ugarch

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, that is, the residuals or innovations (ɛt), 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.

    Note:   The latest value of residuals is the last element of vector U.

P

Nonnegative, 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 ([[KAPPA]]) 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.

The time-conditional variance, σt2, of a GARCH(P,Q) process is modeled as

σt2=K+i=1Pαiσti2+j=1Qβjεtj2,

where α represents the argument Alpha, β represents Beta, and the GARCH(P, Q) coefficients {Κ, α, β} are subject to the following constraints.

i=1Pαi+j=1Qβj<1K>0αi0i=1,2,,Pβj0j=1,2,,Q.

Note that U is a vector of residuals or innovations (ɛt) of an econometric model, representing a mean-zero, discrete-time stochastic process.

Although σt2 is generated using the equation above, ɛt and σt2 are related as

εt=σtυt,

where {υt} is an independent, identically distributed (iid) sequence ~ N(0,1).

    Note   The Econometrics Toolbox™ software provides a comprehensive and integrated computing environment for the analysis of volatility in time series. For information, see the Econometrics Toolbox documentation or the financial products Web page at http://www.mathworks.com/products/finprod/.

Examples

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

References

James D. Hamilton, Time Series Analysis, Princeton University Press, 1994

See Also

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