Hey,

I'm trying to implement a GARCH model with Johnson-Su innovations in order to simulate returns of financial asset. The model should look like this:

r_t = alpha + lambda*sqrt(h_t) + sqrt(h_t)*epsilon_t
h_t = alpha0 + alpha1*epsilon_(t-1)^2 + beta1 * h_(t-1).

Alpha refers to a risk-free return, lambda to the risk-premium.

I've implemented it like this:

#specification of the model
spec = ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(1,1), submodel = NULL, external.regressors =
NULL, variance.targeting = FALSE), mean.model = list(
armaOrder = c(0,0), include.mean = TRUE, archm = TRUE, archpow = 1,
arfima = FALSE, external.regressors = NULL, archex = FALSE),
distribution.model = "jsu", start.pars = list(), fixed.pars = list())

#fit the model to historical closing price (prices)
fit = ugarchfit(data = prices, spec = spec)

#save coefficients of the fitted model into 'par'
par <- coef(fit)
m = coef(fit)["mu"]
lambda = coef(fit)["archm"]
gamma = coef(fit)["skew"]
delta = coef(fit)["shape"]
#GARCH parameter
a0 = coef(fit)["omega"]
a1 = coef(fit)["alpha1"]
b1 = coef(fit)["beta1"]

My problem is that I often get negative values for lambda, i.e. for the intended risk-premium. So I'm wondering if I've made a mistake in the implementation, as one would usually expect a positive lambda.
And a second question is about the Johnson-Su distribution: Am I right by extracting the Johnson-Su parameters gamma (delta) by the keywords "skew" ("shape")?

Many thanks in advance,

Patrick


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