Constructing Index: Combining Financial Time Series

[Tosh5457]

Hello,

I'm facing a problem in a project that I'm not being able to solve. I have two different timeseries, and I want to construct an index that represents the two of them, each of variable weights (so I could choose 50% weight for each, or other combination).

These are financial time series, with these properties:
- The returns on these series aren't normally distributed, they're symmetrical heavy-tailed
- One of the series has both negative and positive values. The other only has positive values

How could I approach this task?

 

[andrewkirk]

The construction will depend on what you want the index to represent. For instance the S&P 500 Price Index represents the amount to which one dollar, invested at some long-ago base date, would have accumulated if it was always invested in the associated stock portfolio defined by S&P, assuming the portfolio was rebalanced costlessly every day, and that no dividends were received. The S&P 500 Accumulation Index is the same except that it includes dividends in the accumulation.

What do you want your index to represent?

 

[Tosh5457]

The construction will depend on what you want the index to represent. For instance the S&P 500 Price Index represents the amount to which one dollar, invested at some long-ago base date, would have accumulated if it was always invested in the associated stock portfolio defined by S&P, assuming the portfolio was rebalanced costlessly every day, and that no dividends were received. The S&P 500 Accumulation Index is the same except that it includes dividends in the accumulation.

What do you want your index to represent?

I just want to compare this index to another timeseries, to see how changes in it affect the other one. I think that would be the same as the S&P 500 Price Index.

EDIT: In my case, it would also be important to normalize both timeseries, because they are different in nature unlike S&P components

 

[andrewkirk]

I just want to compare this index to another timeseries, to see how changes in it affect the other one.

In that case the best tool would be to do a regression of that other one against the two time series that you were thinking of combining into an index. That will give you an idea of what impact changes in the two components have on changes in the third.

Constructing an index would confuse rather than clarify the situation.

 

[ssd]

You may use z scores with arbitrary origin and scale. Let X & Y be the two series. Calculate z from (x-μ_x)/σ_x=( z-c)/d and from (y-μ_y)/σ_y=( z-c)/d, for all observed x and y, where c & d are arbitrary. μ,σ are mean and sd etc. The z values are now comparable.

 

[FactChecker]

As @andrewkirk says, you probably should use statistical techniques to determine how to combine two independent time series to estimate the dependent time series. I have never done work with cross correlations of multiple time series, but Lutkepohl's book New Introduction to Multiple Time Series Analysis may be very applicable.