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Univariate GARCH(P,Q) parameter estimation with Gaussian innovations
[Kappa, Alpha, Beta] = ugarch(U, P, Q)
U | Single column vector of random disturbances, that is, the residuals or innovations (ɛ_{t}), 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 | 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. |
[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, , of a GARCH(P,Q) process is modeled as
where α represents the argument Alpha, β represents Beta, and the GARCH(P, Q) coefficients {Κ, α, β} are subject to the following constraints.
Note that U is a vector of residuals or innovations (ɛ_{t}) of an econometric model, representing a mean-zero, discrete-time stochastic process.
Although is generated using the equation above, ɛ_{t} and are related as
where is an independent, identically distributed (iid) sequence ~ N(0,1).
Note ugarch corresponds generally to the Econometrics Toolbox™ function garchfit. 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/. |