[Math] Residual plot in the logarithmic model.

Question

We've got some data containing two variables, where $x$ is the predictor and $y$ is the response variable. We make a model of the form of:
$$y=\alpha+\beta \cdot x + \epsilon$$
Then we see that in the residual plot (residuals vs. $\hat{y}$) the variance is increasing as $\hat{y}$. We then decide to transform our model to a logarithmic form, i.e.:
$$log(y)=\alpha+\beta \cdot x + \epsilon$$
And now my question is: When performing a residual plot analysis, do we plot residuals vs. $\hat{log(y)}$ or $\hat{y}$?

Answer 1

You should plot them against the log (y) as these are the residuals that need to be tested for the logarithmic form