# GARCH_RESID - GARCH Fitted Values of Standardized Residuals

Returns an array of the standardized residuals for the fitted GARCH model.

## Syntax

GARCH_RESID ([x], order, µ, [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.
[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 model's standardized residuals is defined as:$$\epsilon_t = \frac{a_t}{\sigma_t}$$ $$a_t = x_t - \mu$$

Where:

• $\epsilon$ is the GARCH model's standardized residual at time $t$.
• $a_t$ is the GARCH model's residual at time $t$.
• $x_t$ is the value of the time series at time $t$.
• $\mu$ is the GARCH mean.
• $\sigma_t$ is the GARCH conditional volatility at time $t$.
6. 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).
7. 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).