The simul.farx function simulates a FARX(1) process with one
endogeneous variable, one exogeneous variable and a strong white
noise. To do so, the function uses the fact that a FARX(1) model can
be seen as a FAR(1) model in a wider space. Therefore, the method is
very similar to the one used by the function simul.far.
The simulation is realized in two steps.
First step, the function compute a FAR(1) process \(T_n\) in a
functional space (that we call in the sequel H) using a simple
equation and the given parameters. \(T_n\) is of the form
\((T_{1n},T_{2n})\) where \(T_{1n}\) and
\(T_{2n}\) are respectively the endogeneous and the exogeneous
parts of the process.
Second step, the process \(T_n\) is projected in the canonical
basis using the base and base.exo linear projectors to
give the endogeneous (\(X_n\)) and the exogeneous
(\(Z_n\)) variables respectively.
Those two basis need to be orthonormal and wide enought. In the
contrary, the function use the orthonormalization
function to make it so. Notice that the size of this matrix
corresponds to the dimension of the "modelization space" H (let's call
it \(m_2=m1_2+m2_2\)). Of course, the larger m2
the better the functionnal approximation is. Whatever, keep in mind
that m2=2m is a good compromise, in order to avoid the
memory limits.
In H, the linear operator \(\rho\) is expressed as:
$$
\left(\begin{array}{cc}%
d.rho.mod & \code{d.a} \cr%
0 & d.rho.exo.mod%
\end{array}\right)%
$$
Where d.rho.mod and d.rho.exo.mod are modified version of the provided
d.rho and d.rho.exo respectively to avoid 0 on their
diagonal. More precisely, the 0 on their diaginals are replaced by:
$$\left(\varepsilon_{k+1}, \varepsilon_{k+2}, \ldots, %
\varepsilon_{\code{m2}}\right)$$
where
$$\varepsilon_{i}=\frac{\code{cst1}}{i^2}+%
\frac{1-\code{cst1}}{e^i}$$
and k is the position in the d.rho or d.r.ho.exo
diagonal.
In H, \(C^T\), the covariance operator of \(T_n\), is
defined by:
$$
\left(\begin{array}{cc}%
alpha.mod & alpha.conj.mod \cr%
t(alpha.conj.mod) & alpha.exo%
\end{array}\right)%
$$
Where alpha.mod and alpha.exo.mod are modified versions of
\(m1_2 * \)alpha and \(m2_2 * \)alpha.conj respectively to avoid 0 on their diagonal. More
precisely, the 0 on their diaginals are replaced by:
$$\left(\epsilon_{k+1}, \epsilon_{k+2}, \ldots, %
\epsilon_{\code{m2b}}\right)$$
where
$$\epsilon_{i}=\frac{\code{cst1}}{i}$$
alpha.exo is a matrix representation of the covariance operator of
\(T_{2n}\) and is obtained by inverting the following relation:
$$alpha.conj.mod = d.rho.exo.mod * alpha.conj.mod * t(d.rho.mod) +%
d.rho.exo.mod * mod.alpha * t(\code{d.a})$$
The theoretical.coef function is provided to help the user
making comparison. Calling this function with the same parameters that
where used in a simulation (realized with simul.farx or
simul.far), we obtain the parameters used internaly by the
function to make the simulation. Those values can therefore be
compared to those obtained with the estimation function far
(examples are provided below).