# 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. I am stuck trying to figure out the equations.

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.

• Start writing $E[X_{t+1}|X_t,Y_t]$ Oct 30, 2022 at 7:57
• Alternatively, start by $\mathbb{E}(X_{t+k}\mid X_{t+k-1},Y_{t+k-1})$ and then substitute for the unknown $X_{t+k-1},Y_{t+k-1}$ and continue this way (recursively) until you only have data that you already know (indexed by $t$ or lower) on the right hand side. Oct 30, 2022 at 8:47
• Cross-posted on Economics SE: economics.stackexchange.com/questions/53308 Oct 30, 2022 at 12:29
• @RichardHardy sorry for the cross-post, I wasn't sure which platform is better to ask the question on. Oct 30, 2022 at 15:33
• @eddie, I just included a link to the other post so that people could see answers and comments from both threads. E.g. if you get an answer on Economics SE, let the Cross Validated users check it out. Oct 30, 2022 at 16:24