } \,v'_{t_2 } } \right]} \right] = \left[ {\begin{array}{*{20}c} {Q\,} & S \\ {S'} & {\,R} \\ \end{array} } \right]\delta _{t_1 t_2 }$$
we shall go on to estimate the vector θ, containing p parameters, in which we shall include all the unknown elements within the following matrices: Φ, Γ, E, H, D, C, Q, R and S. Recall that vectors x t, u t and z t have dimensions n, r, and m, respectively.
Jaime Terceiro Lomba

Chapter 5. Extensions of the Analysis

The formulation of the econometric model with measurement errors given in Chapter 3 and defined by equations (3.17) to (3.26) can be extended in several directions. There is an immediate generalization following, for example, Harvey and Pierse (1984) and Ansley and Kohn (1983), for cases where observations are missing or where only specific temporal aggregates of the variables are observed. Also see the various studies compiled by Parzen (1984). Recall that the model in state-space form can be written:
$${{X}_{t+1}}=\phi {{X}_{t}}+E{{W}_{t}}$$
where \({{Z}_{t}}={{\left[ y{{_{t}^{*}}^{\prime }}u{{_{t}^{*}}^{\prime }} \right]}^{\prime }}\) is the vector formed by the endogenous and exogenous variables observed with error.
Jaime Terceiro Lomba

Chapter 6. Numerical Results

In this chapter we shall concentrate on an analysis of the estimation problems for the different parametrizations of a model corresponding to a dynamic specification developed naturally under assumptions frequently found in economic theory, such as partial adjustment.
Jaime Terceiro Lomba

Chapter 7. Conclusions

In this monograph we have developed a new formulation for dynamic econometric models with measurement errors. We have also obtained an algorithm for the maximum likelihood estimation of all the parameters of the model. The estimates obtained in this way are consistent and asymptotically normal and efficient.
Jaime Terceiro Lomba


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