sys.path.append("C:/Users/wigr11ab/Dropbox/KU/K3/FE/Python/")
import timeSeriesModule as tsm # Import ARCH simulation
import plotsModule as pltf # Custom plotting
np.set_printoptions(suppress=True) # Disable scientific notation
periods = 1000
x = np.arange(0, periods)
arch = tsm.archFct(1., 0.5, periods)
aarch = tsm.aArchFct(1, 0.6, 0.4, periods)
x1 = arch[0][:periods - 1]
y1 = arch[1][1:]
x2 = aarch[0][:periods - 1]
y2 = aarch[1][1:]
pltf.scatterDuo(x1, x2, y1, y2, 'ARCH', 'A-ARCH', title='News Curve')
# TAR model
tar = tsm.tarFct(0., 0., 0., 1., 0.9, 2.5, periods)
xTar = tar[:periods - 1]
yTar = tar[1:]
pltf.scatterUno(xTar, yTar, xLab='Lagged returns', title='TAR model')
pltf.plotUno(x, tar, title='TAR Process')
pStar = pStarFct(mat, states, a_r, b_r)
pStarT = pStarTFct(mat, states, a_r, a_s, b_r, p)
# Compute the log-likelihood to maximise
logLik = logLikFct(var, p, pStar, pStarT)
# Save parameters for later plotting (redundant wrt optimisation)
vs[:, m] = var
ps[:, m] = p
llh[m] = logLik
# ============================================= #
# ===== Plotting ============================== #
# ============================================= #
if states == 2:
pltm.plotDuo(range(sims), vs[0,:], vs[1,:], 'Var_1', 'Var_2', 'Trials', 'Variance')
pltm.plotDuo(range(sims), ps[0,:], ps[3,:], 'p11', 'p22', 'Trials', 'Probability')
elif states == 3:
pltm.plotTri(range(sims), vs[0,:], vs[1,:], vs[2,:], 'Trials', 'Var_1', 'Var_2', 'Var_3', 'Variance')
pltm.plotTri(range(sims), ps[0,:], ps[4,:], ps[8,:], 'Trials', 'p11', 'p22', 'p33', 'Probability')
pltm.plotUno(d, pStar[0,:], xLab = 'Time', yLab = 'p1', title = 'Smoothed State Probabilities')
pltm.plotUno(d, pStar[1,:], xLab = 'Time', yLab = 'p2', title = 'Smoothed State Probabilities')
pltm.plotUno(d, pStar[2,:], xLab = 'Time', yLab = 'p3', title = 'Smoothed State Probabilities')
elif states == 4:
pltm.plotQuad(range(sims), vs[0,:], vs[1,:], vs[2,:], vs[3,:], 'Trials', 'Var_1', 'Var_2', 'Var_3', 'Var_4', 'Variance')
pltm.plotQuad(range(sims), ps[0,:], ps[5,:], ps[10,:], ps[15,:], 'Trials', 'p11', 'p22', 'p33', 'p44', 'Probability')
pltm.plotUno(range(sims), llh, yLab = 'log-likelihood value')
sig2 = 1.0
alpha = 0.8
periods = 100
n = 10000
scores = np.zeros(n)
for i in np.arange(n):
tArch = tsm.tArchFct(sig2, alpha, 3, periods)
s = score.tArch3(tArch)
scores[i] = s
np.mean(scores)
np.var(scores)
x = np.arange(n)
pltm.plotUno(x, scores, 'Score value', 'Simulation trial',
'Distribution of scores', 'upper right')
x = np.sort(np.random.normal(size=n))
# sm.qqplot(scores, )
pltm.scatterUno(x,
np.sort(scores),
yLab='Emprical quantiles: Score',
xLab='Theoretical quantiles',
title='Normal QQ-Plot')
pltm.hist(scores, title='Histogram of scores')
# Next exercise 1.7
sp500 = pd.DataFrame(
pd.read_excel('C:/Users/wigr11ab/Dropbox/KU/K3/FE/Exercises/SP500.xlsx'))
date = np.array(sp500[['Date']][15096:])
s1, s2, p = 1.0, 0.5, 0.3
# Compute initial pStar given initial parameters
f1 = 1 / np.sqrt(2 * np.pi * s1) * np.exp(-0.5 * y**2 / s1)
f2 = 1 / np.sqrt(2 * np.pi * s2) * np.exp(-0.5 * y**2 / s2)
pStar = p * f1 / (p * f1 + (1 - p) * f2)
for m in range(rep):
# Reevaluate parameters given pStar
s1 = sum(pStar * y**2) / sum(pStar)
s2 = sum((1 - pStar) * y**2) / sum(1 - pStar)
p = sum(pStar) / mat
# Update pStar given new parameters
f1 = 1 / np.sqrt(2 * np.pi * s1) * np.exp(-0.5 * y**2 / s1)
f2 = 1 / np.sqrt(2 * np.pi * s2) * np.exp(-0.5 * y**2 / s2)
pStar = p * f1 / (p * f1 + (1 - p) * f2) # Compute new pStar
# Compute the log-likelihood to maximise
logLik = pStar * np.log(f1 * p) + (1 - pStar) * np.log(f2 * (1 - p))
sVol = pStar * s1 + (1 - pStar) * s2
# Save parameters for later plotting (redundant wrt optimisation)
par[0, m], par[1, m], par[2, m] = s1, s2, p
llh[m] = sum(logLik)
pltm.plotDuo(range(rep), par[0, :], par[1, :], 'Sigma_h', 'Sigma_l', 'Time',
'Volatility')
pltm.plotUno(range(rep), par[2, :])
pltm.plotUno(range(rep), llh)
# Analysing the residuals z = x / sigma
theta = np.exp(aPar)
n = len(y)
x = np.squeeze(y)
xLag2 = x[:n - 1] ** 2
idxP = (x > 0)[:n - 1]
idxN = (x < 0)[:n - 1]
s2 = theta[0] + theta[1] * idxP * xLag2 + theta[2] * idxN * xLag2
z = x[1:] / s2
pltm.qqPlot(z)
pltm.hist(z)
pltm.plotUno(np.arange(n-1), z, yLab='Residuals')
# Use the delta method to find se for GJR through AArch parameters (done manually)
a = np.array([np.exp(aPar[0]), np.exp(aPar[1]), np.exp(aPar[2]) - np.exp(aPar[1])])
A = np.array([[a[0], 0, 0], \
[0, a[1], 0], \
[0, -a[1], a[2] + a[1]]])
se = np.sqrt(np.diag(A.dot(sandwich).dot(np.transpose(A))))
tVal = a / se
gjrResults = pd.DataFrame([a, se, tVal, mlVal], \
columns=['sigma2', 'alpha', 'gamma'], \
index=['estimate', 'se', 't-val', 'ml val'])
gjrResults
mlResults