Combining economic forecasts through information measures

Abstract

The increasing number of prospective sources and methods provides a wide variety of forecasts for a given economic variable. Therefore, the theory suggests the convenience of combining the individual results to obtain a single aggregated prediction. The traditional methods for combining forecasts are based on the relative past performance of the forecasts to be combined. However, the number of forecasters is increasing considerably in the last years so it is not possible to have enough information about their past forecast task. This article focuses on the information theory as a framework to combine experts’ forecasts when information is limited. More specifically, we use the principle of entropy maximization to obtain a combined forecast from Shannon's measure (1948) and we also propose its extension to the quadratic uncertainty measure (Pérez, 1985). The empirical behaviour of both procedures is tested over a pool of forecasts referring to Spanish economic growth.

Notes

¹ Thus, the structural changes can be allowed through time-varying combining weights models (Diebold and Pauly, 1987; Sessions and Chatterjee, 1989) and nonlinear combining regression (Deutsch et al ., 1994), serially correlated errors are considered in dynamic combining regression (Hendry and Mizon, 1978; Coulson and Robins, 1993), and the problem of nonstationarity was also considered by Hallman and Kamstra (1989) among others.

² Moreover, combined forecasts are proved to be helpful in situations where there is not a dominant forecasting technique. Fullerton (1998) studies the predictability of secondary market prices, showing that the best performance corresponds to the combination of forecasts obtained through two different procedures: three stage least squares and a least absolute deviation methodology.

³ There is a controversy regarding the efficacy of placing restrictions on the combining model based on ordinary least squares. In a recent work, Terregrossa (2005) has formed combinations of component earnings–growth forecasts with restricted and unrestricted OLS suggesting that combinations formed with weights generated by OLS with the constant suppressed and the sum-of-the-coefficients constrained to equal one generally perform best.

⁴ Some economic applications of this measure can be found among others in Pérez (1985), Pérez et al . (1986), Río and Pérez (1987), Alvargonzález et al . (2004) and Moreno et al . (2005).

⁵ This restriction is not necessary in the maximization process of the Shannon entropy since in this case the obtained is always positive.

⁶ More specifically, Ashiya (2005) evaluates the accuracy and rationality of real GDP forecasts made by 38 Japanese private institutions and reveals that the consensus forecast outperforms typical institution's forecast. Also Reitz and Stadtmann (2005) found evidence in favour of a consensus among foreign exchange rate forecasters and Ashiya (2003) focuses on the IMF forecasts for the G7 countries finding that the combination of the long-term and the short-term forecasts significantly improves the directional accuracy.

⁷ However Pons-Novell (2004) shows evidence, based on the Livingston Survey, that these strategic behaviours are less likely to occur when the accuracy of the economic forecasts can be easily and quickly assessed. Sharpe (1997) compares economic forecasts by industry type inferring that intentional bias may be a component of the forecasts provided by some industries. The accuracy of the institutional panel of forecasters for the UK economy is also analysed by Bridge and Whyman (1997).

⁸ In case an organism does not provide a prediction in the period t, the corresponding weights in t − 1 are distributed between the remaining organisms in proportion to their respective weights.

⁹ Some simulations (Moreno, 2005) have shown that the wider the V, the nearer are the estimated weights to those corresponding to the arithmetic mean. Therefore, it is necessary to consider a vector V that does not annul the information contained in the individual forecasts. Moreover, in agreement with the previous empirical evidence in the first stage of prediction the individual forecasts show a greater bias than in the last one (where they have more information), thus justifying the decreasing of vectors V amplitude in the final stages.

¹⁰ However, when studying the 1-year percentage change of GDP we found a considerable similarity between forecasts of the considered institutions, and therefore no significant differences are found between the predictive ability of the combined forecast and the estimated weights. Moreover, although the use of prior bias can improve forecast accuracy (Shaffer, 2003) our sample are not long enough to study the accuracy forecasts in this context.