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EGARCH conditional variance time series model

Use `egarch`

to specify a univariate exponential
generalized autoregressive conditional heteroscedastic (EGARCH) model. The
`egarch`

function returns an `egarch`

object
specifying the functional form of an EGARCH(*P*,*Q*) model, and stores its
parameters values.

The key components of an `egarch`

model include:

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

*P*.The ARCH polynomial, which is composed of the magnitudes of lagged standardized innovations.

The leverage polynomial, which is composed of lagged standardized innovations.

*Q*is the maximum of the ARCH and leverage polynomial degrees.

*P* and *Q* are the maximal nonzero lags in the GARCH, and
ARCH and leverage polynomials, respectively. Other model components include the innovation
mean model offset, a conditional variance model constant, and 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.

`Mdl = egarch`

`Mdl = egarch(P,Q)`

`Mdl = egarch(Name,Value)`

returns a zero-degree
conditional variance `Mdl`

= egarch`egarch`

object.

returns an `Mdl`

= egarch(`P`

,`Q`

)`egarch`

object that has a GARCH polynomial with a
degree of `P`

, and ARCH and leverage polynomials both 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 allows for easy model template creation 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 pairs. You can specify multiple values. Enclose each name in single
quotes. For example, `Mdl`

= egarch(`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 an

`egarch`

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

.An EGARCH(1,1) specification is complex enough for most applications. Typically in these models, the GARCH and ARCH coefficients are positive, and the leverage coefficients are negative. If you get these signs, then large unanticipated downward shocks increase the variance. If you get signs opposite to those signs that are expected, you can encounter difficulties inferring volatility sequences and forecasting. A negative ARCH coefficient is problematic. In this case, an EGARCH model might not be the best choice for your application.

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

- Specify EGARCH Models
- Modify Properties of Conditional Variance Models
- Specify Conditional Mean and Variance Models
- Infer Conditional Variances and Residuals
- Compare Conditional Variance Models Using Information Criteria
- Assess EGARCH Forecast Bias Using Simulations
- Forecast a Conditional Variance Model
- Conditional Variance Models
- EGARCH Model