Embedded in this thread are multiple questions. I'm currently im the process of implementing a DCC GARCH forecast model on quantopian (a python-powered trading platform).
The two step consists of first estimating the conditional volatility over time $D_t$ (as canonicalized by Engle). I apply the traditional log-likelihood with the minimize function from scipy package. For 2nd step, it is the same except I run into a bit of ambiguity:
Consider the log-likelihood for the 2nd step $L(\phi|\hat{\theta})\propto \sum_{t=1}^{T}log(|R_{t}|)+\epsilon_t^{'}R_t^{-1}\epsilon_t$. The first term evaluates to an N by N matrix while the second term evaluates to a scalar. Thus, the likelihood for each timestep is an N by N matrix. In implementation, only a scalar is expected to be return, do I just sum all the terms in $log(|R_t|)$ when calculating the actual likelihood?
Furthermore, the current time it takes for the minimize function to converge takes too long, any advice on faster estimation techniques is appreciated.
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