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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.,

$E ( y_{t + n} |_{} y_t, y_{t - 1}, \ldots ., y_1, x_t, x_{t - 1}, \ldots ., x_1, S_t)$.

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    $\begingroup$ Is this homework? And the plus sign in the first equation is unnecessary or something is missing? $\endgroup$
    – mpiktas
    Oct 8, 2013 at 13:42

1 Answer 1

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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}

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