garchinfer (GARCH Toolbox)

Inverse filter to infer GARCH innovations and conditional standard deviations from an observed return series

Syntax

[Innovations, Sigma, LogLikelihood] = garchinfer(Spec, Series, X)

Arguments

Spec
GARCH specification structure that contains the conditional mean and variance specifications, as well as the optimization parameters of a GARCH model. You can create Spec by calling the function garchset or the estimation function garchfit.

Series
Matrix of observations of the underlying univariate return series of interest for which garchinfer infers the innovations and corresponding conditional standard deviations.
Each column of Series is an independent realization (i.e., path). The last row of Series holds the most recent observation of each realization.

X
(optional) Time series regression matrix of observed explanatory data. Typically, X is a matrix of asset returns (e.g., the return series of an equity index), and represents the past history of the explanatory data. Each column of X is an individual time series used as an explanatory variable in the regression component of the conditional mean. In each column, the first row contains the oldest observation and the last row the most recent.
The number of valid (non-NaN) most recent observations in each column of X must equal or exceed the number of valid most recent observations in Series. If the number of valid observations in a column of X exceeds that of Series, garchinfer uses only the most recent observations of X. If X = [] or is not specified, the conditional mean has no regression component.

`Spec`	GARCH specification structure that contains the conditional mean and variance specifications, as well as the optimization parameters of a GARCH model. You can create `Spec` by calling the function `garchset` or the estimation function `garchfit`.
`Series`	Matrix of observations of the underlying univariate return series of interest for which `garchinfer` infers the innovations and corresponding conditional standard deviations. Each column of `Series` is an independent realization (i.e., path). The last row of `Series` holds the most recent observation of each realization.
`X`	(optional) Time series regression matrix of observed explanatory data. Typically, `X` is a matrix of asset returns (e.g., the return series of an equity index), and represents the past history of the explanatory data. Each column of `X` is an individual time series used as an explanatory variable in the regression component of the conditional mean. In each column, the first row contains the oldest observation and the last row the most recent. The number of valid (non-`NaN`) most recent observations in each column of `X` must equal or exceed the number of valid most recent observations in `Series`. If the number of valid observations in a column of `X` exceeds that of `Series`, `garchinfer` uses only the most recent observations of `X`. If `X = []` or is not specified, the conditional mean has no regression component.

Description

[Innovations, Sigma, LogLikelihood] = garchinfer(Spec, Series, X) acts as an inverse, or whitening, filter to infer the innovations and conditional standard deviations from an observed return series, using a conditional mean specification of ARMAX form and a conditional variance specification of GARCH form as input. Since garchinfer provides an interface to the appropriate log-likelihood objective function, it also computes the log-likelihood value as a convenience.

Innovations
Innovations matrix inferred from the input Series matrix. The size of Innovations is the same as the size of Series, and its organization is the same as Series.

Sigma
Conditional standard deviation matrix corresponding to Innovations. The size of Sigma is the same as the size of Series.

LogLikelihood
Vector of log-likelihood objective function values for each realization of Series. The length of LogLikelihood is the same as the number of columns in Series.

`Innovations`	Innovations matrix inferred from the input `Series` matrix. The size of `Innovations` is the same as the size of `Series`, and its organization is the same as `Series`.
`Sigma`	Conditional standard deviation matrix corresponding to `Innovations`. The size of `Sigma` is the same as the size of `Series`.
`LogLikelihood`	Vector of log-likelihood objective function values for each realization of `Series`. The length of `LogLikelihood` is the same as the number of columns in `Series`.

See Also
garchfit, garchllfn, garchpred, garchset, garchsim

fmincon (in the Optimization Toolbox)

References

Box, G.E.P., G.M. Jenkins, G.C. Reinsel, Time Series Analysis: Forecasting and Control, third edition, Prentice Hall, 1994.

Hamilton, J.D., Time Series Analysis, Princeton University Press, 1994.

garchget garchllfn