def _kernel_cov(self, z: Float64Array) -> Float64Array:
nobs = z.shape[0]
bw = self.bandwidth
kernel = self._kernel
assert kernel is not None
kernel_estimator = KERNEL_LOOKUP[kernel]
weights = kernel_estimator(bw, nobs - 1)
out = cov_kernel(z, weights)
return (out + out.T) / 2
def cov(self) -> NDArray:
"""Estimated covariance"""
e = np.asarray(self._all_params) - self._params.T
e = e[np.all(np.isfinite(e), 1)]
nobs = e.shape[0]
bw = self.bandwidth
assert self._kernel is not None
w = KERNEL_LOOKUP[self._kernel](bw, nobs - 1)
cov = cov_kernel(e, w)
return cov / (nobs - int(bool(self._debiased)))
def cov(self) -> NDArray:
"""Estimated covariance"""
x = self._x
nobs = x.shape[0]
xpxi = inv(x.T @ x / nobs)
eps = self.eps
xe = x * eps
assert self._time_ids is not None
xe = DataFrame(xe, index=self._time_ids.squeeze())
xe = xe.groupby(level=0).sum()
xe.sort_index(inplace=True)
xe_nobs = xe.shape[0]
bw = self._bandwidth
if self._bandwidth is None:
bw = float(np.floor(4 * (xe_nobs / 100)**(2 / 9)))
assert bw is not None
w = KERNEL_LOOKUP[self._kernel](bw, xe_nobs - 1)
xeex = cov_kernel(xe.values, w) * (xe_nobs / nobs)
xeex *= self._scale
out = (xpxi @ xeex @ xpxi) / nobs
return (out + out.T) / 2
def _single_cov(self, xe: NDArray, bw: float) -> NDArray:
nobs = xe.shape[0]
w = KERNEL_LOOKUP[self._kernel](bw, nobs - 1)
return cov_kernel(xe, w)
def test_linear_model_parameters(data):
mod = LinearFactorModel(data.portfolios, data.factors)
res = mod.fit()
f = mod.factors.ndarray
p = mod.portfolios.ndarray
n = f.shape[0]
moments = np.zeros(
(n, p.shape[1] * (f.shape[1] + 1) + f.shape[1] + p.shape[1]))
fc = np.c_[np.ones((n, 1)), f]
betas = lstsq(fc, p, rcond=None)[0]
eps = p - fc @ betas
loc = 0
for i in range(eps.shape[1]):
for j in range(fc.shape[1]):
moments[:, loc] = eps[:, i] * fc[:, j]
loc += 1
b = betas[1:, :].T
lam = lstsq(b, p.mean(0)[:, None], rcond=None)[0]
pricing_errors = p - (b @ lam).T
for i in range(lam.shape[0]):
lam_error = (p - (b @ lam).T) @ b[:, [i]]
moments[:, loc] = lam_error.squeeze()
loc += 1
alphas = pricing_errors.mean(0)[:, None]
moments[:, loc:] = pricing_errors - alphas.T
mod_moments = mod._moments(eps, b, alphas, pricing_errors)
assert_allclose(res.betas, b)
assert_allclose(res.risk_premia, lam.squeeze())
assert_allclose(res.alphas, alphas.squeeze())
assert_allclose(moments, mod_moments)
m = moments.shape[1]
jac = np.eye(m)
block1 = p.shape[1] * (f.shape[1] + 1)
# 1,1
jac[:block1, :block1] = np.kron(np.eye(p.shape[1]), fc.T @ fc / n)
# 2, 1
loc = 0
nport, nf = p.shape[1], f.shape[1]
block2 = block1 + nf
for i in range(nport):
block = np.zeros((nf, nf + 1))
for j in range(nf): # rows
for k in range(1, nf + 1): # cols
block[j, k] = b[i][j] * lam[k - 1]
if j + 1 == k:
block[j, k] -= alphas[i]
jac[block1:block2, loc:loc + nf + 1] = block
loc += nf + 1
# 2, 2
jac[block1:block2, block1:block2] = b.T @ b
# 3,1
block = np.zeros((nport, nport * (nf + 1)))
row = col = 0
for _ in range(nport):
for j in range(nf + 1):
if j != 0:
block[row, col] = lam[j - 1]
col += 1
row += 1
jac[-nport:, :(nport * (nf + 1))] = block
# 3, 2
jac[-nport:, (nport * (nf + 1)):(nport * (nf + 1)) + nf] = b
# 3, 3: already done since eye
mod_jac = mod._jacobian(b, lam, alphas)
assert_allclose(mod_jac[:block1], jac[:block1])
assert_allclose(mod_jac[block1:block2, :block1],
jac[block1:block2, :block1])
assert_allclose(mod_jac[block1:block2, block1:block2], jac[block1:block2,
block1:block2])
assert_allclose(mod_jac[block1:block2, block2:], jac[block1:block2,
block2:])
assert_allclose(mod_jac[block2:], jac[block2:])
s = moments.T @ moments / (n - (nf + 1))
ginv = np.linalg.inv(jac)
cov = ginv @ s @ ginv.T / n
order = np.zeros((nport, nf + 1), dtype=np.int64)
order[:, 0] = np.arange(block2, block2 + nport)
for i in range(nf):
order[:, i + 1] = (nf + 1) * np.arange(nport) + (i + 1)
order = np.r_[order.ravel(), block1:block2]
cov = cov[order][:, order]
cov = (cov + cov.T) / 2
assert_allclose(cov, res.cov)
acov = cov[:block1:(nf + 1), :block1:(nf + 1)]
jstat = float(alphas.T @ np.linalg.pinv(acov) @ alphas)
assert_allclose(res.j_statistic.stat, jstat)
assert_allclose(res.j_statistic.pval,
1 - stats.chi2(nport - nf).cdf(jstat))
get_all(res)
res = LinearFactorModel(data.portfolios,
data.factors).fit(cov_type="kernel",
debiased=False)
std_mom = moments / moments.std(0)[None, :]
mom = std_mom.sum(1)
bw = kernel_optimal_bandwidth(mom)
w = kernel_weight_bartlett(bw, n - 1)
s = cov_kernel(moments, w)
cov = ginv @ s @ ginv.T / n
cov = cov[order][:, order]
cov = (cov + cov.T) / 2
assert_allclose(cov, res.cov)