Linear Models with Correlated Disturbances

In each chapter of this volume some specific topics in the econometric analysis of time series data are studied. All topics have in common the statistical inference in linear models with correlated disturbances. The main aim of the study is to give a survey of new and old estimation techniques for regression models with disturbances that follow an autoregressive-moving average process. In the final chapter also several test strategies for discriminating between various types of autocorrelation are discussed. In nearly all chapters it is demonstrated how useful the simple geometric interpretation of the well-known ordinary least squares (OLS) method is. By applying these geometric concepts to linear spaces spanned by scalar stochastic variables, it emerges that well-known as well as new results can be derived in a simple geometric manner, sometimes without the limiting restrictions of the usual derivations, e. g. , the conditional normal distribution, the Kalman filter equations and the Cramer-Rao inequality. The outline of the book is as follows. In Chapter 2 attention is paid to a generalization of the well-known first order autocorrelation transformation of a linear regression model with disturbances that follow a first order Markov scheme. Firstly, the appropriate lower triangular transformation matrix is derived for the case that the disturbances follow a moving average process of order q (MA(q». It turns out that the calculations can be carried out either analytically or in a recursive manner.

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.- 2.4.1 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.- 5.4.2.1 The likelihood ratio test.- 5.4.2.2 The modified Lagrange multiplier (MLM) test.- 5.4.2.3 An infinite number of missing observations.- 5.4.2.4 The power of the MLM test.- 5.4.2.5 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 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.- 7.3.1 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.