Description
1. GARCH(1,1) estimated 4 ways. (total 4 points)
(a) (1 point) The GARCH(1,1) likelihood can be written as a function of the four unknown
parameters α, β, σ, and σ1. Use the continuously compounded returns you compute from the
SPX data to estimate the GARCH(1,1) parameters, treating all four as unknown parameters to be
estimated. What are the estimated parameters α, β, σ, and σ1? (Note that the question asks for
the estimates of σ, and σ1, not σ2 and σ1
2
.)
Remark: You should do this using R and one of the general optimization functions such as
optim(). To do this you will have to write your own R function to compute the negative of the
log likelihood.
(b) (1 point) Using the same data, set the long-run variance σ2 equal to the sample variance of the
data used for estimation. If you do this, the GARCH(1,1) likelihood can be written as a function
of the three unknown parameters α, β, and σ1. Use maximum likelihood to estimate the three
parameters α, β, and σ1. What is your estimate of the long-run standard deviation σ? What are
your estimates of the parameters α, β, and σ1?
(c) (1/2 point) Next set the initial variance σ1
2 equal to the sample variance over the data period
used for estimation. If you do this, the GARCH(1,1) likelihood can be written as a function of
the three unknown parameters α, β, and σ. Use maximum likelihood to estimate the three
parameters α, β, and σ. What is your estimate of the initial standard deviation σ1? What are
your estimates of the parameters α, β, and σ?
(d) (1/2 point) Now combine the approaches in (b) and (c), that is set both the long-run variance
σ2 and the initial variance σ1
2 equal to the sample variance of the data used for estimation. If
you do this, the GARCH(1,1) likelihood can be written as a function of the two unknown
parameters α and β. Use maximum likelihood to estimate the two parameters α and β . What
are your estimates of the long-run standard deviation σ and the initial standard deviation σ1?
What are your estimates of the parameters α and β?
2
(e) (1 point) Use each of the four models estimated in parts (a)-(d) to forecast the return standard
deviation for February 24, 2020. What are the four forecasts of the return standard deviation?
2. Using the GARCH(1,1) model to forecast volatility (total 2 points) This question asks you
to use the four sets of estimates from Question (1) to compute four forecasts of the variance
variance = �𝑅𝑅𝑡𝑡
2
21
𝑡𝑡=1
over the first 21 days following February 21, 2020. That is, assume that the current date and
time are just after the close of trading on February 21, 2020, you have data up through the close
of trading on February 21, 2020, and you have estimated the GARCH(1,1) models in Question 1.
You now want to forecast the realized variance over the 21 trading days starting from February
24, 2020.
(a) (1 point) What are your four forecasts of the realized variance?
(b) (1 point) Express your four forecasts in terms of annualized volatilities. (Take the square
root, and then annualize by multiplying by 252 / 21 . Or else annualize by multiplying by
252/21, and then take the square root.)
3. GARCH(1,1) using existing R functions (total 2 points) In this question use the same
continuously compounded returns you used in Question 1.
(a) (1 point) Use the R function garch() from the R package tseries to estimate the parameters of
a GARCH(1,1) model. (https://www.rdocumentation.org/packages/tseries/versions/0.10-
42/topics/garch) Does the function return estimates of all four parameters α, β, σ, and σ1? If
not, which parameters does it estimate? What are the estimated parameters?
Hint: Use your estimates from Question 1(a) as your initial guesses.
(b) (1 point) Now use the R function garchFit() from the R package fGarch to estimate the
parameters of a GARCH(1,1) model. (See https://cran.rproject.org/web/packages/fGarch/fGarch.pdf) Does the function return estimates of all four
parameters α, β, σ, and σ1? If not, which parameters does it estimate? What are the estimated
parameters?
4. NGARCH(1,1) (2 point) The NGARCH(1,1) likelihood can be written as a function of the
four unknown parameters α, β, θ, σ, and σ1. Use the continuously compounded returns you
compute from the SPX data to estimate the NGARCH(1,1) parameters, treating all five as
unknown parameters to be estimated. What are the estimated parameters α, β, θ, σ, and σ1?
Remark: You should do this using R and one of the general optimization functions such as
optim(). To do this you will have to write your own R function to compute the (negative of the)
log likelihood.
