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I read about quantile regression and was confused about one sentence: "We would be way too confident" about confidence interval in this post (under the second figure): here. I think that the confidence interval here is too small, so why does the author says that we could be too confident about estimating the mean of the response variable given the independent variable?

Maryam
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    Large confidence requires larger intervals, caeteris paribus. You ask about the logical converse: if you think you have large confidence but your interval is too small, then you must be mistaken and you should have less confidence. – whuber Dec 30 '21 at 17:12

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The statistical use of the word confidence is different than the colloquial use, which may be the source of some confusion.

More to do with your link, a better claim would have been "we would be too precise". Precision is the inverse of variance, so when the variance is small then the precision is high. The width of a confidence interval relies on the standard error (which is the standard deviation of the estimate, hence the square root of the variance of the estimate). If the confidence interval is too narrow, them the estimate is too precise.

The reason I bring up precision is because the data in figure 2 in your link is heteroskedastic, which geom_smooth(method='lm') does not account for. To properly account for heterogeneity of variance, we need a sandwhich estimator, which has the result of increasing the estimated standard error of the coefficients for the model (hence decreasing the precision).

Demetri Pananos
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