def _var_acf(coefs, sig_u):
"""
Compute autocovariance function ACF_y(h) for h=1,...,p
Notes
-----
Lutkepohl (2005) p.29
"""
p, k, k2 = coefs.shape
assert(k == k2)
A = util.comp_matrix(coefs)
# construct VAR(1) noise covariance
SigU = np.zeros((k*p, k*p))
SigU[:k,:k] = sig_u
# vec(ACF) = (I_(kp)^2 - kron(A, A))^-1 vec(Sigma_U)
vecACF = L.solve(np.eye((k*p)**2) - np.kron(A, A), vec(SigU))
acf = unvec(vecACF)
acf = acf[:k].T.reshape((p, k, k))
return acf
def _var_acf(coefs, sig_u):
"""
Compute autocovariance function ACF_y(h) for h=1,...,p
Notes
-----
Lutkepohl (2005) p.29
"""
p, k, k2 = coefs.shape
assert (k == k2)
A = util.comp_matrix(coefs)
# construct VAR(1) noise covariance
SigU = np.zeros((k * p, k * p))
SigU[:k, :k] = sig_u
# vec(ACF) = (I_(kp)^2 - kron(A, A))^-1 vec(Sigma_U)
vecACF = L.solve(np.eye((k * p)**2) - np.kron(A, A), vec(SigU))
acf = unvec(vecACF)
acf = acf[:k].T.reshape((p, k, k))
return acf
def test_causality(self, equation, variables, kind='f', signif=0.05,
verbose=True):
"""Compute test statistic for null hypothesis of Granger-noncausality,
general function to test joint Granger-causality of multiple variables
Parameters
----------
equation : string or int
Equation to test for causality
variables : sequence (of strings or ints)
List, tuple, etc. of variables to test for Granger-causality
kind : {'f', 'wald'}
Perform F-test or Wald (chi-sq) test
signif : float, default 5%
Significance level for computing critical values for test,
defaulting to standard 0.95 level
Notes
-----
Null hypothesis is that there is no Granger-causality for the indicated
variables. The degrees of freedom in the F-test are based on the
number of variables in the VAR system, that is, degrees of freedom
are equal to the number of equations in the VAR times degree of freedom
of a single equation.
Returns
-------
results : dict
"""
if isinstance(variables, (basestring, int, np.integer)):
variables = [variables]
k, p = self.neqs, self.k_ar
# number of restrictions
N = len(variables) * self.k_ar
# Make restriction matrix
C = np.zeros((N, k ** 2 * p + k), dtype=float)
eq_index = self.get_eq_index(equation)
vinds = mat([self.get_eq_index(v) for v in variables])
# remember, vec is column order!
offsets = np.concatenate([k + k ** 2 * j + k * vinds + eq_index
for j in range(p)])
C[np.arange(N), offsets] = 1
# Lutkepohl 3.6.5
Cb = np.dot(C, vec(self.params.T))
middle = L.inv(chain_dot(C, self.cov_params, C.T))
# wald statistic
lam_wald = statistic = chain_dot(Cb, middle, Cb)
if kind.lower() == 'wald':
df = N
dist = stats.chi2(df)
elif kind.lower() == 'f':
statistic = lam_wald / N
df = (N, k * self.df_resid)
dist = stats.f(*df)
else:
raise Exception('kind %s not recognized' % kind)
pvalue = dist.sf(statistic)
crit_value = dist.ppf(1 - signif)
conclusion = 'fail to reject' if statistic < crit_value else 'reject'
results = {
'statistic' : statistic,
'crit_value' : crit_value,
'pvalue' : pvalue,
'df' : df,
'conclusion' : conclusion,
'signif' : signif
}
if verbose:
summ = output.causality_summary(results, variables, equation, kind)
print summ
return results
def test_vec():
arr = np.array([[1, 2],
[3, 4]])
assert(np.array_equal(vec(arr), [1, 3, 2, 4]))
def test_commutation_matrix():
m = np.random.randn(4, 3)
K = tools.commutation_matrix(4, 3)
assert(np.array_equal(vec(m.T), np.dot(K, vec(m))))
def test_elimination_matrix():
for k in range(2, 10):
m = np.random.randn(k, k)
Lk = tools.elimination_matrix(k)
assert(np.array_equal(vech(m), np.dot(Lk, vec(m))))
def test_duplication_matrix():
for k in range(2, 10):
m = tools.unvech(np.random.randn(k * (k + 1) / 2))
Dk = tools.duplication_matrix(k)
assert(np.array_equal(vec(m), np.dot(Dk, vech(m))))
def test_causality(self,
equation,
variables,
kind='f',
signif=0.05,
verbose=True):
"""Compute test statistic for null hypothesis of Granger-noncausality,
general function to test joint Granger-causality of multiple variables
Parameters
----------
equation : string or int
Equation to test for causality
variables : sequence (of strings or ints)
List, tuple, etc. of variables to test for Granger-causality
kind : {'f', 'wald'}
Perform F-test or Wald (chi-sq) test
signif : float, default 5%
Significance level for computing critical values for test,
defaulting to standard 0.95 level
Notes
-----
Null hypothesis is that there is no Granger-causality for the indicated
variables. The degrees of freedom in the F-test are based on the
number of variables in the VAR system, that is, degrees of freedom
are equal to the number of equations in the VAR times degree of freedom
of a single equation.
Returns
-------
results : dict
"""
if isinstance(variables, (basestring, int, np.integer)):
variables = [variables]
k, p = self.neqs, self.k_ar
# number of restrictions
N = len(variables) * self.k_ar
# Make restriction matrix
C = np.zeros((N, k**2 * p + k), dtype=float)
eq_index = self.get_eq_index(equation)
vinds = mat([self.get_eq_index(v) for v in variables])
# remember, vec is column order!
offsets = np.concatenate(
[k + k**2 * j + k * vinds + eq_index for j in range(p)])
C[np.arange(N), offsets] = 1
# Lutkepohl 3.6.5
Cb = np.dot(C, vec(self.params.T))
middle = L.inv(chain_dot(C, self.cov_params, C.T))
# wald statistic
lam_wald = statistic = chain_dot(Cb, middle, Cb)
if kind.lower() == 'wald':
df = N
dist = stats.chi2(df)
elif kind.lower() == 'f':
statistic = lam_wald / N
df = (N, k * self.df_resid)
dist = stats.f(*df)
else:
raise Exception('kind %s not recognized' % kind)
pvalue = dist.sf(statistic)
crit_value = dist.ppf(1 - signif)
conclusion = 'fail to reject' if statistic < crit_value else 'reject'
results = {
'statistic': statistic,
'crit_value': crit_value,
'pvalue': pvalue,
'df': df,
'conclusion': conclusion,
'signif': signif
}
if verbose:
summ = output.causality_summary(results, variables, equation, kind)
print summ
return results
