Interpreting the residual standard error. How to compare residuals from different models?

Question

I am interested in interpreting the residual standard error.

Does it allow me to build confidence intervals? For example:

I have 2 models:

Modelling $\ln Y$ with rse 0.5

Modelling $Y$ with rse 0.2

Would this imply that the $\ln Y \pm t_{n-p}(0.025) \times 0.5$ is the 95% confidence interval for $\ln Y$?

Similarly $Y \pm t_{n-p}(0.025) \times 0.2$ for the second case?

Would this imply that the $\ln$ model is better?

Since for small $Y$, we have $$\exp\{\ln(Y)+t_{n-p}(0.025) \times 0.5\}-\exp\{\ln(Y)\}\leq \{Y + t_{n-p}(0.025) \times 0.2-Y\}$$

Is this a valid method of comparing residuals?

Answer 1

There is some misunderstandings here. You seem to want to compare linear models (not stated, but I will assume that) for $Y$ or for $\ln Y$ on some predictors $x$ (not stated). That comparison should not be done only on base of rse calculated from the residuals!

You should note that the structure of those two models are very different. Start with $$ \ln Y = \beta_0 + \beta_1 x + \epsilon $$ Exponentiating we get $$ Y = e^{\beta_0 + \beta_1 x + \epsilon} = e^{\beta_0} e^{\beta_1 x} e^{\epsilon} \\ = \gamma_0 z^{\beta_1} \delta, \quad (\text{say, with $z=e^x$}) $$ so this corresponds to a multiplicative model (with multiplicative errors) on the original $Y$ scale. Rather than look at residuals you should first ask yourself which of this models make sense. Then you could look at residuals, but plotting them against the regressor(s) $x$, not only looking at a qqplot.