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I am trying 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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You have to minimize RQ = - ( $I$($r$ < - $q$ ) - $\theta$ ) * ( $r$ + $q$ ). Where $r$,$q$ are the vectors of returns and quantiles found (for a given $\gamma$0, $\gamma$1, $\alpha$ ), * is the matrix multiplication, and $I$( ) is the indicator function. Note that because we have an autoregressive function, you must have a first $q$ (VaR) which in the paper is the choosen $\theta$ quantile of the first 300 observations. The minimization problem is solved by finding which coefficents ($\gamma$0,$\gamma$1 and $\alpha$ ) is the ones that minimizes RQ. In the paper this is done by generating 10000 random coefficients combinations, then using the top 10 $a$ results in a optimizer ( he uses 'fminsearch' and 'fminunc' functions from MATLAB, in R you can use optim() ) and reiterating 5 times to refine the results.

Manganelli has a site where you can find the code for MATLAB.

$a$ - for the simetric absolute CAViaR

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