Density Estimation and Combination under Model Ambiguity

provided by Research Papers in Economics

Density Estimation and Combination under Model Ambiguity via
a Pilot Nonparametric Estimate: an Application to Stock
Returns.

Stefania D’Amico*

First Version: May 2003, This Version: September 2003
Primary Job Market Paper

Abstract

This paper proposes a method to estimate the probability density of a variable of interest in
the presence of model ambiguity. In the first step, each candidate parametric model is estimated
minimizing the Kullback-Leibler ‘distance’ (KLD) from a reference nonparametric density esti-
mate. Given that the KLD represents a measure of uncertainty about the true structure, in the
second step, its information content is used to rank and combine the estimated models. The
paper shows that the resulting parameters estimator is root-n consistent and asymptotically
normally distributed. The KLD between the nonparametric and the parametric density esti-
mates is also shown to be asymptotically normally distributed. This result leads to determine
the weights in the model combination, using the distribution function of a Normal centered on
the average performance of all plausible models. As such, this combination technique does not
require that the true structure belongs to the set of competing models and is computationally
simple. I apply the proposed method to estimate the density function of daily stock returns
under different phases of the business cycle. The results indicate that the double Gamma dis-
tribution is more adequate than the Gaussian distribution in modeling stock returns, and that
the models combination outperforms in- and out-of-sample each individual candidate model. I
also explore the model’s implications for the optimal share to invest in the risky asset.

*I am indebeted to Phoebus Dhrymes for his guidance and many interesting discussions. I wish to thank Jean
Boivin, Xiaohong Chen, Mitali Das, Rajeev Dehejia, Marc Henry and Alexei Onatski for valuable comments. I would
also like to thank Stefano Eusepi and Mira Farka, without their constant support this paper would have never existed.
Address: Department of Economics, Columbia University. E-mail: sd445@columbia.edu

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