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I am trying to interpret these 2 residual vs fitted value plots if anyone can help. I was reading about a cone like structure (heteroskedasticity) but am unsure if this is the case in the first one. Obviously this isnt great but i cant pinpoint what its telling me. The same in the second plot, this is really bad, i think it's saying that my upper and lower values are poor but would be good if someone could confirm.

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  • $\begingroup$ The second plot indicates that your model fails to capture some non-linearity resp. you are missing predictors (such as a squared term). The first plot is indeed an obvious case of heteroskedasticity. Both plots look like they result from simulated data. I show similar plots if I give an Introduction to regression lecture. $\endgroup$
    – Roland
    Commented Jun 14, 2022 at 5:36

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The plots indicate that the model suffers from heteroscedasticity. Generally, any systematic patterns (like a cone shape in the first plot or an inverted-U shape pattern in the second plot) may indicate the presence of heteroscedasticity. The error variance does not seem to be constant, meaning your regression results may not be reliable.

But, it's better not to rely on just graphs. You should try formal tests of heteroscedasticity to make sure.

If you find that heteroscedasticity exists (using formal tests), try applying Robust Standard Errors in your regression model. Then, compare its results with your original model. If the significance of variables does not change, heteroscedasticity is not a serious problem in your model. But if variable significance changes, you cannot rely on your model.

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  • $\begingroup$ Thanks. Would the second one also show a non linearity, or is that what causes heteroscedasticity here? $\endgroup$
    – Joe
    Commented Jun 13, 2022 at 11:44
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    $\begingroup$ I would actually argue in common with Faraway that tests can be a bit too wooden, and that graphical methods are better. No doubt there is some scholarly difference of opinion on this matter. $\endgroup$ Commented Jun 13, 2022 at 13:56
  • $\begingroup$ Every test has its shortcomings of course. In practice, I would say that it is best to supplement with both. Do not rely on just graphs or any one test. In any case, reporting Robust Standard Errors is always a good practice in regressions. $\endgroup$ Commented Jun 14, 2022 at 4:52
  • $\begingroup$ Checking the actual cause of heteroscedasticity will need a further examination of the data/variables. It is possible that you may be excluding some important variables from your regression or due to the effect of outliers in the data. Or, it might just be the nature of your variable. It will depend on the variables involved and their behaviour. $\endgroup$ Commented Jun 14, 2022 at 4:59
  • $\begingroup$ A formal test of heteroscedasticity will either fail to identify it or will confirm what is visually obvious, leaving one no better off for the effort. The graph shows how the residuals vary. Associated with such a graph are derived graphs, such as the spread-vs-level plot, that will reveal what can be done to alleviate the problem. It is in this sense that one can effectively go beyond "just" relying on graphs. $\endgroup$
    – whuber
    Commented Jun 15, 2022 at 20:51

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