Density Estimation and Combination under Model Ambiguity

6.4 In and Out-of-sample performance of model combination

Lets first consider the in-sample performance of model combination using the entire dataset from December
1, 1969 to October 31, 2001. In this case, after normalizing the p(K^dIj), the double Gamma G(1.1104, 146.38)
receives a weight of 0.5359 and the Normal N (0.0004, (0.0082)²) receives a weight of 0.4641. The Kullback-
Leibler distance between the nonparametric density estimate and the model combination equals 0.0256,
attaining a loss almost half of the best minimizer. If I consider the sample including all expansions, to the
Gamma G(1.1212, 160.68) it is assigned a weight equal to 0.5243 and to the Normal N (0.0005, (0.0075)²) a
weight of 0.4757. This model combination delivers a distance from the nonparametric density equal to 0.0179
which is a third of that achieved by the best model. Finally, considering only contraction data, the Gamma
G(1.1237, 97.42) receives a weight of 0.4937, while the Normal N (-0.0008, (0.0123)²) attains a weight equal
to 0.5063. In this case as well, the model combination outperforms the best model by achieving a KI equal
to 0.0137, which is one fourth of the distance achieved by the best model.

Now, to verify the performance of NPQMLE and of the model combination out of sample, we analyze
the previous results in the context of a different dataset, using the series of stock returns observed from
November 1, 2001 to September 30, 2003, for a total number of observations of 479. This sample represents
the most recent case of expansion, or more precisely recovery, according to the latest determination of the
Business Cycle Committee of the NBER. The summary statistics are displayed below.

Using this data, but the parameter estimates and the weights obtained from the expansion sample for
the period December 1, 1969 to October 31, 2001, we evaluate the KI distance between the nonparametric
density estimated in the new sample and the parametric model estimated in the previous sample. I obtain
the following results: the KI between the model combination and f^bn is equal to 0.7649, between the Gamma
distribution and f^bn is equal to 0.7749 and between the Normal and f^bn is 0.9235. That is, the model
combination slightly outperforms both models, including the Gamma that in the case of expansion was the
best minimizer. This result can be further corroborated using a larger out-of sample dataset and bootstrap
methodology.

7 Investors’ optimal asset allocation

7.1 The Optimization Problem

In this section, I first briefly describe the framework to derive the optimal portfolio choice under ‘uncertainty’,
when an investor uses the similarity-weighted predictive distribution as the model on the basis of which to
act. Second, I consider how the estimated model combination affects investor’s optimal asset allocation.

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