statsmodels.tsa.statespace.dynamic_factor.DynamicFactor

class statsmodels.tsa.statespace.dynamic_factor.DynamicFactor(endog, k_factors, factor_order, exog=None, error_order=0, error_var=False, error_cov_type='diagonal', enforce_stationarity=True, **kwargs)

Dynamic factor model

Parameters:

• endog (array_like) – The observed time-series process \(y\)
• exog (array_like, optional) – Array of exogenous regressors for the observation equation, shaped nobs x k_exog.
• k_factors (int) – The number of unobserved factors.
• factor_order (int) – The order of the vector autoregression followed by the factors.
• error_cov_type ({'scalar', 'diagonal', 'unstructured'}, optional) – The structure of the covariance matrix of the observation error term, where “unstructured” puts no restrictions on the matrix, “diagonal” requires it to be any diagonal matrix (uncorrelated errors), and “scalar” requires it to be a scalar times the identity matrix. Default is “diagonal”.
• error_order (int, optional) – The order of the vector autoregression followed by the observation error component. Default is None, corresponding to white noise errors.
• error_var (boolean, optional) – Whether or not to model the errors jointly via a vector autoregression, rather than as individual autoregressions. Has no effect unless error_order is set. Default is False.
• enforce_stationarity (boolean, optional) – Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. Default is True.
• **kwargs – Keyword arguments may be used to provide default values for state space matrices or for Kalman filtering options. See Representation, and KalmanFilter for more details.

exog

array_like, optional – Array of exogenous regressors for the observation equation, shaped nobs x k_exog.

k_factors

int – The number of unobserved factors.

factor_order

int – The order of the vector autoregression followed by the factors.

error_cov_type

{‘diagonal’, ‘unstructured’} – The structure of the covariance matrix of the error term, where “unstructured” puts no restrictions on the matrix and “diagonal” requires it to be a diagonal matrix (uncorrelated errors).

error_order

int – The order of the vector autoregression followed by the observation error component.

error_var

boolean – Whether or not to model the errors jointly via a vector autoregression, rather than as individual autoregressions. Has no effect unless error_order is set.

enforce_stationarity

boolean, optional – Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. Default is True.

Notes

The dynamic factor model considered here is in the so-called static form, and is specified:

\[\begin{split}y_t & = \Lambda f_t + B x_t + u_t \\ f_t & = A_1 f_{t-1} + \dots + A_p f_{t-p} + \eta_t \\ u_t & = C_1 u_{t-1} + \dots + C_1 f_{t-q} + \varepsilon_t\end{split}\]

where there are k_endog observed series and k_factors unobserved factors. Thus \(y_t\) is a k_endog x 1 vector and \(f_t\) is a k_factors x 1 vector.

\(x_t\) are optional exogenous vectors, shaped k_exog x 1.

\(\eta_t\) and \(\varepsilon_t\) are white noise error terms. In order to identify the factors, \(Var(\eta_t) = I\). Denote \(Var(\varepsilon_t) \equiv \Sigma\).

Options related to the unobserved factors:

• k_factors: this is the dimension of the vector \(f_t\), above. To exclude factors completely, set k_factors = 0.
• factor_order: this is the number of lags to include in the factor evolution equation, and corresponds to \(p\), above. To have static factors, set factor_order = 0.

Options related to the observation error term \(u_t\):

• error_order: the number of lags to include in the error evolution equation; corresponds to \(q\), above. To have white noise errors, set error_order = 0 (this is the default).
• error_cov_type: this controls the form of the covariance matrix \(\Sigma\). If it is “dscalar”, then \(\Sigma = \sigma^2 I\). If it is “diagonal”, then \(\Sigma = \text{diag}(\sigma_1^2, \dots, \sigma_n^2)\). If it is “unstructured”, then \(\Sigma\) is any valid variance / covariance matrix (i.e. symmetric and positive definite).
• error_var: this controls whether or not the errors evolve jointly according to a VAR(q), or individually according to separate AR(q) processes. In terms of the formulation above, if error_var = False, then the matrices :math:C_i` are diagonal, otherwise they are general VAR matrices.

References

[*]	Lütkepohl, Helmut. 2007. New Introduction to Multiple Time Series Analysis. Berlin: Springer.

Methods

filter(params[, transformed, complex_step, …])	Kalman filtering
fit([start_params, transformed, cov_type, …])	Fits the model by maximum likelihood via Kalman filter.
from_formula(formula, data[, subset])	Not implemented for state space models
hessian(params, *args, **kwargs)	Hessian matrix of the likelihood function, evaluated at the given parameters
impulse_responses(params[, steps, impulse, …])	Impulse response function
information(params)	Fisher information matrix of model
initialize()	Initialize (possibly re-initialize) a Model instance.
initialize_approximate_diffuse([variance])
initialize_known(initial_state, …)
initialize_statespace(**kwargs)	Initialize the state space representation
initialize_stationary()
loglike(params, *args, **kwargs)	Loglikelihood evaluation
loglikeobs(params[, transformed, complex_step])	Loglikelihood evaluation
observed_information_matrix(params[, …])	Observed information matrix
opg_information_matrix(params[, …])	Outer product of gradients information matrix
predict(params[, exog])	After a model has been fit predict returns the fitted values.
prepare_data()	Prepare data for use in the state space representation
score(params, *args, **kwargs)	Compute the score function at params.
score_obs(params[, method, transformed, …])	Compute the score per observation, evaluated at params
set_conserve_memory([conserve_memory])	Set the memory conservation method
set_filter_method([filter_method])	Set the filtering method
set_inversion_method([inversion_method])	Set the inversion method
set_smoother_output([smoother_output])	Set the smoother output
set_stability_method([stability_method])	Set the numerical stability method
simulate(params, nsimulations[, …])	Simulate a new time series following the state space model
simulation_smoother([simulation_output])	Retrieve a simulation smoother for the state space model.
smooth(params[, transformed, complex_step, …])	Kalman smoothing
transform_jacobian(unconstrained[, …])	Jacobian matrix for the parameter transformation function
transform_params(unconstrained)	Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation
untransform_params(constrained)	Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer.
update(params[, transformed, complex_step])	Update the parameters of the model

Attributes

endog_names	Names of endogenous variables
exog_names
initial_variance
initialization
loglikelihood_burn
param_names	(list of str) List of human readable parameter names (for parameters actually included in the model).
start_params	(array) Starting parameters for maximum likelihood estimation.
tolerance