Replicating an AR-GARCH-M Model with Chaotic and Random Variance Shocks

# Athens Stock Exchange GARCH-M (Paper replication)


# Athens Stock Exchange GARCH-M (Paper replication)












I am trying to replicate one part of a paper which tries to model the Athens Stock Exchange daily returns. I do not have the original dataset, so some differences are expected, but when I fit the model in R with the appxoximate dates, some coefficients are way different. So I am trying to understand what is the mistake of my model compared to the paper. More specifically :

Paper link: https://www.um.edu.mt/library/oar/bitstream/123456789/30874/1/Volatility_behaviour_in_emerging_markets_a_case_study_of_the_Athens%20stock_exchange_2000.pdf

My aim is to replicate section 4.4 and table 5 in the end of the paper.The proposed model is an AR(1)-GARCH-M(1,1) with two extra shocks included in the variance:

Feigenbaum chaotic model : $z_t = 3.57z _{t-1}*(1-z _{t-1}) $

Exogenous normal shock : $e_t ~ N(0,1)$

So the variance mean is $h_t = a_0 + a_1ε^2_{t-1} + β_1 h_{t-1} + γ_1 z_t^2 + γ_2e_t^2$ and the model mean is an AR(1) with mean.

My R code is the following. TS is the time series. First i create the Feigenbaum variable with initial value 0.7, then the extra exogenous shock as normal distribution. Finally I fit the model, setbounds to allow negative values and adding two external regressors in the variance model.

```
TS$FG <- 0.7
for (i in  2:nrow(TS)) {
TS$FG[i]<- (3.57*TS$FG[(i-1)]*(1 - TS$FG[(i-1)]))
}
TS$eshock <- rnorm(nrow(TS))

#Fit a GARCH-M(1,1)
spec <-
  ugarchspec(
    variance.model = list(model = "sGARCH", garchOrder = c(1, 1),
                          external.regressors = cbind(as.matrix(TS$FG^2), as.matrix(TS$eshock^2))),
    mean.model = list(armaOrder = c(1, 0), archm = T, archpow = 1, include.mean = T),
    distribution.model = "norm"
  )

setbounds(spec) <- list(omega = c(-1, 1),vxreg1  = c(-1, 1), vxreg2  = c(-1, 1), alpha1 = c(-1, 1), beta1 = c(-1, 1))

model_fit <-
  ugarchfit(
    spec = spec,
    data = TS$returns,
    solver = "hybrid",
    fit.control = list(rec.init = 0.7)
  )
model_fit
```

Below you can see the results of the regression:

Even allowing for algorithm differences, maybe slightly more data, some coefficients are pretty different.

- The ARCH-M coefficient is almost half of the paper (0.11 vs 0.20)

- alpha and beta seem to be opposite! I have 0.20 and 0.78 while the paper has 0.77 and 0.21. This is the most weird result

- My external regressors are a bit different but acceptable.

I have a feeling that if I resolve the alpha/beta coefficients (not sure what is wrong) maybe my model will be able to replicate closely the paper's results.

Let me know what you think it might be wrong my approach.

Thank you