GARCHM_RESID - GARCH-M fitted values of standardized residuals

Returns an array for the fitted GARCH-M model standardized residuals.

Syntax

GARCHM_RESID (X, Order, Mean, Lambda, Alphas, Betas, Innovation, ν)

X

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

Order

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).

Mean

is the GARCH-M model mean (i.e., mu).

Lambda

is the volatility coefficient for the mean. In finance, lambda is referenced as the risk premium. If missing, a default of 0 is assumed.

Alphas

are the parameters of the ARCH(p) component model (starting with the lowest lag).

Betas

are the parameters of the GARCH(q) component model (starting with the lowest lag).

Innovation

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

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

ν

is the shape parameter (or degrees of freedom) of the innovations/residuals probability distribution function. If missing, a default of 5.0 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. The standardized residuals have a mean of zero and a variance of one (1).
5. The GARCH-M model's standardized residuals is defined as: $$\epsilon_t = \frac{a_t}{\sigma_t}$$ $$a_t = x_t - \mu -\lambda \sigma_t$$ Where:
   • $\epsilon$ is the GARCH-M model's standardized residual at time $t$.
   • $a_t$ is the GARCH-M model's residual at time $t$.
   • $x_t$ is the value of the time series at time $t$.
   • $\mu$ is the GARCH-M mean.
   • $\sigma_t$ is the GARCH-M conditional volatility at time $t$.
   • $\lambda$ is the volatility coefficient in the conditional mean.
6. The number of parameters in the input argument - alpha - determines the order of the ARCH component model.
7. The number of parameters in the input argument - beta - determines the order of the GARCH component model.

Files Examples

Related Links

• Wikipedia - Autoregressive conditional heteroskedasticity.

References

• Hamilton, J.D.; Time Series Analysis, Princeton University Press (1994), ISBN 0-691-04289-6.
• Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740.