Sum of the residuals in multiple linear regression

In summary, the sum of residuals in multiple linear regression is the total amount of error in the regression model, calculated by taking the difference between each observed value and its corresponding predicted value and adding them together. A larger sum of residuals indicates a poorer fit of the model, while a smaller sum suggests a better fit. It can also be used to calculate the R-squared value, which represents the percentage of variation in the dependent variable that is explained by the independent variables in the model. The sum of residuals can be negative in certain cases, but ideally, it should be close to zero for a good fit.
  • #1
kingwinner
1,270
0
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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  • #2
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?
 

1. What is the sum of the residuals in multiple linear regression?

The sum of the residuals in multiple linear regression is the total of all the differences between the actual values and the predicted values of the dependent variable. In other words, it represents the total amount of error in the regression model.

2. How is the sum of residuals calculated in multiple linear regression?

The sum of residuals is calculated by taking the difference between each observed value of the dependent variable and the corresponding predicted value from the regression model, and then adding all of these differences together.

3. What does a larger sum of residuals indicate in multiple linear regression?

A larger sum of residuals indicates a poorer fit of the regression model to the data. It suggests that there is a significant amount of unexplained variation in the dependent variable that is not accounted for by the independent variables in the model.

4. Can the sum of residuals in multiple linear regression be negative?

Yes, the sum of residuals can be negative in multiple linear regression. This can happen when the predicted values are larger than the actual values, resulting in a negative difference between the two. However, the overall sum of residuals should ideally be close to zero for a good fit of the model.

5. How can the sum of residuals be used to evaluate the performance of a multiple linear regression model?

The sum of residuals can be used as a measure of the overall goodness of fit of a multiple linear regression model. A smaller sum of residuals indicates a better fit, while a larger sum indicates a poorer fit. It can also be compared to the sum of squares of the dependent variable to calculate the coefficient of determination (R-squared), which represents the percentage of variation in the dependent variable that is explained by the independent variables in the model.

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