When regressing a variable on a constant of 1, the coefficient of this constant is the mean. However, when I specified that the residuals follow a GARCH(1,1) model, the coefficient of the constant does not anymore represent the mean in my model. In my opinion this makes no sense since the residuals are defined to have a mean of zero. The mu in the example below should be the same as the mean of R_d.
Small Example in R:
library(rugarch) library(quantmod) getSymbols('C', from = '2000-01-01') C = adjustOHLC(C, use.Adjusted = TRUE) R_d = ROC(Cl(C), na.pad = FALSE) mean(R_d) -0.000436420257283668 spec = ugarchspec(mean.model = list(armaOrder = c(0, 0)), variance.model = list(model = 'sGARCH', garchOrder = c(1, 1)), distribution = 'norm') fit = ugarchfit(data = R_d, spec = spec) coef(fit) mu: 0.000430648533256351 omega: 2.11481883824743e-06 alpha1: 0.0871525584932368 beta1: 0.911847414938857
For egarch or gjr, I get also different results for mu..