def test_gmm_kernel_direct(cov_data):
x, z, eps, sigma = cov_data
bandwidth = 12
k = len(x)
nobs = x[0].shape[0]
wm = KernelWeightMatrix(kernel='bartlett', bandwidth=bandwidth)
w = wm.weight_matrix(x, z, eps, sigma=sigma)
cov_est = GMMKernelCovariance(x, z, eps, w, sigma=sigma, debiased=debias, kernel='bartlett',
bandwidth=bandwidth)
xpz_wi_zpxi = _xpz_wi_zpxi(x, z, w)
xpz = _xpz(x, z)
wi = np.linalg.inv(w)
xpz_wi = xpz @ wi
ze = [z[i] * eps[:, i:i + 1] for i in range(k)]
ze = np.concatenate(ze, 1)
zeez = ze.T @ ze / nobs
w = kernel_weight_bartlett(bandwidth)
for i in range(1, bandwidth + 1):
op = ze[:-i].T @ ze[i:] / nobs
zeez += w[i] * (op + op.T)
assert_allclose(zeez, cov_est._omega())
direct = xpz_wi_zpxi @ (xpz_wi @ zeez @ xpz_wi.T) @ xpz_wi_zpxi / nobs
direct = (direct + direct.T) / 2
if debias:
df = [vx.shape[1] * np.ones(vx.shape[1]) for vx in x]
df = np.concatenate(df)[:, None]
df = np.sqrt(df)
adj = nobs / (nobs - df @ df.T)
direct *= adj
direct = (direct + direct.T) / 2
assert_allclose(direct, cov_est.cov)
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)[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])[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, lam, 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 i 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)
def test_kernel_bartlett():
w = kernel_weight_bartlett(0)
assert_equal(w, 1 - np.arange(1))
w = kernel_weight_bartlett(10)
assert_equal(w, 1 - np.arange(11) / 11)