<p><span class="math-container">$$Var[y_t] = E[(u_t+\theta_1u_{t-1}+\dots+\theta_1^{t-1}u_1)^2] = E[u_t^2]+\theta_1^2E[u_{t-1}^2]+\theta_1^4 E[u_{t-2}^2] + \dots +\theta_1^{2t-2}E[u_1^2]$$</span>
the latter equality follows from the assumption that <span class="math-container">$u_t$</span> is not serially correlated (i.e. <span class="math-container">$E[u_{i}u_{j}] = 0\ \forall\ i \neq j$</span>). Then, since <span class="math-container">$E[u_{t}^2] = \sigma^2\ \forall\ t$</span> it follows that
<span class="math-container">$$Var[y_t] = \sigma^2(1+\theta_1^2 + \theta_1^4+\dots+\theta_1^{2t-2})$$</span>
it remains to find <span class="math-container">$(1+\theta_1^2 + \theta_1^4+\dots+\theta_1^{2t-2})$</span> which is equal to <span class="math-container">$\frac{1-\theta_1^{2t}}{1-\theta_1^2}$</span> according to <a href="https://en.wikipedia.org/wiki/Geometric_series" rel="nofollow noreferrer">geometric series sum formula</a>.</p>
<p>Thus, <span class="math-container">$$Var[y_t] = \sigma^2\frac{1-\theta_1^{2t}}{1-\theta_1^2} $$</span></p>

CC BY-SA 4.0
2021-01-12T06:02:57.223Z
2021-01-12T06:02:57.223Z
D F
<p>I wondered if someone could help me in terms of the required algebraic steps from expressions (3) - (4), for the the moving average representation of the AR (1) below?</p>
<p>Would be appreciated.</p>
<p><span class="math-container">$$y_t=a +\theta_1y_{t-1}+u_t$$</span></p>
<p><span class="math-container">$$var[y_t]= E([(u_t+\theta_1u_{t-1}+\theta^2u_{t-2}+...+\theta^{t-1}u_1)]^2)$$</span></p>
<p><span class="math-container">$$var[y_t]= \sigma^2(1+\theta^2+\theta^4+...+\theta^{t-1})$$</span></p>
<p><span class="math-container">$$var[y_t]= \sigma^2\frac{ 1-\theta_1^{2t}}{1-\theta_1^2}$$</span></p>

<mathematical-economics><time-series>
Variance in the Context of a AR (1) Model
EB3112
# Variance in the Context of a AR (1) Model

(Asked by: username_0 on 2021-01-11)

I wondered if someone could help me in terms of the required algebraic steps from expressions (3) - (4), for the the moving average representation of the AR (1) below?

Would be appreciated.

$$y\_t=a +\theta\_1y\_{t-1}+u\_t$$

$$var[y\_t]= E([(u\_t+\theta\_1u\_{t-1}+\theta^2u\_{t-2}+...+\theta^{t-1}u\_1)]^2)$$

$$var[y\_t]= \sigma^2(1+\theta^2+\theta^4+...+\theta^{t-1})$$

$$var[y\_t]= \sigma^2\frac{ 1-\theta\_1^{2t}}{1-\theta\_1^2}$$

**Tags:** mathematical-economics, time-series

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## 1 answers

### Answer by username_1 on 2021-01-12. Score: 1 | Accepted answer

$$Var[y\_t] = E[(u\_t+\theta\_1u\_{t-1}+\dots+\theta\_1^{t-1}u\_1)^2] = E[u\_t^2]+\theta\_1^2E[u\_{t-1}^2]+\theta\_1^4 E[u\_{t-2}^2] + \dots +\theta\_1^{2t-2}E[u\_1^2]$$
the latter equality follows from the assumption that $u\_t$ is not serially correlated (i.e. $E[u\_{i}u\_{j}] = 0\ \forall\ i \neq j$). Then, since $E[u\_{t}^2] = \sigma^2\ \forall\ t$ it follows that
$$Var[y\_t] = \sigma^2(1+\theta\_1^2 + \theta\_1^4+\dots+\theta\_1^{2t-2})$$
it remains to find $(1+\theta\_1^2 + \theta\_1^4+\dots+\theta\_1^{2t-2})$ which is equal to $\frac{1-\theta\_1^{2t}}{1-\theta\_1^2}$ according to [geometric series sum formula](https://en.wikipedia.org/wiki/Geometric_series).

Thus, $$Var[y\_t] = \sigma^2\frac{1-\theta\_1^{2t}}{1-\theta\_1^2} $$


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