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:

Question

BlackTom AI · Accepted Answer

R^2 = 0.01 means only 1% of the variation in Y is explained by X1.

1) Relevant concepts
- TSS = Σ(Yi − Ȳ)² (total variation in Y).
- ESS (also SSR in some texts) = Σ(Ŷi − Ȳ)² (variation explained by the model).
- SSR (sum of squared residuals, sometimes SSE) = Σ(Yi − Ŷi)² (unexplained variation).
- R² = ESS / TSS = 1 − SSR / TSS. R² measures the fraction of Y’s variance explained by the regressors.

2) Step-by-step reasoning and calculation
- Given R² = 0.01 → ESS / TSS = 0.01.
- Using R² = 1 − SSR / TSS, solve for SSR / TSS:
  SSR / TSS = 1 − R² = 1 − 0.01 = 0.99.
- Interpretation: 99% of the total variance remains unexplained by X1; only 1% is explained by X1.

3) Why the correct answer is correct
- Statement c: "it suggests that there are other important factors to influence the left-hand side variable." Because 99% of variance is unexplained by X1, it is reasonable to conclude that other variables (or noise) account for most variation in Y. Thus c is correct.

4) Why the other options are incorrect
- a: "it specifically shows what the other factors can influence the left-hand side variable" — R² gives no information about the identity or nature of omitted factors; it only quantifies unexplained variance. So a is incorrect.
- b: "it suggests that the regression model has good explanatory power" — R² = 0.01 implies poor explanatory power (only 1% explained), so b is incorrect.
- d: "the sum of squared residuals SSR is low relative to TSS" — calculated SSR/TSS = 0.99, i.e., SSR is large relative to TSS (not low). So d is incorrect.

Conclusion: Only option c is correct, which matches the provided answer.

类似问题

Which of the following is the correct interpretation for 0.60 coefficient of determination?