level log regression interpretation?

Question

If I want to estimate a level-log regression by OLS, I do that because I believe that my x value (the independend variable) displays a diminishing marginal return on my y value (the dependend variable).

For example hours = beta0 + beta1*log(wage) where hours = hour worked per week wage = hourly wage

Then OLS fits a linear line. To interpret my beta1 cofficient I divide it by 100 by saying a 1 % increase in wage has a XX effect on hours worked per week.

But from my estimated beta1 cofficient, how can I see the diminishing effect the independend variable has on the dependend now that it is a linear line?

Suddenly after the estimation I cannot see how I can interpret this constant to be a diminishing effect on the dependend variable?

Kind Regards Maria

Answer 1

This should have been posted into the stat version of StackOverflow. Anyways my suggestion is to try this (start with a basic linear model):

1) Check the plot of the residuals. If there is no sign of heteroscedasticity in the linear model, then stop. Otherwise if you can see a pattern in the residuals (funnel, sinusoids or anything else) continue. -> E[sigma_i]!=sigma for i=1..k where k = model dimensions.

2) Try with a squared model. In this case I would do:

Y = beta[0]+beta[1]*X+beta[2]*X^2

Then if your ideas are correct you should get a positive beta[1] and a negative beta[2]. Most likely with abs(beta[1])>abs(beta[2]). This mean that with for small value or X the effect of the squared component (negative) will be little to none, while with for a big value of X the negative squared component will be very strong. Now go back to 1) if you get normal residuals you are done.

3) Try with:

Y = beta[0]+beta[1]*log(X)

and with:

Y = beta[0]+beta[1]*log(X^2)

And see which one gives you the best residuals.

There is only one issue in your reasoning. You don't have anymore a linear line, but a curve, as denoted by the relationship Y = b*LN(X). Therefore the log curve itself explains your "diminishing returns".