Empirical Methods in Finance

Homework 6

Problem 1: ARCH, GARCH, and Realized Variance

From Kenneth Frenchís web page, download the monthly and daily returns to the investment (CMA) factor from 19630701 to 20191231. CMA is the last factor in the FF 5-factor

model. Using the monthly return data:

1. Estimate an ARMA(1,1) for the return series. Report the results. What is the estimated monthly persistence of expected returns to CMA? What is the half-life of the

expected return series in months?

2. Estimate an ARCH(12) and a GARCH(1,1) process for the residuals from this ARMA(1,1).

Report the results and plot the time series of the conditional variance from each model

on the same plot. Are the estimated variance processes stationary?

3. Plot the absolute values of the normalized residuals, t = “t=t

, for each model on two

separate plots. Using eyeball econometrics, do the models do a good job of accounting

for clustering of volatility? Plot the autocorrelation functions of jt

j.

4. Using the daily data on CMA, estimate monthly realized variance for month t as

RVt =

PNt

d=1 r

2

t;d; (1)

where r

2

t;d is the squared return of day d in month t and where Nt

is the number of

days in month t. Plot the resulting monthly time series of RVt

.

1

5. What are the Örst order autocorrelations of RVt and ”

2

t

? What is the correlation

between RVt and ”

2

t

? What is the correlation between RVt and

2

t

from the GARCH

model? What is the correlation between ”

2

t and

2

t

from the GARCH model?

6. Run an ARMA(1,1) on RVt assuming normally distributed errors (which strictly speaking canít be correct). Report the results. Let vt = Et1 [RVt

] where the expectation

is obtained from the estimated ARMA. What is the correlation between vt and RVt?

Plot on the same graph the time series of vt and

2

t

from the GARCH.