def _omega_forc_cov(self, steps):
# Approximate MSE matrix \Omega(h) as defined in Lut p97
G = self._zz
Ginv = L.inv(G)
# memoize powers of B for speedup
# TODO: see if can memoize better
B = self._bmat_forc_cov()
_B = {}
def bpow(i):
if i not in _B:
_B[i] = np.linalg.matrix_power(B, i)
return _B[i]
phis = self.ma_rep(steps)
sig_u = self.sigma_u
omegas = np.zeros((steps, self.neqs, self.neqs))
for h in range(1, steps + 1):
if h == 1:
omegas[h-1] = self.df_model * self.sigma_u
continue
om = omegas[h-1]
for i in range(h):
for j in range(h):
Bi = bpow(h - 1 - i)
Bj = bpow(h - 1 - j)
mult = np.trace(chain_dot(Bi.T, Ginv, Bj, G))
om += mult * chain_dot(phis[i], sig_u, phis[j].T)
omegas[h-1] = om
return omegas
def mse(self, steps):
"""
Compute theoretical forecast error variance matrices
Parameters
----------
steps : int
Number of steps ahead
Notes
-----
.. math:: \mathrm{MSE}(h) = \sum_{i=0}^{h-1} \Phi \Sigma_u \Phi^T
Returns
-------
forc_covs : ndarray (steps x neqs x neqs)
"""
ma_coefs = self.ma_rep(steps)
k = len(self.sigma_u)
forc_covs = np.zeros((steps, k, k))
prior = np.zeros((k, k))
for h in xrange(steps):
# Sigma(h) = Sigma(h-1) + Phi Sig_u Phi'
phi = ma_coefs[h]
var = chain_dot(phi, self.sigma_u, phi.T)
forc_covs[h] = prior = prior + var
return forc_covs
def cum_effect_cov(self, orth=False):
"""
Compute asymptotic standard errors for cumulative impulse response
coefficients
Parameters
----------
orth : boolean
Notes
-----
eq. 3.7.7 (non-orth), 3.7.10 (orth)
Returns
-------
"""
Ik = np.eye(self.neqs)
PIk = np.kron(self.P.T, Ik)
F = 0.
covs = self._empty_covm(self.periods + 1)
for i in range(self.periods + 1):
if i > 0:
F = F + self.G[i - 1]
if orth:
if i == 0:
apiece = 0
else:
Bn = np.dot(PIk, F)
apiece = chain_dot(Bn, self.cov_a, Bn.T)
Bnbar = np.dot(np.kron(Ik, self.cum_effects[i]), self.H)
bpiece = chain_dot(Bnbar, self.cov_sig, Bnbar.T) / self.T
covs[i] = apiece + bpiece
else:
if i == 0:
covs[i] = np.zeros((self.neqs**2, self.neqs**2))
continue
covs[i] = chain_dot(F, self.cov_a, F.T)
return covs
def lr_effect_cov(self, orth=False):
"""
Returns
-------
"""
lre = self.lr_effects
Finfty = np.kron(np.tile(lre.T, self.lags), lre)
Ik = np.eye(self.neqs)
if orth:
Binf = np.dot(np.kron(self.P.T, np.eye(self.neqs)), Finfty)
Binfbar = np.dot(np.kron(Ik, lre), self.H)
return (chain_dot(Binf, self.cov_a, Binf.T) +
chain_dot(Binfbar, self.cov_sig, Binfbar.T))
else:
return chain_dot(Finfty, self.cov_a, Finfty.T)
def _cov_sigma(self):
"""
Estimated covariance matrix of vech(sigma_u)
"""
D_K = tsa.duplication_matrix(self.neqs)
D_Kinv = npl.pinv(D_K)
sigxsig = np.kron(self.sigma_u, self.sigma_u)
return 2 * chain_dot(D_Kinv, sigxsig, D_Kinv.T)
def _orth_cov(self):
# Lutkepohl 3.7.8
Ik = np.eye(self.neqs)
PIk = np.kron(self.P.T, Ik)
H = self.H
covs = self._empty_covm(self.periods + 1)
for i in range(self.periods + 1):
if i == 0:
apiece = 0
else:
Ci = np.dot(PIk, self.G[i-1])
apiece = chain_dot(Ci, self.cov_a, Ci.T)
Cibar = np.dot(np.kron(Ik, self.irfs[i]), H)
bpiece = chain_dot(Cibar, self.cov_sig, Cibar.T) / self.T
# Lutkepohl typo, cov_sig correct
covs[i] = apiece + bpiece
return covs
def H(self):
k = self.neqs
Lk = tsa.elimination_matrix(k)
Kkk = tsa.commutation_matrix(k, k)
Ik = np.eye(k)
# B = chain_dot(Lk, np.eye(k**2) + commutation_matrix(k, k),
# np.kron(self.P, np.eye(k)), Lk.T)
# return np.dot(Lk.T, L.inv(B))
B = chain_dot(Lk,
np.dot(np.kron(Ik, self.P), Kkk) + np.kron(self.P, Ik),
Lk.T)
return np.dot(Lk.T, L.inv(B))
def cov_ybar(self):
r"""Asymptotically consistent estimate of covariance of the sample mean
.. math::
\sqrt(T) (\bar{y} - \mu) \rightarrow {\cal N}(0, \Sigma_{\bar{y}})\\
\Sigma_{\bar{y}} = B \Sigma_u B^\prime, \text{where } B = (I_K - A_1
- \cdots - A_p)^{-1}
Notes
-----
Lutkepohl Proposition 3.3
"""
Ainv = L.inv(np.eye(self.neqs) - self.coefs.sum(0))
return chain_dot(Ainv, self.sigma_u, Ainv.T)
def cov(self, orth=False):
"""
Compute asymptotic standard errors for impulse response coefficients
Notes
-----
Lutkepohl eq 3.7.5
Returns
-------
"""
if orth:
return self._orth_cov()
covs = self._empty_covm(self.periods + 1)
covs[0] = np.zeros((self.neqs ** 2, self.neqs ** 2))
for i in range(1, self.periods + 1):
Gi = self.G[i - 1]
covs[i] = chain_dot(Gi, self.cov_a, Gi.T)
return covs
def cov(self, orth=False):
"""
Compute asymptotic standard errors for impulse response coefficients
Notes
-----
Lütkepohl eq 3.7.5
Returns
-------
"""
if orth:
return self._orth_cov()
covs = self._empty_covm(self.periods + 1)
covs[0] = np.zeros((self.neqs**2, self.neqs**2))
for i in range(1, self.periods + 1):
Gi = self.G[i - 1]
covs[i] = chain_dot(Gi, self.cov_a, Gi.T)
return covs
def forecast_cov(ma_coefs, sig_u, steps):
"""
Compute theoretical forecast error variance matrices
Parameters
----------
Returns
-------
forc_covs : ndarray (steps x neqs x neqs)
"""
k = len(sig_u)
forc_covs = np.zeros((steps, k, k))
prior = np.zeros((k, k))
for h in xrange(steps):
# Sigma(h) = Sigma(h-1) + Phi Sig_u Phi'
phi = ma_coefs[h]
var = chain_dot(phi, sig_u, phi.T)
forc_covs[h] = prior = prior + var
return forc_covs
def swar_transform(subset, position, theta):
'''Apply to a sub-group of observations'''
n = subset.shape[0]
B = np.ones((n, n)) / n
out = subset - chain_dot(np.diag(theta[position]), B, subset)
return out
def a_test_causality(self, caused, causing=None, kind='f', signif=0.05):
if not (0 < signif < 1):
raise ValueError("signif has to be between 0 and 1")
allowed_types = (string_types, int)
if isinstance(caused, allowed_types):
caused = [caused]
if not all(isinstance(c, allowed_types) for c in caused):
raise TypeError("caused has to be of type string or int (or a "
"sequence of these types).")
caused = [self.names[c] if type(c) == int else c for c in caused]
caused_ind = [util.get_index(self.names, c) for c in caused]
if causing is not None:
if isinstance(causing, allowed_types):
causing = [causing]
if not all(isinstance(c, allowed_types) for c in causing):
raise TypeError("causing has to be of type string or int (or "
"a sequence of these types) or None.")
causing = [self.names[c] if type(c) == int else c for c in causing]
causing_ind = [util.get_index(self.names, c) for c in causing]
if causing is None:
causing_ind = [i for i in range(self.neqs) if i not in caused_ind]
causing = [self.names[c] for c in caused_ind]
k, p = self.neqs, self.k_ar
# number of restrictions
num_restr = len(causing) * len(caused) * p
num_det_terms = self.k_exog
# Make restriction matrix
C = np.zeros((num_restr, k * num_det_terms + k ** 2 * p), dtype=float)
cols_det = k * num_det_terms
row = 0
for j in range(p):
for ing_ind in causing_ind:
for ed_ind in caused_ind:
C[row, cols_det + ed_ind + k * ing_ind + k ** 2 * j] = 1
row += 1
# Lutkepohl 3.6.5
Cb = np.dot(C, vec(self.params.T))
middle = scipy.linalg.inv(chain_dot(C, self.cov_params, C.T))
# wald statistic
lam_wald = statistic = chain_dot(Cb, middle, Cb)
if kind.lower() == 'wald':
df = num_restr
dist = stats.chi2(df)
elif kind.lower() == 'f':
statistic = lam_wald / num_restr
df = (num_restr, 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)
# print(pvalue)
# print("---====--")
return pvalue, CausalityTestResults(causing, caused, statistic,
crit_value, pvalue, df, signif,
test="granger", method=kind)
def kalmanfilter(F, A, H, Q, R, y, X, xi10, ntrain, history=False):
"""
Returns the negative log-likelihood of y conditional on the information set
Assumes that the initial state and all innovations are multivariate
Gaussian.
Parameters
-----------
F : array-like
The (r x r) array holding the transition matrix for the hidden state.
A : array-like
The (nobs x k) array relating the predetermined variables to the
observed data.
H : array-like
The (nobs x r) array relating the hidden state vector to the
observed data.
Q : array-like
(r x r) variance/covariance matrix on the error term in the hidden
state transition.
R : array-like
(nobs x nobs) variance/covariance of the noise in the observation
equation.
y : array-like
The (nobs x 1) array holding the observed data.
X : array-like
The (nobs x k) array holding the predetermined variables data.
xi10 : array-like
Is the (r x 1) initial prior on the initial state vector.
ntrain : int
The number of training periods for the filter. This is the number of
observations that do not affect the likelihood.
Returns
-------
likelihood
The negative of the log likelihood
history or priors, history of posterior
If history is True.
Notes
-----
No input checking is done.
"""
# uses log of Hamilton 13.4.1
F = np.asarray(F)
H = np.atleast_2d(np.asarray(H))
n = H.shape[1] # remember that H gets transposed
y = np.asarray(y)
A = np.asarray(A)
X = np.asarray(X)
if y.ndim == 1: # note that Y is in rows for now
y = y[:, None]
nobs = y.shape[0]
xi10 = np.atleast_2d(np.asarray(xi10))
# if xi10.ndim == 1:
# xi10[:,None]
if history:
state_vector = [xi10]
Q = np.asarray(Q)
r = xi10.shape[0]
# Eq. 12.2.21, other version says P0 = Q
# p10 = np.dot(np.linalg.inv(np.eye(r**2)-np.kron(F,F)),Q.ravel('F'))
# p10 = np.reshape(P0, (r,r), order='F')
# Assume a fixed, known intial point and set P0 = Q
#TODO: this looks *slightly * different than Durbin-Koopman exact likelihood
# initialization p 112 unless I've misunderstood the notational translation.
p10 = Q
loglikelihood = 0
for i in range(nobs):
HTPHR = np.atleast_1d(np.squeeze(chain_dot(H.T, p10, H) + R))
# print HTPHR
# print HTPHR.ndim
# print HTPHR.shape
if HTPHR.ndim == 1:
HTPHRinv = 1. / HTPHR
else:
HTPHRinv = np.linalg.inv(HTPHR) # correct
# print A.T
# print X
# print H.T
# print xi10
# print y[i]
part1 = y[i] - np.dot(A.T, X) - np.dot(H.T, xi10) # correct
if i >= ntrain: # zero-index, but ntrain isn't
HTPHRdet = np.linalg.det(np.atleast_2d(HTPHR)) # correct
part2 = -.5 * chain_dot(part1.T, HTPHRinv, part1) # correct
#TODO: Need to test with ill-conditioned problem.
loglike_interm = (-n/2.) * np.log(2*np.pi) - .5*\
np.log(HTPHRdet) + part2
loglikelihood += loglike_interm
# 13.2.15 Update current state xi_t based on y
xi11 = xi10 + chain_dot(p10, H, HTPHRinv, part1)
# 13.2.16 MSE of that state
p11 = p10 - chain_dot(p10, H, HTPHRinv, H.T, p10)
# 13.2.17 Update forecast about xi_{t+1} based on our F
xi10 = np.dot(F, xi11)
if history:
state_vector.append(xi10)
# 13.2.21 Update the MSE of the forecast
p10 = chain_dot(F, p11, F.T) + Q
if not history:
return -loglikelihood
else:
return -loglikelihood, np.asarray(state_vector[:-1])
def kalmanfilter(F, A, H, Q, R, y, X, xi10, ntrain, history=False):
"""
Returns the negative log-likelihood of y conditional on the information set
Assumes that the initial state and all innovations are multivariate
Gaussian.
Parameters
-----------
F : array-like
The (r x r) array holding the transition matrix for the hidden state.
A : array-like
The (nobs x k) array relating the predetermined variables to the
observed data.
H : array-like
The (nobs x r) array relating the hidden state vector to the
observed data.
Q : array-like
(r x r) variance/covariance matrix on the error term in the hidden
state transition.
R : array-like
(nobs x nobs) variance/covariance of the noise in the observation
equation.
y : array-like
The (nobs x 1) array holding the observed data.
X : array-like
The (nobs x k) array holding the predetermined variables data.
xi10 : array-like
Is the (r x 1) initial prior on the initial state vector.
ntrain : int
The number of training periods for the filter. This is the number of
observations that do not affect the likelihood.
Returns
-------
likelihood
The negative of the log likelihood
history or priors, history of posterior
If history is True.
Notes
-----
No input checking is done.
"""
# uses log of Hamilton 13.4.1
F = np.asarray(F)
H = np.atleast_2d(np.asarray(H))
n = H.shape[1] # remember that H gets transposed
y = np.asarray(y)
A = np.asarray(A)
X = np.asarray(X)
if y.ndim == 1: # note that Y is in rows for now
y = y[:,None]
nobs = y.shape[0]
xi10 = np.atleast_2d(np.asarray(xi10))
# if xi10.ndim == 1:
# xi10[:,None]
if history:
state_vector = [xi10]
Q = np.asarray(Q)
r = xi10.shape[0]
# Eq. 12.2.21, other version says P0 = Q
# p10 = np.dot(np.linalg.inv(np.eye(r**2)-np.kron(F,F)),Q.ravel('F'))
# p10 = np.reshape(P0, (r,r), order='F')
# Assume a fixed, known intial point and set P0 = Q
#TODO: this looks *slightly * different than Durbin-Koopman exact likelihood
# initialization p 112 unless I've misunderstood the notational translation.
p10 = Q
loglikelihood = 0
for i in range(nobs):
HTPHR = np.atleast_1d(np.squeeze(chain_dot(H.T,p10,H)+R))
# print HTPHR
# print HTPHR.ndim
# print HTPHR.shape
if HTPHR.ndim == 1:
HTPHRinv = 1./HTPHR
else:
HTPHRinv = np.linalg.inv(HTPHR) # correct
# print A.T
# print X
# print H.T
# print xi10
# print y[i]
part1 = y[i] - np.dot(A.T,X) - np.dot(H.T,xi10) # correct
if i >= ntrain: # zero-index, but ntrain isn't
HTPHRdet = np.linalg.det(np.atleast_2d(HTPHR)) # correct
part2 = -.5*chain_dot(part1.T,HTPHRinv,part1) # correct
#TODO: Need to test with ill-conditioned problem.
loglike_interm = (-n/2.) * np.log(2*np.pi) - .5*\
np.log(HTPHRdet) + part2
loglikelihood += loglike_interm
# 13.2.15 Update current state xi_t based on y
xi11 = xi10 + chain_dot(p10, H, HTPHRinv, part1)
# 13.2.16 MSE of that state
p11 = p10 - chain_dot(p10, H, HTPHRinv, H.T, p10)
# 13.2.17 Update forecast about xi_{t+1} based on our F
xi10 = np.dot(F,xi11)
if history:
state_vector.append(xi10)
# 13.2.21 Update the MSE of the forecast
p10 = chain_dot(F,p11,F.T) + Q
if not history:
return -loglikelihood
else:
return -loglikelihood, np.asarray(state_vector[:-1])
def fit(self, q=.5, vcov='robust', kernel='epa', bandwidth='hsheather',
max_iter=1000, p_tol=1e-6, **kwargs):
'''Solve by Iterative Weighted Least Squares
Parameters
----------
q : float
Quantile must be between 0 and 1
vcov : string, method used to calculate the variance-covariance matrix
of the parameters. Default is ``robust``:
- robust : heteroskedasticity robust standard errors (as suggested
in Greene 6th edition)
- iid : iid errors (as in Stata 12)
kernel : string, kernel to use in the kernel density estimation for the
asymptotic covariance matrix:
- epa: Epanechnikov
- cos: Cosine
- gau: Gaussian
- par: Parzene
bandwidth: string, Bandwidth selection method in kernel density
estimation for asymptotic covariance estimate (full
references in QuantReg docstring):
- hsheather: Hall-Sheather (1988)
- bofinger: Bofinger (1975)
- chamberlain: Chamberlain (1994)
'''
if q < 0 or q > 1:
raise Exception('p must be between 0 and 1')
kern_names = ['biw', 'cos', 'epa', 'gau', 'par']
if kernel not in kern_names:
raise Exception("kernel must be one of " + ', '.join(kern_names))
else:
kernel = kernels[kernel]
if bandwidth == 'hsheather':
bandwidth = hall_sheather
elif bandwidth == 'bofinger':
bandwidth = bofinger
elif bandwidth == 'chamberlain':
bandwidth = chamberlain
else:
raise Exception("bandwidth must be in 'hsheather', 'bofinger', 'chamberlain'")
endog = self.endog
exog = self.exog
nobs = self.nobs
exog_rank = np_matrix_rank(self.exog)
self.rank = exog_rank
self.df_model = float(self.rank - self.k_constant)
self.df_resid = self.nobs - self.rank
n_iter = 0
xstar = exog
beta = np.ones(exog_rank)
# TODO: better start, initial beta is used only for convergence check
# Note the following doesn't work yet,
# the iteration loop always starts with OLS as initial beta
# if start_params is not None:
# if len(start_params) != rank:
# raise ValueError('start_params has wrong length')
# beta = start_params
# else:
# # start with OLS
# beta = np.dot(np.linalg.pinv(exog), endog)
diff = 10
cycle = False
history = dict(params = [], mse=[])
while n_iter < max_iter and diff > p_tol and not cycle:
n_iter += 1
beta0 = beta
xtx = np.dot(xstar.T, exog)
xty = np.dot(xstar.T, endog)
beta = np.dot(pinv(xtx), xty)
resid = endog - np.dot(exog, beta)
mask = np.abs(resid) < .000001
resid[mask] = ((resid[mask] >= 0) * 2 - 1) * .000001
resid = np.where(resid < 0, q * resid, (1-q) * resid)
resid = np.abs(resid)
xstar = exog / resid[:, np.newaxis]
diff = np.max(np.abs(beta - beta0))
history['params'].append(beta)
history['mse'].append(np.mean(resid*resid))
if (n_iter >= 300) and (n_iter % 100 == 0):
# check for convergence circle, shouldn't happen
for ii in range(2, 10):
if np.all(beta == history['params'][-ii]):
cycle = True
break
warnings.warn("Convergence cycle detected", ConvergenceWarning)
if n_iter == max_iter:
warnings.warn("Maximum number of iterations (1000) reached.",
IterationLimitWarning)
e = endog - np.dot(exog, beta)
# Greene (2008, p.407) writes that Stata 6 uses this bandwidth:
# h = 0.9 * np.std(e) / (nobs**0.2)
# Instead, we calculate bandwidth as in Stata 12
iqre = stats.scoreatpercentile(e, 75) - stats.scoreatpercentile(e, 25)
h = bandwidth(nobs, q)
h = min(np.std(endog),
iqre / 1.34) * (norm.ppf(q + h) - norm.ppf(q - h))
fhat0 = 1. / (nobs * h) * np.sum(kernel(e / h))
if vcov == 'robust':
d = np.where(e > 0, (q/fhat0)**2, ((1-q)/fhat0)**2)
xtxi = pinv(np.dot(exog.T, exog))
xtdx = np.dot(exog.T * d[np.newaxis, :], exog)
vcov = chain_dot(xtxi, xtdx, xtxi)
elif vcov == 'iid':
vcov = (1. / fhat0)**2 * q * (1 - q) * pinv(np.dot(exog.T, exog))
else:
raise Exception("vcov must be 'robust' or 'iid'")
lfit = QuantRegResults(self, beta, normalized_cov_params=vcov)
lfit.q = q
lfit.iterations = n_iter
lfit.sparsity = 1. / fhat0
lfit.bandwidth = h
lfit.history = history
return RegressionResultsWrapper(lfit)
def test_chain_dot():
A = np.arange(1, 13).reshape(3, 4)
B = np.arange(3, 15).reshape(4, 3)
C = np.arange(5, 8).reshape(3, 1)
assert_equal(tools.chain_dot(A, B, C), np.array([[1820], [4300], [6780]]))
def fit(self, q=.5, vcov='robust', kernel='epa', bandwidth='hsheather',
max_iter=1000, p_tol=1e-6, **kwargs):
"""
Solve by Iterative Weighted Least Squares
Parameters
----------
q : float
Quantile must be between 0 and 1
vcov : str, method used to calculate the variance-covariance matrix
of the parameters. Default is ``robust``:
- robust : heteroskedasticity robust standard errors (as suggested
in Greene 6th edition)
- iid : iid errors (as in Stata 12)
kernel : str, kernel to use in the kernel density estimation for the
asymptotic covariance matrix:
- epa: Epanechnikov
- cos: Cosine
- gau: Gaussian
- par: Parzene
bandwidth : str, Bandwidth selection method in kernel density
estimation for asymptotic covariance estimate (full
references in QuantReg docstring):
- hsheather: Hall-Sheather (1988)
- bofinger: Bofinger (1975)
- chamberlain: Chamberlain (1994)
"""
if q < 0 or q > 1:
raise Exception('p must be between 0 and 1')
kern_names = ['biw', 'cos', 'epa', 'gau', 'par']
if kernel not in kern_names:
raise Exception("kernel must be one of " + ', '.join(kern_names))
else:
kernel = kernels[kernel]
if bandwidth == 'hsheather':
bandwidth = hall_sheather
elif bandwidth == 'bofinger':
bandwidth = bofinger
elif bandwidth == 'chamberlain':
bandwidth = chamberlain
else:
raise Exception("bandwidth must be in 'hsheather', 'bofinger', 'chamberlain'")
endog = self.endog
exog = self.exog
nobs = self.nobs
exog_rank = np.linalg.matrix_rank(self.exog)
self.rank = exog_rank
self.df_model = float(self.rank - self.k_constant)
self.df_resid = self.nobs - self.rank
n_iter = 0
xstar = exog
beta = np.ones(exog_rank)
# TODO: better start, initial beta is used only for convergence check
# Note the following does not work yet,
# the iteration loop always starts with OLS as initial beta
# if start_params is not None:
# if len(start_params) != rank:
# raise ValueError('start_params has wrong length')
# beta = start_params
# else:
# # start with OLS
# beta = np.dot(np.linalg.pinv(exog), endog)
diff = 10
cycle = False
history = dict(params = [], mse=[])
while n_iter < max_iter and diff > p_tol and not cycle:
n_iter += 1
beta0 = beta
xtx = np.dot(xstar.T, exog)
xty = np.dot(xstar.T, endog)
beta = np.dot(pinv(xtx), xty)
resid = endog - np.dot(exog, beta)
mask = np.abs(resid) < .000001
resid[mask] = ((resid[mask] >= 0) * 2 - 1) * .000001
resid = np.where(resid < 0, q * resid, (1-q) * resid)
resid = np.abs(resid)
xstar = exog / resid[:, np.newaxis]
diff = np.max(np.abs(beta - beta0))
history['params'].append(beta)
history['mse'].append(np.mean(resid*resid))
if (n_iter >= 300) and (n_iter % 100 == 0):
# check for convergence circle, should not happen
for ii in range(2, 10):
if np.all(beta == history['params'][-ii]):
cycle = True
warnings.warn("Convergence cycle detected", ConvergenceWarning)
break
if n_iter == max_iter:
warnings.warn("Maximum number of iterations (" + str(max_iter) +
") reached.", IterationLimitWarning)
e = endog - np.dot(exog, beta)
# Greene (2008, p.407) writes that Stata 6 uses this bandwidth:
# h = 0.9 * np.std(e) / (nobs**0.2)
# Instead, we calculate bandwidth as in Stata 12
iqre = stats.scoreatpercentile(e, 75) - stats.scoreatpercentile(e, 25)
h = bandwidth(nobs, q)
h = min(np.std(endog),
iqre / 1.34) * (norm.ppf(q + h) - norm.ppf(q - h))
fhat0 = 1. / (nobs * h) * np.sum(kernel(e / h))
if vcov == 'robust':
d = np.where(e > 0, (q/fhat0)**2, ((1-q)/fhat0)**2)
xtxi = pinv(np.dot(exog.T, exog))
xtdx = np.dot(exog.T * d[np.newaxis, :], exog)
vcov = chain_dot(xtxi, xtdx, xtxi)
elif vcov == 'iid':
vcov = (1. / fhat0)**2 * q * (1 - q) * pinv(np.dot(exog.T, exog))
else:
raise Exception("vcov must be 'robust' or 'iid'")
lfit = QuantRegResults(self, beta, normalized_cov_params=vcov)
lfit.q = q
lfit.iterations = n_iter
lfit.sparsity = 1. / fhat0
lfit.bandwidth = h
lfit.history = history
return RegressionResultsWrapper(lfit)
def test_chain_dot():
A = np.arange(1, 13).reshape(3, 4)
B = np.arange(3, 15).reshape(4, 3)
C = np.arange(5, 8).reshape(3, 1)
assert_equal(tools.chain_dot(A, B, C), np.array([[1820], [4300], [6780]]))
def swar_transform(subset, position, theta):
'''Apply to a sub-group of observations'''
n = subset.shape[0]
B = np.ones((n,n)) / n
out = subset - chain_dot(np.diag(theta[position]), B, subset)
return out
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