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Q1.) Is there anything wrong in principle with this simple sampling strategy? I mean sampling is a valid strategy, it just may not be the best. WOuld a VWAP style price be better? Would just an average be better? Typically when no trade has happened you can model the price as the last, average of the bid/ask spread, etc. The price you want depends ...

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You may have a look at a list of clustering algos available in sklearn here, but I think all of them are of $O(n^2$) complexity. As well, have a look at the TSNE clustering algo, which is supposed to be $O(log(n)*n)$, but this may not be the fact depending on a particular implementation. A particular case in point is again Python sklearn implementation of ...

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You can use RATS software in which VAR GARCH is inbuilt function with CCC, DCC VECH and BEKK for co-variance estimation.

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Welcome to quant.SE! I do not have specific experience with the CARR Model, however, I had a short look in the paper you mentioned: As far as I understand the model specification you just implement a GARCH(p,q) estimation for the range $R_t:=\max{P_\tau}-\min{P_\tau}$ where $\tau=t-1,t-1+\frac{1}{n},\dots,t$ where $n$ is the number of intervals used in ...

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Your interpretation is wrong. If r>1 (not in absolute value) the series follows and explosive and therefore is not stationary. Reject the null hypothesis does mean that all the roots are out of the unitary circle.

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Time series of 4 years of quarterly data with 1 for 4th quarter: ts(rep(c(0,0,0,1),4), f=4) Qtr1 Qtr2 Qtr3 Qtr4 1 0 0 0 1 2 0 0 0 1 3 0 0 0 1 4 0 0 0 1 Is this what you want? That was a dummy series of ts class. Actually, depending on your needs simple rep(c(0,0,0,1),4) may suffice, e.g. for ARIMA ...

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