How to interpret linear trends in residuals?
When I see it correctly in the top figure, there is some &quot; bunching&quot; in your data, meaning that there are a number of companies with the same (or very similar) number of employees. Since you appear to run a regression with only one independent variable (right hand side), this bunching will be visible in the residual.
Your model is:
$$ y_i = \beta_0 + \beta_1 x_i + u_i$$
Now say $\beta_0 = 1$ and $\beta_1 = 0.1$, and with &quot;bunching&quot; in $y$ you will get something like:
y     x      y_hat  u_hat
--------------------------
5     10     2      3
5     20     3      2
5     30     4      1
5     40     5      0

So the linear nature of the model will be reflected in the residual given that there is &quot;bunching&quot; in the data.
Note that using a log-log approach (which is perfectly okay) will change the natural interpretation of the estimated coefficients. In a log-log case, you will interprete the results as &quot;a one percent change in $x$ will be associated with a $\beta_1$ percent change in $y$&quot; (all other things equal).
Since the number of emplyees is bounded (no negative employees -&gt; &quot;count data&quot;), things like Poisson regression could be worth a try.
Overall your model may be underspecified. If you can, include additional $x$-variables in your model, so to better reflect the data generating process. See page 114 in this book.