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 = ESS/TSS = 1 − SSR/TSS, so an R^2 of 0.01 means the fitted model explains only 1% of the total variation in Y while 99% of the variation remains in the residuals.

1) Relevant knowledge points:
- TSS (total sum of squares) = ESS (explained sum of squares) + SSR (sum of squared residuals).
- R^2 = ESS/TSS = 1 − SSR/TSS and measures the fraction of Y-variance explained by the regressors.
- R^2 does not identify which omitted factors exist, nor does it prove causation.

2) Step-by-step reasoning:
- Given R^2 = 0.01, plug into R^2 = ESS/TSS to see ESS/TSS = 0.01. Equivalently SSR/TSS = 1 − 0.01 = 0.99.
- This implies most of the variation in Y is not explained by X1; only 1% is explained.

3) Calculations / intermediate steps:
- If TSS = 100 (example), then ESS = 0.01 × 100 = 1 and SSR = 100 − 1 = 99.
- Directly from formula: SSR/TSS = 0.99.

4) Why the correct answer is correct:
- Option (c) states the low R^2 suggests other important factors influence the left-hand variable. Because only 1% of variance is explained by X1, it is reasonable to infer that other variables or noise account for the remaining 99% — so (c) is correct.

5) Why the other options are incorrect:
- (a) Claims SSR is low relative to TSS. This is false: SSR/TSS = 0.99, so SSR is large relative to TSS.
- (b) Claims the model has good explanatory power. This is false because R^2 = 0.01 indicates very poor explanatory power (only 1% explained).
- (d) Claims R^2 specifically shows what the other factors are. This is false: R^2 only measures how much variance is explained, not which omitted factors or their effects.

Conclusion: The only correct statement is (c), matching the provided answer.

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