(To be removed) Univariate GARCH(P,Q) parameter estimation with Gaussian innovations
As an alternative to ugarch
, use
the garch
object
to create conditional variance models. For more information, see Specify GARCH Models Using garch.
ugarch
is removed. Use the garch
object
to create conditional variance models and the estimate
function to
fit conditional variance models to data. For more information, see Likelihood Ratio Test for Conditional Variance Models.
[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