Suppose that a simple regression has been estimated. Yi is the dependent variable, X1i is the independent variable and subscript i runs over observations i=1,2,...,n. is the error term. Choose ALL CORRECT definition(s) of the OLS residualsMultiple choice
A
a.
B
b.
C
c.
D
d.

Question

Suppose that a simple regression has been estimated. Yi is the dependent variable, X1i is the independent variable and subscript i runs over observations i=1,2,...,n. is the error term. Choose ALL CORRECT definition(s) of the OLS residualsMultiple choice

A

a.

B

b.

C

c.

D

d.

BlackTom AI · Accepted Answer

The OLS residual for observation i is ε̂_i = Y_i − Ŷ_i = Y_i − (β̂_0 + β̂_1 X_i).

1) Relevant knowledge points and concepts
- Ŷ_i is the fitted (predicted) value from the estimated regression: Ŷ_i = β̂_0 + β̂_1 X_i.
- The residual ε̂_i is the difference between the observed dependent variable and its fitted value: ε̂_i = Y_i − Ŷ_i.
- The true (unobserved) error is ε_i = Y_i − (β_0 + β_1 X_i), which uses the true (non-hat) parameters.

2) Step-by-step reasoning
- Start with definition: ε̂_i ≡ Y_i − Ŷ_i.
- Substitute the fitted value: Ŷ_i = β̂_0 + β̂_1 X_i.
- Thus ε̂_i = Y_i − (β̂_0 + β̂_1 X_i) = Y_i − β̂_0 − β̂_1 X_i.

3) Intermediate steps shown
- Ŷ_i = β̂_0 + β̂_1 X_i
- ε̂_i = Y_i − Ŷ_i
- ε̂_i = Y_i − (β̂_0 + β̂_1 X_i) = Y_i − β̂_0 − β̂_1 X_i

4) Why the correct answer is correct
- Option b, Y_i − Ŷ_i, exactly matches the definition above. It contains observed Y_i and the predicted Ŷ_i computed from estimated coefficients, so it is the OLS residual.

5) Why the other options are incorrect
- a) Ŷ_i − β̂_0 − β̂_1 X_i: substitute Ŷ_i = β̂_0 + β̂_1 X_i, this becomes (β̂_0 + β̂_1 X_i) − β̂_0 − β̂_1 X_i = 0 for every i, so it is identically zero and not the residual.
- c) Y_i − β_0 − β_1 X_i: uses true (unknown) β_0, β_1, so this equals the true error ε_i (unobservable), not the estimated residual ε̂_i.
- d) (Y_i − Ȳ)^2: squared deviation of Y_i from the sample mean, unrelated to the OLS residual (and squared).

6) Conclusion
The correct definition of the OLS residual is Y_i − Ŷ_i (option b). This matches the provided correct answer: Yi - Ŷi.

Suppose that a simple regression has been estimated. Yi is the dependent variable, X1i is the independent variable and subscript i runs over observations i=1,2,...,n. is the error term. Choose ALL CORRECT definition(s) of the OLS residualsMultiple choice

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