Density Estimation and Combination under Model Ambiguity

Table I

Furthermore, Ang and Bekaert (2001,2002) and Guidolin and Timmermann (2002) have stressed the
importance of distinguishing between ‘bear’ and ‘bull’ regimes in modeling stock returns and indicate that
these persistent regimes have important economic implications for investors’ portfolio decisions. Based on
these observations, I have chosen to divide the data in two groups. The first contains all samples relative
to contraction (C) and the second includes all samples relative to expansion (E). These two phases of the
business cycle typically coincide with ‘bear’ and ‘bull’ regimes of the stock market. This implies that the
optimal model for asset returns is conditional on the specific regime, which for simplicity I assume to be
known at the time of the empirical analysis. The next table indicates the periods of contraction and expansion
included in the dataset.¹⁹

Table II

Under the assumption that in each regime all subsamples are drawn from a fixed distribution, it is possible
to create for each state a unique sample that include all contractions and all expansions respectively. Merging
together all the recessions we obtain a sample of 1321 observations, while combining all expansions we obtain a
sample of 5921 observations. The descriptive statistics for these two subsamples are reported in the following
tables.

It is evident from Table I and III, that these data are not consistent with the common assumption that
the true model for X_t is the Gaussian distribution. These values confirm previous studies where daily stock
returns have been found to exhibit high excess Kurtosis and negative Skewness for index returns. Further,

¹⁹ The contractions and expansions are those provided by NBER’s Business Cycle D ating Committee for the US Economy,
available at the website www.nber.org/cycles. I have excluded the recession of 1990-91 because of the limited number of
observations that does not allow to estimte the nonparametric density with precision in that subsample.

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