The GARCH (Generalized ARCH) model

Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) is an important extension of ARCH, by Bollerslev (1986). The GARCH (p,q) process is defined as follows:

Here, is the variance at time t, q is the order for the error terms, p is the order for the variance, is a constant, is the coefficient for the error term at t-i, is the coefficient for the variance at time t-i. Obviously, the simplest GARCH process is when both p and q are set to 1, that is, GARCH (1,1), which has following formula:

Simulating a GARCH process

Based on the previous program related to ARCH (1), we could simulate a GARCH (1,1) process as follows:

import scipy as sp
sp.random.seed(12345)
n=1000          # n is the number of observations
n1=100          # we need to drop the first several observations
n2=n+n1         # sum of two numbers
alpha=(0.1,0.3)     # GARCH (1,1) coefficients alpha0 and alpha1, see Equation (3)
beta=0.2
errors=sp.random.normal(0,1,n2)
t=sp.zeros(n2...