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Financial代写 | Econometrics代写 | EF 5070 Problem Set

2019-09-25 11:09 星期三 所属: 作业代写 浏览:7

Financial代写 Considerthe daily returns of Caterpillar (CAT) stock from January 2, 2010 to Nov 16, Download the Caterpillar data using the ’quanmod’ 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.

EF 5070: Financial Econometrics Problem Set 5     

Due 5:00 pm, Dec 7th, 2018

Financial代写
Financial代写

马尔可夫链

Notes

  1. Due Friday, 5:00pm, Dec
  2. Shuyi Cheng, the TA of this course, will collect problem sets   in her office,   AC2   5022 , onFriday Dec 7th from 3:30 pm-5:00 pm .      
  3. If you cannot hand in your problem set in person, please merge everything into ONEfile

and send that merged file to the TA at  XXX before the deadline.

  1. Hand in your problem set together with the codes that you used to generate the
  2. Each student needs to write his/her own solutions, even though discussions of the assign- ments between students are
  3. If not specifically specified, use 5% significance level (the associated critical value is 1.96 for standardnormal distribution) to draw conclusions in this problem 
  4. For this problem set, you may use the following R packages: (See R demo codes provided in Chapter 3,4 and 5 from Canvas fordetails).

 

  1. Considerthe daily returns of Caterpillar (CAT) stock from January 2, 2010 to Nov 16, Download the Caterpillar data using the ‘quanmod’ 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 r t  be the percentage log returns.

 

  • Is the expected value of r tzero? Write down the null and alternative hypothesis and the test statistics. Write down your conclusion. Consider the following R command:

 

 

  • Are there any serial correlations in r t , r 2 respectively? Why
  • Fit a Gaussian GARCH 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? Do you observe significant GARCH effect?Why?
  • Let z t r t  −  r ¯ t , where  r ¯ t  =   1  Σ n r t  is the sample mean of  r t . Fit an IGARCH(1,1)

model with a constant term to the at series z t . Write down the fitted model.

  • Let σ tbe the fitted volatility of the IGARCH(1,1) model. Define the standardized residuals as s t  = z t . Is there any serial correlation in the standardized residuals? Why? (Hint: consider the LB test). Consider the following R command:  

 

 

  • Using the provided package (garchM.R), fit a GARCH-M model to r t . Write down the fitted model. Do the mean evolutions of log returns statistically significantly depend on conditional volatility?Why?

 

 

  • Usingthe provided package (Tgarch11.R), fit a TGARCH(1,1) model to the log returns r t . Write down the fitted Is the leverage effect statistically significant? Why? I introduce two methods to implement and estimate a threshold GARCH model as follows:

Approach 1:

 

 

Please redo (g) using another R build-in package: Approach 2:

 

 

  1. Consider the daily returns of Starbucks (SBUX) stock from January 2, 2009 to Nov 15, 2017.Download the Starbucks data using the ‘quanmod’ package in 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 r t  be the percentage log returns.

  • Fit a AR model to the r tprocess and write down the fitted model. Is the model adequate?
  • Do you observe statistically significant predictability ability that lagged historical returns have on current returns? Form your hypothesis, test statistics, rejection rule and
  • Redo part (a) manually: 1) create lagged return vectors, r t 1 , · · , r t p , and dependent variable y t ; 2) regress y ton r t 1 , · · · , r t p  using the lm command in R.
  • Download the S&P500 time series from January 2, 2009 to Nov 15, 2017 from Yahoo Finance  via the  quantmod  package in Using daily closing price to construct simple returns of S&P500 index so as to form log returns. Multiple the log returns by 100 to obtain the percentage returns. Let r m,t  be the percentage log returns of S&P500, which is used to as the market return.

 

  • Now, let’s empirically investigate the CAPM theory by running the following simple marketregression:

 

r t  α  βr m,t  s t ,  where  s  ∼  iidN  (0 , σ 2 ) . (1) 

Write down the fitted model.

  • Basedon part (e), are you able to confirm the CAPM theory statistically significantly at 1% significance level? Write down your hypothesis, test statistics, rejection rule and 

Next, we are about to investigate the role nonlinearity play in determining equity prices. 1) Create a dummy variable C 1  that takes on value one if current market return is positive and zero otherwise. 2) Create a variable nsp t  that is equal to the multiplication of market returns r m,t  and the dummy variable C 1 .

  • 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 constant effect in (1). Write down your regression model, fit it to the r tseries 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 

 

 

  • 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 r tseries 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 
  • 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 effectsin (1). Write down your regression model, fit it to the  r t  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 
  • Basedon the model estimated in (i), please interpret model parameters 
  1. Consider the same Starbucks stock returns studied in Question 2, and we are about to investigate the role the market return S & P play in determining equity returns using a Markov-Chain Regime Switching
    • Please describe the Markov-chain regime switching model using three
    • Please name two appealing features that Markov-chain regime switching model has compared to the threshold modelling
    • Now, we consider a simple two-stage Regime Switching model. Please modify the simplemarket return regression in (1) into a two-stage Regime Switching model (2). Fit the model to the r t  series and write down the fitted 

 

 

r t  =



α 1  +  β 1 r m,t  γ 1 nsp t  s 1 ,t , where s 1 ,t  ∼  iidN  (0 , σ 2 ) , α 2  +  β 2 r m,t γ 2 nsp t s 2 ,t , where s 2 ,t ∼  iidN (0 , σ 2 ) ,      

 

(2)

 

  • Please interpret α 1 , β 1 , α 2 , β 2 , σ 2 and  σ 2 .

1 2

  • Do you observe any statistically significant asymmetric pattern in marginal effects in anyregime?
  • Please write down the estimated probability transition matrix, and interpret each element in that
  • Howlong do you expect the Starbucks stock to stay within each regime? Form your answer based on expected durations within each 
  1. Consider the stock returns of Caterpillar (CAT) company and the S & P composite index from Jan 2009 to Nov

 

 

  • Please download the corresponding data using the quantmod command.Construct log returns of CAT ( r t ) and S & P 500 ( r m,t ) and briefly justify your 

Now, we define the direction of price movement of CAT stock as follows: M t  = 1 if

r t > 0 and M t = 0 otherwise. S t = 1 if r m,t > 0 and M m,t = 0 otherwise.     

  • Fit a linear logistic regression model for P ( M t= 1| I t 1 ) using M t 1 , S t 1  as input. Based on your regression output, can you conclude that past price movements of either the stock or the market predict the future CAT price movements? Why?
  • Using( M t i , S t i ) for  i = 1 , 2 , 3 to build a 6-2-1 forward network with direct link for    

P ( M t  = 1). Write down the fitted model.

 

  1. Considerthe trading of Starbucks stock, data ‘taq-t-sbuxdec2031-2014.txt’ as 
    • Using the data within the normal trading hours only, say from 9:30 am to 4:00 pm Eastern time, to construct a series of intraday 0 . 1-minimute, 1-minute and 5-minute log returns. Plot the log return
    • Are there any serial correlation in the intraday log returns, which are constructedin (a)?
    • Usingthe 0 . 1-minute and 5-minute returns to compute the realized volatility for each trading 

The following questions rely on the dataset taq-sbux-pch-dec22-2014.txt, which con- tains the categorical information of price change, duration between trades, and size (volume/100). The variables are price-change category, price-change, duration, and size. Specifically, the price changes are divided into 7 categories, namely

 

< −0 . 02 ,  [−0 . 02 ,  −0 . 01) ,  [−0 . 01 ,  0) ,  0 ,  (0 ,  0 . 01] ,  (0 . 01 ,  0 . 02) , > 0 . 02 . (3)  

  • Using the ordered probit model with the following explanatory variables: lag-1 cate- gory,lag-2 category, lag-1 price change, lag-2 price change, lag-1 size, lag-2 size, lag-3 size and lag-4 size. Write down the fitted 
  • Please compute the probability of the no price change given that both lag-1, lag-2 price changes are in Category 2, actual price previous price changes are 0,0,0.001,0.02and all previous sizes are 
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