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ECON1195 Financial Econometrics Assignment 3
This is an individual assignment comprises 50% of the overall assessment. It consists of Eight questions.
You need to attempt All the Questions. This assignment is based on the relevant course materials
(lectures, practice exercises, R exercises, etc). It covers the lecture materials between week 1 and week 12. This assignment is due for submission on Canvas by Friday, 11 June 2021, 11.59PM Melbourne time. Answers can be typed or handwritten and scanned. You also need submit your R script on Canvas. Academic Integrity/plagiarism: You can achieve academic integrity by honestly submitting work that is your own. Presenting work that fails to ac- knowledge other people’s work within yours can compromise academic integrity. Submission guidelines: All work for Assessable Tasks is required to be submitted on the due date and time as outlined in the Assessment Briefs. The exception to this is where an approved ELS plan, an application for Special Consideration or an approved Extension of Time is in place, submitted before the task’s due date with appropriate documentation. Re-submission: can only be authorised in specific circumstances by formal RMIT committees.
for all information regarding adjustments to assessable work. Late Submission: Work submitted within 7 calendar days of a due (or an approved amended due) date may be accepted in exceptional circumstances but will only be assessed as Pass (50%) or Fail. Work submitted beyond 7 calendar days of a due date will be assessed as 0%. 1 Question 1 Let rt denotes the return of a financial asset and σt denotes the standard deviation of returns at time t. If rt follows an ARMA(2,2) model, rt = φ0 + φ1rt−1 + φ2rt−2 + et + θ1et−1 + θ2et−2, (a) Derive the unconditional mean of rt, E(rt) (show all necessary steps and conditions). (b) For given information available at time t, derive the 1-step, 2-step and 3-step ahead forecasts of rt (show all necessary steps and conditions). (c) If we estimate the ARMA(2,2) model, we obtain φ0 = 0, φ1 = 0.5, φ2 = 0.2, θ1 = −0.8 and θ2 = 0.6, compute the E(rt=3|It=2), E(rt=4|It=2) and E(rt=5|It=2) based on the information provided in the Table 1. Table 1: Monthly returns t rt et 1 3.1 0.3 2 -1.1 0.5 3 ? - 4 ? - 5 ? - 2 Question 2 Consider the following ARMA(1,1)-ARCH(3) model, rt = φ0 + φ1rt−1 + et + θ1et−1, et = σtzt, σ2t = ω + α1e 2 t−1 + α2e 2 t−2 + α3e 2 t−3, (a) Derive the unconditional variance of rt, var(rt) (show all necessary steps and conditions). (b) For given information available at time t, derive the 1-step, 2-step and 3- step ahead forecasts of variance of rt (show all necessary steps and conditions). (c) If we estimate the model, we obtain φ0 = 0.1, φ1 = 0.3, θ1 = 0.6, ω = 0.15, α1 = 0.3, α2 = 0.2 and α3 = 0.1, compute the var(rt=4|It=3), var(rt=5|It=3) and var(rt=6|It=3) based on the information provided in the Table 2. Table 2: Monthly returns t rt et 1 1.5 1 2 -1.3 - 3 2.2 - Consider the following GARCH(2,2) model, rt = µ+ et, et = σtzt, σ2t = ω + α1e 2 t−1 + α2e 2 t−2 + β1σ 2 t−1 + β2σ 2 t−2, (d) Derive the unconditional variance of rt, var(rt) (show all necessary steps and conditions). (e) For given information available at time t, derive the 1-step, 2-step and 3- step ahead forecasts of variance of rt (show all necessary steps and conditions). (f) If we estimate the model, we obtain µ = 0.95, φ0 = 0.1, φ1 = 0.3, θ1 = 0.6, ω = 0.15, α1 = 0.3, α2 = 0.2 and β1 = 0.15, β2 = 0.1. And we have σ21 = 3.5, σ 2 2 = 2.5 and σ 2 3 = 4. Compute the var(rt=4|It=3), var(rt=5|It=3) and var(rt=6|It=3) based on the information provided in the Table 3. Table 3: Monthly returns t rt et 1 1.5 - 2 -1.3 - 3 2.2 - 3 Question 3 The dataset ’nasdaq.csv’ contains only adjusted closing price of Nasdaq in- dex. You are employed as an analyst by an investment bank in the U.S. Assume that you bought a certain amount of Nasdaq index with US$600,000 for the bank. (a) According to daily returns of Nasdaq index, decide the percentile of the daily return, below which there are 5% observed Nasdaq daily returns. What is the value at risk (VaR) for the bank’s holding of Nasdaq index during the next 24 hours at the 99% level of confidence? Assume that the daily return follows the standard normal distribution. (b)What is the one-day VaR for the bank’s holding of Nasdaq index at the 99% level of confidence? Assume that the daily return follows the Student’s t distribution. (c) What is the one-day VaR for the bank’s holding of Nasdaq index at the 99% level of confidence? (d) Discuss and compare the VaR estimated in (b) and (c). (e) What is the one-week VaR for the investor’s holding of Nasdaq index at the 95% level of confidence? (f) What is the one-month VaR for the investor’s holding of Nasdaq index at the 99% level of confidence? Assume that the daily return follows the standard normal distribution. Fit an AR(1)-GARCH(1,1) model, rt = φ0 + φ1rt−1 + et, et = σtzt, σ2t = ω + α1e 2 t−1 + β1σ 2 t−1, where zt are independent and identically distributed as the standard normal distribution. (g) Estimate the model and write down the estimated model; (h) Calculate the one-day conditional value-at-risk for the bank’s holding of Nasdaq index at the 95% confidence level; (i) Is this estimation of VaR reasonable? Briefly explain; Assume that the daily return follows the Student’s t distribution. Fit the RiskMetrics model, rt = µ+ et, et = σtzt, σ2t = (1− λ)r2t−1 + λσ2t−1, where λ = 0.94; and zt are independent and identically distributed as the Stu- dent t distribution with 6 degrees of freedom. (j) Calculate the one-day conditional value-at-risk for the bank’s holding of Nasdaq index at the 99% confidence level. 4 Question 4 We have obtained four plots in the graph, they are: (a) line plot of rt; (b) histogram of rt; (c) Acf of rt; (d) Acf of r 2 t ; (a) What does the line plot of rt tell you? Briefly explain; (b) What does the histogram of rt tell you? Briefly explain; (c) What does the acf of rt tell you? Briefly explain; (d) What test can be used as an alternative to the question (c)? Explain how it works? (e) What does the acf of r2t tell you? Briefly explain; (f) What test can be used as an alternative to the question (e)? Explain how it works? 5 Question 5 We have obtained the correlogram of rt in the graph, based on this correlo- gram, (a) Is it reasonable to fit an AR(1) model? Briefly explain; (b) Is it reasonable to fit an AR(2) model? Briefly explain; (c) Is it reasonable to fit an MA(1) model? Briefly explain; (d) Is it reasonable to fit an MA(2) model? Briefly explain; (e) Is it reasonable to fit an ARMA(1,1) model? Briefly explain; (f) Is it reasonable to fit an ARMA(1,2) model? Briefly explain; (g) Is it reasonable to fit an ARMA(2,1) model? Briefly explain; 6 Question 6 Based on the news impact curve in the graph (left panel); (a) What does this plot tell? (b) What model is most likely to produce this type of news impact curve? (c) Write down the model and explain how this model captures this partic- ular pattern. We have obtained and plotted 100-step ahead forecasts of σt in the graph (right panel); Consider the following two ARCH models, ARCH(1) : σ2t = ω + α1e 2 t−1; (1) ARCH(4) : σ2t = ω + α1e 2 t−1 + α2e 2 t−2 + α3e 2 t−3 + +α4e 2 t−4; (2) (d) Which model is more likely to produce the forecast plots in the graph, ARCH(1) or ARCH(4)? Explain briefly. 7 Question 7 In this question, please use α = 5% to make decision in all hypothesis test questions. The dataset ’index returns.csv’ contains daily returns of 4 different market indices: SP500: the continuously compound return of the US standard&Poor 500 index; Nikkei: the continuously compound return of the Japan Nikkei index; HS: the continuously compounded return of the Hong Kong Heng Seng index; ASX : the continuously compounded returns of the Australia ASX 200 index; (a) Obtain the line plots of all Four time series in one graph, make sure the labels for x- and y- axis are readable and appropriate. Comment on the graph. (b) Perform hypothesis test to check whether we can apply the VAR to all four time series; (c) Write down a VAR(2) model in the matrix form for the four time series; (d) Is it reasonable to fit the VAR(2) model? (e) If it is reasonable to fit the VAR(2) model, estimate the VAR(2) model and write down the estimated model. If it is not reasonable to fit the VAR(2) model, estimate an appropriate model and write down the estimated model. (f) Test for Granger causality between four time series and interpret the results; (g) Is the result consistent with your expectation? Provide an economic interpretation of the results. (h) Obtain plots of impulse response analysis for 8 periods (ie, n.ahead=8) and interpret the results. 8 Question 8 In this question, please use α = 10% to make decision in all hypothesis test questions. The Purchasing Power Parity (PPP) states that an amount of money should purchase the same quantity of goods in any country when the amount of money is converted to that countries domestic currency at the market exchange rate. For example, if the exchange rate is AUD 1= US$0.88 and a Big Mac is priced at $3.57 in the US, then the price of a Big Mac should be at the price of $3.57/0.88=AUD 4.06 in Australia according to the PPP theory; The above example is also referred as the Law of One Price, Pt = EtP ∗ t , where Pt is the domestic currency price, P ∗ t is the foreign currency and Et is the exchange rate at time t. After the re-arrangement, we have Et = Pt/P ∗ t . In the dataset ’ppp aus us.csv’,
we have Pt =Australia, P ∗ t =US, Et =ner. (a) Read the data into R, and obtain the log version of Et and
log version of the ratio Pt/P ∗ t , denoted as log(Et) and log(Pt/P ∗ t ), respectively. Generate the line plots of log(Et) and log(Pt/P ∗ t ). Comment on the plots. (b) Check whether we can apply the cointegration to analysis of the rela- tionship between log(Et) and log(Pt/P ∗ t )? As an implication, PPP theory asserts that there is a long-run relationship between the nominal exchange rate (NER) and aggregate price indices. This long-run relationship can be demonstrated by using the following model, log(Et) = α+ β log(P ∗ t /Pt) + et, (3) where we know α = 0 and β = 1. (c) Conduct a co-integration test to check whether there is a long run rela- tionship in the model; (d) Let assume the coefficients α and β are unknown in the model, conduct a co-integration test. Is the result consistent with your answer in (c)? (e) Use the regression method to check whether the relationship between log(Et) and log(P ∗ t /Pt) is spurious. (f) If there’s a disequilibrium between the nominal exchange rate (NER) and aggregate price indices, an adjustment will take place in order to restore the long run equilibrium according to the PPP theory. Use the error correction model (ECM) to study this adjustment in the short run. Discuss your finding in the ECM. 9
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