.. module:: statsmodels.tsa.statespace :synopsis: Statespace models for time-series analysis .. currentmodule:: statsmodels.tsa.statespace .. _statespace: Time Series Analysis by State Space Methods :mod:statespace ============================================================= :mod:statsmodels.tsa.statespace contains classes and functions that are useful for time series analysis using state space methods. A general state space model is of the form .. math:: y_t & = Z_t \alpha_t + d_t + \varepsilon_t \\ \alpha_t & = T_t \alpha_{t-1} + c_t + R_t \eta_t \\ where :math:y_t refers to the observation vector at time :math:t, :math:\alpha_t refers to the (unobserved) state vector at time :math:t, and where the irregular components are defined as .. math:: \varepsilon_t \sim N(0, H_t) \\ \eta_t \sim N(0, Q_t) \\ The remaining variables (:math:Z_t, d_t, H_t, T_t, c_t, R_t, Q_t) in the equations are matrices describing the process. Their variable names and dimensions are as follows Z : design :math:(k\_endog \times k\_states \times nobs) d : obs_intercept :math:(k\_endog \times nobs) H : obs_cov :math:(k\_endog \times k\_endog \times nobs) T : transition :math:(k\_states \times k\_states \times nobs) c : state_intercept :math:(k\_states \times nobs) R : selection :math:(k\_states \times k\_posdef \times nobs) Q : state_cov :math:(k\_posdef \times k\_posdef \times nobs) In the case that one of the matrices is time-invariant (so that, for example, :math:Z_t = Z_{t+1} ~ \forall ~ t), its last dimension may be of size :math:1 rather than size nobs. This generic form encapsulates many of the most popular linear time series models (see below) and is very flexible, allowing estimation with missing observations, forecasting, impulse response functions, and much more. **Example: AR(2) model** An autoregressive model is a good introductory example to putting models in state space form. Recall that an AR(2) model is often written as: .. math:: y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \epsilon_t This can be put into state space form in the following way: .. math:: y_t & = \begin{bmatrix} 1 & 0 \end{bmatrix} \alpha_t \\ \alpha_t & = \begin{bmatrix} \phi_1 & \phi_2 \\ 1 & 0 \end{bmatrix} \alpha_{t-1} + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \eta_t Where .. math:: Z_t \equiv Z = \begin{bmatrix} 1 & 0 \end{bmatrix} and .. math:: T_t \equiv T & = \begin{bmatrix} \phi_1 & \phi_2 \\ 1 & 0 \end{bmatrix} \\ R_t \equiv R & = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\ \eta_t & \sim N(0, \sigma^2) There are three unknown parameters in this model: :math:\phi_1, \phi_2, \sigma^2. Models and Estimation --------------------- The following are the main estimation classes, which can be accessed through statsmodels.tsa.statespace.api and their result classes. Seasonal Autoregressive Integrated Moving-Average with eXogenous regressors (SARIMAX) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The SARIMAX class is an example of a fully fledged model created using the statespace backend for estimation. SARIMAX can be used very similarly to :ref:tsa  models, but works on a wider range of models by adding the estimation of additive and multiplicative seasonal effects, as well as arbitrary trend polynomials. .. autosummary:: :toctree: generated/ sarimax.SARIMAX sarimax.SARIMAXResults For an example of the use of this model, see the SARIMAX example notebook __ or the very brief code snippet below: .. code-block:: python # Load the statsmodels api import statsmodels.api as sm # Load your dataset endog = pd.read_csv('your/dataset/here.csv') # We could fit an AR(2) model, described above mod_ar2 = sm.tsa.SARIMAX(endog, order=(2,0,0)) # Note that mod_ar2 is an instance of the SARIMAX class # Fit the model via maximum likelihood res_ar2 = mod_ar2.fit() # Note that res_ar2 is an instance of the SARIMAXResults class # Show the summary of results print(res_ar2.summary()) # We could also fit a more complicated model with seasonal components. # As an example, here is an SARIMA(1,1,1) x (0,1,1,4): mod_sarimax = sm.tsa.SARIMAX(endog, order=(1,1,1), seasonal_order=(0,1,1,4)) res_sarimax = mod_sarimax.fit() # Show the summary of results print(res_sarimax.summary()) The results object has many of the attributes and methods you would expect from other Statsmodels results objects, including standard errors, z-statistics, and prediction / forecasting. Behind the scenes, the SARIMAX model creates the design and transition matrices (and sometimes some of the other matrices) based on the model specification. Unobserved Components ^^^^^^^^^^^^^^^^^^^^^ The UnobservedComponents class is another example of a statespace model. .. autosummary:: :toctree: generated/ structural.UnobservedComponents structural.UnobservedComponentsResults For examples of the use of this model, see the example notebook __ or a notebook on using the unobserved components model to decompose a time series into a trend and cycle __ or the very brief code snippet below: .. code-block:: python # Load the statsmodels api import statsmodels.api as sm # Load your dataset endog = pd.read_csv('your/dataset/here.csv') # Fit a local level model mod_ll = sm.tsa.UnobservedComponents(endog, 'local level') # Note that mod_ll is an instance of the UnobservedComponents class # Fit the model via maximum likelihood res_ll = mod_ll.fit() # Note that res_ll is an instance of the UnobservedComponentsResults class # Show the summary of results print(res_ll.summary()) # Show a plot of the estimated level and trend component series fig_ll = res_ll.plot_components() # We could further add a damped stochastic cycle as follows mod_cycle = sm.tsa.UnobservedComponents(endog, 'local level', cycle=True, damped_cycle=true, stochastic_cycle=True) res_cycle = mod_cycle.fit() # Show the summary of results print(res_cycle.summary()) # Show a plot of the estimated level, trend, and cycle component series fig_cycle = res_cycle.plot_components() Vector Autoregressive Moving-Average with eXogenous regressors (VARMAX) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The VARMAX class is an example of a multivariate statespace model. .. autosummary:: :toctree: generated/ varmax.VARMAX varmax.VARMAXResults For an example of the use of this model, see the VARMAX example notebook __ or the very brief code snippet below: .. code-block:: python # Load the statsmodels api import statsmodels.api as sm # Load your (multivariate) dataset endog = pd.read_csv('your/dataset/here.csv') # Fit a local level model mod_var1 = sm.tsa.VARMAX(endog, order=(1,0)) # Note that mod_var1 is an instance of the VARMAX class # Fit the model via maximum likelihood res_var1 = mod_var1.fit() # Note that res_var1 is an instance of the VARMAXResults class # Show the summary of results print(res_var1.summary()) # Construct impulse responses irfs = res_ll.impulse_responses(steps=10) Dynamic Factor Models ^^^^^^^^^^^^^^^^^^^^^ The DynamicFactor class is another example of a multivariate statespace model. .. autosummary:: :toctree: generated/ dynamic_factor.DynamicFactor dynamic_factor.DynamicFactorResults For an example of the use of this model, see the Dynamic Factor example notebook __ or the very brief code snippet below: .. code-block:: python # Load the statsmodels api import statsmodels.api as sm # Load your dataset endog = pd.read_csv('your/dataset/here.csv') # Fit a local level model mod_dfm = sm.tsa.DynamicFactor(endog, k_factors=1, factor_order=2) # Note that mod_dfm is an instance of the DynamicFactor class # Fit the model via maximum likelihood res_dfm = mod_dfm.fit() # Note that res_dfm is an instance of the DynamicFactorResults class # Show the summary of results print(res_ll.summary()) # Show a plot of the r^2 values from regressions of # individual estimated factors on endogenous variables. fig_dfm = res_ll.plot_coefficients_of_determination() Custom state space models ^^^^^^^^^^^^^^^^^^^^^^^^^ The true power of the state space model is to allow the creation and estimation of custom models. Usually that is done by extending the following two classes, which bundle all of state space representation, Kalman filtering, and maximum likelihood fitting functionality for estimation and results output. .. autosummary:: :toctree: generated/ mlemodel.MLEModel mlemodel.MLEResults For a basic example demonstrating creating and estimating a custom state space model, see the Local Linear Trend example notebook __. For a more sophisticated example, see the source code for the SARIMAX and SARIMAXResults classes, which are built by extending MLEModel and MLEResults. In simple cases, the model can be constructed entirely using the MLEModel class. For example, the AR(2) model from above could be constructed and estimated using only the following code: .. code-block:: python import numpy as np from scipy.signal import lfilter import statsmodels.api as sm # True model parameters nobs = int(1e3) true_phi = np.r_[0.5, -0.2] true_sigma = 1**0.5 # Simulate a time series np.random.seed(1234) disturbances = np.random.normal(0, true_sigma, size=(nobs,)) endog = lfilter([1], np.r_[1, -true_phi], disturbances) # Construct the model class AR2(sm.tsa.statespace.MLEModel): def __init__(self, endog): # Initialize the state space model super(AR2, self).__init__(endog, k_states=2, k_posdef=1, initialization='stationary') # Setup the fixed components of the state space representation self['design'] = [1, 0] self['transition'] = [[0, 0], [1, 0]] self['selection', 0, 0] = 1 # Describe how parameters enter the model def update(self, params, transformed=True, **kwargs): params = super(AR2, self).update(params, transformed, **kwargs) self['transition', 0, :] = params[:2] self['state_cov', 0, 0] = params[2] # Specify start parameters and parameter names @property def start_params(self): return [0,0,1] # these are very simple # Create and fit the model mod = AR2(endog) res = mod.fit() print(res.summary()) This results in the following summary table:: Statespace Model Results ============================================================================== Dep. Variable: y No. Observations: 1000 Model: AR2 Log Likelihood -1389.437 Date: Wed, 26 Oct 2016 AIC 2784.874 Time: 00:42:03 BIC 2799.598 Sample: 0 HQIC 2790.470 - 1000 Covariance Type: opg ============================================================================== coef std err z P>|z| [0.025 0.975] ------------------------------------------------------------------------------ param.0 0.4395 0.030 14.730 0.000 0.381 0.498 param.1 -0.2055 0.032 -6.523 0.000 -0.267 -0.144 param.2 0.9425 0.042 22.413 0.000 0.860 1.025 =================================================================================== Ljung-Box (Q): 24.25 Jarque-Bera (JB): 0.22 Prob(Q): 0.98 Prob(JB): 0.90 Heteroskedasticity (H): 1.05 Skew: -0.04 Prob(H) (two-sided): 0.66 Kurtosis: 3.02 =================================================================================== Warnings: [1] Covariance matrix calculated using the outer product of gradients (complex-step). The results object has many of the attributes and methods you would expect from other Statsmodels results objects, including standard errors, z-statistics, and prediction / forecasting. More advanced usage is possible, including specifying parameter transformations, and specifing names for parameters for a more informative output summary. State space representation and Kalman filtering ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ While creation of custom models will almost always be done by extending MLEModel and MLEResults, it can be useful to understand the superstructure behind those classes. Maximum likelihood estimation requires evaluating the likelihood function of the model, and for models in state space form the likelihood function is evaluted as a byproduct of running the Kalman filter. There are two classes used by MLEModel that facilitate specification of the state space model and Kalman filtering: Representation and KalmanFilter. The Representation class is the piece where the state space model representation is defined. In simple terms, it holds the state space matrices (design, obs_intercept, etc.; see the introduction to state space models, above) and allows their manipulation. FrozenRepresentation is the most basic results-type class, in that it takes a "snapshot" of the state space representation at any given time. See the class documentation for the full list of available attributes. .. autosummary:: :toctree: generated/ representation.Representation representation.FrozenRepresentation The KalmanFilter class is a subclass of Representation that provides filtering capabilities. Once the state space representation matrices have been constructed, the :py:meth:filter  method can be called, producing a FilterResults instance; FilterResults is a subclass of FrozenRepresentation. The FilterResults class not only holds a frozen representation of the state space model (the design, transition, etc. matrices, as well as model dimensions, etc.) but it also holds the filtering output, including the :py:attr:filtered state  and loglikelihood (see the class documentation for the full list of available results). It also provides a :py:meth:predict  method, which allows in-sample prediction or out-of-sample forecasting. A similar method, :py:meth:predict , provides additional prediction or forecasting results, including confidence intervals. .. autosummary:: :toctree: generated/ kalman_filter.KalmanFilter kalman_filter.FilterResults kalman_filter.PredictionResults The KalmanSmoother class is a subclass of KalmanFilter that provides smoothing capabilities. Once the state space representation matrices have been constructed, the :py:meth:filter  method can be called, producing a SmootherResults instance; SmootherResults is a subclass of FilterResults. The SmootherResults class holds all the output from FilterResults, but also includes smoothing output, including the :py:attr:smoothed state  and loglikelihood (see the class documentation for the full list of available results). Whereas "filtered" output at time t refers to estimates conditional on observations up through time t, "smoothed" output refers to estimates conditional on the entire set of observations in the dataset. .. autosummary:: :toctree: generated/ kalman_smoother.KalmanSmoother kalman_smoother.SmootherResults Statespace diagnostics ---------------------- Three diagnostic tests are available after estimation of any statespace model, whether built in or custom, to help assess whether the model conforms to the underlying statistical assumptions. These tests are: - :py:meth:test_normality  - :py:meth:test_heteroskedasticity  - :py:meth:test_serial_correlation  A number of standard plots of regression residuals are available for the same purpose. These can be produced using the command :py:meth:plot_diagnostics . Statespace Tools ---------------- There are a variety of tools used for state space modeling or by the SARIMAX class: .. autosummary:: :toctree: generated/ tools.companion_matrix tools.diff tools.is_invertible tools.constrain_stationary_univariate tools.unconstrain_stationary_univariate tools.constrain_stationary_multivariate tools.unconstrain_stationary_multivariate tools.validate_matrix_shape tools.validate_vector_shape