Using our standard notation, which of these describe the residuals for the simple linear regression model and for the multiple linear regression model.

c005.p0560.q001

Question

Using our standard notation, which of these describe the residuals for the simple linear regression model and for the multiple linear regression model.

c005.p0560.q001

BlackTom AI · Accepted Answer

The residual for observation i is e_i = y_i − ŷ_i, so the correct symbolic descriptions are e (or the residual vector e) and Y − Ŷ.

1) Relevant concepts
- Residual: the observed minus the fitted (predicted) value; it is an estimate of the unobserved error term ε_i.
- Simple linear regression (SLR) fitted value: ŷ_i = b0 + b1 x_i.
- Multiple linear regression (MLR) fitted value: ŷ_i = b0 + b1 x_{i1} + ... + bp x_{ip}.
- Vector/matrix form (MLR): Ŷ = X β̂, so residual vector e = Y − Ŷ = Y − X β̂.

2) Step-by-step reasoning
- Compute the fitted value for each observation:
  - SLR: ŷ_i = b0 + b1 x_i.
  - MLR: ŷ_i = b0 + Σ_{j=1}^p b_j x_{ij}.
- Subtract the fitted value from the observed response to get the residual:
  - e_i = y_i − ŷ_i.
- In matrix notation (all observations at once): e = Y − Ŷ = Y − X β̂.

3) Intermediate steps / formulas
- Example SLR: if b0 = 2, b1 = 0.5, x_i = 4, then ŷ_i = 2 + 0.5(4) = 4. Residual e_i = y_i − 4.
- MLR matrix form: given Y (n×1), X (n×(p+1)), β̂ ((p+1)×1), compute Ŷ = X β̂, then e = Y − X β̂.

4) Why these are correct
- "e" is the standard symbol for residuals; "Y − Ŷ" explicitly shows residuals equal observed minus predicted. Both are equivalent representations (scalar per observation or vector form for all observations).

5) Why other choices are incorrect
- Any choice that is not e or Y − Ŷ (for example "None of these" or some other symbol not representing observed minus fitted) does not represent the residual definition and is therefore incorrect.
- Residuals are not the true error ε_i (they estimate ε_i), so an option equating residuals exactly to ε would be incorrect.

Conclusion: The correct descriptions of the residuals are e and Y − Ŷ, which matches the provided answer.

Using our standard notation, which of these describe the residuals for the simple linear regression model and for the multiple linear regression model. c005.p0560.q001 多项选择题

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