The average weekly earnings of google shares are $1.005 per share .docx

The average weekly earnings of google shares are $1.005 per
share with a standard deviation of $0.045. The distribution of
returns is more or less symmetric but have high peak. Normality
test (Jarque Bera Test) rejects hypothesis of normality for
earnings data at 5% level.
Jarque Bera Test
data: returns
X-squared = 54.1642, df = 2, p-value = 1.731e-12
The distribution of log(returns) is also not closer to normal
distribution, with symmetry close to 0 but kurtosis statistics is
1.5 which is far from normal distribution statistic. The Jarque-
Bera test is significant (p-value < 0.05) indicating that
hypothesis of normal distribution for log(returns) can be
rejected.
Jarque Bera Test
data: rets
X-squared = 49.4632, df = 2, p-value = 1.816e-11
The following plot is time series plot of weekly price of google
shares which shows increasing trend.
Time series Plot of Google Prices
Time plot of returns show that google stock returns shows
periods of high volatility at various times. Most returns are
between +/- 10%. Sample moments show that distribution of log
returns is somewhat symmetric with very thin tails (kurtosis =
1.56).
To check null hypothesis of non stationarity:
Dickey Fuller Test is expressed as H0: φ1=1 vs Ha: φ1<>1

where the null hypothesis indicates unit-root non-stationarity.
The Dickey-Fuller test shows that the earnings time series is
unit-root stationary. The test p-values for lags 7 is less than
0.05, therefore the null hypothesis of non stationarity can be
rejected (small p-values< 0.05).
Augmented Dickey-Fuller Test
data: rets
Dickey-Fuller = -7.1264, Lag order = 7, p-value = 0.01
alternative hypothesis: stationary
To check serial correlations in the log returns:
Log returns are not serially correlated as shown by the Ljung-
Box test with p-values > 0.05 and the autocorrelation plot.
Box-Pierce test
data: coredata(rets)
X-squared = 0.686, df = 1, p-value = 0.4075
To look for evidence of ARCH effects in the log returns:
The analysis below shows a strong ARCH effect. The squared
returns are strongly correlated. The Ljung-Box tests on squared
residuals are highly significant with p-values < 0.005, and
autocorrelation plots shows large autocorrelations for the first
15 lags.
Box-Pierce test
data: coredata(rets^2)
X-squared = 8.1483, df = 1, p-value = 0.00431
Plot of PACF: There is no correlation till lag 10 which shows
there is no AR lag.
To fit an ARMA(0,0)-GARCH(2,1) model for the log returns
using a normal- distribution for the error terms:
Fitted Model:

Residual Analysis:
Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
group statistic p-value(g-1)
1 20 28.78 0.06948
2 30 38.57 0.11019
3 40 46.39 0.19380
4 50 51.15 0.38929
Above output shows that error terms are normally distributed as
all the P values are greater than 0.05.
To fit ARMA(0,0)-eGARCH(1,1) model with Gaussian
distribution:
Fitted Model is:
Residual Analysis:
------------------------------------
1 20 28.95 0.06673
2 30 27.82 0.52739
3 40 46.91 0.18000
4 50 50.72 0.40546
To fit ARMA(0,0)-TGARCH(2,1) model with norm-distribution:
Fitted Model:
Residual Analysis:

------------------------------------
1 20 18.29 0.5030
2 30 31.28 0.3525
3 40 29.17 0.8742
4 50 51.36 0.3813
BEST FIT FOR THE MODEL:
garch21 egarch11 gjrgarch21
Akaike -3.436225 -3.451379 -3.463469
Bayes -3.391975 -3.407129 -3.401519
Shibata -3.436449 -3.451603 -3.463905
Hannan-Quinn -3.418814 -3.433968 -3.439094
From above output it can be said that AR(0)-EGARCH(1,1) and
AR(0) – GJR(1,1) both have almost equal (smallest) values for
all best fit criterion and hence can be considered as best model.
Forecast Analysis:
RE-FIT MODELS LEAVING 100 OUT-OF-SAMPLE
OBSERVATIONS FOR FORECAST and EVALUATE
STATISTICS for 5 Step Ahead Forecast.
Model
garch21
egarch11
tgarch21
MSE
0.0009497851
0.0009497741
0.0009499554
MAE
0.0224897000
0.0225260500
0.0225533000

DAC
0.6000000000
0.6000000000
0.6000000000
Since MSE is smallest for GJRGARCH model. We should
consider this model for forecasting.
CODE:
library(e1071)
library(tseries)
library(xts)
library(rugarch)
price = scan("clipboard")
pricets=ts(price,frequency=52,start=c(2004,45))
returns=pricets/lag(pricets, -1)
summary(price)
skewness(price)
kurtosis(price)
sd(price)
summary(returns)
skewness(returns)
kurtosis(returns)
sd(returns)
#Normality check
hist(returns, prob=TRUE)
lines(density(returns, adjust=2), type="l")
jarque.bera.test(returns)
#log return time series
rets = log(pricets/lag(pricets, -1))
#Normality check
hist(rets, prob=TRUE)
lines(density(rets, adjust=2), type="l")

jarque.bera.test(rets)
plot.ts(price.ts)
# strip off the dates and just create a simple numeric object
(require:xts)
ret = coredata(rets);
# creates time plot of log returns
plot(rets)
summary(rets)
skewness(rets)
kurtosis(rets)
sd(returns)
#ADF test for checking null hypothesis of non stationarity
adf.test(rets)
# Computes Ljung-Box test on returns
Box.test( coredata(rets))
# Computes Ljung-Box test on squared returns to test non-linear
independence
Box.test(coredata(rets^2))
# Computes Ljung-Box test on absolute returns to test non-
linear independence
Box.test(abs(coredata(rets)))
par(mfrow=c(3,1))
# Plots ACF function of vector data
acf(ret)
# Plot ACF of squared returns to check for ARCH effect
acf(ret^2)
# Plot ACF of absolute returns to check for ARCH effect
acf(abs(ret))
#plot returns, square returns and abs(returns)
par(mfrow=c(3,1))
plot(rets, type='l')

plot(rets^2,type='l')
plot(abs(rets),type='l')
par(mfrow=c(1,1))
# plots PACF of squared returns to identify order of AR model
pacf(coredata(rets),lag=10)
#specify model using functions in rugarch package
#Fit ARMA(0,0)-GARCH(2,1) model
garch21.spec=ugarchspec(variance.model=list(garchOrder=c(2,1
)), mean.model=list(armaOrder=c(0,0)))
#estimate model
garch21.fit=ugarchfit(spec=garch21.spec, data=rets)
garch21.fit
#Fit ARMA(0,0)-eGARCH(1,1) model with Gaussian
distribution
egarch11.spec=ugarchspec(variance.model=list(model =
"eGARCH",
garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0)))
#estimate model
egarch11.fit=ugarchfit(spec=egarch11.spec, data=ret)
egarch11.fit
#Fit ARMA(0,0)-TGARCH(2,1) model with norm-distribution
gjrgarch21.spec=ugarchspec(variance.model=list(model =
"gjrGARCH",
garchOrder=c(2,1)), mean.model=list(armaOrder=c(0,0)),
distribution.model = "norm")
#estimate model
gjrgarch21.fit=ugarchfit(spec=gjrgarch21.spec, data=ret)
gjrgarch21.fit

# compare information criteria
model.list = list(garch21 = garch21.fit,
egarch11 = egarch11.fit,
gjrgarch21 = gjrgarch21.fit)
info.mat = sapply(model.list, infocriteria)
rownames(info.mat) = rownames(infocriteria(garch21.fit))
info.mat
# RE-FIT MODELS LEAVING 100 OUT-OF-SAMPLE
OBSERVATIONS FOR FORECAST
# EVALUATION STATISTICS
garch21.fit = ugarchfit(spec=garch21.spec, data=ret,
out.sample=100)
egarch11.fit = ugarchfit(egarch11.spec, data=ret,
out.sample=100)
tgarch21.fit = ugarchfit(spec=gjrgarch21.spec, data=ret,
out.sample=100)
garch21.fcst = ugarchforecast(garch21.fit, n.roll=100,
n.ahead=5)
egarch11.fcst = ugarchforecast(egarch11.fit, n.roll=100,
n.ahead=5)
tgarch21.fcst = ugarchforecast(tgarch21.fit, n.roll=100,
n.ahead=5)
fcst.list = list(garch21=garch21.fcst,
egarch11=egarch11.fcst,
tgarch21=tgarch21.fcst)
fpm.mat = sapply(fcst.list, fpm)
fpm.mat
EXPLORATORY ANALYSIS

nobs 469.000000
NAs 0.000000
Minimum -0.166652
Maximum 0.164808
1. Quartile -0.022391
3. Quartile 0.029000
Mean 0.003843
Median 0.005614
Sum 1.802469
SE Mean 0.002079
LCL Mean -0.000242
UCL Mean 0.007929
Variance 0.002027
Stdev 0.045027
Skewness -0.040829
Kurtosis 1.569306
Title: Jarque - Bera Normalality TestTest Results: STATISTIC:
X-squared: 49.4632 P VALUE: Asymptotic p Value: 1.816e-
11
The distribution of log(returns) is not normal distribution, with
symmetry close to 0 but kurtosis statistics is 1.5 which is far
from normal distribution statistic. The Jarque-Bera test is
significant (p-value < 0.05) indicating that hypothesis of
normal distribution for log(returns) can be rejected.
The following plot is time series plot of weekly price of Google
shares which shows increasing trend.
(
Time series Plot of Google Prices
)
Time plot of returns show that google stock returns shows

periods of high volatility at various times. Most returns are
returns is somewhat symmetric with very thin tails (kurtosis =
1.56). There is various times where the returns produce +/-15%
with higher volatility also. Volatility doesn’t seem to die down
quickly and it continues for a while after a volatile period.
To check null hypothesis of non stationarity:
Dickey Fuller Test is expressed as H0: φ1=1 vs Ha: φ1<>1
where the null hypothesis indicates unit-root non-stationarity.
The Dickey-Fuller test shows that the earnings time series is
unit-root stationary. The test p-values for lags 7 is less than
0.05, therefore the null hypothesis of non stationarity can be
rejected (small p-values< 0.05).
Augmented Dickey-Fuller TestTest Results: PARAMETER:
Lag Order: 7 STATISTIC: Dickey-Fuller: -7.1331 P
VALUE: 0.01
alternative hypothesis: stationary
ACF TESTS
The ACF plots below show the Google stock returns are not
correlated indicating a constant mean model for rt. Both the
squared returns time series and the absolute time series show
large autocorrelations. We can conclude that the log returns
process has a strong non-linear dependence.
Ljung Box testing for ARCH effects for Lags of 6-12-18 for
squared returns and absolute returns
Box-Ljung testdata: coredata(rets^2)X-squared = 48.3339, df =
6, p-value = 1.013e-08Box-Ljung testdata: coredata(rets^2)X-
squared = 71.9976, df = 12, p-value = 1.352e-10Box-Ljung
testdata: coredata(rets^2)X-squared = 98.7945, df = 18, p-value
= 3.68e-13

The Ljung Box tests confirm that the Ljung Box tests on the
squared returns are autocorrelated. The analysis shows a strong
ARCH effect. The squared returns are strongly correlated. The
Ljung-Box tests on squared residuals are highly significant with
p-values < 0.005, and autocorrelation plots shows large
autocorrelations for the first 18 lags.
Plot of PACF: There is no correlation till lag 13 which shows
there is no AR lag.
MODEL FITTING
*---------------------------------** GARCH Model Fit
**---------------------------------*Conditional Variance Dynamics
-----------------------------------GARCH Model:
sGARCH(1,1)Mean Model: ARFIMA(0,0,0)Distribution: std
Optimal Parameters------------------------------------
Estimate Std. Error t value Pr(>|t|)mu 0.004432 0.001778
2.4924 0.012689omega 0.000103 0.000053 1.9574
0.050295alpha1 0.094169 0.032716 2.8784 0.003997beta1
0.856326 0.044635 19.1850 0.000000shape 6.797338
2.000925 3.3971 0.000681
To fit an ARMA(0,0)-GARCH(2,1) model for the log returns
using a normal distribution for the error terms:
Fitted Model:
with 7 degrees of freedom
Residual AnalysisQ-Statistics on Standardized Residuals---------
--------------------------- statistic p-valueLag[1]

0.0339 0.8539Lag[p+q+1][1] 0.0339 0.8539Lag[p+q+5][5]
2.4766 0.7800d.o.f=0H0 : No serial correlationQ-Statistics on
Standardized Squared Residuals------------------------------------
statistic p-valueLag[1] 0.1274 0.7212Lag[p+q+1][3]
1.4245 0.2327Lag[p+q+5][7] 3.4346 0.6333d.o.f=2
Above output shows there is no evidence of autocorrelation in
residuals. They behave like white noise. Also, there is no
evidence of serial correlation in squared residuals. They also
behave like white noise. Adjusted Pearson Goodness-of-Fit
Test:------------------------------------ group statistic p-value(g-
1)1 20 8465 02 30 13148 03 40
17834 04 50 22521 0
Test for Goodness of fit. Normal Distribution rejected.
To fit ARMA(0,0)-eGARCH(1,1) model with Gaussian
distribution:
Conditional Variance Dynamics -----------------------------------
GARCH Model: eGARCH(1,1)Mean Model:
ARFIMA(0,0,0)Distribution: norm Optimal Parameters-----------
------------------------- Estimate Std. Error t value
Pr(>|t|)mu 0.004814 0.001837 2.6205 0.008780omega -
0.427600 0.165861 -2.5781 0.009936alpha1 -0.086690
0.033886 -2.5583 0.010518beta1 0.931657 0.026273
35.4604 0.000000gamma1 0.177187 0.048248 3.6724
0.000240
Fitted Model is:
Residual Analysis:Q-Statistics on Standardized Residuals--------
---------------------------- statistic p-valueLag[1]
0.03655 0.8484Lag[p+q+1][1] 0.03655 0.8484Lag[p+q+5][5]

2.6812 0.1015Lag[p+q+5][7] 4.1362 0.5300d.o.f=2ARCH
LM Tests------------------------------------ Statistic DoF
P-ValueARCH Lag[2] 2.224 2 0.3290ARCH Lag[5]
4.682 5 0.4559ARCH Lag[10] 7.053 10 0.7204
1)1 20 26.65 0.11312 30 32.04 0.31793 40
43.67 0.27984 50 47.74 0.5243
To fit ARMA(0,0)-tGARCH(1,1) model with Gaussian
distribution:Conditional Variance Dynamics -----------------------
------------GARCH Model: gjrGARCH(2,1)Mean Model:
ARFIMA(0,0,0)Distribution: norm Optimal Parameters-----------
------------------------- Estimate Std. Error t value
Pr(>|t|)mu 0.004928 0.001860 2.649926 0.008051omega
0.000120 0.000053 2.257334 0.023987alpha1 0.000000
0.000877 0.000011 0.999991alpha2 0.072854 0.037335
1.951388 0.051011beta1 0.828559 0.053752 15.414339
0.000000gamma1 0.397469 0.144242 2.755564
0.005859gamma2 -0.315459 0.131445 -2.399922 0.016399
Fitted Model is:
Residual Analysis:Q-Statistics on Standardized Residuals--------
---------------------------- statistic p-valueLag[1]
0.2311 0.6307Lag[p+q+1][1] 0.2311 0.6307Lag[p+q+5][5]
4.191 0.04063Lag[p+q+5][8] 4.680 0.45620d.o.f=3ARCH LM
Tests------------------------------------ Statistic DoF P-

ValueARCH Lag[2] 2.461 2 0.2921ARCH Lag[5] 4.904
5 0.4277ARCH Lag[10] 7.140 10 0.7122
1)1 20 20.00 0.39472 30 32.43 0.30143 40
30.87 0.82034 50 54.56 0.2714
BEST FIT FOR THE MODEL: garch11 egarch11
gjrgarch11Akaike -3.479720 -3.455538 -3.474405Bayes
-3.435471 -3.411288 -3.412455Shibata -3.479944 -3.455762
-3.474841Hannan-Quinn -3.462310 -3.438127 -3.450030
From above output it can be said that AR(0)-EGARCH(1,1)
have the (smallest) values for all best fit criterion and hence can
be considered as best model.
Forecast Analysis:
RE-FIT MODELS LEAVING 100 OUT-OF-SAMPLE
OBSERVATIONS FOR FORECAST and EVALUATE
STATISTICS for 5 Step Ahead Forecast.
garch11 egarch11 tgarch21 MSE 0.000950199
0.0009498531 0.0009499473MAE 0.02257449 0.0225407
0.02255243 DAC 0.6 0.6 0.6
Since MSE is smallest for eGARCH model. We should consider
this model for forecasting.
CODE:
#Tomas Georgakopoulos CSC 425 Final Project
setwd("C:/Course/CSC425")

# Analysis of Google weekly returns from
library(forecast)
library(TSA)
# import data in R and compute log returns
# import libraries for TS analysis
library(zoo)
library(tseries)
myd= read.table('Weekly-GOOG-TSDATA.csv', header=T,
sep=',')
pricets = zoo(myd$price, as.Date(as.character(myd$date),
format=c("%m/%d/%Y")))
#log return time series
rets = log(pricets/lag(pricets, -1))
# strip off the dates and just create a simple numeric object
ret = coredata(rets);
#compute statistics
library(fBasics)
basicStats(rets)
#HISTOGRAM
par(mfcol=c(1,2))
hist(rets, xlab="Weekly Returns", prob=TRUE,
main="Histogram")
#Add approximating normal density curve
xfit<-
seq(min(rets,na.rm=TRUE),max(rets,na.rm=TRUE),length=40)
yfit<-
dnorm(xfit,mean=mean(rets,na.rm=TRUE),sd=sd(rets,na.rm=TR
UE))
lines(xfit, yfit, col="blue", lwd=2)
#CREATE NORMAL PROBABILITY PLOT

qqnorm(rets)
qqline(rets, col = 'red', lwd=2)
# creates time plot of log returns
par(mfrow=c(1,1))
plot(rets)
#Perform Jarque-Bera normality test.
normalTest(rets,method=c('jb'))
#SKEWNESS TEST
skew_test=skewness(rets)/sqrt(6/length(rets))
skew_test
print("P-value = ")
2*(1-pnorm(abs(skew_test)))
#FAT-TAIL TEST
k_stat = kurtosis(rets)/sqrt(24/length(rets))
print("Kurtosis test statistic")
k_stat
print("P-value = ")
2*(1-pnorm(abs(k_stat)))
#COMPUTE DICKEY-FULLER TEST
library(fUnitRoots)
adfTest(rets, lags=7, type=c("c"))
independence at lag 6 and 12
Box.test(coredata(rets^2),lag=6,type='Ljung')
linear independence at lag 6 and 12

Box.test(abs(coredata(rets)),lag=6,type='Ljung')
par(mfrow=c(3,1))
acf(ret)
acf(ret^2)
acf(abs(ret))
#plot returns, square returns and abs(returns)
par(mfrow=c(3,1))
plot(rets, type='l')
plot(rets^2,type='l')
plot(abs(rets),type='l')
par(mfrow=c(1,1))
# plots PACF of squared returns to identify order of AR model
pacf(coredata(rets),lag=30)
#GARCH Models
library(rugarch)
#Fit AR(0,0)-GARCH(1,1) model
)), mean.model=list(armaOrder=c(0,0)),distribution.model =
"std")
#estimate model

garch11.fit=ugarchfit(spec=garch11.spec, data=rets)
garch11.fit
#Fit AR(0,0)-eGARCH(1,1) model with Gaussian distribution
"eGARCH",garchOrder=c(1,1)),
mean.model=list(armaOrder=c(0,0)))
#estimate model
egarch11.fit=ugarchfit(spec=egarch11.spec, data=rets)
egarch11.fit
#Fit AR(0,0)-TGARCH(1,1) model with norm-distribution
gjrgarch11.spec=ugarchspec(variance.model=list(model =
"gjrGARCH",garchOrder=c(2,1)),
mean.model=list(armaOrder=c(0,0)), distribution.model =
"norm")
#estimate model
gjrgarch11.fit=ugarchfit(spec=gjrgarch11.spec, data=rets)
gjrgarch11.fit
model.list = list(garch11 = garch11.fit,
egarch11 = egarch11.fit,
gjrgarch11 = gjrgarch11.fit)
info.mat
garch11.fit = ugarchfit(spec=garch11.spec, data=rets,
out.sample=100)
egarch11.fit = ugarchfit(egarch11.spec, data=rets,
out.sample=100)

tgarch11.fit = ugarchfit(spec=gjrgarch11.spec, data=rets,
out.sample=100)
n.ahead=1)
egarch11.fcst = ugarchforecast(egarch11.fit, n.roll=100,
n.ahead=1)
tgarch11.fcst = ugarchforecast(tgarch11.fit, n.roll=100,
n.ahead=1)
fcst.list = list(garch11=garch11.fcst,
egarch11=egarch11.fcst,
tgarch21=tgarch11.fcst)
fpm.mat
CSC425 – Time series analysis and forecasting
Homework 5 – not be submitted
The goal of this assignment is to provide students with some
practical training on
GARCH/EGARCH/TGARCH models for the analysis of the
volatility of a stock return.
Solution

s will be posted on Thursday November 7th 2013
Reading assignment:
1. Read Chapter 4 in short book and Chapter 3 in long book on
volatility models
2. Review course documents posted under week 7 and 8.
Problem
Use the datafile nordstrom_w_00_13.csv that contains the
Nordstrom (JWN ) stock weekly prices
from January 2000 to October 2013. The data file contains
dates (date), daily prices (price). You
can also use the code and the analysis of the S&P500 returns
used in week 7 and 8 lectures as
your reference for the analysis of this data. Analyze the
Nordstrom stock log returns following
the steps below.

1. Compute log returns, and analyze their time plot.
Moments
N 693 Sum Weights
693
Mean 0.00268299 Sum Observations
1.85931229
Std Deviation 0.06068019 Variance
0.00368209
Skewness -0.2804891 Kurtosis
7.55351876
Uncorrected SS 2.55299156 Corrected SS
2.54800304
Coeff Variation 2261.66262 Std Error Mean
0.00230505

Time plot of returns show that Nordstrom stock returns shows
periods of high volatility at
various times, and consistently high volatility from 2007 to
2010. Most returns are
returns is somewhat
symmetric with very fat tails (kurtosis = 7.55).
return
-0.5
-0.4
-0.3
-0.2
-0.1

0.0
0.1
0.2
0.3
0.4
date
01/01/2000 01/01/2002 01/01/2004 01/01/2006 01/01/2008
01/01/2010 01/01/2012 01/01/2014 01/01/2016
2. Is there evidence of serial correlations in the log returns? Use
autocorrelations and 5%
significance level to answer the question.
Log returns are not serially correlated as shown by the Ljung-
Box test with p-values >

0.05 and the autocorrelation plot. .
Name of Variable = return
Mean of Working Series 0.002683
Standard Deviation 0.060636
Number of Observations 693
Autocorrelation Check for White Noise
To Chi- Pr >
Lag Square DF ChiSq --------------------
Autocorrelations--------------------
6 6.16 6 0.4050 -0.009 0.011 -0.048 -0.009
0.078 -0.006
12 9.08 12 0.6957 -0.042 -0.038 -0.019 0.011
-0.015 0.016

18 26.17 18 0.0960 0.033 -0.026 0.088 -0.032
0.101 -0.056
24 31.58 24 0.1377 0.029 -0.037 0.025 -0.042
-0.026 -0.048
30 33.84 30 0.2871 -0.029 -0.003 -0.037 0.023
0.011 -0.016
3. Is there evidence of ARCH effects in the log returns? Use
appropriate tests at 5%
significance level to answer this question.
The analysis below shows a strong ARCH effect. The squared
returns are strongly
correlated. The Ljung-Box tests on squared residuals are highly
significant with p-values
< 0.001, and autocorrelation plots shows large autocorrelations
for the first 10 lags.

Name of Variable = returnsq
To Chi- Pr >
6 331.56 6 <.0001 0.534 0.281 0.123 0.101
0.167 0.240
12 407.05 12 <.0001 0.246 0.187 0.053 0.036
0.032 0.082

18 469.70 18 <.0001 0.069 0.072 0.128 0.127
0.156 0.145
24 494.90 24 <.0001 0.149 0.090 0.050 0.035
0.014 0.031
30 501.01 30 <.0001 0.033 0.064 0.051 0.023
0.010 -0.004
4. Fit an GARCH(1,1) model for the log returns using a t-
distribution for the error terms.
Perform model checking (analyze if residuals are white noise, if
squared residuals are
white noise, and check if t-distribution is a good fit for the
data) and write down the fitted
model.
The GARCH model with t-distribution is an adequate model for
the volatility of the

returns. All parameters are significant and the squared residuals
are white noise (LB
tests on squared residuals are non significant).
Fitted GARCH model can be expressed as follows:
rt=0.005 +at
at=σtet with σt
2
=0.1483 a
2
t-1+0.836 σ
2
t-1
(Intercept ARCH0 can be considered equal to zero)
Where error term et has t-distribution with 1/0.185=5 degrees
of freedom.

The AUTOREG Procedure
GARCH Estimates
SSE 2.55069233 Observations
693
MSE 0.00368 Uncond Var
0.00531549
Log Likelihood 1102.73026 Total R-Square
.
SBC -2172.7554 AIC -
2195.4605
MAE 0.04086373 AICC -
2195.3732
MAPE 113.334007 HQC -
2186.6796

Normality Test 126.0152
Pr > ChiSq <.0001
Parameter Estimates
Standard Approx
Variable DF Estimate Error t Value Pr >
|t|
Intercept 1 0.004653 0.001570 2.96
0.0030
ARCH0 1 0.0000848 0.0000429 1.98
0.0482
ARCH1 1 0.1483 0.0369 4.01
<.0001
GARCH1 1 0.8358 0.0359 23.30
<.0001
TDFI 1 0.1851 0.0448 4.14 <.0001

Inverse of t DF
R output
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.004729 0.001465 3.2281 0.001246
omega 0.000086 0.000050 1.7302 0.083590
alpha1 0.145749 0.052318 2.7858 0.005340
beta1 0.836471 0.051967 16.0962 0.000000
shape 5.345569 1.009241 5.2966 0.000000
5. Fit and EGARCH(1,1) model for the NDX log returns using a
normal distribution for the
error terms. Perform model checking and write down the fitted
model.
The EGARCH model is adequate although the Gaussian
distribution on the error terms is

not sufficient to describe most extreme events. No ARCH
effects are shown in residuals.
The leverage parameter “theta” is significant showing that
volatility of Nordstrom stock
returns is affected more heavily by negative shocks.
The fitted model can be written as follows:
rt=0.0012 +at
at=σtet with ln σt
2
=-0.194+0.213 g(et-1) +0.996 ln σ
2
t-1
g(et-1)= -0.47et-1+[|et-1|-E(et-1)]
The AUTOREG Procedure

Exponential GARCH Estimates
SSE 2.54945444 Observations
693
MSE 0.00368 Uncond Var
.
Log Likelihood 1093.55049 Total R-Square
.
stres
-7
-6
-5
-4
-3
-2

-1
0
1
2
3
4
5
6
7
quantiles
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
t-distribution QQplot for residuals of GARCH model

SBC -2154.3958 AIC -
2177.101
MAE 0.04095052 AICC -
2177.0136
MAPE 101.222881 HQC -
2168.32
Normality Test 71.9939
Pr > ChiSq <.0001
Parameter Estimates
Standard Approx
Variable DF Estimate Error t Value
Pr > |t|
Intercept 1 0.001236 0.001752 0.71
0.4806

EARCH0 1 -0.1940 0.0636 -3.05
0.0023
EARCH1 1 0.2133 0.0316 6.75
<.0001
EGARCH1 1 0.9662 0.0106 91.41
<.0001
THETA 1 -0.4748 0.1121 -4.23
<.0001
R output
mu 0.001271 0.001774 0.71655 0.473654
omega -0.169047 0.110169 -1.53443 0.124924
alpha1 -0.095670 0.035615 -2.68620 0.007227
beta1 0.970507 0.018820 51.56682 0.000000
gamma1 0.194542 0.059579 3.26526 0.001094
In R model is written as

rt=0.0012 +at
2
=-0.169-0.095( et-1 +0.194(|et-1|-E(et-1))+0.970 ln σ
2
t-1
where error term has t-distribution with 6 degrees of freedom.
Note that the leverage coefficients of et-1 in SAS and R are
equivalent.
6. Fit and EGARCH(1,1) model for the Nordstrom log returns
using a t- distribution for the
error terms. Perform model checking and write down the fitted
model.

The EGARCH model t-distributed errors is a good model for the
data. No ARCH effects
are shown in residuals. The leverage parameter “theta” is
significant and negative
showing that volatility of Nordstrom stock returns is affected
more heavily by negative
shocks.
rt=0.0032 +at
2
=-0.191+0.158 g(et-1) +0.974 ln σ
2
t-1
g(et-1)= -0.55et-1+[|et-1|-E(et-1)]

et has t-distribution with 6 degrees of freedom.
The MODEL Procedure
Nonlinear Liklhood Summary of Residual
Errors
DF DF Adj
Equation Model Error SSE MSE Root MSE
R-Square R-Sq
7 686 2.5482 0.00371 0.0609 -0.0001
-0.0088
nresid.y 686 1028.4 1.4991 1.2244
Nonlinear Liklhood Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr

> |t|
intercept 0.003273 0.00160 2.05
0.0410
earch0 -0.19123 0.0932 -2.05
0.0405
earch1 0.158378 0.0490 3.23
0.0013
egarch1 0.974881 0.0139 70.01
<.0001
theta -0.55013 0.1969 -2.79
0.0053
df 5.993367 1.3083 4.58 <.0001
R output
mu 0.003374 0.001552 2.1743 0.029681
omega -0.137932 0.098212 -1.4044 0.160191
alpha1 -0.104435 0.034086 -3.0638 0.002185

beta1 0.977603 0.016446 59.4433 0.000000
gamma1 0.180592 0.075474 2.3928 0.016722
shape 5.783774 1.169335 4.9462 0.000001
In R model is written as
rt=0.0033 +at
2
=-0.138-0.104( et-1 +0.180(|et-1|-E(et-1))+0.978 ln σ
2
t-1
where error term has t-distribution with 6 degrees of freedom.
Note that the leverage coefficients of et-1 in SAS and R are
equivalent.
Name of Variable = resid2

To Chi- Pr >
6 5.11 6 0.5299 0.026 0.020 -0.044 -0.061
0.024 -0.004
12 13.61 12 0.3261 0.025 0.088 0.014 -0.041
0.017 -0.039
18 19.28 18 0.3748 0.043 -0.032 -0.038 -0.028
-0.049 0.020

24 23.55 24 0.4873 0.004 0.028 -0.025 -0.051
-0.039 -0.021
7. Find a GJK-TGARCH(1,1) model for the Nordstrom log
returns using a t-distribution for
the innovations. Perform model checking and write down the
fitted model.
Similar to the AR(0)- EGARCH(1,1) model, the AR(0)-
GJR(1,1) model is also a good
model for the data. No ARCH effects are shown in residuals.
The leverage parameter is
significant and positive showing that volatility of Nordstrom
stock returns is affected
more heavily by negative shocks.
rt=0.0036 +at

at=σtet with σt
2
=(0.034 +0.13Nt-1)a
2
t-1 +0.800 σ
2
t-1
with Nt-1 = 1 if at-1<0, and Nt-1 = 0 if at-1>0
The MODEL Procedure
Nonlinear Liklhood Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr
> |t|
intercept 0.003589 0.00157 2.28

0.0226
df 5.028263 0.9762 5.15
<.0001
arch0 0.000095 0.000044 2.18
0.0297
arch1 0.034154 0.0259 1.32
0.1869
garch1 0.800444 0.0584 13.71
<.0001
gamma 0.129939 0.0496 2.62
0.0090
R output
mu 0.003880 0.001519 2.5535 0.010665
omega 0.000083 0.000053 1.5798 0.114152
alpha1 0.039211 0.032406 1.2100 0.226283
beta1 0.858372 0.058370 14.7058 0.000000

gamma1 0.153341 0.074560 2.0566 0.039723
shape 5.767186 1.177356 4.8984 0.000001
rt=0.0039 +at
at=σtet with σt
2
=(0.039 +0.15Nt-1)a
2
t-1 +0.858 σ
2
t-1
with Nt-1 = 1 if at-1<0, and Nt-1 = 0 if at-1>0

8. Is the leverage effect significant at the 5% level?
The leverage parameters in both the EGARCH and the GJR
models are significant, and
have values that indicate that volatility react more heavily to
negative shocks.
9. What model provides the best fit for the data? Explain.
Since the leverage parameters in the AR(0)-EGARCH(1,1) and
AR(0) – GJR(1,1) models
are significant, and residuals are white noise with no ARCH
effect, we can conclude that
both models provide a good explanation of the volatility
behavior for Nordstrom stock
returns. Either models can be chosen, there is no clear winner.
(Note that in SAS, PROC
MODEL does not compute BIC criterion – backtesting can be

used to compare models.)
In R the egarch(1,1) and the GJR-GARCH(1,1) models have
very similar BIC values-
garch11.t egarch11 gjrgarch11
Akaike -3.172122 -3.186143 -3.183976
Bayes -3.139358 -3.146827 -3.144660
10. Use the selected model to compute up to 5 step-ahead
forecasts of the simple returns and
its volatility.
The following predictions are based on the AR(0)-
EGARCH(1,1) model. sigma is the
predicted conditional standard deviation or volatility, and series
is the predicted return
(note that predicted returns are the sample mean returns since
returns are white noise)

R output for forecasts
sigma series
2013-10-29 0.02732 0.003374
2013-10-30 0.02764 0.003374
2013-10-31 0.02796 0.003374
2013-11-01 0.02827 0.003374
2013-11-04 0.02858 0.003374
SAS code
*import data from file;
proc import datafile='nordstrom_w_00_13.csv' out=myd
replace;
resid
-7
-6

-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7

quantiles
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
t-distribution QQplot of residuals for GJRGARCH model
S&P500 analysis
Fitting an AR(1)-GARCH(1,1) model with Gaussian errors
> # Fitting a ARMA(0,0)-GARCH(1,1) model using the rugarch
package
> # log returns show weak serial autocorrelations in the mean
and an ARCH
effect
> # Fitting an GARCH(1,1) model

> # Use ugarchspec() function to specify model
>
)),
> #estimate model
> garch11.fit=ugarchfit(spec=garch11.spec, data=ret)
> garch11.fit
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------

GARCH Model : sGARCH(1,1)
Mean Model : ARFIMA(0,0,0)
Distribution : norm
Optimal Parameters
------------------------------------
mu 0.006814 0.001798 3.7889 0.000151
omega 0.000090 0.000049 1.8302 0.067224
alpha1 0.118390 0.031384 3.7723 0.000162
beta1 0.844396 0.034387 24.5556 0.000000

The fitted ARMA(0,0)-GARCH(1,1) model with Gaussian errors
can be written as
222
8444.01184.000009.0
0068.0
tttttt
tt
aea
ar

mu 0.006814 0.001914 3.5592 0.000372
omega 0.000090 0.000070 1.2858 0.198502
alpha1 0.118390 0.041165 2.8760 0.004028
beta1 0.844396 0.040113 21.0503 0.000000
LogLikelihood : 828.5681
Information Criteria
------------------------------------
Akaike -3.4286

Bayes -3.3938
Shibata -3.4287
Hannan-Quinn -3.4149
Q-Statistics on Standardized Residuals
------------------------------------
statistic p-value
Lag[1] 0.5261 0.4682
Lag[p+q+1][1] 0.5261 0.4682
Lag[p+q+5][5] 5.9001 0.3161
d.o.f=0
H0 : No serial correlation
Q-Statistics on Standardized Squared Residuals

------------------------------------
statistic p-value
Lag[1] 0.2657 0.6062
Lag[p+q+1][3] 0.8252 0.3637
Lag[p+q+5][7] 3.0889 0.6863
d.o.f=2
ARCH LM Tests
------------------------------------
Statistic DoF P-Value
ARCH Lag[2] 0.5084 2 0.7755
ARCH Lag[5] 0.9231 5 0.9685
ARCH Lag[10] 6.3220 10 0.7875

Nyblom stability test
------------------------------------
Joint Statistic: 1.3254
Individual Statistics:
mu 0.07929
omega 0.20176
alpha1 0.02459
beta1 0.09300
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 2.1382 0.0330120 **
Negative Sign Bias 0.5723 0.5673614
Positive Sign Bias 1.3403 0.1807809
Joint Effect 17.4223 0.0005786 ***
------------------------------------
1 20 43.12 0.0012496
2 30 53.82 0.0033902

3 40 75.76 0.0003804
4 50 79.60 0.0037117
Ljung-Box test for serial correlation
computed on residuals, and Ljung-
Box test for ARCH/GARCH effect
computed on squared residuals.
Goodness of fit test for distribution of error
term. The null hypothesis states that the
distribution for the error terms in the model
is adequate. Thus a small p-value < α,
indicates that null hypothesis can be
rejected and the distribution assumption is

not adequate.
> #create selection list of plots for garch(1,1) fit
> plot(garch11.fit)
> #to display all subplots on one page
> plot(garch11.fit, which="all")
Residual analysis of GARCH model shows that the model fits
the data adequately. Ljung Box test (Q-statistic) on
residuals is not significant, showing that hypothesis of no
correlation for residuals cannot be rejected. Similarly the
Ljung Box test (Q-statistic) on the squared standardized
residuals is not significant suggesting that residuals show
no ARCH/GARCH effect. The Adjusted Pearson goodness of fit
test is significant indicating that the normal
distribution assumed for the error term is not appropriate, as
also shown by the QQ plot of the residuals.

Fitting an AR(1)-GARCH(1,1) model with t-distributed errors
Elapsed time : 0.260026
> plot(garch11.fit, which="all")
Error in plot.new() : figure margins too large
> # use Student-t innovations
> #specify model using functions in rugarch package
> #Fit ARMA(0,0)-GARCH(1,1) model with t-distribution
>
garch11.t.spec=ugarchspec(variance.model=list(garchOrder=c(1
,1)),
mean.model=list(armaOrder=c(0,0)), distribution.model = "std")

> #estimate model
> garch11.t.fit=ugarchfit(spec=garch11.t.spec, data=ret)
> garch11.t.fit
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
-----------------------------------
GARCH Model : sGARCH(1,1)
Distribution : std
Optimal Parameters

------------------------------------
mu 0.008012 0.001746 4.5892 0.000004
omega 0.000122 0.000068 1.7901 0.073444
alpha1 0.123683 0.038919 3.1780 0.001483
beta1 0.821471 0.050706 16.2008 0.000000
shape 6.891814 1.978494 3.4834 0.000495
The fitted ARMA(0,0)-GARCH(1,1) model with Gaussian errors
can be written as
222
8215.01237.000012.0

0080.0
tttttt
tt
aea
ar
Error term {et} has t-distribution with 7 degrees of freedom

mu 0.008012 0.001817 4.4090 0.000010
omega 0.000122 0.000071 1.7216 0.085136
alpha1 0.123683 0.038112 3.2453 0.001173
beta1 0.821471 0.041476 19.8061 0.000000
shape 6.891814 1.976630 3.4866 0.000489
------------------------------------
Akaike -3.4689
Bayes -3.4255
Shibata -3.4691

------------------------------------
statistic p-value
Lag[1] 0.4662 0.4947
Lag[p+q+1][1] 0.4662 0.4947
Lag[p+q+5][5] 5.6618 0.3405
d.o.f=0
------------------------------------
statistic p-value

Lag[1] 0.2474 0.6189
Lag[p+q+1][3] 0.8569 0.3546
Lag[p+q+5][7] 2.9570 0.7066
d.o.f=2
ARCH LM Tests
------------------------------------
ARCH Lag[2] 0.5959 2 0.7423
ARCH Lag[5] 0.9740 5 0.9646
ARCH Lag[10] 6.0871 10 0.8079
------------------------------------

mu 0.17184
omega 0.23451
alpha1 0.04303
beta1 0.10209
shape 0.07244
Sign Bias Test

------------------------------------
t-value prob sig
Sign Bias 2.121 0.0343991 **
Joint Effect 16.352 0.0009603 ***
------------------------------------
1 20 22.66 0.25269
2 30 44.47 0.03312

3 40 39.50 0.44759
4 50 43.64 0.68967
Residual analysis of GARCH model with t-distributed error
terms shows that the model fits the data adequately.
Ljung Box test (Q-statistic) on residuals is not significant,
showing that hypothesis of no correlation for residuals
cannot be rejected. Similarly the Ljung Box test (Q-statistic) on
the squared standardized residuals is not significant
suggesting that residuals show no ARCH/GARCH effect. The
Adjusted Pearson goodness of fit test is not significant
indicating that the distribution of the error terms can be
described by a t-distribution with 7 degrees of freedom,
as also shown by the QQ plot of the residuals (created using
plot(garch11.t.fit, which=9).
Fitting an ARMA(0,0)-EGARCH(1,1) model

The EGARCH model fitted by the rugarch package has a
slightly different form than the model in the textbook.
Here is the general expression of the EGARCH(1,1) model:
)ln(|))(||(|()ln(
,
2
1111111
2
ttttt
tttttt
eEee

eaar
where µt can follow an ARMA process, but most typically it
will be constant. Gamma1 (γ1) is the leverage
parameter. So if gamma1 in output is significant, then we can
conclude that the volatility process has an
asymmetric behavior.
Fitting an ARMA(0,0)-EGARCH(1,1) model with Gaussian
distribution (similar to SAS example)
> #Fit ARMA(0,0)-eGARCH(1,1) model with Gaussian
distribution
> egarch11.spec=ugarchspec(variance.model=list(model =
"eGARCH",

garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0)))
> #estimate model
> egarch11.fit=ugarchfit(spec=egarch11.spec, data=ret)
> egarch11.fit
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
-----------------------------------
GARCH Model : eGARCH(1,1)
Distribution : norm

Optimal Parameters
------------------------------------
mu 0.005937 0.001860 3.1915 0.001415
omega -0.610497 0.263901 -2.3134 0.020703
alpha1 -0.111069 0.042614 -2.6064 0.009150
beta1 0.902462 0.041695 21.6442 0.000000
gamma1 0.209405 0.050542 4.1432 0.000034
Using the R output above, the ARMA(0,0)-EGARCH(1,1) model
can be written as follows.

Fitted model:
rt = 0.0059 + at, at=σtet
ln(σ2t) = -0.610 + (-0.111 et-1 + 0.209(|et-1| - E(|et-1|)) +
0.9024 ln(σ
2
t-1)
Note that since et has Gaussian distribution, the
E(|et|)=sqrt(2/pi) = 0.7979 or approx 0.80 (see page 143 in
textbook). Thus we can rewrite the expression above as

07979.0209.0209.0111.0
07979.0209.0209.0111.0
)ln(9024.0610.0)ln(
111
1112
1
2
ttt
ttt
tt
eifee

eifee
And after some algebra, we can write:
0320.0
0098.0
)ln(9024.07767.0)ln(

11
112
1
2
tt
tt
tt
eife
eife
Taking the antilog transformation we have

0)320.0exp(
0)098.0exp(
)7767.0exp(
11
119024.02
1
2

tt
ttx
tt
eife
eife
For a standardized shock with magnitude 2, (i.e. two standard
deviations), we have
56.1
)2098.0exp(
))2(320.0exp(
)2(
)2(
1

e
e
Therefore, the impact of negative shock of size two-standard
deviations is about 56% higher than the impact of a
positive shock of the same size.
mu 0.005937 0.001958 3.0328 0.002423
omega -0.610497 0.364615 -1.6744 0.094060
alpha1 -0.111069 0.066343 -1.6742 0.094096
beta1 0.902462 0.057336 15.7400 0.000000

gamma1 0.209405 0.048127 4.3511 0.000014
------------------------------------
Akaike -3.4553
Bayes -3.4119
Shibata -3.4555

------------------------------------
statistic p-value
Lag[1] 0.3351 0.5627
Lag[p+q+1][1] 0.3351 0.5627
Lag[p+q+5][5] 5.1570 0.3970
d.o.f=0
------------------------------------
statistic p-value
Lag[1] 0.7262 0.3941
Lag[p+q+1][3] 1.0203 0.3124

Lag[p+q+5][7] 2.8738 0.7194
d.o.f=2
ARCH LM Tests
------------------------------------
ARCH Lag[2] 1.018 2 0.6010
ARCH Lag[5] 1.165 5 0.9482
ARCH Lag[10] 6.026 10 0.8131
------------------------------------

mu 0.1829
omega 0.1224
alpha1 0.1527
beta1 0.1301
gamma1 1.0125
Sign Bias Test
------------------------------------
t-value prob sig

Sign Bias 1.973 0.04902 **
Joint Effect 11.259 0.01041 **
------------------------------------
1 20 22.49 0.2604
2 30 35.86 0.1777
3 40 50.31 0.1060
4 50 55.07 0.2558

Residual analysis of EGARCH model shows that the model fits
the data adequately – residuals are white noise and
show no ARCH effect. The goodness of fit test supports the
choice of a Gaussian distribution for the error term.
Although the qqplot shows that the error distribution has thicker
left tail than the normal distribution. We will fit
the t-distribution to check if that’s a better fit for the behavior
of extreme values.
Fitting an ARMA(0,0)-EGARCH(1,1) model with t-distribution
> #Fit ARMA(0,0)-eGARCH(1,1) model with t-distribution
> egarch11.t.spec=ugarchspec(variance.model=list(model =
"eGARCH",
distribution.model = "std")

> #estimate model
> egarch11.t.fit=ugarchfit(spec=egarch11.t.spec, data=ret)
> egarch11.t.fit
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
-----------------------------------
GARCH Model : eGARCH(1,1)
Distribution : std

Optimal Parameters
------------------------------------
mu 0.007093 0.001757 4.0369 0.000054
omega -0.677075 0.246000 -2.7523 0.005917
alpha1 -0.145367 0.045752 -3.1773 0.001487
beta1 0.893975 0.038715 23.0912 0.000000
gamma1 0.202717 0.055934 3.6242 0.000290
shape 7.926664 2.405188 3.2957 0.000982
Using the R output above, the ARMA(0,0)-EGARCH(1,1) model
can be written as follows.
Fitted model:

rt = 0.007 + at, at=σtet
ln(σ2t) = -0.677 + (-0.145 et-1 + 0.203(|et-1| - E(|et-1|)) + 0.894
ln(σ
2
t-1)
with t-distribution with 8 degrees of freedom (nearest integer to
shape value)
Note that since et has t-distribution, the E(|et|)=
)2/()1(
]2/)1[(22
-
distribution (denoted by shape in R output).

Note that since gamma1 is significant, the volatility has an
asymmetric behavior. Therefore a negative shock has a
stronger impact on the volatility compared to a positive shock
of the same size.
mu 0.007093 0.001888 3.7574 0.000172
omega -0.677075 0.212770 -3.1822 0.001462
alpha1 -0.145367 0.042936 -3.3857 0.000710
beta1 0.893975 0.034112 26.2071 0.000000
gamma1 0.202717 0.045692 4.4366 0.000009
shape 7.926664 2.720128 2.9141 0.003567

------------------------------------
Akaike -3.4965
Bayes -3.4445
Shibata -3.4969
------------------------------------
statistic p-value
Lag[1] 0.1872 0.6653

Lag[p+q+1][1] 0.1872 0.6653
Lag[p+q+5][5] 4.7776 0.4436
d.o.f=0
------------------------------------
statistic p-value
Lag[1] 0.6769 0.4107
Lag[p+q+1][3] 0.9526 0.3291
Lag[p+q+5][7] 2.5747 0.7652
d.o.f=2

ARCH LM Tests
------------------------------------
ARCH Lag[2] 0.9725 2 0.6149
ARCH Lag[5] 1.0934 5 0.9547
ARCH Lag[10] 5.4934 10 0.8559
------------------------------------
mu 0.20752
omega 0.12081

alpha1 0.15261
beta1 0.13304
gamma1 0.93741
shape 0.08736
Sign Bias Test
------------------------------------
t-value prob sig
Sign Bias 1.7639 0.07840 *

Joint Effect 8.8291 0.03165 **
------------------------------------
1 20 20.66 0.35571
2 30 39.23 0.09736
3 40 54.47 0.05098
4 50 66.51 0.04860
Residual analysis of EGARCH model shows that the model fits

show no ARCH effect. However the goodness of fit test shows
that the t-distribution is not a good choice for the
error terms (test p-values are small and rejects the null
hypothesis of error term having a t-distribution. The qqplot
shows that the error distribution has thicker tails than the t-
distribution. (Further analysis shows that the ged
distribution does a better job representing the extreme values
distribution)
Fitting an ARMA(0,0)-GJRGARCH(1,1) model or TGARCH
model
The GJRGARCH(1,1) model fitted by the rugarch package has
the following expression
)ln()()ln(
,
2

11
2
1111
2
tttt
tttttt
aN
eaar

where µt can follow an ARMA process, but most typically it
will be constant. Nt-1 is the indicator variable s.t.
Nt-1= 1 when at-1 (shock at time t-1) is negative, and Nt-1 = 0
otherwise. Gamma1 (γ1) is the leverage parameter. So
if gamma1 in output is significant, then we can conclude that
the volatility process has an asymmetric behavior.
> #Fit ARMA(0,0)-TGARCH(1,1) model with t-distribution
> gjrgarch11.t.spec=ugarchspec(variance.model=list(model =
"gjrGARCH",
> #estimate model
> gjrgarch11.t.fit=ugarchfit(spec=gjrgarch11.t.spec, data=ret)
Warning message:
In .makefitmodel(garchmodel = "gjrGARCH", f = .gjrgarchLLH,
T = T, :

NaNs produced
> gjrgarch11.t.fit
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
-----------------------------------
GARCH Model : gjrGARCH(1,1)
Distribution : std
Optimal Parameters

------------------------------------
mu 0.007202 0.001766 4.077739 0.000045
omega 0.000212 0.000094 2.262826 0.023646
alpha1 0.000001 0.001394 0.000791 0.999369
beta1 0.781111 0.065122 11.994488 0.000000
gamma1 0.215386 0.069611 3.094141 0.001974
shape 7.179979 2.031526 3.534279 0.000409
Using the R output above, the ARMA(0,0)-TGARCH(1,1) model
can be written as follows. Note that the arch(1)
coefficient is zero and we remove it from the model.
2

1
2
11
2
781.0)215.00000.0(00021.0
,0072.0
tttt
ttttt
aN
eaar

Where error term is assumed to have t-distribution with 7
degrees of freedom.
or
2
1
2
11
2
781.0215.000021.0
,0072.0

tttt
ttttt
aN
eaar
Where Nt-1 is an indicator variable for negative innovations
such that

aif
N
For a standardized shock with magnitude 2, (i.e. two standard
deviations), we have
966.1
002.0781.0)002.04()000.0(00021.0
002.0781.0)002.04()215.00000.0(00021.0
)2(
)2(
1
2
1
2

Where the value for
2
a is computed as
2
1
22
ttt
. Since
2
2

2
mu 0.007202 0.001921 3.749767 0.000177
omega 0.000212 0.000112 1.902791 0.057068
alpha1 0.000001 0.000054 0.020238 0.983853
beta1 0.781111 0.063980 12.208691 0.000000
gamma1 0.215386 0.056979 3.780097 0.000157
shape 7.179979 2.274951 3.156102 0.001599

------------------------------------
Akaike -3.4870
Bayes -3.4350
Shibata -3.4873
------------------------------------
statistic p-value

Lag[1] 0.1348 0.7135
Lag[p+q+1][1] 0.1348 0.7135
Lag[p+q+5][5] 4.7791 0.4434
d.o.f=0
------------------------------------
statistic p-value
Lag[1] 0.7412 0.3893
Lag[p+q+1][3] 1.0145 0.3138
Lag[p+q+5][7] 2.5887 0.7631

d.o.f=2
ARCH LM Tests
------------------------------------
ARCH Lag[2] 1.015 2 0.6020
ARCH Lag[5] 1.239 5 0.9411
ARCH Lag[10] 5.120 10 0.8830
------------------------------------

mu 0.18155
omega 0.27666
alpha1 0.05641
beta1 0.20413
gamma1 0.03198
shape 0.06486
Sign Bias Test
------------------------------------
t-value prob sig

Sign Bias 2.117 0.03474 **
Joint Effect 10.173 0.01715 **
------------------------------------
1 20 19.83 0.4048
2 30 29.37 0.4457
3 40 37.17 0.5535
4 50 52.78 0.3300

Elapsed time : 0.458046
> plot(gjrgarch11.t.fit, which="all")
Residual analysis of TGARCH model shows that the model fits
show no ARCH effect. The goodness of fit test also shows that
the t-distribution is adequate.
Apply information criteria for model selection
> # MODEL COMPARISON
> # compare information criteria
> model.list = list(garch11 = garch11.fit, garch11.t =
garch11.t.fit,
+ egarch11 = egarch11.t.fit,

+ gjrgarch11 = gjrgarch11.t.fit)
> info.mat = sapply(model.list, infocriteria)
> rownames(info.mat) = rownames(infocriteria(garch11.fit))
> info.mat
garch11 garch11.t egarch11 gjrgarch11
Akaike -3.428558 -3.468892 -3.480644 -3.487042
Bayes -3.393831 -3.425483 -3.428554 -3.434952
Shibata -3.428694 -3.469105 -3.480950 -3.487348
Hannan-Quinn -3.414909 -3.451830 -3.460170 -3.466568
Best model according to model selection criteria is the GJR-
GARCH(1,1) model with t-distribution

R CODE:
# Analysis of daily S&P500 index
#
library(fBasics)
library(tseries)
library(rugarch)
# import data in R
# import libraries for TS analysis
myd= read.table('sp500_feb1970_Feb2010.txt', header=T)
# create time series object •
rts= ts(myd$return, start = c(1970, 1), frequency=12)
# create a simple numeric object

ret =myd$return;
# CREATE TIME PLOT
plot(rts)
acf(ret)
# Plot ACF of squared data to check for non-linear dependence
acf(ret^2)
independence at lag 6
and 12
Box.test(ret^2,lag=6,type='Ljung')
Box.test(ret^2,lag=12,type='Ljung')

linear independence at lag 6
and 12
Box.test(abs(ret),lag=6,type='Ljung')
Box.test(abs(ret),lag=12,type='Ljung')
# FITTING AN GARCH(1,1) MODEL WITH GAUSSIAN
DISTRIBUTION
# Use ugarchspec() function to specify model
)),
#estimate model
garch11.fit=ugarchfit(spec=garch11.spec, data=ret)

garch11.fit
#persistence = alpha1+beta1
persistence(garch11.fit)
#half-life: ln(0.5)/ln(alpha1+beta1)
halflife(garch11.fit)
#create selection list of plots for garch(1,1) fit
plot(garch11.fit)
#to display all subplots on one page
plot(garch11.fit, which="all")
#FIT ARMA(0,0)-GARCH(1,1) MODEL WITH T-
DISTRIBUTION
# specify model using functions in rugarch package

garch11.t.spec=ugarchspec(variance.model=list(garchOrder=c(1
,1)),
#estimate model
garch11.t.fit=ugarchfit(spec=garch11.t.spec, data=ret)
garch11.t.fit
plot(garch11.t.fit)
#FIT ARMA(0,0)-EGARCH(1,1) MODEL WITH GAUSSIAN
DISTRIBUTION
"eGARCH", garchOrder=c(1,1)),
#estimate model

egarch11.fit=ugarchfit(spec=egarch11.spec, data=ret)
egarch11.fit
plot(egarch11.fit, which="all")
#FIT ARMA(0,0)-EGARCH(1,1) MODEL WITH T-
DISTRIBUTION
egarch11.t.spec=ugarchspec(variance.model=list(model =
"eGARCH", garchOrder=c(1,1)),
mean.model=list(armaOrder=c(0,0)), distribution.model =
"ged")
#estimate model
egarch11.t.fit=ugarchfit(spec=egarch11.t.spec, data=ret)
egarch11.t.fit
plot(egarch11.t.fit, which="all")

# compute expected value E(|e|)
shape=coef(egarch11.t.fit)[6]
exp.abse=(2*sqrt(shape-2)*gamma((shape+1)/2))/((shape-
1)*gamma(shape/2)*sqrt(pi))
#FIT ARMA(0,0)-TGARCH(1,1) MODEL WITH T-
DISTRIBUTION
gjrgarch11.t.spec=ugarchspec(variance.model=list(model =
"gjrGARCH",
#estimate model
gjrgarch11.t.fit=ugarchfit(spec=gjrgarch11.t.spec, data=ret)
gjrgarch11.t.fit
plot(gjrgarch11.t.fit, which="all")

#FIT ARMA(0,0)-IGARCH(1,1) MODEL WITH SKEWED T-
DISTRIBUTION
igarch11.t.spec=ugarchspec(variance.model=list(model =
"iGARCH", garchOrder=c(1,1)),
#estimate model
igarch11.t.fit=ugarchfit(spec=igarch11.t.spec, data=ret)
igarch11.t.fit
plot(igarch11.t.fit, which="all")
# MODEL COMPARISON
model.list = list(garch11 = garch11.fit, garch11.t =
garch11.t.fit,

egarch11 = egarch11.t.fit,
gjrgarch11 = gjrgarch11.t.fit)
info.mat
garch11.fit = ugarchfit(spec=garch11.spec, data=ret,
out.sample=100)
garch11.t.fit = ugarchfit(spec=garch11.t.spec, data=ret,
out.sample=100)
egarch11.t.fit = ugarchfit(egarch11.t.spec, data=ret,
out.sample=100)

tgarch11.t.fit = ugarchfit(spec=gjrgarch11.t.spec, data=ret,
out.sample=100)
# COMPUTE 100 1-STEP AHEAD ROLLING FORECASTS
W/O RE-ESTIMATING
n.ahead=1)
garch11.t.fcst = ugarchforecast(garch11.t.fit, n.roll=100,
n.ahead=1)
egarch11.t.fcst = ugarchforecast(egarch11.t.fit, n.roll=100,
n.ahead=1)
tgarch11.t.fcst = ugarchforecast(tgarch11.t.fit, n.roll=100,
n.ahead=1)
# COMPUTE FORECAST EVALUATION STATISTICS USING
FPM() FUNCTION

fcst.list = list(garch11=garch11.fcst, garch11.t=garch11.t.fcst,
egarch11.t=egarch11.t.fcst,
tgarch11.t=tgarch11.t.fcst)
fpm.mat
CSC425 – Time series analysis and forecasting
Homework 4 -

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