Good morning, I have a huge problem that I can't solve, I hope someone can help me. I have the following process: $X=(X_t)$ with $t ∈N$ that is a time-discrete stochastic process defined as $X_t=0.5X_{t-1}+e_t$ with $e_t$ i.i. distributed following a normal with parameters N(0,B) each t=1,2,... with B>0. I have to verify:
- 1) if $E(X_t)$ cannot be computed each $t>=0$; I have computed $E(X_t)$ and I find that is equal to $0.5E(X_{t-1})$, but I don't know if this shows that is possibile to compute $E(X_t)$.
- 2) X is not weakly stationary: I have to prove that $E(X_t)$ is costant, but for me is not, and that $cov(s,s+h)=cov(t,t+h)$ but I have no idea how to compute it (this point is very important, I hope you will give me some advice in computing cov)
- 3)$Cov(X_t,X_{t-1})$ is different from B, same problem of point 2, I don't know how to compute cov
- 4) X is strictly stationary: I have to prove that X's at time $t_i$ are distributed as X's at time $t_i+h$ Thank to everyone, your help will be very fruitful.
Answers could be found in books on time-series analysis. The equation $X_t=0.5X_{t-1}+e_t$ is an auto-regressive model of order 1. The notation used is AR(1).
In general, if you do not specify $X_0$, then it is not possible to compute the required quantities. Rephrasing the question: Does there exist a stationary process that satisfies the recursive equation? If so, compute its variance and auto-correlation.
Define $Y_t:=\sum_{i=0}^\infty 0.5^je_{t-j}$ where $e_j\sim N(0,\sigma_{e}^2)$ iid. Then $Y$ satisfies the recursive equation $Y_t=0.5Y_{t-1}+e_t$. The process $Y$ is strictly stationary as well. Note that $E[Y_t]=0$. For the variance you have $var(Y_t)=\sum_{j=0}^\infty 0.5^{2j}\sigma_{e}^2 = \frac43\sigma_{e}^2$. $$cov(Y_t,Y_{t-1})=cov(0.5Y_{t-1}+e_t,Y_{t-1})=0.5 cov(Y_{t-1},Y_{t-1})+cov(e_t,Y_{t-1})=0.5 \frac43\sigma_{e}^2+0$$