PS 2 EF 5070: Financial EconometricsEF 5070: Financial EconometricsProblem Set 2Due 5:00 pm, Nov 1st, 2019Notes1. Due Friday, 5:00pm, Nov 1st.2. Han Zhang, the TA of this course, will collect problem sets in her office, (AC3) 9-233,on Friday Oct 11th from 3:30 pm-5:00 pm.3. If you cannot hand in your problem set in person, please scan and send your problem set tothe TA at hzhang368@cityu.edu.hk before the deadline.4. Hand in your problem set together with the i) R codes that you used to generate the results,ii) the associated R log file, and iii) your written solution.5. Each student needs to write his/her own solutions, even though discussions of the assignmentsbetween students are encouraged.6. If not specifically specified, use 5% significance level (the associated critical value is 1.96 forstandard normal distribution) to draw conclusions in this problem set.7. Some useful R strategies:(1) For LB test, first using the R commandtsdiag(model)to examine whether there are any serial dependency in residuals.(2) For LB test, now use another R build-in command,Box.test(model$residuals,lag=12,type=’Ljung’)Compare the reported p-value with 5% significance level.(3) To fit the data with an AR model, consider the build-in R codem=ar(data,method=mle)m_order=m$orderm_aic=m$aicm_1=arima(data,order=c(m_order,0,0))m_1Page 1PS 2 EF 5070: Financial Econometricspar(mfrow=c(2,1))plot(data,xlab=’Time’,ylab=’Returns’)plot(m_aic)(4) To plot the ACF and PACF of a series, consider the build-in commanddata_acfdata_pacf(5) To fit the data with an MA model, consider the build-in R codem_1=arima(data,order=c(0,0,m_order))(6) To perform forecast and construct the 95% confidence interval, consider the following commandR codes:predlines(pred$pred,col=blue, lwd=5)lines(pred$pred+2*pred$se,col=red,lty=3, lwd=5)lines(pred$pred-2*pred$se,col=red,lty=3, lwd=5)(7) Plot several graphs, say m × n, in one page with the same scale and arrange them into mrows and n columns.par(mfcol=c(m,n))plot(...)plot(...)Page 2PS 2 EF 5070: Financial Econometrics(8) To detect a unit root process in the series, consider the following R code (the value of m inthe option k=m can be any reasonable positive integer):library(tseries)adf.test(data, k=10)1. Suppose that simple return of a monthly bond index follows an MA(2) model,Rt = 0.1 + νt − 0.8νt−1 + 0.1νt−2, (1)where νt ∼ N(0, 4).(a) What is the mean of the simple return of this monthly bond?(b) What is the variance of the simple return of this monthly bond?(c) Consider the forecast origin h = 100 with ν100 = 0.2, ν99 = −0.1 and ν98 = 1.Compute the 1-step-ahead forecast of the simple return at the forecast origin h = 100and the variance of your forecast error.(d) Based on part (c), compute the 2-step-ahead forecast of the simple return at theforecast origin h = 100 and the variance of your forecast error.(e) Based on part (c), how long do you expect the forecast value converge to its meanlevel? Explain it briefly.2. SEF 5070代写、代做Financial Econometuppose the simple daily log return of a stock follows the dynamics,rt = −0.3 + 0.1rt−3 + �t, (2)where �t ∼ N(0, 2).(a) What is the mean of the simple daily log return of this stock?(b) What is the variance of the simple daily log return of this stock?(c) Based on Part (a) and (b), is the stock simple log return stationary (with E(rt) =µ 2 (d) Consider the forecast origin h = 100 with r100 = 0.5, r99 = 0.5, r98 = −1 andr97 = −0.2. Compute the 1-step-ahead forecast (hint: rh(1) = E(rh+1|Ih))of thesimple return at the forecast origin h = 100 and the variance of your forecast error.Page 3PS 2 EF 5070: Financial Econometrics(e) Based on part (d), compute the 2-step-ahead forecast (hint: ˆrh(2) = E(rh+2|Ih)) ofthe simple return at the forecast origin h = 100 and the variance of your forecasterror.(f) Based on part (d), compute the 3-step-ahead forecast (hint: ˆrh(3) = E(rh+3|Ih)) ofthe simple return at the forecast origin h = 100 and the variance of your forecasterror.3. Consider the simple log returns of Starbucks stock from 2009 Jan 1st to 2019 Oct 1st.(a) Download the according data using the quantmod command in R.(b) Report summary statistics, including sample mean, sample variance, skewness, kurtosis,minimum and maximum of the raw data.(c) Build an AR model for the series and decide the best order p using AIC and PACF,respectively.(d) Estimate your proposed AR(p) model and provide forecasts from Oct to Oct 17th andcompare your forecast with the actual stock returns during that period.(e) Build an MA model for the series and decide the best order q using ACF. Brieflyexplain why we can choose the best order using ACF?(f) Estimate your proposed MA(q) model and provide forecast from Oct to Oct 17th andcompare your forecast with the actual stock returns during that period.(g) Is your AR(p) or MA(q) model adequate? Perform Ljung-Box test separately.4. Consider the daily VIX index. VIX, calculated and published by the Chicago Board OptionsExchange (CBOE), is widely used as a measure for market level uncertainty.(a) Please download the daily VIX index from January 1, 2009 to Oct 1, 2018 using thequantmod command in R.hint#To use a specific column of your dataset, say the 6th column inthis question, and transform it into numeric format, considerthe following command:vix(b) Plot the daily VIX index, the distribution of VIX index and its ACF in one page.(c) Is the series a nonstationary process? Why? (Consider the ADF test)Page 4PS 2 EF 5070: Financial Econometrics(d) Is the differenced series a stationary process? Why? (Consider the ADF test)(e) Build a ARMA model for the differenced VIX index, including your analysis on modeladequacy.Now, we introduce another way to identify the best order for a ARIMA model byusing a build-in command in R:auto.arima(vix)(f) Write down the fitted model.(g) Obtain 1-step to 20-step ahead forecast of the VIX index based on your model in part(c) at the forecast origin May 01, 2019. Plot your forecasting result and compare withthe true data.Page 5转自：http://www.3daixie.com/contents/11/3444.html