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GJR conditional variance time series model

Use `gjr`

to specify a univariate GJR (Glosten, Jagannathan, and Runkle) model. The `gjr`

function returns a `gjr`

object specifying the functional form of a GJR(*P*,*Q*) model, and stores its parameter values.

The key components of a `gjr`

model include the:

GARCH polynomial, which is composed of lagged conditional variances. The degree is denoted by

*P*.ARCH polynomial, which is composed of the lagged squared innovations.

Leverage polynomial, which is composed of lagged squared, negative innovations.

Maximum of the ARCH and leverage polynomial degrees, denoted by

*Q*.

*P* is the maximum nonzero lag in the GARCH polynomial, and
*Q* is the maximum nonzero lag in the ARCH and leverage
polynomials. Other model components include an innovation mean model offset, a
conditional variance model constant, and the innovations distribution.

All coefficients are unknown (`NaN`

values) and estimable unless you specify their values using name-value pair argument syntax. To estimate models containing all or partially unknown parameter values given data, use `estimate`

. For completely specified models (models in which all parameter values are known), simulate or forecast responses using `simulate`

or `forecast`

, respectively.

returns a zero-degree conditional variance `Mdl`

= gjr`gjr`

object.

creates a GJR conditional variance model object (`Mdl`

= gjr(`P`

,`Q`

)`Mdl`

) with a GARCH polynomial with a degree of `P`

and ARCH and leverage polynomials each with a degree of `Q`

. All polynomials contain all consecutive lags from 1 through their degrees, and all coefficients are `NaN`

values.

This shorthand syntax enables you to create a template in which you specify the polynomial degrees explicitly. The model template is suited for unrestricted parameter estimation, that is, estimation without any parameter equality constraints. However, after you create a model, you can alter property values using dot notation.

sets properties or additional options using name-value pair arguments. Enclose each property name in quotes. For example, `Mdl`

= gjr(`Name,Value`

)`'ARCHLags',[1 4],'ARCH',{0.2 0.3}`

specifies the two ARCH coefficients in `ARCH`

at lags `1`

and `4`

.

This longhand syntax enables you to create more flexible models.

`estimate` | Fit conditional variance model to data |

`filter` | Filter disturbances through conditional variance model |

`forecast` | Forecast conditional variances from conditional variance models |

`infer` | Infer conditional variances of conditional variance models |

`simulate` | Monte Carlo simulation of conditional variance models |

`summarize` | Display estimation results of conditional variance model |

You can specify a `gjr`

model as part of a composition of conditional mean and variance models. For details, see `arima`

.

[1] Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.”
*The Journal of Finance*. Vol. 48, No. 5, 1993, pp. 1779–1801.

[2] Tsay, R. S. *Analysis of Financial Time Series*. 3rd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2010.