In addition a farreaching agreement has been formed that returns cannot be regarded as i.i.d. and at most as being uncorrelated. This argument holds at least for financial time series of relatively high frequency, for example for daily data. In Figure 12.1 we show a normally distributed white noise, a GARCH(1,1) process in Figure 12.2 and the DAFOX index (199396) in Figure 12.3, see http://finance.wiwi.unikarlsruhe.de/Forschung/dafox.html . It can be seen from the figure that the GARCH process is obviously more appropriate for modelling stock returns than white noise.
However the ARCH model is only the starting point of the empirical study and relies on a wide range of specification tests. Some practically relevant disadvantages of the ARCH model have been discovered recently, for example, the definition and modelling of the persistence of shocks and the problem of modelling asymmetries. Thus a large number of extensions of the standard ARCH model have been suggested. We will discuss them in detail later.
Let be a discrete stochastic process and from Definition
10.15
the relative
increase or the return of the process . If the returns
are independent and identically distributed, then follows a
geometric random walk. It is assumed in ARCH models that the
returns depend on past information with a specific form.
As mentioned before denotes the information set at time , which encompasses and all the past realizations of the process . This means in a general model
(13.1) 
The ARCH model of order 1, ARCH(1), is defined as follows:
where is the best linear projection described in Section 11.4. Obviously a strong ARCH(1) process is also semistrong and a semistrong also weak. On the other hand the conditional variance of a weak ARCH(1) process can be nonlinear (unequal to ). In this case it can not be a semistrong ARCH process.
Setting , it holds for the semistrong and the strong ARCH models that and . In strong ARCH models is i.i.d. so that no dependence can be modelled in higher moments than the second moment. It is frequently assumed that is normally distributed, which means is conditionally normally distributed:
Originally only strong and semistrong ARCH models are discussed in the literature. Weak ARCH models are important because they are closed under temporal aggregation. If, for example, daily returns follow a weak ARCH process, then the weekly and monthly returns are also weak ARCH with corresponding parameter adjustments. This phenomenon holds in general for strong and semistrong models.
According to Definition 12.1 the process is a martingale difference and therefore white noise.
If the innovation is symmetrically distributed around zero, then all odd moments of are equal to zero. Under the assumption of normal distribution (12.3) the conditions for the existence of higher even moments can be derived.
For the boundary case and the normally distributed innovations , while for it holds that . The unconditional distribution is also leptokurtic under conditional heteroscedasticity, i.e., the curvature is high in the middle of the distribution and the tails are fatter than those of a normal distribution, which is frequently observed in financial markets.
The thickness of the tails and thus the existence of moments depend on the parameters of the ARCH models. The variance of the ARCH(1) process is finite when (Theorem 12.2), while the fourth moment in the case of normally distributed error terms exists when (Theorem 12.3). Already in the sixties Mandelbrot had questioned the existence of the variance of several financial time series. Frequently empirical distributions have so fat tails that one can not conclude a finite variance. In order to make empirical conclusions on the degree of the tail's thickness of the unconditional distribution, one can assume, for example, that the distribution is a Pareto type, i.e., for large :

Hill (1975) has suggested an estimator using the maximum likelihood method:

0 for 
The dynamics of the volatility process in the case of ARCH(1) is essentially determined by the parameter . In Theorem 12.5 it was shown that the square of an ARCH(1) process follows an AR(1) process. The correlation structure of the empirical squared observations of returns are frequently more complicated than a simple AR(1) process. In Section 12.1.3 we will consider an ARCH model of order with , which allows a more flexible modelling of the correlation structure.
The volatility is a function of the past squared observations in ARCH models in a narrow sense. In the more general GARCH models (Section 12.1.5) it may depend on the past squared volatilities in addition. These models belong to the large group of unpredictable time series with stochastic volatility. In the strong form, they have where is measurable, i.e. the volatility depends only on the information to the time point and the i.i.d. innovations with . For such a time series it holds , i.e. is unpredictable and, except in the special case that const. conditionally heteroscedastic. The stylized facts 24 are only fulfilled under certain qualitative assumptions. For example, in order to produce volatility cluster must tend to be large when the squared observations or volatilities of the recent past observations are large. The generalizations of the ARCH models observed in this section fulfill the corresponding conditions.
This result is often used reversely in order to estimate the parameter of financial models in the continuous time where one approximates the corresponding diffusion processes through discrete GARCH time series and estimates its parameter. Nelson (1990) shows only the convergence of GARCH processes against diffusion processes in a weak sense (convergence on the distribution). A recent work of Wang (2002) shows however that the approximation does not hold in a stronger sense, especially the likelihood process is not asymptotically equivalent. In this sense the maximum likelihood estimators for the discrete time series do not converge against the parameters of the diffusion limit process.
Theorem 12.5 says that an ARCH(1) process can be represented as an AR(1) process in . A simple YuleWalker estimator uses this property:
The estimation of ARCH models is normally done using the maximum likelihood (ML) method. Assuming that the returns have a conditionally normal distribution, we have:
Figure 12.5 shows the conditional likelihood of a generated ARCH(1) process with . The parameter is chosen so that the unconditional variance is everywhere constant, i.e., with a variance of , . The optimization of the likelihood of an ARCH(1) model can be found by analyzing the graph. Most often we would like to know the precision of the estimator as well. Essentially it is determined by the second derivative of the likelihood at the optimization point by the asymptotic properties of the ML estimator (see Section 12.1.6). Furthermore one has to use numerical methods such as the score algorithm introduced in Section 11.8 to estimate the parameters of the models with a larger order. In this case the first and second partial derivatives of the likelihood must be calculated.

With the ARCH(1) model these are
(13.10)  
(13.11)  
(13.12)  
(13.13)  
(13.14) 
Proof:
Obviously Theorem 12.6 also holds for the parameter in place of . In addition it essentially holds for more general models, for example the estimation of GARCH models in Section 12.1.6. In more complicated models one can replace the second derivative with the square of the first derivative, which is easier to calculate. It is assumed, however, that the likelihood function is correctly specified, i.e., the true distribution of the error terms is normal.
Under the two conditions
(13.15) 
If the true distribution of is normal, then and the asymptotic covariance matrix is simplified to , i.e., the inverse of the Fischer Information matrix. If the true distribution is instead leptokurtic, then the maximum of (12.9) is still consistent, but no longer efficient. In this case the ML method is interpreted as the `Quasi Maximum Likelihood' (QML) method.
In a Monte Carlo simulation study in Shephard (1996) 1000 ARCH(1) processes with and were generated and the parameters were estimated using QML. The results are given in Table 12.2. Obviously with the moderate sample sizes () the bias is negligible. The variance, however, is still so large that a relatively large proportion (10%) of the estimators are larger than one, which would imply covariance nonstationarity. This, in turn, has a considerable influence on the volatility prediction.

The definition of an ARCH(1) model will be extended for the case that lags, on which the conditional variance depends.
The conditional variance in an ARCH() model is also a linear function of the squared lags.
If instead , then the unconditional variance does not exist and the process is not covariancestationary.
It is problematic with the ARCH() model that for some applications a larger order must be used, since large lags only lose their influence on the volatility slowly. It is suggested as an empirical rule of thumb to use a minimum order of . The disadvantage of a large order is that many parameters have to be estimated under restrictions. The restrictions can be categorized as conditions for stationarity and the strictly positive parameters. If efficient estimation methods are to be used, for example, the maximum likelihood method, the estimation of large dimensional parameter spaces can be numerically quite complicated to obtain.
One possibility of reducing the number of parameters while including a long history is to assume linearly decreasing weights on the lags, i.e.,
For the general ARCH() model from (12.16) the
conditional likelihood is
(13.18) 
The ARCH() model can be generalized by extending it with autoregressive terms of the volatility.
The sufficient but not necessary conditions for
If follows a GARCH process, then from Theorem 12.9 we can see that follows an ARMA model with conditional heteroscedastic error terms . As we know if all the roots of the polynomial lie outside the unit circle, then the ARMA process (12.21) is invertible and can be written as an AR() process. Moveover it follows from Theorem 12.8 that the GARCH() model can be represented as an ARCH() model. Thus one can deduce analogous conclusions from the ARMA models in determining the order of the model. There are however essential differences in the definition of the persistence of shocks.
General conditions for the existence of higher moments of the GARCH() models are given in He and Teräsvirta (1999). For the smaller order models and under the assumption of distribution we can derive:
The function (12.22) is illustrated in Figure 12.6 for all , , i.e., the distribution of is leptokurtic. We can observe that the kurtosis equals 3 only in the case of the boundary value where the conditional heteroscedasticity disappears and a Gaussian white noise takes place. In addition it can be seen in the figure that the kurtosis increases in slowly for a given . On the contrary it increases in much faster for a given .

In practical applications it is frequently shown that models with smaller order sufficiently describe the data. In most cases GARCH(1,1) is sufficient.
A substantial disadvantage of the standard ARCH and GARCH models exists since they can not model asymmetries of the volatility with respect to the sign of past shocks. This results from the squared form of the lagged shocks in (12.16) and (12.19). Therefore they have an effect on the level but no effect on the sign. In other words, bad news (identified by a negative sign) has the same influence on the volatility as good news (positive sign) if the absolute values are the same. Empirically it is observed that bad news has a larger effect on the volatility than good news. In Section 12.2 and 13.1 we will take a closer look at the extensions of the standard models which can be used to calculate these observations.
Based on the ARMA representation of GARCH processes (see Theorem 12.9) YuleWalker estimators are considered once again. These estimators are, as can be shown, consistent and asymptotically normally distributed, . However in the case of GARCH models they are not efficient in the sense that the matrix is positively definite, where is the asymptotic covariance matrix of the QML estimator, see (12.25). In the literature there are several experiments on the efficiency of the YuleWalker and QML estimators in finite samples, see Section 12.4. In most cases maximum likelihood methods are chosen in order to get the efficiency.
The likelihood function of the general GARCH() model (12.19) is identical to (12.17) with the extended parameter vector . Figure 12.7 displays the likelihood function of a generated GARCH(1,1) process with , , and . The parameter was chosen so that the unconditional variance is everywhere constant, i.e., with a variance of , . As one can see, the function is flat on the right, close to the optimum, thus the estimation will be relatively imprecise, i.e., it will have a larger variance. In addition, Figure 12.8 displays the contour plot of the likelihood function.


The first partial derivatives of (12.17) are
(13.23) 
Under the conditions
(13.26)  
If the distribution of is specified correctly, then and the asymptotic variance can be simplified to , i.e., the inverse of the Fisher Information matrix. If this is not the case and it is instead leptokurtic, for example, the maximum of (12.9) is still consistent but no longer efficient. In this case the ML method is interpreted as the `Quasi Maximum Likelihood' (QML) method.
Consistent estimators for the matrices and can be obtained
by replacing the expectation with the simple average.