Time Series analysis tsa¶

statsmodels.tsa contains model classes and functions that are useful for time series analysis. Basic models include univariate autoregressive models (AR), vector autoregressive models (VAR) and univariate autoregressive moving average models (ARMA). Non-linear models include Markov switching dynamic regression and autoregression. It also includes descriptive statistics for time series, for example autocorrelation, partial autocorrelation function and periodogram, as well as the corresponding theoretical properties of ARMA or related processes. It also includes methods to work with autoregressive and moving average lag-polynomials. Additionally, related statistical tests and some useful helper functions are available.

Estimation is either done by exact or conditional Maximum Likelihood or conditional least-squares, either using Kalman Filter or direct filters.

Currently, functions and classes have to be imported from the corresponding module, but the main classes will be made available in the statsmodels.tsa namespace. The module structure is within statsmodels.tsa is

• stattools : empirical properties and tests, acf, pacf, granger-causality, adf unit root test, kpss test, bds test, ljung-box test and others.

• ar_model : univariate autoregressive process, estimation with conditional and exact maximum likelihood and conditional least-squares

• arima.model : univariate ARIMA process, estimation with alternative methods

• statespace : Comprehensive statespace model specification and estimation. See the statespace documentation.

• vector_ar, var : vector autoregressive process (VAR) and vector error correction models, estimation, impulse response analysis, forecast error variance decompositions, and data visualization tools. See the vector_ar documentation.

• arma_process : properties of arma processes with given parameters, this includes tools to convert between ARMA, MA and AR representation as well as acf, pacf, spectral density, impulse response function and similar

• sandbox.tsa.fftarma : similar to arma_process but working in frequency domain

• tsatools : additional helper functions, to create arrays of lagged variables, construct regressors for trend, detrend and similar.

• filters : helper function for filtering time series

• regime_switching : Markov switching dynamic regression and autoregression models

Some additional functions that are also useful for time series analysis are in other parts of statsmodels, for example additional statistical tests.

Some related functions are also available in matplotlib, nitime, and scikits.talkbox. Those functions are designed more for the use in signal processing where longer time series are available and work more often in the frequency domain.

Descriptive Statistics and Tests¶

Estimation¶

The following are the main estimation classes, which can be accessed through statsmodels.tsa.api and their result classes

Univariate Autoregressive Processes (AR)¶

The basic autoregressive model in Statsmodels is:

The ar_model.AutoReg model estimates parameters using conditional MLE (OLS), and supports exogenous regressors (an AR-X model) and seasonal effects.

AR-X and related models can also be fitted with the arima.ARIMA class and the SARIMAX class (using full MLE via the Kalman Filter).

See the notebook Autoregressions for an overview.

Autoregressive Moving-Average Processes (ARMA) and Kalman Filter¶

Basic ARIMA model and results classes are as follows:

This model allows estimating parameters by various methods (including conditional MLE via the Hannan-Rissanen method and full MLE via the Kalman filter). It is a special case of the SARIMAX model, and it includes a large number of inherited features from the state space models (including prediction / forecasting, residual diagnostics, simulation and impulse responses, etc.).

See the notebooks ARMA: Sunspots Data and ARMA: Artificial Data for an overview.

Exponential Smoothing¶

Linear and non-linear exponential smoothing models are available:

Separately, linear and non-linear exponential smoothing models have also been implemented based on the “innovations” state space approach. In addition to the usual support for parameter fitting, in-sample prediction, and out-of-sample forecasting, these models also support prediction intervals, simulation, and more.

Finally, linear exponential smoothing models have also been separately implemented as a special case of the general state space framework (this is separate from the “innovations” state space approach described above). Although this approach does not allow for the non-linear (multiplicative) exponential smoothing models, it includes all features of state space models (including prediction / forecasting, residual diagnostics, simulation and impulse responses, etc.).

See the notebook Exponential Smoothing for an overview.

ARMA Process¶

The following are tools to work with the theoretical properties of an ARMA process for given lag-polynomials.

Autoregressive Distributed Lag (ARDL) Models¶

Autoregressive Distributed Lag models span the space between autoregressive models (AutoReg) and vector autoregressive models (VAR).

The ardl.ARDL model estimates parameters using conditional MLE (OLS) and allows for both simple deterministic terms (trends and seasonal dummies) as well as complex deterministics using a DeterministicProcess.

AR-X and related models can also be fitted with SARIMAX class (using full MLE via the Kalman Filter).

See the notebook Autoregressive Distributed Lag Models for an overview.

Error Correction Models (ECM)¶

Error correction models are reparameterizations of ARDL models that regress the difference of the endogenous variable on the lagged levels of the endogenous variables and optional lagged differences of the exogenous variables.

Statespace Models¶

See the statespace documentation.

Vector ARs and Vector Error Correction Models¶

See the vector_ar documentation.

Regime switching models¶

See the notebooks Markov switching dynamic regression and Markov switching autoregression for an overview.

Time Series Filters¶

See the notebook Time Series Filters for an overview.

TSA Tools¶

VARMA Process¶

Interpolation¶

Deterministic Processes¶

Deterministic processes simplify creating deterministic sequences with time trend or seasonal patterns. They also provide methods to simplify generating deterministic terms for out-of-sample forecasting. A DeterministicProcess can be directly used with AutoReg to construct complex deterministic dynamics and to forecast without constructing exogenous trends.

Users who wish to write custom deterministic terms must use subclass DeterministicTerm.

See the notebook Deterministic Terms in Time Series Models for an overview.

Forecasting Models¶

The Theta Model¶

The Theta model is a simple forecasting method that combines a linear time trend with a Simple Exponential Smoother (Assimakopoulos & Nikolopoulos). An estimator for the parameters of the Theta model and methods to forecast are available in:

Forecasting after STL Decomposition¶

statsmodels.tsa.seasonal.STL is commonly used to remove seasonal components from a time series. The deseasonalized time series can then be modeled using a any non-seasonal model, and forecasts are constructed by adding the forecast from the non-seasonal model to the estimates of the seasonal component from the final full-cycle which are forecast using a random-walk model.

See the notebook Seasonal Decomposition for an overview.

Prediction Results¶

Most forecasting methods support a get_prediction method that return a PredictionResults object that contains both the prediction, its variance and can construct a prediction interval.

Results Class¶