Interpretation of $\beta$ in case of log-lin model for relationship between $X$ and $Y$

Question

In many papers, the dependent variable is transformed by taking natural log. For instance, consider the following model:

$$\newcommand{\Cov}{{\rm Cov}} \ln(\text{Y}) = \alpha + \beta\, X_1 + \epsilon $$

I understand that the interpretation of $\beta$ is the percentage change in $Y$ for a given unit change in $X$. However, some papers comment on the direction of the relationship between $X$ and $Y$ (and not $\ln(Y)$) on the basis of the sign of $\beta$.

Through a simple simulation exercise, I have found out that it may be erroneous to comment on the direction of the relationship between $Y$ and $X$ on the basis of the sign of $\beta$ in the above equation. Specifically, the signs of $\Cov(X,Y)$ and $\Cov(X, \ln(Y))$ may be different.

I ask this question specifically in the context of regressions (particularly multiple regression). I am actually concerned with listing down different scenarios in which the beta coefficient may change sign from $X$ and $Y$ to $\log(Y)$ and $X$. I am working on a review paper in the context of corporate finance, and I would like to highlight this issue of the log transformation of $Y$ (or $X$ for that matter) that may further result in different signs of beta coefficients when compared to relationship between $X$ and $Y$.

Answer 1

It's possible for the correlation to change sign from $X$ and $Y$ to $\log Y$ and $X$. One easy class of examples is when a transformation dampens an outlier that was exerting major leverage. Another class of examples is whenever an outlier is created by a transformation.

The Stata code used to produce this example should be fairly transparent.

clear
input y x
1       2   
2       3
3       4
4       5
5       6
6       7 
10      1
end 
gen log_y = log(y) 
corr y log_y x 

Correlations:

             |        y    log_y        x
-------------+---------------------------
           y |   1.0000
       log_y |   0.9352   1.0000
           x |  -0.0516   0.2424   1.0000

The example is not especially blatant, but that is much of the point. You don't need extraordinary or outrageously pathological datasets to see this effect.

The question is unclear, as is that on Statalist, on precisely what kind of model is being considered, but talk of covariance to me implies plain correlation and regression, so there is no obvious need to invoke any wider framework.

Answer 2

Since the logarithm is a monotonic function, if $Cov(X,Y) > 0$, then $Cov(X, log(Y)) > 0$.

That said, it's possible in a regression model to obtain different findings. Unlike least squares, the sign of the regression coefficient in a bivariate GLM is not necessarily of the same sign as the covariance. Rather, a GLM obtains a regression covariate by iteratively reweighting the least squares parameter to find the maximum likelihood, at least in the case of Poisson regression (which is also called a log linear model). The reason for this is: in a Poisson probability model, the variance of the response is higher when the mean is higher, so in domains where the mean response achieves larger values, the Poisson model downweights large residuals.

It's possible, therefore, to generate bimodal data so that the OLS slope is positive but the Poisson GLM (log linear model) slope is negative.