Univariate GARCH(P,Q) parameter estimation with Gaussian innovations
[Kappa, Alpha, Beta] = ugarch(U, P, Q)
| 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
| |
| Nonnegative, scalar integer representing a model order
of the GARCH process. | |
| Positive, scalar integer representing a model order of
the GARCH process. |
[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, $${\sigma}_{t}^{2}$$, of a GARCH(P,Q) process is modeled as
$${\sigma}_{t}^{2}=K+{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}{\sigma}_{t-i}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}{\epsilon}_{t-j}^{2}},$$
where α represents the argument Alpha
, β represents Beta
,
and the GARCH(P, Q) coefficients {Κ, α, β}
are subject to the following constraints.
$$\begin{array}{l}{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}}<1\\ K>0\\ \begin{array}{cc}{\alpha}_{i}\ge 0& i=1,2,\dots ,P\\ {\beta}_{j}\ge 0& j=1,2,\dots ,Q.\end{array}\end{array}$$
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 $${\sigma}_{t}^{2}$$ is generated using the equation above, ɛ_{t} and $${\sigma}_{t}^{2}$$ are related as
$${\epsilon}_{t}={\sigma}_{t}{\upsilon}_{t},$$
where $$\left\{{\upsilon}_{t}\right\}$$ 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 |
James D. Hamilton, Time Series Analysis, Princeton University Press, 1994