Density Estimation and Combination under Model Ambiguity

The economic agents use the personal model g_r|_s as if it represented the actual model. The problem
is that assuming g_r|_s to be the density function of a Normal random variable, is likely that the model is
misspecified.

What is described next is a characterization of expected utility maximization under ‘risk’ very similar
to Gilboa-Schmilder(2001), that is the decision process is one in which the “decision maker first forms
probabilistic beliefs and then uses them to reduce a decision problem under uncertainty to a decision problem
under risk”. In other words, the use of models selection and combination described in previous sections
helps reducing the degree of model ambiguity, because it shrinks the set of candidate models into a unique
distribution that characterizes the risk of the decision problem.

Example 2: CARA investors with probabilistic belief M (θM_j ). In the context of example 1, lets suppose
that the investor instead of assuming that the returns are Normally distributed, builds his probabilistic belief
g_r/_s as described in section II and III. Moreover, lets assume that the current regime is known. Therefore,
his model for the asset return is equal to M(θM_j ).

EM(bM_j)[^exP(^-δar)] = ^-ʃ^eχp(^-δar) ∙ (XPj(KCI)^fj(r^,b)))^dr

= - ^Xpj(KcI)  exp(-δar)fj (r, ^bθ))dr = ^X pj (KcI)(-Ef_j (exp(-δar).

jj

If we define t = δa we can rewrite the expected utility function as -E[exp(-tr)], which is the Moment
Generating Function²⁰ (MGF) after we account for the change in signs. This implies that the expected
utility function, when the expectation is taken under the model M (θ_Mj ) is equal to

-    pj(KCI)MGFfj(-t),

j

which is nothing more than the weighted average of the Moment Generating Function of each model included
in M(^bθMj ).

It is important to notice that the existence of a closed form solution for the optimal share a is still
guaranteed. However, in this framework the optimal choice not only depends on the risk aversion and on
the moments of the probability law of stock returns, but it also depends on the weights contained in the
model combination. As such, it is affected by the measure of uncertainty about the true structure. Explicit
formulas for each MGF and for the expected utility are provided in the Appendix.

²⁰The Moment Generating Function MGF_r(t) = E_r(exptr).

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