# realized correlation estimation

I'm trying to implement the Hayashi - Yoshida estimator for correlation (T. Hayashi, N. Yoshida: On covariance estimation of non-synchronously observed diffusion processes, 2005) and there's something I'm missing with respect to realized correlation and realized variance. Assuming that $$Y = \log P$$, and that I have access to $$N$$ discrete (asynchronous) observations for two assets on a time grid $$[0,t]$$, the HY estimator is as follows:

$$RV^{(i)}_{[0,t]}=\sum_{\tau \in[0,t]}\left(Y^{(i)}_{\tau}-Y^{(i)}_{\tau-1}\right)^2$$

$$RC^{(1,2)}_{[0,t]}=\frac{\sum_{\tau_1 \in[0,t]}\sum_{\tau_2 \in[0,t]}\left(Y^{(1)}_{\tau_1}-Y^{(1)}_{\tau_2-1}\right)\left(Y^{(2)}_{\tau_2}-Y^{(2)}_{\tau_2-1}\right)\mathbb{I}_{[\tau_1-1, \tau_1]\cap[\tau_2-1, \tau_2]}}{\sqrt{RV^{(1)}_{[0,t]}\cdot RV^{(2)}_{[0,t]}}}$$

where $$RV$$ estimates the realized variance and $$RC$$ the realized correlation over the period. Now, both these estimators should be unbiased but I was wondering, how do they relate exactly to the variance and correlation (Pearson)? I found empirically that $$\sqrt{RV/n}\approx \sigma$$ (sample std), which I guess makes sense (I'm not completely sure why), but I can't seem to find any similar scaling relationship between $$RC$$ and $$\rho$$.

The data is tick-by-tick over one day (not sure if it's relevant).

• Try a simple example where the intervals are all the same and see how it works in that case (how it compares to the Pearson calculation). Looking at the equations I agree with $RV\approx n \sigma^2$, my guess is either $RC\approx \rho$ or $RC\approx n\rho$. May 15 '20 at 12:54
• There might also be a factor of 1/2 involved. My current idea is $RC\approx n\ \rho/2$ but I have to think about it... May 15 '20 at 14:07