Comparing two regressions that differ by a few data points

Question

I have built a model that explains how much risk of the stock market (S&P 500 index) is attributable to each sector, where each sector is independent from each other (correlation coefficients among the sectors are all 0).

For example, I have the following data:

Period                      Consumer Discretionary    Consumer Staples    Energy    Financials    Health Care    Industrials    Information Tech    Materials    Telecommunication    Utilities
2018-04-26 to 2018-05-25    10.32%                      7.13%             7.75%      12.61%        15.21%           7.34%          27.62%            7.79%            1.76%           2.45%
2018-04-27 to 2018-05-29    7.90%                      5.22%            4.96%        20.67%        12.27%           10.97%         21.85%           9.72%             5.41%         1.01%

where I am using rolling-regression (using daily return, 31 days rolling-period) to calculate the values. It is interpretated that, during 2018-04-26 and 2018-05-25, 10.32% of the total market risk is explained by the Consumer Discretionary sector, independent from other sectors. Likewise, during 2018-04-27 to 2018-05-29, 21.85% of the total market risk is explained by the Information Tech sector, independent from other sectors.

The input data for those two regressions are exactly the same, except the first one contains 2018-04-26 and does not contain 2018-05-29 while the second one does not contain 2018-04-26 but does contain 2018-05-29. So any two consecutive rows differ by two data points.

My question is, using the regressions output and input data, is it possible to identify which input data are responsible for changes in the values?

For example, you can see that the risk contribution of Financials increases from 12.61% to 20.67% between the two rows. Is it because something happened on 2018-05-29, which is present in the second regression but not the first regression? Or is it because of 2018-04-26, which is present in the first regression but not the second regression? If neither, does it mean that those two data points (2018-04-26 and 2018-05-29) are not that important in explaining the change?

I am interested in this analysis because, given the output data, I want to know what actually took place in the market and explain it qualitatively. What is the name of this kind of analysis?

Thank you very much.

Answer 1

Given your current setup, it would be hard to tell whether 2018-04-26 or 2018-05-29 is the cause of the change.

There are a variety of regression diagnostics that can be used to determine which point is significant. I would recommend starting with Cook's Distance. This is a measure of the influence of an individual point on the final regression. It works by calculating the effect of deleting an individual observation from the regression.

If you don't have access to a library for calculating Cook's Distance, then you could simply delete each observation from your regression and recalculate your stats. This is very close to a technique called JackKnife Resampling.

I would also recommend starting from a regression sample that includes both days, so that you can isolate the effect of removing one or other of the days when using these techniques. BTW, I also want to echo that you might have more luck on Cross Validated or Stats (see, e.g., https://stats.stackexchange.com/questions/8344/influence-functions-and-ols).