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I am tring to replicate results from Engle and Manganelli (2004). The following is one of their specifications, $q_t(\theta)=\gamma_0+\gamma_1q_{t-1}(\theta)+\alpha|r_{t-1}|$, q is the quantile of return distribution, r is the return.

I do not know how to use quantile regression to estimate this process, since we have quantiles on both sides of equation. Any suggesions?

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I don't know how to solve your problem, but a good start might be to look up quantile autoregressions which aren't too hard AFAIK. – user13253 Aug 23 '12 at 4:15

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