# Differences vs ratios

High, I am working on an exercise which involves performing a regression analysis to predict market direction (e.g. up or down). I am using daily OHLCV data. I've created various factors from the price data so I can run my regression. One of the factors is high - low. However, a similar factor can be high/low. Of course they are very highly correlated (R=.97). Which one would you choose and why? I feel I would go for the ratio because it is more stable in time (e.g. high-low from 10 years ago is not comparable to this metric today).

You are correct, $$high-low$$ will be problematic. Fortunately, the $$\frac{high}{low}$$ has a solution. The difficulty is that it will not be analytic, so you are going to have a headache.

The probability distribution of a high value is the Gumbel distribution. The probability distribution of the low value is also a Gumbel distribution. The ratio of the two is a ratio distribution. It will not have a mean or a variance so you will not be able to use tools like ordinary least squares for most problems. The link will show you how to perform the calculations, but as it is not analytic, you will need to build an approximation.

What can save you from the associated nightmare is the formulation of your problem.

Because you are mapping to a simple "up versus down", the solution can converge politely to a nice probability distribution. The ugly, weird probability distribution that your regressors are drawn from won't impact you if you are mapping to a binary case. What you want to be very careful with is being cautious in your interpretation of any prediction.

Let me give you a real-world example. There is a firm, whose name I do not remember, that makes parts for Microfiche machines. If you are young, you may never have seen one. They are an absolutely brilliant technology if no longer in use.

On a split-adjusted basis the firm went from five cents per share to roughly \$35,000 per share to roughly four cents per share. If you were looking at either end as a annual high or low, you could get some pretty extreme numbers.

Indeed, it wouldn't be shocking to get some extreme values very near zero or very large even on daily data.

If you would use logistic regression, you will want to be careful as to how linear this process is in the log-odds format.

I would also be careful to factor in liquidity costs. It is likely that there will be structural breaks throughout the series. However, many of those breaks will be liquidity breaks. The number of true breaks, once liquidity has been accounted for, will be reduced from the naive number of breaks.

If you stray from "up versus down", you will need to change your methods.