Repository: statsmodels/statsmodels

Issue:
BUG: Inconsistencies in underlying state space model of holtwinters.ExponentialSmoothing
#### Short explanation

The statsmodels implementation of exponential smoothing in `holtwinters` implicitly uses the additive error state space models. Fits and calculations of information criteria are based on this and might be wrong when users assume the multiplicative error model.

#### Background

Up to recently, the statsmodels implementation of exponential smoothing in `holtwinters` did not use an explicit state space model. Fitting the model parameters and initial state was and is done by minimizing the sum of squared residuals, where `residuals = data - predicted`. This is currently implemented in the `SMOOTHERS`.

However, minimizing the sum of residuals with `residuals = data - predicted` really only makes sense when using an additive error model,
```
y_t = yhat_t + e_t,
```
 with `e_t = y_t - yhat_t ~ N(0, sigma**2)`. In this case, negative log-likelihood is
```
l = n * log(SSE/n),
```
so minimizing SSE is the same as minimizing negative log-likelihood.

When using a multiplicative error model, the state space model is 
```
y_t = yhat_t * (1 + e_t) = yhat_t + yhat_t * e_t.
```
In this case, `e_t = (y_t - yhat_t) / (yhat_t) ~ N(0, sigma**2)`. Therefore, the residuals should be `residuals = (data - prediction)/prediction`. In this case, negative log-likelihood is (see [Hyndman et al. 2002 (PDF)](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.468.7646&rep=rep1&type=pdf), equations (7) and (9)):
```
l = n * log(SSE/n) + 2 * sum(log(yhat))
```
with `SSE = sum((y - yhat) / yhat)`.

In the R package `forecast` (or the newer version, `fable`), the residuals are calculated as proposed above, that is, they are different for additive and multiplicative error models. Also, the default criterion used for optimizing the parameters is log likelihood.

#### How this affects the current implementation

With the introduction of state space simulations in #6560, it is now possible to choose between an additive and a multiplicative error model for simulations. However, fits are still performed implicitly using the additive model. Similar, information criteria are calculated using the likelihood of the additive model.

Users of `simulate` with `error="mul"` will therefore get imperfect fits and wrong information criteria for the model they assume to be using.

#### How this could be fixed

Fixing would require some changes behind the curtain, especially in the smoothers, but more importantly, it would require the user to specify the error model **before** the fit. An additional keyword argument in the `ExponentialSmoother` constructor with `"add"` as default value would be backwards compatible.


Discussion:
I agree, this is a good idea. Does this imply we need to rewrite the smoothers? One thing to keep in mind is that the versions actually called during fitting are written in Cython, so that that can be compiled for speed. Cython isn't too hard to pick up, so rewriting would still be doable, even if you haven't seen it before.
> Does this imply we need to rewrite the smoothers?

I think so. I'm familiar with Cython, so I guess that won't be a problem. In my opinion, there are two ways to go:

1) Change the smoothers to return negative log-likelihood, and fix the information criteria calculations - this would be pretty straightforward.

2)  I saw that `ExponentialSmoothing` inherits from `base.LikelihoodModel`, which already has a `fit` implementation that is maximizing the `loglike` method, so I thought about implementing `loglike` for the `ExponentialSmoothing` class. Then we could call the super class `fit` inside `ExponentialSmoothing.fit` to do the optimization. This would probably be an almost complete refactoring of the existing code and would probably take me more time, but I think it could reduce the complexity of the code (for example, I could avoid the code duplication that we have now with the smoothers and `_predict`), and make more use of the inheritance structure.

In both cases I would again use the implementation in the `forecast` package as reference for testing.
What is your preferred way to go?



This is somewhat up to you, especially given that (2) seems like more work. I think either one of these options would make a great improvement to our exponential smoothing models.

Personally, I lean towards (2), because I would like to see us move towards a more uniform API and set of features that the current exponential smoothing models don't always implement (e.g. I would like to see it with a similar API / feature set as `sm.tsa.statespace.ExponentialSmoothing`). With that in mind, if you do want to take on (2) then I would encourage creating a completely new model class, since otherwise you would likely yourself restricted by the API design of the current setup.
Although if you prefer to take on (2) and keep it within the existing model structure, that is also fine with me.
I think the route of creating a completely new model class is the best way to go, and I'm quite motivated to do this, even if it might be more effort.

Where should I put the new model class? Conceptually it would make sense to me to put it in `statespace`.

I was reading a bit through your exponential smoothing implementation and the `statsmodels.tsa.statespace.MLEModel` and `statsmodels.tsa.statespace.MLEResults` class and found that there are quite a few methods that I would also want to use and I could just copy, e.g. the information criteria in the results class, or `fix_params`, but maybe it is better to put all this in a new base class?
> I think the route of creating a completely new model class is the best way to go, and I'm quite motivated to do this, even if it might be more effort.

That's great.  I've got a lot on my plate at the moment outside of Statsmodels, but I'll help as much as I can, and hopefully be able to help more in a week or two.

> Where should I put the new model class? Conceptually it would make sense to me to put it in statespace.

I would put it in `tsa/exponential_smoothing`.  I made that directory with the idea in mind that I would eventually do what you're planning to do.

> I was reading a bit through your exponential smoothing implementation and the statsmodels.tsa.statespace.MLEModel and statsmodels.tsa.statespace.MLEResults class and found that there are quite a few methods that I would also want to use and I could just copy, e.g. the information criteria in the results class, or fix_params, but maybe it is better to put all this in a new base class?

This is a good idea. For now, why don't you subclass `MLEModel` and `MLEResults` and override where necessary?  Later we can see how best to factor out the base results.  But if you run into problems with this approach, we can adjust course if necessary.

Resolving patch:
diff --git a/setup.py b/setup.py
--- a/setup.py
+++ b/setup.py
@@ -148,6 +148,7 @@
 exts = dict(
     _stl={'source': 'statsmodels/tsa/_stl.pyx'},
     _exponential_smoothers={'source': 'statsmodels/tsa/_exponential_smoothers.pyx'},  # noqa: E501
+    _ets_smooth={'source': 'statsmodels/tsa/exponential_smoothing/_ets_smooth.pyx'},  # noqa: E501
     _innovations={'source': 'statsmodels/tsa/_innovations.pyx'},
     _hamilton_filter={'source': 'statsmodels/tsa/regime_switching/_hamilton_filter.pyx.in'},  # noqa: E501
     _kim_smoother={'source': 'statsmodels/tsa/regime_switching/_kim_smoother.pyx.in'},  # noqa: E501
diff --git a/statsmodels/tsa/exponential_smoothing/base.py b/statsmodels/tsa/exponential_smoothing/base.py
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/exponential_smoothing/base.py
@@ -0,0 +1,652 @@
+from collections import OrderedDict
+import contextlib
+import warnings
+
+import numpy as np
+import pandas as pd
+from scipy.stats import norm
+
+from statsmodels.base.data import PandasData
+from statsmodels.tools.decorators import cache_readonly
+from statsmodels.tools.eval_measures import aic, aicc, bic, hqic
+from statsmodels.tools.sm_exceptions import PrecisionWarning
+from statsmodels.tools.numdiff import (
+    _get_epsilon,
+    approx_fprime,
+    approx_fprime_cs,
+    approx_hess_cs,
+)
+from statsmodels.tools.tools import pinv_extended
+import statsmodels.tsa.base.tsa_model as tsbase
+
+
+class StateSpaceMLEModel(tsbase.TimeSeriesModel):
+    """
+    This is a temporary base model from ETS, here I just copy everything I need
+    from statespace.mlemodel.MLEModel
+    """
+
+    def __init__(
+        self, endog, exog=None, dates=None, freq=None, missing="none", **kwargs
+    ):
+        # TODO: this was changed from the original, requires some work when
+        # using this as base class for state space and exponential smoothing
+        super().__init__(
+            endog=endog, exog=exog, dates=dates, freq=freq, missing=missing
+        )
+
+        # Store kwargs to recreate model
+        self._init_kwargs = kwargs
+
+        # Prepared the endog array: C-ordered, shape=(nobs x k_endog)
+        self.endog, self.exog = self.prepare_data(self.data)
+        self.use_pandas = isinstance(self.data, PandasData)
+
+        # Dimensions
+        self.nobs = self.endog.shape[0]
+
+        # Setup holder for fixed parameters
+        self._has_fixed_params = False
+        self._fixed_params = None
+        self._params_index = None
+        self._fixed_params_index = None
+        self._free_params_index = None
+
+    @staticmethod
+    def prepare_data(data):
+        raise NotImplementedError
+
+    def clone(self, endog, exog=None, **kwargs):
+        raise NotImplementedError
+
+    def _validate_can_fix_params(self, param_names):
+        for param_name in param_names:
+            if param_name not in self.param_names:
+                raise ValueError(
+                    'Invalid parameter name passed: "%s".' % param_name
+                )
+
+    @property
+    def k_params(self):
+        return len(self.param_names)
+
+    @contextlib.contextmanager
+    def fix_params(self, params):
+        """
+        Fix parameters to specific values (context manager)
+
+        Parameters
+        ----------
+        params : dict
+            Dictionary describing the fixed parameter values, of the form
+            `param_name: fixed_value`. See the `param_names` property for valid
+            parameter names.
+
+        Examples
+        --------
+        >>> mod = sm.tsa.SARIMAX(endog, order=(1, 0, 1))
+        >>> with mod.fix_params({'ar.L1': 0.5}):
+                res = mod.fit()
+        """
+        # Initialization (this is done here rather than in the constructor
+        # because param_names may not be available at that point)
+        if self._fixed_params is None:
+            self._fixed_params = {}
+            self._params_index = OrderedDict(
+                zip(self.param_names, np.arange(self.k_params))
+            )
+
+        # Cache the current fixed parameters
+        cache_fixed_params = self._fixed_params.copy()
+        cache_has_fixed_params = self._has_fixed_params
+        cache_fixed_params_index = self._fixed_params_index
+        cache_free_params_index = self._free_params_index
+
+        # Validate parameter names and values
+        self._validate_can_fix_params(set(params.keys()))
+
+        # Set the new fixed parameters, keeping the order as given by
+        # param_names
+        self._fixed_params.update(params)
+        self._fixed_params = OrderedDict(
+            [
+                (name, self._fixed_params[name])
+                for name in self.param_names
+                if name in self._fixed_params
+            ]
+        )
+
+        # Update associated values
+        self._has_fixed_params = True
+        self._fixed_params_index = [
+            self._params_index[key] for key in self._fixed_params.keys()
+        ]
+        self._free_params_index = list(
+            set(np.arange(self.k_params)).difference(self._fixed_params_index)
+        )
+
+        try:
+            yield
+        finally:
+            # Reset the fixed parameters
+            self._has_fixed_params = cache_has_fixed_params
+            self._fixed_params = cache_fixed_params
+            self._fixed_params_index = cache_fixed_params_index
+            self._free_params_index = cache_free_params_index
+
+    def fit_constrained(self, constraints, start_params=None, **fit_kwds):
+        """
+        Fit the model with some parameters subject to equality constraints.
+
+        Parameters
+        ----------
+        constraints : dict
+            Dictionary of constraints, of the form `param_name: fixed_value`.
+            See the `param_names` property for valid parameter names.
+        start_params : array_like, optional
+            Initial guess of the solution for the loglikelihood maximization.
+            If None, the default is given by Model.start_params.
+        **fit_kwds : keyword arguments
+            fit_kwds are used in the optimization of the remaining parameters.
+
+        Returns
+        -------
+        results : Results instance
+
+        Examples
+        --------
+        >>> mod = sm.tsa.SARIMAX(endog, order=(1, 0, 1))
+        >>> res = mod.fit_constrained({'ar.L1': 0.5})
+        """
+        with self.fix_params(constraints):
+            res = self.fit(start_params, **fit_kwds)
+        return res
+
+    @property
+    def start_params(self):
+        """
+        (array) Starting parameters for maximum likelihood estimation.
+        """
+        if hasattr(self, "_start_params"):
+            return self._start_params
+        else:
+            raise NotImplementedError
+
+    @property
+    def param_names(self):
+        """
+        (list of str) List of human readable parameter names (for parameters
+        actually included in the model).
+        """
+        if hasattr(self, "_param_names"):
+            return self._param_names
+        else:
+            try:
+                names = ["param.%d" % i for i in range(len(self.start_params))]
+            except NotImplementedError:
+                names = []
+            return names
+
+    @classmethod
+    def from_formula(
+        cls, formula, data, subset=None, drop_cols=None, *args, **kwargs
+    ):
+        """
+        Not implemented for state space models
+        """
+        raise NotImplementedError
+
+    def _wrap_data(self, data, start_idx, end_idx, names=None):
+        # TODO: check if this is reasonable for statespace
+        data = np.squeeze(data)
+        if self.use_pandas:
+            _, _, _, index = self._get_prediction_index(start_idx, end_idx)
+            if data.ndim < 2:
+                data = pd.Series(data, index=index, name=names)
+            else:
+                data = pd.DataFrame(data, index=index, columns=names)
+        return data
+
+    def _wrap_results(
+        self,
+        params,
+        result,
+        return_raw,
+        cov_type=None,
+        cov_kwds=None,
+        results_class=None,
+        wrapper_class=None,
+    ):
+        if not return_raw:
+            # Wrap in a results object
+            result_kwargs = {}
+            if cov_type is not None:
+                result_kwargs["cov_type"] = cov_type
+            if cov_kwds is not None:
+                result_kwargs["cov_kwds"] = cov_kwds
+
+            if results_class is None:
+                results_class = self._res_classes["fit"][0]
+            if wrapper_class is None:
+                wrapper_class = self._res_classes["fit"][1]
+
+            res = results_class(self, params, result, **result_kwargs)
+            result = wrapper_class(res)
+        return result
+
+    def _score_complex_step(self, params, **kwargs):
+        # the default epsilon can be too small
+        # inversion_method = INVERT_UNIVARIATE | SOLVE_LU
+        epsilon = _get_epsilon(params, 2., None, len(params))
+        kwargs['transformed'] = True
+        kwargs['complex_step'] = True
+        return approx_fprime_cs(params, self.loglike, epsilon=epsilon,
+                                kwargs=kwargs)
+
+    def _score_finite_difference(self, params, approx_centered=False,
+                                 **kwargs):
+        kwargs['transformed'] = True
+        return approx_fprime(params, self.loglike, kwargs=kwargs,
+                             centered=approx_centered)
+
+    def _hessian_finite_difference(self, params, approx_centered=False,
+                                   **kwargs):
+        params = np.array(params, ndmin=1)
+
+        warnings.warn('Calculation of the Hessian using finite differences'
+                      ' is usually subject to substantial approximation'
+                      ' errors.', PrecisionWarning)
+
+        if not approx_centered:
+            epsilon = _get_epsilon(params, 3, None, len(params))
+        else:
+            epsilon = _get_epsilon(params, 4, None, len(params)) / 2
+        hessian = approx_fprime(params, self._score_finite_difference,
+                                epsilon=epsilon, kwargs=kwargs,
+                                centered=approx_centered)
+
+        # TODO: changed this to nobs_effective, has to be changed when merging
+        # with statespace mlemodel
+        return hessian / (self.nobs_effective)
+
+    def _hessian_complex_step(self, params, **kwargs):
+        """
+        Hessian matrix computed by second-order complex-step differentiation
+        on the `loglike` function.
+        """
+        # the default epsilon can be too small
+        epsilon = _get_epsilon(params, 3., None, len(params))
+        kwargs['transformed'] = True
+        kwargs['complex_step'] = True
+        hessian = approx_hess_cs(
+            params, self.loglike, epsilon=epsilon, kwargs=kwargs)
+
+        # TODO: changed this to nobs_effective, has to be changed when merging
+        # with statespace mlemodel
+        return hessian / (self.nobs_effective)
+
+
+class StateSpaceMLEResults(tsbase.TimeSeriesModelResults):
+    r"""
+    Class to hold results from fitting a state space model.
+
+    Parameters
+    ----------
+    model : MLEModel instance
+        The fitted model instance
+    params : ndarray
+        Fitted parameters
+
+    Attributes
+    ----------
+    model : Model instance
+        A reference to the model that was fit.
+    nobs : float
+        The number of observations used to fit the model.
+    params : ndarray
+        The parameters of the model.
+    """
+
+    def __init__(self, model, params, scale=1.0):
+        self.data = model.data
+        self.endog = model.data.orig_endog
+
+        super().__init__(model, params, None, scale=scale)
+
+        # Save the fixed parameters
+        self._has_fixed_params = self.model._has_fixed_params
+        self._fixed_params_index = self.model._fixed_params_index
+        self._free_params_index = self.model._free_params_index
+        # TODO: seems like maybe self.fixed_params should be the dictionary
+        # itself, not just the keys?
+        if self._has_fixed_params:
+            self._fixed_params = self.model._fixed_params.copy()
+            self.fixed_params = list(self._fixed_params.keys())
+        else:
+            self._fixed_params = None
+            self.fixed_params = []
+        self.param_names = [
+            "%s (fixed)" % name if name in self.fixed_params else name
+            for name in (self.data.param_names or [])
+        ]
+
+        # Dimensions
+        self.nobs = self.model.nobs
+        self.k_params = self.model.k_params
+
+        self._rank = None
+
+    @cache_readonly
+    def nobs_effective(self):
+        raise NotImplementedError
+
+    @cache_readonly
+    def df_model(self):
+        raise NotImplementedError
+
+    @cache_readonly
+    def df_resid(self):
+        return self.nobs_effective - self.df_model
+
+    @cache_readonly
+    def aic(self):
+        """
+        (float) Akaike Information Criterion
+        """
+        return aic(self.llf, self.nobs_effective, self.df_model)
+
+    @cache_readonly
+    def aicc(self):
+        """
+        (float) Akaike Information Criterion with small sample correction
+        """
+        return aicc(self.llf, self.nobs_effective, self.df_model)
+
+    @cache_readonly
+    def bic(self):
+        """
+        (float) Bayes Information Criterion
+        """
+        return bic(self.llf, self.nobs_effective, self.df_model)
+
+    @cache_readonly
+    def fittedvalues(self):
+        # TODO
+        raise NotImplementedError
+
+    @cache_readonly
+    def hqic(self):
+        """
+        (float) Hannan-Quinn Information Criterion
+        """
+        # return (-2 * self.llf +
+        #         2 * np.log(np.log(self.nobs_effective)) * self.df_model)
+        return hqic(self.llf, self.nobs_effective, self.df_model)
+
+    @cache_readonly
+    def llf(self):
+        """
+        (float) The value of the log-likelihood function evaluated at `params`.
+        """
+        raise NotImplementedError
+
+    @cache_readonly
+    def mae(self):
+        """
+        (float) Mean absolute error
+        """
+        return np.mean(np.abs(self.resid))
+
+    @cache_readonly
+    def mse(self):
+        """
+        (float) Mean squared error
+        """
+        return self.sse / self.nobs
+
+    @cache_readonly
+    def pvalues(self):
+        """
+        (array) The p-values associated with the z-statistics of the
+        coefficients. Note that the coefficients are assumed to have a Normal
+        distribution.
+        """
+        pvalues = np.zeros_like(self.zvalues) * np.nan
+        mask = np.ones_like(pvalues, dtype=bool)
+        mask[self._free_params_index] = True
+        mask &= ~np.isnan(self.zvalues)
+        pvalues[mask] = norm.sf(np.abs(self.zvalues[mask])) * 2
+        return pvalues
+
+    @cache_readonly
+    def resid(self):
+        raise NotImplementedError
+
+    @cache_readonly
+    def sse(self):
+        """
+        (float) Sum of squared errors
+        """
+        return np.sum(self.resid ** 2)
+
+    @cache_readonly
+    def zvalues(self):
+        """
+        (array) The z-statistics for the coefficients.
+        """
+        return self.params / self.bse
+
+    def _get_prediction_start_index(self, anchor):
+        """Returns a valid numeric start index for predictions/simulations"""
+        # TODO: once this is the base class for statespace models, use this
+        # method in simulate
+        if anchor is None or anchor == "start":
+            iloc = 0
+        elif anchor == "end":
+            iloc = self.nobs
+        else:
+            iloc, _, _ = self.model._get_index_loc(anchor)
+            if isinstance(iloc, slice):
+                iloc = iloc.start
+
+        if iloc < 0:
+            iloc = self.nobs + iloc
+        if iloc > self.nobs:
+            raise ValueError("Cannot anchor simulation outside of the sample.")
+        return iloc
+
+    def _cov_params_approx(
+        self, approx_complex_step=True, approx_centered=False
+    ):
+        evaluated_hessian = self.nobs_effective * self.model.hessian(
+            params=self.params,
+            transformed=True,
+            includes_fixed=True,
+            method="approx",
+            approx_complex_step=approx_complex_step,
+            approx_centered=approx_centered,
+        )
+        # TODO: Case with "not approx_complex_step" is not hit in
+        # tests as of 2017-05-19
+
+        if len(self.fixed_params) > 0:
+            mask = np.ix_(self._free_params_index, self._free_params_index)
+            if len(self.fixed_params) < self.k_params:
+                (tmp, singular_values) = pinv_extended(evaluated_hessian[mask])
+            else:
+                tmp, singular_values = np.nan, [np.nan]
+            neg_cov = np.zeros_like(evaluated_hessian) * np.nan
+            neg_cov[mask] = tmp
+        else:
+            (neg_cov, singular_values) = pinv_extended(evaluated_hessian)
+
+        self.model.update(self.params, transformed=True, includes_fixed=True)
+        if self._rank is None:
+            self._rank = np.linalg.matrix_rank(np.diag(singular_values))
+        return -neg_cov
+
+    @cache_readonly
+    def cov_params_approx(self):
+        """
+        (array) The variance / covariance matrix. Computed using the numerical
+        Hessian approximated by complex step or finite differences methods.
+        """
+        return self._cov_params_approx(
+            self._cov_approx_complex_step, self._cov_approx_centered
+        )
+
+    def summary(
+        self,
+        alpha=0.05,
+        start=None,
+        title=None,
+        model_name=None,
+        display_params=True,
+    ):
+        """
+        Summarize the Model
+
+        Parameters
+        ----------
+        alpha : float, optional
+            Significance level for the confidence intervals. Default is 0.05.
+        start : int, optional
+            Integer of the start observation. Default is 0.
+        model_name : str
+            The name of the model used. Default is to use model class name.
+
+        Returns
+        -------
+        summary : Summary instance
+            This holds the summary table and text, which can be printed or
+            converted to various output formats.
+
+        See Also
+        --------
+        statsmodels.iolib.summary.Summary
+        """
+        from statsmodels.iolib.summary import Summary
+
+        # Model specification results
+        model = self.model
+        if title is None:
+            title = "Statespace Model Results"
+
+        if start is None:
+            start = 0
+        if self.model._index_dates:
+            ix = self.model._index
+            d = ix[start]
+            sample = ["%02d-%02d-%02d" % (d.month, d.day, d.year)]
+            d = ix[-1]
+            sample += ["- " + "%02d-%02d-%02d" % (d.month, d.day, d.year)]
+        else:
+            sample = [str(start), " - " + str(self.nobs)]
+
+        # Standardize the model name as a list of str
+        if model_name is None:
+            model_name = model.__class__.__name__
+
+        # Diagnostic tests results
+        try:
+            het = self.test_heteroskedasticity(method="breakvar")
+        except Exception:  # FIXME: catch something specific
+            het = np.array([[np.nan] * 2])
+        try:
+            lb = self.test_serial_correlation(method="ljungbox")
+        except Exception:  # FIXME: catch something specific
+            lb = np.array([[np.nan] * 2]).reshape(1, 2, 1)
+        try:
+            jb = self.test_normality(method="jarquebera")
+        except Exception:  # FIXME: catch something specific
+            jb = np.array([[np.nan] * 4])
+
+        # Create the tables
+        if not isinstance(model_name, list):
+            model_name = [model_name]
+
+        top_left = [("Dep. Variable:", None)]
+        top_left.append(("Model:", [model_name[0]]))
+        for i in range(1, len(model_name)):
+            top_left.append(("", ["+ " + model_name[i]]))
+        top_left += [
+            ("Date:", None),
+            ("Time:", None),
+            ("Sample:", [sample[0]]),
+            ("", [sample[1]]),
+        ]
+
+        top_right = [
+            ("No. Observations:", [self.nobs]),
+            ("Log Likelihood", ["%#5.3f" % self.llf]),
+        ]
+        if hasattr(self, "rsquared"):
+            top_right.append(("R-squared:", ["%#8.3f" % self.rsquared]))
+        top_right += [
+            ("AIC", ["%#5.3f" % self.aic]),
+            ("BIC", ["%#5.3f" % self.bic]),
+            ("HQIC", ["%#5.3f" % self.hqic]),
+        ]
+
+        if (
+            hasattr(self, "filter_results")
+            and self.filter_results is not None
+            and self.filter_results.filter_concentrated
+        ):
+            top_right.append(("Scale", ["%#5.3f" % self.scale]))
+
+        if hasattr(self, "cov_type"):
+            top_left.append(("Covariance Type:", [self.cov_type]))
+
+        format_str = lambda array: [  # noqa:E731
+            ", ".join(["{0:.2f}".format(i) for i in array])
+        ]
+        diagn_left = [
+            ("Ljung-Box (Q):", format_str(lb[:, 0, -1])),
+            ("Prob(Q):", format_str(lb[:, 1, -1])),
+            ("Heteroskedasticity (H):", format_str(het[:, 0])),
+            ("Prob(H) (two-sided):", format_str(het[:, 1])),
+        ]
+
+        diagn_right = [
+            ("Jarque-Bera (JB):", format_str(jb[:, 0])),
+            ("Prob(JB):", format_str(jb[:, 1])),
+            ("Skew:", format_str(jb[:, 2])),
+            ("Kurtosis:", format_str(jb[:, 3])),
+        ]
+
+        summary = Summary()
+        summary.add_table_2cols(
+            self, gleft=top_left, gright=top_right, title=title
+        )
+        if len(self.params) > 0 and display_params:
+            summary.add_table_params(
+                self, alpha=alpha, xname=self.param_names, use_t=False
+            )
+        summary.add_table_2cols(
+            self, gleft=diagn_left, gright=diagn_right, title=""
+        )
+
+        # Add warnings/notes, added to text format only
+        etext = []
+        if hasattr(self, "cov_type") and "description" in self.cov_kwds:
+            etext.append(self.cov_kwds["description"])
+        if self._rank < (len(self.params) - len(self.fixed_params)):
+            cov_params = self.cov_params()
+            if len(self.fixed_params) > 0:
+                mask = np.ix_(self._free_params_index, self._free_params_index)
+                cov_params = cov_params[mask]
+            etext.append(
+                "Covariance matrix is singular or near-singular,"
+                " with condition number %6.3g. Standard errors may be"
+                " unstable." % np.linalg.cond(cov_params)
+            )
+
+        if etext:
+            etext = [
+                "[{0}] {1}".format(i + 1, text) for i, text in enumerate(etext)
+            ]
+            etext.insert(0, "Warnings:")
+            summary.add_extra_txt(etext)
+
+        return summary
diff --git a/statsmodels/tsa/exponential_smoothing/ets.py b/statsmodels/tsa/exponential_smoothing/ets.py
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/exponential_smoothing/ets.py
@@ -0,0 +1,1675 @@
+r"""
+ETS models for time series analysis.
+
+The ETS models are a family of time series models. They can be seen as a
+generalization of simple exponential smoothing to time series that contain
+trends and seasonalities. Additionally, they have an underlying state space
+model.
+
+An ETS model is specified by an error type (E; additive or multiplicative), a
+trend type (T; additive or multiplicative, both damped or undamped, or none),
+and a seasonality type (S; additive or multiplicative or none).
+The following gives a very short summary, a more thorough introduction can be
+found in [1]_.
+
+Denote with :math:`\circ_b` the trend operation (addition or
+multiplication), with :math:`\circ_d` the operation linking trend and dampening
+factor :math:`\phi` (multiplication if trend is additive, power if trend is
+multiplicative), and with :math:`\circ_s` the seasonality operation (addition
+or multiplication).
+Furthermore, let :math:`\ominus` be the respective inverse operation
+(subtraction or division).
+
+With this, it is possible to formulate the ETS models as a forecast equation
+and 3 smoothing equations. The former is used to forecast observations, the
+latter are used to update the internal state.
+
+.. math::
+
+    \hat{y}_{t|t-1} &= (l_{t-1} \circ_b (b_{t-1}\circ_d \phi))\circ_s s_{t-m}\\
+    l_{t} &= \alpha (y_{t} \ominus_s s_{t-m})
+             + (1 - \alpha) (l_{t-1} \circ_b (b_{t-1} \circ_d \phi))\\
+    b_{t} &= \beta/\alpha (l_{t} \ominus_b l_{t-1})
+             + (1 - \beta/\alpha) b_{t-1}\\
+    s_{t} &= \gamma (y_t \ominus_s (l_{t-1} \circ_b (b_{t-1}\circ_d\phi))
+             + (1 - \gamma) s_{t-m}
+
+The notation here follows [1]_; :math:`l_t` denotes the level at time
+:math:`t`, `b_t` the trend, and `s_t` the seasonal component. :math:`m` is the
+number of seasonal periods, and :math:`\phi` a trend damping factor.
+The parameters :math:`\alpha, \beta, \gamma` are the smoothing parameters,
+which are called ``smoothing_level``, ``smoothing_trend``, and
+``smoothing_seasonal``, respectively.
+
+Note that the formulation above as forecast and smoothing equation does not
+distinguish different error models -- it is the same for additive and
+multiplicative errors. But the different error models lead to different
+likelihood models, and therefore will lead to different fit results.
+
+The error models specify how the true values :math:`y_t` are updated. In the
+additive error model,
+
+.. math::
+
+    y_t = \hat{y}_{t|t-1} + e_t,
+
+in the multiplicative error model,
+
+.. math::
+
+    y_t = \hat{y}_{t|t-1}\cdot (1 + e_t).
+
+Using these error models, it is possible to formulate state space equations for
+the ETS models:
+
+.. math::
+
+   y_t &= Y_t + \eta \cdot e_t\\
+   l_t &= L_t + \alpha \cdot (M_e \cdot L_t + \kappa_l) \cdot e_t\\
+   b_t &= B_t + \beta \cdot (M_e \cdot B_t + \kappa_b) \cdot e_t\\
+   s_t &= S_t + \gamma \cdot (M_e \cdot S_t+\kappa_s)\cdot e_t\\
+
+with
+
+.. math::
+
+   B_t &= b_{t-1} \circ_d \phi\\
+   L_t &= l_{t-1} \circ_b B_t\\
+   S_t &= s_{t-m}\\
+   Y_t &= L_t \circ_s S_t,
+
+and
+
+.. math::
+
+   \eta &= \begin{cases}
+               Y_t\quad\text{if error is multiplicative}\\
+               1\quad\text{else}
+           \end{cases}\\
+   M_e &= \begin{cases}
+               1\quad\text{if error is multiplicative}\\
+               0\quad\text{else}
+           \end{cases}\\
+
+and, when using the additve error model,
+
+.. math::
+
+   \kappa_l &= \begin{cases}
+               \frac{1}{S_t}\quad
+               \text{if seasonality is multiplicative}\\
+               1\quad\text{else}
+           \end{cases}\\
+   \kappa_b &= \begin{cases}
+               \frac{\kappa_l}{l_{t-1}}\quad
+               \text{if trend is multiplicative}\\
+               \kappa_l\quad\text{else}
+           \end{cases}\\
+   \kappa_s &= \begin{cases}
+               \frac{1}{L_t}\quad\text{if seasonality is multiplicative}\\
+               1\quad\text{else}
+           \end{cases}
+
+When using the multiplicative error model
+
+.. math::
+
+   \kappa_l &= \begin{cases}
+               0\quad
+               \text{if seasonality is multiplicative}\\
+               S_t\quad\text{else}
+           \end{cases}\\
+   \kappa_b &= \begin{cases}
+               \frac{\kappa_l}{l_{t-1}}\quad
+               \text{if trend is multiplicative}\\
+               \kappa_l + l_{t-1}\quad\text{else}
+           \end{cases}\\
+   \kappa_s &= \begin{cases}
+               0\quad\text{if seasonality is multiplicative}\\
+               L_t\quad\text{else}
+           \end{cases}
+
+When fitting an ETS model, the parameters :math:`\alpha, \beta`, \gamma,
+\phi` and the initial states `l_{-1}, b_{-1}, s_{-1}, \ldots, s_{-m}` are
+selected as maximizers of log likelihood.
+
+References
+----------
+.. [1] Hyndman, R.J., & Athanasopoulos, G. (2019) *Forecasting:
+   principles and practice*, 3rd edition, OTexts: Melbourne,
+   Australia. OTexts.com/fpp3. Accessed on April 19th 2020.
+"""
+
+from collections import OrderedDict
+import contextlib
+import numpy as np
+import pandas as pd
+from scipy.stats import _distn_infrastructure, rv_continuous, rv_discrete
+
+from statsmodels.base.covtype import descriptions
+import statsmodels.base.wrapper as wrap
+from statsmodels.iolib.table import SimpleTable
+from statsmodels.iolib.tableformatting import fmt_params
+from statsmodels.iolib.summary import forg
+from statsmodels.tools.decorators import cache_readonly
+from statsmodels.tools.tools import Bunch
+from statsmodels.tools.validation import (
+    bool_like,
+    int_like,
+    string_like,
+    array_like,
+)
+import statsmodels.tsa.base.tsa_model as tsbase
+from statsmodels.tsa.exponential_smoothing import base
+import statsmodels.tsa.exponential_smoothing._ets_smooth as smooth
+from statsmodels.tsa.exponential_smoothing.initialization import (
+    _initialization_heuristic,
+)
+from statsmodels.tsa.tsatools import freq_to_period
+
+# Implementation details:
+
+# * The smoothing equations are implemented only for models having all
+#   components (trend, dampening, seasonality). When using other models, the
+#   respective parameters (smoothing and initial parameters) are set to values
+#   that lead to the reduced model (often zero).
+#   The internal model is needed for smoothing (called from fit and loglike),
+#   forecasts, and simulations.
+# * Somewhat related to above: There are 2 sets of parameters: model/external
+#   params, and internal params.
+#   - model params are all parameters necessary for a model, and are for
+#     example passed as argument to the likelihood function or as start_params
+#     to fit
+#   - internal params are what is used internally in the smoothing equations
+# * Regarding fitting, bounds, fixing parameters, and internal parameters, the
+#   overall workflow is the following:
+#   - get start parameters in the form of external parameters (includes fixed
+#     parameters)
+#   - transform external parameters to internal parameters, bounding all that
+#     are missing -> now we have some missing parameters, but potentially also
+#     some user-specified bounds
+#   - set bounds for fixed parameters
+#   - make sure that starting parameters are within bounds
+#   - set up the constraint bounds and function
+# * Since the traditional bounds are nonlinear for beta and gamma, if no bounds
+#   are given, we internally use beta_star and gamma_star for fitting
+
+
+class ETSModel(base.StateSpaceMLEModel):
+    """
+    ETS models.
+
+    Parameters
+    ----------
+    endog : array_like
+        The observed time-series process :math:`y`
+    error: str, optional
+        The error model. "add" (default) or "mul".
+    trend : str or None, optional
+        The trend component model. "add", "mul", or None (default).
+    damped_trend : bool, optional
+        Whether or not an included trend component is damped. Default is False.
+    seasonal : str, optional
+        The seasonality model. "add", "mul", or None (default).
+    seasonal_periods: int, 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. Required if
+        `seasonal` is not None.
+    initialization_method : str, optional
+        Method for initialization of the state space model. One of:
+
+        * 'estimated' (default)
+        * 'heuristic'
+        * 'known'
+
+        If 'known' initialization is used, then `initial_level` must be
+        passed, as well as `initial_trend` and `initial_seasonal` if
+        applicable.
+        'heuristic' uses a heuristic based on the data to estimate initial
+        level, trend, and seasonal state. 'estimated' uses the same heuristic
+        as initial guesses, but then estimates the initial states as part of
+        the fitting process.  Default is 'estimated'.
+    initial_level : float, optional
+        The initial level component. Only used if initialization is 'known'.
+    initial_trend : float, optional
+        The initial trend component. Only used if initialization is 'known'.
+    initial_seasonal : array_like, optional
+        The initial seasonal component. An array of length `seasonal`.  Only
+        used if initialization is 'known'.
+    bounds : dict or None, optional
+        A dictionary with parameter names as keys and the respective bounds
+        intervals as values (lists/tuples/arrays).
+        The availabe parameter names are, depeding on the model and
+        initialization method:
+            * "smoothing_level"
+            * "smoothing_trend"
+            * "smoothing_seasonal"
+            * "damping_trend"
+            * "initial_level"
+            * "initial_trend"
+            * "initial_seasonal.0", ..., "initial_seasonal.<m-1>"
+        The default option is ``None``, in which case the traditional
+        (nonlinear) bounds as described in [1]_ are used.
+    """
+
+    def __init__(
+        self,
+        endog,
+        error="add",
+        trend=None,
+        damped_trend=False,
+        seasonal=None,
+        seasonal_periods=None,
+        initialization_method="estimated",
+        initial_level=None,
+        initial_trend=None,
+        initial_seasonal=None,
+        bounds=None,
+        dates=None,
+        freq=None,
+        missing="none",
+    ):
+
+        super().__init__(
+            endog, exog=None, dates=dates, freq=freq, missing=missing
+        )
+
+        # MODEL DEFINITION
+        # ================
+        options = ("add", "mul", "additive", "multiplicative")
+        # take first three letters of option -> either "add" or "mul"
+        self.error = string_like(error, "error", options=options)[:3]
+        self.trend = string_like(
+            trend, "trend", options=options, optional=True
+        )
+        if self.trend is not None:
+            self.trend = self.trend[:3]
+        self.damped_trend = bool_like(damped_trend, "damped_trend")
+        self.seasonal = string_like(
+            seasonal, "seasonal", options=options, optional=True
+        )
+        if self.seasonal is not None:
+            self.seasonal = self.seasonal[:3]
+
+        self.has_trend = self.trend is not None
+        self.has_seasonal = self.seasonal is not None
+
+        if self.has_seasonal:
+            self.seasonal_periods = int_like(
+                seasonal_periods, "seasonal_periods", optional=True
+            )
+            if seasonal_periods is None:
+                self.seasonal_periods = freq_to_period(self._index_freq)
+            if self.seasonal_periods <= 1:
+                raise ValueError("seasonal_periods must be larger than 1.")
+        else:
+            # in case the model has no seasonal component, we internally handle
+            # this as if it had an additive seasonal component with
+            # seasonal_periods=1, but restrict the smoothing parameter to 0 and
+            # set the initial seasonal to 0.
+            self.seasonal_periods = 1
+
+        # reject invalid models
+        if np.any(self.endog <= 0) and (
+            self.error == "mul"
+            or self.trend == "mul"
+            or self.seasonal == "mul"
+        ):
+            raise ValueError(
+                "endog must be strictly positive when using"
+                "multiplicative error, trend or seasonal components."
+            )
+        if self.damped_trend and not self.has_trend:
+            raise ValueError("Can only dampen the trend component")
+
+        # INITIALIZATION METHOD
+        # =====================
+        self.set_initialization_method(
+            initialization_method,
+            initial_level,
+            initial_trend,
+            initial_seasonal,
+        )
+
+        # BOUNDS
+        # ======
+        self.set_bounds(bounds)
+
+        # SMOOTHER
+        # ========
+        if self.trend == "add" or self.trend is None:
+            if self.seasonal == "add" or self.seasonal is None:
+                self._smoothing_func = smooth._ets_smooth_add_add
+            else:
+                self._smoothing_func = smooth._ets_smooth_add_mul
+        else:
+            if self.seasonal == "add" or self.seasonal is None:
+                self._smoothing_func = smooth._ets_smooth_mul_add
+            else:
+                self._smoothing_func = smooth._ets_smooth_mul_mul
+
+    def set_initialization_method(
+        self,
+        initialization_method,
+        initial_level=None,
+        initial_trend=None,
+        initial_seasonal=None,
+    ):
+        """
+        Sets a new initialization method for the state space model.
+
+        Parameters
+        ----------
+        initialization_method : str, optional
+            Method for initialization of the state space model. One of:
+
+            * 'estimated' (default)
+            * 'heuristic'
+            * 'known'
+
+            If 'known' initialization is used, then `initial_level` must be
+            passed, as well as `initial_trend` and `initial_seasonal` if
+            applicable.
+            'heuristic' uses a heuristic based on the data to estimate initial
+            level, trend, and seasonal state. 'estimated' uses the same
+            heuristic as initial guesses, but then estimates the initial states
+            as part of the fitting process. Default is 'estimated'.
+        initial_level : float, optional
+            The initial level component. Only used if initialization is
+            'known'.
+        initial_trend : float, optional
+            The initial trend component. Only used if initialization is
+            'known'.
+        initial_seasonal : array_like, optional
+            The initial seasonal component. An array of length `seasonal`. Only
+            used if initialization is 'known'.
+        """
+        self.initialization_method = string_like(
+            initialization_method,
+            "initialization_method",
+            options=("estimated", "known", "heuristic"),
+        )
+        if self.initialization_method == "known":
+            if initial_level is None:
+                raise ValueError(
+                    "`initial_level` argument must be provided"
+                    ' when initialization method is set to "known".'
+                )
+            if self.has_trend and initial_trend is None:
+                raise ValueError(
+                    "`initial_trend` argument must be provided"
+                    " for models with a trend component when"
+                    ' initialization method is set to "known".'
+                )
+            if self.has_seasonal and initial_seasonal is None:
+                raise ValueError(
+                    "`initial_seasonal` argument must be provided"
+                    " for models with a seasonal component when"
+                    ' initialization method is set to "known".'
+                )
+        elif (
+            self.initialization_method == "heuristic"
+            or self.initialization_method == "estimated"
+        ):
+            (
+                initial_level,
+                initial_trend,
+                initial_seasonal,
+            ) = _initialization_heuristic(
+                self.endog,
+                trend=self.trend,
+                seasonal=self.seasonal,
+                seasonal_periods=self.seasonal_periods,
+            )
+        if not self.has_trend:
+            initial_trend = 0
+        if not self.has_seasonal:
+            initial_seasonal = 0
+        self.initial_level = initial_level
+        self.initial_trend = initial_trend
+        self.initial_seasonal = initial_seasonal
+
+        # we also have to reset the params index dictionaries
+        self._internal_params_index = OrderedDict(
+            zip(self._internal_param_names, np.arange(self._k_params_internal))
+        )
+        self._params_index = OrderedDict(
+            zip(self.param_names, np.arange(self.k_params))
+        )
+
+    def set_bounds(self, bounds):
+        """
+        Set bounds for parameter estimation.
+
+        Parameters
+        ----------
+        bounds : dict or None, optional
+            A dictionary with parameter names as keys and the respective bounds
+            intervals as values (lists/tuples/arrays).
+            The available parameter names are in ``self.param_names``.
+            The default option is ``None``, in which case the traditional
+            (nonlinear) bounds as described in [1]_ are used.
+        """
+        if bounds is None:
+            self.bounds = {}
+        else:
+            if not isinstance(bounds, (dict, OrderedDict)):
+                raise ValueError("bounds must be a dictionary")
+            for key in bounds:
+                if key not in self.param_names:
+                    raise ValueError(
+                        f"Invalid key: {key} in bounds dictionary"
+                    )
+                bounds[key] = array_like(
+                    bounds[key], f"bounds[{key}]", shape=(2,)
+                )
+            self.bounds = bounds
+
+    @staticmethod
+    def prepare_data(data):
+        """
+        Prepare data for use in the state space representation
+        """
+        endog = np.array(data.orig_endog, order="C")
+        if endog.ndim != 1:
+            raise ValueError("endog must be 1-dimensional")
+        return endog, None
+
+    @property
+    def nobs_effective(self):
+        return self.nobs
+
+    @property
+    def short_name(self):
+        name = "".join(
+            [
+                str(s)[0].upper()
+                for s in [self.error, self.trend, self.seasonal]
+            ]
+        )
+        if self.damped_trend:
+            name = name[0:2] + "d" + name[2]
+        return name
+
+    @property
+    def _param_names(self):
+        param_names = ["smoothing_level"]
+        if self.has_trend:
+            param_names += ["smoothing_trend"]
+        if self.has_seasonal:
+            param_names += ["smoothing_seasonal"]
+        if self.damped_trend:
+            param_names += ["damping_trend"]
+
+        # Initialization
+        if self.initialization_method == "estimated":
+            param_names += ["initial_level"]
+            if self.has_trend:
+                param_names += ["initial_trend"]
+            if self.has_seasonal:
+                param_names += [
+                    f"initial_seasonal.{i}"
+                    for i in range(self.seasonal_periods)
+                ]
+        return param_names
+
+    @property
+    def state_names(self):
+        names = ["level"]
+        if self.has_trend:
+            names += ["trend"]
+        if self.has_seasonal:
+            names += ["seasonal"]
+        return names
+
+    @property
+    def initial_state_names(self):
+        names = ["initial_level"]
+        if self.has_trend:
+            names += ["initial_trend"]
+        if self.has_seasonal:
+            names += [
+                f"initial_seasonal.{i}" for i in range(self.seasonal_periods)
+            ]
+        return names
+
+    @property
+    def _smoothing_param_names(self):
+        return [
+            "smoothing_level",
+            "smoothing_trend",
+            "smoothing_seasonal",
+            "damping_trend",
+        ]
+
+    @property
+    def _internal_initial_state_names(self):
+        param_names = [
+            "initial_level",
+            "initial_trend",
+        ]
+        param_names += [
+            f"initial_seasonal.{i}" for i in range(self.seasonal_periods)
+        ]
+        return param_names
+
+    @property
+    def _internal_param_names(self):
+        return self._smoothing_param_names + self._internal_initial_state_names
+
+    @property
+    def _k_states(self):
+        return 1 + int(self.has_trend) + int(self.has_seasonal)  # level
+
+    @property
+    def _k_states_internal(self):
+        return 2 + self.seasonal_periods
+
+    @property
+    def _k_smoothing_params(self):
+        return self._k_states + int(self.damped_trend)
+
+    @property
+    def _k_initial_states(self):
+        return (
+            1
+            + int(self.has_trend)
+            + +int(self.has_seasonal) * self.seasonal_periods
+        )
+
+    @property
+    def k_params(self):
+        k = self._k_smoothing_params
+        if self.initialization_method == "estimated":
+            k += self._k_initial_states
+        return k
+
+    @property
+    def _k_params_internal(self):
+        return 4 + 2 + self.seasonal_periods
+
+    def _internal_params(self, params):
+        """
+        Converts a parameter array passed from outside to the internally used
+        full parameter array.
+        """
+        # internal params that are not needed are all set to zero, except phi,
+        # which is one
+        internal = np.zeros(self._k_params_internal, dtype=params.dtype)
+        for i, name in enumerate(self.param_names):
+            internal_idx = self._internal_params_index[name]
+            internal[internal_idx] = params[i]
+        if not self.damped_trend:
+            internal[3] = 1  # phi is 4th parameter
+        if self.initialization_method != "estimated":
+            internal[4] = self.initial_level
+            internal[5] = self.initial_trend
+            internal[6:] = self.initial_seasonal
+        return internal
+
+    def _model_params(self, internal):
+        """
+        Converts internal parameters to model parameters
+        """
+        params = np.empty(self.k_params)
+        for i, name in enumerate(self.param_names):
+            internal_idx = self._internal_params_index[name]
+            params[i] = internal[internal_idx]
+        return params
+
+    @property
+    def _seasonal_index(self):
+        return 1 + int(self.has_trend)
+
+    def _get_states(self, xhat):
+        states = np.empty((self.nobs, self._k_states))
+        all_names = ["level", "trend", "seasonal"]
+        for i, name in enumerate(self.state_names):
+            idx = all_names.index(name)
+            states[:, i] = xhat[:, idx]
+        return states
+
+    def _get_internal_states(self, states, params):
+        """
+        Converts a state matrix/dataframe to the (nobs, 2+m) matrix used
+        internally
+        """
+        internal_params = self._internal_params(params)
+        if isinstance(states, (pd.Series, pd.DataFrame)):
+            states = states.values
+        internal_states = np.zeros((self.nobs, 2 + self.seasonal_periods))
+        internal_states[:, 0] = states[:, 0]
+        if self.has_trend:
+            internal_states[:, 1] = states[:, 1]
+        if self.has_seasonal:
+            for j in range(self.seasonal_periods):
+                internal_states[j:, 2 + j] = states[
+                    0:self.nobs - j, self._seasonal_index
+                ]
+                internal_states[0:j, 2 + j] = internal_params[6:6 + j][::-1]
+        return internal_states
+
+    @property
+    def _default_start_params(self):
+        return {
+            "smoothing_level": 0.1,
+            "smoothing_trend": 0.01,
+            "smoothing_seasonal": 0.01,
+            "damping_trend": 0.98,
+        }
+
+    @property
+    def _start_params(self):
+        """
+        Returns default start params in the format of external parameters.
+        This should not be called directly, but by calling
+        ``self.start_params``.
+        """
+        params = []
+        for p in self._smoothing_param_names:
+            if p in self.param_names:
+                params.append(self._default_start_params[p])
+
+        if self.initialization_method == "estimated":
+            params += [self.initial_level]
+            if self.has_trend:
+                params += [self.initial_trend]
+            if self.has_seasonal:
+                params += self.initial_seasonal.tolist()
+
+        return np.array(params)
+
+    def _convert_and_bound_start_params(self, params):
+        """
+        This converts start params to internal params, sets internal-only
+        parameters as bounded, sets bounds for fixed parameters, and then makes
+        sure that all start parameters are within the specified bounds.
+        """
+        internal_params = self._internal_params(params)
+        # set bounds for missing and fixed
+        for p in self._internal_param_names:
+            idx = self._internal_params_index[p]
+            if p not in self.param_names:
+                # any missing parameters are set to the value they got from the
+                # call to _internal_params
+                self.bounds[p] = [internal_params[idx]] * 2
+            elif self._has_fixed_params and p in self._fixed_params:
+                self.bounds[p] = [self._fixed_params[p]] * 2
+            # make sure everything is within bounds
+            if p in self.bounds:
+                internal_params[idx] = np.clip(
+                    internal_params[idx]
+                    + 1e-3,  # try not to start on boundary
+                    *self.bounds[p],
+                )
+        return internal_params
+
+    def _setup_bounds(self):
+        # By default, we are using the traditional constraints for the
+        # smoothing parameters if nothing else is specified
+        #
+        #    0 <     alpha     < 1
+        #    0 <   beta/alpha  < 1
+        #    0 < gamma + alpha < 1
+        #  0.8 <      phi      < 0.98
+        #
+        # For initial states, no bounds are the default setting.
+        #
+        # Since the bounds for beta and gamma are not in the simple form of a
+        # constant interval, we will use the parameters beta_star=beta/alpha
+        # and gamma_star=gamma+alpha during fitting.
+
+        lb = np.zeros(self._k_params_internal) + 1e-4
+        ub = np.ones(self._k_params_internal) - 1e-4
+
+        # other bounds for phi and initial states
+        lb[3], ub[3] = 0.8, 0.98
+        if self.initialization_method == "estimated":
+            lb[4:] = -np.inf
+            ub[4:] = np.inf
+
+        # set lb and ub for parameters with bounds
+        for p in self._internal_param_names:
+            idx = self._internal_params_index[p]
+            if p in self.bounds:
+                lb[idx], ub[idx] = self.bounds[p]
+
+        return [(lb[i], ub[i]) for i in range(self._k_params_internal)]
+
+    def fit(
+        self,
+        start_params=None,
+        maxiter=1000,
+        full_output=True,
+        disp=True,
+        callback=None,
+        return_params=False,
+        **kwargs,
+    ):
+        r"""
+        Fit an ETS model by maximizing log-likelihood.
+
+        Log-likelihood is a function of the model parameters :math:`\alpha,
+        \beta, \gamma, \phi` (depending on the chosen model), and, if
+        `initialization_method` was set to `'estimated'` in the constructor,
+        also the initial states :math:`l_{-1}, b_{-1}, s_{-1}, \ldots, s_{-m}`.
+
+        The fit is performed using the L-BFGS algorithm.
+
+        Parameters
+        ----------
+        start_params : array_like, optional
+            Initial values for parameters that will be optimized. If this is
+            ``None``, default values will be used.
+            The length of this depends on the chosen model. This should contain
+            the parameters in the following order, skipping parameters that do
+            not exist in the chosen model.
+
+            * `smoothing_level` (:math:`\alpha`)
+            * `smoothing_trend` (:math:`\beta`)
+            * `smoothing_seasonal` (:math:`\gamma`)
+            * `damping_trend` (:math:`\phi`)
+
+            If ``initialization_method`` was set to ``'estimated'`` (the
+            default), additionally, the parameters
+
+            * `initial_level` (:math:`l_{-1}`)
+            * `initial_trend` (:math:`l_{-1}`)
+            * `initial_seasonal.0` (:math:`s_{-1}`)
+            * ...
+            * `initial_seasonal.<m-1>` (:math:`s_{-m}`)
+
+            also have to be specified.
+        maxiter : int, optional
+            The maximum number of iterations to perform.
+        full_output : bool, optional
+            Set to True to have all available output in the Results object's
+            mle_retvals attribute. The output is dependent on the solver.
+            See LikelihoodModelResults notes section for more information.
+        disp : bool, optional
+            Set to True to print convergence messages.
+        callback : callable callback(xk), optional
+            Called after each iteration, as callback(xk), where xk is the
+            current parameter vector.
+        return_params : bool, optional
+            Whether or not to return only the array of maximizing parameters.
+            Default is False.
+        **kwargs
+            Additional keyword arguments to pass to the optimizer.
+
+        Returns
+        -------
+        results : ETSResults
+        """
+
+        if start_params is None:
+            start_params = self.start_params
+        else:
+            start_params = np.asarray(start_params)
+
+        if self._has_fixed_params and len(self._free_params_index) == 0:
+            final_params = np.asarray(list(self._fixed_params.values()))
+            mlefit = Bunch(
+                params=start_params, mle_retvals=None, mle_settings=None
+            )
+        else:
+            internal_start_params = self._convert_and_bound_start_params(
+                start_params
+            )
+            kwargs["bounds"] = self._setup_bounds()
+
+            # check if we need to use the starred parameters
+            use_beta_star = "smoothing_trend" not in self.bounds
+            if use_beta_star:
+                internal_start_params[1] /= internal_start_params[0]
+            use_gamma_star = "smoothing_seasonal" not in self.bounds
+            if use_gamma_star:
+                internal_start_params[2] /= 1 - internal_start_params[0]
+
+            # pre-allocate memory for smoothing results
+            yhat = np.zeros(self.nobs)
+            xhat = np.zeros((self.nobs, self._k_states_internal))
+
+            kwargs["approx_grad"] = True
+            with self.use_internal_loglike():
+                mlefit = super().fit(
+                    internal_start_params,
+                    fargs=(yhat, xhat, use_beta_star, use_gamma_star),
+                    method="lbfgs",
+                    maxiter=maxiter,
+                    full_output=full_output,
+                    disp=disp,
+                    callback=callback,
+                    skip_hessian=True,
+                    **kwargs,
+                )
+            # convert params back
+            if use_beta_star:
+                mlefit.params[1] *= mlefit.params[0]
+            if use_gamma_star:
+                mlefit.params[2] *= 1 - mlefit.params[0]
+            final_params = self._model_params(mlefit.params)
+
+        if return_params:
+            return final_params
+        else:
+            result = self.smooth(final_params)
+            result.mlefit = mlefit
+            result.mle_retvals = mlefit.mle_retvals
+            result.mle_settings = mlefit.mle_settings
+            return result
+
+    def _loglike_internal(
+        self, params, yhat, xhat, use_beta_star=False, use_gamma_star=False
+    ):
+        """
+        Log-likelihood function to be called from fit to avoid reallocation of
+        memory.
+
+        Parameters
+        ----------
+        params : np.ndarray of np.float
+            Model parameters: (alpha, beta, gamma, phi, l[-1],
+            b[-1], s[-1], ..., s[-m]). This must be in the format of internal
+            parameters.
+        yhat : np.ndarray
+            Array of size (n,) where fitted values will be written to.
+        xhat : np.ndarray
+            Array of size (n, _k_states_internal) where fitted states will be
+            written to.
+        use_beta_star : boolean
+            Whether to internally use beta_star as parameter
+        use_gamma_star : boolean
+            Whether to internally use gamma_star as parameter
+        """
+        if np.iscomplexobj(params):
+            data = np.asarray(self.endog, dtype=np.complex)
+        else:
+            data = self.endog
+        self._smoothing_func(
+            params, data, yhat, xhat, use_beta_star, use_gamma_star
+        )
+        res = self._residuals(yhat, data=data)
+        logL = -self.nobs / 2 * (np.log(2 * np.pi * np.mean(res ** 2)) + 1)
+        if self.error == "mul":
+            logL -= np.sum(np.log(yhat))
+            if np.isnan(logL):
+                logL = np.inf
+        return logL
+
+    @contextlib.contextmanager
+    def use_internal_loglike(self):
+        external_loglike = self.loglike
+        self.loglike = self._loglike_internal
+        try:
+            yield
+        finally:
+            self.loglike = external_loglike
+
+    def loglike(self, params, **kwargs):
+        r"""
+        Log-likelihood of model.
+
+        Parameters
+        ----------
+        params : np.ndarray of np.float
+            Model parameters: (alpha, beta, gamma, phi, l[-1],
+            b[-1], s[-1], ..., s[-m])
+
+        Notes
+        -----
+        The log-likelihood of a exponential smoothing model is [1]_:
+
+        .. math::
+
+           l(\theta, x_0|y) = - \frac{n}{2}(\log(2\pi s^2) + 1)
+                              - \sum\limits_{t=1}^n \log(k_t)
+
+        with
+
+        .. math::
+
+           s^2 = \frac{1}{n}\sum\limits_{t=1}^n \frac{\hat{y}_t - y_t}{k_t}
+
+        where :math:`k_t = 1` for the additive error model and :math:`k_t =
+        y_t` for the multiplicative error model.
+
+        References
+        ----------
+        .. [1] J. K. Ord, A. B. Koehler R. D. and Snyder (1997). Estimation and
+           Prediction for a Class of Dynamic Nonlinear Statistical Models.
+           *Journal of the American Statistical Association*, 92(440),
+           1621-1629
+        """
+        params = self._internal_params(np.asarray(params))
+        yhat = np.zeros(self.nobs, dtype=params.dtype)
+        xhat = np.zeros(
+            (self.nobs, self._k_states_internal), dtype=params.dtype
+        )
+        return self._loglike_internal(np.asarray(params), yhat, xhat)
+
+    def _residuals(self, yhat, data=None):
+        """Calculates residuals of a prediction"""
+        if data is None:
+            data = self.endog
+        if self.error == "mul":
+            return (data - yhat) / yhat
+        else:
+            return data - yhat
+
+    def _smooth(self, params):
+        """
+        Exponential smoothing with given parameters
+
+        Parameters
+        ----------
+        params : array_like
+            Model parameters
+
+        Returns
+        -------
+        yhat : pd.Series or np.ndarray
+            Predicted values from exponential smoothing. If original data was a
+            ``pd.Series``, returns a ``pd.Series``, else a ``np.ndarray``.
+        xhat : pd.DataFrame or np.ndarray
+            Internal states of exponential smoothing. If original data was a
+            ``pd.Series``, returns a ``pd.DataFrame``, else a ``np.ndarray``.
+        """
+        internal_params = self._internal_params(params)
+        yhat = np.zeros(self.nobs)
+        xhat = np.zeros((self.nobs, self._k_states_internal))
+        self._smoothing_func(internal_params, self.endog, yhat, xhat)
+
+        # remove states that are only internal
+        states = self._get_states(xhat)
+
+        if self.use_pandas:
+            _, _, _, index = self._get_prediction_index(0, self.nobs - 1)
+            yhat = pd.Series(yhat, index=index)
+            statenames = ["level"]
+            if self.has_trend:
+                statenames += ["trend"]
+            if self.has_seasonal:
+                statenames += ["seasonal"]
+            states = pd.DataFrame(states, index=index, columns=statenames)
+        return yhat, states
+
+    def smooth(self, params, return_raw=False):
+        """
+        Exponential smoothing with given parameters
+
+        Parameters
+        ----------
+        params : array_like
+            Model parameters
+        return_raw : bool, optional
+            Whether to return only the state space results or the full results
+            object. Default is ``False``.
+
+        Returns
+        -------
+        result : ETSResultsWrapper or tuple
+            If ``return_raw=False``, returns a ETSResultsWrapper
+            object. Otherwise a tuple of arrays or pandas objects, depending on
+            the format of the endog data.
+        """
+        params = np.asarray(params)
+        results = self._smooth(params)
+        return self._wrap_results(params, results, return_raw)
+
+    @property
+    def _res_classes(self):
+        return {"fit": (ETSResults, ETSResultsWrapper)}
+
+    def hessian(
+        self, params, approx_centered=False, approx_complex_step=True, **kwargs
+    ):
+        r"""
+        Hessian matrix of the likelihood function, evaluated at the given
+        parameters
+
+        Parameters
+        ----------
+        params : array_like
+            Array of parameters at which to evaluate the hessian.
+        approx_centered : bool
+            Whether to use a centered scheme for finite difference
+            approximation
+        approx_complex_step : bool
+            Whether to use complex step differentiation for approximation
+
+        Returns
+        -------
+        hessian : ndarray
+            Hessian matrix evaluated at `params`
+
+        Notes
+        -----
+        This is a numerical approximation.
+        """
+        method = kwargs.get("method", "approx")
+
+        if method == "approx":
+            if approx_complex_step:
+                hessian = self._hessian_complex_step(params, **kwargs)
+            else:
+                hessian = self._hessian_finite_difference(
+                    params, approx_centered=approx_centered, **kwargs
+                )
+        else:
+            raise NotImplementedError("Invalid Hessian calculation method.")
+
+        return hessian
+
+    def score(
+        self, params, approx_centered=False, approx_complex_step=True, **kwargs
+    ):
+        method = kwargs.get("method", "approx")
+
+        if method == "approx":
+            if approx_complex_step:
+                score = self._score_complex_step(params, **kwargs)
+            else:
+                score = self._score_finite_difference(
+                    params, approx_centered=approx_centered, **kwargs
+                )
+        else:
+            raise NotImplementedError("Invalid score method.")
+
+        return score
+
+    def update(params, *args, **kwargs):
+        # Dummy method to make methods copied from statespace.MLEModel work
+        ...
+
+
+class ETSResults(base.StateSpaceMLEResults):
+    def __init__(self, model, params, results):
+        super().__init__(model, params)
+        yhat, xhat = results
+        self._llf = self.model.loglike(params)
+        self._residuals = self.model._residuals(yhat)
+        self._fittedvalues = yhat
+
+        # get model definition
+        model_definition_attrs = [
+            "short_name",
+            "error",
+            "trend",
+            "seasonal",
+            "damped_trend",
+            "has_trend",
+            "has_seasonal",
+            "seasonal_periods",
+            "initialization_method",
+        ]
+        for attr in model_definition_attrs:
+            setattr(self, attr, getattr(model, attr))
+        self.param_names = [
+            "%s (fixed)" % name if name in self.fixed_params else name
+            for name in (self.model.param_names or [])
+        ]
+
+        # get fitted states and parameters
+        internal_params = self.model._internal_params(params)
+        self.states = xhat
+        if self.model.use_pandas:
+            states = self.states.iloc
+        else:
+            states = self.states
+        self.initial_state = np.zeros(model._k_initial_states)
+
+        self.level = states[:, 0]
+        self.initial_level = internal_params[4]
+        self.initial_state[0] = self.initial_level
+        self.alpha = self.params[0]
+        self.smoothing_level = self.alpha
+        if self.has_trend:
+            self.slope = states[:, 1]
+            self.initial_trend = internal_params[5]
+            self.initial_state[1] = self.initial_trend
+            self.beta = self.params[1]
+            self.smoothing_trend = self.beta
+        if self.has_seasonal:
+            self.season = states[:, self.model._seasonal_index]
+            self.initial_seasonal = internal_params[6:]
+            self.initial_state[self.model._seasonal_index:] = (
+                self.initial_seasonal
+            )
+            self.gamma = self.params[self.model._seasonal_index]
+            self.smoothing_seasonal = self.gamma
+        if self.damped_trend:
+            self.phi = internal_params[3]
+            self.damping_trend = self.phi
+
+        # Setup covariance matrix notes dictionary
+        # For now, only support "approx"
+        if not hasattr(self, "cov_kwds"):
+            self.cov_kwds = {}
+        self.cov_type = "approx"
+
+        # Setup the cache
+        self._cache = {}
+
+        # Handle covariance matrix calculation
+        self._cov_approx_complex_step = True
+        self._cov_approx_centered = False
+        approx_type_str = "complex-step"
+        try:
+            self._rank = None
+            if self.k_params == 0:
+                self.cov_params_default = np.zeros((0, 0))
+                self._rank = 0
+                self.cov_kwds["description"] = "No parameters estimated."
+            else:
+                self.cov_params_default = self.cov_params_approx
+                self.cov_kwds["description"] = descriptions["approx"].format(
+                    approx_type=approx_type_str
+                )
+        except np.linalg.LinAlgError:
+            self._rank = 0
+            k_params = len(self.params)
+            self.cov_params_default = np.zeros((k_params, k_params)) * np.nan
+            self.cov_kwds["cov_type"] = (
+                "Covariance matrix could not be calculated: singular."
+                " information matrix."
+            )
+
+    @cache_readonly
+    def nobs_effective(self):
+        return self.nobs
+
+    @cache_readonly
+    def df_model(self):
+        return self.model.k_params
+
+    @cache_readonly
+    def fittedvalues(self):
+        return self._fittedvalues
+
+    @cache_readonly
+    def resid(self):
+        return self._residuals
+
+    @cache_readonly
+    def llf(self):
+        """
+        log-likelihood function evaluated at the fitted params
+        """
+        return self._llf
+
+    def _get_prediction_params(self, start_idx):
+        """
+        Returns internal parameter representation of smoothing parameters and
+        "initial" states for prediction/simulation, that is the states just
+        before the first prediction/simulation step.
+        """
+        internal_params = self.model._internal_params(self.params)
+        if start_idx == 0:
+            return internal_params
+        else:
+            internal_states = self.model._get_internal_states(
+                self.states, self.params
+            )
+            start_state = np.empty(6 + self.seasonal_periods)
+            start_state[0:4] = internal_params[0:4]
+            start_state[4:] = internal_states[start_idx - 1, :]
+            return start_state
+
+    def simulate(
+        self,
+        nsimulations,
+        anchor=None,
+        repetitions=1,
+        random_errors=None,
+        random_state=None,
+    ):
+        r"""
+        Random simulations using the state space formulation.
+
+        Parameters
+        ----------
+        nsimulations : int
+            The number of simulation steps.
+        anchor : int, str, or datetime, optional
+            First period for simulation. The simulation will be conditional on
+            all existing datapoints prior to the `anchor`.  Type depends on the
+            index of the given `endog` in the model. Two special cases are the
+            strings 'start' and 'end'. `start` refers to beginning the
+            simulation at the first period of the sample, and `end` refers to
+            beginning the simulation at the first period after the sample.
+            Integer values can run from 0 to `nobs`, or can be negative to
+            apply negative indexing. Finally, if a date/time index was provided
+            to the model, then this argument can be a date string to parse or a
+            datetime type. Default is 'start'.
+        repetitions : int, optional
+            Number of simulated paths to generate. Default is 1 simulated path.
+        random_errors : optional
+            Specifies how the random errors should be obtained. Can be one of
+            the following:
+
+            * ``None``: Random normally distributed values with variance
+              estimated from the fit errors drawn from numpy's standard
+              RNG (can be seeded with the `random_state` argument). This is the
+              default option.
+            * A distribution function from ``scipy.stats``, e.g.
+              ``scipy.stats.norm``: Fits the distribution function to the fit
+              errors and draws from the fitted distribution.
+              Note the difference between ``scipy.stats.norm`` and
+              ``scipy.stats.norm()``, the latter one is a frozen distribution
+              function.
+            * A frozen distribution function from ``scipy.stats``, e.g.
+              ``scipy.stats.norm(scale=2)``: Draws from the frozen distribution
+              function.
+            * A ``np.ndarray`` with shape (`nsimulations`, `repetitions`): Uses
+              the given values as random errors.
+            * ``"bootstrap"``: Samples the random errors from the fit errors.
+
+        random_state : int or np.random.RandomState, optional
+            A seed for the random number generator or a
+            ``np.random.RandomState`` object. Only used if `random_errors` is
+            ``None``. Default is ``None``.
+
+        Returns
+        -------
+        sim : pd.Series, pd.DataFrame or np.ndarray
+            An ``np.ndarray``, ``pd.Series``, or ``pd.DataFrame`` of simulated
+            values.
+            If the original data was a ``pd.Series`` or ``pd.DataFrame``, `sim`
+            will be a ``pd.Series`` if `repetitions` is 1, and a
+            ``pd.DataFrame`` of shape (`nsimulations`, `repetitions`) else.
+            Otherwise, if `repetitions` is 1, a ``np.ndarray`` of shape
+            (`nsimulations`,) is returned, and if `repetitions` is not 1 a
+            ``np.ndarray`` of shape (`nsimulations`, `repetitions`) is
+            returned.
+        """
+
+        r"""
+        Implementation notes
+        --------------------
+        The simulation is based on the state space model of the Holt-Winter's
+        methods. The state space model assumes that the true value at time
+        :math:`t` is randomly distributed around the prediction value.
+        If using the additive error model, this means:
+
+        .. math::
+
+            y_t &= \hat{y}_{t|t-1} + e_t\\
+            e_t &\sim \mathcal{N}(0, \sigma^2)
+
+        Using the multiplicative error model:
+
+        .. math::
+
+            y_t &= \hat{y}_{t|t-1} \cdot (1 + e_t)\\
+            e_t &\sim \mathcal{N}(0, \sigma^2)
+
+        Inserting these equations into the smoothing equation formulation leads
+        to the state space equations. The notation used here follows
+        [1]_.
+
+        Additionally,
+
+        .. math::
+
+           B_t = b_{t-1} \circ_d \phi\\
+           L_t = l_{t-1} \circ_b B_t\\
+           S_t = s_{t-m}\\
+           Y_t = L_t \circ_s S_t,
+
+        where :math:`\circ_d` is the operation linking trend and damping
+        parameter (multiplication if the trend is additive, power if the trend
+        is multiplicative), :math:`\circ_b` is the operation linking level and
+        trend (addition if the trend is additive, multiplication if the trend
+        is multiplicative), and :math:'\circ_s` is the operation linking
+        seasonality to the rest.
+
+        The state space equations can then be formulated as
+
+        .. math::
+
+           y_t = Y_t + \eta \cdot e_t\\
+           l_t = L_t + \alpha \cdot (M_e \cdot L_t + \kappa_l) \cdot e_t\\
+           b_t = B_t + \beta \cdot (M_e \cdot B_t+\kappa_b) \cdot e_t\\
+           s_t = S_t + \gamma \cdot (M_e \cdot S_t + \kappa_s) \cdot e_t\\
+
+        with
+
+        .. math::
+
+           \eta &= \begin{cases}
+                       Y_t\quad\text{if error is multiplicative}\\
+                       1\quad\text{else}
+                   \end{cases}\\
+           M_e &= \begin{cases}
+                       1\quad\text{if error is multiplicative}\\
+                       0\quad\text{else}
+                   \end{cases}\\
+
+        and, when using the additve error model,
+
+        .. math::
+
+           \kappa_l &= \begin{cases}
+                       \frac{1}{S_t}\quad
+                       \text{if seasonality is multiplicative}\\
+                       1\quad\text{else}
+                   \end{cases}\\
+           \kappa_b &= \begin{cases}
+                       \frac{\kappa_l}{l_{t-1}}\quad
+                       \text{if trend is multiplicative}\\
+                       \kappa_l\quad\text{else}
+                   \end{cases}\\
+           \kappa_s &= \begin{cases}
+                       \frac{1}{L_t}\quad
+                       \text{if seasonality is multiplicative}\\
+                       1\quad\text{else}
+                   \end{cases}
+
+        When using the multiplicative error model
+
+        .. math::
+
+           \kappa_l &= \begin{cases}
+                       0\quad
+                       \text{if seasonality is multiplicative}\\
+                       S_t\quad\text{else}
+                   \end{cases}\\
+           \kappa_b &= \begin{cases}
+                       \frac{\kappa_l}{l_{t-1}}\quad
+                       \text{if trend is multiplicative}\\
+                       \kappa_l + l_{t-1}\quad\text{else}
+                   \end{cases}\\
+           \kappa_s &= \begin{cases}
+                       0\quad\text{if seasonality is multiplicative}\\
+                       L_t\quad\text{else}
+                   \end{cases}
+
+        References
+        ----------
+        .. [1] Hyndman, R.J., & Athanasopoulos, G. (2018) *Forecasting:
+           principles and practice*, 2nd edition, OTexts: Melbourne,
+           Australia. OTexts.com/fpp2. Accessed on February 28th 2020.
+        """
+
+        # Get the starting location
+        start_idx = self._get_prediction_start_index(anchor)
+
+        # set initial values and obtain parameters
+        start_params = self._get_prediction_params(start_idx)
+        x = np.zeros((nsimulations, self.model._k_states_internal))
+        (
+            alpha,
+            beta_star,
+            gamma_star,
+            phi,
+            m,
+            _,
+        ) = smooth._initialize_ets_smooth(start_params, x)
+        beta = alpha * beta_star
+        gamma = (1 - alpha) * gamma_star
+        # make x a 3 dimensional matrix: first dimension is nsimulations
+        # (number of steps), next is number of states, innermost is repetitions
+        nstates = x.shape[1]
+        x = np.tile(np.reshape(x, (nsimulations, nstates, 1)), repetitions)
+        y = np.empty((nsimulations, repetitions))
+
+        # get random error eps
+        sigma = np.sqrt(np.sum(self.resid ** 2) / self.df_resid)
+        if isinstance(random_errors, np.ndarray):
+            if random_errors.shape != (nsimulations, repetitions):
+                raise ValueError(
+                    "If random is an ndarray, it must have shape "
+                    "(nsimulations, repetitions)!"
+                )
+            eps = random_errors
+        elif random_errors == "bootstrap":
+            eps = np.random.choice(
+                self.resid, size=(nsimulations, repetitions), replace=True
+            )
+        elif random_errors is None:
+            if random_state is None:
+                eps = np.random.randn(nsimulations, repetitions) * sigma
+            elif isinstance(random_state, int):
+                rng = np.random.RandomState(random_state)
+                eps = rng.randn(nsimulations, repetitions) * sigma
+            elif isinstance(random_state, np.random.RandomState):
+                eps = random_state.randn(nsimulations, repetitions) * sigma
+            else:
+                raise ValueError(
+                    "Argument random_state must be None, an integer, "
+                    "or an instance of np.random.RandomState"
+                )
+        elif isinstance(random_errors, (rv_continuous, rv_discrete)):
+            params = random_errors.fit(self.resid)
+            eps = random_errors.rvs(*params, size=(nsimulations, repetitions))
+        elif isinstance(random_errors, _distn_infrastructure.rv_frozen):
+            eps = random_errors.rvs(size=(nsimulations, repetitions))
+        else:
+            raise ValueError("Argument random_errors has unexpected value!")
+
+        # get model settings
+        mul_seasonal = self.seasonal == "mul"
+        mul_trend = self.trend == "mul"
+        mul_error = self.error == "mul"
+
+        # define trend, damping and seasonality operations
+        if mul_trend:
+            op_b = np.multiply
+            op_d = np.power
+        else:
+            op_b = np.add
+            op_d = np.multiply
+        if mul_seasonal:
+            op_s = np.multiply
+        else:
+            op_s = np.add
+
+        # x translation:
+        # - x[t, 0, :] is level[t]
+        # - x[t, 1, :] is trend[t]
+        # - x[t, 2, :] is seasonal[t]
+        # - x[t, 3, :] is seasonal[t-1]
+        # - x[t, 2+j, :] is seasonal[t-j]
+        # - similarly: x[t-1, 2+m-1, :] is seasonal[t-m]
+        for t in range(nsimulations):
+            B = op_d(x[t - 1, 1, :], phi)
+            L = op_b(x[t - 1, 0, :], B)
+            S = x[t - 1, 2 + m - 1, :]
+            Y = op_s(L, S)
+            if self.error == "add":
+                eta = 1
+                kappa_l = 1 / S if mul_seasonal else 1
+                kappa_b = kappa_l / x[t - 1, 0, :] if mul_trend else kappa_l
+                kappa_s = 1 / L if mul_seasonal else 1
+            else:
+                eta = Y
+                kappa_l = 0 if mul_seasonal else S
+                kappa_b = (
+                    kappa_l / x[t - 1, 0, :]
+                    if mul_trend
+                    else kappa_l + x[t - 1, 0, :]
+                )
+                kappa_s = 0 if mul_seasonal else L
+
+            y[t, :] = Y + eta * eps[t, :]
+            x[t, 0, :] = L + alpha * (mul_error * L + kappa_l) * eps[t, :]
+            x[t, 1, :] = B + beta * (mul_error * B + kappa_b) * eps[t, :]
+            x[t, 2, :] = S + gamma * (mul_error * S + kappa_s) * eps[t, :]
+            # update seasonals by shifting previous seasonal right
+            x[t, 3:, :] = x[t - 1, 2:-1, :]
+
+        # Wrap data / squeeze where appropriate
+        if repetitions > 1:
+            names = ["simulation.%d" % num for num in range(repetitions)]
+        else:
+            names = "simulation"
+        return self.model._wrap_data(
+            y, start_idx, start_idx + nsimulations - 1, names=names
+        )
+
+    def forecast(self, steps=1):
+        """
+        Out-of-sample forecasts
+
+        Parameters
+        ----------
+        steps : int, str, or datetime, optional
+            If an integer, the number of steps to forecast from the end of the
+            sample. Can also be a date string to parse or a datetime type.
+            However, if the dates index does not have a fixed frequency, steps
+            must be an integer. Default
+
+        Returns
+        -------
+        forecast : ndarray
+            Array of out of sample forecasts. A (steps x k_endog) array.
+        """
+        return self._forecast(steps, "end")
+
+    def _forecast(self, steps, anchor):
+        """
+        Dynamic prediction/forecasting
+        """
+        # forecast is the same as simulation without errors
+        return self.simulate(
+            steps, anchor=anchor, random_errors=np.zeros((steps, 1))
+        )
+
+    def predict(self, start=None, end=None, dynamic=False, index=None):
+        """
+        In-sample prediction and out-of-sample forecasting
+
+        Parameters
+        ----------
+        start : int, str, or datetime, optional
+            Zero-indexed observation number at which to start forecasting,
+            i.e., the first forecast is start. Can also be a date string to
+            parse or a datetime type. Default is the the zeroth observation.
+        end : int, str, or datetime, optional
+            Zero-indexed observation number at which to end forecasting, i.e.,
+            the last forecast is end. Can also be a date string to
+            parse or a datetime type. However, if the dates index does not
+            have a fixed frequency, end must be an integer index if you
+            want out of sample prediction. Default is the last observation in
+            the sample.
+        dynamic : bool, int, str, or datetime, optional
+            Integer offset relative to `start` at which to begin dynamic
+            prediction. Can also be an absolute date string to parse or a
+            datetime type (these are not interpreted as offsets).
+            Prior to this observation, true endogenous values will be used for
+            prediction; starting with this observation and continuing through
+            the end of prediction, forecasted endogenous values will be used
+            instead.
+        index : pd.Index, optional
+            Optionally an index to associate the predicted results to. If None,
+            an attempt is made to create an index for the predicted results
+            from the model's index or model's row labels.
+
+        Returns
+        -------
+        forecast : array_like or pd.Series.
+            Array of out of in-sample predictions and / or out-of-sample
+            forecasts. An (npredict,) array. If original data was a pd.Series
+            or DataFrame, a pd.Series is returned.
+        """
+        if start is None:
+            start = 0
+
+        # Handle start, end, dynamic
+        start, end, out_of_sample, _ = self.model._get_prediction_index(
+            start, end, index
+        )
+
+        # Handle `dynamic`
+        if isinstance(dynamic, (bytes, str)):
+            dynamic, _, _ = self.model._get_index_loc(dynamic)
+
+        start_smooth = start
+        end_smooth = min(dynamic, end + 1)
+        nsmooth = end_smooth - start_smooth
+        start_forecast = end_smooth
+        end_forecast = end + out_of_sample + 1
+        nforecast = end_forecast - start_forecast
+
+        y = np.empty(nsmooth + nforecast)
+
+        # In sample nondynamic prediction: smoothing
+        y[0:nsmooth] = self.fittedvalues[start_smooth:end_smooth]
+
+        # Out of sample/dynamic prediction: forecast
+        y[nsmooth:] = self._forecast(nforecast, start_forecast)
+
+        # Wrap data / squeeze where appropriate
+        return self.model._wrap_data(y, start, end_forecast - 1)
+
+    def summary(self, alpha=0.05, start=None):
+        """
+        Summarize the fitted model
+
+        Parameters
+        ----------
+        alpha : float, optional
+            Significance level for the confidence intervals. Default is 0.05.
+        start : int, optional
+            Integer of the start observation. Default is 0.
+
+        Returns
+        -------
+        summary : Summary instance
+            This holds the summary table and text, which can be printed or
+            converted to various output formats.
+
+        See Also
+        --------
+        statsmodels.iolib.summary.Summary
+        """
+        model_name = f"ETS({self.short_name})"
+
+        summary = super().summary(
+            alpha=alpha,
+            start=start,
+            title="ETS Results",
+            model_name=model_name,
+        )
+
+        if self.model.initialization_method != "estimated":
+            params = np.array(self.initial_state)
+            if params.ndim > 1:
+                params = params[0]
+            names = self.model.initial_state_names
+            param_header = [
+                "initialization method: %s" % self.model.initialization_method
+            ]
+            params_stubs = names
+            params_data = [
+                [forg(params[i], prec=4)] for i in range(len(params))
+            ]
+
+            initial_state_table = SimpleTable(
+                params_data, param_header, params_stubs, txt_fmt=fmt_params
+            )
+            summary.tables.insert(-1, initial_state_table)
+
+        return summary
+
+
+class ETSResultsWrapper(wrap.ResultsWrapper):
+    _attrs = {
+        "fittedvalues": "rows",
+        "level": "rows",
+        "resid": "rows",
+        "season": "rows",
+        "slope": "rows",
+    }
+    _wrap_attrs = wrap.union_dicts(
+        tsbase.TimeSeriesResultsWrapper._wrap_attrs, _attrs
+    )
+    _methods = {"predict": "dates", "forecast": "dates"}
+    _wrap_methods = wrap.union_dicts(
+        tsbase.TimeSeriesResultsWrapper._wrap_methods, _methods
+    )
+
+
+wrap.populate_wrapper(ETSResultsWrapper, ETSResults)


Test patch:
diff --git a/statsmodels/tsa/tests/results/fit_ets.R b/statsmodels/tsa/tests/results/fit_ets.R
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/results/fit_ets.R
@@ -0,0 +1,72 @@
+library(fpp2)
+library(forecast)
+library(jsonlite)
+
+concat <- function(...) {
+  return(paste(..., sep=""))
+}
+
+# get variable or NaN
+get_var <- function(named, name) {
+  if (name %in% names(named))
+    val <- c(named[name])
+  else
+    val <- c(NaN)
+  names(val) <- c(name)
+  return(val)
+}
+
+# innov from np.random.seed(0); np.random.randn(4)
+innov <- c(1.76405235, 0.40015721, 0.97873798, 2.2408932)
+
+
+
+# fit all models and add fit results to list
+obtain_model_results <- function(models, data) {
+  results = list()
+  for (damped in c(TRUE, FALSE)) {
+    results_damped = list()
+    for (model in models) {
+      fitted <- tryCatch((function(){
+        fit <- ets(data, model = model, damped = damped)
+        pars <- list()
+        # alpha, beta, gamma, phi
+        for (name in c("alpha", "beta", "gamma", "phi")) {
+          pars[name] <- get_var(fit$par, name)
+        }
+        pars$initstate <- fit$initstate
+        pars$states <- fit$states
+        pars$residuals <- fit$residuals
+        pars$fitted <- fit$fitted
+        pars$sigma2 <- fit$sigma2
+        pars$loglik <- fit$loglik
+        pars$forecast <- forecast(fit, 4, PI = FALSE)$mean
+        pars$simulation <- simulate(fit, 4, innov = innov)
+        return(pars)
+      })(),
+      error = function(e) list())
+      results_damped[[model]] <- fitted
+    }
+    results[[as.character(damped)]] <- results_damped
+  }
+  return(results)
+}
+
+error <- c("A", "M")
+trend <- c("A", "M", "N")
+seasonal <- c("A", "M")
+
+models_seasonal <- outer(error, trend, FUN = "concat") %>%
+  outer(seasonal, FUN = "concat") %>% as.vector
+results <- obtain_model_results(models_seasonal, austourists)
+sink("fit_ets_results_seasonal.json")
+cat(toJSON(results, pretty = TRUE, digits = 8))
+sink()
+
+models_nonseasonal <- outer(error, trend, FUN = "concat") %>%
+  outer(c("N"), FUN = "concat") %>% as.vector
+results <- obtain_model_results(models_nonseasonal, oil)
+sink("fit_ets_results_nonseasonal.json")
+cat(toJSON(results, pretty = TRUE, digits = 8))
+sink()
+
diff --git a/statsmodels/tsa/tests/results/fit_ets_results.json b/statsmodels/tsa/tests/results/fit_ets_results.json
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/results/fit_ets_results.json
@@ -0,0 +1,1460 @@
+{
+  "TRUE": {
+    "AAA": {
+      "alpha": [0.35445427],
+      "beta": [0.03200749],
+      "gamma": [0.39993387],
+      "phi": [0.97999997],
+      "initstate": [24.01278357, 0.97770147, 1.76951063, -0.50735902, -6.61171798, 5.34956637],
+      "states": [
+        [24.01278357, 0.97770147, 1.76951063, -0.50735902, -6.61171798, 5.34956637],
+        [24.87594278, 0.94956991, 5.24239035, 1.76951063, -0.50735902, -6.61171798],
+        [25.79010745, 0.9290963, -6.63023783, 5.24239035, 1.76951063, -0.50735902],
+        [26.39027198, 0.88248951, -0.85752942, -6.63023783, 5.24239035, 1.76951063],
+        [26.74711302, 0.81896703, 1.19633128, -0.85752942, -6.63023783, 5.24239035],
+        [27.29604079, 0.77968198, 4.95618366, 1.19633128, -0.85752942, -6.63023783],
+        [28.78962071, 0.82996196, -5.80714604, 4.95618366, 1.19633128, -0.85752942],
+        [29.50745302, 0.80473623, -0.96531719, -5.80714604, 4.95618366, 1.19633128],
+        [31.58323395, 0.90487115, 2.64862203, -0.96531719, -5.80714604, 4.95618366],
+        [32.26169263, 0.86796269, 4.72114001, 2.64862203, -0.96531719, -5.80714604],
+        [32.29771297, 0.77704594, -6.72624726, 4.72114001, 2.64862203, -0.96531719],
+        [32.5730234, 0.71760125, -1.51389479, -6.72624726, 4.72114001, 2.64862203],
+        [30.71316568, 0.4717987, -0.24335402, -1.51389479, -6.72624726, 4.72114001],
+        [31.43935494, 0.48618645, 5.01881783, -0.24335402, -1.51389479, -6.72624726],
+        [31.43200125, 0.43277371, -7.27214155, 5.01881783, -0.24335402, -1.51389479],
+        [31.49083957, 0.39113317, -1.92604331, -7.27214155, 5.01881783, -0.24335402],
+        [31.92360103, 0.38777595, -0.18755806, -1.92604331, -7.27214155, 5.01881783],
+        [31.54347218, 0.31137836, 4.16113474, -0.18755806, -1.92604331, -7.27214155],
+        [30.14680385, 0.15147522, -9.19231901, 4.16113474, -0.18755806, -1.92604331],
+        [30.73212996, 0.18789635, -1.43310732, -9.19231901, 4.16113474, -0.18755806],
+        [32.26678583, 0.30609117, 1.33624265, -1.43310732, -9.19231901, 4.16113474],
+        [34.17800118, 0.44546607, 5.97911777, 1.33624265, -1.43310732, -9.19231901],
+        [35.05182089, 0.47604192, -8.69895135, 5.97911777, 1.33624265, -1.43310732],
+        [33.46060176, 0.28070556, -3.75487366, -8.69895135, 5.97911777, 1.33624265],
+        [33.77795022, 0.27890728, 1.38392163, -3.75487366, -8.69895135, 5.97911777],
+        [34.82231194, 0.34295395, 6.84908061, 1.38392163, -3.75487366, -8.69895135],
+        [34.51674386, 0.27815221, -9.42294529, 6.84908061, 1.38392163, -3.75487366],
+        [35.36811282, 0.32485341, -3.10183133, -9.42294529, 6.84908061, 1.38392163],
+        [35.75273907, 0.32434056, 1.45869456, -3.10183133, -9.42294529, 6.84908061],
+        [36.89391453, 0.39220031, 7.77804175, 1.45869456, -3.10183133, -9.42294529],
+        [37.80823516, 0.43221248, -8.82498187, 7.77804175, 1.45869456, -3.10183133],
+        [38.66219424, 0.46243284, -2.61621754, -8.82498187, 7.77804175, 1.45869456],
+        [39.54356215, 0.49184948, 1.94181803, -2.61621754, -8.82498187, 7.77804175],
+        [40.54767941, 0.52915893, 8.36713716, 1.94181803, -2.61621754, -8.82498187],
+        [40.72389773, 0.4876606, -9.21126675, 8.36713716, 1.94181803, -2.61621754],
+        [40.94129466, 0.45438308, -2.91015376, -9.21126675, 8.36713716, 1.94181803],
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diff --git a/statsmodels/tsa/tests/results/fit_ets_results_nonseasonal.json b/statsmodels/tsa/tests/results/fit_ets_results_nonseasonal.json
new file mode 100644
--- /dev/null
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+}
\ No newline at end of file
diff --git a/statsmodels/tsa/tests/results/fit_ets_results_seasonal.json b/statsmodels/tsa/tests/results/fit_ets_results_seasonal.json
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/results/fit_ets_results_seasonal.json
@@ -0,0 +1,943 @@
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+        [46.7602509, 1.01350018, 0.96116989, 0.77116966, 1.23036005, 1.0372612],
+        [46.24062446, 1.01017465, 1.03725658, 0.96116989, 0.77116966, 1.23036005],
+        [47.84178276, 1.01347846, 1.23036551, 1.03725658, 0.96116989, 0.77116966],
+        [49.18039716, 1.01543783, 0.77117168, 1.23036551, 1.03725658, 0.96116989],
+        [49.3575246, 1.01383859, 0.96116784, 0.77117168, 1.23036551, 1.03725658],
+        [49.42402779, 1.01215085, 1.03725424, 0.96116784, 0.77117168, 1.23036551],
+        [50.06822892, 1.0122702, 1.2303657, 1.03725424, 0.96116784, 0.77117168],
+        [50.83453641, 1.01268028, 0.7711721, 1.2303657, 1.03725424, 0.96116784],
+        [53.29210726, 1.01749902, 0.96117404, 0.7711721, 1.2303657, 1.03725424],
+        [53.21784623, 1.01494638, 1.03725071, 0.96117404, 0.7711721, 1.2303657],
+        [54.18115776, 1.01537265, 1.23036641, 1.03725071, 0.96117404, 0.7711721],
+        [53.91358634, 1.01262833, 0.77116928, 1.23036641, 1.03725071, 0.96117404],
+        [54.22729671, 1.01170826, 0.96117285, 0.77116928, 1.23036641, 1.03725071],
+        [55.17073458, 1.01247701, 1.03725178, 0.96117285, 0.77116928, 1.23036641],
+        [54.31649733, 1.00869914, 1.23036018, 1.03725178, 0.96117285, 0.77116928],
+        [55.36229436, 1.01012522, 0.77117075, 1.23036018, 1.03725178, 0.96117285],
+        [56.49622254, 1.01152456, 0.96117466, 0.77117075, 1.23036018, 1.03725178],
+        [57.44124784, 1.01222752, 1.03725276, 0.96117466, 0.77117075, 1.23036018],
+        [58.90674949, 1.01402258, 1.23036314, 1.03725276, 0.96117466, 0.77117075],
+        [60.88872198, 1.01667397, 0.77117349, 1.23036314, 1.03725276, 0.96117466],
+        [62.81140916, 1.01868759, 0.96117724, 0.77117349, 1.23036314, 1.03725276],
+        [63.82038663, 1.01833305, 1.03725227, 0.96117724, 0.77117349, 1.23036314]
+      ],
+      "residuals": [-0.00091448, -0.02976639, 0.00119013, -0.03317463, -0.07431107, 0.08714535, -0.0326297, 0.08144376, -0.12577452, -0.02014529, -0.05731995, -0.18356981, -0.0400334, 0.00341243, -0.02583263, -0.02022816, -0.08562037, -0.14403191, 0.12214083, 0.13365685, 0.05055469, 0.03576999, -0.15192608, 0.0576928, 0.02396523, -0.0715449, 0.01825693, 0.05462281, 0.0335408, 0.03300612, -0.01104272, 0.0406024, -0.00917131, -0.0123059, -0.05337343, -0.0875184, 0.05592532, 0.00219859, 0.02826144, -0.04331769, 0.06188176, -0.03391385, -0.01411115, -0.03074968, 0.08418949, -0.04971691, -0.0050292, -0.04447222, 0.04432723, 0.02620311, -0.02134583, -0.02256255, 0.00159818, 0.00549066, 0.06449307, -0.03400226, 0.00569241, -0.036632, -0.01231465, 0.01029858, -0.0505723, 0.01916168, 0.01877588, 0.00941894, 0.02403552, 0.03543872, 0.026844, -0.00471708],
+      "fitted": [30.08002071, 19.73596435, 25.28759654, 28.53818066, 34.65144166, 21.60517085, 29.43644243, 32.47857556, 42.13842507, 25.52114805, 32.59030455, 35.14539202, 38.1690218, 23.74358564, 30.08896005, 32.42623195, 38.47185125, 23.10278179, 26.37971023, 30.466752, 39.28743502, 25.73531031, 33.34598369, 33.27162005, 41.21786832, 26.54858745, 32.08162322, 35.32765816, 43.7742375, 28.41268862, 36.75002751, 40.15182097, 49.73271248, 31.66506861, 39.98474848, 42.56200244, 48.52322351, 31.76870512, 40.18766007, 44.73690266, 52.46192251, 34.57988199, 42.92232129, 46.58974201, 54.94987797, 36.6599495, 45.06833855, 49.15738772, 57.47147966, 37.39140737, 48.00047413, 51.90490338, 61.54850742, 39.0849671, 49.48008629, 56.24476581, 66.45606297, 42.42531267, 52.47474385, 56.90586046, 68.72716022, 42.25159673, 53.75152511, 59.27615628, 71.53758457, 46.06416846, 59.50053575, 66.36882862],
+      "sigma2": [0.00389297],
+      "loglik": [-200.21401202],
+      "forecast": [79.9618039, 51.03771265, 64.77873021, 71.18740926],
+      "simulation": [221.01861197, 158.55914169, 403.22625116, 1389.33384434]
+    },
+    "ANM": [],
+    "MNM": {
+      "alpha": [0.60899314],
+      "beta": ["NaN"],
+      "gamma": [0.00016726],
+      "phi": ["NaN"],
+      "initstate": [29.67481548, 1.03842344, 0.96140206, 0.76838081, 1.2317937],
+      "states": [
+        [29.67481548, 1.03842344, 0.96140206, 0.76838081, 1.2317937],
+        [26.46088069, 1.23175705, 1.03842344, 0.96140206, 0.76838081],
+        [25.52284856, 0.76837333, 1.23175705, 1.03842344, 0.96140206],
+        [26.01691671, 0.96140717, 0.76837333, 1.23175705, 1.03842344],
+        [26.35404929, 1.03842713, 0.96140717, 0.76837333, 1.23175705],
+        [26.16353818, 1.23175461, 1.03842713, 0.96140717, 0.76837333],
+        [28.84608284, 0.76839497, 1.23175461, 1.03842713, 0.96140717],
+        [29.31679743, 0.96141148, 0.76839497, 1.23175461, 1.03842713],
+        [32.06164917, 1.03845383, 0.96141148, 0.76839497, 1.23175461],
+        [30.74968032, 1.23174077, 1.03845383, 0.96141148, 0.76839497],
+        [31.84270289, 0.76840247, 1.23174077, 1.03845383, 0.96141148],
+        [31.91129771, 0.96141205, 0.76840247, 1.23174077, 1.03845383],
+        [29.30476715, 1.03843054, 0.96141205, 0.76840247, 1.23174077],
+        [29.57427845, 1.23174388, 1.03843054, 0.96141205, 0.76840247],
+        [30.44580827, 0.76840869, 1.23174388, 1.03843054, 0.96141205],
+        [30.47159936, 0.96141227, 0.76840869, 1.23174388, 1.03843054],
+        [30.54647191, 1.03843124, 0.96141227, 0.76840869, 1.23174388],
+        [29.33636415, 1.23173048, 1.03843124, 0.96141227, 0.76840869],
+        [27.14335339, 0.76839291, 1.23173048, 1.03843124, 0.96141227],
+        [29.36405141, 0.96143387, 0.76839291, 1.23173048, 1.03843124],
+        [31.73702037, 1.03845429, 0.96143387, 0.76839291, 1.23173048],
+        [32.81591761, 1.23174198, 1.03845429, 0.96143387, 0.76839291],
+        [33.95746813, 0.76840025, 1.23174198, 1.03845429, 0.96143387],
+        [31.19068117, 0.96141236, 0.76840025, 1.23174198, 1.03845429],
+        [32.83333811, 1.03846931, 0.96141236, 0.76840025, 1.23174198],
+        [33.70522277, 1.23175096, 1.03846931, 0.96141236, 0.76840025],
+        [32.7145945, 0.76839405, 1.23175096, 1.03846931, 0.96141236],
+        [33.48429498, 0.96141857, 0.76839405, 1.23175096, 1.03846931],
+        [34.94154773, 1.03848172, 0.96141857, 0.76839405, 1.23175096],
+        [36.03082528, 1.23176151, 1.03848172, 0.96141857, 0.76839405],
+        [37.35011683, 0.76840178, 1.23176151, 1.03848172, 0.96141857],
+        [37.62573021, 0.96142052, 0.76840178, 1.23176151, 1.03848172],
+        [39.21403592, 1.03849376, 0.96142052, 0.76840178, 1.23176151],
+        [39.69571744, 1.23176566, 1.03849376, 0.96142052, 0.76840178],
+        [40.30846708, 0.76840503, 1.23176566, 1.03849376, 0.96142052],
+        [39.73663013, 0.96141677, 0.76840503, 1.23176566, 1.03849376],
+        [38.31210052, 1.03848354, 0.96141677, 0.76840503, 1.23176566],
+        [40.31215863, 1.23178332, 1.03848354, 0.96141677, 0.76840503],
+        [40.99571484, 0.76840861, 1.23178332, 1.03848354, 0.96141677],
+        [42.20522568, 0.96142456, 0.76840861, 1.23178332, 1.03848354],
+        [41.60095615, 1.03847945, 0.96142456, 0.76840861, 1.23178332],
+        [43.80844745, 1.23180127, 1.03847945, 0.96142456, 0.76840861],
+        [43.60584027, 0.76840764, 1.23180127, 1.03847945, 0.96142456],
+        [43.8547217, 0.96142607, 0.76840764, 1.23180127, 1.03847945],
+        [43.62888558, 1.03847798, 0.96142607, 0.76840764, 1.23180127],
+        [46.5131524, 1.23182364, 1.03847798, 0.96142607, 0.76840764],
+        [45.7969099, 0.76840439, 1.23182364, 1.03847798, 0.96142607],
+        [46.31083295, 0.96142903, 0.76840439, 1.23182364, 1.03847798],
+        [45.65313527, 1.03847393, 0.96142903, 0.76840439, 1.23182364],
+        [47.52310074, 1.2318375, 1.03847393, 0.96142903, 0.76840439],
+        [48.99264843, 0.76841091, 1.2318375, 1.03847393, 0.96142903],
+        [48.91214612, 0.9614286, 0.76841091, 1.2318375, 1.03847393],
+        [48.87684751, 1.03847373, 0.9614286, 0.76841091, 1.2318375],
+        [49.58802965, 1.23184242, 1.03847373, 0.9614286, 0.76841091],
+        [50.53557449, 0.76841495, 1.23184242, 1.03847373, 0.9614286],
+        [53.12302946, 0.96144212, 0.76841495, 1.23184242, 1.03847373],
+        [52.63362146, 1.0384711, 0.96144212, 0.76841495, 1.23184242],
+        [53.62139963, 1.23184877, 1.0384711, 0.96144212, 0.76841495],
+        [53.35804419, 0.76841391, 1.23184877, 1.0384711, 0.96144212],
+        [53.6924023, 0.96144377, 0.76841391, 1.23184877, 1.0384711],
+        [54.70921862, 1.0384765, 0.96144377, 0.76841391, 1.23184877],
+        [53.65026423, 1.23184222, 1.0384765, 0.96144377, 0.76841391],
+        [55.10503638, 0.76841963, 1.23184222, 1.0384765, 0.96144377],
+        [56.23274541, 0.96144918, 0.76841963, 1.23184222, 1.0384765],
+        [57.07608381, 1.03848078, 0.96144918, 0.76841963, 1.23184222],
+        [58.53365122, 1.23185086, 1.03848078, 0.96144918, 0.76841963],
+        [60.68791098, 0.7684274, 1.23185086, 1.03848078, 0.96144918],
+        [62.42942945, 0.96145676, 0.7684274, 1.23185086, 1.03848078],
+        [63.14721533, 1.03848406, 0.96145676, 0.7684274, 1.23185086]
+      ],
+      "residuals": [-0.17784294, -0.05821046, 0.03178669, 0.02127808, -0.01187027, 0.16835968, 0.02679529, 0.15374109, -0.06719319, 0.05836818, 0.00353728, -0.13412383, 0.01510172, 0.04839001, 0.00139101, 0.00403473, -0.06505049, -0.12275016, 0.13434255, 0.13269779, 0.0558215, 0.05712131, -0.13379134, 0.08647879, 0.04360453, -0.04826153, 0.03863383, 0.07146301, 0.05118987, 0.06012487, 0.01211702, 0.06931652, 0.02017001, 0.02534703, -0.02329504, -0.05886648, 0.08572239, 0.02784363, 0.04844611, -0.02350997, 0.08713312, -0.00759424, 0.00937207, -0.00845599, 0.10855477, -0.02528552, 0.01842678, -0.02332015, 0.06725903, 0.05077694, -0.00269814, -0.00118503, 0.0238927, 0.03137694, 0.08407429, -0.0151278, 0.03081653, -0.00806477, 0.01028963, 0.03109691, -0.03178369, 0.0445257, 0.03360418, 0.02462636, 0.04193359, 0.06043382, 0.04712089, 0.01887962],
+      "fitted": [36.55325061, 20.33203296, 24.53771912, 27.01657605, 32.46178613, 20.10336495, 27.73283084, 30.44335788, 39.49208414, 23.62789959, 30.61394004, 33.13840948, 36.09587634, 22.72494856, 29.27096683, 31.64263934, 37.62542975, 22.54231707, 26.095953, 30.4925483, 39.09145519, 25.21551847, 32.6478601, 32.39009659, 40.44220075, 25.89910169, 31.45221545, 34.77241264, 43.03928492, 27.68587176, 35.90909594, 39.07363304, 48.30233991, 30.50225983, 38.75338735, 41.26624245, 47.19152981, 30.97606565, 39.41396788, 43.82943198, 51.24336395, 33.66278836, 41.92372596, 45.54222736, 53.7421168, 35.74106154, 44.03034313, 48.09278041, 56.2366112, 36.51695912, 47.10295465, 50.79398874, 60.20833347, 38.10398316, 48.58634663, 55.16687038, 64.83632758, 41.20348491, 51.30067112, 55.75800804, 67.39348357, 41.22560929, 52.98039417, 58.39638467, 70.30872977, 44.97840674, 58.34834215, 64.83176246],
+      "sigma2": [0.00475041],
+      "loglik": [-206.42027478],
+      "forecast": [77.78795149, 48.52405046, 60.7133168, 65.57737638],
+      "simulation": [215.00997011, 140.93034998, 309.92465112, 875.07984985]
+    }
+  }
+}
\ No newline at end of file
diff --git a/statsmodels/tsa/tests/test_exponential_smoothing.py b/statsmodels/tsa/tests/test_exponential_smoothing.py
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/test_exponential_smoothing.py
@@ -0,0 +1,624 @@
+"""
+Author: Samuel Scherrer
+"""
+from itertools import product
+import json
+import pathlib
+
+import numpy as np
+from numpy.testing import assert_allclose, assert_almost_equal
+import pandas as pd
+import pytest
+import scipy.stats
+
+from statsmodels.tsa.exponential_smoothing.ets import ETSModel
+
+# This contains tests for the exponential smoothing implementation in
+# tsa/exponential_smoothing/ets.py.
+#
+# Tests are mostly done by comparing results with the R implementation in the
+# package forecast for the datasets `oildata` (non-seasonal) and `austourists`
+# (seasonal).
+#
+# Therefore, a parametrized pytest fixture ``setup_model`` is provided, which
+# returns a constructed model, model parameters from R in the format expected
+# by ETSModel, and a dictionary of reference results. Use like this:
+#
+#     def test_<testname>(setup_model):
+#         model, params, results_R = setup_model
+#         # perform some tests
+#         ...
+
+###############################################################################
+# UTILS
+###############################################################################
+
+# Below I define parameter lists for all possible model and data combinations
+# (for data, see below). These are used for parametrizing the pytest fixture
+# ``setup_model``, which should be used for all tests comparing to R output.
+
+
+def remove_invalid_models_from_list(modellist):
+    # remove invalid models (no trend but damped)
+    for i, model in enumerate(modellist):
+        if model[1] is None and model[3]:
+            del modellist[i]
+
+
+ERRORS = ("add", "mul")
+TRENDS = ("add", "mul", None)
+SEASONALS = ("add", "mul", None)
+DAMPED = (True, False)
+
+MODELS_DATA_SEASONAL = list(
+    product(ERRORS, TRENDS, ("add", "mul"), DAMPED, ("austourists",),)
+)
+MODELS_DATA_NONSEASONAL = list(
+    product(ERRORS, TRENDS, (None,), DAMPED, ("oildata",),)
+)
+remove_invalid_models_from_list(MODELS_DATA_SEASONAL)
+remove_invalid_models_from_list(MODELS_DATA_NONSEASONAL)
+
+ALL_MODELS_AND_DATA = MODELS_DATA_NONSEASONAL + MODELS_DATA_SEASONAL
+
+
+def short_model_name(error, trend, seasonal):
+    short_name = {"add": "A", "mul": "M", None: "N"}
+    return short_name[error] + short_name[trend] + short_name[seasonal]
+
+
+@pytest.fixture(params=ALL_MODELS_AND_DATA)
+def setup_model(
+    request,
+    austourists,
+    oildata,
+    ets_austourists_fit_results_R,
+    ets_oildata_fit_results_R,
+):
+    params = request.param
+    error, trend, seasonal, damped = params[0:4]
+    data = params[4]
+    if data == "austourists":
+        data = austourists
+        seasonal_periods = 4
+        results = ets_austourists_fit_results_R[damped]
+    else:
+        data = oildata
+        seasonal_periods = None
+        results = ets_oildata_fit_results_R[damped]
+
+    name = short_model_name(error, trend, seasonal)
+    if name not in results:
+        pytest.skip(f"model {name} not implemented or not converging in R")
+
+    results_R = results[name]
+    params = get_params_from_R(results_R)
+
+    model = ETSModel(
+        data,
+        seasonal_periods=seasonal_periods,
+        error=error,
+        trend=trend,
+        seasonal=seasonal,
+        damped_trend=damped,
+    )
+
+    return model, params, results_R
+
+
+@pytest.fixture
+def austourists_model(austourists):
+    return ETSModel(
+        austourists,
+        seasonal_periods=4,
+        error="add",
+        trend="add",
+        seasonal="add",
+        damped_trend=True,
+    )
+
+
+@pytest.fixture
+def oildata_model(oildata):
+    return ETSModel(
+        oildata,
+        error="add",
+        trend="add",
+        damped_trend=True,
+    )
+
+
+#############################################################################
+# DATA
+#############################################################################
+
+
+@pytest.fixture
+def austourists():
+    # austourists dataset from fpp2 package
+    # https://cran.r-project.org/web/packages/fpp2/index.html
+    data = [
+        30.05251300,
+        19.14849600,
+        25.31769200,
+        27.59143700,
+        32.07645600,
+        23.48796100,
+        28.47594000,
+        35.12375300,
+        36.83848500,
+        25.00701700,
+        30.72223000,
+        28.69375900,
+        36.64098600,
+        23.82460900,
+        29.31168300,
+        31.77030900,
+        35.17787700,
+        19.77524400,
+        29.60175000,
+        34.53884200,
+        41.27359900,
+        26.65586200,
+        28.27985900,
+        35.19115300,
+        42.20566386,
+        24.64917133,
+        32.66733514,
+        37.25735401,
+        45.24246027,
+        29.35048127,
+        36.34420728,
+        41.78208136,
+        49.27659843,
+        31.27540139,
+        37.85062549,
+        38.83704413,
+        51.23690034,
+        31.83855162,
+        41.32342126,
+        42.79900337,
+        55.70835836,
+        33.40714492,
+        42.31663797,
+        45.15712257,
+        59.57607996,
+        34.83733016,
+        44.84168072,
+        46.97124960,
+        60.01903094,
+        38.37117851,
+        46.97586413,
+        50.73379646,
+        61.64687319,
+        39.29956937,
+        52.67120908,
+        54.33231689,
+        66.83435838,
+        40.87118847,
+        51.82853579,
+        57.49190993,
+        65.25146985,
+        43.06120822,
+        54.76075713,
+        59.83447494,
+        73.25702747,
+        47.69662373,
+        61.09776802,
+        66.05576122,
+    ]
+    index = pd.date_range("1999-01-01", "2015-12-31", freq="Q")
+    return pd.Series(data, index)
+
+
+@pytest.fixture
+def oildata():
+    # oildata dataset from fpp2 package
+    # https://cran.r-project.org/web/packages/fpp2/index.html
+    data = [
+        111.0091346,
+        130.8284341,
+        141.2870879,
+        154.2277747,
+        162.7408654,
+        192.1664835,
+        240.7997253,
+        304.2173901,
+        384.0045673,
+        429.6621566,
+        359.3169299,
+        437.2518544,
+        468.4007898,
+        424.4353365,
+        487.9794299,
+        509.8284478,
+        506.3472527,
+        340.1842374,
+        240.2589210,
+        219.0327876,
+        172.0746632,
+        252.5900922,
+        221.0710774,
+        276.5187735,
+        271.1479517,
+        342.6186005,
+        428.3558357,
+        442.3945534,
+        432.7851482,
+        437.2497186,
+        437.2091599,
+        445.3640981,
+        453.1950104,
+        454.4096410,
+        422.3789058,
+        456.0371217,
+        440.3866047,
+        425.1943725,
+        486.2051735,
+        500.4290861,
+        521.2759092,
+        508.9476170,
+        488.8888577,
+        509.8705750,
+        456.7229123,
+        473.8166029,
+        525.9508706,
+        549.8338076,
+        542.3404698,
+    ]
+    return pd.Series(data, index=pd.date_range("1965", "2013", freq="AS"))
+
+
+#############################################################################
+# REFERENCE RESULTS
+#############################################################################
+
+
+def obtain_R_results(path):
+    with path.open("r") as f:
+        R_results = json.load(f)
+
+    # remove invalid models
+    results = {}
+    for damped in R_results:
+        new_key = damped == "TRUE"
+        results[new_key] = {}
+        for model in R_results[damped]:
+            if len(R_results[damped][model]):
+                results[new_key][model] = R_results[damped][model]
+
+    # get correct types
+    for damped in results:
+        for model in results[damped]:
+            for key in ["alpha", "beta", "gamma", "phi", "sigma2"]:
+                results[damped][model][key] = float(
+                    results[damped][model][key][0]
+                )
+            for key in [
+                "states",
+                "initstate",
+                "residuals",
+                "fitted",
+                "forecast",
+                "simulation",
+            ]:
+                results[damped][model][key] = np.asarray(
+                    results[damped][model][key]
+                )
+    return results
+
+
+@pytest.fixture
+def ets_austourists_fit_results_R():
+    """
+    Dictionary of ets fit results obtained with script ``results/fit_ets.R``.
+    """
+    path = (
+        pathlib.Path(__file__).parent
+        / "results"
+        / "fit_ets_results_seasonal.json"
+    )
+    return obtain_R_results(path)
+
+
+@pytest.fixture
+def ets_oildata_fit_results_R():
+    """
+    Dictionary of ets fit results obtained with script ``results/fit_ets.R``.
+    """
+    path = (
+        pathlib.Path(__file__).parent
+        / "results"
+        / "fit_ets_results_nonseasonal.json"
+    )
+    return obtain_R_results(path)
+
+
+def fit_austourists_with_R_params(model, results_R, set_state=False):
+    """
+    Fit the model with params as found by R's forecast package
+    """
+    params = get_params_from_R(results_R)
+    with model.fix_params(dict(zip(model.param_names, params))):
+        fit = model.fit(disp=False)
+
+    if set_state:
+        states_R = get_states_from_R(results_R, model._k_states)
+        fit.states = states_R
+    return fit
+
+
+def get_params_from_R(results_R):
+    # get params from R
+    params = [results_R[name] for name in ["alpha", "beta", "gamma", "phi"]]
+    # in R, initial states are order l[-1], b[-1], s[-1], s[-2], ..., s[-m]
+    params += list(results_R["initstate"])
+    params = list(filter(np.isfinite, params))
+    return params
+
+
+def get_states_from_R(results_R, k_states):
+    if k_states > 1:
+        xhat_R = results_R["states"][1:, 0:k_states]
+    else:
+        xhat_R = results_R["states"][1:]
+        xhat_R = np.reshape(xhat_R, (len(xhat_R), 1))
+    return xhat_R
+
+
+#############################################################################
+# BASIC TEST CASES
+#############################################################################
+
+
+def test_fit_model_austouritsts(setup_model):
+    model, params, results_R = setup_model
+    model.fit(disp=False)
+
+
+#############################################################################
+# TEST OF MODEL EQUATIONS VS R
+#############################################################################
+
+
+def test_smooth_vs_R(setup_model):
+    model, params, results_R = setup_model
+
+    yhat, xhat = model.smooth(params, return_raw=True)
+
+    yhat_R = results_R["fitted"]
+    xhat_R = get_states_from_R(results_R, model._k_states)
+
+    assert_allclose(xhat, xhat_R, rtol=1e-5, atol=1e-5)
+    assert_allclose(yhat, yhat_R, rtol=1e-5, atol=1e-5)
+
+
+def test_residuals_vs_R(setup_model):
+    model, params, results_R = setup_model
+
+    yhat = model.smooth(params, return_raw=True)[0]
+
+    residuals = model._residuals(yhat)
+    assert_allclose(residuals, results_R["residuals"], rtol=1e-5, atol=1e-5)
+
+
+def test_loglike_vs_R(setup_model):
+    model, params, results_R = setup_model
+
+    loglike = model.loglike(params)
+    # the calculation of log likelihood in R is only up to a constant:
+    const = -model.nobs / 2 * (np.log(2 * np.pi / model.nobs) + 1)
+    loglike_R = results_R["loglik"][0] + const
+
+    assert_allclose(loglike, loglike_R, rtol=1e-5, atol=1e-5)
+
+
+def test_forecast_vs_R(setup_model):
+    model, params, results_R = setup_model
+
+    fit = fit_austourists_with_R_params(model, results_R, set_state=True)
+
+    fcast = fit.forecast(4)
+    expected = np.asarray(results_R["forecast"])
+
+    assert_allclose(expected, fcast.values, rtol=1e-3, atol=1e-4)
+
+
+def test_simulate_vs_R(setup_model):
+    model, params, results_R = setup_model
+
+    fit = fit_austourists_with_R_params(model, results_R, set_state=True)
+
+    innov = np.asarray([[1.76405235, 0.40015721, 0.97873798, 2.2408932]]).T
+    sim = fit.simulate(4, anchor="end", repetitions=1, random_errors=innov)
+    expected = np.asarray(results_R["simulation"])
+
+    assert_allclose(expected, sim.values, rtol=1e-5, atol=1e-5)
+
+
+def test_fit_vs_R(setup_model):
+    model, params, results_R = setup_model
+    fit = model.fit(disp=False)
+
+    # check log likelihood: we want to have a fit that is better, i.e. a fit
+    # that has a **higher** log-likelihood
+    const = -model.nobs / 2 * (np.log(2 * np.pi / model.nobs) + 1)
+    loglike_R = results_R["loglik"][0] + const
+    loglike = fit.llf
+    assert loglike >= loglike_R - 1e-4
+
+
+def test_predict_vs_R(setup_model):
+    model, params, results_R = setup_model
+    fit = fit_austourists_with_R_params(model, results_R, set_state=True)
+
+    n = fit.nobs
+    prediction = fit.predict(end=n + 3, dynamic=n)
+
+    yhat_R = results_R["fitted"]
+    assert_allclose(prediction[: n], yhat_R, rtol=1e-5, atol=1e-5)
+
+    forecast_R = results_R["forecast"]
+    assert_allclose(prediction[n:], forecast_R, rtol=1e-3, atol=1e-4)
+
+
+#############################################################################
+# OTHER TESTS
+#############################################################################
+
+
+def test_initialization_known(austourists):
+    initial_level, initial_trend = [36.46466837, 34.72584983]
+    model = ETSModel(
+        austourists,
+        error="add",
+        trend="add",
+        damped_trend=True,
+        initialization_method="known",
+        initial_level=initial_level,
+        initial_trend=initial_trend,
+    )
+    internal_params = model._internal_params(model._start_params)
+    assert initial_level == internal_params[4]
+    assert initial_trend == internal_params[5]
+    assert internal_params[6] == 0
+
+
+def test_initialization_heuristic(oildata):
+    model_estimated = ETSModel(
+        oildata,
+        error="add",
+        trend="add",
+        damped_trend=True,
+        initialization_method="estimated",
+    )
+    model_heuristic = ETSModel(
+        oildata,
+        error="add",
+        trend="add",
+        damped_trend=True,
+        initialization_method="heuristic",
+    )
+    fit_estimated = model_estimated.fit(disp=False)
+    fit_heuristic = model_heuristic.fit(disp=False)
+    yhat_estimated = fit_estimated.fittedvalues.values
+    yhat_heuristic = fit_heuristic.fittedvalues.values
+
+    # this test is mostly just to see if it works, so we only test whether the
+    # result is not totally off
+    assert_allclose(yhat_estimated[10:], yhat_heuristic[10:], rtol=0.5)
+
+
+def test_bounded_fit(oildata):
+    beta = [0.99, 0.99]
+    model1 = ETSModel(
+        oildata,
+        error="add",
+        trend="add",
+        damped_trend=True,
+        bounds={"smoothing_trend": beta}
+    )
+    fit1 = model1.fit(disp=False)
+    assert fit1.smoothing_trend == 0.99
+
+    # same using with fix_params semantic
+    model2 = ETSModel(
+        oildata,
+        error="add",
+        trend="add",
+        damped_trend=True,
+    )
+    with model2.fix_params({"smoothing_trend": 0.99}):
+        fit2 = model2.fit(disp=False)
+    assert fit2.smoothing_trend == 0.99
+    assert_allclose(fit1.params, fit2.params)
+
+    # using fit_constrained
+    fit3 = model2.fit_constrained({"smoothing_trend": 0.99})
+    assert fit3.smoothing_trend == 0.99
+    assert_allclose(fit1.params, fit3.params)
+
+
+def test_seasonal_periods(austourists):
+    # test auto-deduction of period
+    model = ETSModel(austourists, error="add", trend="add", seasonal="add")
+    assert model.seasonal_periods == 4
+
+    # test if seasonal period raises error
+    try:
+        model = ETSModel(austourists, seasonal="add", seasonal_periods=0)
+    except ValueError:
+        pass
+
+
+def test_simulate_keywords(austourists_model):
+    """
+    check whether all keywords are accepted and work without throwing errors.
+    """
+    fit = austourists_model.fit(disp=False)
+
+    # test anchor
+    assert_almost_equal(
+        fit.simulate(4, anchor=0, random_state=0).values,
+        fit.simulate(4, anchor="start", random_state=0).values,
+    )
+    assert_almost_equal(
+        fit.simulate(4, anchor=-1, random_state=0).values,
+        fit.simulate(4, anchor="2015-12-31", random_state=0).values,
+    )
+    assert_almost_equal(
+        fit.simulate(4, anchor="end", random_state=0).values,
+        fit.simulate(4, anchor="2016-03-31", random_state=0).values,
+    )
+
+    # test different random error options
+    fit.simulate(4, repetitions=10)
+    fit.simulate(4, repetitions=10, random_errors=scipy.stats.norm)
+    fit.simulate(4, repetitions=10, random_errors=scipy.stats.norm())
+    fit.simulate(4, repetitions=10, random_errors=np.random.randn(4, 10))
+    fit.simulate(4, repetitions=10, random_errors="bootstrap")
+
+    # test seeding
+    res = fit.simulate(4, repetitions=10, random_state=10).values
+    res2 = fit.simulate(
+        4, repetitions=10, random_state=np.random.RandomState(10)
+    ).values
+    assert np.all(res == res2)
+
+
+def test_summary(austourists_model):
+    # just try to run summary to see if it works
+    fit = austourists_model.fit(disp=False)
+    fit.summary()
+
+    # now without estimated initial states
+    austourists_model.set_initialization_method("heuristic")
+    fit = austourists_model.fit(disp=False)
+    fit.summary()
+
+    # and with fixed params
+    fit = austourists_model.fit_constrained({"smoothing_trend": 0.9})
+    fit.summary()
+
+
+def test_score(austourists_model):
+    fit = austourists_model.fit(disp=False)
+    score_cs = austourists_model.score(fit.params)
+    score_fd = austourists_model.score(
+        fit.params,
+        approx_complex_step=False,
+        approx_centered=True,
+    )
+    assert_almost_equal(score_cs, score_fd, 4)
+
+
+def test_hessian(austourists_model):
+    # The hessian approximations are not very consistent, but the test makes
+    # sure they run
+    fit = austourists_model.fit(disp=False)
+    austourists_model.hessian(fit.params)
+    austourists_model.hessian(
+        fit.params,
+        approx_complex_step=False,
+        approx_centered=True,
+    )