Density Estimation and Combination under Model Ambiguity

because of its asymmetry, the fat tails and the sharp peak around zero. Third, Diebold-Gunther and Tay
(1998) in their application to density forecasting of daily S&P 500 returns indicate that the Normal forecasts
are severely deficient. Finally, Knight-Satchell and Tran (1995) show that scale Gamma distributions are a
very good model for UK FT100 index.

Given these facts, let assume that the set of candidate models for the risky asset’s returns consists of three
distributions: a Normal (N(μ, σ²)), a Fisher-Tippet¹⁸ (F(α, β)) and a mixture of general Gamma (G(ς, λ)).
The first model, as described above, derives from the ‘convenient’ version of random walk hypothesis. The
second model is suggested by the empirical evidence reported in the first two points which advocates the use
of extreme value distribution with more probability mass in the tail areas, and the third model is a direct
consequence of the study by Finally, Knight-Satchell and Tran (1995).

Let Xt be the log of asset return for day t, it will be modelled using the following densities:

1) f (Xt; μ,σ) ≡

exp -

2) f (Xt; α,β) ≡ ¹ exp(Xt ^α) exp(- exp(Xt ^α)).
ββ    β

The third model requires some more details since Gamma distribution is defined only for 0 ≤ Xt ≤∞
, as such the distribution for X_t will be a mixture of two Gammas. Following the authors, lets define the
variable:

1 with probability p

^t 0 with probability 1-p

where p is the proportion of returns that are less than a specified benchmark γ. It then follows that Xt is
defined

Xt =γ+X1t(1-Zt)-X2tZt

where Xj_t are independent random variables with density fj (∙). Hence if Z_t = 1, X_t ≤ γ and we sample
from the X₂ distribution; if Z_t = 0, X_t > γ and we sample from the X₁ distribution. f₁(∙) and f₂(∙) are

defined as follow:

λ^ς

^fι(Xιt;^{ς, λ}) ≡ r(ς) ^(Xit ^- γ)^ς 1 ^exP(^-λ(Xιt ^- Y))

λ^ς

f2(X2t; ς,λ) ≡ Γ(ς)(γ - X2t)^ς-1 exp(-λ(γ - X_2t))

6.2 The Data

To implement the empirical application I use daily closing price observations on the US S&P500 index over
the period from December 1, 1969 to October 31, 2001, for a total of 7242 observations. The source of the
data is DRI. Stock return X_t is computed as log(1 + R_t) where R_t = ^tpt ^t1^-1. Descriptive statistics for the
entire sample are provided in the following table.

¹⁸It is also known as double exponential distribution and a particular case of it is the Gumbel distribution.

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