XploRe Help : genarch

Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Library:	times
See also:	genarma

Quantlet:		genarch
Description:		generates a time series e_t=u_t*s_t, where u_t is standard normal distributed and the variances s^2_t follow the GARCH process s^2_t = a_0 + a(B)e^2_t + b(B)s^2_t Here, B denotes the backshift (or lag) operator. You have to deliver the coefficient vectors a and b and the length T of the series e. The sum a(1) + b(1) must be less than one and all coefficients should be non-negative (see the note).

Reference(s):

Patterson, Kerry (2000): An Introduction to Applied Econometrics. A Time Series Approach. Houndmills (Mac Millan Press), pp.711-719

Usage:	`{e,s2} = genarch(a,b,T)`
Input:
	`a`	vector of the ARCH coefficients (that are a_0 and the coefficients of the lag polynom a(B))
	`T`	integer; length of the generated sample
	`b`	vector of GARCH parameters (that are the coefficients of the lag polynom b(B))
Output:
	`e`	(T x 1) vector of the generated time series e_t
	`s2`	(T x 1) vector of the generated variances s^2_t

Note:

A necessary and sufficient condition for stationarity is a(1) + b(1)<1. If all coefficients are non-negative, the variances s^2_t are always positive. However, one can relax the last restriction slightly (see Patterson, p.718).

Example:

library("times") ; loads the quantlets from times library randomize(0) ; sets the seed {e,s2} = genarch(#(2.5,0.1),0.65,5000) acfplot(e^2,12) ; display of the sample acf for e^2_t

Result:

Display that shows the sample autocorrelation function
of the squared series e^2_t. The first correlation
coefficients are significant. It is easy to check
that the simple series e_t is uncorrelated.

Author: C. Hafner, W. Haerdle, R. Schulz, 20010430 license MD*Tech