<ol>
<li>You need to know the properties of conditional expectations. Learn them if you don't.</li>
<li>Write the expressions for <span class="math-container">$Y_{t+k}$</span> and for <span class="math-container">$X_{t+k}$</span>.</li>
<li>Apply the conditional expectation in the expressions in 2. and use its properties per 1.</li>
<li>If you have a stochastic assumption on the error terms <span class="math-container">$e_t$</span> and <span class="math-container">$u_t$</span> in relation to <span class="math-container">$Y_t$</span> and <span class="math-container">$X_t$</span>, use it to arrive at a final result.</li>
</ol>

CC BY-SA 4.0
2022-10-30T16:00:52.850Z
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 $B$
eddie
CC BY-SA 4.0
2022-10-30T17:09:35.650Z
@eddie Just a larger sigma algebra.
Alecos Papadopoulos
CC BY-SA 4.0
2022-10-30T13:43:27.157Z
2022-10-30T14:42:27.110Z
2022-10-30T14:42:27.110Z
Alecos Papadopoulos
<p>I am working with a VAR and trying to understand the dynamics of it for forecasting.</p>
<p>Currently, I am trying to generate conditional forecasts by expressing the equations in the form of conditional expectations.</p>
<p>So, I need to find <span class="math-container">$E[Y_{t+k}|X_t,Y_t]$</span> and <span class="math-container">$E[X_{t+k}|X_t,Y_t]$</span> for <span class="math-container">$k=1,2,3$</span>,</p>
<p>where <span class="math-container">$Y_t = a + bY_{t-1} + cX_{t-1} + e_t$</span> and <span class="math-container">$X_t = m + nY_{t-1} + pX_{t-1} + u_t$</span>.</p>
<p>I am having a hard time writing out the equations for <span class="math-container">$E[Y_{t+k}|X_t,Y_t]$</span> and <span class="math-container">$E[X_{t+k}|X_t,Y_t]$</span> because I am not sure what formula to apply or how to start.</p>

CC BY-SA 4.0
2022-10-30T12:30:19.630Z
Cross-posted on Cross Validated: https://stats.stackexchange.com/questions/594006 (and received a couple of hints there).
Richard Hardy
<macroeconomics><econometrics><mathematical-economics><regression><forecasting>
Write equations for $E[Y_{t+k}|X_t,Y_t]$ and $E[X_{t+k}|X_t,Y_t]$
eddie
# Write equations for $E[Y_{t+k}|X_t,Y_t]$ and $E[X_{t+k}|X_t,Y_t]$

(Asked by: username_0 on 2022-10-30)

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.

**Tags:** macroeconomics, econometrics, mathematical-economics, regression, forecasting

> **Comments:**
> **username_1**: Cross-posted on Cross Validated: https://stats.stackexchange.com/questions/594006 (and received a couple of hints there).  

---

## 1 answers

### Answer by username_2 on 2022-10-30. Score: 2

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.

> **Comments:**
> **username_0**: 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 $B$  
> **username_2**: @username_0 Just a larger sigma algebra.  

---

