Stephen G. Hall and James Mitchell 233
Notes
- However, Granger and Machina (2006) show that, under conditions on the second deriva-
tive of the loss function, there is always some point forecast which leads to the same loss
as if the decision maker had minimized loss given the density forecast.
- See Clements and Hendry (1998, Ch. 7) for a complete taxonomy of forecast errors. See
also Garrattet al.(2006, Ch. 7).
- We fail to distinguish between random variables and their realizations to avoid introduc-
ing yet more notation; but the meaning should be clear from the context. When it is not,
we clarify.
- Diebold, Gunther and Tay (1998) show that the principle generalizes to the case whenytis
multivariate rather than univariate.
- Alternatively, graphical means of exploratory data analysis are often used to examine the
quality of density forecasts (see Diebold, Gunther and Tay, 1998; Diebold, Tay and Wallis,
1999).
- This also means we do not have to worry about any additional uncertainty introduced
because estimation of the loss differential seriesdt|t−hitself requires parameters (e.g.,μ,
ρandσ) to be estimated.
- Related expressions decomposing the aggregate density (5.40), based on the “law of con-
ditional variances,” are seen in Giordani and Söderlind (2003). This law states that for the
random variablesytandi:V(yt)=E[V(yt|i)]+V[E(yt|i)]. For criticism see Wallis (2005).
- For further discussion of the relationship, if any, between dispersion/disagreement and
individual uncertainty see Bomberger (1996).
- The individual forecasts ofg(yt|it−h)are treated as given. Alternatively, one might esti-
mate a finite mixture where the moments of theg(·)are estimated simultaneously with
wi; see Rafferyet al.(2005) and Geweke and Amisano (2008).
- The marginal likelihood is the product of the one-step-ahead densities: Pr∏ (T|Si)=
T
t= 1 g(yt|it−^1 ).
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