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EXTENSIONS OF GARCH PROCESSES

1. Integrated GARCH Process

Recall from Topic 4 (p. 8) that if tε follows a GARCH(1,1) process, then it can
be shown that 2tε has an ARMA(1,1) representation, namely,

2 20 1 1 1 1 1( )t t t tv vε α α β ε β− −= + + − + (1)

where 2 2t t tv ε σ= − is the difference between the squared innovation and the conditional
variance at time t. In many applications, we find that 1 1α β+ is approximately one.
When 1 1 1α β+ = , equation (1) becomes

2 20 1 1 1t t t tv vε α ε β− −= + − + (2)

so that there is a unit root in the squared residuals 2tε . Equation (2) can be written as:

2 2 2 20 1 1 1t t t t t tv v whereε α β ε ε ε− −∆ = − + ∆ = −

Because the there is a unit root in the squared residuals (they are stationary in first
differences), the model is called an Integrated GARCH(1,1), also known as the
IGARCH(1,1) model.
Recall from Topic 4 notes (p. 8), that the h-step ahead forecast of the conditional
variance from a GARCH(1,1) model is:

0 1 1 1 1 1 1 1 1 1( | ) [1 ( ) ( ) ( ) ] ( )

t h t tE σ α α β α β α β α β σ

+ +Ω = + + + + + + + + +

lim ( | ) var( )

When 1 1 1α β+ = ,

2 20 1( | ) ( 1)t h t tE hσ α σ+ +Ω = − + (4)

so that the forecast of the conditional variance becomes larger and larger as h increases.
In the limit, as h → ∞ , the forecast of the conditional variance becomes infinitely large,
meaning that the unconditional variance of the process is infinite (or undefined) as can be
seen from equation (3) upon substituting 1 1 1α β+ = .

2. Asymmetric GARCH Models

In the GARCH (or ARCH) models that we have discussed so far, a positive or
negative shock last period (that is, 1tε − ) will have the same impact on today’s volatility
because the squared of 1tε − enters the model only. However, negative shocks appear to
contribute more to stock market volatility than do positive shocks. This is called the
leverage effect. A negative shock to aggregate stock prices reduces the aggregate market
value of equity relative to the aggregate market value of corporate debt. Thus the
likelihood of corporate bankruptcy increases as firms are more highly leveraged. This
increases the risk of holding stocks.

The simplest GARCH model allowing for asymmetric response is the threshold
GARCH or the TGARCH model. In this model the GARCH(1,1) conditional variance
function is replaced with:

0 1 1 1 1 1 1

0, 0, 0, 0

σ α α ε γε β σ

The dummy variable tD keeps track of whether the lagged residual is positive or
negative. When 1 0tε − ≥ , the effect of the lagged squared residual on the current
conditional variance 2( )tσ is simply 1α . In contrast, when 1 0tε − < , 1D = so that the effect of the lagged squared residual on the current conditional variance is 1α γ+ . If 0γ = , the response is symmetric and we have the standard GARCH(1,1) model. If 0γ ≠ , there is an asymmetric response of the conditional variance to “news”, the lagged residual. If there are leverage effects, 0γ > so that negative shocks have a bigger impact
on the conditional variance than do positive shocks.
Asymmetric response may also be introduced by way of the exponential GARCH
or EGARCH model:

2 21 10 1 1 1

ln( ) ln( )t tt t

σ α α γ β σ

= + + + (6)

There are three important characteristics of the EGARCH model. First, the log of the
conditional variance is being modeled not the conditional variance itself. Regardless of
the magnitude of 2ln( )tσ , the implied value of

tσ can never be negative. Thus, it is

permissible for the coefficients (in equation (6)) to be negative. In other words, the log
specification ensures that the conditional variance is always positive because 2tσ is
obtained by exponentiating 2ln( )tσ . Second, instead of using the value of

1tε − , the

EGARCH model uses the absolute value of the standardized value of 1tε − (that is, 1tε −
divided by it standard error 1tσ − ) as the measure of the size of a shock. Note that the
standardized value of 1tε − is a unit free measure. Third, the EGARCH model allows for
asymmetric response of the log of the conditional variance to “news”. The sign of the
“news” is captured by the term 1 1/t tε σ− − . If 1 1/t tε σ− − is positive, the effect of the
standardized shock on the conditional variance is 1α γ+ . If 1 1/t tε σ− − is negative, the
effect of the standardized shock on the conditional variance is 1α γ− . If 0γ < , the effect of a negative standardized shock is larger than that of a positive shock so that there is evidence for a leverage effect. 3. Tests for Leverage Effects First estimate the mean equation with, say, a GARCH(1,1) specification for the variance equation, by maximum likelihood methods and form the standardized residuals To test for leverage effects, one could estimate a regression of the form η− − −= + + + + + 0 1 1 2 2t t t n t k ts a a s a s a s (7) where tη is the regression disturbance. If there are no leverage effects, the squared standardized residuals should be uncorrelated with the levels of the standardized residuals. If the regression slope coefficients were negative and statistically significant, that would indicate negative shocks are associated with large values of the conditional variance and, thus, there are leverage effects. Engle and Ng (1993) developed a second way to determine whether positive and negative shocks have different effects on the conditional variance. Let The Sign Bias test uses the regression equation of the form 2 0 1 1t t ts a a D η−= + + (8) where tη is the regression disturbance. If a t-test indicates that 1a is statistically different from zero, the sign of the current period shock is helpful in predicting volatility. In particular, if 1a is positive and statistically different from zero, negative shocks tend to increase the conditional variance. To generalize the test, one could estimate the regression: 2 0 1 1 2 1 1 3 1 1(1 )t t t t t t ts a a D a D s a D s η− − − − −= + + + − + Note that 1(1 )tD −− assigns a value of one to positive or zero shocks. The presence of 1 1t tD s− − and 1 1(1 )t tD s− −− is designed to determine whether the effects of positive and negative shocks on the conditional variance depend on their size. Statistical significance of 2a and 3a would suggest the presence of size bias, where not only the sign (indicated by the statistical significance of 1a ) but also the magnitude or size of the shock is important for predicting the conditional variance. 4. Leverage Effects in the Composite NYSE Index Recall from Topic 4 notes that we estimated an MA(1)-GARCH(1,1) model for the percentage daily logarithmic change in the NYSE index, denoted tsr , over the period January 3, 1995 to August 30, 2002, a total of 1,931 observations. Having done this, we now save the standardized residuals from this model (denoted ts ) and estimate the regression given by equation (7) for three lags. The results are shown in Table 1. Table 1: Estimation of Regression Equation for Leverage Effects Dependent Variable: S2 Method: Least Squares Sample (adjusted): 5 1931 Included observations: 1927 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 0.974715 0.046069 21.15789 0 S(-1) -0.15996 0.045941 -3.48179 0.0005 S(-2) -0.25772 0.045936 -5.6104 0 S(-3) -0.0882 0.045937 -1.92 0.055 R-squared 0.024002 Mean dependent var 1.000512 Adjusted R-squared 0.022479 S.D. dependent var 2.037487 S.E. of regression 2.014457 Akaike info criterion 4.24065 Sum squared resid 7803.602 Schwarz criterion 4.252199 Log likelihood -4081.87 F-statistic 15.76358 Durbin-Watson stat 2.075408 Prob(F-statistic) 0 The coefficients on 1ts − , 2ts − and 3ts − are negative and statistically significant. Thus, negative shocks are associated with large values of the conditional variance, suggesting the presence of leverage effects. Table 2 reports the results of the sign bias test given by equation (8). Table 2: Results of the Sign Bias Test Dependent Variable: S2 Method: Least Squares Sample (adjusted): 3 1931 Included observations: 1929 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 0.637753 0.126028 5.060401 0 D(-1) 0.418212 0.135489 3.086678 0.0021 R-squared 0.00492 Mean dependent var 0.999597 Adjusted R-squared 0.004404 S.D. dependent var 2.036632 S.E. of regression 2.032143 Akaike info criterion 4.257095 Sum squared resid 7957.747 Schwarz criterion 4.262864 Log likelihood -4103.97 F-statistic 9.527584 Durbin-Watson stat 1.967774 Prob(F-statistic) 0.002053 Since the coefficient on ( 1)D − is positive and significant, we again conclude that negative shocks tend to increase the conditional variance of tsr . In view of these findings, we estimated the MA(1)-TGARCH(1,1) model. The results are reported in table 3. The coefficient on the asymmetric term is 0.1948. It is positive and statistically significant. Thus, there is evidence for leverage effects in the returns to the NYSE Composite Index. It is interesting to compare the value of the likelihood function from the MA(1)- TGARCH model, which is 2475.02− with that from the MA(1)-GARCH(1,1) model, which is 2516.63− . It is valid to make such a comparison since the MA(1)- TGARCH(1,1) model nests the MA(1)-GARCH(1,1). In other words, the MA(1)- GARCH(1,1) model can be viewed as a restricted model with respect to the MA(1)- TGARCH(1,1) model since it is obtained from the latter when the coefficient on the asymmetric term is restricted to be zero. Clearly the maximized value of the likelihood function from the MA(1)-TGARCH(1,1) model is larger than that from the MA(1)- GARCH(1,1) model. We would expect this since the coefficient on the asymmetric term in the TGARCH model is highly statistically significant. Nevertheless, we could perform a likelihood ratio test of the restriction that the coefficient on the asymmetric term is zero as follows: 2( 2516.63 ( 2475.02)) R ULR LL LL= − − The LR statistic is distributed as a 2(1)χ since there is only one restriction here. Since 0.0583.2 (1) 3.841χ> = , we reject the null that there is no asymmetric response and

conclude that the MA(1)-TGARCH(1,1) model is better than the MA(1)-GARCH(1,1)

Table 3: Results of Estimation of MA(1)-TGARCH(1,1) Model

Dependent Variable: SR
Method: ML – ARCH (Marquardt) – Normal distribution
Sample (adjusted): 2 1931
Included observations: 1930 after adjustments
Convergence achieved after 19 iterations
MA backcast: 1, Variance backcast: ON
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0) + C(6)*GARCH(-1) Mean Equation Coefficient Std. Error z-Statistic Prob. C 0.041635 0.020227 2.05835 0.0396 MA(1) 0.114205 0.023915 4.775395 0 Variance Equation C 0.018143 0.002799 6.482478 0 RESID(-1)^2 -0.00644 0.009726 -0.66206 0.5079 RESID(-1)^2*(RESID(-1)<0) 0.194754 0.015796 12.32918 0 GARCH(-1) 0.893656 0.009227 96.85522 0 R-squared 0.003704 Mean dependent var 0.0353 Adjusted R-squared 0.001115 S.D. dependent var 1.006207 S.E. of regression 1.005646 Akaike info criterion 2.571005 Sum squared resid 1945.788 Schwarz criterion 2.588306 Log likelihood -2475.02 F-statistic 1.430784 Durbin-Watson stat 2.081274 Prob(F-statistic) 0.209991 The results of estimating the MA(1)-EGARCH(1,1) model are shown in table 4. The coefficient on the asymmetric term (shown in the table as C(5)) is 0.15524− . Since this coefficient is negative and statistically significant, there is evidence for a leverage effect, that is negative shocks have a bigger impact on the log of the conditional variance than do positive shocks. We cannot compare the maximized log-likelihood value from the MA(1)-EGARCH(1,1) model with that from the MA(1)-GARCH(1,1) model since the models are not nested: the GARCH model cannot be viewed as a restricted EGARCH model since in the EGARCH the log of the conditional variance is being modeled whereas in the GARCH, the level of the conditional variance is being modeled. Table 4: Results of Estimating MA(1)-EGARCH(1,1) Model Dependent Variable: SR Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 2 1931 Included observations: 1930 after adjustments Convergence achieved after 16 iterations MA backcast: 1, Variance backcast: ON LOG(GARCH) = C(3) + + + C(6)*LOG(GARCH(-1)) Mean Equation Coefficient Std. Error z-Statistic Prob. C 0.027938 0.019166 1.457706 0.1449 MA(1) 0.11858 0.023734 4.996317 0 Variance Equation C(3) -0.10572 0.013741 -7.69377 0 C(4) 0.126307 0.016369 7.716313 0 C(5) -0.15524 0.011328 -13.7042 0 C(6) 0.96458 0.004179 230.8339 0 R-squared 0.003355 Mean dependent var 0.0353 Adjusted R-squared 0.000765 S.D. dependent var 1.006207 S.E. of regression 1.005822 Akaike info criterion 2.555701 Sum squared resid 1946.47 Schwarz criterion 2.573003 Log likelihood -2460.25 F-statistic 1.295357 Durbin-Watson stat 2.089712 Prob(F-statistic) 0.263051 5. Exogenous Variables in the GARCH Specification Sometimes it is useful to include an exogenous variable in the variance equation. For example, financial market volume often helps to explain financial market volatility. In this case, the standard GARCH(1,1) model would be augmented in the following way 2 2 20 1 1 1 1t t t txσ α α ε β σ γ− −= + + + where γ is a parameter and tx is a positive exogenous variable, for example, the volume of trades on the NYSE today. 6. GARCH-in-Mean Models The GARCH(1,1)-in-Mean model (which is written in abbreviated form as GARCH(1,1)-M) is: σ α α ε β σ Let ty be the return on a financial asset or portfolio. Then 1 0 1( | )t t tE y a a σ−Ω = + . Thus, the conditional mean return depends on the conditional standard deviation. Since the conditional standard deviation can be viewed as a measure of the risk associated with the asset or portfolio, the specification for the mean equation captures the notion in finance of a trade-off between mean return and risk. The mean return is time-varying since tσ is time-varying. Only in the case of where 1 0a = is the mean return constant, although there is time-varying volatility in the model given by the GARCH(1,1) specification. Note that in some empirical applications the conditional variance rather than the conditional standard deviation appears in the mean equation. As a practical matter, if there appears to be a shift in the conditional mean of ty in response to changing volatility, then that is indicative of a GARCH-M process. Table 5 presents a GARCH-in-Mean model for the term premium between the three and six month U.S. zero coupon bonds. Table 5: GARCH(1,1)-in-Mean Model for the Term Premium Dependent Variable: TERM Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 1947M01 1987M02 Included observations: 482 after adjustments Convergence achieved after 32 iterations Variance backcast: OFF GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1) Mean Equation Coefficient Std. Error z-Statistic Prob. STDDEV 0.380607 0.076123 4.999884 0 C 0.003369 0.001901 1.772456 0.0763 TERM(-1) 0.712542 0.039053 18.24564 0 Variance Equation C 2.75E-06 6.21E-06 0.443379 0.6575 RESID(-1)^2 0.385388 0.038328 10.05507 0 GARCH(-1) 0.756158 0.015696 48.17537 0 R-squared 0.484561 Mean dependent var 0.223071 Adjusted R-squared 0.479147 S.D. dependent var 0.220262 S.E. of regression 0.158963 Akaike info criterion -1.72331 Sum squared resid 12.0282 Schwarz criterion -1.6713 Log likelihood 421.3169 F-statistic 89.49705 Durbin-Watson stat 2.176771 Prob(F-statistic) 0 The term premium is defined as the yield to maturity on six month bills less the yield to maturity on three month bills. The data cover the period December 1946 to February 1987. The coefficient on the conditional standard deviation is positive and statistically significant as expected since the higher the risk, the higher the term premium required on the long bond relative to the short bond. Also included in the mean equation is the lagged term premium to account for serial correlation. It is apparent that the term premium is quite persistent. Finally, 1 1 1.15α β+ = , which is quite a bit larger than one, violating the sign restrictions on the model. 7. Maximum Likelihood Estimation of the ARMA-GARCH Models Consider the ARMA(1,1)-GARCH(1,1) model: The only observed series we have is { } y . Thus we will have to reconstruct { } σ from observed { } y . We do so iteratively and need to assume values for t=0: .σ Usually, we set 0ε = and 2 2 σ σ= , where 2σ is the (unconditional) sample variance. Then, for given values of parameters , , , , ,cγ φ θ α β and given 0ε = , 2 2 y we can compute All subsequent values of { } ε and 2{ } σ are reconstructed in a similar way: Next step is to specify the likelihood, that is the joint probability to observe specific values of { } y for given , , , , ,cγ φ ϕ α β and 0ε = , 2 2 σ σ= and maximize it for given 0ε = , 2 2 σ σ= with respect to the parameters , , , , ,cγ φ ϕ α β . In order to specify the likelihood we need to know the joint (unconditional) distribution y . However what we are given, instead, is the conditional distributions of Ω . Moreover { } y are not independent. There are two way around this problem both of which lead to the same solution. One way it to consider maximizing the joint likelihood of the standardized innovations = . By assumption { } ξ are iid standard normal random variables and their joint pdf is ( , , , | , , , , , , , ) ( | , ) f c fξ ξ ξ γ φ θ α β ε σ ξ ε σ σ are computed iteratively as in Eq. (9). The other (I would say more proper) way is to use the following decomposition (applying the Bayes formula for conditional probability): 1 1 0 1 1 0 1 1 0 1 1 0 1 2 1 0 2 1 0 1 1 2 1 0 0 ( , ,..., , ) ( | ,,..., , ) ( ,..., , ) ( | ,,..., , ) ( | ,..., , ) ( ,..., , ) ( | ) ( | ) ( | ) ( ) f y y y y f y y y y f y y y f y y y y f y y y y f y y y f y f y f y f = Ω Ω Ω Ω and specify pdf conditional on parameters as 1 1 0 0 0 1 0 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com