Autoregressions

This notebook introduces autoregression modeling using the AutoReg model. It also covers aspects of ar_select_order assists in selecting models that minimize an information criteria such as the AIC. An autoregressive model has dynamics given by

$$ y_t = \delta + \phi_1 y_{t-1} + \ldots + \phi_p y_{t-p} + \epsilon_t. $$

AutoReg also permits models with:

• Deterministic terms (trend)
  • n: No deterministic term
  • c: Constant (default)
  • ct: Constant and time trend
  • t: Time trend only
• Seasonal dummies (seasonal)
  • True includes $s-1$ dummies where $s$ is the period of the time series (e.g., 12 for monthly)
• Exogenous variables (exog)
  • A DataFrame or array of exogenous variables to include in the model
• Omission of selected lags (lags)
  • If lags is an iterable of integers, then only these are included in the model.

The complete specification is

$$ y_t = \delta_0 + \delta_1 t + \phi_1 y_{t-1} + \ldots + \phi_p y_{t-p} + \sum_{i=1}^{s-1} \gamma_i d_i + \sum_{j=1}^{m} \kappa_j x_{t,j} + \epsilon_t. $$

where:

• $d_i$ is a seasonal dummy that is 1 if $mod(t, period) = i$. Period 0 is excluded if the model contains a constant (c is in trend).
• $t$ is a time trend ($1,2,\ldots$) that starts with 1 in the first observation.
• $x_{t,j}$ are exogenous regressors. Note these are time-aligned to the left-hand-side variable when defining a model.
• $\epsilon_t$ is assumed to be a white noise process.

This first cell imports standard packages and sets plats to appear inline.

This cell sets the plotting style, registers pandas date converters for matplotlib, and sets the default figure size.

The first set of examples uses the month-over-month growth rate in U.S. Housing starts that has not been seasonally adjusted. The seasonality is evident by the regular pattern of peaks and troughs. We set the frequency for the time series to "MS" (month-start) to avoid warnings when using AutoReg.

We can start with an AR(3). While this is not a good model for this data, it demonstrates the basic use of the API.

AutoReg supports the same covariance estimators as OLS. Below, we use cov_type="HC0", which is White's covariance estimator. While the parameter estimates are the same, all of the quantities that depend on the standard error change.

plot_predict visualizes forecasts. Here we produce a large number of forecasts which show the string seasonality captured by the model.

plot_diagnositcs indicates that the model captures the key features in the data.

Seasonal Dummies

AutoReg supports seasonal dummies which are an alternative way to model seasonality. Including the dummies shortens the dynamics to only an AR(2).

The seasonal dummies are obvious in the forecasts which has a non-trivial seasonal component in all periods 10 years in to the future.

Seasonal Dynamics

While AutoReg does not directly support Seasonal components since it uses OLS to estimate parameters, it is possible to capture seasonal dynamics using an over-parametrized Seasonal AR that does not impose the restrictions in the Seasonal AR.

We start by selecting a model using the simple method that only chooses the maximum lag. All lower lags are automatically included. The maximum lag to check is set to 13 since this allows the model to next a Seasonal AR that has both a short-run AR(1) component and a Seasonal AR(1) component, so that

$$ (1-\phi_s L^{12})(1-\phi_1 L)y_t = \epsilon_t $$

which becomes $$ y_t = \phi_1 y_{t-1} +\phi_s Y_{t-12} - \phi_1\phi_s Y_{t-13} + \epsilon_t $$

when expanded. AutoReg does not enforce the structure, but can estimate the nesting model

$$ y_t = \phi_1 y_{t-1} +\phi_{12} Y_{t-12} - \phi_{13} Y_{t-13} + \epsilon_t. $$

We see that all 13 lags are selected.

It seems unlikely that all 13 lags are required. We can set glob=True to search all $2^{13}$ models that include up to 13 lags.

Here we see that the first three are selected, as is the 7th, and finally the 12th and 13th are selected. This is superficially similar to the structure described above.

After fitting the model, we take a look at the diagnostic plots that indicate that this specification appears to be adequate to capture the dynamics in the data.

We can also include seasonal dummies. These are all insignificant since the model is using year-over-year changes.

Industrial Production

We will use the industrial production index data to examine forecasting.

We will start by selecting a model using up to 12 lags. An AR(13) minimizes the BIC criteria even though many coefficients are insignificant.

We can also use a global search which allows longer lags to enter if needed without requiring the shorter lags. Here we see many lags dropped. The model indicates there may be some seasonality in the data.

plot_predict can be used to produce forecast plots along with confidence intervals. Here we produce forecasts starting at the last observation and continuing for 18 months.

The forecasts from the full model and the restricted model are very similar. I also include an AR(5) which has very different dynamics

The diagnostics indicate the model captures most of the the dynamics in the data. The ACF shows a patters at the seasonal frequency and so a more complete seasonal model (SARIMAX) may be needed.

Forecasting

Forecasts are produced using the predict method from a results instance. The default produces static forecasts which are one-step forecasts. Producing multi-step forecasts requires using dynamic=True.

In this next cell, we produce 12-step-heard forecasts for the final 24 periods in the sample. This requires a loop.

Note: These are technically in-sample since the data we are forecasting was used to estimate parameters. Producing OOS forecasts requires two models. The first must exclude the OOS period. The second uses the predict method from the full-sample model with the parameters from the shorter sample model that excluded the OOS period.

Comparing to SARIMAX

SARIMAX is an implementation of a Seasonal Autoregressive Integrated Moving Average with eXogenous regressors model. It supports:

• Specification of seasonal and nonseasonal AR and MA components
• Inclusion of Exogenous variables
• Full maximum-likelihood estimation using the Kalman Filter

This model is more feature rich than AutoReg. Unlike SARIMAX, AutoReg estimates parameters using OLS. This is faster and the problem is globally convex, and so there are no issues with local minima. The closed-form estimator and its performance are the key advantages of AutoReg over SARIMAX when comparing AR(P) models. AutoReg also support seasonal dummies, which can be used with SARIMAX if the user includes them as exogenous regressors.