1. The autoregressive model
There are many autoregressive or AR processes.
2. The autoregressive model - I
You will focus on the simplest, first-order case, in which Today's observation is regressed on Yesterday's observation, at all times t.
In R you will work with the mean centered version of the model. Today minus the mean is equal to a slope parameter time yesterday minus the mean, plus noise.
3. The autoregressive model - II
More formally, if epsilon is mean zero White Noise, then Y is an AR process if it follows the displayed equation, for some constant parameters mu and phi.
Mu is the mean, Phi is the slope coefficient, and there is a third parameter, sigma-squared, the variance of the white noise.
4. AR processes - I
If the slope phi is zero, Y is simply a white noise process.
If the slope phi is NOT zero, then Y depends on both the current noise and the previous observation, and the process Y is autocorrelated.
Large values of the slope phi lead to greater autocorrelation. And as you will see on the next slide, negative values of the slope phi result in oscillatory time series.
5. AR examples
Now, let's look at four examples of AR time series. In the top left you see an AR series with phi equal to 0-point-98, and below it, one with phi equal to 1. In both cases the series show strong persistence in level, meaning each observations is close to its neighbors.
In the top right you see an AR series with phi equal to -0-point-6. AR series with negative slope coefficients will exhibit oscillatory patterns, as you see here.
Finally, in the bottom right you see an AR series with phi equal to 1-point-01. Here, there is so much dependence that the series is quickly diverging, downward in this case.
6. Autocorrelations
The AR process has autocorrelation. This is determined by the slope parameter phi. In the first three panels, you can compare the shape of the autocorrelation function for different positive values of phi, 0-point-95, 0-point-75, and 0-point-2, respectively. Note a slower decay when phi is larger. When phi is negative, the autocorrelation function has an alternating sign, but it still decays to zero in magnitude. This is the pattern we see at the bottom right figure where the coefficient is -0-point-9.
7. Random walk
Finally, note that when mu is zero and phi is equal to 1, the AR process becomes the random walk process, and you should recall that random walks are not stationary.
8. Let's practice!
Great! Now you will practice simulating AR processes and estimating their autocorrelation functions.