On the independence of errors in time series

I recently started to learn statistics, and I've stumbled upon the idea of autoregressive time series. My question relates to the assumption of independence of the time series white noise process: let $$y_t = \alpha \, y_{t-1} + u_t$$ be an (at least) second order stationary time series, where $u$ is white noise. Reading online, it seems like all the $u$'s are to be taken as i.i.d., $u\sim \mathcal N(0, \sigma^2)$, so I'd expect $\langle u_t\,u_s\rangle = \sigma\,\delta_{i,j}$. However, playing around with the defining equation above, I get $$\langle u_t\,u_{t+1}\rangle = \langle(y_t - \alpha\,y_{t-1})(y_{t+1} - \alpha\,y_{t})\rangle\\ =(1+\alpha^2)\langle y_t\,y_{t+1}\rangle - \alpha\langle y_t\,y_{t+2}\rangle\\ =\sigma^2\,\frac{\alpha}{1-\alpha^2}$$ where I used the standard formulae for the correlation of $y_t\,y_s$, the fact that $\langle y_t \rangle=0$ and the wide-sense stationariety of the process. From this few equality it looks like the errors $u$ are really independent only if alpha vanishes, in which case the entire process is white noise. In the more general case where $|\alpha|<1,\,\alpha\neq 0$ there seems to be correlation. Looking at this on a different angle, it is true that I can also write $$y_t = \alpha^2\,y_{t-2}+\alpha\,u_{t-1}+u_t$$ and thus I have an equation that (when alpha is non-zero) involves both $u_t$ and $u_{t-1}$. Could you please help me understand my error in the above reasoning? Thank you all

• What is your definition of $\langle . \rangle$? – Matthew Gunn Apr 16 '17 at 23:30
• Hi, it's the expected value of whatever sits inside, so $\langle y_t\rangle$ would just be the mean value of the process – tg_89 Apr 16 '17 at 23:34
• What happened to $\alpha \langle y_t^2 \rangle$? That $\{u\}$ is a white noise process is an assumption that's used to calculate the auto-covariance function. If you're showing that $E[u_ju_i] \neq 0$ for $i\neq j$, then there's some error in your algebra. – Matthew Gunn Apr 16 '17 at 23:35
• ... I forgot to write it down and wasted everyone's time asking this question! Thank you very much for pointing that out! Since we're at it, could you please comment on the second part of the question? – tg_89 Apr 16 '17 at 23:44
• What's the last part of the question? In an AR(1) model, $\{u_t\}$ is a white noise process (i.e. $\operatorname{E}[u_t] = 0$ and $\operatorname{E}[u_tu_{t+j}] = 0$ for $j \neq 0$). Process $\{y_t\}$ by construction exhibits auto-correlation. Your last bit of algebra is going down the road of writing the MA($\infty$), Wold Representation, of an AR(1). – Matthew Gunn Apr 16 '17 at 23:52

• I think you had a small typo / error and dropped a $\alpha E[y_t^2]$ term.
• Your second bit of algebra is moving towards writing an AR(1) as an MA($\infty$) process, that is, the AR(1)'s World Representation.