Question 1 (10p) CAPM vs. Dynamic CAPM Using
Kalman Filter
The file m sp500ret 3mtcm.txt contains three columns. The second column gives the
monthly returns of the SP500 index from January 1994 to December 2006. The third
column gives the monthly rates of the 3-month U.S. Treasury bill in the secondary market,
which are obtained from the Federal Reserve Bank of St. Louis and used as the risk-free
rate here. Consider the ten monthly log returns in the file m logret 10stocks.txt.
(a) (5p) For each stock, fit CAPM for the period from January 1994 to June 1998 and for
the subsequent period from July 1998 to December 2006. Are your estimated betas
significantly di↵erent for the two periods?
(b) (5p) Consider the dynamic linear model
in the Kalman filter. Plot, compare, and discuss your sequential
estimates with the estimate of beta in (a) for the period July 1998 to December 2006.
Question 2 (10p) ARMA-GARCH Modeling and Fore-
casting
The file sp500 d logret.txt contains the daily log returns on the SP500 index from
January 3 1980 to June 28, 2007.
(a) (5p) Fit an AR(1)-GARCH(1,1) model with Gaussian innovations to the data, and
give standard errors of the parameter estimates.
(b) (5p) Compute k-days-ahead forecasts (k =1,...,5) of the log returns and its volatility,
using the fitted model and June 28, 2007 as the forecast origin.
Question 3 (10p) Canonical Correlation Analysis Re-
duced Rank Regression
The file m logret 4auto.txt contains the monthly log returns of four automobile man-
ufacturers (General Motors Corp., Toyota Motor Corp., Ford Motor Co., and Honda Mo-
tor Co.) from January 1994 to June 2007. The file m logret 4soft.txt contains the
monthly log returns of four application software companies (Adobe Systems Inc., Microsoft
Corp., Oracle Corp., and SPSS Inc.) from January 1994 to June 2007.
(a) (5p) Perform. a canonical correlation analysis for these two sets of returns. Give the first
two estimated canonical variate pairs and the corresponding canonical correlations.
(b) (5p) Perform. reduced-rank regression of the log returns of automobile stocks on those
Question 4 (10p) PCA for Implied Volatility
The file impvol sp500 atm tom.txt contains at-the-money implied volatilities (i.e.,
(1,⌧)) with di↵erent times to maturity (⌧ = T � t)ofEuropeancallsontheSP500
index for the period from January 3, 2005 to April 10, 2006.
(a) (5p) Plot the implied volatility surface versus di↵erent dates and di↵erent times to
maturity. You can use wireframe. in the R package lattice or surf in MATLAB to
plot functions of two variables as surfaces.
(b) (5p) Perform. PCA for the di↵erenced series �
Question 5 (20p) Unit-Root Nonstatinarity, VAR(p)
Model Cointegration Analysis
The file m cofi 4rates.txt contains the monthly rates of the 11th District Cost of Funds
Index (COFI), the prime rate of U.S. banks, 1-year and 5-year U.S. Treasury constant
maturity rates, and U.S. Treasury 3-month secondary market rates from September 1989
to June 2007. The COFI rates are obtained from the Federal Home Loan Bank of San
Francisco, and the other rates are obtained from the Federal Reserve Bank of St. Louis.
COFI is a weighted-average interest rate paid by savings institutions headquartered in
Arizona, California, and Nevada and is one of the most popular adjustable-rate mortgage
(ARM) indices. The prime rate is the interest rate at which banks lend to their most
creditworthy customers.
(a) (5p) Perform. the augmented Dickey-Fuller test of the unit-root hypothesis for each of
these rates.
(b) (5p) Assuming the VAR(2) model for the multivariate time series of these five rates,
perform. Johansen’s test for the number of cointegration vectors.
(c) (5p) Estimate the cointegration vectors and use them to describe the equilibrium rela-
tionship between the five rates.
(d) (5p) Regress COFI on the four other rates. Discuss the economic meaning of this
regression relationship and whether the regression is spurious.
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Question 6 (20p) High Frequency Analysis
The file ibm intrastrade 200306.txt contains the transaction data on the New York
Stock Exchange for IBM stock in June 2003. The data are obtained from Wharton Research
Data Services.
(a) (5p) Let x
denote the number of trades in the ith 5-minute interval. Ignoring the time
gaps between trading days, this gives the time series x
of the number of trades on
IBM stock in 5-minute intervals on the NYSE in June, 2003. Plot the time series and
its ACF. Determine if there are intraday period patterns in the series.
(b) (5p) Using the last transaction price in the ith 5-minute interval as the stock price in
that interval, plot the time series y
of 5-minute log returns during the period and the
corresponding ACF.
(c) (5p) Consider the bivariate time series (x
(d) (5p) Tabulate the relative frequencies of price changes in multiples of the tick size
$0.0625.
Question 7 (20p) Value-At-Risk Prediction
The file intel d logret.txt contains daily log returns of Intel stock from July 9, 1986
to June 29, 2007. Compute the 99% 1-day and 10-day VaR for a long position of $1 million
using the following methods (as internal models in Section 12.1.2 of the textbook):
(a) GARCH(1,1) model with standard normal ✏
(b) EGARCH(1, 1) model with standard normal ✏
(c) ARMA(1, 1)-GARCH(1, 1)model with ✏
havingthestandardizedStudent t-distribution
whose degrees of freedom are to be estimated from the data;
(d) the GEV distribution for extreme (negative) returns with subperiod length of 20 trad-
ing days.