R-Square question : What is R2

we define

total sum of squares (TSS)

the explained sum of squares (ESS)

the residual sum of squares (RSS)

Question

R-Square question : What is R2

we define

total sum of squares (TSS)

the explained sum of squares (ESS)

the residual sum of squares (RSS)

BlackTom AI · Accepted Answer

Question restatement: R-Square question: What is R2? We define the following sums of squares in regression: total sum of squares (TSS), the explained sum of squares (ESS), and the residual sum of squares (RSS).

Option 1: 1 - RSS/TSS. This corresponds to the typical definition of the coefficient of determination, R^2, in linear regression, where R^2 equals the proportion of total variation explained by the model. Since RSS/TSS is the proportion of unexplained variation, subtracting it from 1 yields the explained proportion. This aligns with standard statistical practice.

Option 2: 1 + ESS/TSS. Adding ESS/TSS to 1 would yield a value greater than 1 if ESS/TSS is positive, which is impossible for R^2, since R^2 should lie between 0 and 1 in ordinary least squares contexts. This formulation misrepresents how the explained variation relates to the total variation.

Option 3: 1 + RSS/TSS. Similar to Option 2, this would produce values larger than 1 when RSS is positive, which contradicts the conventional 0 to 1 range for R^2. It also conceptually mixes a sum of squares ratio with an additive constant in an inappropriate way for R^2.

Option 4: ESS/RSS. This ratio compares explained variation to residual variation, rather than to total variation. It does not yield a dimensionless proportion of explained variance of the total variance, and it does not match the standard definition of R^2.

Additional note: In multiple contexts, adjusted R^2 or other metrics modify the basic form, but the classic R^2 definition remains 1 minus the ratio of RSS to TSS. The options here do not reflect those adjusted forms, except for Option 1 which matches the fundamental relationship between RSS, TSS, and explained variation.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 12. Quiz for Class 12

R-Square question : What is R2 we define total sum of squares (TSS) the explained sum of squares (ESS) the residual sum of squares (RSS)

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