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9h
comment GARCH model, expectation of volatility?
Okay, so I did some grinding with a Taylor expansion, and interestingly, the first two terms are known irrespective of the distribution of $\epsilon_t$, so we can get a pretty good approximate solution to the question. See my answer for more detail.
10h
comment GARCH model, expectation of volatility?
Also, using a Taylor expansion, we can get quite a good approximation of what the OP is after (although I think an exact solution is not feasible) . See my answer for more detail.
10h
answered GARCH model, expectation of volatility?
10h
comment GARCH model, expectation of volatility?
Hi vitaly. The nice thing about $\mathbb{E} \sigma_t^2$ and $\mathbb{V} \sigma_t^2$ in a GARCH framework is that their values are fixed w.r.t. the parameters of the model irrespective of the distribution of $\epsilon_t$ (as long as it is mean zero). In contrast, I think (although would need to grind through a Taylor expansion to be sure) that $\mathbb{E} \sigma_t$ will depend on the distribution of $\epsilon_t$ and so this parameter is generally not that interesting as it requires a full parametric assumption to pin down.
10h
comment GARCH model, expectation of volatility?
I get your point in this answer, but can I suggest you edit out the second paragraph? As it stands, your second paragraph suggests that $\sigma_t^2$ is not really a random variable in a GARCH framework. This is incorrect. $\sigma_t^2$ has a meaningful unconditional distribution in a GARCH framework, with known unconditional mean and variance.
10h
awarded  Commentator
10h
comment GARCH model, expectation of volatility?
@volcompt For a GARCH(1,1), $\mathbb{V} \sigma_t^2$ is known, as is $\mathbb{E} \sigma_t^2$. But this doesn't help answer OP's question, since the relevant equation that you are referring to is $\mathbb{V} \sigma_t = \mathbb{E} \sigma_t^2 - (\mathbb{E} \sigma_t)^2$, and it still contains two unknowns.
1d
comment Are all stocks and stock indexes just white noise
@Barnaby I'm not sure I understand the question. I'll try and find the time to come back soon and give my interpretation of an answer. But I'll need to have a quick skim of the paper you've referenced first, which is why I need a bit of time :-) I do think the discussion needs a formal (mathematical) definition of white noise, which I'm assuming I'll get from the paper.
Jul
23
comment Are all stocks and stock indexes just white noise
@Barnaby Understood. The problem I refer to can potentially apply to any portfolio of more than one asset, if one examines returns constructed from end-of-day prices. I think your question is an interesting one, by the way, and I might come back and have a crack at a second answer if I get some spare time over the next week or two. Cheers.
Jul
22
comment Are all stocks and stock indexes just white noise
@Barnaby A stock index such as the S&P500 is a portfolio. The "spurious" autocorrelation problem due to thin trading definitely exists in indices prior to the 70's. Also, if you want me to see your comment, you need to preface it with the "at" symbol followed immediately by my username (no spaces) or else I won't receive a notification. I just happened to come back to this page for a different reason and saw your comment. Cheers -colin.
Jul
22
revised Negative high frequency intraday volatility - Zhou estimator
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Jul
22
revised How to interpret Realized Volatility and TSRV using R
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Jul
22
comment Are all stocks and stock indexes just white noise
@Barnaby Also, a lot of the tests pre 70s were done on stock indices or portfolios. Thin trading in the underlying components of the indices led to "spurious" autocorrelation in the index series itself. I think the classic paper on this is Dimson (1979) "Risk Measurement When Shares are Subject to Infrequent Trading"
Jul
22
comment Spot price and volatility has a correlation of -1, why?
Yes, I agree, possibly referring to the leverage effect. I guess the real issue is that the statement doesn't make a whole lot of sense, so we're stuck with trying to guess what the trader friend was talking about... :-) +1
Jul
22
revised How to interpret Realized Volatility and TSRV using R
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Jul
22
comment How to interpret Realized Volatility and TSRV using R
Thanks, no need to apologise :-) There are so many acronyms for these things, I agree it can get a bit confusing.
Jul
21
awarded  Teacher
Jul
21
revised Negative high frequency intraday volatility - Zhou estimator
added 327 characters in body
Jul
21
answered Negative high frequency intraday volatility - Zhou estimator
Jul
21
revised How to interpret Realized Volatility and TSRV using R
added 261 characters in body