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  • Coefficient interpretation in log-level regression- Actual impact of independent variable on dependent variable.

    Hi all,

    I have a regression where the dependent variable is in log form. I was wondering if we can find out the actual impact of the independent variable on the natural form of the dependent variable? Or do we have to rely on percentage change? Here is an example:

    The coefficient of the variable I'm interested is -0.299. First, I take the exponential:

    Code:
    . di exp(0.299)-1
    .34850962
    This means one unit increase in the independent variable decreases the dependent variable by 34%, right?

    But can we say what the actual numerical change is? If the mean value of the natural form of dependent variable is 1,000,000 can we say that it leads to 340.000 decrease?

    Thanks!

  • #2
    Basically what you say is correct. I think that you should round.34850962 to 35%, not 34%, and I would avoid causal language unless you have external reasons for believing the effect is causal. But otherwise, that is correct.

    In reporting this to others, make sure you are clear that the 340,000 decrease applies only to the mean value of 1,000,000 and that what your model actually estimates is a relative decrease of 34%, and the absolute decrease depends on the starting value.

    Correction: The above is wrong--I did not pay attention to the negative sign of the coefficient.

    A unit increase in X is associated with a decrease of 0.299 in log y. A decrease of 0.299 in log y means y itself decreases by a factor of exp(-0.299) = 0.74155941, which corresponds to a decrease of 26%, not 34%. The corresponding absolute decrease in y from a starting value of 1,000,000 would be 260,000.
    Last edited by Clyde Schechter; 15 Jul 2019, 11:54.

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    • #3
      Hakan: Your statement, and Clyde's in #2, will hold if you are considering the (conditional) median of the outcome, but not necessarily the (conditional) mean. One classic reference is https://www.jstor.org/stable/1909517

      If you are interested in the conditional mean E[y|x] then since E[log(y)|x]≠logE[y|x] you are then confronted with the so-called retransformation problem. One way to get a sense (though not a formal test) of whether this matters in your application is to compare your existing estimates with those obtained using a generalized linear model where the outcome is measured in natural, not log, units, e.g.
      Code:
      glm y x, link(log) fam(gamma) robust
      margins, dydx(*)
      If the results from margins are much different than those you reported then there are retransformation issues you might wish to consider.

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