y_in = x_in @ c_hat_sel * 10000
# out-of-sample cointegrated series (basis points)
y_out = x_out @ c_hat_sel * 10000
# -
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_cointegration_detection-implementation-step04): AR(1) long term parameters
# +
exp_infty = np.zeros(3)
sd_infty = np.zeros(3)
tau_halflife = np.zeros(3)
for k in range(3):
# AR1 fit
b_hat, mu_hat_epsi, sig2_hat_epsi = fit_var1(y_in[:, [k]])
# long-run expectation
exp_infty[k] = mu_hat_epsi / (1 - b_hat)
# long-run standard deviation
sd_infty[k] = np.sqrt(sig2_hat_epsi / (1 - b_hat**2))
# half life
tau_halflife[k] = -np.log(2) / np.log(abs(b_hat))
# -
# ## Plots
# +
plt.style.use('arpm')
index_col=0,
parse_dates=True)
db_riskdrivers = db_riskdrivers.loc[t_start:t_end]
dates = pd.to_datetime(np.array(db_riskdrivers.index))[1:]
tau_implvol = np.array(
[col[col.find(' tau=') + 5:] for col in db_riskdrivers.columns])
ind_select = np.in1d(tau_implvol, tau_select)
# -
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_fit_var1_implvol-implementation-step01): Compute risk drivers
x = np.log(db_riskdrivers.iloc[:, ind_select].values)
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_fit_var1_implvol-implementation-step02): Perform VAR1 fit
b_hat, mu_epsi_hat, _ = fit_var1(x)
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_fit_var1_implvol-implementation-step03): Extract invariants realizations
epsi_var1 = x[1:, :] - x[:-1, :] @ b_hat.T
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_fit_var1_implvol-implementation-step04): Save database
# +
out = pd.DataFrame(
{
d: epsi_var1[:, d1]
for d1, d in enumerate(db_riskdrivers.columns.values[ind_select])
},
index=dates)
t_ = y.shape[0] # length of the time series of rolling values
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_fit_yields_var1-implementation-step01): Realized risk drivers (yield or shadow rates)
if yields:
x = y[:]
else:
x = ytm_shadowrates(y, eta=0.013)
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_fit_yields_var1-implementation-step02): Flexible probabilities
p = exp_decay_fp(t_, tau_hl) # exponential decay
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_fit_yields_var1-implementation-step03): Perform VAR(1) fit
b_hat, mu_epsi_hat, sig2_epsi_hat = fit_var1(x, p, nu=nu)
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_fit_yields_var1-implementation-step04): Recovered values of the risk drivers from the fit
x_fit = x[-1, :] @ b_hat.T + mu_epsi_hat.reshape((1, -1))
# ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_fit_yields_var1-implementation-step05): Expectation and covariance of the conditional next step prediction
ind = (np.array([
np.where(tau == tau_select[i])[0] for i, item in enumerate(tau_select)
]).reshape((-1)))
# next-step expectation for all times to maturity
mu_ns = x[-1, :] @ b_hat.T + mu_epsi_hat
# next-step expectation for selected times to maturity
mu_select_ns = mu_ns[ind]
# next-step covariance for all times to maturity
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_checklist_scenariobased_step02-implementation-step04): AR(1) fit of Nelson-Siegel parameters
# +
# the fit is performed only on non-nan entries
t_bonds = np.sum(np.isfinite(x[:, ind_ns_bonds[0]]))
x_obligor = np.zeros((t_bonds, i_bonds))
epsi_obligor = np.zeros((t_bonds - 1, i_bonds))
b_ar_obligor = np.zeros(i_bonds) # initialize AR(1) parameter
for i in range(i_bonds):
# risk driver (non-nan entries)
x_obligor[:, i] = x[t_ - t_bonds + 1:, ind_ns_bonds[i]]
# fit parameter
b_ar_obligor[i], _, _ = fit_var1(x_obligor[:, i])
# invariants
epsi_obligor[:, i] = x_obligor[1:, i] - b_ar_obligor[i] * x_obligor[:-1, i]
# store the next-step function and the extracted invariants
k = 0
for i in ind_ns_bonds:
db_invariants[i] = np.r_[np.full(t_ - t_bonds + 1, np.nan),
epsi_obligor[:, k]]
db_nextstep[i] = 'AR(1)'
tmp = np.full(4, np.nan)
tmp[1] = b_ar_obligor[k]
db_param[i] = tmp
k = k + 1
# -
# +
tau = np.array([tau])
path = '../../../databases/global-databases/fixed-income/db_yields'
y = pd.read_csv(path + '/data.csv', header=0, index_col=0)
y = y[tau.astype(float).astype(str)]
x = y.values # yield to maturity
x = x[::20, :]
x = x[-t_:, :].reshape(-1)
# -
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_elltest_ytm_monthly-implementation-step01): AR(1) fit
# +
p = np.ones(x.shape) / x.shape
b_hat, _, _ = fit_var1(x)
# realized invariant
epsi = x[1:] - x[:-1] * b_hat
# -
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_elltest_ytm_monthly-implementation-step02): ellipsoid tests
# +
plt.style.use('arpm')
# perform and show ellipsoid test for invariance on monthly yield
name1 = 'Invariance test on monthly yield'
acf_x, conf_int_x = \
invariance_test_ellipsoid(x, l_, conf_lev=conf_lev, fit=0, r=2,
title=name1)