Type 0
The measurement model is given by
$$
\mathbf{y}_{i, t}
=
\boldsymbol{\nu}
+
\boldsymbol{\Lambda}
\boldsymbol{\eta}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Theta}
\right)
$$
where
\(\mathbf{y}_{i, t}\),
\(\boldsymbol{\eta}_{i, t}\),
and
\(\boldsymbol{\varepsilon}_{i, t}\)
are random variables
and
\(\boldsymbol{\nu}\),
\(\boldsymbol{\Lambda}\),
and
\(\boldsymbol{\Theta}\)
are model parameters.
\(\mathbf{y}_{i, t}\)
represents a vector of observed random variables,
\(\boldsymbol{\eta}_{i, t}\)
a vector of latent random variables,
and
\(\boldsymbol{\varepsilon}_{i, t}\)
a vector of random measurement errors,
at time \(t\) and individual \(i\).
\(\boldsymbol{\nu}\)
denotes a vector of intercepts,
\(\boldsymbol{\Lambda}\)
a matrix of factor loadings,
and
\(\boldsymbol{\Theta}\)
the covariance matrix of
\(\boldsymbol{\varepsilon}\).
An alternative representation of the measurement error
is given by
$$
\boldsymbol{\varepsilon}_{i, t}
=
\boldsymbol{\Theta}^{\frac{1}{2}}
\mathbf{z}_{i, t},
\quad
\mathrm{with}
\quad
\mathbf{z}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\mathbf{I}
\right)
$$
where
\(\mathbf{z}_{i, t}\) is a vector of
independent standard normal random variables and
\(
\left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)
\left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)^{\prime}
=
\boldsymbol{\Theta} .
\)
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
\(
\left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)
\left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)^{\prime}
=
\boldsymbol{\Psi} .
\)
Type 1
The measurement model is given by
$$
\mathbf{y}_{i, t}
=
\boldsymbol{\nu}
+
\boldsymbol{\Lambda}
\boldsymbol{\eta}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Theta}
\right) .
$$
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.
Type 2
The measurement model is given by
$$
\mathbf{y}_{i, t}
=
\boldsymbol{\nu}
+
\boldsymbol{\Lambda}
\boldsymbol{\eta}_{i, t}
+
\boldsymbol{\kappa}
\mathbf{x}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Theta}
\right)
$$
where
\(\boldsymbol{\kappa}\) represents the coefficient matrix
linking the covariates to the observed variables.
The dynamic structure is given by
$$
\boldsymbol{\eta}_{i, t}
=
\boldsymbol{\alpha}
+
\boldsymbol{\beta}
\boldsymbol{\eta}_{i, t - 1}
+
\boldsymbol{\Gamma}
\mathbf{x}_{i, t}
+
\boldsymbol{\zeta}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\zeta}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Psi}
\right) .
$$