GARCH_FORESD - Volatility Forecast of GARCH Model

(Deprecated) Calculates the estimated error/standard deviation of the conditional mean forecast.

Syntax

GARCH_FORESD ([x], [σ], order, µ, [α], [β], t, return)

[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).

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.

Return

Optional. Is the type of desired volatility output (Term Structure = 0, Local/Step = 1). If missing, local volatility is assumed.

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. For GARCH (1, 1), the squared of the forecast standard error (i.e., conditional variance) is expressed as follows:$$E[\sigma_{T+k}^2]=\alpha_o\times\frac{1-(\alpha_1+\beta_1)^k}{1-(\alpha_1+\beta_1)}+(\alpha_1+\beta_1)^k\sigma_T^2$$
7. The forecast standard error (i.e., conditional volatility) converges monotonically to its long-run average. For the case of GARCH (1, 1):$$E[\sigma_{T+k\rightarrow\infty}^2]=\frac{\alpha_o}{1-(\alpha_1+\beta_1)}$$

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.