What Is the Relationship Between R² and Adjusted R²? Why Can Adjusted R² Sometimes Be Less Than 0?
Everyone who has run a regression has seen these two numbers side by side in Stata output:
1. Introduction: R² Has a "More Honest Younger Brother"
Everyone who has run a regression has seen these two numbers side by side in Stata output:
R-squared = 0.3521
Adj R-squared = 0.3387
Adjusted R² is always slightly smaller than R². If the model has many variables and the sample size is small, it will be much smaller. In some extreme cases, it can even become negative.
What exactly does this "adjustment" adjust? Why is it needed? Behind the emergence of adjusted R² lies a fundamental flaw of R² itself—R² is an "optimistic" statistic: as long as you add variables to the model, R² will never decrease, even if what you add is pure noise.
Adjusted R² is the remedy for this flaw. It puts a "price" on model complexity—for every additional parameter used to "explain" the data, you must pay a price in terms of goodness of fit. If the explanatory power brought by a new variable is insufficient to cover this cost, adjusted R² will decline, or even fall into negative territory.
Core message: R² = explained variation / total variation, which is inherently biased toward complex models—adding variables means R² never falls. Adjusted R² "taxes" model complexity through degrees of freedom—each added variable reduces the residual degrees of freedom by 1, and you need higher explanatory power to compensate. When a model's explanatory power is extremely low and its complexity is extremely high, adjusted R² can be negative.
2. The "Optimistic Bias" of R²—Why Is Adjustment Needed?
2.1 The Exact Formula for R²
SSR is the residual sum of squares. Adding a variable—even one completely unrelated to Y (with a population coefficient of zero)—will, in this particular sample, "happen" to be slightly correlated with Y due to sampling error, thereby shaving off a small amount of SSR.
The decline in SSR is inevitable, regardless of whether the new variable has any real explanatory power. Therefore, the rise in R² is also inevitable.
2.2 An Extreme Example: Fitting 10 Data Points with 9 Independent Variables
k = 9, n = 10. Throw 9 independent variables into the model, and OLS will fit these 10 points perfectly—the residual sum of squares SSR = 0, and R² = 1.
But this does not mean you have "explained" Y. You have merely used enough parameters to "memorize" these 10 data points. With a different sample, the same 9 variables might yield an R² that falls to 0.3.
R² cannot distinguish between "genuine explanatory power" and "overfitting to sample noise."
3. Adjusted R²—Attaching a "Price Tag" to Every Parameter
3.1 The Formula
3.2 What Is It Doing?—Replacing "Sum of Squares" with "Mean Squares"
The ordinary R² directly compares two sums of squares (SSR and SST), without considering the degrees of freedom associated with each.
Adjusted R² divides both SSR and SST by their respective degrees of freedom, converting them into mean squares:
- SSR / (n - k - 1): the unexplained variation per residual degree of freedom (the noise mean square, i.e., ).
- SST / (n - 1): the total variation per total degree of freedom.
The ratio of the two measures—in the sense of "average per degree of freedom"—the proportion of variation the model fails to explain relative to the original variation.
3.3 Why Is Adjusted R² < R²?
This can be seen more clearly from another equivalent formula:
The key lies in the last factor: .
When k > 0 (at least one independent variable in the model), n - 1 > n - k - 1 → this ratio is greater than 1. Therefore:
This factor > 1 is the "complexity tax"—the larger k is, the heavier the penalty. Each additional independent variable reduces the denominator n - k - 1 by 1, making the penalty factor larger.
4. Why Can Adjusted R² Be Less Than 0?
4.1 The Mathematical Condition
Adjusted R² < 0 if and only if:
That is, the model's R² falls below the extremely low threshold of k/(n-1).
4.2 An Example
n = 30 observations, k = 10 independent variables. k/(n-1) = 10/29 ≈ 0.345.
If these 10 variables yield an R² of only 0.20 (below 0.345), then:
Adjusted R² = -0.22. What does this mean? In the sense of "average per degree of freedom," the model's residual noise mean square is greater than the mean square of the original Y—indicating that these 10 variables not only failed to help, but actually worsened the average noise level by consuming too many degrees of freedom.
4.3 When Do You Actually Encounter Negative Adjusted R² in Practice?
- Extremely low n/k ratio: For example, n=20, k=8. You are using 8 parameters to chase 20 data points.
- Independent variables have almost no association with Y: You have included a bunch of pure noise variables (such as randomly generated numbers, or survey questions completely unrelated to Y).
- Although R² > 0, it is only slightly above 0: Pulled into negative territory by the degrees-of-freedom penalty.
4.4 How Do Software Packages Typically Handle This?
Software such as Stata does not report adjusted R² as a negative number—if the computed value is less than 0, it is typically reported as 0 (or simply flagged as < 0). This is technically a truncation, not an indication that it cannot happen mathematically. In the original formula, adjusted R² can be negative. Stata's truncation is based on the consideration that "if a model's explanatory power is worse than noise, then from an inferential perspective the model contains no explainable information."
5. R² vs. Adjusted R²—When to Use Which?
| Use Case | Should You Use R² or Adjusted R²? | Reason |
|---|---|---|
| Describing in-sample fit of a single model | Either R² or adjusted R² | Both convey useful information: R² gives the absolute fit, adjusted R² gives the penalized fit |
| Comparing nested models (same sample, different numbers of independent variables) | Adjusted R² | R² favors complex models; adjusted R² penalizes complexity—more fair |
| Comparing models across different samples (same Y) | Adjusted R² | R² is affected by sample size and model complexity; adjusted R² normalizes for these |
| Causal identification research—reporting core results | Neither is the focus | Causal identification does not rely on R² to support credibility |
| Prediction models—variable selection | Adjusted R² or cross-validated RMSE | Do not use R² for variable selection—it will lead you to include too many noise variables |
6. In Empirical Research, What Kinds of R² Values Might Be Problematic?
R² itself has no "correct" range of values. But in the context of empirical research, certain R² values do warrant further scrutiny.
6.1 Micro Survey Data with R² Above 0.7—Anomalous
If you are analyzing individual-level outcomes such as wages, happiness, or academic performance using household survey data, R² rarely exceeds 0.3–0.5. Human behavior contains a large amount of unobservable heterogeneity—personality, life experiences, genes, chance events—which cannot be fully captured by the dozen or so variables you list on a questionnaire.
When you see R² > 0.7 in micro data, one of the following problems is likely present:
- You have controlled for a lag of Y ()—this is legitimate in dynamic panels, but it will substantially inflate R², and what it reflects is mainly the serial correlation of Y rather than the explanatory power of the independent variables.
- You have controlled for mediators or reverse-causal variables—these variables "absorb" a large amount of variation, but at the cost of stripping your core coefficients of clear causal meaning.
- Your dependent and independent variables contain components of an identity—for example, Y = total expenditure and X = categorical expenditures (food expenditure, housing expenditure, etc.). In this case, an R² close to 1 is not a discovery but an accounting identity.
6.2 Time Series Regressions with R² > 0.90—Check for Stationarity
In macro time series regressions, high R² is often the result of two non-stationary variables drifting along the same trend line. GDP and per-capita telephone ownership both grew substantially between 1980 and 2020—their regression R² will exceed 0.95. But this does not mean "more phones → better economy"; it simply reflects that both variables rise over time.
In time series, a high R² is a signal to check for unit roots and conduct cointegration tests, not a signal of model quality.
6.3 Very Low Within R²—Usually Not a Problem in Panel Data
As discussed earlier, a within R² between 0.01 and 0.10 in a fixed effects model is a very common range—this does not indicate a problem with the model. Rather, after fixed effects remove between-unit variation, the remaining "within-unit year-to-year fluctuations" inherently have a larger noise ratio. Compared to the R² of cross-sectional OLS regression, the expected value should naturally be lower.
6.4 R² Equal to 0 or Very Close to 0—Pay Attention to Sample Size
If n is very small (n < 30) and you have a sufficient number of independent variables, R² may still be 0—especially when there is severe multicollinearity among the independent variables. Seeing R² = 0.000 in the output does not mean "the effect is indeed zero"; it may mean "the model fit has completely failed." Check the sample size, check whether the F-test is significant, and check whether there is perfect linear dependence among the independent variables.
6.5 R² Jumps from 0.2 to 0.8 After Adding a Certain Variable—Ask "Why"
A single variable causing a massive jump in R² indicates that this variable has an extremely strong correlation with Y—strong enough to warrant asking: "Is this a causal relationship, or is it tautological?"
7. Summary: Three Formulas + One Judgment Principle
One judgment principle: R² is your model's "report card" on this sample—but it is a report card that looks only at "absolute accuracy." Adjusted R² is a report card that "deducts for lucky guesses." If the adjusted score is negative, it means your model's "explanatory power" is worse than pure noise—your independent variables combined are no better than a handful of random numbers.
One sentence to close:
"R² never minds you adding more variables—it welcomes all comers, and each addition makes it a little happier. Adjusted R² is the cool-headed accountant—for every variable you add, it records a 'complexity tax' in the ledger. When the tax exceeds the benefit, adjusted R² falls below zero—it is telling you: stop adding. You are just memorizing data, not learning regularities."
8. Presentation Suggestions for Bilibili/WeChat Official Account
- Bilibili video: Consider using a "balance scale animation" to demonstrate the relationship between R² and adjusted R²—on the left side, R² rises a bit each time a variable is added (SSR declines); on the right side, adjusted R² first adds a weight to the "complexity tax" end in the same step, then looks at the net benefit. When the complexity tax exceeds the benefit from the SSR decline, adjusted R² drops.
- WeChat Official Account: For the mathematical derivation and example of adjusted R² < 0, it is recommended to demonstrate with numerical substitution. The four anomalous R² scenarios (high R² in micro data, high R² in time series, low within R², and R² jumps) are best presented as a comparison card of "anomaly signal × what questions to ask."
- Recommended titles:
- Main title: 《What Is the Relationship Between R² and Adjusted R²? Why Can Adjusted R² Sometimes Become Negative?》
- Alternative title: 《R² Never Minds How Many Variables You Add—Adjusted R² Is the Cool-Headed Accountant》
- Key quote:
"R² welcomes all comers—each variable added makes it a little happier. Adjusted R² is the cool-headed accountant—for every variable added, it records a 'complexity tax.' When the tax exceeds the benefit, it falls below zero, telling you: stop adding. You are memorizing data, not learning regularities."