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. The estimate of the slope coefficient =0 .
Choose ALL CORRECT statement(s) below about the coefficient of determination R2.

Question

BlackTom AI · Accepted Answer

When the estimated slope is zero, the fitted equation reduces to an intercept-only model with Ŷi equal to the sample mean Ȳ.

1) Relevant knowledge points and concepts
- OLS intercept-only fit: if β̂1 = 0, then the predicted value Ŷi = β̂0. Minimizing squared errors gives β̂0 = Ȳ.
- Definitions: TSS = Σ(Yi − Ȳ)² (total sum of squares). The problem statement defines SSR as the sum of squared residuals (commonly denoted SSE): SSR (residual) = Σ(Yi − Ŷi)². R² = 1 − (SSR/TSS) under this notation.

2) Step-by-step reasoning
- Given β̂1 = 0, the fitted model is Ŷi = β̂0 for all i.
- Minimize Σ(Yi − b0)² with respect to b0: derivative −2Σ(Yi − b0) = 0 ⇒ b0 = Ȳ. So Ŷi = Ȳ for every observation.

3) Calculations and intermediate steps
- Residuals: eî = Yi − Ŷi = Yi − Ȳ.
- Residual sum of squares (SSR as defined in the question) = Σ(Yi − Ȳ)².
- But Σ(Yi − Ȳ)² is exactly TSS. Hence SSR = TSS.
- R² = 1 − (SSR/TSS) = 1 − (TSS/TSS) = 1 − 1 = 0.

4) Why the correct answer is correct
- Because the model with β̂1 = 0 predicts every Yi by the mean Ȳ, there is no explained variation beyond the mean; the proportion of variance explained, R², is zero.

5) Why the other options are incorrect
- b (0 < R² < 1): False because R² = 0, not strictly between 0 and 1.
- c (R² > (SSR/TSS), with SSR the sum of squared residuals): False because, using the question’s SSR notation, R² = 1 − (SSR/TSS), so R² is not generally greater than SSR/TSS; in the present case SSR/TSS = 1 so R² = 0 < 1 = SSR/TSS.
- d (R² = Ȳ): Dimensionally and conceptually wrong; R² is a proportion (0 to 1), not equal to the sample mean.

6) Conclusion
- Thus the correct choice is (a) R² = 0, which matches the provided answer.

类似问题

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Suppose that a simple regression has Yi is the dependent variable, X1i is the independent variable and subscript i runs over observations i=1,2,...,n. is the error term. The value of coefficient of determination is calculated as R2=0.01. Choose ALL CORRECT statement(s) about this low value of R2:

Suppose that a simple regression has Yi is the dependent variable, X1i is the independent variable and subscript i runs over observations i=1,2,...,n. is the error term. The value of coefficient of determination is calculated as R2=0.01. Choose ALL CORRECT statement(s) about this low value of R2:

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