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The solution of an n-dimensional
stochastic differential equation driven by Gaussian white noises is a Markov vector.
In this way, the transition joint probability density function (JPDF) of this
vector is given by a deterministic parabolic partial differential
equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few
exact solutions of this equation so that the analyst must resort to approximate
or numerical procedures. The finite element method (FE) is among the latter,
and is reviewed in this paper. Suitable computer codes are written for the two
fundamental versions of the FE method, the Bubnov-Galerkin and the
Petrov-Galerkin method. In order to reduce the computational effort, which is
to reduce the number of nodal points, the following refinements to the method
are proposed: 1) exponential (Gaussian) weighting functions different from the
shape functions are tested; 2) quadratic and cubic splines are used to
interpolate the nodal values that are known in a limited number of points. In
the applications, the transient state is studied for first order systems only,
while for second order systems, the steady-state JPDF is determined, and it is
compared with exact solutions or with simulative solutions: a very good
agreement is found.
In the paper, a
general framework for large scale modeling of macroeconomic and financial time
series is introduced. The proposed approach is characterized by simplicity of
implementation, performing well independently of persistence and
heteroskedasticity properties, accounting for common deterministic and
stochastic factors. Monte Carlo results strongly support the proposed
methodology, validating its use also for relatively small cross-sectional and
The paper introduces a new simple semiparametric estimator of the conditional variance-covariance and correlation matrix (SP-DCC). While sharing a similar sequential approach to existing dynamic conditional correlation (DCC) methods, SP-DCC has the advantage of not requiring the direct parameterization of the conditional covariance or correlation processes, therefore also avoiding any assumption on their long-run target. In the proposed framework, conditional variances are estimated by univariate GARCH models, for actual and suitably transformed series, in the first step; the latter are then nonlinearly combined in the second step, according to basic properties of the covariance and correlation operator, to yield nonparametric estimates of the various conditional covariances and correlations. Moreover, in contrast to available DCC methods, SP-DCC allows for straightforward estimation also for the non-symultaneous case, i.e. for the estimation of conditional cross-covariances and correlations, displaced at any time horizon of interest. A simple expost procedure to ensure well behaved conditional variance-covariance and correlation matrices, grounded on nonlinear shrinkage, is finally proposed. Due to its sequential implementation and scant computational burden, SP-DCC is very simple to apply and suitable for the modeling of vast sets of conditionally heteroskedastic time series.
The paper introduces a new Frequentist model averaging estimation procedure, based on a stacked OLS estimator across models, implementable on cross-sectional, panel, as well as time series data. The proposed estimator shows the same optimal properties of the OLS estimator under the usual set of assumptions concerning the population regression model. Relatively to available alternative approaches, it has the advantage of performing model averaging exante in a single step, optimally selecting models’ weight according to the MSE metric, i.e. by minimizing the squared Euclidean distance between actual and predicted value vectors. Moreover, it is straightforward to implement, only requiring the estimation of a single OLS augmented regression. By exploiting exante a broader information set and benefiting of more degrees of freedom, the proposed approach yields more accurate and (relatively) more efficient estimation than available expost methods.