I’m trying to derive another value from the coefficients of two regression equations.

I have 2 regression equations (for example)

Miles traveled/year = α1 + β1X + δ1Y

Fuel used/year = α2 + β2X + + δ2Y

X = engine size and Y = tire pressure

The co-variance between the miles traveled/year and fuel used is pretty high.

The standard interpretation of the coefficients is that an increase of one unit of engine size results in an increase of β1 miles traveled and an increase of one unit of engine size results in an increase of β2 fuel used.

Now fuel economy = miles traveled/fuel used. With that in mind can I used the coefficients to produce β1/ β2 which would be the fuel economy associated with engine size changes and something similar for tire pressure, δ1/ δ2 ?

I’m not trying to combine the regression equations but to generate other values for further research.

Also since the coefficients are normally distributed would the ratio have a Cauchy distribution?

From the standard interpretation of the coefficients what I am proposing seems sensible but I’m not entirely convinced.

I’d really appreciate any help,

Thank you