# Write equations for $E[Y_{t+k}|X_t,Y_t]$ and $E[X_{t+k}|X_t,Y_t]$

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.

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.
• 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 $B$ Oct 30, 2022 at 16:00
• @eddie Just a larger sigma algebra. Oct 30, 2022 at 17:09