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rvolatilep-valueautoregressive-models

GARCH MODEL with Exogenous Variables in the conditional mean and variance

I need help writing a GARCH equation with exogenous variables by hand. I can write conditional mean and conditional variance equations, but not with exogenous variables. The fitted GARCH model is a AR(1)-GARCH(1,1) model. This is what I have so far:

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I need help adding mxreg1 and mxreg2 (the significant exogenous variables) in the conditional mean equation. Thanks!

If it is hard for you to answer the question because it is like a hand written equation, you could do your best and upload a picture of a readable version of the conditional mean equation. Thanks!

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : fGARCH(1,1)
fGARCH Sub-Model    : GARCH
Mean Model  : ARFIMA(1,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.006505    0.009810 -0.66311 0.507258
ar1     0.149542    0.044502  3.36030 0.000779
mxreg1  0.372466    0.054183  6.87422 0.000000
mxreg2 -0.000754    0.000249 -3.02629 0.002476
mxreg3  0.000749    0.000514  1.45800 0.144840
mxreg4  0.000003    0.000006  0.50995 0.610085
mxreg5  0.062401    0.080066  0.77937 0.435760
omega   0.000276    0.000017 15.95469 0.000000
alpha1  0.119026    0.023418  5.08266 0.000000
beta1   0.785459    0.032615 24.08273 0.000000
vxreg1  0.000000    0.001602  0.00000 1.000000
vxreg2  0.000000    0.000010  0.00000 1.000000
vxreg3  0.000000    0.000021  0.00000 1.000000
vxreg4  0.000000    0.000000  0.00000 1.000000
vxreg5  0.001912    0.003569  0.53563 0.592212
Solution

  • I was working on an ARMAX model and I believe the same coefficient could be applied to the conditional return equation of the garch model with exogenous variables. To add one covariate, add ßYt to the conditional return equation. For more than one covariate it would be in the form ß(Yt + Yt-1 + … Yt-b) where “b” is the number of exogenous variables.