Recursive Substitution in Time Series Would appreciate some guidance on a matter of recursive substitution, where we have the AR model:
$$y_t = \alpha +\theta_1y_{t-1}+ u_t$$
And
$$E(y_t)= \mu_t$$
Where:
$$\mu_t = (1+\theta_1 + \theta_1^2+..+\theta^{t-1})\alpha+\theta^ty_0$$
It follows by recursive substitution we get:
$$y_t = \mu_t +(u_t +\theta_1u_{t-1}+\theta^2u_{t-2}+...+\theta^{t-1}u_1)$$
And subsequently:
$$E[y_t] = E[\mu_t]+ E[(u_t +\theta_1u_{t-1}+\theta^2u_{t-2}+...+\theta^{t-1}u_1)]= \mu_t$$
Would someone be able to explain how the step concerning recursive substitution is obtained, moving from line 2,3,4?
\begin{align}y_t &amp;= \alpha + \theta_1y_{t-1}+u_t \\
&amp;= \alpha+\theta_1(\alpha + \theta_1y_{t-2}+u_{t-1}) + u_{t} \\
&amp;= (1+\theta_1) \alpha + \theta_1^2y_{t-2} + \theta_1u_{t-1}+u_{t} \\
&amp;=  (1+\theta_1) \alpha + \theta_1^2(\alpha + \theta_1y_{t-3}+u_{t-3}) + \theta_1u_{t-1}+u_{t} \\
&amp;=  (1+\theta_1 + \theta_1^2) \alpha + \theta_1^3y_{t-3} + \theta_1^3u_{t-3}+\theta_1u_{t-1}+u_{t} \\
&amp;= \dots \\
&amp;= (1+\theta_1+\dots+\theta_1^{t-1})\alpha+\theta_1^ty_0+\theta_1^{t}u_{0}+\dots+\theta_1u_{t-1}+u_{t} \\
&amp;= \mu_t+\theta_1^{t}u_{0}+\dots+\theta_1u_{t-1}+u_{t}
\end{align}
Hence
$$E[y_t] = E[\mu_t]+\theta_1^{t}E[u_{0}]+\dots+\theta_1E[u_{t-1}]+E[u_{t}] = E[\mu_t]$$ since $E[u_{t}] = 0\ \forall\ t$
Finally, $$E[\mu_t] = (1+\theta_1+\dots+\theta_1^{t-1})\alpha+\theta_1^tE[y_0] = (1+\theta_1+\dots+\theta_1^{t-1})\alpha+\theta_1^ty_0 = \mu_t$$ where we assume that $y_0$ is non-random.