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I am estimating a model of the following form:

log(y) =  b1 x + b2 x^2 + b3 log(z1) + b4 z2

This is an econometric model with a focus on the impact of x x^2 (the inverted U relationship), where x ranges from 0 to 1. I included the other variables z1 and z2 just to be clear that there are other controls (some which are log-transformed and some not).

I get a clear inverted U relationship - slope check at end points and a marginal plot confirms that.

My question is the following: How can I say something about the change in x impacting y (not log(y))? This article (https://library.virginia.edu/data/articles/interpreting-log-transformations-in-a-linear-model) provides a nice way to interpret in case I don't have the x^2 term. How to do this correctly when I have the x^2 term?

For example, if I am able to do one of the following that will be wonderful:

  • If x increases by 0.1 from its current sample mean what will be impact on y?
  • If x increases by 10% from its current sample mean, what will be the impact on y?

Thanks!

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I think the following approach will work. Comments and suggestions would be greatly appreciated.

Now, if I evaluate log(y1) = log(y) evaluated at x=x1 and log(y2) = log(y) evaluated at x=x2, then

log(y2) - log(y1) = -b1 x1 - b2 x1^2 + b1 x2 + b2 x2^2

Taking exponents on both sides and simplifying:

y2/y1 = Exp(-b1 x1 - b2 x1^2 + b1 x2 + b2 x2^2)

So, if I now substitute b1, b2 coefficients and select two points x1 and x2 (x2>x1, and both are smaller than the inflection point) in the RHS, can I then interpret that as the ratio between y2/y1?

Say, that the ratio is evaluated as 1.4, then can I state that moving x from x1 to x2, yields a 40% increase in y?

From my experiments with predicted values, this does work out. However, I am concerned that I am making any errors by not considering standard errors.

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