GARCH_SIM - Simulated Values of a GARCH Model

Returns an array of cells for the model simulated values.

Syntax

GARCH_SIM ([x], [σ], order, µ, [α], [β], f, ν, t, seed)

[X]

Required. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).

[σ]

Optional. Is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)) of the last q realized volatilities.

Order

Optional. Is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).

Value	Order
1	Ascending (the first data point corresponds to the earliest date) (default).
0	Descending (the first data point corresponds to the latest date).

µ

Optional. Is the GARCH model long-run mean (i.e., mu). If missing, the process mean is assumed to be zero.

[α]

Required. Are the parameters of the ARCH(p) component model: [αo α1, α2 … αp] (starting with the lowest lag).

[β]

Optional. Are the parameters of the GARCH(q) component model: [β1, β2 … βq] (starting with the lowest lag).

F

Optional. Is the probability distribution function of the innovations/residuals (1 = Gaussian (default), 2 = t-Distribution, 3 = GED).

Value	Probability Distribution
1	Gaussian or Normal Distribution (default).
2	Student's t-Distribution.
3	Generalized Error Distribution (GED).

ν

Optional. Is the shape parameter (or degrees of freedom) of the innovations/residuals’ probability distribution function.

T

Optional. Is the forecast time/horizon (expressed in terms of steps beyond the end of the time series X). If missing, t = 1 is assumed.

Seed

Required. Is an unsigned integer for setting up the random number generator(s).

Remarks

1. The underlying model is described here.
2. The time series is homogeneous or equally spaced.
3. The time series may include missing values (e.g., #N/A) at either end.
4. For the input argument - ([α]) (parameters of the ARCH component):
   • The input argument is not optional.
   • The value in the first element must be positive.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
   • In the case where alpha has one non-missing entry/element (first), no ARCH component is included.
   • The order of the ARCH component model is solely determined by the order (minus one) of the last value in the array with a numeric value (vs. missing or error).
5. For the input argument - ([β]) (parameters of the GARCH component):
   • The input argument is optional and can be omitted, in which case no GARCH component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values or error codes (i.e., #NUM!, #VALUE!, etc.).
   • The order of the GARCH component model is solely determined by the order of the last value in the array with a numeric value (vs. missing or error).
6. By definition, the GARCH_FORE function returns a constant value equal to the model mean (μ) for all horizons.
7. The function GARCH_SIM was added in version 1.63 SHAMROCK.

Files Examples

Related Links

• Wikipedia - Autoregressive conditional heteroskedasticity.

References

• James Douglas Hamilton; Time Series Analysis, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
• Tsay, Ruey S.; Analysis of Financial Time Series, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.