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GARCH_AIC (Pro.)
Calculates the Akaike's information criterion (AICi) of a given estimated GARCHi model (with corrections to small sample sizes).
Syntax
GARCH_AIC(X, Order, mean, 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 of the data series (i.e. whether the first data point corresponds to the earliest or latest date (earliest date=1 (default), latest date=0)).
| Order | Description |
|---|---|
| 1 | ascending (the first data point corresponds to the earliest date) |
| 0 | descending (the first data point corresponds to the latest date) |
mean
is the GARCH model mean (i.e. mu).
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, 2=t-Distribution, 3=GEDi). If missing, a gaussian distribution is assumed.
| 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
- The underlying model is described here.
- Akaike's Information Criterion (AIC) is described here.
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g. #N/A) at either end.
- Given a fixed data set, several competing models may be ranked according to their AIC, the model with the lowest AIC being the best.
- The number of parameters in the input argument - alpha - determines the order of the ARCH component model.
- The number of parameters in the input argument - beta - determines the order of the GARCH component model.
- GARCH(p,q) model has p+q+2 parameters to estimate.
-
The AIC for a GARCH model is defined as:
Where:
-
is the log-likelihood function.
-
is the number of non-missing values.
-
is the order of the ARCH component model.
-
is the order of the GARCH component model.
-
Examples
Example 1:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | Date | Data | ||
| 2 | January 10, 2008 | -2.827 |
GARCH(1,1) |
|
| 3 | January 11, 2008 | -0.947 | Mean | -0.160 |
| 4 | January 12, 2008 | -0.877 | Alpha_0 | 0.608 |
| 5 | January 14, 2008 | 1.209 | Alpha_1 | 0.00 |
| 6 | January 13, 2008 | -1.669 | Beta_1 | 0.391 |
| 7 | January 15, 2008 | 0.835 | ||
| 8 | January 16, 2008 | -0.266 | ||
| 9 | January 17, 2008 | 1.361 | ||
| 10 | January 18, 2008 | -0.343 | ||
| 11 | January 19, 2008 | 0.475 | ||
| 12 | January 20, 2008 | -1.153 | ||
| 13 | January 21, 2008 | 1.144 | ||
| 14 | January 22, 2008 | -1.070 | ||
| 15 | January 23, 2008 | -1.491 | ||
| 16 | January 24, 2008 | 0.686 | ||
| 17 | January 25, 2008 | 0.975 | ||
| 18 | January 26, 2008 | -1.316 | ||
| 19 | January 27, 2008 | 0.125 | ||
| 20 | January 28, 2008 | 0.712 | ||
| 21 | January 29, 2008 | -1.530 | ||
| 22 | January 30, 2008 | 0.918 | ||
| 23 | January 31, 2008 | 0.365 | ||
| 24 | February 1, 2008 | -0.997 | ||
| 25 | February 2, 2008 | -0.360 | ||
| 26 | February 3, 2008 | 1.347 | ||
| 27 | February 4, 2008 | -1.339 | ||
| 28 | February 5, 2008 | 0.481 | ||
| 29 | February 6, 2008 | -1.270 | ||
| 30 | February 7, 2008 | 1.710 | ||
| 31 | February 8, 2008 | -0.125 | ||
| 32 | February 9, 2008 | -0.940 |
| Formula | Description (Result) | |
|---|---|---|
| =GARCH_AIC($B$2:$B$32,1,$D$3,$D$4:$D$5,$D$6) | Akaike's information criterion (AIC) for Guassian Distribution (96.013) | |
| =GARCH_AIC($B$2:$B$32,1,$D$3,$D$4:$D$5,$D$6,2,5) | Akaike's information criterion (AIC) for t-Distribution with Freedom = 5 (103.0997) | |
| =GARCH_AIC($B$2:$B$32,1+$D$14,$D$3,$D$4:$D$5,$D$6,3,2) | Akaike's information criterion (AIC) for GED with Freedom = 2 (96.013) |
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
