Markov Switching Forecast. How can I derive this? Consider the autoregressive model,
$\left[ \begin{array}{l}
  y^{\ast}_t\\
  x_t^{\ast}
\end{array} \right] =  \left[ \begin{array}{l}
  a_{11}\\
  a_{21}
\end{array} \begin{array}{l}
  a_{12}\\
  a_{22}
\end{array} \right] \left[ \begin{array}{l}
  y^{\ast}_{t - 1}\\
  x^{\ast}_{t - 1}
\end{array} \right] + \text{} \left[ \begin{array}{l}
  \varepsilon_t\\
  \upsilon_t
\end{array} \right],$
where $\{ \varepsilon_t \}$ and $\{ v_t \}$ are white-noise processes with
zero mean, $y_t^{\ast}$ and $x_t^{\ast}$ are given by
$\begin{array}{lll}
  y^{\ast}_t & = & y_t - \alpha_1 - \alpha_2 S_t,\\
  x^{\ast}_t & = & x_t - \alpha_3 - \alpha_4 S_t,
\end{array}$
and $\{ S_t \}$ follows a two-state Markov process with transition
probabilities
$\begin{array}{lll}
  p & = & P ( S_t = 1 |  S_{t - 1} = 1),\\
  q & = & P ( S_t = 0 | S_{t - 1} = 0 ) .
\end{array}$
Derive the expected value of $y_{t + n}$ conditional on information available
at time $t$ about the current and past values of $( y_t, x_t)$ and the current
value of $S_t$, i.e., 
$E ( y_{t + n} |_{}  y_t, y_{t - 1}, \ldots .,
y_1, x_t, x_{t - 1}, \ldots ., x_1, S_t)$.
 A: My attemp is the following:
From the system i derived
$\begin{array}{lll}
  y^{\ast}_{t + n} & = & a_{12} \sum_{j = 0}^{\infty} a_{11}^j x_{t + n - j -
  1}^{\ast} + \sum_{j = 0}^{\infty} a_{11}^j \varepsilon_{t + n - j}\\
  & = & a_{11}^n y^{\ast}_t + a_{12} \sum_{j = 0}^{n - 1} a_{11}^j x_{t + n -
  j - 1}^{\ast} + \sum_{j = 0}^{n - 1} a_{11}^j \varepsilon_{t + n - j}\\
  y_{t + n} - \alpha_1 - \alpha_2 S_{t + n} & = & a_{11}^n ( y_t - \alpha_1 -
  \alpha_2 S_t) + a_{12} \sum_{j = 0}^{n - 1} a_{11}^j ( x_{t + n - j - 1} -
  \alpha_3 - \alpha_4 S_{t + n - j - 1}) + \sum_{j = 0}^{n - 1} a_{11}^j
  \varepsilon_{t + n - j}
\end{array}$
Then,
$y_{t + n} = \alpha_1 ( 1 - a_{11}^n) + \alpha_2 ( S_{t + n} - a_{11}^n S_t) +
a_{11}^n y_t + a_{12} \sum_{j = 0}^{n - 1} a_{11}^j ( x_{t + n - j - 1} -
\alpha_3 - \alpha_4 S_{t + n - j - 1}) + \sum_{j = 0}^{n - 1} a_{11}^j
\varepsilon_{t + n - j}$
Taking expectations conditional on information at time t:
\begin{equation}
  E ( y_{t + n} |  I_t) = \alpha_1 ( 1 - a_{11}^n) + \alpha_2 ( E (
  S_{t + n} |  I_t) - a_{11}^n S_t) + a_{11}^n y_t + a_{12} \sum_{j
  = 0}^{n - 1} a_{11}^j ( E ( x_{t + n - j - 1} |  I_t) - \alpha_3 -
  \alpha_4 E ( S_{t + n - j - 1} | I_t ))
\end{equation}
