# Testing which index is a better benchmark to track stock prices

Let's say a Hedge Fund is tracking a stock price. Now the fund has three columns of data, Stock price, Index 1, and Index 2. All of these have data from 2016/01/01 - 2017/01/01. If the fund is to decide which Index it is going to use as the main benchmark for the price of the stock, then which is a better way to make such a decision?

1. Simple Regression for P = beta0 + beta1 * Index1 + error and P = beta0 + beta1 * Index2 + error

However, I feel this approach is wrong, because the data we are using is time series data, and this violates many assumptions of OLS model, so we should not use this, am I understanding this correctly?

1. Time series regression for the above model.

Could we run this as a time series model?

1. Correlation Coefficient between (P, Index1), and (P, Index2).

Could you please inform me which is better and the reasons behind it?

Thank you very much!

To avoid "violating the assumptions of the OLS model" it is important to do the regression using returns $$\frac{P_t-P_{t-1}}{P_{t-1}}$$ (or logarithmic returns $$\ln P_t - \ln P_{t-1}$$) for both the index and the stock and not the price level $$P_t$$. A regression in levels is statistically invalid (so called unit root problem).

Whether you look at the $$R^2$$ from the regression or calculate the correlation $$\rho$$ directly makes little difference, the concept is the same.

• Is it correct to say that using prices have autocorrelation/serial correlation issue? I still don't get it why using returns in the regression doesn't violate the OLS assumptions. Thank you for your help! – commentallez-vous Dec 5 '18 at 17:00
• Yes, each price is close to the previous price. To solve this the classical remedy it to take first differences of the data, the first differences are random identically distributed. in finance we more commonly take the differences of the logarithms, or the returns, but the idea is the same. – noob2 Dec 5 '18 at 17:03
• Thank you very much for your clarification! It really helps! – commentallez-vous Dec 5 '18 at 17:06

I have seen this done mostly qualitatively. For example, this is a large cap tech name, so I will benchmark it against the QQQ.

With short time periods of data and a single stock, running a historical regression and choosing a benchmark without taking into consideration your intuition could lead to sporadic results.

Now, if you are doing this wholesale across thousands of securities and trying to algorithmically pick the best benchmark, then running a regression and looking at the R^2 OR running a correlation is a very simple solution to the problem, with full understanding that you could land on outlandish relationships because of market regime, idiosyncratic performance masking itself as some unrelated systematic performance, or dumb luck. I would likely start with the OLS regression as it provides a bit more information as to the relationship of the stock vs. index.

I don't think this violates the OLS model - it is exactly how CAPM calculates stock betas / alphas / residuals.