How do ARMA and ARMA-GARCH models relate?

Question

I would like to understand how GARCH models work but I'm having some problems. I have a highly persistent AR time series and I would like to model the conditional mean as well as its conditional variance. I thought of 2 possible ways:

Estimate an AR(1) model, obtain the residuals, fit a GARCH(1,1) to the residuals. The first model models the mean, the second model the variance.
Estimate an ARMA-GARCH model.

I assumed, that both ways should yield the same results in terms of coefficients but that's not the case. Why do the AR coefficients differ?

I have created a reprex in R. You only need the fGarch and tseries packages besides base.

library(fGarch)
library(tseries)

# Simulate time series with heteroscedasticity
set.seed(1)
y <- c(0)
for(i in 2:5000) {
  y[i] <- 0.99*y[i-1] + rnorm(1, sd = (sin(i/300)+1)^2) 
}
plot(y, type = "l")

# Model Estimation

# Estimate AR 1

model_arma <- arma(y, order = c(1,0,0))
model_arma$coef
#>        ar1  intercept 
#>  0.9820884 -0.0391143

resids <- residuals(model_arma)[-1]

# Estimate Garch(1,1)
model_garch <- garchFit(~garch(1,1), trace = FALSE)
model_garch@fit$matcoef
#>            Estimate  Std. Error    t value     Pr(>|t|)
#> mu     -0.006190316 0.008461996 -0.7315433 4.644474e-01
#> omega   0.010761385 0.002837514  3.7925399 1.491143e-04
#> alpha1  0.153134061 0.026421658  5.7957780 6.800519e-09
#> beta1   0.805973743 0.033381268 24.1444916 0.000000e+00

# Estimate Arma-Garch
model_armagarch <- garchFit(~arma(1,0) + garch(1,1), trace = FALSE)
model_armagarch@fit$matcoef
#>            Estimate  Std. Error    t value     Pr(>|t|)
#> mu     -0.006097132 0.008400905 -0.7257709 4.679792e-01
#> ar1     0.051377942 0.025641637  2.0036920 4.510307e-02
#> omega   0.011189102 0.002820163  3.9675370 7.261924e-05
#> alpha1  0.157402606 0.026259893  5.9940306 2.047028e-09
#> beta1   0.799952222 0.032890817 24.3214454 0.000000e+00

^{Created on 2019-12-20 by the reprex package (v0.3.0)}

Thank you Richard, I just forgot to accept your answer. Thank you very much. — BerriJ, Dec 26 '19 at 11:59

score 1 · Accepted Answer · answered Dec 21 '19 at 08:02

There are two questions in your post:

How do ARMA and ARMA-GARCH models relate?
Why do the AR coefficients differ?

The first question is answered extensively here.

The second has been discussed multiple times, see e.g. these threads. The basic insight is that the parameter estimates of the conditional mean part are related to the those of the conditional variance part. You change one, and all others change. This is similar to the omitted variable bias in regression: if you omit a variable, the coefficients on other variables change (except if the omitted one is orthogonal to the space spanned by others). In the present context, you are omitting not a variable but an equation for the conditional variance. No wonder the conditional mean parameter estimates change.

How do ARMA and ARMA-GARCH models relate?

1 Answers1