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.- 2.2 Nonstationary Processes.- 2.3 Vector Autoregressive Moving Average Processes.- 2.4 Estimation.- 2.5 Model Specification.- 2.6 Summary.- 3. Forecasting Vector Stochastic Processes.- 3.1 Forecasting Known Processes.- 3.2 Forecasting Vector ARMA Processes with Estimated Coefficients.- 3.3 Forecasting Autoregressive Processes of Unknown Order.- 3.4 Forecasting Nonstationary Processes.- 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.- 4.3 Forecasting Linearly Transformed Nonstationary Processes.- 4.4 Linearly Transformed Vector ARMA Processes.- 4.5 Summary and Comments.- 5. Forecasting Contemporaneously Aggregated Estimated Processes.- 5.1 Summary of Assumptions and Predictors.- 5.2 Estimated Coefficients.- 5.3 Unknown Orders and Estimated Coefficients.- 5.4 Nonstationary Processes.- 5.5 Small Sample Results.- 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.- 6.5 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.- 7.6 Empirical Examples.- 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.- 8.6 Examples.- 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.- 9.3 Prediction Based on Estimated Processes with Unknown Orders.- 9.4 Monte Carlo Comparison of Predictors.- 9.5 Forecasts of U.S. Gross Private Domestic Investment.- 9.6 Summary and Conclusions.- 10. Epilogue.- 10.1 Summary and Conclusions.- 10.2 Some Remaining Problems.- Appendix. Data Used for Examples.