Estimating an EGARCH Model with Seasonal and Regime Effects
Summary
The document presents an EGARCH specification for changes in a series and its conditional log variance. The mean equation includes seasonal indicators and two regime indicators interacting with seasonal effects. The variance equation includes lagged standardized residual magnitudes, a signed standardized residual term, lagged log variance, and seasonal terms that also vary with the regimes.
The author describes developing a stable maximum-likelihood estimation scheme in AD Model Builder and provides a fitted parameter vector with an objective value and gradient diagnostic. This illustrates that the model was numerically estimated, but the data, parameter uncertainty, convergence checks beyond the reported diagnostic, and out-of-sample performance are not discussed. The document mainly asks whether existing GARCH software can estimate this customized model, so it offers a specification and implementation experience rather than a validated forecasting or trading result.
Key ideas
- The EGARCH variance equation models log variance using lagged standardized shocks and lagged log variance.
- Seasonal indicators and two regime indicators enter both the mean and variance specifications.
- The author reports fitting the nonlinear model by maximum likelihood with AD Model Builder.
- A parameter vector and optimization diagnostics are provided, but uncertainty and predictive performance are not assessed.
- The document asks whether standard GARCH software can handle the customized specification.
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Full text
# Comments on solution to a rather complicated EGARCH model
# Comments on solution to a rather complicated EGARCH model
I do a lot of nonlinear parameter estimation, but do not have any experience in finance. I posted the following answer to a question about a EGARCH model on cross validated. I have copied the model description and solution to here so that I have a self-contained question.
I have the data if anyone is interested. My junk email address is xbobolson@gmail.com.
$$\Delta y_{t} = \alpha + \sum^{12}_{j}\beta_{j}S_{j,t}(1 + \gamma_{1}D_{1,t} + \gamma_{2}D_{2,t}) + \varepsilon_{t}$$
$$\ln(\sigma^2_{t}) = \omega_{0} + \sum^{2}_{i=1} \omega_{i} |\frac{\varepsilon_{t-i}}{\sigma_{t-i}}| + \lambda \frac{\varepsilon_{t-1}}{\sigma_{t-1}} + \omega_{3} \ln(\sigma^{2}_{t-1}) + \sum^{12}_{j}\delta_{j}|S_{j,t}|(1 + \rho_{1}D_{1,t} + \rho_{2}D_{2,t})$$
https://stats.stackexchange.com/questions/250274/additional-nonlinear-terms-in-the-egarch-model-maximum-likelihood-estimation.
It took me a while to develop a stable estimation scheme using AD Model Builder. I am curious to know from the GARCH experts if this is simple to do with some existing software.
```
# Number of parameters = 35 Objective function value = -89.3242 Maximum gradient component = 7.45641e-05
# alpha:
-0.0184773755820
# beta:
0.0776272250120 -0.0282086700728 0.0105870775566 -0.796727950111 -0.0329451864914 0.0574720791322 -0.00769006016788 0.0634996913172 -0.271844797702 -0.00148927676168 -0.107749917570 -0.0235315136652
# gamma:
-1.28418932915 0.253708104907
# omega0:
-0.0288892906469
# omega:
-0.0390792976332 0.0693366331859
# lambda:
-0.0174406066104
# ls1: log standard deviations for periods 1 and 2
0.521640582874
# omega3:
0.995013839900
# delta:
-0.0454107838910 0.0442755011514 0.0848044432830 -0.202324778947 -0.0184050276271 0.00891208520744 0.00954099100492 0.00610579281349 -0.0181269442505 0.0502826182040 -0.0135121954742 -0.143376988696
# rho:
0.144999774670 1.88445900772
```Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.