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I regressed a model in R with the following functional form/code (where sqrt, log, and squared refer to the transformations):

model<-lm(sqrt(y)~x1 + x2 + x3 + x4_squared + log_x5 + x6 + x7 + log_x8 + x9 + x10 + x11 + x12 + x13` + x14 + x15, data = data)

I got a residual standard error of 19.77 on 265 degrees of freedom. I am trying to put this RSE in context, I feel it should be lower but I want to better understand where to draw the line and why. My residuals vs. fitted plot is below.My residuals vs. fitted plot is below.

I looked at the individual standard errors of my variables, and the two logged variables (x5 and x8) have high standard errors, of 142 and 24. I plotted the relationships of these variables with the dependent variable and there appears to be a logarithmic relationship (below)? I am a bit stuck on how to linearize this without overfitting, and hopefully fix my RSE in the process. Thank you all very much for the help!

enter image description here enter image description here

Jay J
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  • Welcome, Jay J! A couple of quick comments/questions: 1)your residuals do not necessarily look catastrophic, in the light that the fitted values range from 200 to 600, but given the many variables, yes, you would probably want a closer fit. 2) you should relate the magnitude of standard errors to the magnitude of the estimated coefficient. An se of 0.3 can be huge when the estimate is 0.1. 3) you have many variables - did you select them from even more? Are there any grouping variables (factors) among them? – Ute Jul 13 '23 at 17:06
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    This is possibly helpful: https://stats.stackexchange.com/q/222179/1352. Sometimes it is simply not possible to achieve the error we would like to achieve - if you toss a fair coin, you won't get an MSE below 0.25. – Stephan Kolassa Jul 13 '23 at 17:14
  • Thanks for the quick responses, everyone, I really appreciate the help! @Ute- For x5 I got a se of 142 and a coefficient of -822, which is far too high (x5 is CPI points). I am wondering if this has to do with a violation of the linearity assumption? In the plot I attached above there appears to be a nonlinear relationship between the two. However, given the residuals vs fitted and QQ plot looks good, is this nonlinearity actually compromising the integrity of the model? Thanks for the help! – Jay J Jul 13 '23 at 19:33
  • The standard deviation of your Y variable tells you how accurate the mean is to predict Y values. The RSE tells you how accurate the regression line is to predict Y values. So can interpret RSE by comparing it with the ordinary standard deviation of Y (irrespective of the X variables) to see how much improvement you get when you use regression. – BigBendRegion Jul 14 '23 at 00:07

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I don't see any obvious problems with your model based on the information you've provided. For any fixed values of the predictors, your response seems to have a high degree of variability -- which is reflected in the high residual SE. There's nothing you can do -- except collect data on additional predictors that have more explanatory power!

Rachel Altman
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    Thanks for your quick response Rachel! I am glad to hear this, I think I've just been staring at the residuals for too long haha. – Jay J Jul 13 '23 at 19:20