4Strategies for addressing the challengesThis section describes a variety of strategies that have been used to addressthe challenges outlined in the previous section. We start with forecast combi-nations, turn to °ltering (unobserved components) methods and approaches forcapturing model change, before covering Bayesian methods, machine learningtechniques and theory-induced constraints on the forecasting models.4.1Forecast combinationForecast combination methods have been used extensively in economic forecast-ing, but are less widespread in °nancial forecasting. This is somewhat surprisinggiven that a large literature has established substantial bene°ts from combiningforecasts in a wide set of areas.9Why combine °nancial forecasts? One answer is that forecasters often em-ploy many models with similar predictive performance making it di¢ cult to9See, e.g., Clements (1989) and Timmermann (2006).9

identify a single, superior model. Another reason lies in state-dependent fore-casting performance: certain models may work well under some market condi-tions but not in other and it can be di¢ cult to tell, ex ante, which conditionswill prevail in the future. Alternatively, the forecasting environment may sim-ply be unstable, rendering individual forecasting models± past track recordsunreliable for their future performance. A third reason is simply that of "diver-si°cation": all models are misspeci°ed and combining forecasts, e.g., by usingan equally-weighted average of forecasts, has the e/ect of diversifying acrossmodel uncertainty.10To see how forecast combination works, consider a vector ofnindividualforecasts of some variable,ft+1jt= (f1t+1jt; f2t+1jt; :::; fnt+1jt)0, wherefjt+1jtisthe one-step-ahead forecast ofyt+1given information at timet,°t, generatedby thejth model.The simple equal-weighted forecast combination takes theformfct+1jt=1nnXj=1fjt+1jt:(13)This "1=n" strategy has proven highly successful in empirical applications, in-cluding to form portfolios (DeMiguel et al.(2007)).There are no weights toestimate from the data in this combination scheme and the weight on eachforecast does not depend on the individual forecasts±past performance.A more general approach to forecast combination that accounts for the indi-vidual models±past forecasting performance estimates the combination weightsfrom a linear regression of the outcome,yt+1, on the predictorsyt+1=±0+nXj=1±jfjt+1jt+"t+1:(14)An intercept term (±0) is often included so as to ensure that the combinedforecast is (unconditionally) unbiased.If each of the individual forecasts isbelieved to be unbiased, alternatively one can impose the constraints±0= 0andPnj=1±j= 1in (14) so as to preserve unbiasedness of the combined forecast.Rapach, Strauss, and Zhou (2010) is a notable exception to the relativeshortage of papers in °nancial forecasting that use forecast combination meth-ods. They °t univariate forecasting models to returns on the US stock market,using a set of predictors from Welch and Goyal (2008), and form an equal-weighted average of these forecasts.Suppose the univariate return predictionmodels take the formrt+1=´0j+´1jxjt+ujt+1;j= 1; :::; n(15)10As a case in point, the accuracy of individual forecasts of asset returns is often reducedby the e/ects of estimation error.

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Winter

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Jared

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