On the efficacy of constraints on the linear combination forecast model

Abstract

Combination forecasting has been demonstrated to be a successful technique for enhanced forecast accuracy of economic and financial variables. An established method to generate the component-forecast weights is the ordinary-least-squares (OLS) regression technique: Actual values of a variable are regressed on within-sample values of forecasts generated by alternative forecast sources. The estimated regression coefficients then serve as weights for out-of-sample combination forecasts. The present study addresses the controversy regarding the efficacy of placing restrictions on the combining model when generating weights for out-of-sample forecasts. Combinations are formed of component earnings-growth forecasts generated separately by financial analysts and a statistical model. Both restricted and unrestricted OLS are used in turn to generate the component-forecast weights. The findings suggest that combinations formed with weights generated by OLS with the constant suppressed and the sum-of-the-coefficients constrained to equal one, lead to enhanced forecast-accuracy and generally perform best. This study differs from a previous related study appearing in Applied Financial Economics 1 in at least three main ways: (1) Combination forecasts are formed using actual regression-coefficients as forecast weights; (2) Forecast weights are generated using unrestricted OLS, as well as restricted OLS; (3) All combination forecasts are strictly ex-ante simulated.

Acknowledgements

The author is grateful for the technical support of Lary Jones, Cindy Kester, Ken Lynch, and Pinar Kose.

Notes

 See Terregrossa (1999).

 Early work in this area focused on combination forecasts of macroeconomic variables. See, for example, Newbold and Granger (1974), Cooper and Nelson (1975), Granger and Newbold (1975), Makridakis and Winkler (1983), Bischoff (1989) and Fair and Shiller (1990). For examples of combination forecasts of financial variables (e.g. firm earnings) see Malkiel and Cragg (1980), Fried and Givoly (1982), Ashton and Ashton (1985), Conroy and Harris (1987), Guerard (1987), Newbold et al. (1987), Lobo (1991), Lobo (1992) and Terregrossa (1999).

 See, for example, Nelson (1972), Cooper and Nelson (1975), Clemen (1986), Guerard (1987), Lobo (1991), and Terregrossa (1999).

 The OLS technique is appropriate when there is believed to be temporal stability of the weights underlying the combination forecast. If this is not the case then other weighting schemes may be more appropriate, such as the Granger-Newbold (1975) method which gives more weight to forecasts which have performed better in the recent past and which allows for a non-stationary relationship between respective forecast performances. See Bischoff (1989) for a thorough discussion of this issue.

 Thus, if the process of constraining the linear combination leads to somewhat biased estimators, it may be worthwhile to trade off some incurred bias for more efficient estimators to enhance the accuracy of the out-of-sample forecasts. An estimator with lower dispersion about the mean (more efficient) and some bias will more closely approximate the true parameter than will an unbiased estimator with a larger dispersion about the mean.

 In a related paper (Terregrossa [1999]) all employed forecast weights were rounded approximations, based on estimated coefficients from constrained OLS; actual regression-coefficients were not utilized as forecast weights. In the present study actual regression-coefficients generated from in-sample regressions are employed as forecast weights for out-of-sample combination forecasts. Forecast weights are generated from both unconstrained and constrained OLS, to allow more direct and meaningful comparison with previous studies in this area.

 See Terregrossa (1999).

 The Capital Asset Pricing Model (CAPM) was jointly developed by Markowitz (1959), Sharpe (1964) and Lintner (1965).

 A major reason for this approach is that risk-adjusted expected-return models have been shown to generate more accurate forecasts of earnings variables than a well-known representative modeller of the of the time-series behaviour of reported annual earnings, the submartingale (see Rozeff, 1983).The implication, as explained in a previous study (see Terregrossa, 1999), is that a risk-adjusted expected-return model may embody more independent information regarding the movements of an earnings forecast variable (and thus is more useful for combination forecasting) than a time-series model.

 The justification of the appropriateness of the use of OLS to generate forecast-weights, presented in Terregrossa (1999), is also applicable in the present study.

 See Terregrossa (1999).

 See Terregrossa (1999) for a detailed description and explanation of the CAPM-based forecasting method that is employed in the present study to generate the statistical component-forecast of EPS growth.

 As noted in a previous study (see Terregrossa, 2001) it may be that firms with higher growth rates of earnings may have different variances of forecast error than firms with smaller growth rates of earnings. Therefore, errors in predicting growth rates may be associated with one of the right-hand variables. The White (1980) procedure corrects for heteroscedasticity caused by variance related to right-hand variables.

 For a detailed list and explanation of the criteria each firm must satisfy to be included in a given sample of firms, chosen from the Center for Research of Security Prices (CRSP) tape, see Terregrossa (1999). The same criterion is exactly applicable in the present study.

 Although the improvements in the forecasting errors are slight, small differences in compound growth rates may translate into large changes in the absolute level of future expected earnings. Current stock-value is a function of the absolute size of future expected earnings.

 This holds true for combination models 2, 3 and 4. (see Tables 1A and 2A.)

This holds true only for combination model 4. (See Tables 1B and 2B.)

 This finding of a temporal stability of the in-sample, restricted OLS regression coefficients employed as out-of-sample combination forecast weights in the present study, supports and strengthens a similar finding in a previous study (see Terregrossa, 1999).