# EF 5070辅导、辅导R编程语言

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EF 5070: Financial Econometrics

Problem Set 3

Due 5:00 pm, Dec 1st, 2022

Notes

1. Due 5:00pm, Dec 1st.

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 Op-

tions Exchange (CBOE), is widely used as a measure for market level uncertainty.

(a) Please download the daily VIX index from January 1, 2006 to Oct 22, 2022 using the

quantmod command in R.

hint

Page 1

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 Walmart stock from January 2, 2017 to Nov 1, 2022. Down-

load the Walmart data using the ’quantmod’ package in R. Using daily closing price to

construct simple returns so as to form log returns. Multiply 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.

Method 1: First, consider the following R command and draw your conclusion.

Page 2

PS 3 EF 5070: Financial Econometrics

t.test(rt)

Method 2: Second, consider a regression model to testify whether rt has zero mean.

rt = α + ?t.

(b) Are there any serial correlations in rt and r2t ? Performance a test and justify your

conclusion.

(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 ? rˉt, where rˉt = 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 = ztσt . 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)

(g) Using the provided package (Tgarch11.R), fit a TGARCH(1,1) model to the log re-

turns rt. Write down the fitted model. Is the leverage effect statistically significant?

Why? I introduce two methods to implement and estimate a threshold GARCH model

as follows:

Approach 1:

Page 3

PS 3 EF 5070: Financial Econometrics

> source("Tgarch11.R")

> model=Tgarch11(data)

Please redo (g) using another R build-in package:

Approach 2:

model=garchFit(~garch(1,1),data=data,trace=F,leverage=T)

3. Consider the daily returns of Walmart stock from January 2, 2018 to Nov 15, 2022 again.

Download the Walmart data using the ’quanmod’ package in R. Using daily closing price

to construct simple returns so as to form log returns. Multiply the log returns by 100 to

obtain the percentage returns. Let rt be the percentage log returns.

Download the S&P500 time series during the same sampling period from Yahoo Finance

via the quantmod package in R. Using daily closing price to construct simple returns of

S&P500 index so as to form log returns. Multiply the log returns by 100 to obtain the

percentage returns. Let rm,t be the percentage log returns of S&P500, which is used to as

the market return.

getSymbols('^GSPC',env=sp500,src='yahoo',from='',end='')

(a) Now, let’s empirically investigate the CAPM theory by running the following simple

market regression:

rt = α + βrm,t + ?t,where ? ～ i.i.d.N(0, σ2). (1)

Write down the fitted model.

(b) Based on part (a), are you able to confirm the CAPM theory statistically significantly

at 1% significance level? Write down your hypothesis, test statistics, rejection rule

and conclusion.

Next, we are about to investigate the role nonlinearity play in determining equity

prices. 1) Create a dummy variable C1 that takes on value one if current market

return is positive and zero otherwise. 2) Create a variable nspt that is equal to the

multiplication of market returns rm,t and the dummy variable C1.

Page 4

PS 3 EF 5070: Financial Econometrics

(c) Based on the simple market regression model (1), while holding other factors un-

changed, please design a new threshold market regression model that allows you to

examine whether nonlinearity only affects the abnormal return in (1). Write down

your regression model, fit it to the rt series and write down your fitted model. Is there

a statistically significant asymmetric pattern in mean at 5% level? Write down your

hypothesis, test statistics, rejection rule and conclusion.

(d) Based on the simple market regression model (1), while holding other factors un-

changed, please design another threshold market regression model that allows you

to examine whether there is any asymmetric pattern only in marginal effects in (1).

Write down your regression model, fit it to the rt series and write down your fitted

model. Is there a statistically significant asymmetric pattern in mean at 5% level?

Write down your hypothesis, test statistics, rejection rule and conclusion.

(e) Based on the simple market regression model (1), while holding other factors un-

changed, please design another threshold market regression model that allows you

to examine whether there is any asymmetric pattern in both constant and marginal

effects in (1). Write down your regression model, fit it to the rt series and write down

your fitted model. Is there a statistically significant asymmetric pattern in mean at

5% level? Write down your hypothesis, test statistics, rejection rule and conclusion.

4. Consider the same Walmart stock returns studied in Question 3, and we are about to

investigate the role the market return S&P play in determining equity returns using a

Markov-Chain Regime Switching model.

(a) Please describe the Markov-chain regime switching model using two sentences.

(b) Please name two appealing features that Markov-chain regime switching model has

compared to the threshold modelling procedure.

(c) Now, we consider a simple two-stage Regime Switching model. Please modify the

simple market return regression in (1) into a two-stage Regime Switching model (2).

Fit the model to the rt series and write down the fitted model.

rt =α1 + β1rm,t + γ1nspt + ?1,t, where?1,t ～ i.i.d.N(0, σ21),α2 + β2rm,t + γ2nspt + ?2,t, where?2,t ～ i.i.d.N(0, σ22), (2)

(d) Please interpret α1, β1, α2, β2, σ21 and σ22.

(e) Do you observe any statistically significant asymmetric pattern in marginal effects in

any regime?

Page 5

PS 3 EF 5070: Financial Econometrics

(f) Please write down the estimated probability transition matrix, and interpret each

element in that matrix.

(g) How long do you expect the Walmart stock to stay within each regime? Form your

answer based on expected duration within each regime.

EF 5070: Financial Econometrics

Problem Set 3

Due 5:00 pm, Dec 1st, 2022

Notes

1. Due 5:00pm, Dec 1st.

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 Op-

tions Exchange (CBOE), is widely used as a measure for market level uncertainty.

(a) Please download the daily VIX index from January 1, 2006 to Oct 22, 2022 using the

quantmod command in R.

hint

Page 1

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 Walmart stock from January 2, 2017 to Nov 1, 2022. Down-

load the Walmart data using the ’quantmod’ package in R. Using daily closing price to

construct simple returns so as to form log returns. Multiply 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.

Method 1: First, consider the following R command and draw your conclusion.

Page 2

PS 3 EF 5070: Financial Econometrics

t.test(rt)

Method 2: Second, consider a regression model to testify whether rt has zero mean.

rt = α + ?t.

(b) Are there any serial correlations in rt and r2t ? Performance a test and justify your

conclusion.

(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 ? rˉt, where rˉt = 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 = ztσt . 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)

(g) Using the provided package (Tgarch11.R), fit a TGARCH(1,1) model to the log re-

turns rt. Write down the fitted model. Is the leverage effect statistically significant?

Why? I introduce two methods to implement and estimate a threshold GARCH model

as follows:

Approach 1:

Page 3

PS 3 EF 5070: Financial Econometrics

> source("Tgarch11.R")

> model=Tgarch11(data)

Please redo (g) using another R build-in package:

Approach 2:

model=garchFit(~garch(1,1),data=data,trace=F,leverage=T)

3. Consider the daily returns of Walmart stock from January 2, 2018 to Nov 15, 2022 again.

Download the Walmart data using the ’quanmod’ package in R. Using daily closing price

to construct simple returns so as to form log returns. Multiply the log returns by 100 to

obtain the percentage returns. Let rt be the percentage log returns.

Download the S&P500 time series during the same sampling period from Yahoo Finance

via the quantmod package in R. Using daily closing price to construct simple returns of

S&P500 index so as to form log returns. Multiply the log returns by 100 to obtain the

percentage returns. Let rm,t be the percentage log returns of S&P500, which is used to as

the market return.

getSymbols('^GSPC',env=sp500,src='yahoo',from='',end='')

(a) Now, let’s empirically investigate the CAPM theory by running the following simple

market regression:

rt = α + βrm,t + ?t,where ? ～ i.i.d.N(0, σ2). (1)

Write down the fitted model.

(b) Based on part (a), are you able to confirm the CAPM theory statistically significantly

at 1% significance level? Write down your hypothesis, test statistics, rejection rule

and conclusion.

Next, we are about to investigate the role nonlinearity play in determining equity

prices. 1) Create a dummy variable C1 that takes on value one if current market

return is positive and zero otherwise. 2) Create a variable nspt that is equal to the

multiplication of market returns rm,t and the dummy variable C1.

Page 4

PS 3 EF 5070: Financial Econometrics

(c) Based on the simple market regression model (1), while holding other factors un-

changed, please design a new threshold market regression model that allows you to

examine whether nonlinearity only affects the abnormal return in (1). Write down

your regression model, fit it to the rt series and write down your fitted model. Is there

a statistically significant asymmetric pattern in mean at 5% level? Write down your

hypothesis, test statistics, rejection rule and conclusion.

(d) Based on the simple market regression model (1), while holding other factors un-

changed, please design another threshold market regression model that allows you

to examine whether there is any asymmetric pattern only in marginal effects in (1).

Write down your regression model, fit it to the rt series and write down your fitted

model. Is there a statistically significant asymmetric pattern in mean at 5% level?

Write down your hypothesis, test statistics, rejection rule and conclusion.

(e) Based on the simple market regression model (1), while holding other factors un-

changed, please design another threshold market regression model that allows you

to examine whether there is any asymmetric pattern in both constant and marginal

effects in (1). Write down your regression model, fit it to the rt series and write down

your fitted model. Is there a statistically significant asymmetric pattern in mean at

5% level? Write down your hypothesis, test statistics, rejection rule and conclusion.

4. Consider the same Walmart stock returns studied in Question 3, and we are about to

investigate the role the market return S&P play in determining equity returns using a

Markov-Chain Regime Switching model.

(a) Please describe the Markov-chain regime switching model using two sentences.

(b) Please name two appealing features that Markov-chain regime switching model has

compared to the threshold modelling procedure.

(c) Now, we consider a simple two-stage Regime Switching model. Please modify the

simple market return regression in (1) into a two-stage Regime Switching model (2).

Fit the model to the rt series and write down the fitted model.

rt =α1 + β1rm,t + γ1nspt + ?1,t, where?1,t ～ i.i.d.N(0, σ21),α2 + β2rm,t + γ2nspt + ?2,t, where?2,t ～ i.i.d.N(0, σ22), (2)

(d) Please interpret α1, β1, α2, β2, σ21 and σ22.

(e) Do you observe any statistically significant asymmetric pattern in marginal effects in

any regime?

Page 5

PS 3 EF 5070: Financial Econometrics

(f) Please write down the estimated probability transition matrix, and interpret each

element in that matrix.

(g) How long do you expect the Walmart stock to stay within each regime? Form your

answer based on expected duration within each regime.