Do I want to overfit, when doing outlier detection based on regression? Imagine, we have speed data of car and we would like to detect, if car speeds up or down more than it should.
Do I want to just overfit my model, so the outlier (higher or lower speed) would lead me to higher residual easing the process of outlier detection?
Edit: To clarify, training data have no outliers.

If your goal is to detect "if car speeds up or down more than it should", then your regression model should capture the kinds of acceleration the car "should" be performing, but not capture the kinds it "shouldn't". If your model is very over-fit, it will capture all kinds of acceleration, and nothing will be an outlier. If it's under-fit, it won't capture some acceleration that you think of as normal, so will show too many outliers.
All that aside, linear regression on it's own is not generally appropriate for time series data, and not necessarily the best way of identifying outliers in any kind of data. There are specific approaches for identifying outliers in time series data you should look into.
