Case EXT_GARCH: Volatility Dynamics - Asymmetric Response

After the introduction of ARCH and GARCH models, a number of extensions and modifications have been proposed. Among them, two important ones are discussed in this lesson. The first one is TARCH (Threshold ARCH) and the second is EGARCH (Exponential GARCH).

In a well known empirical study of stock volatility, Black (1978) observed that “when things go badly for the firm, its stock price will fall, and the volatility of the stock will go up. When things go well, the stock price will rise and the volatility of the stock will go down. A negative return will be tied to a rise in volatility, and a positive return will be tied to a fall in volatility.” Translating this quotation into volatility modeling, >0 will reduce the volatility and will increase it. Both TARCH and EGARCH models allow asymmetric effect of the residuals.

A. TARCH (Asymmetric)

TARCH (1,1) model:

where d = 1 if >0 and d = 0 if , so that:

when >0, and

when .

is called the leverage effect.

Example:

We continue to work on the CISCO stock return.

Dependent Variable: CSCO_R
Method: ML - ARCH (Marquardt)
Sample(adjusted): 6/26/1996 8/18/1997
Included observations: 299 after adjusting endpoints
Convergence achieved after 12 iterations
Bollerslev-Wooldrige robust standard errors & covariance
Variance backcast: ON
Coefficient / Std. Error / z-Statistic / Prob.
Variance Equation
C / 1.19E-05 / 6.92E-06 / 1.711585 / 0.0870
ARCH(1) / -0.037183 / 0.017981 / -2.067891 / 0.0387
(RESID<0)*ARCH(1) / 0.127564 / 0.031171 / 4.092385 / 0.0000
GARCH(1) / 0.960923 / 0.022371 / 42.95407 / 0.0000
R-squared / -0.002878 / Mean dependent var / 0.001467
Adjusted R-squared / -0.013077 / S.D. dependent var / 0.027386
S.E. of regression / 0.027565 / Akaike info criterion / -4.442796
Sum squared resid / 0.224144 / Schwarz criterion / -4.393292
Log likelihood / 668.1981 / Durbin-Watson stat / 1.902019

Note: (RESID<0)*ARCH(1) is the estimated .

B. EGARCH (Exponential)

EGARCH (1,1) model in Eview.

Dependent Variable: CSCO_R
Method: ML - ARCH (Marquardt)
Sample(adjusted): 6/26/1996 8/18/1997
Included observations: 299 after adjusting endpoints
Convergence achieved after 20 iterations
Variance backcast: ON
Coefficient / Std. Error / z-Statistic / Prob.
C / 0.000308 / 0.001400 / 0.219967 / 0.8259
Variance Equation
C / -0.105459 / 0.072564 / -1.453323 / 0.1461
|RES|/SQR[GARCH](1) / -0.018766 / 0.034270 / -0.547599 / 0.5840
RES/SQR[GARCH](1) / -0.129056 / 0.024733 / -5.217943 / 0.0000
EGARCH(1) / 0.983057 / 0.007633 / 128.7926 / 0.0000
R-squared / -0.001796 / Mean dependent var / 0.001467
Adjusted R-squared / -0.015426 / S.D. dependent var / 0.027386
S.E. of regression / 0.027597 / Akaike info criterion / -4.463431
Sum squared resid / 0.223902 / Schwarz criterion / -4.401551
Log likelihood / 672.2830 / Durbin-Watson stat / 1.904073

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