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 timeinvariant (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 MovingAverage 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.

Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors model 

Class to hold results from fitting an SARIMAX model. 
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, zstatistics, 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.

Univariate unobserved components time series model 
Class to hold results from fitting an unobserved components 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 MovingAverage with eXogenous regressors (VARMAX)¶
The VARMAX class is an example of a multivariate statespace model.

Vector Autoregressive Moving Average with eXogenous regressors model 

Class to hold results from fitting an VARMAX 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¶
The DynamicFactor class is another example of a multivariate statespace model.

Dynamic factor model 

Class to hold results from fitting an DynamicFactor model. 
For an example of the use of this model, see the Dynamic Factor 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')
# 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()
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.
Linear exponential smoothing models 


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()
# HoltWinters' 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.

State space model for maximum likelihood estimation 

Class to hold results from fitting a state space model. 
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
===================================================================================
LjungBox (Q): 24.25 JarqueBera (JB): 0.22
Prob(Q): 0.98 Prob(JB): 0.90
Heteroskedasticity (H): 1.05 Skew: 0.04
Prob(H) (twosided): 0.66 Kurtosis: 3.02
===================================================================================
Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complexstep).
The results object has many of the attributes and methods you would expect from other statsmodels results objects, including standard errors, zstatistics, 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:
Construct a model instance with an input dataset
Apply parameters to the model (for example, using fit) to construct a results instance
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) QQ plot
# (4) correlogram
results.plot_diagnostics()
#  Examine diagnostic hypothesis tests
# JarqueBera: [test_statistic, pvalue, skewness, kurtosis]
print(results.test_normality(method='jarquebera'))
# GoldfeldQuandt type test: [test_statistic, pvalue]
print(results.test_heteroskedasticity(method='breakvar'))
# LjungBox 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 insample dynamic prediction starting in 2005Q1
# and outofsample 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 mostused 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 smoothingloglike
 compute the loglikelihood 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 modelstate_names
 names of the elements of the (unobserved) state vectorstart_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 outputpredict
 compute insample predictions and outofsample forecasts (point estimates only)get_prediction
 compute insample predictions and outofsample forecasts, including confidence intervalsforecast
 compute outofsample forecasts (point estimates only) (this is a convenience wrapper around predict)get_forecast
 compute outofsample forecasts, including confidence intervals (this is a convenience wrapper around get_prediction)simulate
 simulate new data according to the state space modelimpulse_responses
 compute impulse responses from the state space model
Commonly used attributes include:
params
 estimated parametersbse
 standard errors of estimated parameterspvalues
 pvalues associated with estimated parametersllf
 loglikelihood of the data at the estimated parameterssse
,mse
, andmae
 sum of squared errors, mean square error, and mean absolute errorfittedvalues
 fitted values from the model (note that these are onestepahead predictions)resid
 residuals from the model (note that these are onestepahead 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 (onesided) 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 (twosided) 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 (onesided) covariance of the state vector
states.smoothed_cov  smoothed (twosided) 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:
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 crossvalidation exercise on timeseries 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)
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 highdimensional (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, insample (nondynamic) prediction, and outofsample forecasting are all still available.
Lowlevel 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 resultstype 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.

State space representation of a time series process 
Frozen Statespace Model 
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
insample prediction or outofsample forecasting. A similar method,
predict
, provides
additional prediction or forecasting results, including confidence intervals.

State space representation of a time series process, with Kalman filter 

Results from applying the Kalman filter to a state space model. 

Results of insample and outofsample prediction for state space models generally 
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.

State space representation of a time series process, with Kalman filter and smoother. 
Results from applying the Kalman smoother and/or filter to a state space model. 
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.
State space representation of a time series process, with Kalman filter and smoother, and with simulation smoother. 

Results from applying the Kalman smoother and/or filter to a state space model. 
Statespace Tools¶
There are a variety of tools used for state space modeling or by the SARIMAX class:

Create a companion matrix 

Difference a series simply and/or seasonally along the zeroth axis. 

Determine if a polynomial is invertible. 
Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation 

Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer 

Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression. 

Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer 


Validate the shape of a possibly timevarying matrix, or raise an exception 

Validate the shape of a possibly timevarying vector, or raise an exception 