GARCHM_RESID - GARCH-M Fitted Standardized Residuals.

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

Syntax

GARCHM_RESID ([x], order, µ, λ, [α], [β], f, ν)

[X]

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

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.

λ

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

[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.

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 are 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 - [α_o α_1, α₂ … α_p] - determines the order of the ARCH component model.
7. The number of parameters in the input argument - [β_1, β₂ … β_q] - determines the order of the GARCH component model.

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.