# When should one use the Reduced Major Axis regression, aka Geometric Mean Functional Relationship?

I recently learnt about the Reduced Major Axis regression, a.k.a Geometric Mean Functional Relationship, least products regression, diagonal regression, line of organic correlation, and least areas line (see wikipedia).

The idea is that for a traditional univariate OLS regression (y=a+bx), the errors that are minimized is the "vertical" distance (along the y axis) of observations to the regression line.

The reduced major axis regression assumes that there are errors to both y and x. What is minimized is the product of the "y-distance" and "x-distance" of observations to the line. So it's actually the area of the triangle between the observation and the line that's minimized.

Paul Samuelson apparently said it was a cool regression.

However I have a hard time really understanding why and when I should use this. So, do you use it? Why?

• Reread the assumption! It tells you precisely when one would consider this approach. :-) (Of course there's more to say about it, for otherwise this procedure would be used far more often than it is. The key is to consider whether accounting for the errors in $x$ might make a difference in the analysis.) – whuber Jun 11 '12 at 21:07

## 1 Answer

It is used when the symmetrical relationship between two variables is required i.e. the underlying relationship between the two variables OLS defines the estimated relationship between the dependent and independent variables and is therefore asymmetrical i.e. x regressed on y or y regressed on x. The slopes of these two regressions are different. The geometric mean functional relationship regression provides an estimated slope that is the same no matter if x is regressed on y or y is regressed on x – hence the symmetry. The geometric mean functional relationship is essentially the difference of the two above regression slopes (OLS regression slope and reverse OLS regression slope). In the finance world, Dr. Chris Tofallis has proposed this measure as more appropriate for the estimation of Beta.