I have a model whose specification is

$$R_{t+1} = \sigma_{t+1} \varepsilon_{t+1} \text{ with } \ \ \ \ \varepsilon_{t+1} \sim N(0,1)$$ $$r_{t+1} \sim N(0.43 + \log \sigma_{t+1}, 0.29^2)$$ $$\log\sigma_{t+1} - \log \sigma_t = \kappa(\theta - \log \sigma_t) + \phi\frac{r_t - 0.43 - \log \sigma_t}{0.29} + \delta \varepsilon_t$$

I have to calibrate this model (i.e. finding estimate for $\kappa$, $\theta$, $\delta$, $\phi$) using the Maximum Likelihood method. I only observe $r_t$ (and we assume $\varepsilon_t$ independent)

I am having trouble finding the maximum likelihood function for $\log \sigma_t$.

I tried doing the following: I know that $\log \sigma_t$ is normally distributed, so I only need to find the mean and the variance; for the mean, calling $\alpha_t = E[\log \sigma_t]$, I get the recursive equation

$$\alpha_{t+1} = \alpha_t + \kappa(\theta - \alpha_t)$$

and I could so something similar for the variance $\beta_t = \text{Var}(\log \sigma_t)$; at this point I could solve the recursive formulas to find $\alpha_t$ and $\beta_t$, but then how can I use this information to if I only get $r_t$?

My idea was: I estimate the mean of $r_t$ (a normally distributed random variable) with $\tilde r_{t}$, the observed value (since it is only one point, this is the MLE) and then estimate the parameters maximizing $\log \sigma_t$ using the fact that $\log \sigma_t = \tilde r_t - 0.43$.

This does not feel rigorous though and I am not an expert in using the MLE. Could you please provide some references / help?

P.S. This is taken from "Volatility forecasting and explanatory variables: a tractable and easy approach to stochastic volatility" by Chapados and Dorion, 2013. In the paper they just said that finding the MLE is straightforward.

  • $\begingroup$ You have $R_{t+1}$ in the beginning but nowhere else. Should it be $r_{t+1}$? $\endgroup$ – Richard Hardy Jun 27 '16 at 8:04
  • $\begingroup$ @RichardHardy I reported it for completeness as it appears in the paper, but honestly I have no idea why it's there.. $\endgroup$ – Ant Jun 27 '16 at 8:10
  • $\begingroup$ @Ant, I am having the exact problem. Have you solved the problem? $\endgroup$ – oercim Mar 10 at 16:56

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