Hello everyone, I have a query regarding Time series analysis. I have two series namely TS1 and TS2. I want to know which one of these series is dependent and which is independent. I was wondering what tests should be performed to find that out? Thank you for your help. Regards,

Great question. Check out "The Mathematics of Technical Analysis" by the late Cliff Sherry. It's somewhat of a disaster in terms of organization, but it has the information you want. There are a great many different kinds of tests for different kinds of dependencies. My favorite is to count the frequency of + or - price changes following + or - price changes and compare it to a random, independent "time series". The "golden rule" is: compare to a random, independent time series.

Hello Petadollar, Thank you for your help. Will definitely try out the method you suggested. Regards,

Here is an example from the S&P 500 (e-mini futures) last fall, using some of the techniques from Sherry's book. The graph across the top is the price. Histogram "N" is the 5 minute prices changes (histogram of first differences To the right of that is the differential spectrum test. It tells you if dependencies exist or not (but not what kind of dependencies). It tests for symmetry of the first-difference histogram. Directly under the differential spectrum test is the digram test for serial price dependency. "1" means price decrease and "2" means increase. So "11" means a decrease followed by a decrease, "12" means decrease then increase, etc. "obs" is how many were seen in this time series and "exp" is how many are expected from a random, independent time series. The two cdfs (cumulative distribution functions) plotted along the left-bottom are tests for stationarity (labelled stn) and randomness (labelled ran) These results showed the price changes over 5 min intervals, for the S&P emini between Sept. 21 and Oct 14, were stationary, random, but dependent. The dependency means that trends are in play and chart patterns can be useful. Random means that the historical prices don't completely determine the future prices. Good traders already knew all this. But it's nice to see proof. Furthermore, you can watch the markets change their behavior and also see difference in time frames with this kind of analysis. The problem, I found, with studying dependencies there are not many statisticians who study dependencies. Useful material is difficult to come by.

If you import any financial time series into Excel, and run the descriptive statistics function, you will find all distibutions have some degree of skew and kurtosis. There have been plenty of folks with Ph.D.s and even Nobel Prize winners who have failed to develop an economic model that comes close to accurately accounting for this skew and kurtosis.

How is that relevant to searching for dependencies in price changes? Don't you have to fit to a distribution to get skew and kurtosis? edit: Answered my own question. No, we can compute skew and kurtosis without fitting: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm And it seems to provide similar information to the "differential spectrum test" (DST). Since statistical packages compute skew and kurtosis automatically (usually, during the course of a fit to a normal or other distribution), this is a good alternative to the DST to test for dependencies. As usual, compare to random, independent data.

Eviews is by far a superior product compared to Excel. What you can do with this software is almost limitless. Read all the features it has to offer. It's not cheap but it's worth every cent if you are serious about analyzing the market from a statistical standpoint. http://www.eviews.com/EViews7/ev7features.html

Price change depenencies are a result of human actions in the financial markets (or computerized actions as programmed by humans.) These dependencies introduce skew and kurtosis in the distributions of the time series. Many very smart people have failed to develop an econometric model of markets that accurately reflects the effects of all the underlying fundamental forces that cause skew and kurtosis to appear. All this means is that markets are more complex than we can accurately model currently. The result of this complexity makes it extremely hard to find statistically meaningful dependencies. That's why traditional indicators perform so poorly when tested against out-of-sample data.