I develop a method of prediction that ranks different probabilistic models and determines the optimal
value of their parameters, maximizing the sum of their similarities to relevant past cases. The similarity
is measured by the opposite of the distance, that is the Kullback-Leibler Information (KI), between the
probabilistic model and the estimated nonparametric density. The final weights used to combine models are
function of these distances which embody the ignorance about the true structure. The cognitive plausibility
of my methodology is founded on case-based decision theory (CBDT). In particular the behavioral axioms of
Inductive Inference2 developed by Gilboa and Schmeilder (2001) provide support for my prediction method.
This estimation approach, being based only on an objective measure of the proximity between multiple
candidate models and actual data, aims to overcome the necessity to have a specific prior over the set of
models and about parameters belonging to each of the models under consideration. It refers only to the
analogy between past samples (actually encountered cases) and models at hand3 . This requires a limited
amount of hypothetical reasoning since it relies directly on data that are available to any observer without
ambiguity.
I apply the proposed method to determine the predictive density of daily stock returns under different
phases of the business cycle and I use the latter to investigate the implications of the model on portfolio
choice under uncertainty. This empirical application is motivated both by the difficulty in estimating the
probability law of asset returns which usually are modelled with misspecified density function, and by the
large availability of data for financial series which facilitates the use of nonparametric techniques.
This way of implementing probabilistic prediction is essential to improve econometric modeling and to
decision making. In fact, my method like others in the literature, can be considered as a preliminary step
to account explicitly for model ambiguity in econometrics. One of the first studies that uses information
criteria to identify the most adequate regression model among a set of alternatives is due to Sawa (1978).
Later contributions by White (1980,‘82) examine the detection and consequences of model misspecification in
Nonlinear Regression Model and MLE. A Subsequent work by Sin and White (1996) uses information criteria
for selecting misspecified parametric models. More recently, a paper by Skouras (2001), investigates the
determination of a predictive density by exploiting the discrepancy between expected utilities under the true
and the misspecified model. Nevertheless, none of these studies makes use of a preliminary nonparametric
estimation to estimate and distinguish among alternative models4 . On the other hand, a study by Cristobal,
Roca and Manteiga (1987) which describes linear regression parameter estimators based on preliminary
nonparametric estimation does not incorporate the assumption of model uncertainty. To my knowledge,
this is the first work which develops an estimation technique via a pilot nonparametric estimate under the
assumption of model ambiguity. Furthermore and more importantly, none of these papers focuses on model
combination.
There are three additional strands of literature related to this work. The first includes Bayesian Model
Averaging (BMA) and its application to stock returns predictability and to the investment opportunity
set, see for example Avramov (2002) and Cremers (2002). Differently from the Bayesian approach, in
this study it is not necessary to assume that the true structure belongs to the set of candidate models.
Further, the implementation of the model combination is computationally extremely easy because it does
not require numerical integration to obtain for each model the ‘probability’ of being correct. The second vein,
though characterized by a completely different approach, represents the studies about forecast evaluation
2As shown in Gilboa-Schmeidler (2001) this is also the same principle at the base of Maximum Likelihood Estimation.
3 However also these models are suggested by past experience or by economic theory.
4 The literature on nonparametric testing provides me the technical machinery to derive the asymptotic distribution of the
KLI. See for example Hall(1984, 1987), Robinson(1991), Fan(1994), Zheng (1996, 2000), and Hong and White(2000).