Explanation of Dependent log variable

[Tariq Abdullah]

I'm using a vector of log dependent variables due to a change in the main variable of interest ( which is a difference in the percentage of income ). Should it be explained like follows:

“if we change x by one percent, we’d expect y to change by β1 percent ( Like Log-log regression ).

I know it's a basic question, But I'm slightly confused. Any kind of direction is welcome!

[Clyde Schechter]

I'm not entirely clear on what you did. If you regressed log y as the dependent variable against x as a dependent variable (which would be log-linear, not log-log), then you would say that a difference of one unit in x is approximately associated with a β1 percent difference in y. If x itself is a percentage, then the correct wording would be that a 1 percentage point difference (not one percent difference) in x is approximately associated with a β1 percent difference in y.

Caveat: This rule of thumb is just an approximation. If β1 is less than about 0.1 it is a good approximation. But it deteriorates rapidly after that. If β1 exceeds 0.2, I would not use this approximation at all. In between, it's probably OK for most purposes, but should not be taken literally.

[Tariq Abdullah]

I'm not entirely clear on what you did. If you regressed log y as the dependent variable against x as a dependent variable (which would be log-linear, not log-log), then you would say that a difference of one unit in x is approximately associated with a β1 percent difference in y. If x itself is a percentage, then the correct wording would be that a 1 percentage point difference (not one percent difference) in x is approximately associated with a β1 percent difference in y.

Caveat: This rule of thumb is just an approximation. If β1 is less than about 0.1 it is a good approximation. But it deteriorates rapidly after that. If β1 exceeds 0.2, I would not use this approximation at all. In between, it's probably OK for most purposes, but should not be taken literally.

Thanks so much for clarifying my doubt. I was personally struggling with it but your explanation is quite clear to me now. Much appreciated!