Live Autoregressive Process

This post join two things that I like: Highcharts and the Time series. So I have a lot of fun making this minipost to show how works an autoregressive process. Well, let’s remember the structure of an autoregresive process. $y_t = varphi,y_{t-1}+epsilon_t$, where $epsilon_t$ is a white noise, i.e, a $cov(epsilon_j, epsilon_i) = 0$ if $i neq j$, $Var(epsilon_t) = sigma^2_epsilon$, and $varphi$ is a real parameter.

Now, let’s see first how a white noise looks like. The next chart show the process $y_t = epsilon_t$, so if we see for a moment the chart we’ll notice the process look likes an independent random sequence. There’s no structure of any dependence.

The two charts above show autoregressive process for two distinct values of $varphi$. It’s easy to see the difference. In the first process the next value of the serie tends to have the same sign of the past value. In the other procces ocurrs the oposite. That’s the effect of the sign of the parameter.

It’s interesting to note: the two process are diferents but they have the same mean and variance. In fact, in the $AR(1)$ process $Var(y_t) = frac{sigma_epsilon^2}{1-varphi^2}$, only if the process are stationary $|varphi| < 1$.