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)$.

**Answer**

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}

**Attribution***Source : Link , Question Author : Julian Lopez Baasch , Answer Author : Julian Lopez Baasch*