Sum of the residuals in multiple linear regression

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SUMMARY

The sum of the residuals in multiple linear regression is zero when an intercept is included in the model. This is consistent with the results from simple linear regression, where ∑e_i = 0 and ∑(e_i)(Y_i hat) = 0. However, these results do not hold true for regression models that do not include an intercept, such as regression through the origin. Utilizing a matrix approach in multiple regression can further clarify these concepts.

PREREQUISITES
  • Understanding of multiple linear regression concepts
  • Familiarity with residuals and their properties
  • Knowledge of matrix algebra as applied to regression analysis
  • Experience with statistical software for regression modeling
NEXT STEPS
  • Study the properties of residuals in multiple linear regression
  • Learn about regression models without intercepts and their implications
  • Explore matrix methods for multiple linear regression analysis
  • Investigate statistical software options for performing multiple linear regression
USEFUL FOR

Statisticians, data analysts, and anyone involved in regression modeling who seeks to deepen their understanding of residuals in multiple linear regression.

kingwinner
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In my textbook, the following results are proved in the context of SIMPLE linear regression:
∑e_i = 0
∑(e_i)(Y_i hat)= 0

I tried to modify the proofs to mutliple linear regression, but I am unable to do so, so I am puzzled...

Are these results also true in MULTIPLE linear regression?

Thanks!
 
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They are true as long as you include an intercept - they aren't for regression through the origin.

Are you using a matrix approach in your multiple regression?
 

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