# TSRV parameters selection

I'm thinking about how to select the $$J$$, and particularly, $$K$$ parameters for the Two Scale Realized Volatility estimation? I cannot find any reference for that in the original paper - there it says $$K=cn^{\frac{2}{3}}$$, but I couldn't find what $$c$$ exactly is, $$n$$ is number of observations.

I've been playing around with the highfrequency package in R which offers a test set of of 8100 1-minute stock prices. If you estimate TSRV with rRTSCov, it seems that the estimation is quite sensitive to $$K$$. My ultimate goal is to estimate 10 or 30-minute volatility with 10-second prices. How would I pick parameters for that? Is it large enough sample?

Many thanks for suggestions.

• Hi Jan. I know this was an old question of yours and you probably found the answer before my post below. However, could you please consider accepting my answer, if you find it correct? If not, please specify how it doesn't answer your question, I will be glad to help. Accepting the answer will stop community from bumping up your question on the front page in the future. Sorry for any inconvenience.
– Pleb
Jun 15, 2021 at 22:46
• Hi @Pleb, thanks for that! Jun 16, 2021 at 5:56

### How to find optimal $$K$$?

If we let $$i$$ for $$i=1,\ldots,n$$ be the amount of intraday returns over a fixed interval $$T$$ (one day in empirical applications), then in Zhang et al. (2005) (p. 1397) specified above, they tell you that the optimal $$c$$, can be obtained via the equation:

$$\begin{equation} c^{*} = \left(\frac{T}{12 \cdot \left(\mathbb{E}\left[\varepsilon_T^2\right]\right)^2} \cdot \int_{0}^{T} \sigma_s^4 \: ds\right)^{-\frac{1}{3}}, \end{equation}$$

where $$\mathbb{E}\left[\varepsilon_T^2\right]$$ is the variance of the noise process and can be found a couple of pages afterwards (p. 1404):

$$\begin{equation} \widehat{\mathbb{E}\left[\varepsilon_T^2\right]} = \frac{1}{2n}\sum_{i=1}^n r_{i,T}^2, \end{equation}$$

which was originally derived in the paper of Hansen and Lunde (2006). Furthermore, the integrated quarticity, $$\int_{0}^T \sigma_s^4 \: ds$$, can be estimated using the realized counterpart (be wary: noise and jumps under second-frequencies will affect the realized quarticity estimator):

$$\begin{equation} RQ_T = \frac{n}{3}\sum_{i=1}^n r_{i,T}^4. \end{equation}$$

### How to find optimal J?

Reiterating from one of your own questions, the original paper does not work with any J subscript. However, their updated paper, Zhang et al. (2011) (p. 165) extend the TSRV estimator for dependent noise, where they further define the average lag j realized volatility and argue that it reduces to their original TSRV estimator for $$J=1$$:

We will continue to call this estimator the TSRV estimator, noting that the estimator we proposed in Zhang et al.(2005) is the special case where $$J=1$$ and $$K\rightarrow \infty$$ as $$n\rightarrow\infty$$.

To conclude, $$J=1$$ if you follow along Zhang et al. (2011) and $$K=c n^{\frac{2}{3}}$$ for $$c$$ being estimated as described above.