class statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothing(endog, trend=False, damped_trend=False, seasonal=None, initialization_method='estimated', initial_level=None, initial_trend=None, initial_seasonal=None, bounds=None, concentrate_scale=True, dates=None, freq=None, missing='none')[source]

Linear exponential smoothing models


The observed time-series process \(y\)

trendbool, optional

Whether or not to include a trend component. Default is False.

damped_trendbool, optional

Whether or not an included trend component is damped. Default is False.

seasonalint, optional

The number of periods in a complete seasonal cycle for seasonal (Holt-Winters) models. For example, 4 for quarterly data with an annual cycle or 7 for daily data with a weekly cycle. Default is no seasonal effects.

initialization_methodstr, optional

Method for initialize the recursions. One of:

  • ‘estimated’

  • ‘concentrated’

  • ‘heuristic’

  • ‘known’

If ‘known’ initialization is used, then initial_level must be passed, as well as initial_slope and initial_seasonal if applicable. Default is ‘estimated’.

initial_levelfloat, optional

The initial level component. Only used if initialization is ‘known’.

initial_trendfloat, optional

The initial trend component. Only used if initialization is ‘known’.

initial_seasonalarray_like, optional

The initial seasonal component. An array of length seasonal or length seasonal - 1 (in which case the last initial value is computed to make the average effect zero). Only used if initialization is ‘known’.

boundsiterable[tuple], optional

An iterable containing bounds for the parameters. Must contain four elements, where each element is a tuple of the form (lower, upper). Default is (0.0001, 0.9999) for the level, trend, and seasonal smoothing parameters and (0.8, 0.98) for the trend damping parameter.

concentrate_scalebool, optional

Whether or not to concentrate the scale (variance of the error term) out of the likelihood.


The parameters and states of this model are estimated by setting up the exponential smoothing equations as a special case of a linear Gaussian state space model and applying the Kalman filter. As such, it has slightly worse performance than the dedicated exponential smoothing model, sm.tsa.ExponentialSmoothing, and it does not support multiplicative (nonlinear) exponential smoothing models.

However, as a subclass of the state space models, this model class shares a consistent set of functionality with those models, which can make it easier to work with. In addition, it supports computing confidence intervals for forecasts and it supports concentrating the initial state out of the likelihood function.


[1] Hyndman, Rob, Anne B. Koehler, J. Keith Ord, and Ralph D. Snyder.

Forecasting with exponential smoothing: the state space approach. Springer Science & Business Media, 2008.


clone(endog[, exog])

filter(params[, cov_type, cov_kwds, …])

Kalman filtering

fit([start_params, transformed, …])

Fits the model by maximum likelihood via Kalman filter.

fit_constrained(constraints[, start_params])

Fit the model with some parameters subject to equality constraints.


Fix parameters to specific values (context manager)

from_formula(formula, data[, subset])

Not implemented for state space models

handle_params(params[, transformed, …])

hessian(params, *args, **kwargs)

Hessian matrix of the likelihood function, evaluated at the given parameters

impulse_responses(params[, steps, impulse, …])

Impulse response function


Fisher information matrix of model.


Initialize (possibly re-initialize) a Model instance.


Initialize approximate diffuse

initialize_known(initial_state, …)

Initialize known


Initialize the state space representation


Initialize stationary

loglike(params, *args, **kwargs)

Loglikelihood evaluation

loglikeobs(params[, transformed, …])

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 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 the memory conservation method


Set the filtering method


Set the inversion method


Set the smoother output


Set the numerical stability method

simulate(params, nsimulations[, …])

Simulate a new time series following the state space model


Retrieve a simulation smoother for the state space model.

smooth(params[, cov_type, cov_kwds, …])

Kalman smoothing

transform_jacobian(unconstrained[, …])

Jacobian matrix for the parameter transformation function


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

update(params[, transformed, …])

Update the parameters of the model



Names of endogenous variables.


The names of the exogenous variables.






(list of str) List of human readable parameter names (for parameters actually included in the model).


(array) Starting parameters for maximum likelihood estimation.


(list of str) List of human readable names for unobserved states.