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Fast same-step forecast in SUTSE model and its theoretical properties

Wataru Yoshida, Kei Hirose

TL;DR

This work considers the problem of forecasting multivariate time series by a Seemingly Unrelated Time Series Equations (SUTSE) model and proposes a two-stage procedure for forecasting that is much faster than the ordinary SUTSE model because it does not require a large matrix computation.

Abstract

We consider the problem of forecasting multivariate time series by a Seemingly Unrelated Time Series Equations (SUTSE) model. The SUTSE model usually assumes that error variables are correlated. A crucial issue is that the model estimation requires heavy computational loads because of a large matrix computation, especially for high-dimensional data. To alleviate the computational issue, we propose a two-stage procedure for forecasting. First, we perform the Kalman filter as if error variables are uncorrelated; that is, univariate time-series analyses are conducted separately to avoid a large matrix computation. Next, the forecast value is computed by using a distribution of forecast error. The proposed algorithm is much faster than the ordinary SUTSE model because we do not require a large matrix computation. Some theoretical properties of our proposed estimator are presented. Monte Carlo simulation is performed to investigate the effectiveness of our proposed method. The usefulness of our proposed procedure is illustrated through a bus congestion data application.

Fast same-step forecast in SUTSE model and its theoretical properties

TL;DR

This work considers the problem of forecasting multivariate time series by a Seemingly Unrelated Time Series Equations (SUTSE) model and proposes a two-stage procedure for forecasting that is much faster than the ordinary SUTSE model because it does not require a large matrix computation.

Abstract

We consider the problem of forecasting multivariate time series by a Seemingly Unrelated Time Series Equations (SUTSE) model. The SUTSE model usually assumes that error variables are correlated. A crucial issue is that the model estimation requires heavy computational loads because of a large matrix computation, especially for high-dimensional data. To alleviate the computational issue, we propose a two-stage procedure for forecasting. First, we perform the Kalman filter as if error variables are uncorrelated; that is, univariate time-series analyses are conducted separately to avoid a large matrix computation. Next, the forecast value is computed by using a distribution of forecast error. The proposed algorithm is much faster than the ordinary SUTSE model because we do not require a large matrix computation. Some theoretical properties of our proposed estimator are presented. Monte Carlo simulation is performed to investigate the effectiveness of our proposed method. The usefulness of our proposed procedure is illustrated through a bus congestion data application.
Paper Structure (19 sections, 11 theorems, 96 equations, 4 figures)

This paper contains 19 sections, 11 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Forecasts using the SUTSE model
  3. SUTSE model
  4. One-step-ahead forecast
  5. Same-step forecast
  6. Fast method
  7. Kalman filter with disregarding correlation
  8. Fast one-step-ahead forecast
  9. Fast same-step forecast
  10. Estimation of $V(v_{n+1}')$ with the graphical lasso
  11. Convergence of $E(v_{t}')$ and $V(v_{t}')$
  12. Consistency
  13. Numerical example
  14. Monte Carlo simulation
  15. Bus congestion forecasting
  16. ...and 4 more sections

Key Result

Lemma 4.1

$\\ \text{Under assumptions 1,2,3, }\text{$P_{t}\to ^\exists P$, $P_{t}'\to ^\exists P'(t\to \infty )$.} \\ \text{And thus }F_{t}\to ^\exists F,~F_{t}'\to ^\exists F',~K_{t}\to ^\exists K,~K_{t}'\to ^\exists K',~L_{t}\to ^\exists L,~L_{t}'\to ^\exists L'. \\ \text{Also, }^\exists M>0,\ 0<^\exists r<

Figures (4)

  • Figure 1.1 : Example: bus congestion same-step forecast
  • Figure 6.2 : Comparison of MSE between existing and fast method
  • Figure 6.3 : Comparison of computation time between existing and fast method
  • Figure 6.4 : Comparison of MSE of the same-step forecast between the existing and fast methods

Theorems & Definitions (21)

  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.1
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Lemma 4.7
  • proof
  • Lemma 5.1
  • ...and 11 more