# How to estimate the following model?

Suppose I have the following model:

$$r_t=\sigma_t * \epsilon_t$$

where $r_t$ is the return at time t, $\sigma_t$ is the volatility, the model used to model this volatility is an exponentially weighted moving average with known parameter $\lambda$. $\epsilon_t$ is a random variable distributed according to the hyperbolic distribution with parameters $\alpha, \beta , \mu, \delta$.

First question: How do I estimate this model?

Do I

1. Since $\lambda$ is known, calculate the $\hat{\sigma}_t$.
2. Calculate $r_t/\hat{\sigma}_t$ which give the so called standardized residuals.
3. Using the standardized residuals estimate the parameters of the hyperbolic distribution with classical ML.

Or

1. Include the $\hat{\sigma}_t$ in the log-likelihood of the hyperbolic distribution and maximize this, so this could be called a "joint" estimation. Since not the normal ML is done, but the ML with the estimated $\sigma$ included.

Second question: Suppose the volatility is modeled by an ARCH process.

Do I have to use an R package which estimates all parameters jointly, so the output give me the values of the ARCH process AND the values of the hyperbolic distribution?

Or can I use a "normal" ARCH command (which will assume the $\epsilon$ to be N(0,1) distributed (I guess) calculate the $\sigma$. Then do like above, calculate the standardized residuals by calculating $r_t/\sigma_t$ and use these to estimate an hyperbolic distribution using ML. What do you think about this "divided" approach?

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