A researcher adds 15 additional control variables to a regression, and R² rises from 0.41 to 0.68. A colleague says this proves the new model is better. What is wrong with this reasoning?
ANothing — higher R² always indicates a better model, since more variation is explained
BR² mechanically rises when variables are added, even irrelevant ones; in-sample fit improvement says nothing about out-of-sample prediction or causal validity
CThe colleague should have used adjusted R² only if the added variables were categorical
DR² above 0.5 is a sign of overfitting, so the original model was preferable
OLS minimizes residuals for the sample in hand, so adding any variable — even pure noise — can never make R² decrease. The rise from 0.41 to 0.68 may partly reflect genuine explanatory power, but could also reflect overfitting. To evaluate whether the additions improve the model, you need adjusted R² (which penalizes for added parameters) or out-of-sample validation. R² alone cannot distinguish a better model from a more overfitted one.
Question 2 Multiple Choice
A randomized controlled trial estimates the effect of a job training program on earnings with R² = 0.04. An observational study of the same program achieves R² = 0.71 by including many demographic controls. Which estimate is more causally trustworthy?
AThe observational study — its R² is far higher, meaning the model fits the data much better
BThe RCT — randomization eliminates confounding, making the treatment effect estimate unbiased regardless of R²
CThey are equivalent — both report regression estimates, so the causal validity is the same
DThe observational study — more controls always reduce omitted variable bias
R² measures how much variation in the outcome the model explains — a purely descriptive statistic. A low-R² RCT produces an unbiased estimate of the causal effect because randomization ensures treatment assignment is independent of all confounders, observed or not. The high-R² observational estimate may still be heavily biased if selection into the program is correlated with unobserved characteristics. Causal validity comes from the identification strategy, not from R².
Question 3 True / False
Adding a variable to an OLS regression model can never decrease R².
TTrue
FFalse
Answer: True
True. OLS finds the coefficients that minimize the residual sum of squares for the given sample. With an additional variable, the model has more flexibility — at worst, it sets the new coefficient to zero and achieves the same fit as before. It cannot do worse. This is why R² always weakly increases with more variables, making it an unreliable criterion for model selection (unlike adjusted R² or information criteria like AIC/BIC, which penalize for complexity).
Question 4 True / False
A regression model with R² = 0.90 produces coefficient estimates that are more likely to be unbiased than a model with R² = 0.30.
TTrue
FFalse
Answer: False
False. R² is a measure of descriptive fit — how much of the outcome's variance is explained in-sample. Unbiasedness of coefficient estimates depends on whether the model's identification assumptions hold: no omitted variables correlated with regressors, no reverse causality, no measurement error. A model with R² = 0.90 can be severely confounded, producing biased estimates. A model with R² = 0.30 from a clean randomized experiment produces unbiased estimates. These two properties are independent.
Question 5 Short Answer
Why is R² an inadequate criterion for evaluating the causal validity of a regression model, and what should researchers care about instead?
Think about your answer, then reveal below.
Model answer: R² measures only how much of the in-sample variation in Y is explained by the regressors — a descriptive property. A model can have high R² while producing badly biased coefficient estimates if important confounders are omitted or if regressors are endogenous. Causal validity depends on the identification strategy: whether assignment to the key regressor is as good as random (RCT, instrumental variables, regression discontinuity, etc.). Econometricians prioritize consistency and unbiasedness of estimates over fit, because a precise answer to the wrong question is worse than a noisy answer to the right one.
This is why econometrics puts such emphasis on endogeneity, omitted variable bias, and instrumental variables — these are threats to causal identification that R² cannot detect. An R² near zero can be perfectly acceptable in a well-identified causal study; an R² near one in an observational study should raise suspicion that the model is either overfitted or confounded by variables correlated with both Y and the regressors.