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Detrending,Transforms and Time Series Analysis

October 12, 2009

I have much pleasure of being quoted by Prof Dhananjay, regarding the volatility.

Thy author in his limited wisdom, decided to enlighten his readers in his own small ways, the potential challenges and rewards one might get in this approach.

Considereth, O Thine Readership,thou hast a price series named p_{t} and you want to analyse this. So the idea is, to first make it stationary.

A question might arise, why stationary and what is stationary?

Lets start our quest with the second question, what in the god’s name  is stationary? But before I broach that subject, I want, to hold your precious patience, and let me go about this in a holistic manner,

Do you remember, how easy is to solve an equation like this?

3x=9

Its easy, simple and we can easily do it by transposing x=9/3; But what if you are asked to solve a system of equation like this:

a_{1}x+b_{1}y=c_{1}
a_{2}x+b_{2}y=c_{2}

So if we dont know how to solve a simultaneous equation, and what we know instead is how to solve a simple linear equation in one variable. Hence lets leverage that knowledge to solve our work. Lets decompose this equation into a simple linear equation and try to solve it like that.

Now, coming back to price series and return series etc, its very easy and we know how to deal with finite mean, finite standard deviations, nice looking bell curves, not so nice looking kurtic distributions etc. We know how to deal with it. But what we dont know, is how to deal with arbitrarily large variances etc,which is present highly in the day to day observation series we have.

Dont get me wrong, we do have finite variance, finite mean distributions as well. But when a series tends to trend(pun unintended) our work simply becomes unweildly. So we decided lets reduce it to something we know, we have worked with. Lets reduce it to those general characteristics? Why? Because its not stationary!

What? What is not stationary? The price series, if you try to investigate, it has time varying characteristics i.e the price can be thought as

p_{t}=\mu_{0}(t)+\sigma(t) {d}X

Here the mean \mu_{0}(t) is not a constant but a time varying metric. Similarly \sigma(t) And of ten in a trend the variance becomes arbitrarily large. That it means it will take an arbitrarily large observations for us to notice a reversion to the average.

So to cut these things off, lets detrend it. That is ,take away the time dependent components off and form a return series instead r_{t}, where r(t) is the return of the price series from t-1st day to t th day.

It serves essentially two purpose: a. we essentially make the mean as finite from an unknown number(potentially large).  And secondly, if we assume that the variance is finite, then we have something we can work with.Of course how we treat the unknowns in our hand, underpins the challenge of the system.From here onwards our paths diverge into various forays, and I will try to bring forward some more development in the next few posts. I would be glad, to know if I have got your appetite whetted.

P.S: I will be trying to touch up more on these subjects, with some simulations and real life charts. At the very outset, it might not give that great a “feel” of things to come or how to use it, but trust me 🙂

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2 Comments leave one →
  1. October 13, 2009 1:52 am

    Hi Soham,

    I wholeheartedly welcome and appreciate your post . This is the need of the hour in the world of mumbo – jumbo of technical indicators and trendline magic spewed out by market experts daily. I believe only quantitative/ Statistical approach is predictive others are merely descriptive .

    I am eager to learn from your real time understanding of application of quant models to markets.

  2. October 13, 2009 1:56 am

    The pleasure and honour is all mine, Gentle Reader 😀

    P.S: For those, who are irritated by this archaic English, I am much moved by watching period films of 13th Century Britain and from here onwards, you will be either Gentle Sire/Fair Maiden 😀

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