Forecasting Aggregated Vector ARMA Processes

This study is concerned with forecasting time series variables and the impact of the level of aggregation on the efficiency of the forecasts. Since temporally and contemporaneously disaggregated data at various levels have become available for many countries, regions, and variables during the last decades the question which data and procedures to use for prediction has become increasingly important in recent years. This study aims at pointing out some of the problems involved and at pro­ viding some suggestions how to proceed in particular situations. Many of the results have been circulated as working papers, some have been published as journal articles, and some have been presented at conferences and in seminars. I express my gratitude to all those who have commented on parts of this study. They are too numerous to be listed here and many of them are anonymous referees and are therefore unknown to me. Some early results related to the present study are contained in my monograph "Prognose aggregierter Zeitreihen" (Lutkepohl (1986a)) which was essentially completed in 1983. The present study contains major extensions of that research and also summarizes the earlier results to the extent they are of interest in the context of this study.

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.