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Resources › Econometrics › 4. R² and adjusted R²
Econometrics · Lesson 4 of 18

R², adjusted R² and comparing models

R² is the most quoted number in a regression table and the most misread. It says how much of the variation in y the regressors account for. It does not say the model is right, the coefficients are unbiased, or that adding another variable was a good idea. This lesson shows exactly what R² measures, why it can only go up as variables are added, how adjusted R² corrects that, and when two R² values may be compared at all.

← Econometrics moduleLesson 4 of 18 · about 45 minutes
1Multiple regression2Matrix form and Gauss-Markov3t and F tests4R² and adjusted R²5Dummies and interactions6Functional forms7Misspecification8Multicollinearity9Heteroscedasticity10Autocorrelation11Instrumental variables12Simultaneous equations13Panel data14Logit and probit15Unit roots16ARMA, VAR and cointegration17Test Atlas18Hard problems
A question to hold on to
Which of the following statements describe(s) the relationship between R² and adjusted R² (written R̄²)? (A) If R² = 1, then R̄² = 1. (B) If R² = 0, then R̄² can be negative. (C) If R² = 1, then R̄² = 0. (D) If R² = 0, then R̄² = 1.
IIT JAM 2025Solved at the end ↓
New here?What is the Ashanagar survey?The invented household survey behind this page's examples: 600 households, generated from a model whose true coefficients we publish.

The Ashanagar survey is invented. The statistics modules used one district of 60 households. Econometrics needs more households and more variables, so this is a new, larger survey of 600 households in the same imaginary district: income, spending, household size, whether the household is in the town or a village, the head’s schooling and earnings.

Because we wrote the data-generating process, we know the true answers, and every lab can show whether a regression recovers them. Household spending, in thousands of rupees a month, was generated as 1 + 0.45 × income + 0.9 × household size + 1.5 × town + an error with standard deviation 2.5. A real survey never tells you its true coefficients. This one does, so you can see how close an estimate comes.

It also holds a variable, the head’s ability, that a real survey could not observe. It is kept so that the lessons on omitted variables can show what happens when it is left out. Every figure here is illustrative, not official statistics.

Households
600
Share in the town
41%
Mean monthly income
₹15.4 thousand
Mean monthly spending
₹12.6 thousand
Mean schooling of the head
8.2 years

How the module uses it →

In this guide
Lesson · about 45 minutes
  1. Splitting the variation
  2. R² never falls
  3. Adjusted R²
  4. Comparing models
  5. R² is not the point
  6. The opening question, solved
  7. The lesson on one screen
  8. Check yourself
  9. Past papers, by level
01
The idea

Splitting the variation

Every household’s spending differs from the average, ȳ. Some of that gap is accounted for by the regression, the part ŷ − ȳ, and some is left over, the residual e. Square and add over all households and the two parts do not interfere, because the residuals are orthogonal to the fitted values (lesson 1):

The variation splits, and R² is a share
SST∑(y−yˉ​)2​​=SSR∑(y^​−yˉ​)2​​+SSE∑e2​​,R2=SSTSSR​=1−SSTSSE​
SST: total, SSR (also called ESS, the explained sum of squares): explained, SSE (also RSS, the residual sum of squares): left over. With an intercept, R² lies between 0 and 1.

For the survey regression of spending on income, household size and town, SST = 12,313.3, SSR = 8,395 and SSE = 3,918.2, so R² = 8,395 ÷ 12,313.3 = 0.682. The three regressors account for about 68% of the variation in spending. The leftover share, 1 − R² = 0.318, is sometimes called the coefficient of non-determination.

Four facts to carry into the exam:

  • R² is a squared correlation. It equals the squared correlation between y and the fitted values: 0.682 here. In a simple regression it is the square of r between y and x, so r = 0.6 gives R² = 0.36.
  • R² = 0 exactly when every slope estimate is zero. The fitted line is flat at ȳ (GATE 2025 asks it for a slope of 0). The intercept is then ȳ, not zero.
  • R² is unit-free. Rescaling y or any x leaves R² unchanged. That is why it can be compared across variables measured in different units.
  • The degrees of freedom of the residuals are n − k − 1, in a regression with k slopes and an intercept (CUET 2023).
02
Interactive

R² never falls

Add any variable to a regression, however useless, and R² cannot go down. The reason is simple: least squares chooses the coefficients that make SSE as small as possible. With a new regressor it can always set the new coefficient to zero and recover the old fit, so its best SSE is at most the old one. And a new variable that is only noise usually fits some of the sample’s random wobble, so R² rises.

Here is the proof by experiment. y really depends on one variable. Add columns of pure noise, one at a time, and watch both R² and adjusted R².

R²0.094
Adjusted R²0.054
t of the last one addednone yet
|t| above 1?none yet
Adjusted R² rose?none yet
246810121416182022−0.6−0.4−0.20.20.40.60.81

Horizontal: the number of pure-noise regressors added. Vertical: R² and adjusted R².

R²Adjusted R²
What it shows

With 0 junk regressors added, R² is 0.094, up from 0.094 for x alone, although nothing real was added. Slide to 22 (the most this sample allows) and R² nears 1: with enough columns, least squares can explain any data. Adjusted R² is 0.054. It does not climb with the junk. Step through one at a time: it goes up exactly when the last regressor’s |t| is above 1, and down when it is below. R² cannot tell junk from signal. Adjusted R² charges for every column it uses, so it rises only when a column earns its place.

Push the slider all the way. With n − 3 junk columns plus x and the intercept, the regression has n − 1 parameters, and R² approaches 1: with enough regressors, least squares can match any sample exactly. A high R² built from many regressors on few observations is evidence of nothing.

Watch out for

The mistakes examiners are fishing for

  • Judging a model by R² alone. More regressors always give a higher R². Compare models with adjusted R² or with F tests of the added variables.
  • Saying R² “decreases” when a variable is dropped and calling it worse. It does fall, by the amount the variable explained of the sample, whether or not it belongs.
03
The correction

Adjusted R²

Adjusted R² charges a price for every parameter. It replaces the two sums of squares by estimates of variance, each divided by its degrees of freedom:

Adjusted R²
Rˉ2=1−SST/(n−1)SSE/(n−k−1)​=1−(1−R2)n−k−1n−1​
k is the number of slopes here, so n − k − 1 is the residual degrees of freedom. Some papers write k for all parameters including the intercept, and then the formula reads n − k. Read what k means before you substitute.

Four properties, each of which has been asked:

  • It is never above R². The factor (n − 1) ÷ (n − k − 1) is at least 1, so R̄² ≤ R², with equality only when k = 0 or R² = 1.
  • It can be negative. If R² = 0, R̄² = 1 − (n − 1) ÷ (n − k − 1), which is negative for any k ≥ 1. A regression that explains less than chance would is penalised below zero (IIT JAM 2022 and 2025).
  • If R² = 1, R̄² = 1. The numerator 1 − R² is zero, so nothing is subtracted.
  • It does not always rise when a good variable is added. The rule is exact, and it is what the lab showed: adding a regressor raises R̄² if and only if its t statistic exceeds 1 in absolute value.
Derive it

Adjusted R² rises exactly when the added variable has |t| above 1

We want to show
Adding one regressor raises adjusted R² if and only if the t statistic on the new regressor is larger than 1 in absolute value.
  1. 1
    Adjusted R² as a ratio of variances.

    With k coefficients (the intercept counts), adjusted R² compares the estimated error variance with the variance of y:

    Rˉ2=1−SST/(n−1)SSE/(n−k)​

    SST and n do not change when a regressor is added. So adjusted R² rises exactly when SSE ÷ (n − k) falls.

  2. 2
    State the condition.

    Let the old model have sum of squared residuals SSE and k coefficients, and the new one SSE′ and k + 1. Adjusted R² rises when

    n−k−1SSE′​<n−kSSE​
  3. 3
    Rearrange it into an F.

    Multiply out and rearrange: SSE ÷ (n − k) > SSE′ ÷ (n − k − 1) is the same as (SSE − SSE′) > SSE′ ÷ (n − k − 1), that is

    SSE′/(n−k−1)SSE−SSE′​>1

    The left side is exactly the F statistic for adding one regressor.

  4. 4
    And F for one regressor is t squared.

    By the previous lesson’s result, that F equals the square of the new regressor’s t. So the condition is t² > 1, that is |t| > 1. Check it on the survey: adding the head’s schooling to the spending regression gives it a t of 0.086, so |t| is below 1, and adjusted R² goes from 0.68019 to 0.67965, which is a fall, as the rule says. The same number from the formula: 1 − (1 − 0.68179) × 599 ÷ 595 = 0.67965.

Step 1 of 4
So
Adjusted R² rises when |t| of the new variable exceeds 1, a much weaker standard than the |t| above about 2 needed for significance at 5%. A variable can therefore raise adjusted R² and still be insignificant. R² itself always rises, because SSE′ is never larger than SSE.
Pause and predict
A regressor has t = 1.4, which is not significant at 5%. If it is added to the model, what happens to adjusted R²?
04
Choosing between models

Comparing models

Four survey regressions of spending, each adding one variable to the last:

RegressorsParameters, k + 1R²Adjusted R²SSE
Income20.60590.60534,852.3
Income and size30.6650.66394,124.5
Income, size and town40.68180.68023,918.2
Income, size, town and schooling50.68180.67973,918.2

R² climbs at every step, and it must. Adjusted R² climbs from the first to the third model, where the added variables earn their place (size and town), and the fourth step, schooling, adds almost nothing: its t is 0.09, so adjusted R² moves from 0.6802 to 0.6797.

Rules for comparing R². Two models’ R² (or adjusted R²) can be compared only if they have:

  • The same dependent variable, in the same form (JNU 2021). A regression of ln y and a regression of y explain different things, and their R² values measure different variation. Compare them by converting predictions back to the same scale, not by their R².
  • The same observations. If one model drops incomplete rows, its sample differs.
  • The same treatment of the intercept. A regression through the origin reports an uncentred R², a different statistic.

Beyond adjusted R², economists use information criteria (Akaike and Schwarz, often written AIC and BIC), which add a penalty for each parameter to a measure of lack of fit and are smaller for better models. The Schwarz penalty is heavier, so it picks smaller models. You need to know that they exist and what they do, but the exams in the bank ask only for the adjusted R².

Watch out for

The mistakes examiners are fishing for

  • Comparing R² of a log model with R² of a level model. They are not comparable, even though both are called R².
  • Choosing the model with the highest adjusted R² as if it were a test. It is a rule of thumb that errs towards keeping weak variables (it rises at |t| > 1). Test the variables you doubt with an F test (lesson 3).
05
A warning

R² is not the point

A high R² is neither necessary nor sufficient for a good regression. Three reasons it misleads:

  • It says nothing about bias. In the survey, the head’s earnings regressed on schooling and town has R² = 0.779. Add the head’s ability, and R² rises to 0.891, while the schooling coefficient falls from 1.7 to 1.3, close to the true 1.3. The model with the lower R² had the biased schooling coefficient. A high R² did not protect it (lesson 7).
  • It rises mechanically with trending variables. Two unrelated series that both grow over time have an R² close to one. This is spurious regression (lesson 15).
  • A low R² is not an error. Cross-section regressions of individual behaviour often have R² of 0.1 or 0.2 because people differ for reasons no survey records. The coefficients can still be precisely estimated and well worth knowing.
What to say in an exam

R² measures fit, not validity. State what it is (the share of the variation in y that the regressors explain), what it ignores (bias, causality, the right functional form), and that adjusted R² adds only a penalty for the number of regressors.

06
Solving the opening question

The opening question, solved

IIT JAM 2025 · hard
Which statements describe the relationship between R² and adjusted R²? (A) R² = 1 gives R̄² = 1. (B) R² = 0 allows a negative R̄². (C) R² = 1 gives R̄² = 0. (D) R² = 0 gives R̄² = 1.
  1. 1
    Write the formula.
    R̄² = 1 − (1 − R²)(n − 1) ÷ (n − k − 1), with n − k − 1 less than n − 1 whenever there is at least one slope.
  2. 2
    Test R² = 1.
    The bracket is 0, so R̄² = 1 − 0 = 1. Statement A is true and C is false.
  3. 3
    Test R² = 0.
    R̄² = 1 − (n − 1) ÷ (n − k − 1), and the fraction is more than 1, so R̄² is negative. With n = 30 and k = 3, for instance, it is 1 − 29 ÷ 26 = −0.115. Statement B is true and D is false.
Answer
A and B. The surprise to remember is that adjusted R² is not confined to the interval from 0 to 1, whereas R² (with an intercept) is.
In economics

Goodness of fit is rarely the objective in economics. A forecasting model is judged on how well it predicts new data, where R² on the estimation sample is a poor guide (it only rises as variables are added). A policy model is judged on whether its coefficient can be read as an effect, which has nothing to do with R². Report R², but argue from the identification of the coefficient you care about.

07
Revise

The lesson on one screen

The lesson on one screen
Splitting
SST = SSR + SSE (explained and residual sums of squares). R² = SSR ÷ SST = 1 − SSE ÷ SST.
R² facts
Squared correlation of y and ŷ. In simple regression, r². Unit-free. Zero exactly when every slope is zero. Between 0 and 1 with an intercept.
Never falls
Adding any regressor cannot lower R². With n − 1 parameters R² = 1.
Adjusted R²
1 − (1 − R²)(n − 1) ÷ (n − k − 1). At most R², can be negative, equals 1 when R² = 1.
When it rises
Adding a regressor raises adjusted R² exactly when its |t| > 1.
Comparing
Same dependent variable (same form), same observations, same treatment of the intercept. Not a log model against a level model.
Information criteria
AIC and BIC add a parameter penalty to lack of fit, and BIC's is heavier.
Warning
R² is about fit. It does not detect bias, spurious trends, or the wrong functional form.
08
Practice

Check yourself

0 of 13 correct so far

Compute

R² from the sums of squares
SSE = 30 and SST = 120. Find R².
Adjusted R²
R² = 0.60 with 26 observations and 3 slopes (plus an intercept). Find adjusted R².
A negative adjusted R²
R² = 0.05 with 20 observations and 4 slopes (plus an intercept). Find adjusted R².
R² from a correlation
In a simple regression the correlation between y and x is −0.6. Find R².
The residual sum of squares
SST = 200 and R² = 0.35. Find SSE.
Does the new variable earn its place?
A model with 3 parameters (including the intercept) fitted to 44 observations has SSE = 100. Adding one regressor lowers SSE to 96. Find the F statistic for adding it, and check whether it exceeds 1.
A large sample
R² = 0.5, n = 101 and one slope. Find adjusted R².
The regression of y on its residuals
A regression with an intercept has R² = 0.8. If y is regressed on the residuals e of that regression, what is the R²?

Quick questions

1When a regressor is added to a regression on the same data, R²
2Which of these can be negative?
3To compare the R² of two regressions, they must have the same
4In a regression with an intercept, R² = 0 means that
5Adding a regressor raises adjusted R² if and only if its t statistic is, in absolute value,
09
Real questions

Past papers, by level

Foundation

Definitions and one-step calculations: a t statistic, an R², which assumption is which. Most CUET and JNU questions sit here.

2 questionsPractise →

Standard

Restricted against unrestricted regressions, reading output, and statement questions, up to GATE and IIT JAM level.

6 questionsPractise →

Delhi School level

This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 18: Tests and goodness of fit from a regression table.

5 questionsLesson 18 →
All 8 past-paper questions for this lesson, easiest first: practise the full set →

✦These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

The Delhi School’s questions on this material (an adjusted R² from a home-sales table, an explained sum of squares from a dummy regression) are in the hard-problems lesson at the end of the module. The squared correlation and the simple-regression case are in the regression primer.

← Lesson 3t tests, F tests and reading regression outputLesson 5 →Dummy variables, the dummy trap and interactions

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