The OLS residuals

Question

The OLS residuals

BlackTom AI · Accepted Answer

OLS residuals are defined by the difference between observed outcomes and fitted values: e_i = y_i − ŷ_i.

1. Relevant knowledge points and concepts
- Observed value y_i: the actual dependent variable value in the sample.
- Fitted value ŷ_i: the value predicted by the estimated OLS regression, ŷ = Xβ̂.
- Residual e_i: sample quantity computed as e_i = y_i − ŷ_i; it is observable and used for diagnostics.
- Population error u_i (true disturbance): unobserved; residuals estimate these but are not equal in general.
- Matrix form: e = y − ŷ = y − Xβ̂ = (I − P)y, where P = X(X′X)^{-1}X′ is the projection matrix. Orthogonality: X′e = 0.

2. Step-by-step reasoning
- Estimate β̂ by OLS: β̂ = (X′X)^{-1}X′y.
- Compute fitted values: ŷ = Xβ̂.
- Compute residuals elementwise: e_i = y_i − ŷ_i for each observation i.

3. Calculations and intermediate steps (numeric illustration)
- Suppose y = [3.0, 5.0], and fitted values ŷ = [2.5, 4.8].
- Then residuals e = y − ŷ = [3.0 − 2.5, 5.0 − 4.8] = [0.5, 0.2].

4. Why the correct answer is correct
- By definition in OLS, residuals are computed directly from observed y and estimated fitted values ŷ; this matches the option “Can be calculated by subtracting the fitted values from the actual values.”

5. Why the other options are incorrect
- “Are unknown since we do not know the population regression function”: incorrect because residuals use the estimated regression, not the (unknown) population function; they are observable.
- “Should not be used in practice since they indicate that your regression does not run through all your observations”: incorrect and misleading—residuals are diagnostic tools used routinely; the regression need not pass through every observation.
- “Can be calculated using the errors from the regression function”: incorrect because the true errors (population disturbances) are unobserved; residuals estimate those errors, not vice versa.

6. Conclusion
The correct answer is “Can be calculated by subtracting the fitted values from the actual values,” which matches the provided answer.

The OLS residuals单项选择题

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