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Suppose that the data has been generated by a GARCH(1,1) model, i.e. \begin{align} y_t &= h_t \epsilon_t, \; \epsilon_t \sim N(0,1) \\ h_t &= \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \beta_1 h_{t-1}. \end{align}

If we estimate the parameters $(\alpha_0, \alpha_1, \beta_1)$ by maximum likelihood and make some assumption about initial values we can filter out an estimated series $\hat{h}_t$ for the conditional volatility process.

My first question would be: what are the statistical properties of $\hat{h}_t$? How is it related to the true series $h_t$?

  • I'm looking for something like $\hat{h}_t \overset{p}{\rightarrow} h_t$ as the sample size diverges. Or, perhaps, $E\left( \hat{h}_{t-1} | \text{some condition} \right) = h_t$. In other words, I'm looking for a legitimate justification to be able to use $\hat{h}_t$ as the true $h_t$ in some further analysis.
  • If you would have references on the topic, it would be great.

Now, what if we have changed the second equation to \begin{align} h_t &= \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \beta_1 h_{t-1} + \eta_t, \; \eta_t \sim N(0,\sigma^2) \end{align} and this becomes our new DGP, but we suppose that I'm estimating the same model as before. In other words, I am ignoring the fact that there are really two sources of uncertainty.

My second question is then what happens to the properties of $\hat{h}_t$?

  • The point here is trying to figure out in the simplest possible example what is lost by simplifying massively the estimation and using GARCH(1,1) instead of accounting explicitly for the measurement problem introduced by the volatility shock during estimation.

Again, if you have a textbook treatment of some of this stuff, it would be appreciated. I'm assuming this sort of concern has been looked into. I just can't seem to find anything, but I might be looking in all the wrong places.

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