Can adjusted R-squared be negative?

Yes, when the degree-of-freedom-adjusted residual variation exceeds the response's sample variation.

Is it always below ordinary R-squared?

Under the standard intercept formula with positive residual degrees of freedom, it is no greater.

Does adding one business variable always add one parameter?

No. A categorical variable or interaction can add several fitted columns.

Is it the same as cross-validated R-squared?

No. Cross-validated scores use predictions for held-out observations and have different construction and interpretation.

AI Glossary / Time series and signal processing / Intermediate

R-bar-squared (R̄2 )

Definition

R-bar-squared, usually called adjusted R-squared, is a regression fit statistic that modifies ordinary R-squared to account for the model's residual degrees of freedom. It penalizes adding predictors that produce too little improvement in fit. It is useful for comparing suitable regression models, but it is not a direct measure of future forecast accuracy.

What it means

Ordinary R-squared cannot decrease when extra predictors are added to a nested ordinary least-squares model on the same data. Even irrelevant predictors can improve the training fit a little. Adjusted R-squared asks whether that improvement is sufficient relative to the additional fitted parameters.
The adjustment changes the interpretation. It should not be presented simply as the literal percentage of observed variation explained, and it can be negative. A higher value suggests a better fit-complexity balance under its convention, not proof of causal explanation, correct specification, or successful deployment.

How it works

For an ordinary full-rank least-squares regression with an intercept, n observations, and p predictors excluding the intercept, adjusted R-squared is 1 - [SSE/(n-p-1)]/[SST/(n-1)]. SSE is the residual sum of squares, while SST is the sum of squared deviations of the response from its sample mean.
Equivalently, it is 1 - (1-R-squared)(n-1)/(n-p-1). Positive residual degrees of freedom and nonzero response variation are needed for this standard expression. For the same response observations, maximizing it is equivalent to minimizing the residual mean square.
Count fitted columns appropriately, including dummy variables and interactions rather than merely named input fields. Rank deficiency, models without an intercept, weighting, or other model families require attention to the software's definitions. The familiar intercept-based formula should not be applied blindly to every reported fit statistic.

Three practical examples

An analyst compares two regression models fitted to the same demand observations and checks whether an added predictor improves adjusted as well as ordinary R-squared.
A business reviewer spots a negative adjusted value and asks whether the model's complexity is supported by the available sample.
A forecasting team reports adjusted R-squared as a descriptive training statistic while using later forecast errors and uncertainty checks to assess operational performance.

Think of it this way

Think of scoring a construction design by how much space it provides after accounting for the resources needed to build it. Adding another room increases floor area, but the extra materials may not justify the gain. Adjusted R-squared similarly rewards fit only after an explicit allowance for fitted complexity.

Worked example

For an original comparison, stipulate two nested, full-rank least-squares models with intercepts fitted to the same 20 observations. The response's total sum of squares is 100. Model A has two predictors and residual sum of squares 30. Model B adds a third predictor and reduces residual sum of squares to 29.
Ordinary R-squared increases from 1 - 30/100 = 0.70 to 1 - 29/100 = 0.71. Model A has 20 - 2 - 1 = 17 residual degrees of freedom; Model B has 16.
Adjusted R-squared for A is 1 - (30/17)/(100/19) = 0.664706. For B it is 1 - (29/16)/(100/19) = 0.655625. The larger model improves ordinary R-squared but worsens adjusted R-squared because its residual mean square rises.
For B to improve the adjusted statistic, its SSE would need to be below 30 times 16/17, approximately 28.235294. Its value of 29 does not meet that threshold. Exact fraction calculations independently verify the rankings and threshold. These are stipulated compatible fit summaries, not empirical evidence that a particular predictor lacks business value.

Illustrative case study

Fictional case: Silver Fern Logistics receives a model proposal that advertises a small increase in R-squared after adding several calendar variables. The review team asks for the number of fitted columns, adjusted R-squared, and performance on later forecast periods.
The extra fit does not by itself justify the additional complexity, so the team keeps both candidates under evaluation. It also checks whether the calendar variables will be known when forecasts are issued. This invented case illustrates a review process, not a claim that adjusted R-squared alone determines which model should be used.

Common misconceptions and mistakes

Do not treat adjusted R-squared as a percentage forecast accuracy score. A value of 0.8 does not mean that 80% of future predictions will be correct or within an acceptable tolerance.
Do not compare values from different response samples, response transformations, or incompatible definitions without justification. Do not assume a high value establishes causality or removes problems from correlated errors and nonstationary time series.
Do not assume its penalty prevents overfitting. Repeated model search can still exploit accidental patterns, and the criterion may favor too many predictors for forecasting.

Business use

Adjusted R-squared can make regression reporting more honest by placing a small training-fit gain alongside its complexity cost. It is particularly helpful when stakeholders might otherwise interpret any increase in ordinary R-squared as improvement.
Report the sample size, response definition, intercept convention, predictor count or design rank, and later forecast performance. Inspect residuals and evaluate decisions at relevant horizons. Use additional selection criteria and time-appropriate validation when the goal is forecasting, rather than optimizing this single training statistic.

Check your understanding

Which example model has higher ordinary R-squared?

Answer: Model B, at 0.71.

Which has higher adjusted R-squared?

Answer: Model A, approximately 0.664706.

Why does the ranking reverse?

Answer: B's small SSE reduction does not compensate for losing one residual degree of freedom.

Does A's higher adjusted value guarantee lower future error?

Answer: No; later predictive evaluation remains necessary.

Related terms

Coefficient of determination
Degrees of freedom
Model selection
Overfitting
adjusted R-squared
residual sum of squares
AIC
BIC
regression diagnostics

Learn next

compare ordinary and adjusted R-squared as predictors are added to a fixed regression sample. Record residual degrees of freedom and evaluate the same candidates on later observations to see whether training rankings persist.

Sources

Sources: statsmodels RegressionResults.rsquared_adj documentation: https://www.statsmodels.org/stable/generated/statsmodels.regression.linear_model.RegressionResults.rsquared_adj.html . Hyndman and Athanasopoulos, selecting predictors: https://otexts.robjhyndman.com/fpp3/selecting-predictors.html . Hyndman glossary: https://robjhyndman.com/mwh3/FG4.pdf . Primary indexed formulas and limitations consulted. Example and case are original.

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