This lecture will cover:
Financial time series data are special because of their features, which include tail heaviness, asymmetry, volatility, and serial dependence without correlation.
Let consider \(P_t\) the price of a stock of other financial asset at time \(t\), then we can define the log return by \(Z_t = log(P_t) − log(P_{t−1})\) A model of the return series should support the fact that this data has a conditional variance \(h_t\) of \(Z_t\) is not independent of past value of \(Z_t\).
The idea of the ARCH model is to incorporate the sequence \(h_t\) in the model by:
\[ Z_t =\sqrt{h_t e_t}, \qquad e_t \sim IIDN(0, 1) \]
where \(h_t\) is known as the volatility and related to the past values of $Z_t$ via the ARCH(p) model:
\[ h_t = \alpha_0 + \sum_{i=1}^{p} \alpha_i Z^2_{t-i} \]
The GARCH(p,q) model is given by:
\[ h_t = \alpha_0 + \sum_{i=1}^{p} \alpha_i Z^2_{t-i} + \sum_{j=1}^{q} \beta_j h_{t-j} \]
\(\alpha_0 > 0 \text{ and } \alpha_i ≥ 0, \beta_j ≥ 0\)
library(rugarch) # run GARCH models
library(quantmod) # upload financial data from Yahoo Finance
library(itsmr)
library(aTSA)
getSymbols("GOOG", from = as.Date("2010-01-01"), to = as.Date("2023-3-30"))[1] "GOOG"# Select Open Price
op=GOOG$GOOG.Open
# plot Time Series
plot(op)
# Return data
R= diff(log(as.numeric(op)))
# Plot Return
plot(R)
# Plot ACF
acf(R,lag.max = 50)
# Plot ACF of R^2
acf(R^2, lag.max = 50)
# Specify the model
spec1 = ugarchspec()
spec1
*---------------------------------*
* GARCH Model Spec *
*---------------------------------*
Conditional Variance Dynamics
------------------------------------
GARCH Model : sGARCH(1,1)
Variance Targeting : FALSE
Conditional Mean Dynamics
------------------------------------
Mean Model : ARFIMA(1,0,1)
Include Mean : TRUE
GARCH-in-Mean : FALSE
Conditional Distribution
------------------------------------
Distribution : norm
Includes Skew : FALSE
Includes Shape : FALSE
Includes Lambda : FALSE # Mean Model AR(1)
spec1 = ugarchspec(mean.model = list(armaOrder=c(1,0)))
spec1
*---------------------------------*
* GARCH Model Spec *
*---------------------------------*
Conditional Variance Dynamics
------------------------------------
GARCH Model : sGARCH(1,1)
Variance Targeting : FALSE
Conditional Mean Dynamics
------------------------------------
Mean Model : ARFIMA(1,0,0)
Include Mean : TRUE
GARCH-in-Mean : FALSE
Conditional Distribution
------------------------------------
Distribution : norm
Includes Skew : FALSE
Includes Shape : FALSE
Includes Lambda : FALSE # Mean Model- EWMA
spec2 = ugarchspec(mean.model = list(armaOrder=c(0,0)),variance.model = list(garchOrder =c(2,0)))
spec2
*---------------------------------*
* GARCH Model Spec *
*---------------------------------*
Conditional Variance Dynamics
------------------------------------
GARCH Model : sGARCH(2,0)
Variance Targeting : FALSE
Conditional Mean Dynamics
------------------------------------
Mean Model : ARFIMA(0,0,0)
Include Mean : TRUE
GARCH-in-Mean : FALSE
Conditional Distribution
------------------------------------
Distribution : norm
Includes Skew : FALSE
Includes Shape : FALSE
Includes Lambda : FALSE # Fit GARCH
ugfit = ugarchfit(spec = spec2, data = R)
ugfit
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(2,0)
Mean Model : ARFIMA(0,0,0)
Distribution : norm
Optimal Parameters
------------------------------------
Estimate Std. Error t value Pr(>|t|)
mu 0.000618 0.000249 2.4819 0.013068
omega 0.000159 0.000008 21.0675 0.000000
alpha1 0.242481 0.031290 7.7495 0.000000
alpha2 0.301324 0.037833 7.9645 0.000000
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.000618 0.000293 2.1104 0.034820
omega 0.000159 0.000016 10.1275 0.000000
alpha1 0.242481 0.051411 4.7165 0.000002
alpha2 0.301324 0.082867 3.6363 0.000277
LogLikelihood : 9034.563
Information Criteria
------------------------------------
Akaike -5.4221
Bayes -5.4148
Shibata -5.4221
Hannan-Quinn -5.4195
Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic p-value
Lag[1] 0.7328 0.3920
Lag[2*(p+q)+(p+q)-1][2] 0.8391 0.5535
Lag[4*(p+q)+(p+q)-1][5] 2.3090 0.5474
d.o.f=0
H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic p-value
Lag[1] 0.7798 0.3772
Lag[2*(p+q)+(p+q)-1][5] 4.7131 0.1776
Lag[4*(p+q)+(p+q)-1][9] 7.2305 0.1806
d.o.f=2
Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale P-Value
ARCH Lag[3] 0.1857 0.500 2.000 0.6665
ARCH Lag[5] 0.8537 1.440 1.667 0.7768
ARCH Lag[7] 2.4760 2.315 1.543 0.6174
Nyblom stability test
------------------------------------
Joint Statistic: 1.3026
Individual Statistics:
mu 0.2283
omega 0.7062
alpha1 0.5477
alpha2 0.1634
Asymptotic Critical Values (10% 5% 1%)
Joint Statistic: 1.07 1.24 1.6
Individual Statistic: 0.35 0.47 0.75
Sign Bias Test
------------------------------------
t-value prob sig
Sign Bias 0.56811 0.5700
Negative Sign Bias 0.02728 0.9782
Positive Sign Bias 0.89695 0.3698
Joint Effect 2.72729 0.4356
Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
group statistic p-value(g-1)
1 20 164.0 4.305e-25
2 30 192.0 5.784e-26
3 40 197.8 4.106e-23
4 50 219.3 2.128e-23
Elapsed time : 0.2528391 # Forecast with GARCH
ugf = ugarchforecast(ugfit, n.ahead = 10)
ugf
*------------------------------------*
* GARCH Model Forecast *
*------------------------------------*
Model: sGARCH
Horizon: 10
Roll Steps: 0
Out of Sample: 0
0-roll forecast [T0=1979-02-13 18:00:00]:
Series Sigma
T+1 0.0006181 0.01788
T+2 0.0006181 0.01548
T+3 0.0006181 0.01770
T+4 0.0006181 0.01753
T+5 0.0006181 0.01811
T+6 0.0006181 0.01819
T+7 0.0006181 0.01838
T+8 0.0006181 0.01845
T+9 0.0006181 0.01853
T+10 0.0006181 0.01857plot(ugf@forecast$sigmaFor,type="l")
plot(ugf@forecast$seriesFor,type="l")
library(astsa)
plot(nyse,type="o")
plota(nyse)
plota(nyse^2)
spec2 = ugarchspec(mean.model = list(armaOrder=c(1,0)),
variance.model = list(garchOrder =c(1,1)))
spec2
*---------------------------------*
* GARCH Model Spec *
*---------------------------------*
Conditional Variance Dynamics
------------------------------------
GARCH Model : sGARCH(1,1)
Variance Targeting : FALSE
Conditional Mean Dynamics
------------------------------------
Mean Model : ARFIMA(1,0,0)
Include Mean : TRUE
GARCH-in-Mean : FALSE
Conditional Distribution
------------------------------------
Distribution : norm
Includes Skew : FALSE
Includes Shape : FALSE
Includes Lambda : FALSE # Fit GARCH
ugfit = ugarchfit(spec = spec2, data = nyse)
ugfit
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model : ARFIMA(1,0,0)
Distribution : norm
Optimal Parameters
------------------------------------
Estimate Std. Error t value Pr(>|t|)
mu 0.000734 0.000197 3.7272 0.000194
ar1 0.107539 0.025129 4.2795 0.000019
omega 0.000006 0.000000 22.5392 0.000000
alpha1 0.108783 0.007194 15.1220 0.000000
beta1 0.815154 0.010097 80.7321 0.000000
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.000734 0.000200 3.6781 0.000235
ar1 0.107539 0.022202 4.8436 0.000001
omega 0.000006 0.000001 6.2411 0.000000
alpha1 0.108783 0.014164 7.6803 0.000000
beta1 0.815154 0.027102 30.0767 0.000000
LogLikelihood : 6732.066
Information Criteria
------------------------------------
Akaike -6.7271
Bayes -6.7131
Shibata -6.7271
Hannan-Quinn -6.7219
Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic p-value
Lag[1] 1.536 0.2152
Lag[2*(p+q)+(p+q)-1][2] 1.560 0.4047
Lag[4*(p+q)+(p+q)-1][5] 2.102 0.6816
d.o.f=1
H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic p-value
Lag[1] 1.264 0.2609
Lag[2*(p+q)+(p+q)-1][5] 1.327 0.7823
Lag[4*(p+q)+(p+q)-1][9] 1.760 0.9316
d.o.f=2
Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale P-Value
ARCH Lag[3] 0.06405 0.500 2.000 0.8002
ARCH Lag[5] 0.10165 1.440 1.667 0.9865
ARCH Lag[7] 0.46563 2.315 1.543 0.9814
Nyblom stability test
------------------------------------
Joint Statistic: 15.6521
Individual Statistics:
mu 0.07411
ar1 0.18618
omega 2.10064
alpha1 0.18819
beta1 0.21744
Asymptotic Critical Values (10% 5% 1%)
Joint Statistic: 1.28 1.47 1.88
Individual Statistic: 0.35 0.47 0.75
Sign Bias Test
------------------------------------
t-value prob sig
Sign Bias 0.1864 0.852153
Negative Sign Bias 2.8413 0.004539 ***
Positive Sign Bias 0.9201 0.357652
Joint Effect 12.4901 0.005880 ***
Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
group statistic p-value(g-1)
1 20 94.48 5.282e-12
2 30 96.52 3.499e-09
3 40 127.48 2.524e-11
4 50 115.75 2.522e-07
Elapsed time : 0.2217271 # Forecast with GARCH
ugf = ugarchforecast(ugfit, n.ahead = 10)
ugf
*------------------------------------*
* GARCH Model Forecast *
*------------------------------------*
Model: sGARCH
Horizon: 10
Roll Steps: 0
Out of Sample: 0
0-roll forecast [T0=2000-01-01]:
Series Sigma
T+1 0.0012251 0.009906
T+2 0.0007872 0.009839
T+3 0.0007401 0.009777
T+4 0.0007350 0.009719
T+5 0.0007345 0.009665
T+6 0.0007344 0.009616
T+7 0.0007344 0.009569
T+8 0.0007344 0.009527
T+9 0.0007344 0.009487
T+10 0.0007344 0.009450plot(ugf@forecast$sigmaFor,type="l")
plot(ugf@forecast$seriesFor,type="l")