ECON1195 Financial Econometrics Assignment 3 March 2021 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
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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.
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