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:
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...