Ejemplo n.º 1
0
def test_missing():
    # Datasets
    endog = np.arange(10).reshape(10,1)
    endog_pre_na = np.ascontiguousarray(np.c_[
        endog.copy() * np.nan, endog.copy() * np.nan, endog, endog])
    endog_post_na = np.ascontiguousarray(np.c_[
        endog, endog, endog.copy() * np.nan, endog.copy() * np.nan])
    endog_inject_na = np.ascontiguousarray(np.c_[
        endog, endog.copy() * np.nan, endog, endog.copy() * np.nan])

    # Base model
    mod = KalmanFilter(np.ascontiguousarray(np.c_[endog, endog]), k_states=1,
                       initialization='approximate_diffuse')
    mod['design', :, :] = 1
    mod['obs_cov', :, :] = np.eye(mod.k_endog)*0.5
    mod['transition', :, :] = 0.5
    mod['selection', :, :] = 1
    mod['state_cov', :, :] = 0.5
    llf = mod.loglikeobs()

    # Model with prepended nans
    mod = KalmanFilter(endog_pre_na, k_states=1,
                       initialization='approximate_diffuse')
    mod['design', :, :] = 1
    mod['obs_cov', :, :] = np.eye(mod.k_endog)*0.5
    mod['transition', :, :] = 0.5
    mod['selection', :, :] = 1
    mod['state_cov', :, :] = 0.5
    llf_pre_na = mod.loglikeobs()

    assert_allclose(llf_pre_na, llf)

    # Model with appended nans
    mod = KalmanFilter(endog_post_na, k_states=1,
                       initialization='approximate_diffuse')
    mod['design', :, :] = 1
    mod['obs_cov', :, :] = np.eye(mod.k_endog)*0.5
    mod['transition', :, :] = 0.5
    mod['selection', :, :] = 1
    mod['state_cov', :, :] = 0.5
    llf_post_na = mod.loglikeobs()

    assert_allclose(llf_post_na, llf)

    # Model with injected nans
    mod = KalmanFilter(endog_inject_na, k_states=1,
                       initialization='approximate_diffuse')
    mod['design', :, :] = 1
    mod['obs_cov', :, :] = np.eye(mod.k_endog)*0.5
    mod['transition', :, :] = 0.5
    mod['selection', :, :] = 1
    mod['state_cov', :, :] = 0.5
    llf_inject_na = mod.loglikeobs()

    assert_allclose(llf_inject_na, llf)
Ejemplo n.º 2
0
def test_simulate():
    # Test for simulation of new time-series
    from scipy.signal import lfilter

    # Common parameters
    nsimulations = 10
    sigma2 = 2
    measurement_shocks = np.zeros(nsimulations)
    state_shocks = np.random.normal(scale=sigma2**0.5, size=nsimulations)

    # Random walk model, so simulated series is just the cumulative sum of
    # the shocks
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.

    actual = mod.simulate(nsimulations,
                          measurement_shocks=measurement_shocks,
                          state_shocks=state_shocks)[0].squeeze()
    desired = np.r_[0, np.cumsum(state_shocks)[:-1]]

    assert_allclose(actual, desired)

    # Local level model, so simulated series is just the cumulative sum of
    # the shocks plus the measurement shock
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.

    actual = mod.simulate(nsimulations,
                          measurement_shocks=np.ones(nsimulations),
                          state_shocks=state_shocks)[0].squeeze()
    desired = np.r_[1, np.cumsum(state_shocks)[:-1] + 1]

    assert_allclose(actual, desired)

    # Local level-like model with observation and state intercepts, so
    # simulated series is just the cumulative sum of the shocks minus the state
    # intercept, plus the observation intercept and the measurement shock
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['obs_intercept', 0, 0] = 5.
    mod['design', 0, 0] = 1.
    mod['state_intercept', 0, 0] = -2.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.

    actual = mod.simulate(nsimulations,
                          measurement_shocks=np.ones(nsimulations),
                          state_shocks=state_shocks)[0].squeeze()
    desired = np.r_[1 + 5, np.cumsum(state_shocks - 2)[:-1] + 1 + 5]

    assert_allclose(actual, desired)

    # Model with time-varying observation intercept
    mod = KalmanFilter(k_endog=1, k_states=1, nobs=10)
    mod['obs_intercept'] = (np.arange(10) * 1.).reshape(1, 10)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.

    actual = mod.simulate(nsimulations,
                          measurement_shocks=measurement_shocks,
                          state_shocks=state_shocks)[0].squeeze()
    desired = np.r_[0, np.cumsum(state_shocks)[:-1] + np.arange(1, 10)]

    assert_allclose(actual, desired)

    # Model with time-varying observation intercept, check that error is raised
    # if more simulations are requested than are nobs.
    mod = KalmanFilter(k_endog=1, k_states=1, nobs=10)
    mod['obs_intercept'] = (np.arange(10) * 1.).reshape(1, 10)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    assert_raises(ValueError, mod.simulate, nsimulations + 1,
                  measurement_shocks, state_shocks)

    # ARMA(1,1): phi = [0.1], theta = [0.5], sigma^2 = 2
    phi = np.r_[0.1]
    theta = np.r_[0.5]
    mod = sarimax.SARIMAX([0], order=(1, 0, 1))
    mod.update(np.r_[phi, theta, sigma2])

    actual = mod.ssm.simulate(nsimulations,
                              measurement_shocks=measurement_shocks,
                              state_shocks=state_shocks)[0].squeeze()
    desired = lfilter([1, theta], [1, -phi], np.r_[0, state_shocks[:-1]])

    assert_allclose(actual, desired)

    # SARIMAX(1,0,1)x(1,0,1,4), this time using the results object call
    mod = sarimax.SARIMAX([0.1, 0.5, -0.2],
                          order=(1, 0, 1),
                          seasonal_order=(1, 0, 1, 4))
    res = mod.filter([0.1, 0.5, 0.2, -0.3, 1])

    actual = res.simulate(nsimulations,
                          measurement_shocks=measurement_shocks,
                          state_shocks=state_shocks).squeeze()
    desired = lfilter(res.polynomial_reduced_ma, res.polynomial_reduced_ar,
                      np.r_[0, state_shocks[:-1]])

    assert_allclose(actual, desired)
Ejemplo n.º 3
0
def test_impulse_responses():
    # Test for impulse response functions

    # Random walk: 1-unit impulse response (i.e. non-orthogonalized irf) is 1
    # for all periods
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    mod['state_cov', 0, 0] = 2.

    actual = mod.impulse_responses(steps=10)
    desired = np.ones((11, 1))

    assert_allclose(actual, desired)

    # Random walk: 1-standard-deviation response (i.e. orthogonalized irf) is
    # sigma for all periods (here sigma^2 = 2)
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    mod['state_cov', 0, 0] = 2.

    actual = mod.impulse_responses(steps=10, orthogonalized=True)
    desired = np.ones((11, 1)) * 2**0.5

    assert_allclose(actual, desired)

    # Random walk: 1-standard-deviation cumulative response (i.e. cumulative
    # orthogonalized irf)
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    mod['state_cov', 0, 0] = 2.

    actual = mod.impulse_responses(steps=10,
                                   orthogonalized=True,
                                   cumulative=True)
    desired = np.cumsum(np.ones((11, 1)) * 2**0.5)[:, np.newaxis]

    assert_allclose(actual, desired)

    # Random walk: 1-unit impulse response (i.e. non-orthogonalized irf) is 1
    # for all periods, even when intercepts are present
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['state_intercept', 0] = 100.
    mod['design', 0, 0] = 1.
    mod['obs_intercept', 0] = -1000.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    mod['state_cov', 0, 0] = 2.

    actual = mod.impulse_responses(steps=10)
    desired = np.ones((11, 1))

    assert_allclose(actual, desired)

    # Univariate model (random walk): test that an error is thrown when
    # a multivariate or empty "impulse" is sent
    mod = KalmanFilter(k_endog=1, k_states=1)
    assert_raises(ValueError, mod.impulse_responses, impulse=1)
    assert_raises(ValueError, mod.impulse_responses, impulse=[1, 1])
    assert_raises(ValueError, mod.impulse_responses, impulse=[])

    # Univariate model with two uncorrelated shocks
    mod = KalmanFilter(k_endog=1, k_states=2)
    mod['design', 0, 0:2] = 1.
    mod['transition', :, :] = np.eye(2)
    mod['selection', :, :] = np.eye(2)
    mod['state_cov', :, :] = np.eye(2)

    desired = np.ones((11, 1))

    actual = mod.impulse_responses(steps=10, impulse=0)
    assert_allclose(actual, desired)

    actual = mod.impulse_responses(steps=10, impulse=1)
    assert_allclose(actual, desired)

    # In this case (with sigma=sigma^2=1), orthogonalized is the same as not
    actual = mod.impulse_responses(steps=10, impulse=0, orthogonalized=True)
    assert_allclose(actual, desired)

    # Univariate model with two correlated shocks
    mod = KalmanFilter(k_endog=1, k_states=2)
    mod['design', 0, 0:2] = 1.
    mod['transition', :, :] = np.eye(2)
    mod['selection', :, :] = np.eye(2)
    mod['state_cov', :, :] = np.array([[1, 0.5], [0.5, 1.25]])

    desired = np.ones((11, 1))

    # Non-orthogonalized (i.e. 1-unit) impulses still just generate 1's
    actual = mod.impulse_responses(steps=10, impulse=0)
    assert_allclose(actual, desired)

    actual = mod.impulse_responses(steps=10, impulse=1)
    assert_allclose(actual, desired)

    # Orthogonalized (i.e. 1-std-dev) impulses now generate different responses
    actual = mod.impulse_responses(steps=10, impulse=0, orthogonalized=True)
    assert_allclose(actual, desired + desired * 0.5)

    actual = mod.impulse_responses(steps=10, impulse=1, orthogonalized=True)
    assert_allclose(actual, desired)

    # Multivariate model with two correlated shocks
    mod = KalmanFilter(k_endog=2, k_states=2)
    mod['design', :, :] = np.eye(2)
    mod['transition', :, :] = np.eye(2)
    mod['selection', :, :] = np.eye(2)
    mod['state_cov', :, :] = np.array([[1, 0.5], [0.5, 1.25]])

    ones = np.ones((11, 1))
    zeros = np.zeros((11, 1))

    # Non-orthogonalized (i.e. 1-unit) impulses still just generate 1's, but
    # only for the appropriate series
    actual = mod.impulse_responses(steps=10, impulse=0)
    assert_allclose(actual, np.c_[ones, zeros])

    actual = mod.impulse_responses(steps=10, impulse=1)
    assert_allclose(actual, np.c_[zeros, ones])

    # Orthogonalized (i.e. 1-std-dev) impulses now generate different
    # responses, and only for the appropriate series
    actual = mod.impulse_responses(steps=10, impulse=0, orthogonalized=True)
    assert_allclose(actual, np.c_[ones, ones * 0.5])

    actual = mod.impulse_responses(steps=10, impulse=1, orthogonalized=True)
    assert_allclose(actual, np.c_[zeros, ones])

    # AR(1) model generates a geometrically declining series
    mod = sarimax.SARIMAX([0.1, 0.5, -0.2], order=(1, 0, 0))
    phi = 0.5
    mod.update([phi, 1])

    desired = np.cumprod(np.r_[1, [phi] * 10])[:, np.newaxis]

    # Test going through the model directly
    actual = mod.ssm.impulse_responses(steps=10)
    assert_allclose(actual, desired)

    # Test going through the results object
    res = mod.filter([phi, 1.])
    actual = res.impulse_responses(steps=10)
    assert_allclose(actual, desired)
Ejemplo n.º 4
0
def test_missing():
    # Datasets
    endog = np.arange(10).reshape(10, 1)
    endog_pre_na = np.ascontiguousarray(np.c_[endog.copy() * np.nan,
                                              endog.copy() * np.nan, endog,
                                              endog])
    endog_post_na = np.ascontiguousarray(np.c_[endog, endog,
                                               endog.copy() * np.nan,
                                               endog.copy() * np.nan])
    endog_inject_na = np.ascontiguousarray(np.c_[endog,
                                                 endog.copy() * np.nan, endog,
                                                 endog.copy() * np.nan])

    # Base model
    mod = KalmanFilter(np.ascontiguousarray(np.c_[endog, endog]),
                       k_states=1,
                       initialization='approximate_diffuse')
    mod['design', :, :] = 1
    mod['obs_cov', :, :] = np.eye(mod.k_endog) * 0.5
    mod['transition', :, :] = 0.5
    mod['selection', :, :] = 1
    mod['state_cov', :, :] = 0.5
    llf = mod.loglikeobs()

    # Model with prepended nans
    mod = KalmanFilter(endog_pre_na,
                       k_states=1,
                       initialization='approximate_diffuse')
    mod['design', :, :] = 1
    mod['obs_cov', :, :] = np.eye(mod.k_endog) * 0.5
    mod['transition', :, :] = 0.5
    mod['selection', :, :] = 1
    mod['state_cov', :, :] = 0.5
    llf_pre_na = mod.loglikeobs()

    assert_allclose(llf_pre_na, llf)

    # Model with appended nans
    mod = KalmanFilter(endog_post_na,
                       k_states=1,
                       initialization='approximate_diffuse')
    mod['design', :, :] = 1
    mod['obs_cov', :, :] = np.eye(mod.k_endog) * 0.5
    mod['transition', :, :] = 0.5
    mod['selection', :, :] = 1
    mod['state_cov', :, :] = 0.5
    llf_post_na = mod.loglikeobs()

    assert_allclose(llf_post_na, llf)

    # Model with injected nans
    mod = KalmanFilter(endog_inject_na,
                       k_states=1,
                       initialization='approximate_diffuse')
    mod['design', :, :] = 1
    mod['obs_cov', :, :] = np.eye(mod.k_endog) * 0.5
    mod['transition', :, :] = 0.5
    mod['selection', :, :] = 1
    mod['state_cov', :, :] = 0.5
    llf_inject_na = mod.loglikeobs()

    assert_allclose(llf_inject_na, llf)
Ejemplo n.º 5
0
def test_simulate():
    # Test for simulation of new time-series
    from scipy.signal import lfilter

    # Common parameters
    nsimulations = 10
    sigma2 = 2
    measurement_shocks = np.zeros(nsimulations)
    state_shocks = np.random.normal(scale=sigma2**0.5, size=nsimulations)

    # Random walk model, so simulated series is just the cumulative sum of
    # the shocks
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.

    actual = mod.simulate(
        nsimulations, measurement_shocks=measurement_shocks,
        state_shocks=state_shocks)[0].squeeze()
    desired = np.r_[0, np.cumsum(state_shocks)[:-1]]

    assert_allclose(actual, desired)

    # Local level model, so simulated series is just the cumulative sum of
    # the shocks plus the measurement shock
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.

    actual = mod.simulate(
        nsimulations, measurement_shocks=np.ones(nsimulations),
        state_shocks=state_shocks)[0].squeeze()
    desired = np.r_[1, np.cumsum(state_shocks)[:-1] + 1]

    assert_allclose(actual, desired)

    # Local level-like model with observation and state intercepts, so
    # simulated series is just the cumulative sum of the shocks minus the state
    # intercept, plus the observation intercept and the measurement shock
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['obs_intercept', 0, 0] = 5.
    mod['design', 0, 0] = 1.
    mod['state_intercept', 0, 0] = -2.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.

    actual = mod.simulate(
        nsimulations, measurement_shocks=np.ones(nsimulations),
        state_shocks=state_shocks)[0].squeeze()
    desired = np.r_[1 + 5, np.cumsum(state_shocks - 2)[:-1] + 1 + 5]

    assert_allclose(actual, desired)

    # Model with time-varying observation intercept
    mod = KalmanFilter(k_endog=1, k_states=1, nobs=10)
    mod['obs_intercept'] = (np.arange(10)*1.).reshape(1, 10)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.

    actual = mod.simulate(
        nsimulations, measurement_shocks=measurement_shocks,
        state_shocks=state_shocks)[0].squeeze()
    desired = np.r_[0, np.cumsum(state_shocks)[:-1] + np.arange(1,10)]

    assert_allclose(actual, desired)

    # Model with time-varying observation intercept, check that error is raised
    # if more simulations are requested than are nobs.
    mod = KalmanFilter(k_endog=1, k_states=1, nobs=10)
    mod['obs_intercept'] = (np.arange(10)*1.).reshape(1, 10)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    assert_raises(ValueError, mod.simulate, nsimulations+1, measurement_shocks,
                  state_shocks)

    # ARMA(1,1): phi = [0.1], theta = [0.5], sigma^2 = 2
    phi = np.r_[0.1]
    theta = np.r_[0.5]
    mod = sarimax.SARIMAX([0], order=(1,0,1))
    mod.update(np.r_[phi, theta, sigma2])

    actual = mod.ssm.simulate(
        nsimulations, measurement_shocks=measurement_shocks,
        state_shocks=state_shocks)[0].squeeze()
    desired = lfilter([1, theta], [1, -phi], np.r_[0, state_shocks[:-1]])

    assert_allclose(actual, desired)

    # SARIMAX(1,0,1)x(1,0,1,4), this time using the results object call
    mod = sarimax.SARIMAX([0.1, 0.5, -0.2], order=(1,0,1),
                          seasonal_order=(1,0,1,4))
    res = mod.filter([0.1, 0.5, 0.2, -0.3, 1])

    actual = res.simulate(
        nsimulations, measurement_shocks=measurement_shocks,
        state_shocks=state_shocks).squeeze()
    desired = lfilter(
        res.polynomial_reduced_ma, res.polynomial_reduced_ar,
        np.r_[0, state_shocks[:-1]])

    assert_allclose(actual, desired)
Ejemplo n.º 6
0
def test_impulse_responses():
    # Test for impulse response functions

    # Random walk: 1-unit impulse response (i.e. non-orthogonalized irf) is 1
    # for all periods
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    mod['state_cov', 0, 0] = 2.

    actual = mod.impulse_responses(steps=10)
    desired = np.ones((11, 1))

    assert_allclose(actual, desired)

    # Random walk: 1-standard-deviation response (i.e. orthogonalized irf) is
    # sigma for all periods (here sigma^2 = 2)
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    mod['state_cov', 0, 0] = 2.

    actual = mod.impulse_responses(steps=10, orthogonalized=True)
    desired = np.ones((11, 1)) * 2**0.5

    assert_allclose(actual, desired)

    # Random walk: 1-standard-deviation cumulative response (i.e. cumulative
    # orthogonalized irf)
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['design', 0, 0] = 1.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    mod['state_cov', 0, 0] = 2.

    actual = mod.impulse_responses(steps=10, orthogonalized=True,
                                   cumulative=True)
    desired = np.cumsum(np.ones((11, 1)) * 2**0.5)[:, np.newaxis]

    assert_allclose(actual, desired)

    # Random walk: 1-unit impulse response (i.e. non-orthogonalized irf) is 1
    # for all periods, even when intercepts are present
    mod = KalmanFilter(k_endog=1, k_states=1)
    mod['state_intercept', 0] = 100.
    mod['design', 0, 0] = 1.
    mod['obs_intercept', 0] = -1000.
    mod['transition', 0, 0] = 1.
    mod['selection', 0, 0] = 1.
    mod['state_cov', 0, 0] = 2.

    actual = mod.impulse_responses(steps=10)
    desired = np.ones((11, 1))

    assert_allclose(actual, desired)

    # Univariate model (random walk): test that an error is thrown when
    # a multivariate or empty "impulse" is sent
    mod = KalmanFilter(k_endog=1, k_states=1)
    assert_raises(ValueError, mod.impulse_responses, impulse=1)
    assert_raises(ValueError, mod.impulse_responses, impulse=[1,1])
    assert_raises(ValueError, mod.impulse_responses, impulse=[])

    # Univariate model with two uncorrelated shocks
    mod = KalmanFilter(k_endog=1, k_states=2)
    mod['design', 0, 0:2] = 1.
    mod['transition', :, :] = np.eye(2)
    mod['selection', :, :] = np.eye(2)
    mod['state_cov', :, :] = np.eye(2)

    desired = np.ones((11, 1))

    actual = mod.impulse_responses(steps=10, impulse=0)
    assert_allclose(actual, desired)

    actual = mod.impulse_responses(steps=10, impulse=1)
    assert_allclose(actual, desired)

    # In this case (with sigma=sigma^2=1), orthogonalized is the same as not
    actual = mod.impulse_responses(steps=10, impulse=0, orthogonalized=True)
    assert_allclose(actual, desired)

    # Univariate model with two correlated shocks
    mod = KalmanFilter(k_endog=1, k_states=2)
    mod['design', 0, 0:2] = 1.
    mod['transition', :, :] = np.eye(2)
    mod['selection', :, :] = np.eye(2)
    mod['state_cov', :, :] = np.array([[1, 0.5], [0.5, 1.25]])

    desired = np.ones((11, 1))

    # Non-orthogonalized (i.e. 1-unit) impulses still just generate 1's
    actual = mod.impulse_responses(steps=10, impulse=0)
    assert_allclose(actual, desired)

    actual = mod.impulse_responses(steps=10, impulse=1)
    assert_allclose(actual, desired)

    # Orthogonalized (i.e. 1-std-dev) impulses now generate different responses
    actual = mod.impulse_responses(steps=10, impulse=0, orthogonalized=True)
    assert_allclose(actual, desired + desired * 0.5)

    actual = mod.impulse_responses(steps=10, impulse=1, orthogonalized=True)
    assert_allclose(actual, desired)

    # Multivariate model with two correlated shocks
    mod = KalmanFilter(k_endog=2, k_states=2)
    mod['design', :, :] = np.eye(2)
    mod['transition', :, :] = np.eye(2)
    mod['selection', :, :] = np.eye(2)
    mod['state_cov', :, :] = np.array([[1, 0.5], [0.5, 1.25]])

    ones = np.ones((11, 1))
    zeros = np.zeros((11, 1))

    # Non-orthogonalized (i.e. 1-unit) impulses still just generate 1's, but
    # only for the appropriate series
    actual = mod.impulse_responses(steps=10, impulse=0)
    assert_allclose(actual, np.c_[ones, zeros])

    actual = mod.impulse_responses(steps=10, impulse=1)
    assert_allclose(actual, np.c_[zeros, ones])

    # Orthogonalized (i.e. 1-std-dev) impulses now generate different
    # responses, and only for the appropriate series
    actual = mod.impulse_responses(steps=10, impulse=0, orthogonalized=True)
    assert_allclose(actual, np.c_[ones, ones * 0.5])

    actual = mod.impulse_responses(steps=10, impulse=1, orthogonalized=True)
    assert_allclose(actual, np.c_[zeros, ones])

    # AR(1) model generates a geometrically declining series
    mod = sarimax.SARIMAX([0.1, 0.5, -0.2], order=(1,0,0))
    phi = 0.5
    mod.update([phi, 1])

    desired = np.cumprod(np.r_[1, [phi]*10])[:, np.newaxis]
    
    # Test going through the model directly
    actual = mod.ssm.impulse_responses(steps=10)
    assert_allclose(actual, desired)

    # Test going through the results object
    res = mod.filter([phi, 1.])
    actual = res.impulse_responses(steps=10)
    assert_allclose(actual, desired)