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GARCHM_LLF (Pro.)
Calculates the log-likelihood function for the fitted GARCH-Mi model.
Syntax
GARCHM_LLFi(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 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-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 lagi).
betas
are the parameters of the GARCH(q) component model (starting with the lowest lag).
innovation
is the probability distribution model for the innovations/residuals (1=Gaussian, 2=t-Distribution, 3=GEDi). If missing, a gaussian distribution is assumed.
| value | Description |
|---|---|
| 1 | Gaussian or Normal Distribution (default) |
| 2 | Student's t-Distribution |
| 3 | Generalized Error Distribution (GED) |
v
is the shape parameter (or degrees of freedom) of the innovations/residuals probability distribution function.
Remarks
- The underlying model is described here.
- The Log-Likelihood Function (LLF) 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.
- The maximum likelihood estimation (MLE) is a statistical method for fitting a model to the data and provides estimates for the model's parameters.
- 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.
Examples
Example 1:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | Date | Data | ||
| 2 | January 10, 2008 | -2.827 |
GARCH-M(1,1) |
|
| 3 | January 11, 2008 | -0.947 | Mean | -0.076 |
| 4 | January 12, 2008 | -0.877 | Lambda | 0.145 |
| 5 | January 14, 2008 | 1.209 | Alpha_0 | 0.593 |
| 6 | January 13, 2008 | -1.669 | Alpha_1 | 0.000 |
| 7 | January 15, 2008 | 0.835 | Beta_1 | 0.403 |
| 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) | |
|---|---|---|
| =GARCHM_LLF($B$2:$B$32,1,$D$3,$D$4,$D$5:$D$6,$D$7) | Log-Likelihood Function for Guassian Distribution (-45.482) | |
| =GARCHM_LLF($B$2:$B32,1,$D$3,$D$4,$D$5:$D$6,$D$7,2,5) | Log-Likelihood Function for t-Distribution when v = 5 (-49.122) | |
| =GARCHM_LLF($B$2:$B32,1,$D$3,$D$4,$D$5:$D$6,$D$7,3,2) | Log-Likelihood Function for GED when v = 2 (-45.482) | |
| =GARCHM_AICi($B$2:$B$32,1,$D$3,$D$4,$D$5:$D$6,$D$7) | Akaike's information criterion (96.964) | |
| =GARCHM_CHECK($D$3,$D$4,$D$5:$D$6,$D$7) | The GARCH-M(1,1) model is stable? (1) |
Files Examples
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
