Research Techniques EC903: Financial Time Series Models 98/99 Notes(3)
GARCH models
Generalised Auto-Regressive Conditional Heteroscedastic process
(Bollerslev, 1986, A Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307-327).
In the first applications of ARCH models it was found that a large order of the lag q was required in the conditional variance function
ht= 0+ 12t-1 + ... + q2t-q
This would imply the estimation of a large number of coefficients, subject to inequality restrictions: 00 and i0, i=1,...,q, to ensure that the conditional variance is positive.
Bollerslev, 1986, proposed a generalisation of ARCH, consisting in the following specification for the conditional variance:
ht= 0+ 12t-1 + ... + q2t-q + 1 ht-1 + ...+ p ht-p
This process is called GARCH of order p and q, and is defined as GARCH(p,q).
To ensure that the conditional variance is positive, the following restrictions are imposed:
00 , i0 for i=1,...,q, and i0 for i=1,...,p.
When p=0, the process reduces to the ARCH(q) process, and for p=q=0, t is a white noise.
The GARCH(p,q) extension is similar to the extension of a time series AR process to the general ARMA process, and to that of a Distributed Lag model to an Autoregressive Distributed Lag.
To see these similarities, rewrite the GARCH in terms of the lag polynomials
(L)= 1 L+ 2 L2 + ...+ q Lq and (L)= 1 L+ 2 L2 + ...+ p Lp
ht= 0+ (L)2t + (L) ht
If the roots of the characteristic equation 1-(z)=0 all lie outside the unit circle, the GARCH (p,q) process has the inverted representation:
or
where the coefficients i’s are obtained from the power series expansion:
d(L)=(L)(1-(L))-1.
So, a GARCH(p,q) process is an infinite order ARCH (a distributed lag of past 2t).
The inequality restrictions required when considering the inverted representation of a GARCH model are weaker than those required in the coefficients of the GARCH(p,q). These are:
0 , and i0 for all i’s.
For example, in the GARCH(1,2) process:
ht= 0+ 12t-1 +22t-2 + 1 ht-1
the inequalities 00 , 10, 10 , and 11+20 are sufficient to ensure that ht is positive. So, 2 may be negative.
To conclude, many studies have reported negative coefficients, and yet satisfy the conditions for a positive conditional variance based on the inverted representation of the GARCH process. So, inequality constraints should not be imposed in estimation, as their violation does not necessarily imply misspecification of the conditional variance.
Unconditional moments of the ARCH-GARCH process
The unconditional moments characterise formally the properties of the ARCH and GARCH process.
Engle, 1982, gives expressions for the moments of the ARCH process and stated necessary and sufficient conditions for their existence.
Results for the GARCH process can be found in Bollerslev, 1986.
Using the law of iterated expectations, it is proved that the unconditional mean of a GARCH(p,q) is zero:
E(t)=E[E(t/t-1)]=E[0]=0
and the unconditional variance is constant. For a GARCH(1,1) this is equal to
E(t)=E[E(t/t-1)]
=E(ht)
= 0+1E(2t-1)+1E(ht-1)
= 0+(1 +1)E(2t-1)
which converges to
2= 0/(1-1 -1)
if 1 +1<1
For a GARCH(p,q) this result generalises to
2= 0/(1-(1)-(1))
where (1)= 1+2+...+q and (1)= 1+2+...+p , and (1)+(1)<1 are necessary and sufficient conditions for the existence of the variance.
Note: in empirical applications of GARCH, the condition for the existence of the variance often is not satisfied.
Third moment is zero E(t)=0
Fourth moment for an ARCH(1) process is
which is >3, the kurtosis coefficient of a normal distribution. Conditions for the existence of fourth moment is that 321<1.
Persistence and Stationarity
The Stationarity conditions for the ARCH(Q) process, with Q a sufficiently large value are given by: d(1)<1, or equivalently (1)+(1)<1.
The proof can be found in Bollerslev, 1986.
The GARCH(1,1) model is the most commonly used model in the GARCH class. By adding and subtracting 1ht-1 from the original equation
ht=0+12t-1+1ht-1
gives
ht=0+1(2t-1-ht-1)+(1+1)ht-1
Here (2t-1-ht-1) can be viewed as the shock to volatility, so the coefficient 1 measures the effect of a volatility shock to next period’s volatility.
The coefficient (1+1) is the equivalent of the autoregressive coefficient in an ARMA(1,1) process, and measures the rate at which the effect of a volatility shock dies out over time.
A GARCH(1,1) model with 1+1=1 has a unit autoregressive root. This means that today’s volatility affects forecasts of volatility into the indefinite future. This is known as an integrated GARCH, or IGARCH(1,1) process.
Although the IGARCH(1,1) process looks very much like a linear random walk with drift, there are some important differences between the two processes.
A linear random walk is non stationary in two ways:
i) it has no stationary distribution, therefore it is not strictly stationary;
ii) it has no unconditional first or second moments, therefore it is not covariance stationary.
In the IGARCH(1,1) process, ht is strictly stationary but not generally covariance stationary (its stationary distribution generally lacks unconditional moments).
Dependence of the t‘s
The GARCH process implies an ARMA representation for the squared errors. This can be seen by considering the surprise in squared returns vt = 2t - ht, andrewriting the GARCH(p,q) process as
where vt is serially uncorrelated with mean zero, or equivalently as
where m=max(p,q), i =0 for i>q, and i =0 for i>p. So, 2t has an ARMA(m,p) representation, which differs from a standard ARMA model as here the shocks themselves (vt) are heteroscedastic.
Example: the GARCH(1,1) process can be rewritten as
2t = 0+ (1+1)2t-1 + 1vt-1 + vt
= 0+ (1+1)2t-1 + 1(2t-1-ht-1)+ (2t-ht)
Note: in Eviews the coefficients i and i are called ARCH and GARCH respectively, while, in Microfit, these are called the MA and the AR coefficients.
Alternative functional forms
1. Multiplicative ARCH (or log ARCH)
log(ht) = + log(2t-1)+ ... + qlog(2t-q)
ensures that the conditional variance is strictly positive. In fact, taking the exponential of both sides, gives: ht = exp( ), so no inequality restrictions are required on the i‘s.
2. TARCH -Threshold heteroscedastic model (GJR, 1989, and Zakoian, 1990) (see Eviews)
ht= 0+ 12t-1 + 2t-1dt-1 + ht-1
where dt is a dummy variable equal 1 if t <0, and equal 0 otherwise, and is the asymmetric coefficient. Good news (t>0) have an effect of 1 , while the impact of bad news (t<0) is given by 1+ .
3. EGARCH model (exponential GARCH) (see Eviews and Microfit)
log(ht) = +log(ht-1)+t-1/ht-1+(t-1/ht-1)
where is the asymmetric coefficient. If is significantly different from zero, then there is an asymmetric effect. Note that here the sign of is of opposite sign to that in the TARCH model, which uses a dummy for negative errors.
Like in the multiplicative model, the log transformation ensures that the conditional variance is positive. The impact of past errors is exponential rather than quadratic, like in standard GARCH models. This specification differs from the original one proposed by Nelson, 1989, where the third term on the r.h.s. is (t-1/ht-1 )-2/. So, the estimate of the constant given in EViews differs from that obtained from the original model by -2/
4. Absolute value GARCH (Microfit)
The absolute value GARCH(1,1) model specifies the conditional standard deviation function as
ht= 0+ 1t-1+ 1ht-1
i.e. the forecasts of future standard deviation are linear in current and past standard deviations and absolute values of returns drive revisions in the forecasts.
In standard GARCH models, instead, forecasts of future variance are linear in current and past variances and squared returns drive corrections in the forecasts.
Testing for asymmetry in Eviews
It is a test to determine whether there is predictability from the level to the square of the series. It is based on the Ljung Box test for skew cross correlation between the level and the square of the series. A significant test on the standardised residuals indicates that the model is not well specified.
ARCH and GARCH in mean processes
Engle, Lilien and Robins, 1987, Estimating Time Varying Risk Premia in the Term Structure: the ARCH-M model, Econometrica, 55, 391-407, also in Engle, 1995.
The basic ARCH model was extended by ELR by introducing the conditional variance in the mean equation.
yt = x’t + ht + t ,t=1,...,T
where
tt-1N(0,ht)
and ht is an ARCH or a GARCH process.
This model is particularly attractive for the study of asset markets, where the conditional variance is used to measure expected risk (time varying risk premia).
Interpretation: the expected return (yt) of an asset is proportional to the expected risk (ht) of the asset. The coefficient measures the risk return trade off. As there is little suggestion from economic theory about the nature of this trade off (depending on the risk preferences of the traders), in empirical ARCH-M models the conditional variance enters the mean equation in various forms: log(ht), ht and ht.
The ARCH-M model provides a framework to test and estimate a time varying risk premium (0). ELR apply the model to 3 interest rate data sets, and explain failures of the expectations hypothesis of the term structure.
Example:
yt = ht + t ,
ht= 0+ 12t-1
yt = (0+12t-1) + t , where t is an ARCH(1) process.
Other extensions:
The conditional variance equation can be extended with the inclusion of exogenous or predetermined variables, z :
ht= 0+ 12t-1 + 1ht-1 + zt
provided that z0 and 0, this model still constrains volatility to be positive.
Estimation
The GARCH model is estimated by Maximum Likelihood
Note: Eviews and Microfit estimate the following ARCH models
EViews:
EGARCH : Exponential GARCH
log(ht) = +log(ht-1)+t-1/ht-1+(t-1/ht-1)
TARCH : Asymmetric or Threshold ARCH
ht= 0+ 12t-1 + 2t-1dt-1 + ht-1
where dt is a dummy variable equal 1 if t <0, and equal 0 otherwise, and is the asymmetric coefficient. Good news (t>0) have an effect of 1 , while the impact of bad news (t<0) is given by 1+ . So, the impact on volatility changes as t crosses the threshold of zero.
C : Component (permanent and transitory) ARCH
Distinguishes between a transitory component (short run movements in volatility) and a permanent component (long run levels of volatility), and allows a varying long run component. The model is given by
ht= qt+ (2t-1-qt-1) + ht-1-qt-1)
qt= 0+(qt-1- 0)+ (2t-1-ht-1)
where qt is the long run component, which will converge to 0 with powers of The transitory component is ht-qt which will converge to zero with powers of +.
A : Asymmetric Component (permanent and transitory) ARCH
This is the component model combined with the TARCH model
V : ARCH-M, with variance in the mean equation
S : ARCH-M, with standard deviation in the mean equation
Microfit:
GARCH
GARCH-M
AGARCH Absolute value GARCH
AGARCH-M Absolute value GARCH in Mean
EGARCH : Exponential GARCH
EGARCH-M : Exponential GARCH in Mean
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