ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A Q.NO. 8
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ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A
Regression Residuals Question – Solution
Question 8 :
Which of the following statements are associated with residuals of a regression model?
The sum of the residuals in any regression model that contains an intercept is always zero.
The sum of the observed values yiy_iyi is not equal to the sum of the fitted values y^i\hat{y}_iy^i.
The sum of the residuals weighted by the corresponding regressor variable is zero.
The sum of the residuals weighted by the corresponding fitted values is zero.
Select the correct answer using the code given below:(a) 1, 2 and 3(b) 1, 3 and 4(c) 1, 2 and 4(d) 2, 3 and 4
Solution
Let the residual be:
ei=yi−y^ie_i = y_i - \hat{y}_iei=yi−y^i
where
yiy_iyi = observed value
y^i\hat{y}_iy^i = fitted (predicted) value
Statement 1
“The sum of residuals in any regression model containing an intercept is always zero.”
Property of Ordinary Least Squares (OLS):
∑ei=0\sum e_i = 0∑ei=0
Therefore, Statement 1 is TRUE.
Statement 2
“The sum of observed values yiy_iyi is not equal to the sum of fitted values y^i\hat{y}_iy^i.”
Since
∑ei=∑(yi−y^i)=0\sum e_i = \sum (y_i - \hat{y}_i) = 0∑ei=∑(yi−y^i)=0∑yi=∑y^i\sum y_i = \sum \hat{y}_i∑yi=∑y^i
So the statement saying they are not equal is FALSE.
Statement 3
“The sum of residuals weighted by the corresponding regressor variable is zero.”
In OLS, residuals are orthogonal to regressors:
∑xiei=0\sum x_i e_i = 0∑xiei=0
Therefore, Statement 3 is TRUE.
Statement 4
“The sum of residuals weighted by the fitted values is zero.”
OLS property:
∑y^iei=0\sum \hat{y}_i e_i = 0∑y^iei=0
Therefore, Statement 4 is TRUE.
Final Answer
True statements:✔ 1✔ 3✔ 4
Correct Option: (b) 1, 3 and 4
ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A Click Here to Download
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