Russell 2000 weekly returns

Russell 2000 is a value-weighted index for small cap companies and a common benchmark for mutual funds that identify themselves as "small-cap".

Objective

For our purpose, we try to model the weekly returns (and volatility) using econmetric techniques

Data

The graph above shows the weekly returns, 4 weeks equally-weighted moving average (WMAⁱ) and exponentially weighted volatility (EWMAⁱ)

Note: The WMA shown above appears relatively constant over time, while the EWMA (volatility) is time-varying.

Preliminary Analysis

Modeling

We examine GARCHⁱ(1,1) model as our first candiadate to fit the data. We assume gaussian-type of innovations:
r_t=\mu + a_t
a_t= \sigma_{t}\epsilon
\epsilon\sim N(0,1)
\sigma_t^2=\alpha_0+\alpha_1\times\sigma_{t-1}^2+\beta_1a_{t-1}^2

The GARCH model appears to fit the data pretty well, the residuals are white noise and the excess kurtosis dropped pretty well. Nevertheless, we need to try a fat-tailed (non-gaussian) type of innovation (\epsilon) to take care of the remaining excess-kurtosis.