first model is Mi = G(ς, λ), the second model M2 = N(μ, σ2) is a Normal and the third model M3 = W(α, β)
is a Weibull distribution.
We report the results for sample size n = 400, 800 and 1200, when β = 4l5, ɪ, in order to show the
sensitivity of the parameter estimates to the choice of the smoothing parameter.
Table 1 indicates that the estimates of Gamma parameters using NPQMLE perform worse than QMLE,
obviously because under the correct structure QMLE is MLE. In fact, in the case of NPQMLE KI(g, fj (bθ)) =
0.0084 for MLE KI (g, fj (θ)) = 0.0047. This is due to estimation error introduced by the use of the nonpara-
metric density as reference estimate. But, when the assumed model is not the correct ones, as in the case
of Normal and Weibull, the NPQMLE method delivers an estimate of the parametric density that is closer
to the true structure than those estimated by QMLE. As shown in the table, in the case of the Normal,
using NPQMLE KI (g, fj (bθ)) is equal to 0.0233, while for QMLE it equals 0.044. A very similar result is
obtained for the Weibull distribution. Table 2 displays analogous results but, since the bandwidth h is
smaller, NPQMLE provides parameters’ estimates that are characterized by a lower bias and slightly higher
variance. This overall causes a reduction of the distance between misspecified and true model, as it can be
noticed by the lower values of KI(g, fj (θ)).
As the sample size increases, from Table 3 we can notice that NPQMLE delivers an estimate of the
misspecified model that gets closer to the true one. This was expected, since as n increases the nonparametric
density gets closer to the true model and this helps improving the estimation results. Further, in NPQMLE
the distance between g and fj (θ) reduces approximately at the same rate as QMLE. This can be clearly seen
observing for example, the reduction of the KI (g, fj (θ)) for both estimation methods in the case of Gamma.
Nevertheless, under the misspecified models, NPQMLE still outperforms QMLE, in the sense that it still
delivers a KI which is half of that obtained by QMLE.
6 Application to stock returns
6.1 A Set of simple models
I now apply the described prediction method to determine stock returns predictive density, that will subse-
quently be used by an investor to choose the optimal share to invest in the risky asset. Typically, due to the
hypothesis of asset market efficiency, stock prices are assumed to follow a random walk, that is:
pt = μ + pt-i + et, etIID, where pt = Iog(Pt).
Further, since the most widespread assumption for the innovations et is normality, stock returns are
normally distributed with mean μ and variance σ2. While contrasting evidence exists on the predictability
of stock returns, there is substantial support against the normality assumption.
First, as reported by Campbell-Lo-Mackinay (1997)16, the skewness for daily US stock returns tend to
be negative for stock index and positive for individual stocks. Second, the excess Kurtosis for daily US
stock returns is large and positive for both index and individual stocks. Both characteristics are further
documented in Ullah-Pagan17 (1999) using non-parametric estimation of monthly stock returns’ density
from 1834 to 1925. In their analysis is clearly shown that the density departs significantly from a normal,
16 The Econometrics of Financial Markets, 1997, pag. 16 and 17.
17 Nonparametric Econometrics, 1999, pag 71-74.
12