GARCH Toolbox 
Using GARCH to Model Financial Time Series
GARCH models account for certain characteristics that are commonly associated with financial time series:
Probability distributions for asset returns often exhibit fatter tails than the standard normal, or Gaussian, distribution. The fat tail phenomenon is known as excess kurtosis. Time series that exhibit a fat tail distribution are often referred to as leptokurtic. The blue (or dashed) line in Figure 11 illustrates excess kurtosis. The red (or solid) line illustrates a Gaussian distribution.
Figure 11: A Plot Showing Excess Kurtosis
In addition, financial time series usually exhibit a characteristic known as volatility clustering, in which large changes tend to follow large changes, and small changes tend to follow small changes (see Figure 12). In either case, the changes from one period to the next are typically of unpredictable sign. Volatility clustering, or persistence, suggests a time series model in which successive disturbances, although uncorrelated, are nonetheless serially dependent.
Figure 12: A Plot Showing Volatility Clustering
Volatility clustering (a type of heteroskedasticity) accounts for some but not all of the fat tail effect (or excess kurtosis) typically observed in financial data. A part of the fat tail effect can also result from the presence of nonGaussian asset return distributions that just happen to have fat tails.
Correlation in Financial Time Series
If you treat a financial time series as a sequence of random observations, this random sequence, or stochastic process, may exhibit some degree of correlation from one observation to the next. You can use this correlation structure to predict future values of the process based on the past history of observations. Exploiting the correlation structure, if any, allows you to decompose the time series into a deterministic component (i.e., the forecast), and a random component (i.e., the error, or uncertainty, associated with the forecast).
Eq. (11) uses these components to represent a univariate model of an observed time series y_{t}.
(11) 
Conditional Variances
The key insight of GARCH lies in the distinction between conditional and unconditional variances of the innovations process {_{t}}. The term conditional implies explicit dependence on a past sequence of observations. The term unconditional is more concerned with longterm behavior of a time series and assumes no explicit knowledge of the past.
GARCH models characterize the conditional distribution of _{t} by imposing serial dependence on the conditional variance of the innovations. Specifically, the variance model imposed by GARCH, conditional on the past, is given by

(12) 

(13) 
Given the form of Eq. (12) and Eq. (13), you can see that _{t}^{2} is the forecast of the next period's variance, given the past sequence of variance forecasts, _{ti}^{2}, and past realizations of the variance itself, _{tj}^{2}.
When P = 0, the GARCH(0,Q) model of Eq. (13) becomes Eq. (14), the original ARCH(Q) model introduced by Engle [8].

(14) 
Eq. (13) and Eq. (14) are referred to as GARCH(P,Q) and ARCH(Q) variance models, respectively. Note that when P = Q = 0, the variance process is simply white noise with variance .
Parsimonious Parameterization. In practice, you often need a large lag Q for ARCH modeling, and this requires that you estimate a large number of parameters. To reduce the computational burden, Bollerslev [4] extended Engle's ARCH model by including past conditional variances. This results in a more parsimonious representation of the conditional variance process.
Volatility Clustering. Eq. (13) and Eq. (14) mimic the volatility clustering phenomenon. Large disturbances, positive or negative, become part of the information set used to construct the variance forecast of the next period's disturbance. In this manner, large shocks of either sign are allowed to persist, and can influence the volatility forecasts for several periods. The lag lengths P and Q, as well the magnitudes of the coefficients G_{i} and A_{j, }determine the degree of persistence. Note that the basic GARCH(P,Q) model of Eq. (13) is a symmetric variance process, in that the sign of the disturbance is ignored.
Serial Dependence in Innovations
A common assumption when modeling financial time series is that the forecast errors (i.e., the innovations) are zeromean random disturbances uncorrelated from one period to the next.
As mentioned above, although successive innovations are uncorrelated, they are not independent. In fact, an explicit generating mechanism for a GARCH(P,Q) innovations process, {_{t}}, is
(15) 
where _{t} is the conditional standard deviation given by the square root of Eq. (13), and z_{t} is a standardized, independent, identically distributed (i.i.d.) random draw from some specified probability distribution. The GARCH literature uses several distributions to model GARCH processes, but the vast majority of research assumes the standard normal (i.e., Gaussian) density such that _{t} ~ N(0
, _{t}^{2}). Reflecting this, Eq. (15) illustrates that a GARCH innovations process {_{t}} simply rescales an i.i.d process {z_{t}} such that the conditional standard deviation incorporates the serial dependence of Eq. (13). Equivalently, Eq. (15) also states that a standardized GARCH disturbance, _{t}/_{t}, is itself an i.i.d. random variable z_{t}.
Notice that GARCH models are consistent with various forms of efficient market theory, which state that asset returns observed in the past cannot improve the forecasts of asset returns in the future. Since GARCH innovations {_{t}} are serially uncorrelated, GARCH modeling does not violate efficient market theory.
Homoskedasticity of the Unconditional Variance
The GARCH Toolbox imposes the following parameter constraints on the conditional variance parameters.

(16) 
The first constraint, a stationarity constraint, is necessary and sufficient for the existence of a finite, timeindependent variance of the innovations process {_{t}}. The remaining constraints are sufficient to ensure that the conditional variance {_{t}^{2}} is strictly positive.
When the conditional variance parameters satisfy the inequalities in Eq. (16), the unconditional variance (i.e., timeindependent, or longrun variance expectation) of the innovations process {_{t}} is

(17) 
Although Eq. (13) shows that the conditional variance of _{t} changes with time, Eq. (17) shows that the unconditional variance is constant (i.e., homoskedastic).
Introducing GARCH  GARCH Toolbox Overview 