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Let's say I perform multiple regression where y = income, x1 = educaiton, x2 = sex, and x3 = religion from 2003 to 2018, where the data is measured daily.

Is there any way to quantify an impact of a single day data (e.x. 2005-07-01) on the regression result?

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    $\begingroup$ If you want to quantify an impact of a single day data you can create a boolean feature that will be false for all days except you chosen one. $\endgroup$
    – Daniel Chepenko
    Commented Jul 19, 2018 at 5:39
  • $\begingroup$ Maybe stating the obvious - this new feature would need to be included in the regression. $\endgroup$
    – bradS
    Commented Jul 19, 2018 at 8:15

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It is hard to quantify exactly what a single point is doing, but you would benefit from plotting data to look for suspicious points and then measuring differences in statistics if you remove it.

Don't forget about the value of plotting data! Anscombe's quartet is a great reminder of how important this is. https://en.wikipedia.org/wiki/Anscombe%27s_quartet

enter image description here

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I suppose you are trying to understand those data points in your sample that could impact your regression results to a large extent. These are called influential and high leverage observations. It is a practice to ensure that you do not have these sort of observations in your training sample as it would impact the final regression result significantly. The way to identify these observations is through the means of Cook's distance. You would need to further analyze those data points that high a high Cook's distance.

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  • $\begingroup$ Hi, I know that people usually remove the outliers and high leverage observations from the dataset. However, in my dataset, those points actually have meanings and I do not want to remove them. Instead, I want to perform regression and identify those data points that have huge impact on the regression result. $\endgroup$
    – JungleDiff
    Commented Jul 19, 2018 at 11:56
  • $\begingroup$ What do you mean when you say ‘impact’? Do you mean how much the estimates change with and without that data point? $\endgroup$
    – Srikrishna
    Commented Jul 19, 2018 at 12:08
  • $\begingroup$ I dont have the precise definition of impact, but yes that would be one of the definitions. And actually, I am performing constrained regression, with coefficients being between 0 and 1, and the sum of the coefficients being 1. So I don't think I can use the usual Cook's distance. $\endgroup$
    – JungleDiff
    Commented Jul 19, 2018 at 12:19

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