A Complete Introduction To Time Series Analysis (with R):: AR(1)

Last time, we defined a characterization of stationarity, and explored three important examples: the IID Noise, the White Noise, and Random Walks, along with how to produce and inspect their data plots and ACF plots to determine stationarity. This time, we will explore two examples of the most important processes for Time Series Analysis: the First-order autoregressive process AR(1)

First-order autoregressive process AR(1)

Dependence graph of the AR(1) process

Let’s see the definition of the AR(1):

Then the AR(1) process satisfies

You can think of this process as follows: the value of the present observation has some dependence on the present noise and some portion of the previous time step value, but not on values further. Inspecting the mean and variance, we can easily see that

Wait, what? That’s not obvious at all! Let’s see why this is true. If we take the expectation, we have

However, in order for X_t to be stationary, this can only be true if the mean is zero, that is