# 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, v)

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)).

Order Description
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.

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

v is the shape factor (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 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.

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
A B C D E
Date Data
January 10, 2008 -2.827 -2.760 GARCH-M(1,1)
January 11, 2008 -0.947 -1.019 Mean -0.076
January 12, 2008 -0.877 -0.949 Lambda 0.145
January 13, 2008 1.209 1.144 Alpha_0 0.593
January 14, 2008 -1.669 -1.743 Alpha_1 0.000
January 15, 2008 0.835 0.769 Beta_1 0.403
January 16, 2008 -0.266 -0.336
January 17, 2008 1.361 1.297
January 18, 2008 -0.343 -0.413
January 19, 2008 0.475 0.408
January 20, 2008 -1.153 -1.226
January 21, 2008 1.144 1.079
January 22, 2008 -1.070 -1.142
January 23, 2008 -1.491 -1.565
January 24, 2008 0.686 0.620
January 25, 2008 0.975 0.910
January 26, 2008 -1.316 -1.389
January 27, 2008 0.125 0.057
January 28, 2008 0.712 0.646
January 29, 2008 -1.530 -1.604
January 30, 2008 0.918 0.852
January 31, 2008 0.365 0.297
February 1, 2008 -0.997 -1.069
February 2, 2008 -0.360 -0.430
February 3, 2008 1.347 1.283
February 4, 2008 -1.339 -1.412
February 5, 2008 0.481 0.414
February 6, 2008 -1.270 -1.343
February 7, 2008 1.710 1.647
February 8, 2008 -0.125 -0.194
February 9, 2008 -0.940 -1.012