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I am regressing a log-normally distributed dependent variable (wage) on a heavily skewed independent variable and I want to make sure I handle it in the best possible way. The independent variable award is the percentage point (measured 0-100) likelihood that a given person will win a specific award in their career. The measure is heavily skewed to zero (~70.00% of the data takes a 0 value), however, it does not look log normal and not quite bimodal. There is no other value with more than 1.00% of the population density, but many values are around 0.25%, and a good number are between 0.50% and 1.00%.

The current model is an OLS regression, log_wages = b0 + b1*award + b2*control_1 + b3*control_2 + b4*control_3. Is this the best model? Do I need to transform award in any way? Are there any tests I should run to test the distribution?

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  • $\begingroup$ (1) How do you know wage is lognormal, rather than some similar distribution? (2) the marginal distribution of log-wage is not assumed to be normal in OLS, so its distribution is not directly relevant. (3) the distribution of $x$'s is not part of the assumptions of regression -- they're not random variables in an OLS model, but fixed. If they're subject to error you should consider the problems this causes in OLS. You should not test a non-assumption. (4) We can't tell you what model is best for your data, for several reasons (not least that all we have to go on is a few sentences). $\endgroup$
    – Glen_b
    Mar 25, 2014 at 21:58
  • $\begingroup$ Items (2) and (3) are discussed in many questions on the site. $\endgroup$
    – Glen_b
    Mar 25, 2014 at 21:59
  • $\begingroup$ @Glen_b I have the wage data so I can directly see that the raw data is log-normal. However, I see what you are getting at, but it is not my cause for concern. Also, the measurement error of my award variable is not the issue. The issue is given that award is distributed in the above way, what should I do in an OLS framework or in another framework. Is there more information about the award data that is needed? What I wrote is exactly what I have. $\endgroup$
    – LF12
    Mar 25, 2014 at 22:09
  • $\begingroup$ "I have the wage data so I can directly see that the raw data is log-normal." -- then you must be a lot cleverer than me. I seriously doubt I could just look at a set of numbers (or even a Q-Q plot) and distinguish lognormal from say a 50-50 mix of gamma and inverse gamma, let alone something even closer to lognormal. The rest of your comment is already answered by my first comment, is it not? If you don't think so, could you clarify what is lacking? $\endgroup$
    – Glen_b
    Mar 25, 2014 at 22:14
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    $\begingroup$ @Glen_b Not sure why you are focusing on the part that is NOT the question, but that is fine. My comment was just to move on to the actual question and I am relying on the shape in the data and the mounds of previous literature. Your "answer" basically says do not test a non-assumption, fine. But, I am asking: Given my variable and its distribution have others dealt with it in the past, and if so what is recommended. $\endgroup$
    – LF12
    Mar 25, 2014 at 22:53

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