Recursive Substitution in Time Series

Question

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?

D F · Answer

begin{align}y_t &= alpha + theta_1y_{t-1}+u_t 
&= alpha+theta_1(alpha + theta_1y_{t-2}+u_{t-1}) + u_{t} 
&= (1+theta_1) alpha + theta_1^2y_{t-2} + theta_1u_{t-1}+u_{t} 
&=  (1+theta_1) alpha + theta_1^2(alpha + theta_1y_{t-3}+u_{t-3}) + theta_1u_{t-1}+u_{t} 
&=  (1+theta_1 + theta_1^2) alpha + theta_1^3y_{t-3} + theta_1^3u_{t-3}+theta_1u_{t-1}+u_{t} 
&= dots 
&= (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} 
&= 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.

Recursive Substitution in Time Series

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