garchinfer - Infer GARCH innovation processes from return series

Spec

GARCH specification structure that contains the conditional mean and variance specifications. It also contains the optimization parameters needed for the estimation. Create this structure by calling garchset, or by using the Coeff output structure returned by garchfit.

Series

Time-series matrix or column vector of observations of the underlying univariate return series of interest. Series is the response variable representing the time series fitted to conditional mean and variance specifications. Each column of Series in an independent realization (that is, path). The last row of Series holds the most recent observation of each realization.

X

Time-series regression matrix of explanatory variables. Typically, X is a regression matrix of asset returns (for example, the return series of an equity index). 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) observations below the last NaN in each column of X must equal or exceed the number of valid observations below the last NaN in Series. If the number of valid observations in a column of X exceeds that of Series, garchinfer uses only the most recent. If X = [] or is unspecified, the conditional mean has no regression component.

PreInnovations

Time-series matrix or column vector of presample innovations on which the recursive mean and variance models are conditioned. This array can have any number of rows, provided it contains sufficient observations to initialize the mean and variance equations. That is, if M and Q are the number of lagged innovations required by the conditional mean and variance equations, respectively, then PreInnovations must have at least max(M,Q) rows.

If the number of rows exceeds max(M,Q), then garchinfer uses only the last (that is, most recent) max(M,Q) rows. If PreInnovations is a matrix, then the number of columns must be the same as the number of columns in Series. If PreInnovations is a column vector, then garchinfer applies the vector to each column (realization) of Series.

PreSigmas

Time-series matrix or column vector of positive presample conditional standard deviations on which the recursive variance model is conditioned. This array can have any number of rows, provided it contains sufficient observations to initialize the conditional variance equation. For example, if P and Q are the number of lagged conditional standard deviations and lagged innovations required by the conditional variance equation, respectively, then PreSigmas must have:

• At least P rows for GARCH and GJR models, and

• At least max(P,Q) rows for EGARCH models.

If the number of rows exceeds the requirement, then garchinfer uses only the last ( most recent) rows. If PreSigmas is a matrix, then the number of columns must be the same as the number of columns in Series. If PreSigmas is a column vector, then garchinfer applies the vector to each column (realization) of Series.

PreSeries

Time-series matrix or column vector of presample observations of the return series of interest on which the recursive mean model is conditioned. This array can have any number of rows, provided it contains sufficient observations to initialize the conditional mean equation. Thus, if R is the number of lagged observations of the return series required by the conditional mean equation, then PreSeries must have at least R rows. If the number of rows exceeds R, then garchinfer uses only the last (most recent) R rows. If PreSeries is a matrix, then the number of columns must be the same as the number of columns in Series. If PreSeries is a column vector, then garchinfer applies the vector to each column (realization) of Series.

Innovations

Innovations time-series matrix inferred from Series. The size of Innovations is the same as the size of Series.

Sigmas

Conditional standard deviation time-series matrix corresponding to Innovations. The size of Sigmas is the same as the size of Series.

LLF

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