Forecasting aggregated vector ARMA processes

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.