Code
library(TSA)
# Load the CREF data
data(CREF)
plot(CREF)
439/639: ARCH and GARCH Models
1 ARCH models
ARCH (Autoregressive Conditional Heteroskedasticity) models are used to model time series data with changing variance over time. They are particularly useful in financial applications where volatility is not constant. The basic idea is to model the variance of the error term as a function of past squared errors.
For comparison, the ARMA models are conditionally homoscedastic, meaning that the variance of the error term is constant over time. In contrast, ARCH models allow for the variance to change over time, which is more realistic for many financial time series. The ARCH model was introduced by Robert Engle in 1982. The basic ARCH(q) model can be expressed as:
\[ Y_t = \mu + \sigma^2_t\epsilon_t \]
where \(\epsilon_t\) is the error term, and it is assumed to be normally distributed with mean 0 and variance 1. The variance \(\sigma^2_t\) is modeled as a function of past squared errors:
\[ \sigma^2_t = \alpha_0 + \alpha_1 \epsilon^2_{t-1} + \alpha_2 \epsilon^2_{t-2} + ... + \alpha_q \epsilon^2_{t-q} \]
where \(\alpha_0 > 0\) and \(\alpha_i \geq 0\) for \(i=1,2,...,q\). The parameters \(\alpha_0, \alpha_1, ..., \alpha_q\) are estimated from the data.
For example, ARCH(1) can be expressed as: \[ Y_t = \mu + \sigma^2_t\epsilon_t \]
\[ \sigma^2_t = \alpha_0 + \alpha_1 \epsilon^2_{t-1} \]
where \(\alpha_0 > 0\) and \(\alpha_1 \geq 0\). The ARCH(1) model captures the idea that the current variance is a function of the previous squared error.
Code
# Calculate the returns
r.cref <- diff(log(CREF))*100
plot(r.cref)
abline(h=0)
title(main="CREF Returns")
Code
# Plot ACF and PACF
acf(r.cref, main="ACF of CREF Returns")
Code
pacf(r.cref, main="PACF of CREF Returns")
Code
# Plot ACF and PACF of absolute returns
acf(abs(r.cref))
title(main="ACF of Absolute CREF Returns")
Code
pacf(abs(r.cref))
title(main="PACF of Absolute CREF Returns")
Code
# Plot ACF and PACF of squared returns
acf(r.cref^2, main="ACF of Squared CREF Returns")
Code
pacf(r.cref^2, main="PACF of Squared CREF Returns")
Note, that the ACF and PACF of the squared returns show significant spikes, indicating the presence of ARCH effects, but ACF and PACF of the original returns shows white noise.
Code
# Perform McLeod-Li test for ARCH effects
McLeod.Li.test(y=r.cref)
Code
qqnorm(r.cref)
qqline(r.cref)
Code
shapiro.test(r.cref)
Shapiro-Wilk normality test
data: r.cref
W = 0.99324, p-value = 0.024122 Simulated ARCH(1) process
Code
set.seed(1235678); library(tseries)
garch01.sim=garch.sim(alpha=c(.01,.9),n=500)
plot(garch01.sim,type='l',ylab=expression(r[t]), xlab='t')
Code
set.seed(1234567)
garch11.sim=garch.sim(alpha=c(0.02,0.05),beta=.9,n=500)
plot(garch11.sim,type='l',ylab=expression(r[t]), xlab='t')
Code
# ACF and PACF of the simulated ARCH(1) process
acf(garch11.sim, main="ACF of Simulated ARCH(1) Process")
Code
pacf(garch11.sim, main="PACF of Simulated ARCH(1) Process")
Code
# ACF and PACF of the absolute values of the simulated ARCH(1) process
acf(abs(garch11.sim), main="ACF of Absolute Simulated ARCH(1) Process")
Code
pacf(abs(garch11.sim), main="PACF of Absolute Simulated ARCH(1) Process")
title(main="PACF of Absolute Simulated ARCH(1) Process")
Code
# ACF and PACF of the squared values of the simulated ARCH(1) process
acf(garch11.sim^2, main="ACF of Squared Simulated ARCH(1) Process")
Code
pacf(garch11.sim^2, main="PACF of Squared Simulated ARCH(1) Process")
Code
# Extended ACF of squared and absolute values
eacf((garch11.sim)^2)AR/MA
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 o o x x o o x o o o o o o o
1 x o o o x o x x o o o o o o
2 x o o o o o x o o o o o o o
3 x x x o o x o o o o o o o o
4 x x o x x o o o o o o o o o
5 x o x x o o o o o o o o o o
6 x o x x o x o o o o o o o o
7 x x x x x x o o o o o o o o Code
eacf(abs(garch11.sim))AR/MA
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 o o x x o o x o o o o o o o
1 x o o o x o o o o o o o o o
2 x x o o o o o o o o o o o o
3 x x o o o x o o o o o o o o
4 x x o x o x o o o o o o o o
5 x o x x x o o o o o o o o o
6 x o x x x x o o o o o o o o
7 x x x x x o x o o o o o o o Plotting Extended ACF for the absolute and squared returns of the CREF data.
Code
eacf(abs(r.cref))AR/MA
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 o o o o o o o o o x x o o o
1 x o o o o o o o o o o o o o
2 x o o o o o o o o o o o o o
3 x o x o o o o o o o o o o o
4 x o x o o o o o o o o o o o
5 x x x x o o o o o o o o o o
6 x x x x o o o o o o o o o o
7 x x x x o o o o o o o o o o Code
eacf((r.cref)^2)AR/MA
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 o o x x o o o x o x o x x o
1 x o o o o o o x o x o o x o
2 x x o o o o o o o o o o o o
3 x x o x o o o o o o o x o o
4 o x x x o o o o o o o x x o
5 x x o x o o o o o o o x o o
6 x x o x x o o o o o o o x o
7 x x o x o o x o o o o o x o 2.1 Fitting ARIMA and GARCH models to simulated GARCH(1,1)
Code
arima(abs(garch11.sim),order=c(1,0,1))
Call:
arima(x = abs(garch11.sim), order = c(1, 0, 1))
Coefficients:
ar1 ma1 intercept
0.9821 -0.9445 0.5077
s.e. 0.0134 0.0220 0.0499
sigma^2 estimated as 0.1486: log likelihood = -232.97, aic = 471.94Code
g1 <- garch(garch11.sim,order=c(2,2))
***** ESTIMATION WITH ANALYTICAL GRADIENT *****
I INITIAL X(I) D(I)
1 3.294364e-01 1.000e+00
2 5.000000e-02 1.000e+00
3 5.000000e-02 1.000e+00
4 5.000000e-02 1.000e+00
5 5.000000e-02 1.000e+00
IT NF F RELDF PRELDF RELDX STPPAR D*STEP NPRELDF
0 1 2.661e+01
1 4 2.641e+01 7.54e-03 1.69e-02 3.7e-02 5.1e+02 3.0e-02 4.31e+00
2 5 2.609e+01 1.21e-02 1.87e-02 4.3e-02 2.0e+00 3.0e-02 1.50e+01
3 6 2.573e+01 1.40e-02 1.46e-02 3.3e-02 2.0e+00 3.0e-02 6.06e+00
4 9 2.279e+01 1.14e-01 1.06e-01 3.4e-01 1.8e+00 2.4e-01 6.46e+00
5 11 2.225e+01 2.39e-02 2.80e-02 6.9e-02 2.0e+00 4.7e-02 2.82e+02
6 12 2.157e+01 3.03e-02 3.31e-02 5.4e-02 2.0e+00 4.7e-02 3.50e+02
7 13 2.111e+01 2.17e-02 2.66e-02 5.4e-02 2.0e+00 4.7e-02 1.76e+02
8 15 2.105e+01 2.88e-03 1.23e-02 1.9e-02 2.0e+00 1.7e-02 3.37e+01
9 16 2.080e+01 1.15e-02 1.42e-02 2.0e-02 2.0e+00 1.7e-02 5.77e+00
10 17 2.056e+01 1.15e-02 2.28e-02 4.4e-02 2.0e+00 3.5e-02 4.10e+00
11 20 2.052e+01 1.88e-03 3.51e-03 2.3e-03 6.1e+00 2.7e-03 1.16e+01
12 21 2.049e+01 1.79e-03 1.79e-03 2.5e-03 2.2e+00 2.7e-03 1.86e+01
13 22 2.042e+01 3.35e-03 3.41e-03 6.1e-03 2.0e+00 5.4e-03 1.91e+01
14 25 2.002e+01 1.98e-02 2.64e-02 5.0e-02 2.0e+00 5.4e-02 1.85e+01
15 29 1.997e+01 2.07e-03 3.54e-03 1.8e-03 4.2e+00 2.0e-03 8.73e-01
16 30 1.994e+01 1.59e-03 1.61e-03 2.1e-03 2.7e+00 2.0e-03 5.52e-01
17 31 1.989e+01 2.73e-03 2.79e-03 3.8e-03 2.0e+00 4.0e-03 5.91e-01
18 34 1.958e+01 1.53e-02 2.11e-02 3.6e-02 1.9e+00 3.7e-02 5.99e-01
19 35 1.930e+01 1.45e-02 2.02e-02 3.3e-02 1.9e+00 3.7e-02 6.31e-01
20 36 1.907e+01 1.21e-02 3.55e-02 2.9e-02 1.8e+00 3.7e-02 3.27e-01
21 37 1.865e+01 2.17e-02 3.50e-02 2.5e-02 1.9e+00 3.7e-02 4.75e-01
22 39 1.857e+01 4.21e-03 5.14e-03 2.5e-03 2.0e+00 3.7e-03 1.23e-01
23 42 1.847e+01 5.60e-03 5.64e-03 7.2e-03 1.6e+00 9.6e-03 1.21e-01
24 44 1.826e+01 1.13e-02 1.17e-02 1.7e-02 1.3e+00 1.9e-02 1.66e-01
25 47 1.825e+01 3.99e-04 4.53e-04 2.1e-04 2.9e+00 3.8e-04 2.11e-01
26 51 1.808e+01 9.32e-03 1.40e-02 1.3e-02 3.3e+00 2.3e-02 2.30e-01
27 53 1.803e+01 2.97e-03 5.38e-03 3.1e-03 2.0e+00 4.6e-03 1.77e-01
28 54 1.798e+01 2.94e-03 3.01e-03 4.0e-03 2.0e+00 4.6e-03 1.72e-01
29 56 1.797e+01 6.34e-04 6.77e-04 7.7e-04 2.2e+00 9.2e-04 1.66e-01
30 57 1.795e+01 1.10e-03 1.12e-03 1.6e-03 2.0e+00 1.8e-03 1.62e-01
31 59 1.792e+01 1.49e-03 1.59e-03 3.0e-03 2.0e+00 3.7e-03 1.58e-01
32 61 1.791e+01 5.48e-04 5.63e-04 6.0e-04 2.0e+00 7.3e-04 1.30e-01
33 62 1.790e+01 6.99e-04 7.34e-04 1.1e-03 2.0e+00 1.5e-03 1.22e-01
34 64 1.789e+01 2.65e-04 2.50e-04 2.3e-04 2.3e+00 2.9e-04 1.18e-01
35 66 1.789e+01 5.43e-05 5.50e-05 5.2e-05 1.5e+01 5.9e-05 1.17e-01
36 68 1.789e+01 1.03e-04 1.03e-04 1.0e-04 4.4e+00 1.2e-04 1.19e-01
37 70 1.789e+01 2.08e-05 2.03e-05 2.0e-05 4.4e+02 2.3e-05 1.19e-01
38 72 1.789e+01 4.14e-05 4.05e-05 4.1e-05 4.7e+01 4.7e-05 1.28e-01
39 74 1.789e+01 8.27e-06 8.07e-06 8.1e-06 9.6e+02 9.4e-06 1.28e-01
40 76 1.789e+01 1.65e-06 1.61e-06 1.6e-06 4.8e+03 1.9e-06 1.30e-01
41 78 1.789e+01 3.30e-07 3.23e-07 3.3e-07 2.4e+04 3.8e-07 1.30e-01
42 80 1.789e+01 6.61e-07 6.45e-07 6.5e-07 3.0e+03 7.5e-07 1.30e-01
43 82 1.789e+01 1.32e-07 1.29e-07 1.3e-07 6.0e+04 1.5e-07 1.30e-01
44 84 1.789e+01 2.64e-07 2.58e-07 2.6e-07 7.6e+03 3.0e-07 1.30e-01
45 86 1.789e+01 5.29e-07 5.16e-07 5.2e-07 3.8e+03 6.0e-07 1.30e-01
46 88 1.789e+01 1.06e-07 1.03e-07 1.0e-07 7.6e+04 1.2e-07 1.30e-01
47 90 1.789e+01 2.11e-08 2.06e-08 2.1e-08 3.8e+05 2.4e-08 1.30e-01
48 92 1.789e+01 4.23e-08 4.13e-08 4.2e-08 4.7e+04 4.8e-08 1.30e-01
49 94 1.789e+01 8.46e-09 8.26e-09 8.3e-09 9.4e+05 9.6e-09 1.30e-01
50 96 1.789e+01 1.69e-08 1.65e-08 1.7e-08 1.2e+05 1.9e-08 1.30e-01
51 98 1.789e+01 3.38e-09 3.30e-09 3.3e-09 2.4e+06 3.8e-09 1.30e-01
52 100 1.789e+01 6.77e-10 6.61e-10 6.7e-10 1.2e+07 7.7e-10 1.30e-01
53 102 1.789e+01 1.35e-09 1.32e-09 1.3e-09 1.5e+06 1.5e-09 1.30e-01
54 104 1.789e+01 2.71e-10 2.64e-10 2.7e-10 3.0e+07 3.1e-10 1.30e-01
55 106 1.789e+01 5.41e-10 5.28e-10 5.3e-10 3.7e+06 6.2e-10 1.30e-01
56 108 1.789e+01 1.08e-10 1.06e-10 1.1e-10 7.4e+07 1.2e-10 1.30e-01
57 111 1.789e+01 8.66e-10 8.46e-10 8.5e-10 2.3e+06 9.9e-10 1.30e-01
58 114 1.789e+01 1.73e-11 1.69e-11 1.7e-11 4.6e+08 2.0e-11 1.30e-01
59 116 1.789e+01 3.46e-11 3.38e-11 3.4e-11 5.8e+07 3.9e-11 1.30e-01
60 119 1.789e+01 6.91e-13 6.76e-13 6.8e-13 1.2e+10 7.9e-13 1.30e-01
61 121 1.789e+01 1.39e-12 1.35e-12 1.4e-12 1.4e+09 1.6e-12 1.30e-01
62 123 1.789e+01 2.75e-13 2.71e-13 2.7e-13 2.9e+10 3.2e-13 1.30e-01
63 125 1.789e+01 5.34e-14 5.41e-14 5.5e-14 1.4e+11 6.3e-14 1.30e-01
64 127 1.789e+01 1.12e-13 1.08e-13 1.1e-13 1.8e+10 1.3e-13 1.30e-01
65 129 1.789e+01 2.03e-14 2.16e-14 2.2e-14 3.6e+11 2.5e-14 1.30e-01
66 131 1.789e+01 6.16e-15 4.33e-15 4.4e-15 1.8e+12 5.0e-15 1.31e-01
67 133 1.789e+01 8.54e-15 8.66e-15 8.7e-15 2.3e+11 1.0e-14 1.36e-01
68 134 1.789e+01 -5.59e+08 1.73e-14 1.7e-14 4.5e+11 2.0e-14 1.29e-01
***** FALSE CONVERGENCE *****
FUNCTION 1.788786e+01 RELDX 1.746e-14
FUNC. EVALS 134 GRAD. EVALS 68
PRELDF 1.732e-14 NPRELDF 1.288e-01
I FINAL X(I) D(I) G(I)
1 1.835300e-02 1.000e+00 -3.499e+00
2 1.728230e-16 1.000e+00 5.063e+00
3 1.135831e-01 1.000e+00 1.355e+01
4 3.368745e-01 1.000e+00 -2.752e+00
5 5.099761e-01 1.000e+00 -2.577e+00Code
summary(g1)
Call:
garch(x = garch11.sim, order = c(2, 2))
Model:
GARCH(2,2)
Residuals:
Min 1Q Median 3Q Max
-3.346827 -0.631881 0.008473 0.736112 3.202344
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 1.835e-02 1.515e-02 1.211 0.2257
a1 1.728e-16 4.723e-02 0.000 1.0000
a2 1.136e-01 5.855e-02 1.940 0.0524 .
b1 3.369e-01 3.696e-01 0.911 0.3621
b2 5.100e-01 3.575e-01 1.426 0.1538
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Diagnostic Tests:
Jarque Bera Test
data: Residuals
X-squared = 0.41859, df = 2, p-value = 0.8112
Box-Ljung test
data: Squared.Residuals
X-squared = 0.005298, df = 1, p-value = 0.942Code
g2 <- garch(garch11.sim,order=c(1,1))
***** ESTIMATION WITH ANALYTICAL GRADIENT *****
I INITIAL X(I) D(I)
1 3.706160e-01 1.000e+00
2 5.000000e-02 1.000e+00
3 5.000000e-02 1.000e+00
IT NF F RELDF PRELDF RELDX STPPAR D*STEP NPRELDF
0 1 2.773e+01
1 5 2.773e+01 2.16e-04 4.35e-04 4.2e-03 7.0e+02 4.2e-03 1.52e-01
2 6 2.772e+01 2.97e-04 3.53e-04 5.2e-03 2.0e+00 4.2e-03 2.85e-01
3 7 2.770e+01 5.63e-04 6.26e-04 8.8e-03 2.0e+00 8.3e-03 1.97e-01
4 11 2.652e+01 4.27e-02 1.90e-02 3.8e-01 9.4e-01 2.7e-01 1.56e-01
5 12 2.385e+01 1.01e-01 1.53e-01 4.4e-01 2.0e+00 5.3e-01 4.52e+01
6 15 2.356e+01 1.24e-02 1.10e-01 6.4e-03 1.3e+01 1.0e-02 3.35e+00
7 16 2.290e+01 2.81e-02 2.74e-02 6.5e-03 2.0e+00 1.0e-02 9.28e+00
8 18 2.060e+01 1.00e-01 1.90e-01 4.7e-02 6.2e+00 7.5e-02 1.05e+00
9 20 1.952e+01 5.22e-02 4.29e-02 3.5e-02 2.0e+00 5.9e-02 5.80e+00
10 22 1.809e+01 7.32e-02 8.53e-02 6.2e-02 2.0e+00 1.2e-01 3.74e+01
11 26 1.785e+01 1.32e-02 2.55e-02 7.9e-04 3.4e+00 1.7e-03 1.17e+01
12 30 1.765e+01 1.13e-02 1.15e-02 6.9e-03 2.0e+00 1.4e-02 9.52e+00
13 31 1.746e+01 1.08e-02 2.06e-02 1.4e-02 2.0e+00 2.8e-02 1.10e+00
14 34 1.744e+01 9.37e-04 2.19e-03 1.4e-04 1.1e+01 2.8e-04 1.37e-01
15 35 1.744e+01 1.00e-04 1.03e-04 1.4e-04 2.0e+00 2.8e-04 3.18e-02
16 36 1.744e+01 7.34e-05 9.97e-05 2.7e-04 2.0e+00 5.6e-04 2.24e-02
17 37 1.743e+01 2.06e-04 3.04e-04 5.1e-04 2.0e+00 1.1e-03 1.40e-02
18 38 1.743e+01 1.16e-04 1.59e-04 4.8e-04 1.9e+00 1.1e-03 3.80e-03
19 39 1.743e+01 4.11e-05 7.69e-05 5.1e-04 1.3e+00 1.1e-03 2.31e-04
20 40 1.743e+01 8.47e-06 1.29e-05 1.3e-04 0.0e+00 3.0e-04 1.29e-05
21 41 1.743e+01 2.90e-06 1.57e-05 9.8e-05 0.0e+00 2.4e-04 1.57e-05
22 42 1.743e+01 7.81e-07 7.47e-07 9.0e-06 0.0e+00 2.1e-05 7.47e-07
23 43 1.743e+01 2.69e-07 4.49e-08 1.6e-06 0.0e+00 4.1e-06 4.49e-08
24 44 1.743e+01 -2.34e-08 5.16e-11 1.9e-07 0.0e+00 4.6e-07 5.16e-11
***** RELATIVE FUNCTION CONVERGENCE *****
FUNCTION 1.743183e+01 RELDX 1.897e-07
FUNC. EVALS 44 GRAD. EVALS 24
PRELDF 5.158e-11 NPRELDF 5.158e-11
I FINAL X(I) D(I) G(I)
1 7.574525e-03 1.000e+00 9.582e-03
2 4.718358e-02 1.000e+00 -5.087e-04
3 9.353769e-01 1.000e+00 9.737e-04Code
summary(g2)
Call:
garch(x = garch11.sim, order = c(1, 1))
Model:
GARCH(1,1)
Residuals:
Min 1Q Median 3Q Max
-3.307030 -0.637977 0.009156 0.741977 3.019441
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 0.007575 0.007590 0.998 0.3183
a1 0.047184 0.022308 2.115 0.0344 *
b1 0.935377 0.035839 26.100 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Diagnostic Tests:
Jarque Bera Test
data: Residuals
X-squared = 0.82911, df = 2, p-value = 0.6606
Box-Ljung test
data: Squared.Residuals
X-squared = 0.53659, df = 1, p-value = 0.4638Code
m1=garch(x=r.cref,order=c(1,1))
***** ESTIMATION WITH ANALYTICAL GRADIENT *****
I INITIAL X(I) D(I)
1 3.744782e-01 1.000e+00
2 5.000000e-02 1.000e+00
3 5.000000e-02 1.000e+00
IT NF F RELDF PRELDF RELDX STPPAR D*STEP NPRELDF
0 1 3.221e+01
1 4 3.210e+01 3.24e-03 4.13e-03 9.8e-03 1.3e+03 1.0e-02 2.74e+00
2 6 3.201e+01 2.94e-03 6.22e-03 2.6e-02 3.4e+02 2.0e-02 3.50e+00
3 7 3.199e+01 6.93e-04 2.39e-03 2.6e-02 1.9e+00 2.0e-02 1.17e-02
4 8 3.194e+01 1.41e-03 1.29e-03 2.2e-02 1.9e+00 2.0e-02 8.47e-03
5 12 2.932e+01 8.20e-02 1.41e-02 7.6e-01 0.0e+00 6.4e-01 1.79e-02
6 14 2.612e+01 1.09e-01 5.98e-02 8.2e-02 2.0e+00 1.3e-01 5.17e+01
7 16 2.593e+01 7.50e-03 4.26e-02 1.6e-02 2.0e+00 2.6e-02 1.15e+04
8 17 2.518e+01 2.89e-02 3.40e-02 1.3e-02 2.0e+00 2.6e-02 3.98e+03
9 19 2.512e+01 2.48e-03 5.40e-03 5.7e-03 2.0e+00 9.9e-03 3.35e+02
10 20 2.504e+01 3.15e-03 3.76e-03 5.4e-03 2.0e+00 9.9e-03 7.80e-01
11 21 2.483e+01 8.37e-03 1.32e-02 1.1e-02 2.0e+00 2.0e-02 1.47e+02
12 23 2.447e+01 1.42e-02 2.22e-02 8.4e-03 2.0e+00 1.7e-02 1.37e+03
13 24 2.446e+01 4.63e-04 5.05e-03 9.0e-03 2.0e+00 1.7e-02 5.17e+00
14 25 2.442e+01 1.63e-03 7.22e-03 3.6e-03 2.0e+00 8.6e-03 1.94e+02
15 26 2.410e+01 1.34e-02 1.50e-02 4.9e-03 2.0e+00 8.6e-03 8.93e+02
16 27 2.398e+01 4.73e-03 5.46e-03 4.4e-03 2.0e+00 8.6e-03 5.30e+02
17 29 2.395e+01 1.12e-03 4.00e-03 3.6e-03 2.0e+00 6.5e-03 1.05e+02
18 30 2.391e+01 1.72e-03 3.84e-03 3.1e-03 2.0e+00 6.5e-03 5.76e+00
19 31 2.386e+01 2.40e-03 2.98e-03 3.2e-03 2.0e+00 6.5e-03 6.55e+00
20 32 2.377e+01 3.76e-03 4.09e-03 6.6e-03 2.0e+00 1.3e-02 1.88e-01
21 35 2.373e+01 1.34e-03 3.22e-03 5.7e-04 2.3e+00 1.1e-03 2.23e-01
22 36 2.370e+01 1.65e-03 1.71e-03 5.3e-04 2.0e+00 1.1e-03 9.28e+00
23 37 2.366e+01 1.35e-03 1.77e-03 8.2e-04 2.0e+00 2.2e-03 4.07e+00
24 38 2.365e+01 6.20e-04 1.88e-03 2.4e-03 2.0e+00 4.3e-03 2.06e-01
25 39 2.361e+01 1.73e-03 1.73e-03 2.3e-03 2.0e+00 4.3e-03 1.72e+00
26 41 2.359e+01 7.23e-04 1.74e-03 1.6e-03 2.0e+00 3.6e-03 1.14e+00
27 42 2.355e+01 1.70e-03 2.69e-03 3.6e-03 2.0e+00 7.1e-03 8.97e-02
28 43 2.345e+01 4.12e-03 4.71e-03 3.7e-03 2.1e+00 7.1e-03 8.91e-01
29 44 2.334e+01 4.99e-03 6.54e-03 7.3e-03 2.0e+00 1.4e-02 4.65e-01
30 46 2.329e+01 2.11e-03 2.64e-03 1.5e-03 5.5e+00 2.9e-03 2.96e-02
31 49 2.328e+01 1.49e-04 2.61e-04 9.1e-05 5.6e+00 1.8e-04 1.32e+00
32 50 2.328e+01 7.36e-05 7.65e-05 9.7e-05 2.9e+00 1.8e-04 7.34e-01
33 51 2.328e+01 1.49e-04 1.55e-04 1.7e-04 2.0e+00 3.7e-04 6.83e-01
34 55 2.323e+01 2.02e-03 3.49e-03 4.3e-03 2.0e+00 9.3e-03 5.61e-01
35 59 2.323e+01 1.39e-04 2.41e-04 7.3e-05 5.1e+00 1.5e-04 2.64e-03
36 60 2.323e+01 1.12e-05 1.37e-05 7.1e-05 1.9e+00 1.5e-04 4.98e-04
37 63 2.323e+01 1.20e-04 1.67e-04 9.3e-04 9.8e-01 1.9e-03 4.28e-04
38 64 2.323e+01 1.90e-05 4.79e-05 9.5e-04 0.0e+00 1.9e-03 4.79e-05
39 65 2.323e+01 4.43e-06 1.61e-06 6.2e-05 0.0e+00 1.5e-04 1.61e-06
40 81 2.323e+01 -6.12e-16 2.77e-16 2.1e-14 4.9e+08 4.1e-14 9.63e-08
***** FALSE CONVERGENCE *****
FUNCTION 2.322510e+01 RELDX 2.067e-14
FUNC. EVALS 81 GRAD. EVALS 40
PRELDF 2.771e-16 NPRELDF 9.635e-08
I FINAL X(I) D(I) G(I)
1 1.632722e-02 1.000e+00 4.747e-02
2 4.414103e-02 1.000e+00 -1.480e-01
3 9.170401e-01 1.000e+00 -3.126e-02Code
summary(m1)
Call:
garch(x = r.cref, order = c(1, 1))
Model:
GARCH(1,1)
Residuals:
Min 1Q Median 3Q Max
-2.78577 -0.61949 0.08695 0.67933 3.30810
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 0.01633 0.01237 1.320 0.1869
a1 0.04414 0.02097 2.105 0.0353 *
b1 0.91704 0.04570 20.066 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Diagnostic Tests:
Jarque Bera Test
data: Residuals
X-squared = 1.0875, df = 2, p-value = 0.5806
Box-Ljung test
data: Squared.Residuals
X-squared = 0.77654, df = 1, p-value = 0.37822.2 Diagnostics of the model
Code
plot(residuals(m1),type='h',ylab='Standardized Residuals')
Code
qqnorm(residuals(m1), main="Normal Q-Q Plot of Residuals")
qqline(residuals(m1))
Code
acf(residuals(m1)^2,na.action=na.omit, main="ACF of Squared Residuals")
Code
acf(abs(residuals(m1)),na.action=na.omit, main="ACF of Absolute Residuals")
We do not detect significant ARCH effects in the residuals. Thus the model is fitted adequately.