r"""
Tests for smoothing and estimation of unobserved states and disturbances

- Predicted states: :math:`E(\alpha_t | Y_{t-1})`
- Filtered states: :math:`E(\alpha_t | Y_t)`
- Smoothed states: :math:`E(\alpha_t | Y_n)`
- Smoothed disturbances :math:`E(\varepsilon_t | Y_n), E(\eta_t | Y_n)`

Tested against R (FKF, KalmanRun / KalmanSmooth), Stata (sspace), and
MATLAB (ssm toolbox)

Author: Chad Fulton
License: Simplified-BSD
"""
import os

import numpy as np
from numpy.testing import assert_allclose, assert_almost_equal, assert_equal
import pandas as pd

import pytest

from statsmodels import datasets
from statsmodels.tsa.statespace import mlemodel, sarimax, varmax
from statsmodels.tsa.statespace.tests.test_impulse_responses import TVSS
from statsmodels.tsa.statespace.kalman_filter import FILTER_UNIVARIATE
from statsmodels.tsa.statespace.kalman_smoother import (
    SMOOTH_CLASSICAL, SMOOTH_ALTERNATIVE,
    SMOOTH_UNIVARIATE)

current_path = os.path.dirname(os.path.abspath(__file__))
import sys
import platform
import re
i386_looser_tolerances=bool(re.match('i.?86|x86',platform.uname()[4])) and np.log2(sys.maxsize)<33
class TestStatesAR3:
    @classmethod
    def setup_class(cls, alternate_timing=False, *args, **kwargs):
        # Dataset / Stata comparison
        path = os.path.join(current_path, 'results',
                            'results_wpi1_ar3_stata.csv')
        cls.stata = pd.read_csv(path)
        cls.stata.index = pd.date_range(start='1960-01-01', periods=124,
                                        freq='QS')
        # Matlab comparison
        path = os.path.join(current_path, 'results',
                            'results_wpi1_ar3_matlab_ssm.csv')
        matlab_names = [
            'a1', 'a2', 'a3', 'detP', 'alphahat1', 'alphahat2', 'alphahat3',
            'detV', 'eps', 'epsvar', 'eta', 'etavar'
        ]
        cls.matlab_ssm = pd.read_csv(path, header=None, names=matlab_names)

        cls.model = sarimax.SARIMAX(
            cls.stata['wpi'], order=(3, 1, 0), simple_differencing=True,
            hamilton_representation=True, *args, **kwargs
        )

        if alternate_timing:
            cls.model.ssm.timing_init_filtered = True

        # Parameters from from Stata's sspace MLE estimation
        params = np.r_[.5270715, .0952613, .2580355, .5307459]
        cls.results = cls.model.smooth(params, cov_type='none')

        # Calculate the determinant of the covariance matrices (for easy
        # comparison to other languages without having to store 2-dim arrays)
        cls.results.det_predicted_state_cov = np.zeros((1, cls.model.nobs))
        cls.results.det_smoothed_state_cov = np.zeros((1, cls.model.nobs))
        for i in range(cls.model.nobs):
            cls.results.det_predicted_state_cov[0, i] = np.linalg.det(
                cls.results.filter_results.predicted_state_cov[:, :, i])
            cls.results.det_smoothed_state_cov[0, i] = np.linalg.det(
                cls.results.smoother_results.smoothed_state_cov[:, :, i])

        # Perform simulation smoothing
        nobs = cls.model.nobs
        k_endog = cls.model.k_endog
        k_posdef = cls.model.ssm.k_posdef
        cls.sim = cls.model.simulation_smoother(filter_timing=0)
        cls.sim.simulate(
            measurement_disturbance_variates=np.zeros(nobs * k_endog),
            state_disturbance_variates=np.zeros(nobs * k_posdef),
            initial_state_variates=np.zeros(cls.model.k_states)
        )

    def test_predict_obs(self):
        assert_almost_equal(
            self.results.filter_results.predict().forecasts[0],
            self.stata.iloc[1:]['dep1'], 4
        )

    def test_standardized_residuals(self):
        assert_almost_equal(
            self.results.filter_results.standardized_forecasts_error[0],
            self.stata.iloc[1:]['sr1'], 4
        )

    def test_predicted_states(self):
        assert_almost_equal(
            self.results.filter_results.predicted_state[:, :-1].T,
            self.stata.iloc[1:][['sp1', 'sp2', 'sp3']], 4
        )
        assert_almost_equal(
            self.results.filter_results.predicted_state[:, :-1].T,
            self.matlab_ssm[['a1', 'a2', 'a3']], 4
        )

    def test_predicted_states_cov(self):
        assert_almost_equal(
            self.results.det_predicted_state_cov.T,
            self.matlab_ssm[['detP']], 4
        )

    def test_filtered_states(self):
        assert_almost_equal(
            self.results.filter_results.filtered_state.T,
            self.stata.iloc[1:][['sf1', 'sf2', 'sf3']], 4
        )

    def test_smoothed_states(self):
        assert_almost_equal(
            self.results.smoother_results.smoothed_state.T,
            self.stata.iloc[1:][['sm1', 'sm2', 'sm3']], 4
        )
        assert_almost_equal(
            self.results.smoother_results.smoothed_state.T,
            self.matlab_ssm[['alphahat1', 'alphahat2', 'alphahat3']], 4
        )

    def test_smoothed_states_cov(self):
        assert_almost_equal(
            self.results.det_smoothed_state_cov.T,
            self.matlab_ssm[['detV']], 4
        )

    def test_smoothed_measurement_disturbance(self):
        assert_almost_equal(
            self.results.smoother_results.smoothed_measurement_disturbance.T,
            self.matlab_ssm[['eps']], 4
        )

    def test_smoothed_measurement_disturbance_cov(self):
        res = self.results.smoother_results
        assert_almost_equal(
            res.smoothed_measurement_disturbance_cov[0].T,
            self.matlab_ssm[['epsvar']], 4
        )

    def test_smoothed_state_disturbance(self):
        assert_almost_equal(
            self.results.smoother_results.smoothed_state_disturbance.T,
            self.matlab_ssm[['eta']], 4
        )

    def test_smoothed_state_disturbance_cov(self):
        assert_almost_equal(
            self.results.smoother_results.smoothed_state_disturbance_cov[0].T,
            self.matlab_ssm[['etavar']], 4
        )


class TestStatesAR3AlternateTiming(TestStatesAR3):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            alternate_timing=True, *args, **kwargs)


class TestStatesAR3AlternativeSmoothing(TestStatesAR3):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            smooth_method=SMOOTH_ALTERNATIVE, *args, **kwargs)

    def test_smoothed_states(self):
        # Initialization issues can change the first few smoothed states
        assert_almost_equal(
            self.results.smoother_results.smoothed_state.T[2:],
            self.stata.iloc[3:][['sm1', 'sm2', 'sm3']], 4
        )
        assert_almost_equal(
            self.results.smoother_results.smoothed_state.T[2:],
            self.matlab_ssm.iloc[2:][['alphahat1', 'alphahat2', 'alphahat3']],
            4
        )

    def test_smoothed_states_cov(self):
        assert_almost_equal(
            self.results.det_smoothed_state_cov.T[1:],
            self.matlab_ssm.iloc[1:][['detV']], 4
        )

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method,
                     SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_ALTERNATIVE)


class TestStatesAR3UnivariateSmoothing(TestStatesAR3):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            filter_method=FILTER_UNIVARIATE, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_UNIVARIATE)


class TestStatesMissingAR3:
    @classmethod
    def setup_class(cls, alternate_timing=False, *args, **kwargs):
        # Dataset
        path = os.path.join(current_path, 'results',
                            'results_wpi1_ar3_stata.csv')
        cls.stata = pd.read_csv(path)
        cls.stata.index = pd.date_range(start='1960-01-01', periods=124,
                                        freq='QS')
        # Matlab comparison
        path = os.path.join(current_path, 'results',
                            'results_wpi1_missing_ar3_matlab_ssm.csv')
        matlab_names = [
            'a1', 'a2', 'a3', 'detP', 'alphahat1', 'alphahat2', 'alphahat3',
            'detV', 'eps', 'epsvar', 'eta', 'etavar'
        ]
        cls.matlab_ssm = pd.read_csv(path, header=None, names=matlab_names)
        # KFAS comparison
        path = os.path.join(current_path, 'results',
                            'results_smoothing3_R.csv')
        cls.R_ssm = pd.read_csv(path)

        # Create missing observations
        cls.stata['dwpi'] = cls.stata['wpi'].diff()
        cls.stata.loc[cls.stata.index[10:21], 'dwpi'] = np.nan

        cls.model = sarimax.SARIMAX(
            cls.stata.loc[cls.stata.index[1:], 'dwpi'], order=(3, 0, 0),
            hamilton_representation=True, *args, **kwargs
        )
        if alternate_timing:
            cls.model.ssm.timing_init_filtered = True

        # Parameters from from Stata's sspace MLE estimation
        params = np.r_[.5270715, .0952613, .2580355, .5307459]
        cls.results = cls.model.smooth(params, return_ssm=True)

        # Calculate the determinant of the covariance matrices (for easy
        # comparison to other languages without having to store 2-dim arrays)
        cls.results.det_predicted_state_cov = np.zeros((1, cls.model.nobs))
        cls.results.det_smoothed_state_cov = np.zeros((1, cls.model.nobs))
        for i in range(cls.model.nobs):
            cls.results.det_predicted_state_cov[0, i] = np.linalg.det(
                cls.results.predicted_state_cov[:, :, i])
            cls.results.det_smoothed_state_cov[0, i] = np.linalg.det(
                cls.results.smoothed_state_cov[:, :, i])

        # Perform simulation smoothing
        nobs = cls.model.nobs
        k_endog = cls.model.k_endog
        k_posdef = cls.model.ssm.k_posdef
        cls.sim = cls.model.simulation_smoother()
        cls.sim.simulate(
            measurement_disturbance_variates=np.zeros(nobs * k_endog),
            state_disturbance_variates=np.zeros(nobs * k_posdef),
            initial_state_variates=np.zeros(cls.model.k_states)
        )

    def test_predicted_states(self):
        assert_almost_equal(
            self.results.predicted_state[:, :-1].T,
            self.matlab_ssm[['a1', 'a2', 'a3']], 4
        )

    def test_predicted_states_cov(self):
        assert_almost_equal(
            self.results.det_predicted_state_cov.T,
            self.matlab_ssm[['detP']], 4
        )

    def test_smoothed_states(self):
        assert_almost_equal(
            self.results.smoothed_state.T,
            self.matlab_ssm[['alphahat1', 'alphahat2', 'alphahat3']], 4
        )

    def test_smoothed_states_cov(self):
        assert_almost_equal(
            self.results.det_smoothed_state_cov.T,
            self.matlab_ssm[['detV']], 4
        )

    def test_smoothed_measurement_disturbance(self):
        assert_almost_equal(
            self.results.smoothed_measurement_disturbance.T,
            self.matlab_ssm[['eps']], 4
        )

    def test_smoothed_measurement_disturbance_cov(self):
        assert_almost_equal(
            self.results.smoothed_measurement_disturbance_cov[0].T,
            self.matlab_ssm[['epsvar']], 4
        )

    # There is a discrepancy between MATLAB ssm toolbox and
    # statsmodels.tsa.statespace on the following variables in the case of
    # missing data. Tests against the R package KFAS confirm our results

    def test_smoothed_state_disturbance(self):
        # See note above about why this assertion is invalid
        # assert_almost_equal(
        #     self.results.smoothed_state_disturbance.T,
        #     self.matlab_ssm[['eta']], 4
        # )
        assert_almost_equal(
            self.results.smoothed_state_disturbance.T,
            self.R_ssm[['etahat']], 9
        )

    def test_smoothed_state_disturbance_cov(self):
        # See note above about why this assertion is invalid
        # assert_almost_equal(
        #     self.results.smoothed_state_disturbance_cov[0].T,
        #     self.matlab_ssm[['etavar']], 4
        # )
        assert_almost_equal(
            self.results.smoothed_state_disturbance_cov[0, 0, :],
            self.R_ssm['detVeta'], 9
        )


class TestStatesMissingAR3AlternateTiming(TestStatesMissingAR3):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(alternate_timing=True, *args, **kwargs)


class TestStatesMissingAR3AlternativeSmoothing(TestStatesMissingAR3):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            smooth_method=SMOOTH_ALTERNATIVE, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method,
                     SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_ALTERNATIVE)


class TestStatesMissingAR3UnivariateSmoothing(TestStatesMissingAR3):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            filter_method=FILTER_UNIVARIATE, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_UNIVARIATE)


class TestMultivariateMissing:
    """
    Tests for most filtering and smoothing variables against output from the
    R library KFAS.

    Note that KFAS uses the univariate approach which generally will result in
    different predicted values and covariance matrices associated with the
    measurement equation (e.g. forecasts, etc.). In this case, although the
    model is multivariate, each of the series is truly independent so the
    values will be the same regardless of whether the univariate approach
    is used or not.
    """
    @classmethod
    def setup_class(cls, **kwargs):
        # Results
        path = os.path.join(current_path, 'results', 'results_smoothing_R.csv')
        cls.desired = pd.read_csv(path)

        # Data
        dta = datasets.macrodata.load_pandas().data
        dta.index = pd.date_range(start='1959-01-01', end='2009-7-01',
                                  freq='QS')
        obs = dta[['realgdp', 'realcons', 'realinv']].diff().iloc[1:]
        obs.iloc[0:50, 0] = np.nan
        obs.iloc[19:70, 1] = np.nan
        obs.iloc[39:90, 2] = np.nan
        obs.iloc[119:130, 0] = np.nan
        obs.iloc[119:130, 2] = np.nan

        # Create the model
        mod = mlemodel.MLEModel(obs, k_states=3, k_posdef=3, **kwargs)
        mod['design'] = np.eye(3)
        mod['obs_cov'] = np.eye(3)
        mod['transition'] = np.eye(3)
        mod['selection'] = np.eye(3)
        mod['state_cov'] = np.eye(3)
        mod.initialize_approximate_diffuse(1e6)
        cls.model = mod
        cls.results = mod.smooth([], return_ssm=True)

        # Calculate the determinant of the covariance matrices (for easy
        # comparison to other languages without having to store 2-dim arrays)
        cls.results.det_scaled_smoothed_estimator_cov = (
            np.zeros((1, cls.model.nobs)))
        cls.results.det_predicted_state_cov = np.zeros((1, cls.model.nobs))
        cls.results.det_smoothed_state_cov = np.zeros((1, cls.model.nobs))
        cls.results.det_smoothed_state_disturbance_cov = (
            np.zeros((1, cls.model.nobs)))

        for i in range(cls.model.nobs):
            cls.results.det_scaled_smoothed_estimator_cov[0, i] = (
                np.linalg.det(
                    cls.results.scaled_smoothed_estimator_cov[:, :, i]))
            cls.results.det_predicted_state_cov[0, i] = np.linalg.det(
                cls.results.predicted_state_cov[:, :, i+1])
            cls.results.det_smoothed_state_cov[0, i] = np.linalg.det(
                cls.results.smoothed_state_cov[:, :, i])
            cls.results.det_smoothed_state_disturbance_cov[0, i] = (
                np.linalg.det(
                    cls.results.smoothed_state_disturbance_cov[:, :, i]))

    def test_loglike(self):
        assert_allclose(np.sum(self.results.llf_obs), -205310.9767)

    def test_scaled_smoothed_estimator(self):
        assert_allclose(
            self.results.scaled_smoothed_estimator.T,
            self.desired[['r1', 'r2', 'r3']]
        )

    def test_scaled_smoothed_estimator_cov(self):
        assert_allclose(
            self.results.det_scaled_smoothed_estimator_cov.T,
            self.desired[['detN']]
        )

    def test_forecasts(self):
        assert_allclose(
            self.results.forecasts.T,
            self.desired[['m1', 'm2', 'm3']]
        )

    def test_forecasts_error(self):
        assert_allclose(
            self.results.forecasts_error.T,
            self.desired[['v1', 'v2', 'v3']]
        )

    def test_forecasts_error_cov(self):
        assert_allclose(
            self.results.forecasts_error_cov.diagonal(),
            self.desired[['F1', 'F2', 'F3']]
        )

    def test_predicted_states(self):
        assert_allclose(
            self.results.predicted_state[:, 1:].T,
            self.desired[['a1', 'a2', 'a3']]
        )

    def test_predicted_states_cov(self):
        assert_allclose(
            self.results.det_predicted_state_cov.T,
            self.desired[['detP']]
        )

    def test_smoothed_states(self):
        assert_allclose(
            self.results.smoothed_state.T,
            self.desired[['alphahat1', 'alphahat2', 'alphahat3']]
        )

    def test_smoothed_states_cov(self):
        assert_allclose(
            self.results.det_smoothed_state_cov.T,
            self.desired[['detV']]
        )

    def test_smoothed_forecasts(self):
        assert_allclose(
            self.results.smoothed_forecasts.T,
            self.desired[['muhat1', 'muhat2', 'muhat3']]
        )

    def test_smoothed_state_disturbance(self):
        assert_allclose(
            self.results.smoothed_state_disturbance.T,
            self.desired[['etahat1', 'etahat2', 'etahat3']]
        )

    def test_smoothed_state_disturbance_cov(self):
        assert_allclose(
            self.results.det_smoothed_state_disturbance_cov.T,
            self.desired[['detVeta']]
        )

    def test_smoothed_measurement_disturbance(self):
        assert_allclose(
            self.results.smoothed_measurement_disturbance.T,
            self.desired[['epshat1', 'epshat2', 'epshat3']]
        )

    def test_smoothed_measurement_disturbance_cov(self):
        assert_allclose(
            self.results.smoothed_measurement_disturbance_cov.diagonal(),
            self.desired[['Veps1', 'Veps2', 'Veps3']]
        )


class TestMultivariateMissingClassicalSmoothing(TestMultivariateMissing):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            smooth_method=SMOOTH_CLASSICAL, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, SMOOTH_CLASSICAL)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method,
                     SMOOTH_CLASSICAL)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_CLASSICAL)


class TestMultivariateMissingAlternativeSmoothing(TestMultivariateMissing):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            smooth_method=SMOOTH_ALTERNATIVE, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method,
                     SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_ALTERNATIVE)


class TestMultivariateMissingUnivariateSmoothing(TestMultivariateMissing):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            filter_method=FILTER_UNIVARIATE, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_UNIVARIATE)


class TestMultivariateVAR:
    """
    Tests for most filtering and smoothing variables against output from the
    R library KFAS.

    Note that KFAS uses the univariate approach which generally will result in
    different predicted values and covariance matrices associated with the
    measurement equation (e.g. forecasts, etc.). In this case, although the
    model is multivariate, each of the series is truly independent so the
    values will be the same regardless of whether the univariate approach is
    used or not.
    """
    @classmethod
    def setup_class(cls, *args, **kwargs):
        # Results
        path = os.path.join(current_path, 'results',
                            'results_smoothing2_R.csv')
        cls.desired = pd.read_csv(path)

        # Data
        dta = datasets.macrodata.load_pandas().data
        dta.index = pd.date_range(start='1959-01-01', end='2009-7-01',
                                  freq='QS')
        obs = np.log(dta[['realgdp', 'realcons', 'realinv']]).diff().iloc[1:]

        # Create the model
        mod = mlemodel.MLEModel(obs, k_states=3, k_posdef=3, **kwargs)
        mod['design'] = np.eye(3)
        mod['obs_cov'] = np.array([
            [0.0000640649,  0.,            0.],
            [0.,            0.0000572802,  0.],
            [0.,            0.,            0.0017088585]])
        mod['transition'] = np.array([
            [-0.1119908792,  0.8441841604,  0.0238725303],
            [0.2629347724,   0.4996718412, -0.0173023305],
            [-3.2192369082,  4.1536028244,  0.4514379215]])
        mod['selection'] = np.eye(3)
        mod['state_cov'] = np.array([
            [0.0000640649,  0.0000388496,  0.0002148769],
            [0.0000388496,  0.0000572802,  0.000001555],
            [0.0002148769,  0.000001555,   0.0017088585]])
        mod.initialize_approximate_diffuse(1e6)
        cls.model = mod
        cls.results = mod.smooth([], return_ssm=True)

        # Calculate the determinant of the covariance matrices (for easy
        # comparison to other languages without having to store 2-dim arrays)
        cls.results.det_scaled_smoothed_estimator_cov = (
            np.zeros((1, cls.model.nobs)))
        cls.results.det_predicted_state_cov = np.zeros((1, cls.model.nobs))
        cls.results.det_smoothed_state_cov = np.zeros((1, cls.model.nobs))
        cls.results.det_smoothed_state_disturbance_cov = (
            np.zeros((1, cls.model.nobs)))

        for i in range(cls.model.nobs):
            cls.results.det_scaled_smoothed_estimator_cov[0, i] = (
                np.linalg.det(
                    cls.results.scaled_smoothed_estimator_cov[:, :, i]))
            cls.results.det_predicted_state_cov[0, i] = np.linalg.det(
                cls.results.predicted_state_cov[:, :, i+1])
            cls.results.det_smoothed_state_cov[0, i] = np.linalg.det(
                cls.results.smoothed_state_cov[:, :, i])
            cls.results.det_smoothed_state_disturbance_cov[0, i] = (
                np.linalg.det(
                    cls.results.smoothed_state_disturbance_cov[:, :, i]))

    def test_loglike(self):
        assert_allclose(np.sum(self.results.llf_obs), 1695.34872)

    def test_scaled_smoothed_estimator(self):
        assert_allclose(
            self.results.scaled_smoothed_estimator.T,
            self.desired[['r1', 'r2', 'r3']], atol=1e-4
        )

    def test_scaled_smoothed_estimator_cov(self):
        # Last obs is zero, so exclude it
        assert_allclose(
            np.log(self.results.det_scaled_smoothed_estimator_cov.T[:-1]),
            np.log(self.desired[['detN']][:-1]), atol=1e-6
        )

    def test_forecasts(self):
        assert_allclose(
            self.results.forecasts.T,
            self.desired[['m1', 'm2', 'm3']], atol=1e-6
        )

    def test_forecasts_error(self):
        assert_allclose(
            self.results.forecasts_error.T[:, 0],
            self.desired['v1'], atol=1e-6
        )

    def test_forecasts_error_cov(self):
        assert_allclose(
            self.results.forecasts_error_cov.diagonal()[:, 0],
            self.desired['F1'], atol=1e-6
        )

    def test_predicted_states(self):
        assert_allclose(
            self.results.predicted_state[:, 1:].T,
            self.desired[['a1', 'a2', 'a3']], atol=1e-6
        )

    def test_predicted_states_cov(self):
        assert_allclose(
            self.results.det_predicted_state_cov.T,
            self.desired[['detP']], atol=1e-16
        )

    def test_smoothed_states(self):
        assert_allclose(
            self.results.smoothed_state.T,
            self.desired[['alphahat1', 'alphahat2', 'alphahat3']], atol=1e-6
        )

    def test_smoothed_states_cov(self):
        assert_allclose(
            self.results.det_smoothed_state_cov.T,
            self.desired[['detV']], atol=1e-16
        )

    def test_smoothed_forecasts(self):
        assert_allclose(
            self.results.smoothed_forecasts.T,
            self.desired[['muhat1', 'muhat2', 'muhat3']], atol=1e-6
        )

    def test_smoothed_state_disturbance(self):
        assert_allclose(
            self.results.smoothed_state_disturbance.T,
            self.desired[['etahat1', 'etahat2', 'etahat3']], atol=1e-6
        )

    def test_smoothed_state_disturbance_cov(self):
        assert_allclose(
            self.results.det_smoothed_state_disturbance_cov.T,
            self.desired[['detVeta']], atol=1e-18
        )

    def test_smoothed_measurement_disturbance(self):
        assert_allclose(
            self.results.smoothed_measurement_disturbance.T,
            self.desired[['epshat1', 'epshat2', 'epshat3']], atol=1e-6
        )

    def test_smoothed_measurement_disturbance_cov(self):
        assert_allclose(
            self.results.smoothed_measurement_disturbance_cov.diagonal(),
            self.desired[['Veps1', 'Veps2', 'Veps3']], atol=1e-6
        )


class TestMultivariateVARAlternativeSmoothing(TestMultivariateVAR):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            smooth_method=SMOOTH_ALTERNATIVE, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method,
                     SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_ALTERNATIVE)


class TestMultivariateVARClassicalSmoothing(TestMultivariateVAR):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            smooth_method=SMOOTH_CLASSICAL, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, SMOOTH_CLASSICAL)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method,
                     SMOOTH_CLASSICAL)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_CLASSICAL)


class TestMultivariateVARUnivariate:
    """
    Tests for most filtering and smoothing variables against output from the
    R library KFAS.

    Note that KFAS uses the univariate approach which generally will result in
    different predicted values and covariance matrices associated with the
    measurement equation (e.g. forecasts, etc.). In this case, although the
    model is multivariate, each of the series is truly independent so the
    values will be the same regardless of whether the univariate approach is
    used or not.
    """
    @classmethod
    def setup_class(cls, *args, **kwargs):
        # Results
        path = os.path.join(current_path, 'results',
                            'results_smoothing2_R.csv')
        cls.desired = pd.read_csv(path)

        # Data
        dta = datasets.macrodata.load_pandas().data
        dta.index = pd.date_range(start='1959-01-01', end='2009-7-01',
                                  freq='QS')
        obs = np.log(dta[['realgdp', 'realcons', 'realinv']]).diff().iloc[1:]

        # Create the model
        mod = mlemodel.MLEModel(obs, k_states=3, k_posdef=3, **kwargs)
        mod.ssm.filter_univariate = True
        mod['design'] = np.eye(3)
        mod['obs_cov'] = np.array([
            [0.0000640649,  0.,            0.],
            [0.,            0.0000572802,  0.],
            [0.,            0.,            0.0017088585]])
        mod['transition'] = np.array([
            [-0.1119908792,  0.8441841604,  0.0238725303],
            [0.2629347724,   0.4996718412, -0.0173023305],
            [-3.2192369082,  4.1536028244,  0.4514379215]])
        mod['selection'] = np.eye(3)
        mod['state_cov'] = np.array([
            [0.0000640649,  0.0000388496,  0.0002148769],
            [0.0000388496,  0.0000572802,  0.000001555],
            [0.0002148769,  0.000001555,   0.0017088585]])
        mod.initialize_approximate_diffuse(1e6)
        cls.model = mod
        cls.results = mod.smooth([], return_ssm=True)

        # Calculate the determinant of the covariance matrices (for easy
        # comparison to other languages without having to store 2-dim arrays)
        cls.results.det_scaled_smoothed_estimator_cov = (
            np.zeros((1, cls.model.nobs)))
        cls.results.det_predicted_state_cov = np.zeros((1, cls.model.nobs))
        cls.results.det_smoothed_state_cov = np.zeros((1, cls.model.nobs))
        cls.results.det_smoothed_state_disturbance_cov = (
            np.zeros((1, cls.model.nobs)))

        for i in range(cls.model.nobs):
            cls.results.det_scaled_smoothed_estimator_cov[0, i] = (
                np.linalg.det(
                    cls.results.scaled_smoothed_estimator_cov[:, :, i]))
            cls.results.det_predicted_state_cov[0, i] = np.linalg.det(
                cls.results.predicted_state_cov[:, :, i+1])
            cls.results.det_smoothed_state_cov[0, i] = np.linalg.det(
                cls.results.smoothed_state_cov[:, :, i])
            cls.results.det_smoothed_state_disturbance_cov[0, i] = (
                np.linalg.det(
                    cls.results.smoothed_state_disturbance_cov[:, :, i]))

    def test_loglike(self):
        assert_allclose(np.sum(self.results.llf_obs), 1695.34872)

    def test_scaled_smoothed_estimator(self):
        assert_allclose(
            self.results.scaled_smoothed_estimator.T,
            self.desired[['r1', 'r2', 'r3']], atol=1e-4
        )

    def test_scaled_smoothed_estimator_cov(self):
        # Last obs is zero, so exclude it
        assert_allclose(
            np.log(self.results.det_scaled_smoothed_estimator_cov.T[:-1]),
            np.log(self.desired[['detN']][:-1])
        )

    def test_forecasts(self):
        assert_allclose(
            self.results.forecasts.T[:, 0],
            self.desired['m1'], atol=1e-6
        )

    def test_forecasts_error(self):
        assert_allclose(
            self.results.forecasts_error.T,
            self.desired[['v1', 'v2', 'v3']], atol=1e-6
        )

    def test_forecasts_error_cov(self):
        assert_allclose(
            self.results.forecasts_error_cov.diagonal(),
            self.desired[['F1', 'F2', 'F3']],rtol=2e-7 if i386_looser_tolerances else 1e-7
        )

    def test_predicted_states(self):
        assert_allclose(
            self.results.predicted_state[:, 1:].T,
            self.desired[['a1', 'a2', 'a3']], atol=1e-8
        )

    def test_predicted_states_cov(self):
        assert_allclose(
            self.results.det_predicted_state_cov.T,
            self.desired[['detP']], atol=1e-18
        )

    def test_smoothed_states(self):
        assert_allclose(
            self.results.smoothed_state.T,
            self.desired[['alphahat1', 'alphahat2', 'alphahat3']], atol=1e-6
        )

    def test_smoothed_states_cov(self):
        assert_allclose(
            self.results.det_smoothed_state_cov.T,
            self.desired[['detV']], atol=1e-18
        )

    def test_smoothed_forecasts(self):
        assert_allclose(
            self.results.smoothed_forecasts.T,
            self.desired[['muhat1', 'muhat2', 'muhat3']], atol=1e-6
        )

    def test_smoothed_state_disturbance(self):
        assert_allclose(
            self.results.smoothed_state_disturbance.T,
            self.desired[['etahat1', 'etahat2', 'etahat3']], atol=1e-6
        )

    def test_smoothed_state_disturbance_cov(self):
        assert_allclose(
            self.results.det_smoothed_state_disturbance_cov.T,
            self.desired[['detVeta']], atol=1e-18
        )

    def test_smoothed_measurement_disturbance(self):
        assert_allclose(
            self.results.smoothed_measurement_disturbance.T,
            self.desired[['epshat1', 'epshat2', 'epshat3']], atol=1e-6
        )

    def test_smoothed_measurement_disturbance_cov(self):
        assert_allclose(
            self.results.smoothed_measurement_disturbance_cov.diagonal(),
            self.desired[['Veps1', 'Veps2', 'Veps3']],rtol=2e-7 if i386_looser_tolerances else 1e-7
        )


class TestMultivariateVARUnivariateSmoothing(TestMultivariateVARUnivariate):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            filter_method=FILTER_UNIVARIATE, *args, **kwargs)

    def test_filter_method(self):
        assert_equal(self.model.ssm.filter_method, FILTER_UNIVARIATE)
        assert_equal(self.model.ssm._kalman_smoother.filter_method,
                     FILTER_UNIVARIATE)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_UNIVARIATE)


class TestVARAutocovariances:
    @classmethod
    def setup_class(cls, which='mixed', *args, **kwargs):
        # Data
        dta = datasets.macrodata.load_pandas().data
        dta.index = pd.date_range(start='1959-01-01', end='2009-7-01',
                                  freq='QS')
        obs = np.log(dta[['realgdp', 'realcons', 'realinv']]).diff().iloc[1:]

        if which == 'all':
            obs.iloc[:50, :] = np.nan
            obs.iloc[119:130, :] = np.nan
        elif which == 'partial':
            obs.iloc[0:50, 0] = np.nan
            obs.iloc[119:130, 0] = np.nan
        elif which == 'mixed':
            obs.iloc[0:50, 0] = np.nan
            obs.iloc[19:70, 1] = np.nan
            obs.iloc[39:90, 2] = np.nan
            obs.iloc[119:130, 0] = np.nan
            obs.iloc[119:130, 2] = np.nan

        # Create the model with typical state space
        mod = mlemodel.MLEModel(obs, k_states=3, k_posdef=3, **kwargs)
        mod['design'] = np.eye(3)
        mod['obs_cov'] = np.array([
            [609.0746647855,  0.,              0.],
            [0.,              1.8774916622,    0.],
            [0.,              0.,            124.6768281675]])
        mod['transition'] = np.array([
            [-0.8110473405,  1.8005304445,  1.0215975772],
            [-1.9846632699,  2.4091302213,  1.9264449765],
            [0.9181658823,  -0.2442384581, -0.6393462272]])
        mod['selection'] = np.eye(3)
        mod['state_cov'] = np.array([
            [1552.9758843938,   612.7185121905,   877.6157204992],
            [612.7185121905,    467.8739411204,    70.608037339],
            [877.6157204992,     70.608037339,    900.5440385836]])
        mod.initialize_approximate_diffuse(1e6)
        cls.model = mod
        cls.results = mod.smooth([], return_ssm=True)

        # Create the model with augmented state space
        kwargs.pop('filter_collapsed', None)
        mod = mlemodel.MLEModel(obs, k_states=6, k_posdef=3, **kwargs)
        mod['design', :3, :3] = np.eye(3)
        mod['obs_cov'] = np.array([
            [609.0746647855,    0.,              0.],
            [0.,                1.8774916622,    0.],
            [0.,                0.,            124.6768281675]])
        mod['transition', :3, :3] = np.array([
            [-0.8110473405,  1.8005304445,  1.0215975772],
            [-1.9846632699,  2.4091302213,  1.9264449765],
            [0.9181658823,  -0.2442384581, -0.6393462272]])
        mod['transition', 3:, :3] = np.eye(3)
        mod['selection', :3, :3] = np.eye(3)
        mod['state_cov'] = np.array([
            [1552.9758843938,  612.7185121905,   877.6157204992],
            [612.7185121905,   467.8739411204,    70.608037339],
            [877.6157204992,    70.608037339,    900.5440385836]])

        mod.initialize_approximate_diffuse(1e6)
        cls.augmented_model = mod
        cls.augmented_results = mod.smooth([], return_ssm=True)

    def test_smoothed_state_autocov(self):
        # Cov(\alpha_{t+1}, \alpha_t)
        # Initialization makes these two methods slightly different for the
        # first few observations
        assert_allclose(self.results.smoothed_state_autocov[:, :, 0:5],
                        self.augmented_results.smoothed_state_cov[:3, 3:, 1:6],
                        atol=1e-4)
        assert_allclose(self.results.smoothed_state_autocov[:, :, 5:-1],
                        self.augmented_results.smoothed_state_cov[:3, 3:, 6:],
                        atol=1e-7)


class TestVARAutocovariancesAlternativeSmoothing(TestVARAutocovariances):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            smooth_method=SMOOTH_ALTERNATIVE, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method,
                     SMOOTH_ALTERNATIVE)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_ALTERNATIVE)


class TestVARAutocovariancesClassicalSmoothing(TestVARAutocovariances):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            smooth_method=SMOOTH_CLASSICAL, *args, **kwargs)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, SMOOTH_CLASSICAL)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method,
                     SMOOTH_CLASSICAL)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_CLASSICAL)


class TestVARAutocovariancesUnivariateSmoothing(TestVARAutocovariances):
    @classmethod
    def setup_class(cls, *args, **kwargs):
        super().setup_class(
            filter_method=FILTER_UNIVARIATE, *args, **kwargs)

    def test_filter_method(self):
        assert_equal(self.model.ssm.filter_method, FILTER_UNIVARIATE)
        assert_equal(self.model.ssm._kalman_smoother.filter_method,
                     FILTER_UNIVARIATE)

    def test_smooth_method(self):
        assert_equal(self.model.ssm.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother.smooth_method, 0)
        assert_equal(self.model.ssm._kalman_smoother._smooth_method,
                     SMOOTH_UNIVARIATE)


class TVSSWithLags(TVSS):
    def __init__(self, endog):
        # TVSS has 2 states, here we will add in 3 lags of those
        super().__init__(endog, _k_states=8)
        self['transition', 2:, :6] = np.eye(6)[..., None]
        # Can't use exact diffuse filtering
        self.ssm.initialize_approximate_diffuse(1e-4)


def get_acov_model(missing, filter_univariate, tvp, oos=None, params=None,
                   return_ssm=True):
    dta = datasets.macrodata.load_pandas().data
    dta.index = pd.date_range(start='1959-01-01', end='2009-7-01',
                              freq='QS')
    endog = np.log(dta[['realgdp', 'realcons']]).diff().iloc[1:]

    if missing == 'all':
        endog.iloc[:5, :] = np.nan
        endog.iloc[11:13, :] = np.nan
    elif missing == 'partial':
        endog.iloc[0:5, 0] = np.nan
        endog.iloc[11:13, 0] = np.nan
    elif missing == 'mixed':
        endog.iloc[0:5, 0] = np.nan
        endog.iloc[1:7, 1] = np.nan
        endog.iloc[11:13, 0] = np.nan

    if oos is not None:
        new_ix = pd.date_range(start=endog.index[0],
                               periods=len(endog) + oos, freq='QS')
        endog = endog.reindex(new_ix)

    if not tvp:
        mod = varmax.VARMAX(endog, order=(4, 0, 0), measurement_error=True,
                            tolerance=0)
        mod.ssm.filter_univariate = filter_univariate
        if params is None:
            params = mod.start_params
        res = mod.smooth(params, return_ssm=return_ssm)
    else:
        mod = TVSSWithLags(endog)
        mod.ssm.filter_univariate = filter_univariate
        res = mod.smooth([], return_ssm=return_ssm)

    return mod, res


@pytest.mark.parametrize('missing', ['all', 'partial', 'mixed', None])
@pytest.mark.parametrize('filter_univariate', [True, False])
@pytest.mark.parametrize('tvp', [True, False])
def test_smoothed_state_autocovariances_backwards(missing, filter_univariate,
                                                  tvp):
    r"""
    Test for Cov(t, t - lag)
    """
    _, res = get_acov_model(missing, filter_univariate, tvp)

    cov = res.smoothed_state_cov.transpose(2, 0, 1)
    desired_acov1 = cov[:, :2, 2:4]
    desired_acov2 = cov[:, :2, 4:6]
    desired_acov3 = cov[:, :2, 6:8]

    # Test all "backward" autocovariances: Cov(t, t-lag)
    acov1 = res.smoothed_state_autocovariance(1).transpose(2, 0, 1)
    assert_allclose(acov1[1:, :2, :2], desired_acov1[1:], rtol=1e-6, atol=1e-6)
    assert_equal(acov1[:1], np.nan)

    acov2 = res.smoothed_state_autocovariance(2).transpose(2, 0, 1)
    assert_allclose(acov2[2:, :2, :2], desired_acov2[2:], rtol=1e-6, atol=1e-6)
    assert_equal(acov2[:2], np.nan)

    acov3 = res.smoothed_state_autocovariance(3).transpose(2, 0, 1)
    assert_allclose(acov3[3:, :2, :2], desired_acov3[3:], rtol=1e-6, atol=1e-6)
    assert_equal(acov3[:3], np.nan)

    # Test for specific autocovariances
    acov1 = res.smoothed_state_autocovariance(1, t=0)
    assert_allclose(acov1, np.nan)
    acov1 = res.smoothed_state_autocovariance(1, t=1)
    assert_allclose(acov1[:2, :2], desired_acov1[1], rtol=1e-6, atol=1e-6)
    acov1 = res.smoothed_state_autocovariance(
        1, start=8, end=9).transpose(2, 0, 1)
    assert_allclose(acov1[:, :2, :2], desired_acov1[8:9], rtol=1e-6, atol=1e-6)

    acov2 = res.smoothed_state_autocovariance(2, t=0)
    assert_allclose(acov2, np.nan)
    acov2 = res.smoothed_state_autocovariance(2, t=1)
    assert_allclose(acov2, np.nan)
    acov2 = res.smoothed_state_autocovariance(2, t=2)
    assert_allclose(acov2[:2, :2], desired_acov2[2], rtol=1e-6, atol=1e-6)
    acov2 = res.smoothed_state_autocovariance(
        2, start=8, end=9).transpose(2, 0, 1)
    assert_allclose(acov2[:, :2, :2], desired_acov2[8:9], rtol=1e-6, atol=1e-6)


@pytest.mark.parametrize('missing', ['all', 'partial', 'mixed', None])
@pytest.mark.parametrize('filter_univariate', [True, False])
@pytest.mark.parametrize('tvp', [True, False])
def test_smoothed_state_autocovariances_forwards(missing, filter_univariate,
                                                 tvp):
    r"""
    Test for Cov(t, t + lag)
    """
    # Out-of-sample model
    # Note: in TVP case, we need to first generate the larger model, and then
    # create the smaller model with the system matrices from the larger model
    # (otherwise they will be different, since the matrices are randomly
    # generated)
    mod_oos, res_oos = get_acov_model(missing, filter_univariate, tvp, oos=3)

    # Basic model
    names = ['obs_intercept', 'design', 'obs_cov', 'transition', 'selection',
             'state_cov']
    if not tvp:
        mod, res = get_acov_model(missing, filter_univariate, tvp,
                                  params=mod_oos.start_params)
    else:
        mod, _ = get_acov_model(missing, filter_univariate, tvp)
        for name in names:
            mod[name] = mod_oos[name, ..., :-3]
        res = mod.ssm.smooth()

    extend_kwargs1 = {}
    extend_kwargs2 = {}
    if tvp:
        keys = ['obs_intercept', 'design', 'obs_cov', 'transition',
                'selection', 'state_cov']
        for key in keys:
            extend_kwargs1[key] = mod_oos[key, ..., -3:-2]
            extend_kwargs2[key] = mod_oos[key, ..., -3:-1]

    assert_allclose(res_oos.llf, res.llf)

    cov = res.smoothed_state_cov.transpose(2, 0, 1)
    desired_acov1 = cov[:, 2:4, :2]
    desired_acov2 = cov[:, 4:6, :2]
    desired_acov3 = cov[:, 6:8, :2]

    oos_cov = np.concatenate(
        (res_oos.smoothed_state_cov, res_oos.predicted_state_cov[..., -1:]),
        axis=2).transpose(2, 0, 1)

    # Test all "forwards" autocovariances: Cov(t, t+lag)
    # For Cov(t, t+lag), the first out-of-sample forward covariance,
    # Cov(T, T+1), is already available, so we dno't need extend kwaargs
    acov1 = res.smoothed_state_autocovariance(-1).transpose(2, 0, 1)
    assert_allclose(acov1[:-1, :2, :2], desired_acov1[1:])
    assert_allclose(acov1[-2:, :2, :2], oos_cov[-5:-3, 2:4, :2])

    acov2 = res.smoothed_state_autocovariance(
        -2, extend_kwargs=extend_kwargs1).transpose(2, 0, 1)
    assert_allclose(acov2[:-2, :2, :2], desired_acov2[2:])
    assert_allclose(acov2[-2:, :2, :2], oos_cov[-4:-2, 4:6, :2])

    acov3 = res.smoothed_state_autocovariance(
        -3, extend_kwargs=extend_kwargs2).transpose(2, 0, 1)
    assert_allclose(acov3[:-3, :2, :2], desired_acov3[3:])
    assert_allclose(acov3[-3:, :2, :2], oos_cov[-4:-1, 6:8, :2])

    # Test for specific autocovariances
    acov1 = res.smoothed_state_autocovariance(
        -1, t=mod.nobs, extend_kwargs=extend_kwargs1)
    assert_allclose(acov1[:2, :2], oos_cov[-3, 2:4, :2])
    acov1 = res.smoothed_state_autocovariance(-1, t=0)
    assert_allclose(acov1[:2, :2], desired_acov1[0 + 1])
    acov1 = res.smoothed_state_autocovariance(
        -1, start=8, end=9).transpose(2, 0, 1)
    assert_allclose(acov1[:, :2, :2], desired_acov1[8 + 1:9 + 1])

    acov2 = res.smoothed_state_autocovariance(
        -2, t=mod.nobs, extend_kwargs=extend_kwargs2)
    assert_allclose(acov2[:2, :2], oos_cov[-2, 4:6, :2])
    acov2 = res.smoothed_state_autocovariance(
        -2, t=mod.nobs - 1, extend_kwargs=extend_kwargs1)
    assert_allclose(acov2[:2, :2], oos_cov[-3, 4:6, :2])
    acov2 = res.smoothed_state_autocovariance(-2, t=0)
    assert_allclose(acov2[:2, :2], desired_acov2[0 + 2])
    acov2 = res.smoothed_state_autocovariance(
        -2, start=8, end=9).transpose(2, 0, 1)
    assert_allclose(acov2[:, :2, :2], desired_acov2[8 + 2:9 + 2])


@pytest.mark.parametrize('missing', ['all', 'partial', 'mixed', None])
@pytest.mark.parametrize('filter_univariate', [True, False])
@pytest.mark.parametrize('tvp', [True, False])
def test_smoothed_state_autocovariances_forwards_oos(missing,
                                                     filter_univariate, tvp):
    # Out-of-sample model
    # Note: in TVP case, we need to first generate the larger model, and then
    # create the smaller model with the system matrices from the larger model
    # (otherwise they will be different, since the matrices are randomly
    # generated)
    mod_oos, res_oos = get_acov_model(missing, filter_univariate, tvp, oos=5)

    # Basic model
    names = ['obs_intercept', 'design', 'obs_cov', 'transition', 'selection',
             'state_cov']
    if not tvp:
        mod, res = get_acov_model(missing, filter_univariate, tvp,
                                  params=mod_oos.start_params)
    else:
        mod, _ = get_acov_model(missing, filter_univariate, tvp)
        for name in names:
            mod[name] = mod_oos[name, ..., :-5]
        res = mod.ssm.smooth()

    assert_allclose(res_oos.llf, res.llf)

    cov = np.concatenate(
        (res_oos.smoothed_state_cov, res_oos.predicted_state_cov[..., -1:]),
        axis=2).transpose(2, 0, 1)
    desired_acov1 = cov[:, 2:4, :2]
    desired_acov2 = cov[:, 4:6, :2]
    desired_acov3 = cov[:, 6:8, :2]

    # Test all "forwards" autocovariances: Cov(t, t+lag)
    extend_kwargs = {}
    if tvp:
        extend_kwargs = {
            'obs_intercept': mod_oos['obs_intercept', ..., -5:],
            'design': mod_oos['design', ..., -5:],
            'obs_cov': mod_oos['obs_cov', ..., -5:],
            'transition': mod_oos['transition', ..., -5:],
            'selection': mod_oos['selection', ..., -5:],
            'state_cov': mod_oos['state_cov', ..., -5:]}

    # Note: we can compute up to Cov(mod_oos.nobs, mod_oos.nobs + 1) using
    # a model that has state space matrices defined up to mod_oos.nobs. Since
    # mod_oos.nobs = mod.nobs + 5, we need to pass in 5 additional time points,
    # and that is what extend_kwargs, above, does.
    acov1 = res.smoothed_state_autocovariance(
        -1, end=mod_oos.nobs, extend_kwargs=extend_kwargs).transpose(2, 0, 1)
    assert_equal(acov1.shape, (mod_oos.nobs, mod.k_states, mod.k_states))
    assert_allclose(acov1[:, :2, :2], desired_acov1[1:])

    # Note: now we can compute up to Cov(mod_oos.nobs - 1, mod_oos.nobs + 1)
    # using a model that has state space matrices defined up to mod_oos.nobs.
    # We still need to pass in 5 additional time points for the extend kwargs.
    # This is why we have end = mod_oos.nobs - 1, because this function returns
    # values through Cov(end, end + 2). Because start=0 (the default), we
    # will have values for Cov(0, 2), Cov(1, 3), ...,
    # Cov(mod_oos.nobs - 1, mod_oos.nobs + 1), and that is a set of
    # mod_oos.nobs - 1 matrices.
    acov2 = res.smoothed_state_autocovariance(
        -2, end=mod_oos.nobs - 1,
        extend_kwargs=extend_kwargs).transpose(2, 0, 1)
    assert_equal(acov2.shape, (mod_oos.nobs - 1, mod.k_states, mod.k_states))
    assert_allclose(acov2[:, :2, :2], desired_acov2[2:])

    # Note: now we can compute up to Cov(mod_oos.nobs - 2, mod_oos.nobs + 1)
    # using a model that has state space matrices defined up to mod_oos.nobs.
    # We still need to pass in 5 additional time points for the extend kwargs.
    acov3 = res.smoothed_state_autocovariance(
        -3, end=mod_oos.nobs - 2,
        extend_kwargs=extend_kwargs).transpose(2, 0, 1)
    assert_equal(acov3.shape, (mod_oos.nobs - 2, mod.k_states, mod.k_states))
    assert_allclose(acov3[:, :2, :2], desired_acov3[3:])


@pytest.mark.parametrize('missing', ['all', 'partial', 'mixed', None])
@pytest.mark.parametrize('filter_univariate', [True, False])
@pytest.mark.parametrize('tvp', [True, False])
def test_smoothed_state_autocovariances_backwards_oos(missing,
                                                      filter_univariate, tvp):
    # Out-of-sample model
    # Note: in TVP case, we need to first generate the larger model, and then
    # create the smaller model with the system matrices from the larger model
    # (otherwise they will be different, since the matrices are randomly
    # generated)
    mod_oos, res_oos = get_acov_model(missing, filter_univariate, tvp, oos=5)

    # Basic model
    names = ['obs_intercept', 'design', 'obs_cov', 'transition', 'selection',
             'state_cov']
    if not tvp:
        mod, res = get_acov_model(missing, filter_univariate, tvp,
                                  params=mod_oos.start_params)
    else:
        mod, _ = get_acov_model(missing, filter_univariate, tvp)
        for name in names:
            mod[name] = mod_oos[name, ..., :-5]
        res = mod.ssm.smooth()

    assert_allclose(res_oos.llf, res.llf)

    cov = np.concatenate(
        (res_oos.smoothed_state_cov, res_oos.predicted_state_cov[..., -1:]),
        axis=2).transpose(2, 0, 1)
    desired_acov1 = cov[:, :2, 2:4]
    desired_acov2 = cov[:, :2, 4:6]
    desired_acov3 = cov[:, :2, 6:8]

    # Test all "backwards" autocovariances: Cov(t, t - lag)
    end = mod_oos.nobs + 1
    extend_kwargs = {}
    if tvp:
        extend_kwargs = {
            'obs_intercept': mod_oos['obs_intercept', ..., -5:],
            'design': mod_oos['design', ..., -5:],
            'obs_cov': mod_oos['obs_cov', ..., -5:],
            'transition': mod_oos['transition', ..., -5:],
            'selection': mod_oos['selection', ..., -5:],
            'state_cov': mod_oos['state_cov', ..., -5:]}

    # Note: we can compute up to Cov(mod_oos.nobs + 1, mod_oos.nobs) using
    # a model that has state space matrices defined up to mod_oos.nobs. Since
    # mod_oos.nobs = mod.nobs + 5, we need to pass in 5 additional time points,
    # and that is what extend_kwargs, above, does.
    acov1 = res.smoothed_state_autocovariance(
        1, end=end, extend_kwargs=extend_kwargs).transpose(2, 0, 1)
    assert_equal(acov1.shape, (mod_oos.nobs + 1, mod.k_states, mod.k_states))
    assert_allclose(acov1[1:, :2, :2], desired_acov1[1:])
    # We cannot compute Cov(1, 0), so this is always NaNs
    assert_equal(acov1[:1], np.nan)

    # Note: we can compute up to Cov(mod_oos.nobs + 1, mod_oos.nobs - 1) using
    # a model that has state space matrices defined up to mod_oos.nobs, which
    # is why we don't need to change `end` here relative to the lag=1 case
    acov2 = res.smoothed_state_autocovariance(
        2, end=end, extend_kwargs=extend_kwargs).transpose(2, 0, 1)
    assert_allclose(acov2[2:, :2, :2], desired_acov2[2:])
    # We cannot compute Cov(1, -1) or Cov(2, 0), so this is always NaNs
    assert_equal(acov2[:2], np.nan)

    # Note: we can compute up to Cov(mod_oos.nobs + 1, mod_oos.nobs - 2) using
    # a model that has state space matrices defined up to mod_oos.nobs, which
    # is why we don't need to change `end` here relative to the lag=1 or lag=2
    # cases
    acov3 = res.smoothed_state_autocovariance(
        3, end=end, extend_kwargs=extend_kwargs).transpose(2, 0, 1)
    assert_allclose(acov3[3:, :2, :2], desired_acov3[3:])
    # We cannot compute Cov(1, -2), Cov(2, -1), or Cov(3, 0), so this is always
    # NaNs
    assert_equal(acov3[:3], np.nan)


def test_smoothed_state_autocovariances_invalid():
    # Tests for invalid calls of `smoothed_state_autocovariance`
    _, res = get_acov_model(missing=False, filter_univariate=False, tvp=False)

    with pytest.raises(ValueError, match='Cannot specify both `t`'):
        res.smoothed_state_autocovariance(1, t=1, start=1)

    with pytest.raises(ValueError, match='Negative `t`'):
        res.smoothed_state_autocovariance(1, t=-1)

    with pytest.raises(ValueError, match='Negative `t`'):
        res.smoothed_state_autocovariance(1, start=-1)

    with pytest.raises(ValueError, match='Negative `t`'):
        res.smoothed_state_autocovariance(1, end=-1)

    with pytest.raises(ValueError, match='`end` must be after `start`'):
        res.smoothed_state_autocovariance(1, start=5, end=4)


@pytest.mark.parametrize('missing', ['all', 'partial', 'mixed', None])
@pytest.mark.parametrize('filter_univariate', [True, False])
@pytest.mark.parametrize('tvp', [True, False])
def test_news_basic(missing, filter_univariate, tvp):
    # Basic tests for news

    # Get the basic model
    mod, res = get_acov_model(missing, filter_univariate, tvp)
    params = [] if tvp else mod.start_params

    # Get an expanded model with one new observation and 9 additional NaN
    # datapoints (so that we can compute the desired value using the
    # `smoothed_forecasts` attribute).
    append = np.zeros((10, 2)) * np.nan
    append[0] = [0.1, -0.2]
    endog2 = np.concatenate((mod.endog, append), axis=0)
    mod2 = mod.clone(endog2)
    res2 = mod2.smooth(params, return_ssm=True)

    # Get an expanded model with only 10 additional NaN datapoints, to compute
    # the baseline `smoothed_forecasts`.
    endog3 = endog2.copy()
    endog3[-10:] = np.nan
    mod3 = mod2.clone(endog3)
    res3 = mod3.smooth(params, return_ssm=True)

    # Test the news computation at the start, middle, and end of the sample, as
    # well as out-of-sample.
    for t in [0, 1, 150, mod.nobs - 1, mod.nobs, mod.nobs + 1, mod.nobs + 9]:
        # Test with a time argument
        out = res2.news(res, t=t)
        desired = (res2.smoothed_forecasts[..., t] -
                   res3.smoothed_forecasts[..., t])
        # The "news" about the t=0 smoothed forecast from new data at
        # observation t=202 is almost identically zero, so we need to set an
        # "atol" to avoid problems with comparing floating point versions of
        # zero.
        assert_allclose(out.update_impacts, desired, atol=1e-14)
        assert_equal(out.revision_impacts, None)

        # Test with start/end arguments
        out = res2.news(res, start=t, end=t + 1)
        assert_allclose(out.update_impacts, desired[None, ...], atol=1e-14)


@pytest.mark.parametrize('missing', ['all', 'partial', 'mixed', None])
@pytest.mark.parametrize('filter_univariate', [True, False])
@pytest.mark.parametrize('tvp', [True, False])
def test_news_revisions(missing, filter_univariate, tvp):
    # Tests for news when there are revisions in the model

    # Get the basic model
    mod, res = get_acov_model(missing, filter_univariate, tvp, oos=10)
    params = [] if tvp else mod.start_params

    endog2 = mod.endog.copy()
    # Revise the last datapoint
    endog2[-11] = [0.0, 0.0]
    # Add a new datapoint
    endog2[-10] = [-0.3, -0.4]
    mod2 = mod.clone(endog2)
    res2 = mod2.smooth(params, return_ssm=True)

    # Test the news computation at the start, middle, and end of the sample, as
    # well as out-of-sample.
    nobs = mod.nobs - 10
    for t in [0, 1, 150, nobs - 1, nobs, nobs + 1, nobs + 9]:
        out = res2.news(res, t=t)

        # Test for the news
        desired = (res2.smoothed_forecasts[..., t] -
                   out.revision_results.smoothed_forecasts[..., t])
        # Relaxed tolerance to 1e-10 after random failures
        assert_allclose(out.update_impacts, desired, atol=1e-10)

        # Test for the revisions
        desired = (out.revision_results.smoothed_forecasts[..., t] -
                   res.smoothed_forecasts[..., t])
        # Relaxed tolerance to 1e-10 after random failures
        assert_allclose(out.revision_impacts, desired, atol=1e-10)


@pytest.mark.parametrize('missing', ['all', 'partial', 'mixed', None])
@pytest.mark.parametrize('filter_univariate', [True, False])
@pytest.mark.parametrize('tvp', [True, False])
def test_news_invalid(missing, filter_univariate, tvp):
    # Tests for invalid calls to news

    # (generic error message used below)
    error_ss = ('This results object has %s and so it does not appear to'
                ' by an extension of `previous`. Can only compute the'
                ' news by comparing this results set to previous results'
                ' objects.')

    # Basic model / results setup
    mod, res = get_acov_model(missing, filter_univariate, tvp, oos=1)
    params = [] if tvp else mod.start_params

    endog2 = mod.endog.copy()
    endog2[-1] = [0.2, 0.5]
    mod2 = mod.clone(endog2)
    res2_filtered = mod2.filter(params, return_ssm=True)
    res2_smoothed = mod2.smooth(params, return_ssm=True)

    # Test that news works with smoothing, but not with only filtering
    res2_smoothed.news(res, t=mod.nobs - 1)
    msg = ('Cannot compute news without having'
           ' applied the Kalman smoother first.')
    with pytest.raises(ValueError, match=msg):
        res2_filtered.news(res, t=mod.nobs - 1)

    # Test that if we want to request news for an out-of-sample period in a
    # time-varying model, then we need to provide a new design matrix
    if tvp:
        msg = ('Cannot compute the impacts of news on periods outside of the'
               ' sample in time-varying models.')
        with pytest.raises(RuntimeError, match=msg):
            res2_smoothed.news(res, t=mod.nobs + 2)

    # Test that news won't work when the calling model is is smaller
    mod, res = get_acov_model(missing, filter_univariate, tvp)
    params = [] if tvp else mod.start_params

    endog2 = mod.endog.copy()[:mod.nobs - 1]
    mod2 = mod.clone(endog2)
    res2 = mod2.smooth(params, return_ssm=True)
    msg = error_ss % 'fewer observations than `previous`'
    with pytest.raises(ValueError, match=msg):
        res2.news(res, t=mod.nobs - 1)

    # Test that news won't work when the state dimensions are different
    mod2 = sarimax.SARIMAX(np.zeros(mod.nobs))
    res2 = mod2.smooth([0.5, 1.], return_ssm=True)
    msg = error_ss % 'different state space dimensions than `previous`'
    with pytest.raises(ValueError, match=msg):
        res2.news(res, t=mod.nobs - 1)

    # Test that news won't work when one of the models is time-varying and one
    # is time-invariant
    mod2, res2 = get_acov_model(missing, filter_univariate, not tvp, oos=1)
    if tvp:
        msg = 'time-invariant design while `previous` does not'
    else:
        msg = 'time-varying design while `previous` does not'
    with pytest.raises(ValueError, match=msg):
        res2.news(res, t=mod.nobs - 1)