Question

5. For a stationary times series with σ,# 10 and au ocorrelations pr.k-: 0.9, k > 0 ob- Var). Also, write down the values of these two variances (Var() and Varrs))) if the autocorrelations are set to zero (psk0, k>0). Contrast the results for the two cases (Pk0.9% vs. Pr0, for k > 0) and point out a significant difference. [10 marks)

Answer 1

Given that, $\sigma_{x}^2=10$ and $\rho _{x,k}=0.9^k$ where k>0

We know that for AR(1) process autocorrelation function is $\phi ^k$

so, we select AR(1) function:

$X_t=\phi*X_{t-1}+\epsilon _t$   where $\epsilon _t\sim N(0,\sigma^2)$ iid

and we also know in this case

$E[x]=\frac{\mu }{1-\phi }$   and $var[x]=\frac{\sigma^2}{(1-\phi)^2 }$

so, $var[\bar{x}]=\frac{\sigma^2}{(1-\phi)^2*n }$

Here, n=5, $\sigma_{x}^2=10$,,$\phi=0.9$

$var[\bar{x}]=200$

$var[x_1]=1000=var[x_5]$

$var[\frac{1}{2}(x_1+x_5)]=1000$

when $\rho =0$ means $\phi =0$

$var[\bar{x}]=2$

$var[\frac{1}{2}(x_1+x_5)]=10$