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4

Very interesting question. I am not an expert on the subject, however, I was able to find a collection of papers on the subject that should get you started. Here is a good and very informative paper that walks you through several tick by tick volatility estimators that seek to reduce the volatility imposed by market micro-structure: Efficient estimation of ...

2

Did you try solving for $w_k$? $$\bar{r}_t = \sum_{k=0}^p w_k r_{t-k}$$ $$\bar R = W R$$ Since you probably have $t>>k$, you can solve for $W$ using OLS $$\bar R = W R +\varepsilon$$ -- UPDATE You can try applying Kalman filter. Here, your state evolution is $$r_t=\mu+\varepsilon_t$$. You introduce new vector $x_t=(r_t, r_{t-1}, \dots, ... 1 Thanks @Aksakal for suggesting Kalman Filter. Here I provide more details. We will view it as a state-space model: $$\begin{split} z_t &= A_t z_{t-1} + B_t u_t + \epsilon_t, \\ y_t &= C_t z_t + D_t u_t + \delta_t, \\ \epsilon_t &\sim \mathcal{N}(0, Q_t),\ \delta_t \sim \mathcal{N}(0, R_t), \end{split}$$ where$z_t$is the latent variable, ... 1 For any continuous distribution we can define $$VaR_{\alpha}=-F^{-1}(1-\alpha)$$ where$F^{-1}$is the inverse of the CDF. Now suppose that you have a distribution which comes from a location-scale family with location parameter$\mu$and scale parameter$\sigma$then $$F^{-1}(1-\alpha)=\mu+\sigma \phi^{-1}(1-\alpha)$$ where$\phi^{-1}\$ is the inverse CDF ...

1

Both approaches can be useful. For stocks, sorting into quantiles is popular because it's easy to understand and explain it's a simple matter to build factor portfolios and track or backtest their performance, while the translation from expected returns to a portfolio is a bit more involved more robust than a single-stock regression, because it is less ...

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