garchsim (GARCH Toolbox)

Univariate GARCH process simulation

Syntax

[Innovations, Sigma, Series] = garchsim(Spec, NumSamples, NumPaths, 
Seed, X)

Arguments

Spec
GARCH specification structure for the conditional mean and variance models. You create Spec by calling the function garchset or the estimation function garchfit. The conditional mean can be of general ARMAX form and conditional variance of general GARCH form.

NumSamples
(optional) Positive integer indicating the number of samples garchsim generates for each path of the Innovations, Sigma, and Series outputs. If NumSamples = [] or is not specified, the default is 100.

NumPaths

(optional) Positive integer indicating the number of sample paths (realizations) garchsim generates for the Innovations, Sigma, and Series outputs. If NumPaths = [] or is not specified, the default is 1, i.e. Innovations, Sigma and Series are column vectors.

Seed

(optional) Scalar random number generator seed. If Seed = [] or is not specified, the default is 0 (the MATLAB initial state).

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.
If X = [] or is not specified, the conditional mean has no regression component. If specified, then at least the most recent NumSamples observations of each return series must be valid (i.e., non-NaN). When the number of valid observations in each series exceeds NumSamples, garchsim uses only the most recent NumSamples observations of X.

`Spec`	GARCH specification structure for the conditional mean and variance models. You create `Spec` by calling the function `garchset` or the estimation function `garchfit`. The conditional mean can be of general ARMAX form and conditional variance of general GARCH form.
`NumSamples`	(optional) Positive integer indicating the number of samples `garchsim` generates for each path of the `Innovations`, `Sigma`, and `Series` outputs. If `NumSamples = []` or is not specified, the default is `100`.
`NumPaths`	(optional) Positive integer indicating the number of sample paths (realizations) `garchsim` generates for the `Innovations`, `Sigma`, and `Series` outputs. If `NumPaths = []` or is not specified, the default is `1`, i.e. `Innovations`, `Sigma` and `Series` are column vectors.
`Seed`	(optional) Scalar random number generator seed. If `Seed = []` or is not specified, the default is `0` (the MATLAB initial state).
`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. If `X = []` or is not specified, the conditional mean has no regression component. If specified, then at least the most recent `NumSamples` observations of each return series must be valid (i.e., non-`NaN`). When the number of valid observations in each series exceeds `NumSamples`, `garchsim` uses only the most recent `NumSamples` observations of `X`.

Description

[Innovations, Sigma, Series] = garchsim(Spec, NumSamples, NumPaths, Seed, X) simulates sample paths for return series, innovations, and conditional standard deviation processes, using specifications for the conditional mean and variance of a univariate time series as input. garchsim samples each of NumPaths sample paths at NumSamples observations.

Innovations
NumSamples by NumPaths matrix of innovations, representing a mean zero, discrete-time stochastic process. The Innovations time series follows the conditional variance (GARCH) specification defined in Spec. Rows are sequential times samples, columns are independent realizations.

Sigma
NumSamples by NumPaths matrix of conditional standard deviations of the corresponding Innovations matrix. Innovations and Sigma are the same size. Rows are sequential times samples. Columns are independent realizations.

Series

NumSamples by NumPaths matrix of the return series of interest. Series is the dependent stochastic process and follows the conditional mean specification of general ARMAX form defined in Spec. Rows are sequential times samples. Columns are independent realizations.

`Innovations`	`NumSamples` by `NumPaths` matrix of innovations, representing a mean zero, discrete-time stochastic process. The `Innovations` time series follows the conditional variance (GARCH) specification defined in `Spec`. Rows are sequential times samples, columns are independent realizations.
`Sigma`	`NumSamples` by `NumPaths` matrix of conditional standard deviations of the corresponding `Innovations` matrix. `Innovations` and `Sigma` are the same size. Rows are sequential times samples. Columns are independent realizations.
`Series`	`NumSamples` by `NumPaths` matrix of the return series of interest. `Series` is the dependent stochastic process and follows the conditional mean specification of general ARMAX form defined in `Spec`. Rows are sequential times samples. Columns are independent realizations.

See Also
garchfit, garchget, garchpred, garchset

References

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

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

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

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

garchset lagmatrix