How to estimate the following model?

Question

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?

Answer 1

I would suggest writing the joint density as the product of the conditional densities then estimate parameters using an optimization package.

The joint density is given by

$$f(r_0, \ldots, r_T) = f(r_0) \prod_{t=1}^T f(r_t|r_0, \ldots, r_{t-1})$$

then the log likelihood function is

$$L = \log(f(r_0)) + \sum_{t=1}^T \log(f(r_t | r_0, \ldots, r_{t-1}) )$$

You may have some issues trying to optimize this function because of the number of parameters in your hyperbolic errors. These parameters may share quite a bit of information and lead to very flat slowly converging likelihood surfaces. This happens frequently with student T distribution when doing joint estimation of $\sigma$ and $\nu$ parameters.

See page 17 of Ruey S Tsay Analysis of Financial Time Series (2nd Ed) for another similarly brief discussion with the normal distribution as an example

Answer 2

Your model is wrong , there are no innovations errors in EWMA models. Indeed the EWMA model belongs to the moving average class of estimators. Then to obtain $\sigma_{t}$ you just need $\lambda$ and $r_{t}$ : The volatility is given by : $\sigma =\sqrt{(1-\lambda)\sum_{t=1}^{T} \lambda^{t-1}(r_{t}-r)^{2}} $.(remark: there is no subscript $t$ associated with the volatility term: it is an unconditional volatility process) . Loosely speaking, EWMA allows us to compute the long term average variance .

It is based on an i.i.d returns model : we obtain a subscript $t$ after estimation because the estimate change over time but in fact there is no conditional variance process.

The way you presented your model is specific for ARCH type models (conditional variance process) such as :

conditional mean process : $r_{t} = \sigma_{t} \epsilon_{t}$

conditional variance process $\sigma_{t} =.... Garch,Figarch,Aparch...$

where $\epsilon_{t}$ follows a specific distirbution.

ARCH and Moving averages models are "concurents" models to estimate volatility and correlations, they can't be applied simultaneously.

Ps:As far as i know, in MLE estimation $\sigma_{t}$ is always integrated in the Likelihood function.