# Relationship between price change and lagged price change

#### Benjameanie

##### Newbie
6 0
Hi All,

I hope this is the right forum for this question.

I've run a test to see if the % change in an index from Day 1 to Day 2 is related to the % change from Day 2 to Day 3. My expectation was that there would be no relationship, but the test suggests I was wrong. The regression diagnostics are as follows:

%Change=-0.0889484 *%ChangeLag + 0.0001481

Prob(t) for %ChangeLag = 0.000315
F=13.04, Prob(F)=0.0003147

My data is 1641 days of FTSE 100 closing price (adjusted) from Yahoo. The order is not randomized (haven't worked out how to do that in R yet).

Would you expect to see this result?

Kind regards,
Ben.

#### Martinghoul

##### Senior member
2,690 276
In a word, yes. There has been a lot of evidence that stock returns might be autocorrelated... You might be observing something along those lines.

#### Calinor

##### Active member
184 20
What exactly is the result? Sorry, but it is a little hard to understand what you have done and where it fits in with statistics.
%Change=-0.0889484 *%ChangeLag + 0.0001481

Prob(t) for %ChangeLag = 0.000315
F=13.04, Prob(F)=0.0003147

Are you just saying that sometimes volatility increases in an index, and the effect of high vol on one day, makes the next day more likely to be more volatile? Well many people will say that the market goes through stages of being more and less volatile, so that wouldn't be shocking. But again, I have probably misunderstood what you are doing.

#### Benjameanie

##### Newbie
6 0
Thanks Martinghoul, that's what I wanted to hear.

Calinor, I've just run a simple linear model with the day-on-day change in closing price as the dependent variable and day-on-day change in closing price at a lag of 1 day as the sole independent variable. The result says that the series are related - so if the FTSE rose by 100 points today, I can expect it to fall by 8.9 points tomorrow (with the caveat that this is based on historical data, obviously).

During the period I'm looking at, the mean of the absolute change in the index was 42.7 points, so my result suggests that if I had simply bet every day that the FTSE would go the opposite way to the previous day, I would have averaged a gain of 3.8 points every day. With a 2 point spread, it looks like this strategy would have actually made money. It feels as if there must be something wrong with my logic, though...

Thanks,
Ben.

#### Jemima

##### Junior member
15 2
A(i) why wouldn't you expect to see results to this effect? There are times when prices can be modelled as random walks (today is a good example in the s&p).. but there are other times when prices DONT behave that way (such as y'day) - OK, in the context of day after day correlation, this is a bad example - but when prices are moving significantly, there is likely a change in fundamentals that is the cause, and so the effects spilling over into the next day isn't absurd.

A(b) Which is the next obvious test, testing for correlation (or autocorrelation) between days and day+1 where the %change in day(1) is deemed to be significant, to get more Signal from your Noise.

What is the DW stat? Can you post your data, I will run it through my software (I know R is meant to be the bees knees but I am computer stupid)

Why would you want to bootstrap the data? Isn't it pretty critical to your hypothesis that the sequence is kept in order?

#### Martinghoul

##### Senior member
2,690 276
What you may want to do is a) deal with the non-trading days (incl weekends), because they often introduce known autocorrelation effects; b) estimate the returns of your strategy in an out-of-sample period; c) estimate the standard deviation of those returns.

In general, just google things and you shall find quite a lot of literature.

EDIT: Jemima, R can do the Durbin-Watson stat, I am pretty sure.

#### Jemima

##### Junior member
15 2
P.S. I think trying to trade this as a strategy is going to cause tears... what is the sharpe ratio? How much heat do you have to take to earn your 3.8 ticks? And so on...

#### Benjameanie

##### Newbie
6 0
I wasn't thinking of trading on this, just getting started looking at this field. And I understand that there are many other things to consider. This is geek heaven!

My expectation wasn't based on anything much other than my novice's assumption that it wouldn't be possible to extract anything useful from the data without doing some serious number-crunching.

Jemima, I'm not sure what you mean by boot-strapping - randomising the order? I guess ordered data will create autocorrelation in the residuals, meaning that the estimate of the coefficient would be more wrong at times of greater volatility. However DW indicates no error autocorrelation (DW = 2.009, p-value = 0.5734), so it seems ordered data is fine.

As you suggested, I tested for autocorrelation in the series of closing price changes at different levels of closing price change and found that the retracement effect found above increased to about 21% (i.e. the regression coefficient is -0.21) where the absolute day-on-day change of the FTSE is less than 1%, so it is definitely possible to get more signal from the noise.

Martinghoul, I'll take up your suggestions, thanks. To include weekends, would I simply insert 2 extra copies of each Friday's data to represent them?

I like the replies I'm getting here- they all seem well informed and are good food for thought. I'll go away now and come back when I've got a slightly less noddy question.

Kind regards,
Ben.

PS I've attached the data in case you want to have a look.

#### melanie911

##### Member
61 6
hard to gain with single serie autocorrelation imo