PS 4 EF 5070: Financial Econometrics
Notes
EF 5070: Financial Econometrics Problem Set 4
1. Due Sat, 5:00pm, Nov 30th.
2. TA will not collect any physical copies. Please submit your electronic version of the problem set into CANVAS and late submission will not be accepted by the system after the deadline.
3. 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.
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, it only contains one questions with several subquestions. Question 2 is completely optimal. You are welcome watching the Panapto Video in Canvas if you are interested in learning how to do Markov Chain Regime Switching Models. Not doing Question 2 will not affect your grade. But you could choose to work on it for bonus points.
7. For this problem set, you may use the following R packages: (See R demo codes provided in Chapter 4 from Canvas for details).
1. Consider the daily returns of the Caterpillar stock from January 2, 2011 to Nov 15, 2019. Download the Starbucks 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.
(a) Fit a AR model to the rt process and write down the fitted model. Is the model adequate?
library(’MSwM’)
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PS 4
(b)
(c) (d)
EF 5070: Financial Econometrics Do you observe statistically significant predictability ability that lagged historical returns have on current returns? Form your hypothesis, test statistics, rejection rule
and conclusion.
Redo part (a) manually: 1) create lagged return vectors, rt−1, · · · , rt−p, and dependent
variable yt; 2) regress yt on rt−1, · · · , rt−p using the lm command in R.
Download the S&P500 time series during the same 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. Multiple 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.
Now, let’s empirically investigate the CAPM theory by running the following simple market regression:
rt =α+βrm,t+εt,whereε∼i.i.d.N(0,σ2). (1) Write down the fitted model.
Based on 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 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.
Basedonthesimplemarketregressionmodel(1),whileholdingotherfactorsunchanged, 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 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.
Basedonthesimplemarketregressionmodel(1),whileholdingotherfactorsunchanged, 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
(e)
(f)
(g)
(h)
getSymbols(’^GSPC’,env=sp500,src=’yahoo’,from=’’,end=’’)
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(i)
(j)
EF 5070: Financial Econometrics a statistically significant asymmetric pattern in mean at 5% level? Write down your
hypothesis, test statistics, rejection rule and conclusion.
Basedonthesimplemarketregressionmodel(1),whileholdingotherfactorsunchanged, 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.
Based on the model estimated in (i), please interpret model parameters respectively.
2. (BonusPoints20pts,seelecturenotesandPanpatoinCanvas)ConsiderthesameCaterpillar stock returns studied in Question 1, 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 three 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 +ε1,t, whereε1,t ∼i.i.d.N(0,σ12), (2)
α2 +β2rm,t +γ2nspt +ε2,t, whereε2,t ∼i.i.d.N(0,σ2),
(d) Please interpret α1, β1, α2, β2, σ12 and σ2.
(e) Do you observe any statistically significant asymmetric pattern in marginal effects in any regime?
(f) Please write down the estimated probability transition matrix, and interpret each element in that matrix.
(g) How long do you expect the Starbucks stock to stay within each regime? Form your answer based on expected durations within each regime.
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