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Linear Models with Correlated Disturbances : Lecture Notes in Economic and Mathematical Systems - Paul Knottnerus

Linear Models with Correlated Disturbances

Lecture Notes in Economic and Mathematical Systems

Paperback ISBN: 9783540539018
Number Of Pages: 196

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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 Cramer-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.

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.- 2.4.1 A recursive algorithm for calculating P.- 2.4.2 The recursive Levinson-Durbin algorithm.- 2.4.3 A supplementary Levinson-Durbin algorithm.- 2.4.4 Inversion of an arbitrary nonsingular matrix.- 2.5 The matrix P in the case of MA(1) disturbances.- 2.5.1 The matrix P.- 2.5.2 A new derivation of the inverse of the autocovariance matrix of an MA(1) process.- 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.- 2.7.1 A derivation of the formula for the autocovariance matrix of an ARMA(p,q) process.- 2.7.2 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.- 4.2.1 Likelihood functions.- 4.2.2 Conditional normal distributions and minimum variance estimators.- 4.3 The Kaiman filter equations.- 4.3.1 The state space model.- 4.3.2 A general geometric derivation of the Kaiman filter equations.- 4.3.3 Comparison with other derivations.- 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.- 5.4.1 Estimation.- 5.4.2 Tests for autocorrelation if observations are missing.- The likelihood ratio test.- The modified Lagrange multiplier (MLM) test.- An infinite number of missing observations.- The power of the MLM test.- An adjusted Lagrange multiplier test.- 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.- 6.5.1 Introduction.- 6.5.2 An efficient 3-step Gauss-Newton estimation method.- 6.5.3 A Gauss-Newton-Prais-Winsten estimation method with small sample adjustments.- 6.6 Dynamic models with several geometric distributed lags.- 6.7 The Cramer-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.- 7.3.1 Introduction.- 7.3.2 The simple AR(2) disturbances model.- 7.3.3 The general disturbances model.- 7.4 Godfrey's test strategy.- 7.5 Comments on Godfrey's test strategy.- References.- Author Index.

ISBN: 9783540539018
ISBN-10: 3540539018
Series: Lecture Notes in Economic and Mathematical Systems
Audience: General
Format: Paperback
Language: English
Number Of Pages: 196
Country of Publication: DE
Dimensions (cm): 24.41 x 16.99  x 1.14
Weight (kg): 0.35