Density Estimation and Combination under Model Ambiguity



it is very striking how these values differ across regimes. First, as found in other studies, contractions and
in general bear regimes are characterized by high volatility and negative mean for stock return, which turns
out to be a problem in determining the optimal share to invest in the risky asset. Second, while during
expansions stock returns show a positive excess kurtosis (even bigger than that displayed in Table I for all
data) and a negative Skewness (three times bigger than that for the entire sample), during contractions
the excess Kurtosis is negative (lower than three) and the Skewness is positive. According to these simple
descriptive statistics, it is reasonable to expect different optimal models for stock returns across these two
regimes.

6.3 Empirical Results.

For each of these samples I estimate the univariate density of stock returns by Nadaraya-Watson kernel
density estimators. For the Kernel function I employ the second-order Gaussian Kernel and the bandwidths
are selected via least-squares cross-validation (Silverman, 1986, p48). Graphs of the nonparametric densities
are reported in the appendix in Figures 1, 2 and 3.

I then use the Kullback-Leibler entropy to measure the distance between the estimated nonparametric
density and each of the models belonging to the set
M. Minimizing this distance I obtain the parameter
estimates for each candidate distribution and a value for KI
j , which allows me to achieve a ranking of all
competing models and the subsequent weight for each of them in the final model combination. The estimated
parameters for each distribution are reported below.

N (μ2)

Entire sample

Expansion

Contraction

^

b______

0.0004*

0.0005*

-0.0008*

^

σ

0.0082*

0.0075*

0.0123*

KLI

0.1897

0.1587

0.0513

*All estimates are significant at 1% level

F (α,β)

Entire sample

Expansion

Contraction

^
α

-0.00179*

-0.0014*

-0.00403*

Jb____

0.008509*

0.00773*

0.01213*

KLI

0.9836

0.9209     '

0.3362    ~

*All estimates are significant at 1% level

G(ς,λ)

Entire sample

Expansion

Contraction

^
ς

1.1104*

1.1212*

1.1237*

b

146.3839*

160.6803*

97.4237*

^

γ_____

0.00031

0.00044

-0.00039

^

p______

0.47878

0.465631

0.53637

1 - p

0.52122

0.5343

0.46363

KLI

0.0468

0.0666

0.0776

*All estimates are significant at 1% level
Table IV

Examining the tables we see that all the estimates are intuitively reasonable and significantly different
from zero. Comparing all the three models over the entire sample, we can notice that the model characterized

15



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