ugarch (Financial Toolbox)

Univariate GARCH(P,Q) parameter estimation with Gaussian innovations.

Syntax

[Kappa, Alpha, Beta] = ugarch(U, P, Q)

Arguments


`U`	Single column vector of random disturbances, i.e., the residuals, or innovations, of an econometric model representing a mean-zero, discrete-time stochastic process. The innovations time series `U` is assumed to follow a GARCH(P,Q) process.
`P`	Non-negative, scalar integer representing a model order of the GARCH process. `P` is the number of lags of the conditional variance. `P` can be zero; when `P = 0`, a GARCH(0,Q) process is actually an ARCH(Q) process.
`Q`	Positive, scalar integer representing a model order of the GARCH process. `Q` is the number of lags of the squared innovations.

Description

[Kappa, Alpha, Beta] = ugarch(U, P, Q)

computes estimated univariate GARCH(P,Q) parameters with Gaussian innovations.

Kappa is the estimated scalar constant term of the GARCH process.

Alpha is a P-by-1 vector of estimated coefficients, where P is the number of lags of the conditional variance included in the GARCH process.

Beta is a Q-by-1 vector of estimated coefficients, where Q is the number of lags of the squared innovations included in the GARCH process.

GARCH(P,Q) coefficients {Kappa, Alpha, Beta} are subject to constraints:

Kappa > 0
Alpha(i) >= 0 for i = 1,2,...P
Beta(i)  >= 0 for i = 1,2,...Q
sum(Alpha(i) + Beta(j)) < 1 for i = 1,2,...P and j = 1,2,...Q

The time-conditional variance, H(t), of a GARCH(P,Q) process is modeled as:

H(t) = Kappa + Alpha(1)*H(t-1) + Alpha(2)*H(t-2) +...+ Alpha(P)*H(t-P)+ Beta(1)*U^2(t-1)+ Beta(2)*U^2(t-2)+...+ Beta(Q)*U^2(t-Q)

Note that U is a vector of innovations, or regression residuals of an econometric model, representing a mean-zero, discrete-time stochastic process. That is, it is assumed that a regression model has already been run, and that U(t) = y(t) - F(X(t),B) is the time series of innovations derived from the model.

Although H is generated via the equation above, U and H are related as

U(t) = sqrt(H(t))*v(t)

where v(t) is an i.i.d. sequence ~ N(0,1).

Example

See ugarchsim for an example of a GARCH(P,Q) process.