Choosing the Right Model

1. Choosing the Right Model

In practice, you will ordinarily not be told the order of the model that you're trying to estimate.

2. Identifying the Order of an AR Model

There are two techniques that can help determine the order of the AR model: The Partial Autocorrelation Function, and the Information Criteria

3. Partial Autocorrelation Function (PACF)

The Partial Autocorrelation Function measures the incremental benefit of adding another lag. Imagine running several regressions, where you regress returns on more and more lagged values. The coefficients in the red boxes represent the values of the partial autocorrelation function for different lags. For example, in the bottom row, the coefficient in the red box, phi 4-4, is the lag-4 value of the Partial Autocorrelation Function, and it represents how significant adding a fourth lag is when you already have three lags.

4. Plot PACF in Python

plot_pacf is the statsmodels function for plotting the partial autocorrelation function. The arguments are the same as that of the plot_acf module that you saw earlier. The input x is a series or array. The argument lags indicates how many lags of the partial autocorrelation function will be plotted. And the alpha argument sets the width of the confidence interval.

5. Comparison of PACF for Different AR Models

These plots show the Partial Autocorrelation Function for AR models of different orders. In the upper left, for an AR(1) model, only the lag-*1* PACF is significantly different from zero. Similarly, for an AR(2) model, two lags are different from zero, and for and AR(3), three lags are different from zero. Finally, for White Noise, there are no lags that are significantly different from zero.

6. Information Criteria

The more parameters in a model, the better the model will fit the data. But this can lead to overfitting of the data. The information criteria adjusts the goodness-of-fit of a model by imposing a penalty based on the number of parameters used. Two common adjusted goodness-of-fit measures are called the Akaike Information Criterion and the Bayesian Information Criterion.

7. Information Criteria

This is the full output from estimating an AR(2) model. The AIC and BIC are highlighted in the red box. To get the AIC and BIC statistics,

8. Getting Information Criteria From statsmodels

you follow the same procedure from the last section to fit the data to a model. In the last section, you learned how to get the full output using summary or just the AR parameters using the params attribute. You can also get the AIC or BIC using those attributes. In practice, the way to use the Bayesian information criterion is to fit several models, each with a different number of parameters, and choose the one with the lowest information criterion.

9. Information Criteria

Suppose we are given a time series of data, and unknown to us, it was simulated from an AR(3) model. Here is a plot of the BIC when we fit the data to an AR(1) up to an AR(6) model. You can see that the lowest BIC occurs for an AR(3).

10. Let's practice!

Now it's your turn.

Create Your Free Account

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