Identification in Dynamic Shock-Error Models

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A. Maravall
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I: The Model and Methodology.- 1. Introduction.- 2. The Model.- 2.1 Equations and Assumptions.- 2.2 Some Notation and Terminology.- 2.3 Identification of the Model when no Errors are Present.- 3. The Parameters and the Admissible Parameter Space.- 4. Analysis of Identification.- 4.1 The Identification Problem.- 4.2 The Covariance Equations.- 4.3 Locally Isolated Solutions of the Covariance Equations.- 4.4 Summary.- 5. A Remark on Estimation.- 6. An Example: Dynamic vs. Contemporaneous Models.- II: White-Noise Shock; White-Noise Exogenous Variables.- 1. The Case of One Exogenous Variable.- 1.1 One Lag per Variable.- 1.2 The Effect of Additional "a Priori" Information.- 1.3 Increasing the Number of Lags of the Variables.- 2. The General Case.- 3. Some Examples and Conclusions.- III: Autocorrelated Shock; White-Noise Exogenous Variables. I..- 1. Moving Average Process.- 1.1 An Example.- 1.2 The General Case.- 1.3 Some Examples and Conclusions.- 2. Autorsgressive Process.- 2.1 The General Case.- 2.2 An Example.- 2.3 A Remark on the Identification of the Autoregressive Process for the Shock.- 2.4 Some Final Remarks.- IV: Autocorrelated Shock; White-Noise Exogenous Variables. II..- 1. Autoregressive-Moving Average Process.- 1.1 The General Case.- 1.2 Some Remarks.- 1.3 Some Examples.- V: Autocorrelated Exogenous Variables; White-Noise Shock.- 1. Some Examples.- 1.1 First Example.- 1.2 Second Example.- 2. Moving Average Processes.- 3. Autoregressive-Moving Average Processes.- 4. Some Final Remarks.- VI: Autocorrelated Shock; Autocorrelated Exogenous Variables; The General Model.- 1. Autocorrelated Shock and Autocorrelated Exogenous Variables.- 1.1 The General Case.- 1.2 Some Examples.- 2. The General Model.- 2.1 The General Result.- 2.2 An Example.- VII: Some Extensions of the General Model.- 1. Correlation Between Exogenous Variables.- 2. Non Stationarity.- 2.1 An Example.- 2.2 The General Case.- 3. A Priori Zero Restrictions in the Coefficients (Seasonal Models).- 4. Autocorrelated Errors of Measurement.- VIII: Summary.- 2. An Example.- References.
Looking at a very simple example of an error-in-variables model, I was surprised at the effect that standard dynamic features (in the form of autocorre 11 lation. in the variables) could have on the state of identification of the model. It became apparent that identification of error-in-variables models was less of a problem when some dynamic features were present, and that the cathegory of "pre determined variables" was meaningless, since lagged endogenous and truly exogenous variables had very different identification properties. Also, for'the models I was considering, both necessary and sufficient conditions for identification could be expressed as simple counting rules, trivial to compute. These results seemed somewhat striking in the context of traditional econometrics literature, and p- vided the original motivation for this monograph. The monograph, therefore, atempts to analyze econometric identification of models when the variables are measured with error and when dynamic features are present. In trying to generalize the examples I was considering, although the final results had very simple expressions, the process of formally proving them became cumbersome and lengthy (in particular for the "sufficiency" part of the proofs). Possibly this was also due to a lack of more high-powered analytical tools and/or more elegant derivations, for which I feel an apology coul be appropiate. With some minor modifications, this monograph is a Ph. D. dissertation presented to the Department of Economics of the University of Wisconsin, Madison. Thanks are due to. Dennis J. Aigner and Arthur S.