Time Series Analysis by State Space Methods statespace¶

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

where \(y_t\) refers to the observation vector at time \(t\), \(\alpha_t\) refers to the (unobserved) state vector at time \(t\), and where the irregular components are defined as

The remaining variables (\(Z_t, d_t, H_t, T_t, c_t, R_t, Q_t\)) in the equations are matrices describing the process. Their variable names and dimensions are as follows

Z : design \((k\_endog \times k\_states \times nobs)\)

d : obs_intercept \((k\_endog \times nobs)\)

H : obs_cov \((k\_endog \times k\_endog \times nobs)\)

T : transition \((k\_states \times k\_states \times nobs)\)

c : state_intercept \((k\_states \times nobs)\)

R : selection \((k\_states \times k\_posdef \times nobs)\)

Q : state_cov \((k\_posdef \times k\_posdef \times nobs)\)

In the case that one of the matrices is time-invariant (so that, for example, \(Z_t = Z_{t+1} ~ \forall ~ t\)), its last dimension may be of size \(1\) 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:

This can be put into state space form in the following way:

Where

and

There are three unknown parameters in this model: \(\phi_1, \phi_2, \sigma^2\).

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 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.

For an example of the use of this model, see the SARIMAX example notebook or the very brief code snippet below:

# 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.

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:

# 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.

For an example of the use of this model, see the VARMAX example notebook or the very brief code snippet below:

# 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¶

Statsmodels has two classes that support dynamic factor models: DynamicFactorMQ and DynamicFactor. Each of these models has strengths, but in general the DynamicFactorMQ class is recommended. This is because it fits parameters using the Expectation-Maximization (EM) algorithm, which is more robust and can handle including hundreds of observed series. In addition, it allows customization of which variables load on which factors. However, it does not yet support including exogenous variables, while DynamicFactor does support that feature.

For an example of the DynamicFactorMQ class, see the very brief code snippet below:

# Load the statsmodels api
import statsmodels.api as sm

# Load your dataset
endog = pd.read_csv('your/dataset/here.csv')

# Create a dynamic factor model
mod_dfm = sm.tsa.DynamicFactorMQ(endog, k_factors=1, factor_order=2)
# Note that mod_dfm is an instance of the DynamicFactorMQ class

# Fit the model via maximum likelihood, using the EM algorithm
res_dfm = mod_dfm.fit()
# Note that res_dfm is an instance of the DynamicFactorMQResults 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()

The DynamicFactor class is suitable for models with a smaller number of observed variables

For an example of the use of the DynamicFactor model, see the Dynamic Factor example notebook

Linear Exponential Smoothing Models¶

The ExponentialSmoothing class is an implementation of linear exponential smoothing models using a state space approach.

Note: this model is available at sm.tsa.statespace.ExponentialSmoothing; it is not the same as the model available at sm.tsa.ExponentialSmoothing. See below for details of the differences between these classes.

A very brief code snippet follows:

# Load the statsmodels api
import statsmodels.api as sm

# Load your dataset
endog = pd.read_csv('your/dataset/here.csv')

# Simple exponential smoothing, denoted (A,N,N)
mod_ses = sm.tsa.statespace.ExponentialSmoothing(endog)
res_ses = mod_ses.fit()

# Holt's linear method, denoted (A,A,N)
mod_h = sm.tsa.statespace.ExponentialSmoothing(endog, trend=True)
res_h = mod_h.fit()

# Damped trend model, denoted (A,Ad,N)
mod_dt = sm.tsa.statespace.ExponentialSmoothing(endog, trend=True,
                                                damped_trend=True)
res_dt = mod_dt.fit()

# Holt-Winters' trend and seasonality method, denoted (A,A,A)
# (assuming that `endog` has a seasonal periodicity of 4, for example if it
# is quarterly data).
mod_hw = sm.tsa.statespace.ExponentialSmoothing(endog, trend=True,
                                                seasonal=4)
res_hw = mod_hw.fit()

Differences between Statsmodels’ exponential smoothing model classes

There are several differences between this model class, available at sm.tsa.statespace.ExponentialSmoothing, and the model class available at sm.tsa.ExponentialSmoothing.

• This model class only supports linear exponential smoothing models, while sm.tsa.ExponentialSmoothing also supports multiplicative models.

• This model class puts the exponential smoothing models into state space form and then applies the Kalman filter to estimate the states, while sm.tsa.ExponentialSmoothing is based on exponential smoothing recursions. In some cases, this can mean that estimating parameters with this model class will be somewhat slower than with sm.tsa.ExponentialSmoothing.

• This model class can produce confidence intervals for forecasts, based on an assumption of Gaussian errors, while sm.tsa.ExponentialSmoothing does not support confidence intervals.

• This model class supports concentrating initial values out of the objective function, which can improve performance when there are many initial states to estimate (for example when the seasonal periodicity is large).

• This model class supports many advanced features available to state space models, such as diagnostics and fixed parameters.

Note: this class is based on a “multiple sources of error” (MSOE) state space formulation and not a “single source of error” (SSOE) formulation.

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.

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:

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 specifying names for parameters for a more informative output summary.

Overview of usage¶

All state space models follow the typical Statsmodels pattern:

1. Construct a model instance with an input dataset

2. Apply parameters to the model (for example, using fit) to construct a results instance

3. Interact with the results instance to examine the estimated parameters, explore residual diagnostics, and produce forecasts, simulations, or impulse responses.

An example of this pattern is as follows:

# Load in the example macroeconomic dataset
dta = sm.datasets.macrodata.load_pandas().data
# Make sure we have an index with an associated frequency, so that
# we can refer to time periods with date strings or timestamps
dta.index = pd.date_range('1959Q1', '2009Q3', freq='QS')

# Step 1: construct an SARIMAX model for US inflation data
model = sm.tsa.SARIMAX(dta.infl, order=(4, 0, 0), trend='c')

# Step 2: fit the model's parameters by maximum likelihood
results = model.fit()

# Step 3: explore / use results

# - Print a table summarizing estimation results
print(results.summary())

# - Print only the estimated parameters
print(results.params)

# - Create diagnostic figures based on standardized residuals:
#   (1) time series graph
#   (2) histogram
#   (3) Q-Q plot
#   (4) correlogram
results.plot_diagnostics()

# - Examine diagnostic hypothesis tests
# Jarque-Bera: [test_statistic, pvalue, skewness, kurtosis]
print(results.test_normality(method='jarquebera'))
# Goldfeld-Quandt type test: [test_statistic, pvalue]
print(results.test_heteroskedasticity(method='breakvar'))
# Ljung-Box test: [test_statistic, pvalue] for each lag
print(results.test_serial_correlation(method='ljungbox'))

# - Forecast the next 4 values
print(results.forecast(4))

# - Forecast until 2020Q4
print(results.forecast('2020Q4'))

# - Plot in-sample dynamic prediction starting in 2005Q1
#   and out-of-sample forecasts until 2010Q4 along with
#   90% confidence intervals
predict_results = results.get_prediction(start='2005Q1', end='2010Q4', dynamic=True)
predict_df = predict_results.summary_frame(alpha=0.10)
fig, ax = plt.subplots()
predict_df['mean'].plot(ax=ax)
ax.fill_between(predict_df.index, predict_df['mean_ci_lower'],
                predict_df['mean_ci_upper'], alpha=0.2)

# - Simulate two years of new data after the end of the sample
print(results.simulate(8, anchor='end'))

# - Impulse responses for two years
print(results.impulse_responses(8))

Basic methods and attributes for estimation / filtering / smoothing¶

The most-used methods for a state space model are:

• fit - estimate parameters via maximum likelihood and return a results object (this object will have also performed Kalman filtering and smoothing at the estimated parameters). This is the most commonly used method.

• smooth - return a results object associated with a given vector of parameters after performing Kalman filtering and smoothing

• loglike - compute the log-likelihood of the data using a given vector of parameters

Some useful attributes of a state space model are:

• param_names - names of the parameters used by the model

• state_names - names of the elements of the (unobserved) state vector

• start_params - initial parameter estimates used a starting values for numerical maximum likelihood optimization

Other methods that are used less often are:

• filter - return a results object associated with a given vector of parameters after only performing Kalman filtering (but not smoothing)

• simulation_smoother - return an object that can perform simulation smoothing

Output and postestimation methods and attributes¶

Commonly used methods include:

• summary - construct a table that presents model fit statistics, estimated parameters, and other summary output

• predict - compute in-sample predictions and out-of-sample forecasts (point estimates only)

• get_prediction - compute in-sample predictions and out-of-sample forecasts, including confidence intervals

• forecast - compute out-of-sample forecasts (point estimates only) (this is a convenience wrapper around predict)

• get_forecast - compute out-of-sample forecasts, including confidence intervals (this is a convenience wrapper around get_prediction)

• simulate - simulate new data according to the state space model

• impulse_responses - compute impulse responses from the state space model

Commonly used attributes include:

• params - estimated parameters

• bse - standard errors of estimated parameters

• pvalues - p-values associated with estimated parameters

• llf - log-likelihood of the data at the estimated parameters

• sse, mse, and mae - sum of squared errors, mean square error, and mean absolute error

• Information criteria, including: aic, aicc, bic, and hquc

• fittedvalues - fitted values from the model (note that these are one-step-ahead predictions)

• resid - residuals from the model (note that these are one-step-ahead prediction errors)

Estimates and covariances of the unobserved state¶

It can be useful to compute estimates of the unobserved state vector conditional on the observed data. These are available in the results object states, which contains the following elements:

• states.filtered - filtered (one-sided) estimates of the state vector. The estimate of the state vector at time t is based on the observed data up to and including time t.

• states.smoothed - smoothed (two-sided) estimates of the state vector. The estimate of the state vector at time t is based on all observed data in the sample.

• states.filtered_cov - filtered (one-sided) covariance of the state vector

• states.smoothed_cov - smoothed (two-sided) covariance of the state vector

Each of these elements are Pandas DataFrame objects.

As an example, in a “local level + seasonal” model estimated via the UnobservedComponents components class we can get an estimates of the underlying level and seasonal movements of a series over time.

fig, axes = plt.subplots(3, 1, figsize=(8, 8))

# Retrieve monthly retail sales for clothing
from pandas_datareader.data import DataReader
clothing = DataReader('MRTSSM4481USN', 'fred', start='1992').asfreq('MS')['MRTSSM4481USN']

# Construct a local level + seasonal model
model = sm.tsa.UnobservedComponents(clothing, 'llevel', seasonal=12)
results = model.fit()

# Plot the data, the level, and seasonal
clothing.plot(ax=axes[0])
results.states.smoothed['level'].plot(ax=axes[1])
results.states.smoothed['seasonal'].plot(ax=axes[2])

Residual 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:

• test_normality

• test_heteroskedasticity

• test_serial_correlation

A number of standard plots of regression residuals are available for the same purpose. These can be produced using the command plot_diagnostics.

Applying estimated parameters to an updated or different dataset¶

There are three methods that can be used to apply estimated parameters from a results object to an updated or different dataset:

• append - retrieve a new results object with additional observations that follow after the end of the current sample appended to it (so the new results object contains both the current sample and the additional observations)

• extend - retrieve a new results object for additional observations that follow after end of the current sample (so the new results object contains only the new observations but NOT the current sample)

• apply - retrieve a new results object for a completely different dataset

One cross-validation exercise on time-series data involves fitting a model’s parameters based on a training sample (observations through time t) and then evaluating the fit of the model using a test sample (observations t+1, t+2, …). This can be conveniently done using either apply or extend. In the example below, we use the extend method.

# Load in the example macroeconomic dataset
dta = sm.datasets.macrodata.load_pandas().data
# Make sure we have an index with an associated frequency, so that
# we can refer to time periods with date strings or timestamps
dta.index = pd.date_range('1959Q1', '2009Q3', freq='QS')

# Separate inflation data into a training and test dataset
training_endog = dta['infl'].iloc[:-1]
test_endog = dta['infl'].iloc[-1:]

# Fit an SARIMAX model for inflation
training_model = sm.tsa.SARIMAX(training_endog, order=(4, 0, 0))
training_results = training_model.fit()

# Extend the results to the test observations
test_results = training_results.extend(test_endog)

# Print the sum of squared errors in the test sample,
# based on parameters computed using only the training sample
print(test_results.sse)

Understanding the Impact of Data Revisions¶

Statespace model results expose a news method that can be used to understand the impact of data revisions – news – on model parameters.

Additional options and tools¶

All state space models have the following options and tools:

Holding some parameters fixed and estimating the rest¶

The fit_constrained method allows fixing some parameters to known values and then estimating the rest via maximum likelihood. An example of this is:

# Construct a model
model = sm.tsa.SARIMAX(endog, order=(1, 0, 0))

# To find out the parameter names, use:
print(model.param_names)

# Fit the model with a fixed value for the AR(1) coefficient:
results = model.fit_constrained({'ar.L1': 0.5})

Alternatively, you can use the fix_params context manager:

# Construct a model
model = sm.tsa.SARIMAX(endog, order=(1, 0, 0))

# Fit the model with a fixed value for the AR(1) coefficient using the
# context manager
with model.fix_params({'ar.L1': 0.5}):
    results = model.fit()

Low memory options¶

When the observed dataset is very large and / or the state vector of the model is high-dimensional (for example when considering long seasonal effects), the default memory requirements can be too large. For this reason, the fit, filter, and smooth methods accept an optional low_memory=True argument, which can considerably reduce memory requirements and speed up model fitting.

Note that when using low_memory=True, not all results objects will be available. However, residual diagnostics, in-sample (non-dynamic) prediction, and out-of-sample forecasting are all still available.

Low-level 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 evaluated 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.

The KalmanFilter class is a subclass of Representation that provides filtering capabilities. Once the state space representation matrices have been constructed, the 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 filtered state and loglikelihood (see the class documentation for the full list of available results). It also provides a predict method, which allows in-sample prediction or out-of-sample forecasting. A similar method, predict, provides additional prediction or forecasting results, including confidence intervals.

The KalmanSmoother class is a subclass of KalmanFilter that provides smoothing capabilities. Once the state space representation matrices have been constructed, the 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 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.

The SimulationSmoother class is a subclass of KalmanSmoother that further provides simulation and simulation smoothing capabilities. The simulation_smoother method can be called, producing a SimulationSmoothResults instance.

The SimulationSmoothResults class has a simulate method, that allows performing simulation smoothing to draw from the joint posterior of the state vector. This is useful for Bayesian estimation of state space models via Gibbs sampling.

Statespace Tools¶

There are a variety of tools used for state space modeling or by the SARIMAX class: