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## FOURIER TRANSFORM OF DING-GRANGER-ENGLE AUTOCORRELATION FUNCTION

The Ding-Granger-Engle autocorrelation function is extremely good at matching autocorrelations seen in actual volatility series (which I have verified with 1900 stocks).  Therefore it’s an interesting question to find it’s Fourier transform for this gives us the spectral density that would be appropriate for stochastic volatility models.  Here is partial solution.

$\rho_t = a \rho_{t-1}^{b_1} b_2^t t^{-b_3}$

Let $G(t) = \sum_{k=0}^\infty a^k b_2^{-k(k+1)/2} b_2^{kt} [(t+k)\cdots t]^{-b_3}$ then

$\hat{\rho}(\xi) = \int_0^1 \rho_t^{b_1} e^{2\pi i \xi t} G(t) dt$

So this is the empirical side of autocorrelations of volatility; on the other side is theory for closed form solutions to long memory stochastic volatility.  On this side Comte-Renault produced a fractional integrated square root process with spectral density

$f(\lambda) = C(\lambda^2 + k^2)^{-2}\lambda^{-2\alpha}$

Here cir2003juin

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