the tests are well defined for nested models, unlike some of the more recent versions of the Diebold and Mariano (1995) test such as West (1996), McCracken (2000), for

example.

T h e p a r t i c u l a r s o f t h i s t e s t i n g p r o c e d u r e a r e a s f o l l o w s . L e t { } 3 3 − = + T R t g t L

denote the

forecast loss function associated with the sequence of forecasts for time

t + 3 , where

R denotes the fixed size of the rolling estimation sample going from t = 1 to T − 3, and let g = 0,1 , that is g is an index that denotes with a 1 the model under

consideration and with a 0 the null model that forms the basis for comparison. Then the test statistic:

GW _{,0 = }

∆L σˆ_{L / }P

⎯^{d }→ N(0,1)

(7)

where

∆L =

1 P

## T −3

# ∑^{(L }

t=R

1 t+3

+ = − 0 3 ˆ ; ) L t L σ

1 p

## T −3

# ∑^{(L }

t=R

1 t+3

−L

0 t+3

)^{2 }

and P refers to the number of observations out-of-sample used in the evaluation. Notice that when it is suspected that there is heterogeneity and/or serial correlation in

the residuals, Giacomini and White (2006) recommend a HAC estimator for σˆ_{L . This }

framework is quite flexible because it permits specification of numerous loss functions. The more traditional loss function is based on the RMSE and can be defined as

L

g t+3

= ( mˆ

g t+3

− m_{t+3 }

)^{2 }; g = 1,0

(8)

However, a trader is probably more concerned about ascertaining that predicted returns are statistically higher with the proposed model in which case the loss function is

L

g t+3

ˆ 3 = g + t µ

;

g = 0,1

(9)

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