Documentation 
Create GJR time series model object
gjr creates model objects for GJR models. The GJR model is a variant of the GARCH conditional variance model, named for Glosten, Jagannathan, and Runkle [1]. The GJR(P,Q) model includes P lagged conditional variances, Q lagged squared innovations, and Q leverage terms.
Create model objects with known or unknown coefficients. Estimate unknown coefficients from data using estimate.
model = gjr creates a conditional variance GJR model of degrees zero.
model = gjr(P,Q) creates a conditional variance GJR model with GARCH degree P and ARCH degree Q.
model = gjr(Name,Value) creates a GJR 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 namevalue pair arguments in any order as Name1,Value1,...,NameN,ValueN.
Specify optional commaseparated 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 GJR model. Default: NaN 
'GARCH' 
Cell vector of nonnegative conditional variance coefficients. When specified without GARCHLags, GARCH is a Pelement cell vector of coefficients at lags 1, 2,...,P. When specified with GARCHLags, GARCH is an equivalentlength cell vector of coefficients associated with the lags in GARCHLags. Default: Cell vector of NaNs with the same length as GARCHLags. 
'ARCH' 
Cell vector of nonnegative squared innovation coefficients. When specified without ARCHLags, ARCH is cell vector of coefficients at lags 1, 2,... to the number of ARCH coefficients in the model. When specified with ARCHLags, ARCH is an equivalentlength cell vector of coefficients associated with the lags in ARCHLags. Default: Cell vector of NaNs with the same length as ARCHLags. 
'Leverage' 
Cell vector of coefficients associated with past signweighted squared innovations. When specified without LeverageLags, Leverage is a cell vector of coefficients at lags 1, 2,... to the number of leverage coefficients in the model. When specified with LeverageLags, Leverage is an equivalentlength cell vector of coefficients associated with the lags in LeverageLags. Default: Cell vector of NaNs with the same length as LeverageLags. 
'Offset' 
Scalar offset, or additive constant, associated with an innovation mean model. Default: 0 
'GARCHLags' 
Vector of positive integer lags associated with the GARCH coefficients. Default: Vector of integers 1, 2,...,P. 
'ARCHLags' 
Vector of positive integer lags associated with the ARCH coefficients. Default: Vector of integers 1, 2,... to the number of ARCH coefficients. 
'LeverageLags' 
Vector of positive integer lags associated with the leverage coefficients. Default: Vector of integers 1, 2,... to the number of leverage coefficients. 
'Distribution' 
Conditional probability distribution of the innovation process. Distribution is a string you specify as 'Gaussian' or 't'. Alternatively, specify it as a data structure with the field Name to store the distribution 'Gaussian' or 't'. If the distribution is 't', then the structure also needs the field DoF to store the degrees of freedom. Default: 'Gaussian' 
Notes:

P 
Number of presample conditional variances needed to initiate 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 gjr(P,Q) syntax. If you use the namevalue syntax, gjr automatically sets the property P equal to the largest lag in GARCHLags. If GARCHLags is not specified, gjr 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 gjr(P,Q) syntax. If you use the namevalue syntax, gjr automatically sets the property Q equal to the largest lag with a nonzero squared innovation coefficient, including ARCH and leverage coefficients and the corresponding lags in ARCHLags and LeverageLags. You cannot modify this property. 
Constant 
Scalar constant in the GJR 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, $${\sigma}_{\epsilon}^{2}=\frac{\kappa}{\left(1{\displaystyle {\sum}_{i=1}^{P}{\gamma}_{i}}{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}}\frac{1}{2}{\displaystyle {\sum}_{j=1}^{Q}{\xi}_{j}}\right)}.$$ This property is readonly. 
Leverage 
Cell vector of coefficients corresponding to the lagged leverage terms. 
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 GJR model parameters 
filter  Filter disturbances with GJR model 
forecast  Forecast GJR process 
infer  Infer GJR model conditional variances 
Display parameter estimation results for GJR models  
simulate  Monte Carlo simulation of GJR 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 GJR(P,Q) conditional variance process, $${\sigma}_{t}^{2}$$, is of the form
$${\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}{\sigma}_{ti}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}{\epsilon}_{tj}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}I\left[{\epsilon}_{tj}<0\right]{\epsilon}_{tj}^{2}},$$
where $$I\left[{\epsilon}_{tj}<0\right]=1$$ if $${\epsilon}_{tj}<0,$$ and 0 otherwise.
The additive constant μ corresponds to the namevalue argument Offset.
The constant κ > 0 corresponds to the namevalue argument Constant.
The coefficients $${\gamma}_{i}\ge 0$$ correspond to the namevalue argument GARCH.
The coefficients $${\alpha}_{j}\ge 0$$ correspond to the namevalue argument ARCH.
The coefficients $${\xi}_{j}$$ correspond to the namevalue argument Leverage.
The distribution of z_{t} (the innovation distribution) corresponds to the namevalue argument Distribution, and can be Gaussian or Student's t.
gjr enforces the constraint $${A}_{j}+{L}_{j}\ge 0.$$
gjr enforces stationarity with the constraint
$$\sum _{i=1}^{P}{\gamma}_{i}}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}}+\frac{1}{2}{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}<1.$$
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB^{®} documentation.