PS 3 EF 5070: Financial Econometrics
EF 5070: Financial Econometrics
Problem Set 3
Due 5:00 pm, Nov 30th, 2020
Notes
1. Due Monday, 5:00pm, Nov 30th.
2. Please submit your problem set zip files which contains all related material into CANVAS by
the deadline. Late submissions will not be accepted.
3. Hand in your problem set together with the i) R codes that you used to generate
the results (print out your script file), ii) the associated R log file (print out your
console window output), and iii) your written (typed) solution.
4. Each student needs to write his/her own solutions, even though discussions of the assignments
between students are encouraged.
5. If not specifically specified, use 5% significance level (the associated critical value is 1.96 for
standard normal distribution) to draw conclusions in this problem set.
6. For this problem set, you may use the following R packages: (See R demo codes provided in
Chapter 4 from Canvas for details).
library(’TSA’)
library(’fGarch’)
library(’parallel’)
library(’rugarch’)
1. Consider the daily VIX index. VIX, calculated and published by the Chicago Board Options
Exchange (CBOE), is widely used as a measure for market level uncertainty.
(a) Please download the daily VIX index from January 1, 2006 to Nov 1, 2020 using the
quantmod command in R.
hint
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PS 3 EF 5070: Financial Econometrics
#To use a specific column of your dataset, say the 6th column in
this question, and transform it into numeric format, consider
the following command:
vix<-as.numeric(VIX[,6])
(b) Use ARCH(q) model with the default Gaussian distribution as a baseline model to fit
the VIX dataset. (1) Write down an equivalent AR representation for the ARCH(q)
process. (2) Explain and show how the best order q can be determined. (3) Write
down the fitted ARCH(q) and its equivalent AR representation.
(c) Now, use GARCH(1,1) model with the default Gaussian distribution to fit the VIX
data. (1) Write down an equivalent ARMA representation for the GARCH(m,s)
process. (2) Write down the fitted GARCH(1,1) model.(3) Do you observe significant
GARCH effect at 5% level?
(d) Next, use GARCH(1,1) model with a student-t distribution to fit the VIX data. Write
down the fitted model. Hint, consider the following R commands.
model<-garchFit(~garch(m,s),data=,cond.dist=’std’,trace=F)
(e) Fit the dataset with a ARMA(p,q)-GARCH(1,1) model. (1) Please explain and show
how to choose the ARMA orders, (p,q). (2) Write down the fitted model. (3) Please
briefly explain why we would prefer GARCH(1,1) over ARCH(q) when modeling the
latent dynamic volatility process here.
(f) Which model would you prefer to explain the evolutions of the market volatility, VIX
dataset? Briefly explain.
2. Consider the daily returns of Pfizer stock from January 2, 2009 to Nov 1, 2020. Download
the Starbucks data using the ’quantmod’ package in R. Using daily closing price to construct
simple returns so as to form log returns. Multiple the log returns by 100 to obtain the
percentage returns. Let rt be the percentage log returns.
(a) Is the expected value of rt zero? Write down the null and alternative hypothesis and
the test statistics. Write down your conclusion. Consider the following R command:
t.test(rt)
(b) Are there any serial correlations in rt and r
2
t ? Why
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PS 3 EF 5070: Financial Econometrics
(c) Fit a Gaussian ARMA(p,q)-GARCH(1,1) model to the rt series. Obtain the normal
QQplot of the standardized residuals (hint: plot(model)), and write down the fitted
model. Is the model adequate? Why?
(d) Let zt = rt − r¯t, where r¯t = 1n
∑n
i=1 rt is the sample mean of rt. Fit an IGARCH(1,1)
model with a constant term to the at series zt. Write down the fitted model.
(e) Let σt be the fitted volatility of the IGARCH(1,1) model. Define the standardized
residuals as t =
zt
σt
. Is there any serial correlation in the standardized residuals?
Why? (Hint: consider the LB test). Consider the following R command:
sresi=zt/model$volatility
(f) Using the provided package (garchM.R), fit a GARCH-M model to rt. Write down the
fitted mode. Do the mean evolutions of log returns statistically significantly depend
on conditional volatility? Why? What is estimated level of risk premium?
> source("garchM.R")
> model=garchM(data)
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