ECON 3350/7350: Applied Econometrics for Macroeconomics and Finance
Tutorial 8: Volatility Models – II
1. Consider the daily share prices of Merck & Co., Inc. (MRK) for the period 2 January 2001 to 23 December 2013 in the data file Merck.csv. Let {yt} denote the time series of the share prices. Recall that we learned how to fit ARMA- ARCH/GARCH models to data last week. Now we consider extensions of these models to capture possible leverage effects and risk premia.
(a) Select and estimate a preferred ARMA(p, q)-TARCH model for the log return series rt = 100 × log(yt/yt−1). Report estimation results and test the existence of leverage effects. Write out the expression for the volatility forecast and compute the predicted volatility for the following four trading days.
(b) Repeat Part (a) but estimate a preferred ARMA(p, q)-EGARCH model.
(c) Fit a GARCH-M model to the data {rt}. Report the estimated model and test the existence of a (time-varying) risk premium. Write out the expression for the volatility forecast and compute the predicted volatility for the following four trading days.
Solution: See the do-file tutorial8.do.
(a) We compare ARMA(p, q)-TARCH models with p, q up to 3 and select the ARMA(2, 2)-TARCH specification as it has the smallest AIC/BIC and all ARMA and TARCH coefficients are significant. The estimated model is
rt = 0.0066 + 0.2514rt−1 − 0.9959rt−2 − 0.2470εt−1 + 1.0002εt−2 + εt (0.0325) (0.0015) (0.0014) (0.0005) (0.0004)
ε = v h ttt
ht = 0.0551 + 0.0404ε2t−1 − 0.0551dt−1ε2t−1 + 0.9676ht−1 (0.0042) (0.0037) (0.0051) (0.0022)
where dt−1 = 1[εt−1 > 0]. Notice that here we employs the same conven- tion in defining the “threshold” dummy dt−1 as Stata, which is different from the textbook (check this!). The “threshold” coefficient is −0.0551 and significantly different from 0. Hence, the t-test result provides strong evidence for the existence of leverage effects. To see this, note that when εt−1 > 0, ht = 0.0551 − 0.0147ε2t−1 + 0.9676ht−1. When εt−1 < 0, ht = 0.0551 + 0.0404ε2t−1 + 0.9676ht−1. The (one-step ahead) forecast of the volatility is based on the following expression:
E(ε2 |ε,ε ,...)=E(v2 h |ε,ε ,...)=h t+1 t t−1 t+1 t+1 t t−1 t+1
= 0.0551 + 0.0404ε2t − 0.0551dtε2t + 0.9676ht
1
(b) We compare ARMA(p, q)-TGARCH models with p, q up to 3 and select the ARMA(2, 1)-EGARCH specification as it has the smallest AIC/BIC. The estimated model is
rt = −0.0439 + 1.0130rt−1 − 0.0135rt−2 − 1.0006εt−1 + εt (0.0117) (0.0163) (0.0163) (0.0005)
ε = v h ttt
ln(ht) = 0.0145 − 0.0728vt−1 + 0.0074(|vt−1| − 2/π) + 0.9879 ln(ht−1) (0.0014) (0.0044) (0.0017) (0.0011)
Notice that we assume vt ∼ N(0,1) here. The coefficient on vt−1 = ε /h1/2 is−0.0728andsignificantlydifferentfrom0,whichindicatesthe
t−1 t−1
existence of leverage effects. The (one-step ahead) forecast of the volatil-
ity is based on the following expression:
E(ε2t+1|εt, εt−1, ...)
=E(v2 h |ε ,ε ,...) = h
t+1 t+1 t t−1 t+1
= exp(0.0145 − 0.0728vt−1 + 0.0074(|vt−1| − 2/π) + 0.9879 ln(ht−1))
(c) Here we fit a standard GARCH-M model to the data. The estimated model can be expressed as
rt = −0.0535 + 0.0213ht + εt (0.0779) (0.0236)
ε =vh ttt
ht = 0.2263 + 0.0537ε2t−1 + 0.8774ht−1 (0.0144) (0.0046) (0.0077)
Notice that here we follow the textbook (and also Stata’s) convention to
ht (as in your lecture slides) in the conditional mean model. The GARCH-M model is not favorable to the existence of a time-varying risk premium as the coefficient on ht in the conditional mean model is not significant. The expression for the volatility forecast is the same as standard GARCH(1, 1) models. You can try some ARMA specifications for the conditional mean model of rt by using the arima
option. Check if you have the same conclusion about the risk premium.
include ht rather than
√
2