Density Estimation and Combination under Model Ambiguity

by the double Gamma outperforms the other two models. Its KI assumes the lowest value (0.0468) which
is four times smaller than that for the Normal and twenty time smaller than that of Fisher-Tippet. Also in
the case of expansion, the double Gamma is clearly better than the other two models; its KcI equals 0.0666
which is half the value for the Normal. In contrast, for the sample including all contractions the Gaussian
distribution performs slightly better than the double Gamma. The value of its KI is equal to 0.0513 which is
smaller than the respective value for the double Gamma (0.0776). Finally, both values are ten times smaller
than the KI for the Fisher-Tippet distribution. These results contradict the common assumption that the
best unique model for the stock returns is the Gaussian distribution, and confirm that the optimal model
changes across regimes. Further, since more than one model performs fairly well, and because each of them
has properties that capture particular characteristics of return distribution, it seems reasonable to combine
them.

It is important to stress some characteristics of the double Gamma, since it is overall the model that
provides the best performance in terms of aggregate similarity to the data. First of all, it is worth mentioning
that in all three samples the values of pb suggest that the sample proportions for negative returns are not
very different from that of positive returns. Second, ς’s estimates in all three samples are greater than unity,
which entails that returns are well described by a bimodal density. All these features of the estimated model
confirm the results that Knight-Satchell and Tran (1995) found in the case of UK stock returns.

The final step to compute the similarity-weighted predictive distribution M (θM_j ) consists in evaluating
for each of the models under consideration the ‘probability’ pj (KI) of being correct. It can be helpful to
first provide the realizations of K^dIj for all models in each of the sample.

Table V

The following table exhibits the value of p(KIj ) for the three models under consideration.

Table VI

As it can be noticed these values represent ‘probabilities’ before normalization since they do not sum up to
unity. Results contained in table VI seem to confirm that this methodology in determining the “probability
of being the correct model” works in the right direction. In fact, in each of the samples the p.d.f. with
the lowest realization of the KI receives the highest pj (KI), and hence it will receive the largest weight in
the model combination. Further, the very poor performance of the Fisher-Tippet distribution with respect
to the other two candidate models, suggests that it would be sensible to discard this model. Thus, in the
next section I present the results obtained combining the Normal and the double Gamma according to the
weights reported in the first two rows of Table VI.

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