Description
0. Gather the data. (0 points, but necessary for the other questions) Login in to WRDS, go to
the data vendor CRSP, and find the daily stock returns data. Download the daily stock returns
for the ticker symbols SPY (SPDR S&P 500 ETF Trust), EEM (iShares MSCI Emerging
Markets ETF), HYG (iShares iBoxx $ High Yield Corporate Bond ETF), FXI (iShares China
Large-Cap ETF), USO (United States Oil Fund, LP), and GLD (SPDR Gold Shares) for the
period January 1, 2016 through December 31, 2019. (The first return should be for January 4,
2016, which was the first trading date in 2016, and the last return should be for December 31,
2019.)
1. Compute continuously compounded or log returns. (1 point) The returns you download
from CRSP will be simple returns. Convert them to continuously compounded returns. What are
the average continuously compounded returns for each of the six series?
2. Estimate a DCC model. (total 5 points) Use the six return series to estimate the DCC(1,1)
model with normally distributed shocks. In doing this you should model the variance of each of
the return series using a GARCH(1,1) model with normally distributed shocks, and assume that
the means of the daily returns are zero.
(a) (3 points) What are the parameter estimates ω, α, and β of each of the six GARCH(1,1)
models? (Your answer should consist of 18 numbers.)
(b) (2 points) What are the parameter estimates α and β of the DCC(1,1) model? (Your answer
should consist of two numbers.)
Remarks. I suggest that you use the R package rmgarch, which includes a function dccfit() that
will estimate the DCC model. To use dccfit(), you must first use dccspec() to specify the model.
To use dccspec(), you should first use multispec() to specify the six GARCH processes for the
six return series. To use multispec(), you should first use ugarchspec() to specify the GARCH
model for each of the return series. That is, you should do something like:
##specify a GARCH(1,1) model assuming the mean returns are zero
uspec <- ugarchspec(variance.model = list(model = “sGARCH”, garchOrder = c(1,1)),
mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
distribution.model = “norm”)
## Combine the univariate specifications of the 6 GARCH models
marginspec <- multispec(replicate(6, uspec))
2
## Create DCC(1,1) specification
mspec <- dccspec(marginspec, dccOrder = c(1,1), model = “DCC”, distribution = “mvnorm”)
## Fit the DCC(1,1) model (returns are in a matrix returns[])
mod <- dccfit(mspec,returns[,1:6])
The package rmgarch also includes functions to simulate GARCH and DCC models, but I will
not ask you to use them on this homework.
3. Principal components. (total 4 points) Now let’s carry out a principal components analysis
of the returns of the six ETFs. Carry out the analysis on the covariance matrix, not the correlation
matrix. In computing the covariance matrix, assume that the expected returns of the stocks are
zero and use the equally-weighted estimator
Σ� = 1
𝑇𝑇�𝑟𝑟𝑡𝑡𝑟𝑟𝑡𝑡′
𝑇𝑇
𝑡𝑡=1
,
where rt ∈ℜ6 is a vector of the returns of the six ETFs on date t.
(a) (1 point) First use the R function eigen() to compute the eigenvectors and eigenvalues. (You
will first need to compute the covariance matrix of the returns.) What is the standard deviation of
the first principal component?
(b) (1 point) Suppose there is a one-standard deviation change in the first principal component,
and the second through sixth principal components remain unchanged. In this scenario, what is
the return on the FXI?
(c) (1 point) Next use the R function prcomp() to carry out the principal components analysis of
the returns. What fraction of the variance of the return on the SPY is explained by the first
principal component?
(d) (1 point) Use the first three principal components to estimate the covariance matrix of the
returns on the six ETFs. Based on this estimate of the covariance matrix, what is the covariance
of the returns between the SPY and FXI?
Remark. To check that you are doing this correctly, note that if you use all six principal
components you will recover the covariance matrix.
