garchfit (GARCH Toolbox)

Univariate GARCH process parameter estimation

Syntax

[Coeff, Errors, LLF, Innovations, Sigma, Summary] = garchfit(Series)
[Coeff, Errors, LLF, Innovations, Sigma, Summary] = garchfit(Spec, 
Series)
[Coeff, Errors, LLF, Innovations, Sigma, Summary] = garchfit(Spec, 
Series, X)
garchfit(...)

Arguments

Series
Vector of observations of the underlying univariate return series for which garchfit estimates the parameters of the conditional mean and variance models. The last element of Series holds the most recent observation.

Spec
(optional) GARCH specification structure that contains the conditional mean and variance models, and optimization parameters. You create the fields in this structure by calling the function garchset, or you can use the Coeff output structure from a previous call to garchfit.

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, garchfit uses only the most recent observations of X. If X = [] or is not specified, the conditional mean has no regression component.

`Series`	Vector of observations of the underlying univariate return series for which `garchfit` estimates the parameters of the conditional mean and variance models. The last element of `Series` holds the most recent observation.
`Spec`	(optional) GARCH specification structure that contains the conditional mean and variance models, and optimization parameters. `Y`ou create the fields in this structure by calling the function `garchset`, or you can use the `Coeff` output structure from a previous call to `garchfit`.
`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`, `garchfit` uses only the most recent observations of `X`. If `X = []` or is not specified, the conditional mean has no regression component.

Description

garchfit estimates the parameters of a conditional mean specification of ARMAX form and a conditional variance specification of GARCH form. If the Display flag (see the function garchset) in the specification structure is set to on (the default), it also displays diagnostic and iterative optimization information in the MATLAB command window (see the function fmincon in the Optimization Toolbox).

[Coeff, Errors, LLF, Innovations, Sigma, Summary] = garchfit(Series) models an observed univariate return series as a constant, C, plus GARCH(1,1) conditionally Gaussian innovations. For models beyond this simplistic (yet common) model, you must provide model parameters in the specification structure, Spec. The C + GARCH(1,1) model is the default model of the GARCH Toolbox.

[Coeff, Errors, LLF, Innovations, Sigma, Summary] = garchfit(Spec, Series) infers the innovations from the return series and fits the model specification, contained in Spec, to the return series by maximum likelihood.

[Coeff, Errors, LLF, Innovations, Sigma, Summary] = garchfit(Spec, Series, X) provides a regression component for the conditional mean.

garchfit(...) (with input arguments as shown above but with no output arguments) prints the final parameter estimates and standard errors to the MATLAB command window. It also produces a tiered plot of the original return series, the inferred innovations (i.e., residuals), and the corresponding conditional standard deviations.

Coeff
GARCH specification structure containing the estimated coefficients. Coeff is of the same form as the Spec input structure. This allows other GARCH Toolbox functions, such as garchset, garchget, garchsim, garchinfer, and garchpred, to accept either Spec or Coeff seamlessly.

Errors
Structure containing the estimation errors (i.e., the standard errors) of the coefficients. The fields of Errors correspond to the coefficient fields (C, AR, MA, Regress, K, GARCH, ARCH) found in Coeff or Spec.

LLF
Optimized log-likelihood objective function value associated with the parameter estimates found in Coeff. garchfit performs the optimization using the fmincon function of the Optimization Toolbox.

Innovations
Innovations vector inferred from Series. The size of Innovations is the same as the size of Series.

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

Summary
Structure of summary information about the optimization process. The fields and their possible values are
warning One of the following strings: 'No Warnings' 'ARMA Model Is Not Stationary/Invertible' converge One of the following strings: 'Function Converged to a Solution' 'Function Did NOT Converge' 'Maximum Function Evaluations or Iterations Reached' covMatrix Covariance matrix of the parameter estimates iterations Number of iterations functionCalls Number of function evaluations constraints One of the following strings: 'No Boundary Constraints' 'Boundary Constraints Active; Errors May Be Inaccurate'

`Coeff`	GARCH specification structure containing the estimated coefficients. `Coeff` is of the same form as the `Spec` input structure. This allows other GARCH Toolbox functions, such as `garchset`, `garchget`, `garchsim`, `garchinfer`, and `garchpred`, to accept either `Spec` or `Coeff` seamlessly.
`Errors`	Structure containing the estimation errors (i.e., the standard errors) of the coefficients. The fields of `Errors` correspond to the coefficient fields (`C`, `AR`, `MA`, `Regress`, `K`, `GARCH`, `ARCH`) found in `Coeff` or `Spec`.
`LLF`	Optimized log-likelihood objective function value associated with the parameter estimates found in `Coeff`. `garchfit` performs the optimization using the `fmincon` function of the Optimization Toolbox.
`Innovations`	Innovations vector inferred from `Series`. The size of `Innovations` is the same as the size of `Series`.
`Sigma`	Conditional standard deviation vector corresponding to `Innovations`. The size of `Sigma` is the same as the size of `Series`.
`Summary`	Structure of summary information about the optimization process. The fields and their possible values are warning One of the following strings: 'No Warnings' 'ARMA Model Is Not Stationary/Invertible' converge One of the following strings: 'Function Converged to a Solution' 'Function Did NOT Converge' 'Maximum Function Evaluations or Iterations Reached' covMatrix Covariance matrix of the parameter estimates iterations Number of iterations functionCalls Number of function evaluations constraints One of the following strings: 'No Boundary Constraints' 'Boundary Constraints Active; Errors May Be Inaccurate'

Note garchfit calculates the error covariance matrix of the parameter estimates, Summary.covMatrix, and the corresponding standard errors found in the Errors output structure, using finite difference approximation. In particular, it calculates the standard errors using the outer-product method (see Hamilton [12], section 5.8, bottom of page 143).

See Also
garchllfn, garchpred, garchset, garchsim,

fmincon (in the Optimization Toolbox)

References

Bollerslev, T., "Generalized Autoregressive Conditional Heteroskedasticity," Journal of Econometrics, Vol. 31, pp. 307-327, 1986.

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

Engle, Robert, "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, vol. 50, pp. 987-1007, 1982.

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

garchdisp garchget