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 = 0.1
theta = 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,
initial_state=np.zeros(
mod.k_states))[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,
initial_state=np.zeros(mod.k_states))
desired = lfilter(res.polynomial_reduced_ma, res.polynomial_reduced_ar,
np.r_[0, state_shocks[:-1]])
assert_allclose(actual, desired)
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: 2-unit impulse response (i.e. non-orthogonalized irf) is 2
# 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, impulse=[2])
desired = np.ones((11, 1)) * 2
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]
actual = mod.impulse_responses(steps=10,
impulse=[1],
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, 0])
assert_allclose(actual, desired)
actual = mod.impulse_responses(steps=10, impulse=1)
assert_allclose(actual, desired)
actual = mod.impulse_responses(steps=10, impulse=[0, 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)
actual = mod.impulse_responses(steps=10,
impulse=[1, 0],
orthogonalized=True)
assert_allclose(actual, desired)
actual = mod.impulse_responses(steps=10,
impulse=[0, 1],
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])
# Test going through the model directly
actual = mod.ssm.impulse_responses(steps=10)
assert_allclose(actual[:, 0], desired)
# Test going through the results object
res = mod.filter([phi, 1.])
actual = res.impulse_responses(steps=10)
assert_allclose(actual, desired)
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: 2-unit impulse response (i.e. non-orthogonalized irf) is 2
# 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, impulse=[2])
desired = np.ones((11, 1)) * 2
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]
actual = mod.impulse_responses(steps=10, impulse=[1], 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=[1,0])
assert_allclose(actual, desired)
actual = mod.impulse_responses(steps=10, impulse=1)
assert_allclose(actual, desired)
actual = mod.impulse_responses(steps=10, impulse=[0,1])
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)
actual = mod.impulse_responses(steps=10, impulse=[1,0], orthogonalized=True)
assert_allclose(actual, desired)
actual = mod.impulse_responses(steps=10, impulse=[0,1], 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])
# Test going through the model directly
actual = mod.ssm.impulse_responses(steps=10)
assert_allclose(actual[:, 0], desired)
# Test going through the results object
res = mod.filter([phi, 1.])
actual = res.impulse_responses(steps=10)
assert_allclose(actual, desired)
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 = 0.1
theta = 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,
initial_state=np.zeros(mod.k_states))[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, initial_state=np.zeros(mod.k_states))
desired = lfilter(
res.polynomial_reduced_ma, res.polynomial_reduced_ar,
np.r_[0, state_shocks[:-1]])
assert_allclose(actual, desired)
# 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)
