4 strategies for addressing the challenges this

4 Strategies for addressing the challenges
This section describes a variety of strategies that have been used to address
the challenges outlined in the previous section. We start with forecast combi-
nations, turn to °ltering (unobserved components) methods and approaches for
capturing model change, before covering Bayesian methods, machine learning
techniques and theory-induced constraints on the forecasting models.
4.1 Forecast combination
Forecast combination methods have been used extensively in economic forecast-
ing, but are less widespread in °nancial forecasting. This is somewhat surprising
given that a large literature has established substantial bene°ts from combining
forecasts in a wide set of areas.
9
Why combine °nancial forecasts? One answer is that forecasters often em-
ploy many models with similar predictive performance making it di¢ cult to
9
See, 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 conditions
will prevail in the future. Alternatively, the forecasting environment may sim-
ply be unstable, rendering individual forecasting models± past track records
unreliable 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 using
an equally-weighted average of forecasts, has the e/ect of diversifying across
model uncertainty.
10
To see how forecast combination works, consider a vector of n individual
forecasts of some variable, f
t +1 j t
= ( f
1 t +1 j t
; f
2 t +1 j t
; :::; f
nt +1 j t
)
0
, where f
jt +1 j t
is
the one-step-ahead forecast of y
t +1
given information at time t , °
t
, generated
by the j th model. The simple equal-weighted forecast combination takes the
form
f
c
t +1 j t
=
1
n
n X
j =1
f
jt +1 j t
: (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 to
estimate from the data in this combination scheme and the weight on each
forecast 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 weights
from a linear regression of the outcome, y
t +1
, on the predictors
y
t +1
= ±
0
+
n X
j =1
±
j
f
jt +1 j t
+ "
t +1
: (14)
An intercept term ( ±
0
) is often included so as to ensure that the combined
forecast is (unconditionally) unbiased. If each of the individual forecasts is
believed to be unbiased, alternatively one can impose the constraints ±
0
= 0
and
P
n
j =1
±
j
= 1 in (14) so as to preserve unbiasedness of the combined forecast.
Rapach, Strauss, and Zhou (2010) is a notable exception to the relative
shortage 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 prediction
models take the form
r
t +1
= ´
0 j
+ ´
1 j
x
jt
+ u
jt +1
; j = 1 ; :::; n (15)
10
As a case in point, the accuracy of individual forecasts of asset returns is often reduced
by the e/ects of estimation error.