The class of autocorrelated conditional heteroscedastic (ARCH) models was intro- duced in the seminal paper by Engle (1982). This type of model has since been modified and extended in several ways. The articles by Engle and Bollerslev (1986), Bollerslev et al. (1992), and Bera and Higgins (1993) provide an overview of the
Financial Risk Modelling and Portfolio Optimization withR, Second Edition. Bernhard Pfaff.
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k k model extensions during the decade or so after the original paper. Today, ARCH
models are not only well established in the academic literature but also widely ap- plied in the domain of risk modelling. In this section the term “ARCH” will be used both for the specific ARCH model and for its extensions and modifications.
The starting point for ARCH models is an expectations equation which only devi- ates from the classical linear regression with respect to the assumption of independent and identically normally distributed errors:
yt =x′t𝜷+𝜖t, (8.1)
𝜖t|Ψt−1 ∼N(0,ht). (8.2) The error process𝜖tis assumed to be normally distributed, but the variance is not con- stant, being allowed to change over time. This variability of the variance is expressed as dependence with respect to the available information at time t−1, denoted by Ψt−1. An alternative notation is commonly encountered in the literature:
𝜖t =𝜂t
√ht, (8.3)
𝜂t ∼𝔇𝜈(0,1), (8.4)
where 𝜂t denotes a random variable with distribution 𝔇with expected value zero and unit variance. Additional parameters of this distribution are subsumed in the parameter vector𝜈. This notation is more general in the sense that now a normally distributed error process is no longer required, but distributions with non-zero excess kurtosis and/or skewness can be interspersed. In the literature one often encounters the Student’s tdistribution, the skewed Student’st distribution, or the generalized error distribution.
The second building block of ARCH models is the variance equation. In an ARCH model of orderqthe conditional variance is explained by the history of the squared errors up to time lagq:
ht =𝛼0+𝛼1𝜖t−12 + ã ã ã +𝛼q𝜖t−q2 , (8.5) where𝛼0>0 and𝛼i≥0,i=1,…,q. These parameter restrictions guarantee a posi- tive conditional variance. The inclusion of the available information up to timet−1 is evident from
𝜖t−i=yt−i−x′t−i𝜷,i=1,…,q. (8.6) It can already be deduced why this model class can capture the stylized fact of volatility clustering: the conditional variance is explained by the errors in previous periods. If these errors are large in absolute value, a large value for the conditional variance results, and vice versa. By way of illustration, Figure 8.1 shows the plots of a white noise process and simulated ARCH(1) and ARCH(4) processes.
A clear pattern of volatility clustering is evident for the ARCH(1) and ARCH(4) processes, and it is more pronounced for the latter. One might be tempted to conclude that the two ARCH processes are more volatile than the white noise. However, all
k k
0 1000 2000 3000 4000 5000
−1005
White Noise
0 1000 2000 3000 4000 5000
−1005
ARCH(1)
0 1000 2000 3000 4000 5000
−1005
ARCH(4)
Figure 8.1 Plots of simulated white noise, ARCH(1), and ARCH(4) processes.
processes have been simulated to have unit unconditional variances. For an ARCH(1) process the unconditional variance is given by
𝜎𝜖2=E(𝜖2t) = 𝛼0
1−𝛼(1), (8.7)
where 𝛼(1) represents the sum of the coefficients for the lagged squared errors.
The processes in Figure 8.1 have been generated according to ht =0.1+0.9𝜖2t and ht =0.1+0.36𝜖t−12 +0.27𝜖t−22 +0.18𝜖2t−3+0.09𝜖2t−4 for the ARCH(1) and ARCH(4) models, respectively. Hence, for both models the unconditional variance equals unity. Note that the fourth moment (i.e., the kurtosis) for ARCH(1) models,
E(𝜖4t) 𝜎𝜖4 =3
(1−𝛼21 1−3𝛼12
)
, (8.8)
is greater than 3 and thus has more probability mass in the tails than the normal distribution. This is further evidence that with ARCH models the stylized facts of financial market returns can be captured well.
Having discussed the baseline ARCH model, the focus is now shifted to its modifi- cation and extensions. Bollerslev (1986) introduced the GARCH(p,q)model into the literature. This differs from the ARCH model in the inclusion of lagged endogenous variables in the variance equation—that is, now the conditional variance depends not only on past squared errors but also on lagged conditional variances:
ht =𝛼0+𝛼1𝜖t−12 + ã ã ã +𝛼q𝜖t−q2 +𝛽1ht−1+ ã ã ã +𝛽pht−p, (8.9)
k k with the restrictions𝛼0>0,𝛼i≥0 fori=1,…,q, and𝛽i≥0 forj=1,…,psuch
that the conditional variance process is strictly positive. The advantage of this ap- proach is that a GARCH model is the same as an ARCH model with an infinite number of lags, if the roots of the lag polynomial 1−𝛽(z)lie outside the unit circle, hence a more parsimonious specification is possible when GARCH-type specifica- tions are used. The unconditional variance for a GARCH(1, 1) model is given by 𝜎𝜖2=E(𝜖t2) = 𝛼0
1−𝛼1−𝛽1. From this, the condition for a stable variance process can be directly deduced, namely,𝛼1+𝛽1<1.
So far it has been assumed that the sign of the shock does not have an impact on the conditional variance. This is because the past errors enter as squares into the variance equation. Nelson (1991) extended the class of ARCH to make it possible to take this effect into account. He proposed the class ofexponentialGARCH (EGARCH) mod- els to capture such asymmetries. The modelling of asymmetric effects can be justified from an economic point of view with leverage effects, in particular when equity re- turns are investigated. For these, a negative relationship between volatility and past returns is postulated (see Black 1976). The variance equation now takes the form
log(ht) =𝛼0+
∑q
i=1
𝛼ig(𝜂t−i) +
∑p
j=1
𝛽jlog(ht−j), (8.10)
where the previously introduced alternative specification of ARCH models is employed and the functiong(𝜂t)is defined as:
g(𝜂t) =𝜃𝜂t+𝛾[|𝜂t|−E(|𝜂t|)]. (8.11) This approach has some peculiarities worth mentioning. First, a multiplicative relationship for explaining the conditional variances is assumed. This becomes evi- dent from the logarithmic form for the variance equation. Hence, there is no need for non-negativity constraints on the parameter space,𝛼i,i=1,…,qand𝛽j,j=1,…,p, because ht=exp (⋅) will always be positive. Second, the impact of the error variables is piecewise linear and takes for a positive shock a value of𝛼i(𝜃+𝛾)and for a negative one 𝛼i(𝜃−𝛾). Here, the sign-dependent impact of past errors with respect to the contemporaneous conditional variance shows up. The first term,g(𝜂t), in the equation depicts the correlation between the error process𝜂t and the future conditional volatility. The ARCH effects themselves are captured by the coefficient 𝛾. The greater the deviation of the variable𝜂tfrom its expected value, the higher the value of the functiong(𝜂t)and hence of the log-value for the conditional variance.
The last of the many extensions of the ARCH model to be discussed isasymmetric powerARCH (APARCH) proposed by Ding et al. (1993). The reason for this choice is that the APARCH model encompasses other ARCH specifications, as will be shown below. Its variance equation is defined as
𝜖t=𝜂tht, (8.12)
𝜂t∼𝔇𝜈(0,1), (8.13)
k k h𝛿t =𝛼0+
∑q
i=1
𝛼i(|𝜖t−i|−𝛾i𝜖t−i)𝛿+
∑p
j=1
𝛽jh𝛿t−j, (8.14)
with parameter restrictions 𝛼0>0, 𝛿≥0, 𝛼i≥0,i=1,…,q, −1< 𝛾i<1,i= 1,…,q, and 𝛽i≥0,i=1,…,p. One peculiarity of this approach for modelling the conditional variance is the exponent 𝛿. For 𝛿=2 the conditional variance results, similarly to the models discussed above. But the parameter is defined for non-negative values in general and hence the long memory characteristic often encountered for absolute and/or squared daily return series can be taken explicitly into account. The long memory characteristic can be described simply by the fact that for absolute and/or squared returns the autocorrelations taper off only slowly and hence dependencies exist between observations that are further apart. Potential asymmetries with respect to positive/negative shocks are reflected in the coefficients 𝛾i,i=1,…,q. The APARCH model includes the following special cases:
•ARCH model, if𝛿=2,𝛾i=0, and𝛽j=0;
•GARCH model, if𝛿=2 and𝛾i=0;
•TS-GARCH (see Schwert 1990; Taylor 1986), if𝛿=1 and𝛾i=0;
•GJR-GARCH (see Glosten et al. 1993), if𝛿=2;
•T-ARCH (see Zakoian 1994), if𝛿=1;
•N-ARCH (see Higgins and Bera 1992), if𝛾i=0 and𝛽j=0;
•Log-ARCH (see Geweke 1986; Pentula 1986), if𝛿→0.
The reader is referred to the citations above for further details on these special cases.
Estimates for the unknown model parameters are often obtained by applying the maximum likelihood or quasi-maximum likelihood principle. Here, numerical opti- mization techniques are applied. Less commonly encountered are Bayesian estima- tion techniques for the unknown parameter in applied research. Whence a model fit is obtained, forecasts for the conditional variances can be computed recursively and the desired risk measures can be deduced together with the quantiles of the assumed distribution for the error process from these.