Documentation |
Create GARCH time series model object
garch creates model objects for generalized autoregressive conditional heteroscedastic (GARCH) models. The GARCH(P,Q) conditional variance model includes P lagged conditional variances and Q lagged squared innovations.
Create model objects with known or unknown coefficients. Estimate unknown coefficients from data using estimate.
model = garch creates a conditional variance GARCH model of degrees zero.
model = garch(P,Q) creates a conditional variance GARCH model with GARCH degree P and ARCH degree Q.
model = garch(Name,Value) creates a GARCH model with additional options specified by one or more Name,Value pair arguments. Name can also be a property name and Value is the corresponding value. Name must appear inside single quotes (''). You can specify several name-value pair arguments in any order as Name1,Value1,...,NameN,ValueN.
Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.
'Constant' |
Positive scalar constant in the GARCH model. Default: NaN |
Note: Each GARCH and ARCH coefficient is associated with an underlying lag operator polynomial and is subject to a near-zero tolerance exclusion test. That is, each coefficient is compared to the default lag operator zero tolerance, 1e-12. A coefficient is only included in the model if its magnitude is greater than 1e-12. Otherwise, it is considered sufficiently close to zero and excluded from the model. See LagOp for additional details. |
P |
Number of presample conditional variances needed to initialize the model. This is equal to the largest lag corresponding to a nonzero conditional variance coefficient. You can only specify this property when using the garch(P,Q) syntax. If you use the name-value syntax, garch automatically sets the property P equal to the largest lag in GARCHLags. If GARCHLags is not specified, garch sets P equal to the number of elements in GARCH. You cannot modify this property. |
Q |
Number of presample innovations needed to initialize the model. This is equal to the largest lag corresponding to a nonzero squared innovation coefficient. You can only specify this property when using the garch(P,Q) syntax. If you use the name-value syntax, garch automatically sets the property Q equal to the largest lag in ARCHLags. If ARCHLags is not specified, garch sets Q equal to the number of elements in ARCH. You cannot modify this property. |
Constant |
Scalar constant in the GARCH model. |
GARCH |
Cell vector of coefficients corresponding to the lagged conditional variance terms. |
ARCH |
Cell vector of coefficients corresponding to the lagged squared innovation terms. |
UnconditionalVariance |
Unconditional variance of the process, $${\sigma}_{\epsilon}^{2}=\frac{\kappa}{(1-{\displaystyle {\sum}_{i=1}^{P}{\gamma}_{i}}-{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}})}.$$ This property is read-only. |
Offset |
Additive constant associated with an innovation mean model. |
Distribution |
Data structure for the conditional probability distribution of the innovation process. The field Name stores the distribution name 'Gaussian' or 't'. If the distribution is 't', then the structure also has the field DoF to store the degrees of freedom. |
estimate | Estimate GARCH model parameters |
filter | Filter disturbances with GARCH model |
forecast | Forecast GARCH process |
infer | Infer GARCH model conditional variances |
Display parameter estimation results for GARCH models | |
simulate | Monte Carlo simulation of GARCH models |
Consider a time series y_{t} with a constant mean offset,
$${y}_{t}=\mu +{\epsilon}_{t},$$
where $${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$ The GARCH(P,Q) conditional variance process, $${\sigma}_{t}^{2}$$, is of the form
$${\sigma}_{t}^{2}=\kappa +{\gamma}_{1}{\sigma}_{t-1}^{2}+\dots +{\gamma}_{P}{\sigma}_{t-P}^{2}+{\alpha}_{1}{\epsilon}_{t-1}^{2}+\dots +{\alpha}_{Q}{\epsilon}_{t-Q}^{2}.$$
The additive constant μ corresponds to the name-value argument Offset.
The constant κ > 0 corresponds to the name-value argument Constant.
The coefficients $${\gamma}_{i}\ge 0$$ correspond to the name-value pair GARCH.
The coefficients $${\alpha}_{j}\ge 0$$ correspond to the name-value pair ARCH.
The toolbox enforces stationarity with the constraint
$$\sum _{i=1}^{P}{\gamma}_{i}}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}}<1.$$
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB^{®} documentation.