ECMT6006 Applied Financial Econometrics Semester 1, 2022
Assignment 4
Due: 11.59PM Monday 30 May 2022
Academic Honesty
Academic honesty is a core value of the University, and all students are required to act honestly,
ethically and with integrity. The consequences of engaging in plagiarism and academic dishonesty,
along with the process by which they are determined and applied, are set out in the Academic
Honesty in Coursework Policy 2015. Under the same policy, as the unit coordinator, I must report
any suspected plagiarism or academic dishonesty.
Instructions
This is an individual assignment which accounts for 8% of your final grade. You may
discuss with your classmates, but please ensure that the submitted work is independent.
You can either hand-write or type your answers, but please compile all your answers
in one PDF file and submit it via a file upload in Canvas. You can only submit your
work once, so please double check before you submit. The page limit of the submission
is 30 pages including appendix (penalty will apply if the page limit is exceeded).
There are 4 questions (with sub-questions) in this assignment and the worth of each
question is given in the square brackets. The total worth of this assignment is 60
points.
Please always show your derivations and reasoning. Answers without intermediate
steps will be considered as incomplete.
For the empirical question, please feel free to use any statistical software to answer
them. Make sure that you present the required results, including figures, and provide
your interpretations if asked. If you use MATLAB live script, you can present your
answers in a document (exported from the live script) which contains your code, output,
and your explanations in texts. If you use separate code, then please attach your code
in an appendix at the end of your submitted PDF file.
Based on the University late policy, a late submission is subject to a penalty of 5% (of
the total points) per calendar day; and work submitted more than 10 days after the
due date will receive a mark of zero.
Patton (2019) refers to the reference textbook by Andrew Patton.
Questions
1. Question 1 in Section 11.7.2 of Patton (2019, p. 408). [6pt]
2. Question 2 in Section 11.7.2 of Patton (2019, p. 409). [6pt]
3. Suppose the log price process P (t) follows the below continuous-time stochastic differ-
ence equation:
dP (t) = μ(t)dt+ σ(t)dW (t) (1)
where μ(t) and σ(t) are, respectively, drift and diffusion (volatility) processes, and
W (t) is a standard Brownian motion. We normalize the time interval with length 1 for
1 day. [18pt]
1
(i) Explain how you would model the one-minute log returns of a stock traded in a
daily six-and-half-hour market using this continuous-time model.
(ii) How are the intra-day one-minute returns distributed on day t. Here t = 1, 2, . . . .
(iii) Why the drift term is often assumed away in the intra-day high-frequency return
analysis Explain the relative magnitudes of drift and volatilty components in
the intra-day return analysis, especially when the sampling interval shrinks to
zero.
(iv) Explain how to use the heterogeneous autoregression (HAR) model for volatility
forecasting.
(v) Both HAR model and GARCH type models can provide volatility forecasts. What
are the key differences between these two methodologies
(vi) Interpret the two tables on Page 431 and 432 of the textbook.
4. (Forecasting evaluation and comparison) You can find monthly term premium on a
10-year zero coupon bond from April 1953 to December 2013 (T = 729 observationss)
in the file “term premium.xlsx”. In this exercise, you will use this dataset to (a)
generate forecasts using an ARMA(1,1) and a “Random Walk” model, (b) evaluate
the optimality of the model forecasts, and (c) compare these two model forecasts.
[30pt]
(i) Generate a time-series plot of the term premium data in the file term premium.xlsx,
with a proper x-label and y-label.
(ii) Use the first half of the sample (first R = 364 observations from April 1953 to
July 1983) to estimate a stationary ARMA(1,1) model
Yt = 0 + Yt 1 + εt + θεt 1, εt ～ i.i.d. WN(0, σ2),
and obtain the parameter estimates 0, , θ .
(iii) What is the feasible one-step-ahead forecast Y t+1|t
(iv) Let ε 0 = 0 and Y0 = 0/(1 ),1 compute Y t+1|t for t = 0, 1, . . . , T 1 and plot
them together with the real data Yt for t = 1, . . . , T .
(v) Note that Y t+1|t for t = R, . . . , T 1 are pseudo out-of-sample forecasts of the
above ARMA(1,1) model using parameter estimates from a fixed window (the
first R observations). Compute the forecast errors et+1|t = Yt+1 Y t+1|t for
t = R, . . . , T 1, and generate a time-series plot of them.
(vi) What would you expect for the serial correlation of the forecast errors if the
above forecasts are optimal Plot the sample ACF of the forecast error series
{et+1|t, t = R, . . . , T 1}, and conduct bot Ljung-Box test and robust test on
the joint serial correlation with lages L = 5, 10, 20. What is your conclusion by
examining the ACF plot and the test results
(vii) The original “Mincer-Zarnowitz” (MZ) regression regresses the realized data on
its forecast and a constant. The null hypothesis is that the intercept is 0 and the
slope is 1. Now consider an alternative and equivalent formulation:
et+1|t = α0 + α1Y t+1|t + ut+1. (2)
1Note that E(εt) = 0 and E(Yt) = 0/(1 ) in this model. Here we just set the initial values of ε and
Y to be their unconditional means.
2
What parameter restrictions would you test in this regression to examine the
optimality of Y t+1|t Run regression (2) using your data on et+1|t and Y t+1|t for
t = R, . . . , T 1, and conduct a test to draw conclusion on the optimality of your
forecasts.
(viii) Now consider an alternative model
Yt = Yt 1 + εt, εt ～ i.i.d. WN(0, σ2).
which is a “Random Walk”. What is the best one-step-ahead forecast Y t+1|t for
this model
(ix) Repeat steps (v), (vi), (vii) for the Random Walk model.
(x) Compare the forecasts from the ARMA(1,1) model and Random Walk model by
a “Diebold-Mariano” (DM) test with a squared error loss function. Specifically,
the difference of the two losses is
dt = (e
a
t+1|t)
2 (ebt+1|t)2, t = R, . . . , T 1
where eat+1|t and e
b
t+1|t denote the forecast errors from the ARMA(1,1) model and
Random Walk model respectively. Conduct a DM test by testing the zero mean
of dt using a robust t-test with Newey-West robust standard errors. Draw your
conclusion on which model is better in terms of forecasting power.
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