Pub. Date:
Springer-Verlag New York, LLC
Forecasting Aggregated Vector ARMA Processes

Forecasting Aggregated Vector ARMA Processes

by Helmut Lütkepohl


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Product Details

ISBN-13: 9780387172088
Publisher: Springer-Verlag New York, LLC
Publication date: 02/28/1987
Series: Lecture Notes in Economics and Mathematical Systems
Pages: 323
Product dimensions: 6.50(w) x 1.50(h) x 9.50(d)

Table of Contents

1. Prologue.- 1.1 Objective of the Study.- 1.2 Survey of the Study.- 2. Vector Stochastic Processes.- 2.1 Discrete-Time, Stationary Vector Stochastic Processes.- 2.1.1 General Assumptions.- 2.1.2 The Wold or Moving Average Representation.- 2.1.3 Autoregressive Representation.- 2.1.4 Spectral Representation.- 2.2 Nonstationary Processes.- 2.3 Vector Autoregressive Moving Average Processes.- 2.3.1 Stationary Processes.- 2.3.2 Nonstationary Processes.- 2.4 Estimation.- 2.4.1 Maximum Likelihood Estimation of Stationary Gaussian Vector ARMA Processes of Known Order.- 2.4.2 Estimation of Vector Autoregressive Processes of Known Order.- 2.4.3 Multivariate Least Squares Estimation of AR Processes with Unknown Order.- 2.4.4 Nonstationary Processes.- 2.5 Model Specification.- 2.5.1 AR Order Determination.- 2.5.2 Subset Autoregressions.- 2.5.3 The Box-Jenkins Approach.- 2.6 Summary.- 3. Forecasting Vector Stochastic Processes.- 3.1 Forecasting Known Processes.- 3.1.1 Predictors Based on the Moving Average Representation.- 3.1.2 Predictors Based on the Autoregressive Representation.- 3.1.3 Forecasting Known Vector ARMA Processes.- 3.2 Forecasting Vector ARMA Processes with Estimated Coefficients.- 3.2.1 The General Case.- 3.2.2 Finite Order AR Processes.- 3.3 Forecasting Autoregressive Processes of Unknown Order.- 3.3.1 The Asymptotic MSE Matrix.- 3.3.2 Proof of Proposition 3.2.- 3.4 Forecasting Nonstationary Processes.- 3.4.1 Known Processes.- 3.4.2 Estimated Coefficients.- 3.4.3 Unknown Order.- 3.5 Comparing Forecasts.- 3.6 Summary.- 4. Forecasting Contemporaneously Aggregated Known Processes.- 4.1 Linear Transformations of Vector Stochastic Processes.- 4.2 Forecasting Linearly Transformed Stationary Vector Stochastic Processes.- 4.2.1 The Predictors.- 4.2.2 Comparison of the Predictors.- 4.2.3 Equality of the Predictors.- 4.2.4 Granger-Causality.- 4.3 Forecasting Linearly Transformed Nonstationary Processes.- 4.4 Linearly Transformed Vector ARMA Processes.- 4.4.1 Finite Order MA Processes.- 4.4.2 ARMA Processes.- 4.5 Summary and Comments.- 5. Forecasting Contemporaneously Aggregated Estimated Processes.- 5.1 Summary of Assumptions and Predictors.- 5.2 Estimated Coefficients.- 5.2.1 Comparison of $${\rm \hat Y}_{\rm t}^{\rm o} ({\rm h})$$ and $${\rm \hat Y}_{\rm t}^{} ({\rm h})$$.- 5.2.2 Comparison of $${\rm \hat Y}_{\rm t}^{\rm o} ({\rm h})$$ and $${\rm \hat Y}_{\rm t}^{\rm u} ({\rm h})$$.- 5.3 Unknown Orders and Estimated Coefficients.- 5.4 Nonstationary Processes.- 5.5 Small Sample Results.- 5.5.1 Design of the Monte Carlo Experiment.- 5.5.2 Simulation Results for AR Process I.- 5.5.3 Simulation Results for AR Process II.- 5.5.4 Simulation Results for MA Process I.- 5.5.5 Simulation Results for MA Process II.- 5.5.6 Simulation Results for MA Process III.- 5.6 An Empirical Example.- 5.7 Conclusions.- 6. Forecasting Temporally and Contemporaneously Aggregated Known Processes.- 6.1 Macro Processes.- 6.2 Six Predictors.- 6.3 Comparison of Predictors.- 6.4 Nonstationary Processes.- 6.4.1 Differencing to Obtain Stationarity.- 6.4.2 Forecasting Aggregated Nonstationary Processes.- 6.5 Temporally and Contemporaneously Aggregated Vector ARMA Processes.- 6.6 Conclusions and Comments.- 7. Temporal Aggregation of Stock Variables - Systematically Missing Observations.- 7.1 Forecasting Known Processes with Systematically Missing Observations.- 7.2 Processes With Estimated Coefficients.- 7.3 Processes With Unknown Orders and Estimated Coefficients.- 7.4 Nonstationary Time Series with Systematically Missing Observations.- 7.5 Monte Carlo Results.- 7.5.1 Univariate AR Processes.- 7.5.2 Bivariate AR Process.- 7.5.3 MA (m) Processes.- 7.5.4 Univariate MA(1) Process.- 7.5.5 Summary of Small Sample Results.- 7.6 Empirical Examples.- 7.6.1 Consumption Expenditures.- 7.6.2 Investment.- 7.7 Concluding Remarks.- 7.A Appendix: Proof of Relation (7.2.18).- 8. Temporal Aggregation of Flow Variables.- 8.1 Forecasting with Known Processes.- 8.2 Forecasts Based on Processes with Estimated Coefficients.- 8.3 Forecasting with Autoregressive Processes of Unknown Order.- 8.4 Temporally Aggregated Nonstationary Processes.- 8.5 Small Sample Comparison.- 8.5.1 A Univariate AR Process.- 8.5.2 A Univariate MA(2) Process.- 8.5.3 A Univariate MA(3) Process.- 8.5.4 A Bivariate MA Process.- 8.5.5 A System with a Stock and a Flow Variable.- 8.6 Examples.- 8.6.1 Consumption.- 8.6.2 Investment.- 8.7 Summary and Conclusions.- 8.A Appendix: Proof of Relation (8.2.23).- 9. Joint tTemporal and Contemporaneous Aggregation.- 9.1 Summary of Processes and Predictors.- 9.2 Prediction Based on Processes with Estimated Coefficients.- 9.2.1 General Results.- 9.2.2 An Example.- 9.2.3 Conclusions for Processes with Estimated Coefficients.- 9.3 Prediction Based on Estimated Processes with Unknown Orders.- 9.3.1 General Comments.- 9.3.2 Comparison of MSEs.- 9.3.3 Summary and Discussion of Results for Processes with Unknown Orders.- 9.4 Monte Carlo Comparison of Predictors.- 9.4.1 Simulation Results for AR Process.- 9.4.2 Simulation Results for MA Process.- 9.4.3 Discussion of Small Sample Results.- 9.5 Forecasts of U.S. Gross Private Domestic Investment.- 9.5.1 First Differences of Investment Data.- 9.5.2 Aggregation of Original Investment Data.- 9.6 Summary and Conclusions.- 10. Epilogue.- 10.1 Summary and Conclusions.- 10.2 Some Remaining Problems.- Appendix. Data Used for Examples.

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