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I am working with a VAR and trying to understand the dynamics of it for forecasting.

Currently, I am trying to generate conditional forecasts by expressing the equations in the form of conditional expectations.

So, I need to find $E[Y_{t+k}|X_t,Y_t]$ and $E[X_{t+k}|X_t,Y_t]$ for $k=1,2,3$,

where $Y_t = a + bY_{t-1} + cX_{t-1} + e_t$ and $X_t = m + nY_{t-1} + pX_{t-1} + u_t$.

I am having a hard time writing out the equations for $E[Y_{t+k}|X_t,Y_t]$ and $E[X_{t+k}|X_t,Y_t]$ because I am not sure what formula to apply or how to start.

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  1. You need to know the properties of conditional expectations. Learn them if you don't.
  2. Write the expressions for $Y_{t+k}$ and for $X_{t+k}$.
  3. Apply the conditional expectation in the expressions in 2. and use its properties per 1.
  4. If you have a stochastic assumption on the error terms $e_t$ and $u_t$ in relation to $Y_t$ and $X_t$, use it to arrive at a final result.
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  • $\begingroup$ the properties of conditional expectations is what I'm having a hard time understanding. I'm not quite sure how they work when computing for $E(X | A, B)$, where $X$ is conditional on $A$ and $B$ $\endgroup$
    – eddie
    Oct 30, 2022 at 16:00
  • $\begingroup$ @eddie Just a larger sigma algebra. $\endgroup$ Oct 30, 2022 at 17:09

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