The whole purpose of this blog is to visualize the ${{Y}_{t}}=\rho {{Y}_{t-1}}+{{e}_{t}}$ time series AR(1) process with various values of $\rho $ and value of variance of ${{e}_{t}}$.

The AR(1) process given by ${{Y}_{t}}=\rho {{Y}_{t-1}}+{{e}_{t}}$. The ${{e}_{t}}$is assumed to be white noise with following properties which I have previously discussed (here).

$E({{e}_{t}})=0\to (i)$

$E(e_{t}^{2})={{\sigma }^{2}}\to (ii)$

$E({{e}_{t}},{{e}_{j}})=0$ for $i\ne j\to (iii)$

I have put an interactive diagram with random seed of 1, for 100 period in which we can play with the value of $\rho $ from -1.1 till 1.1 and value of variance ${{e}_{t}}$ from zero till 1. We know that, if the value of $\left| \rho \right|<1$ then the series will be stationary, if $\left| \rho \right|=1$ then the series will be non-stationary and $\left| \rho \right|>1$ then the series will be explosive. If the following interactive diagram fails to work then click (here) for the external link.

I have put an interactive diagram with random seed of 1, for 100 period in which we can play with the value of $\rho $ from -1.1 till 1.1 and value of variance ${{e}_{t}}$ from zero till 1. We know that, if the value of $\left| \rho \right|<1$ then the series will be stationary, if $\left| \rho \right|=1$ then the series will be non-stationary and $\left| \rho \right|>1$ then the series will be explosive. If the following interactive diagram fails to work then click (here) for the external link.

Here are few snaps!