Linear combination of restrictions and forecasts in time series analysis

Abstract

An important tool in time series analysis is that of combining information in an optimal way. Here we establish a basic combining rule of linear predictors and show that such problems as forecast updating, missing value estimation, restricted forecasting with binding constraints, analysis of outliers and temporal disaggregation can be viewed as problems of optimal linear combination of restrictions and forecasts. A compatibility test statistic is also provided as a companion tool to check that the linear restrictions are compatible with the forecasts generated from the historical data.

Forecasting aggregates and disaggregates with common features

“…The F1 procedure is equivalent to a direct forecast of the aggregate; F2 is equivalent to an indirect forecast of the aggregate using ARIMA models for the components; F3 is equivalent to an indirect forecast based on a multivariate model for the components; and F4 is related to the forecasting procedure, which we propose in this paper. Giacomini & Granger (2004) show that imposing constraints in the fully disaggregated model improves the forecasts. One way to impose constraints is to use their F4 procedure instead of the theoretically optimal F3.…”

“…The theoretical results relating to the advantages of aggregating component forecasts from a multivariate model over forecasting the aggregate directly apply when the DGP is known. Because this is rarely the case in practice, the mean squared error (MSE) of the forecasts includes an additional factor, which is 1/T times a term that depends on the number of parameters to be estimated; see Giacomini & Granger (2004) and the references therein. Then, as it is widely recognized in the literature, the question of which is the best procedure for forecasting the aggregate is mainly empirical.…”

“…However, results from the literature also shed light on this issue. The Giacomini & Granger (2004) results for space-time models argue on the existence of a trade-off between the efficiency gain achieved from specifying the fully disaggregated system and the loss in efficiency that arises from parameter estimation errors. In this context, Giacomini & Granger (2004) also consider four forecasting procedures, F1 to F4, which can be related to the time-series procedures considered in our paper.…”

“…The Giacomini & Granger (2004) results for space-time models argue on the existence of a trade-off between the efficiency gain achieved from specifying the fully disaggregated system and the loss in efficiency that arises from parameter estimation errors. In this context, Giacomini & Granger (2004) also consider four forecasting procedures, F1 to F4, which can be related to the time-series procedures considered in our paper. The F1 procedure is equivalent to a direct forecast of the aggregate; F2 is equivalent to an indirect forecast of the aggregate using ARIMA models for the components; F3 is equivalent to an indirect forecast based on a multivariate model for the components; and F4 is related to the forecasting procedure, which we propose in this paper.…”

“…The disaggregation map can be extended by including overlapping subsets of basic components with common features and by incorporating, for example, additional common-trend restrictions, the types of common cyclical features identified by Cubadda (2007), common seasonality, co-breaks, common non-linearity, common volatility, etc. The second and our most important contribution is to develop, along the lines of Giacomini & Granger (2004), a simple indirect forecasting procedure based on singleequation models that departs from the use of vector models, that imposes important restrictions on those models, and that can in practice produce improved forecasts, as we will see in Section 5. At the same time, our procedure improves on direct forecasting by including additional relevant information, so that it more than compensates for the greater cost arising from estimation errors in more complex models.…”