Microsoft Word – Tutorial 8 T2 2021.docx
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Financial Econometrics T2 2021
Tutorial 8
1. (Miscellaneous questions)
(a) Why do we care about the conditional variance matrix of a vector of returns? What
are required for a matrix to be a “variance matrix”?
(b) What is the daily realised variance of an asset return? What does it estimate?
2. (Realised volatility, data source: http://realized.oxford-man.ox.ac.uk/data)
This exercise is based on on the data in sp500rv-tut11.xls. The time series are related
to the S&P500 index. There are 3 columns in the file: Date, RV (daily realised variance of
the index, 5-minute intervals), Close (daily close price of the index), spanning from 2000-01-
03 to 2012-10-05 with a total of 3181 observations. The goal of this exercise is to examine
the empirical characteristics of the realised variance.
(a) Compute the daily return R
Compute the daily log realised variance LRV
Compute the daily realised standard deviation RSD by typing
and the standardised return SR=R/RSD
Make the time series plots of R, LRV and SR and comment. Find the descriptive
statistics and histograms of R, LRV and SR and comment. Here, “comment” means
“describe the features of the associated objects”.
(b) Find the correlograms of R and LRV and comment.
(c) Is there a unit-root in LRV? If you are to fit an ARMA(𝑝, 𝑞) model to LRV, what
would be the 𝑝 and 𝑞? Estimate your ARMA model, check the residual for autocorrelation
and comment. Also is the residual normally distributed?
(d) Define the residual from the model in part (c) as E, and the fitted (or predicted) value
as F = LRV−E. The conditional variance of R can be approximated by CV = A × exp(F)
where A is the sample mean of exp(E). Then, the conditional standard deviation CSD is
computed as the square root of CV. Make a scatter plot of RSD agains CSD and comment.
(e) Estimate an AR(1)-EGARCH(2,1) model for the return R, which only relies on the
return series. Define the conditional standard deviation from the EGARCH(2,1) as EGSD.
Make time series plots EGSD and CSD, as well as a scatter plot of EGSD against CSD.
Compute the cross correlation of EGSD and CSD and comment.
(f) Re-estimate the AR(1)-EGARCH(2,1) model for R, including LRV(-1) in the
variance equation, ie,
ln(𝜎!”) = 𝛼# + 𝛼$|𝑣!%$| + 𝛾𝑣!%$ + 𝛼”|𝑣!%”| + 𝛽$ ln(𝜎!%$” ) + 𝜓LRV!%$.
Compare the estimation results to those in part (e) and comment.
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