Nelson (1972) and Cooper and Nelson (1975) arouse an issue of forecast encompassing by using exactly the same formula as Bates and Granger (1969). The similarity of forecast encompassing and forecast combinations at first appearance had once induced me to categorize Nelson and Cooper’s work as just an extension of Granger’s work, but the distinguish idea in essence prevents me to do so. In contrast to Bates and Granger (1969) who skipped the evaluation steps and were satisfied with combining information just through combining multiple forecasts, Nelson (1972) and Cooper and Nelson (1975) evaluated the informational increment of an econometric model to the time series model, intending to synthesize model with combined information set. Nelson (1972) concludes that significant weights of both models in the combining regression is due to the inability of a model to include all available information. Cooper and Nelson (1975) followed the previous study, looking into the decreasing prediction errors in a post-sample test through the significance level of t-statistics in the combining regression.
4.3.1 Similar to Bates and Granger (1969) at first appearance
Nelson had written the formula of composite forecasts (Nelson 1972) to evaluate the prediction performance of the FRBMIT-PENN (FMP) econometric model of the U.S.
economy by using the simple time-series models, an empirical representations of individual endogenous variables as stochastic processes of integrated autoregressive moving average (ARIMA) form, to establish standards of accuracy. The formula is,
ܣ௧ ൌ ߚሺܨܯܲሻ௧ ሺͳ െ ߚሻሺܣܴܫܯܣሻ௧ ߝ௧Ǥ Derived ߚመ as
ߚመ ൌ σሾሺܨܯܲሻ௧െ ሺܣܴܫܯܣሻ௧ሿሾܣ௧െ ሺܣܴܫܯܣሻ௧ሿ σሾሺܨܯܲሻ௧െ ሺܣܴܫܯܣሻ௧ሿଶ ǡ
which after transformation was exactly same as the product of variance-covariance method in Bates and Granger (1969),
ߚመ ൌ σ ݑଶ௧ଶ െ σ ݑଵ௧ݑଶ௧ σ ݑଵ௧ଶ σ ݑଶ௧ଶ െ ʹ σ ݑଵ௧ݑଶ௧Ǥ
However, he was not using the ߚመ as weight to optimize the forecasting ability, but rather he interpreted results in some other way.
4.3.2 Inherited the spirit of Markowitz (1952)
Nelson (1972) drew analogues between his work and Markowitz (1952)’s, saying that the ߚመ in encompassing formula is just “as the weight for a minimum variance two-asset portfolio depends on the covariance of returns as well as on return variances” (Nelson 1972, 911). In brief, both of them measures the influence of one thing (forecasting model or portfolio) by its impact to the whole, but not by its individual properties.
Rather than measure individual errors of each model, they use a composite forecast as a benchmark to measure the expected loss reduction in associated with a combined formula. Evidences showed that composite models were largely more accurate than FMP models but only accurate than ARIMA models in some cases, reflecting the inefficiency of certain FMP models in a sense that combining the FMP models with an ARIMA models significantly reduced the forecasting error of FMP.
Nelson (1972)’s motivation is standing from the viewpoint of the decision maker, making an overall evaluation of information contained in models. The question of whether one model or the other is more accurate is irrelevant, for a decision maker, his objective is to minimize expected loss, the contribution of one model should therefore measure by its comparison with the composite model. He had said (Nelson 1972):
[F]rom the viewpoint of the decision maker the question of whether one set of predictions or the other is more accurate is irrelevant. Since his objective is to minimize expected loss, he will purchase any piece of information which reduces expected loss by more than its cost. Thus, the value of the ARIMA predictions, for example, is not measured by their individual errors, but rather by the contribution which they are able to make to the reduction in expected loss associated with a composite prediction or a set of composite predictions (913).
The idea underlying is just the same as Markowitz (1952). Markowitz's work
showed that it is trivial to look at a security's own risk, to an investor, the important thing is the contribution the security brings to the variance of his entire portfolio.
Comparable in idea to Markowitz, Nelson knew the important thing to a decision maker is the overall contribution one model makes to the forecasting ability of the composite forecast rather than individually evaluate the accuracy of each model.
4.3.3 Forecast combinations and encompassing
The connection of forecast combinations and encompassing was discussed by several studies. Note that, however, nether Nelson (1972) nor Cooper and Nelson (1975) used the term ‘forecast encompassing’. It was until 1986, Mizon (1984) coined the term.
This model evaluation technique which essentially coincides with the forecast combination formula hereafter arouse concerns about the connection between it (forecast encompassing) and forecast combinations.
After Mizon (1984), regarding the changing characteristics of economy system and the consequence insufficient validity of each individual model to be passable overall performance, Chong and Hendry (1986) were motivated to investigate the suitable situation of using a system evaluation techniques. Their motivation indicating this paper was in line with Nelson’s work, essentially, efforts to improve the model specification was encouraged. They investigated in 4 methods, among these system evaluation techniques, one of the approaches was forecast encompassing. Comparing the empirical and theoretical fitness of forecast encompassing with that of conventional methods, they concluded the forecast encompassing is both feasible and more promising than others.
Following Chong and Hendry (1986), Diebold (1989) inherited the viewpoint of using combining regression as information encompassing test. In his work, though the pragmatic virtues of forecast combinations was argued, the efficiency of combining forecast was still in doubt to Diebold. Eventually, using combining regression to facilitate the combination of information set was emphasized.
Fang (2003) extended Diebold (1989) not only demonstrated encompassing tests as tools in model specification but reversely demonstrated encompassing tests as tools to explain the accuracy improvement of forecast combinations. Liang and Ryu (2003) showed further encompassing tests as a valuable principle on the choice of the
forecasts in the combining regression. Though without reference to Fang (2003), Liang and Ryu (2003) also established a two-way interaction between forecast combinations and forecasts encompassing.
Recently, more complex econometrics models are connected to the study of forecasts encompassing such as nested model, quintile forecasts, probability forecasts (Clements and Harvey 2010; Clements and Harvey 2011).
Brief summary is provided in Exhibit 6
.
Exhibit 6 Evolution of forecasts encompassing 4.4 A modified exhibit
From the perspective of theoretical development, the evolution of forecast combination is provided in Chapter 1. From perspective of early application of pooling ideas, the impact of portfolio selection and two-stage forecasting model on forecast combination are suggested (Chapter 2). The late development of forecast combination is briefly introduced in this chapter. Also, the impact of portfolio selection on forecast encompassing is suggested.
In sum, a modified exhibit based on Clemen (1989)’s review is provided in Exhibit 7.
Exhibit 7 A modified exhibit based on Clemen (1989)