Reputation
Top tag
Next privilege 150 Rep.
Create new tags
Badges
5
Newest
 Commentator
Impact
~396 people reached

  • 0 posts edited
  • 0 helpful flags
  • 16 votes cast
56m
comment FORECASTING Model AR(1) in an Autoregressive Form The Pi´s Parameters
Hang on, I'll turn this into an answer.
1h
comment FORECASTING Model AR(1) in an Autoregressive Form The Pi´s Parameters
You haven't actually defined $\phi$ or pi in the question, so I can't really answer that question. However, you appear to be using R. So what I can tell you is that the AR1 coefficient estimated by R is the same as the $\phi$ in my previous comment. The intercept estimated by R is not the same as the $\mu$ in my comment. Specifically, R provides estimates in Normal form, which for an AR1 is $(1 - \phi L)(Y_t - c) = \epsilon_t$. So using $\mu$ as defined in my previous comment we have $\mu = c(1 - \phi)$, where $c$ is the intercept estimated output by R. ($L$ is the lag operator)
1d
comment FORECASTING Model AR(1) in an Autoregressive Form The Pi´s Parameters
You state that your model is an AR(1). Therefore it has no MA component. It immediately follows that the first and third form in the linked pdf are equivalent since $\mu(L) = 1$. So if you can do the first form, you can do the third. As near as I can tell, in your description of the third form above, you are confusing the infinite order MA representation of an AR(1) for the actual AR(1). The forecasting equation for an AR(1) is just $\mathbb{E}_t Y_{t+1} = \mu + \phi Y_t$, where $\mu$ is the intercept and $\phi$ is the AR1 coefficient.
1d
comment GARCH model, expectation of volatility?
@ZacharyBlumenfeld Yes, I was a little too hasty to finish my answer. Your concern is a good one. Now that I think about it, I'm also worried about some of the higher moments of $\sigma_t^2$. Existence of these moments is a non-trivial matter. I've adjusted my answer to reflect your concerns. Cheers -colin
2d
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.
2d
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.
2d
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.
2d
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.
2d
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.
Jul
26
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
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
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
comment How to interpret Realized Volatility and TSRV using R
You will usually get RV > TSRV when you use higher frequency returns in your RV calculation, ie return span less than 5 minutes. For lower frequency returns, ie return span 15 minutes, I would not expect RV to be consistently larger than TSRV.
Jul
21
comment How to interpret Realized Volatility and TSRV using R
I think TSRV is consistent for quadratic variation (ie integrated variance plus jump component). The question of jumps for TSRV was never really answered in the original paper of Zhang, Mykland and Ait-Sahalia as the modelling framework explicitly did not include jumps. Then the realised kernels paper came out and everyone kind of forgot about TSRV. Are you sure you're not thinking of bi-power variation or tri-power quarticity in your answer above? Both these estimators are known to be consistent for integrated variance even in the presence of jumps.
Jul
3
comment R: How feasible is it to store — and work with — tick data in a database connected to R?
I currently also do this (for tick data, e.g. transaction, best bid, and best ask), but as my dataset has expanded I've found the number of files on my hard disk to be annoyingly large, i.e. ~2 million. If you'd be willing to expand your answer to discuss this issue I'd be very interested in reading it (as I'm currently looking for alternatives).