Describe AR Model

1. Introducing an AR Model

In an Autoregressive model,

2. Mathematical Description of AR(1) Model

or AR model, today's value equals a mean plus a fraction phi of yesterday's value, plus noise. Since there is only one lagged value on the right hand side, this is called an AR model of order 1, or simply an AR(1) model. If the AR parameter, phi, is one, then the process is a random walk. If phi is zero, then the process is white noise. In order for the process to be stable and stationary, phi has to be between -1 and +1.

3. Interpretation of AR(1) Parameter

As an example, suppose R_t is a time series of stock returns. If phi is negative, then a positive return last period, at time t-1, implies that this period's return is more likely to be negative. This was referred to as "mean reversion" in Chapter 1. On the other hand, if phi is positive, then a positive return last period implies that this period's return is expected to be positive. This was referred to as "momentum" in Chapter 1.

4. Comparison of AR(1) Time Series

Here are four simulated time series with different AR parameters. When phi equals 0-point-9, it looks close to a random walk. When phi equals minus 0-point-9, the process looks more erratic - a large positive value is usually followed by a large negative one. The bottom two are similar, but are less exaggerated and closer to white noise.

5. Comparison of AR(1) Autocorrelation Functions

Here are four autocorrelation functions for different AR parameters. The autocorrelation decays exponentially at a rate of phi. Therefore if phi is 0-point-9, the lag-1 autocorrelation is 0-point-9, the lag-2 autocorrelation is 0-point-9 squared, the lag-3 autocorrelation is 0-point-9 cubed, etc. When phi is negative, the autocorrelation function still decays exponentially, but the signs of the autocorrelation function reverse at each lag.

6. Higher Order AR Models

So far, we've been only looking at AR(1) models. The model can be extended to include more lagged values and more phi parameters. Here we show an AR(1), an AR(2), and an AR(3).

7. Simulating an AR Process

Often, if you want to study and understand a pure AR process, it is useful to work with simulated data. Statsmodels provides modules for simulating AR processes. First, import the class, ArmaProcess. Then define the order and parameters of the AR process. The convention is a little counterintuitive: You must include the zero-lag coefficient of 1, and the sign of the other coefficient is the opposite of what we have been using. For example, for an AR(1) process with phi equal to plus 0-point-9, the second element of the ar array should be the opposite sign, *minus 0-point-9*. This is consistent with the time series literature in the field of signal processing. You also have to input the MA parameters. You will learn about MA models in the next chapter, so for now, just ignore the MA part. Then, you create an instance of the class ArmaProcess. To simulate data, use the method generate_sample, with the number of simulated samples as an argument.

8. Let's practice!

Now let's try some examples.

Create Your Free Account

By continuing, you accept our Terms of Use, our Privacy Policy and that your data is stored in the USA.