# [R-SIG-Finance] statistical features of equity time series

Patrick Burns patrick at burns-stat.com
Sun Oct 28 16:19:35 CET 2012

```If you are assuming a normal (or other
symmetric) distribution for returns,
then those will be log returns rather
than simple returns*.  So the price series
will be generated by:

initialPrice * exp(c(0, cumsum(returnVector)))

I would suggest garch** simulations as a starting
point.  The most obvious feature of market returns
not being IID is volatility clustering.  If what
you care about depends on autocorrelation, then
you could include non-zero ARMA in your model.

But I think you should keep Dirk's caution in mind.
The best models will depend on the question.

* http://www.portfolioprobe.com/2010/10/04/a-tale-of-two-returns/

**
http://www.portfolioprobe.com/2012/07/06/a-practical-introduction-to-garch-modeling/

Pat

On 28/10/2012 13:22, Alex Grund wrote:
> Hi Dirk,
>
>
> 2012/10/28 Dirk Eddelbuettel <edd at debian.org>:
>
>> There are libraries full of papers and dissertations on this.
>
> Okay, could you please mention a few valuable papers? So that I can search more?
>
>> See 1). Which features?
>
> Basically, I started from the naive question: "How to create a time
> series that "looks" like a stock price process over time".
> So, the basic features I came through has been a) the distribution of
> the (daily) returns, b) their auto-correl features and c) binominal
> features.
> To explain what I mean by c):
> Imagine you create normal-distributed (N(0,1)) returns. Then the
> generated time series of prices (price[i] = price[i-1]*(returns[i]+1))
> will slightly tend to fall. This is obviously because of this: Imagine
> you have three returns generated, [-.5; 0; .5], then the series will
> fall. It should be [-.5;0;1] for the series to hold it's level,
> however P(X<-.5) > P(X>1), X~N(0,1), so the series with returns mean 0
> is obviously to fall.
>
> Additionally, one could think of volatility features (such as
> suggested by GARCH).
>
>> | 3) How can I create a time series with statistical features that are
>> | similar to most of the data from a set of given time series?
>>
>> See 1) and 2). Seriously :) The last paper presentation I saw was Diebold who
>> showed how to regenerate trade duration data, as well as high frequency vol,
>> from a "simple" four parameter model.  And simple is a relative term -- he
>> recaptured the features of his (SP100 equity TAQ) data set, but its not a
>> model you can code up in just a few lines.
>
> Okay, are there models to start with? They don't need to be perfect,
> because I want to use them for learning...
>
>> | 4) Is there anything valuable which could make given data more
>> | exhaustible? Something like bootstrapping?
>>
>> Block bootstrap for time series is pretty well established, and the tseries
>> package even had a tsbootstrap() function for over a decade.  You can (fairly
>> easily) extend similar schemes.
>
> Ok, thanks
>
>
> --a
>
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--
Patrick Burns
patrick at burns-stat.com
http://www.burns-stat.com
http://www.portfolioprobe.com/blog