Type 0
The measurement model is given by
$$
\mathbf{y}_{i, t}
=
\boldsymbol{\eta}_{i, t}
$$
where \(\mathbf{y}_{i, t}\)
represents a vector of observed variables
and \(\boldsymbol{\eta}_{i, t}\)
a vector of latent variables
for individual \(i\) and time \(t\).
Since the observed and latent variables are equal,
we only generate data
from the dynamic structure.
The dynamic structure is given by
$$
\boldsymbol{\eta}_{i, t}
=
\boldsymbol{\alpha}
+
\boldsymbol{\beta}
\boldsymbol{\eta}_{i, t - 1}
+
\boldsymbol{\zeta}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\zeta}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Psi}
\right)
$$
where
\(\boldsymbol{\eta}_{i, t}\),
\(\boldsymbol{\eta}_{i, t - 1}\),
and
\(\boldsymbol{\zeta}_{i, t}\)
are random variables,
and
\(\boldsymbol{\alpha}\),
\(\boldsymbol{\beta}\),
and
\(\boldsymbol{\Psi}\)
are model parameters.
Here,
\(\boldsymbol{\eta}_{i, t}\)
is a vector of latent variables
at time \(t\) and individual \(i\),
\(\boldsymbol{\eta}_{i, t - 1}\)
represents a vector of latent variables
at time \(t - 1\) and individual \(i\),
and
\(\boldsymbol{\zeta}_{i, t}\)
represents a vector of dynamic noise
at time \(t\) and individual \(i\).
\(\boldsymbol{\alpha}\)
denotes a vector of intercepts,
\(\boldsymbol{\beta}\)
a matrix of autoregression
and cross regression coefficients,
and
\(\boldsymbol{\Psi}\)
the covariance matrix of
\(\boldsymbol{\zeta}_{i, t}\).
An alternative representation of the dynamic noise
is given by
$$
\boldsymbol{\zeta}_{i, t}
=
\boldsymbol{\Psi}^{\frac{1}{2}}
\mathbf{z}_{i, t},
\quad
\mathrm{with}
\quad
\mathbf{z}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\mathbf{I}
\right)
$$
where
\(
\left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)
\left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)^{\prime}
=
\boldsymbol{\Psi} .
\)
The measurement model is given by
$$
\mathbf{y}_{i, t}
=
\boldsymbol{\eta}_{i, t} .
$$
The dynamic structure is given by
$$
\boldsymbol{\eta}_{i, t}
=
\boldsymbol{\alpha}
+
\boldsymbol{\beta}
\boldsymbol{\eta}_{i, t - 1}
+
\boldsymbol{\Gamma}
\mathbf{x}_{i, t}
+
\boldsymbol{\zeta}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\zeta}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Psi}
\right)
$$
where
\(\mathbf{x}_{i, t}\) represents a vector of covariates
at time \(t\) and individual \(i\),
and \(\boldsymbol{\Gamma}\) the coefficient matrix
linking the covariates to the latent variables.