ECMT6006/ECON4949/ECON4998 Applied Financial
Econometrics 2021 S1 Final Exam
Note: This is an open-book online exam. There are in total five problems
and 25 questions. Please attempt all questions. The total mark is 50 and each
question is worth 2 marks. You will have 3 hours to answer the questions with
handwritten solutions, and 30 minutes to scan and upload your solutions into
Canvas.
Problem 1. [14 marks] Answer the following questions on the return forecasta-
bility.
(i) We introduced three concepts for different levels of dependence which may
exisit in a financial asset return series. What are they?
(ii) How are these three concepts for different levels of dependence related to
each other?
(iii) Take a simple AR(1) model for example, explain the relationship between
return predictability and autocorrelation of the returns.
(iv) If you cannot reject the null hypothesis that a return time series follows a
white noise process at a certain significance level (say 1%), what can you
say about the forecastability of this asset return?
(v) What is the limitation of using ARMA type of models to examine return
forecastability?
(vi) Why do we say that Value-at-Risk (VaR) is often a better risk measure
than volatility in the financial industry? How is return VaR related to the
return volatility?
(vii) Why is it important to forecast the return VaR and Expected Shortfall in
the financial risk management?
Problem 2. [12 marks] Consider the following stationary AR(1)-GARCH(1,1)
model for stock returns:
Rt = φ0 + φ1Rt−1 + εt
εt = σtνt
σ2t = ω + αε
2
t−1 + βσ
2
t−1
νt|Ft−1 ∼ F (0, 1)
where |φ1| < 1, ω > 0, α, β ≥ 0, α+β < 1, σt > 0, Ft−1 denotes the information
set up to time t − 1, and F (0, 1) denotes some distribution with mean 0 and
variance 1.
(i) Show that {ε2t} is an ARMA(1,1) process.
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(ii) Show that the GARCH(1,1) model for {εt} is equivalent to ARCH(∞).
(iii) Decompose the unconditional variance of Rt into two components con-
tributed from its conditional mean and conditional variance. Which com-
ponent dominates the other as you see from the empirical data in class?
(iv) How would you modify the model to capture the potential “leverage effect”
in the stock market?
(v) How would you obtain the feasible one-step ahead two-standard-deviation
interval forecast of the returns at time t?
(vi) How would you construct the feasible two-step ahead two-standard-deviation
interval forecast of the returns at time t?
Problem 3. [6 marks] We decompose an asset return Rt into the conditional
mean µt and the error component εt as
Rt = µt + εt,
and consider choosing a volatility model for εt.
(i) Using the squared residuals εˆ2t as the volatility proxy, we estimate the
following Mincer-Zarnowtiz (MZ) regression
εˆ2t = β0 + β1ht + ut
where ht is the actual volatility forecast from a certain volatility model,
and ut is the regression error term. How would you use a hypothesis test
to evaluate whether a certain forecast ht is optimal, based on the above
regression? Write down the null and alternative hypotheses and explain
how to conduct the test.
(ii) Consider three different volatility models to obtain the volatility forecast
ht. They are (a) pure GARCH model, (b) pure diurnal model, and (c)
GARCH model with diurnal component. The estimation and testing re-
sults are summarized in the below table. Please interpret the results of
the MZ regression on evaluating the optimality of each model.
(iii) The results of the DM tests with MSE or QLIKE loss functions are also
reported in the above table. Interpret the results.
Problem 4. [10 marks] Answer the below questions on Value-at-Risk (VaR)
and Expected Shortfall (ES) forecasting models.
(i) Explain how to use the RiskMetrics model and the Normal-GARCH(1,1)
model to forecast the 1% return VaR and ES.
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(ii) How are the RiskMetrics model and the Normal-GARCH(1,1) model re-
lated, and how are they different from each other?
(iii) How would you use the generalised Mincer-Zarnowitz regression to test
whether the 1% return VaR forecasts V aRRMt+1 for t = 1, . . . , T from the
RiskMetrics model is optimal? Be explicit about the regression model, the
null and alternative hypotheses, and how to test the null hypothesis.
(iv) How would you use the generalised Mincer-Zarnowitz regression to test
whether the 1% return ES forecasts ESRMt+1 for t = 1, . . . , T from the
RiskMetrics model is optimal? Be explicit about the regression model, the
null and alternative hypotheses, and how to test the null hypothesis.
(v) Suppose you have obtained two VaR forecasts, V aRRMt+1 and V aR
GH
t+1 for
t = 1, . . . , T , from the RiskMetrics model and the Normal-Garch(1,1)
model. How would you compare these two forecasts using the Diebold-
Mariano test?
Problem 5. [8 marks] Answer the below questions on volatility forecast eval-
uation.
(i) Explain how you evaluate the one-day ahead forecasts of a daily volatility
model using (pseudo) out-of-sample method.
(ii) Explain why we need to use volatility proxy for evaluating the performance
of a volatility model.
(iii) What is the proposed proxy if we only have daily data for the stock price?
(iv) How can we improve the daily volatility forecast evaluation tests using
intra-day stock price data? Explain why there is a such improvement.