Linear Models with Correlated Disturbances

This book is primarily concerned with the estimation of regression models with correlated disturbances. Topics discussed include maximum likelihood, test strategies, Kalman filtering, conditional normal distributions, the Cramér-Rao inequality, Cholesky decomposition, missing observations and numerical optimization. A simple geometrical approach is used.

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The main aim of this volume is to give a survey of new and old estimation techniques for regression models with correlated disturbances, especially with autoregressive-moving average disturbances. In nearly all chapters the usefulness of the simple geometric interpretation of the classical ordinary Least Squares method is demonstrated. It emerges that both well-known and new results can be derived in a simple geometric manner, e.g., the conditional normal distribution, the Kalman filter equations and the Cramér-Rao inequality. The same geometric interpretation also shows that disturbances which follow an arbitrary correlation process can easily be transformed into a white noise sequence. This is of special interest for Maximum Likelihood estimation. Attention is paid to the appropriate estimation method for the specific situation that observations are missing. Maximum Likelihood estimation of dynamic models is also considered. The final chapter is concerned with several test strategies for detecting the genuine correlation structure among the disturbances. The geometric approach throughout the book provides a coherent insight in apparently different subjects in the econometric field of time series analysis.

Contenu
I Introduction.- II Transformation Matrices and Maximum Likelihood Estimation of Regression Models with Correlated Disturbances.- 2.1 Introduction.- 2.2 The algebraic problem.- 2.3 A dual problem.- 2.4 Recursive methods for calculating the transformation matrix P.- 2.5 The matrix P in the case of MA(1) disturbances.- 2.6 The matrix P in the case of MA(q) disturbances.- 2.7 The matrix P in the case of ARMA(p,q) disturbances.- Appendix 2. A Linear vector spaces.- Appendix 2.B The formula for ßtj if t is small.- III Computational Aspects of data Transformations and Ansley's Algorithm.- 3.1 Introduction.- 3.2 Recursive computations for models with MA(q) disturbances.- 3.3 Recursive computations for models with ARMA(p,q) disturbances.- 3.4 Ansley's method.- IV GLS Estimation by Kalman Filtering.- 4.1 Introduction.- 4.2 Some results from multivariate analysis.- 4.3 The Kaiman filter equations.- 4.4 The likelihood function.- 4.5 Estimation of linear models with ARMA(p,q) disturbances by means of Kaiman filtering.- 4.6 The exact likelihood function for models with ARMA(p,q) disturbances.- 4.7 Predictions and prediction intervals by using Kaiman filtering.- V Estimation of Regression Models with Missing Observations and Serially Correlated Disturbances.- 5.1 Introduction.- 5.2 The model.- 5.3 Derivation of the transformation matrix.- 5.4 Estimation and test procedures.- 5.5 Kaiman filtering with missing observations.- Appendix 5.A Stationarity conditions for an AR(2) process.- VI Distributed lag Models and Correlated Disturbances.- 6.1 Introduction.- 6.2 The geometric distributed lag model.- 6.3 Estimation methods.- 6.4 A simple formula for Koyck's consistent two-step estimator.- 6.5 Efficient estimation of dynamic models.- 6.6 Dynamic models with several geometricdistributed lags.- 6.7 The Cramér-Rao inequality and the Pythagorean theorem.- VII Test Strategies for Discriminating Between Autocorrelation and Misspecification.- 7.1 Introduction.- 7.2 Thursby's test strategy.- 7.3 Comments on Thursby's test strategy.- 7.4 Godfrey's test strategy.- 7.5 Comments on Godfrey's test strategy.- References.- Author Index.