Econometrics · Lesson 3 of 18

t tests, F tests and reading regression output

An estimated coefficient is only a number until you know how far it could be from the truth. This lesson turns the standard errors of lesson 2 into decisions: is one coefficient different from zero, are several different from zero together, and does adding a variable improve the fit enough to be worth keeping? It also trains the skill that most exam questions test: given a regression table with a number missing, find it.

← Econometrics moduleLesson 3 of 18 · about 60 minutes

A question to hold on to

Two regression models are estimated on 350 observations: (1) y = β₀ + β₁x₁ + β₂x₂ + β₃x₃ + β₄x₄ + β₅x₅ + u, and (2) y = α₀ + α₁x₁ + α₂x₂ + v. The R² of model (1) is 0.3521 and of model (2) is 0.2314. What is the value of the test statistic for H₀: β₃ = β₄ = β₅ = 0? (Three decimal places.)

GATE 2021Solved 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 →

Testing

One coefficient: the t test and its interval

Lesson 2 gave the variance of every coefficient. Replace σ² by its estimate s² = SSE ÷ (n − k − 1) and the square root is the standard error. Under assumptions 1 to 6, the standardised coefficient follows a t distribution with n − k − 1 degrees of freedom, where k is the number of slopes:

The t test and interval for one coefficient

t = \frac{b _{j}}{SE ( b _{j} )} \sim t_{n - k - 1}, b_{j} \pm t_{α /2, n - k - 1} SE (b_{j})

Reject H₀: β_j = 0 when |t| exceeds the critical value, equivalently when the interval excludes zero. To test β_j = c, use (b_j − c) ÷ SE.

In the Ashanagar survey regression of spending on income, household size and town, the coefficient on household size is 0.904 with standard error 0.076. So t = 0.904 ÷ 0.076 = 11.91. With 596 degrees of freedom the 5% critical value is 1.964, so size is highly significant. The 95% interval is 0.904 ± 1.964 × 0.076, which is 0.76 to 1.05, and it contains the true 0.9 we built in.

With more than about a hundred degrees of freedom the critical values of t are almost the normal ones: 1.96 at 5%, 2.58 at 1%, 1.65 at 10%. That is why exam questions on large samples quote “2” as the rule of thumb: a t ratio above about 2 in absolute value is significant at 5%. A table that reports t ratios (or stars) for each coefficient lets you read significance off directly (IIT JAM 2024).

Watch out for

The mistakes examiners are fishing for

Using n − 2 degrees of freedom. With several regressors it is n − k − 1 (Delhi School 2019: with 107 observations and six slopes, 100).
Confusing significant with large. A coefficient can be significant and economically tiny in a big sample, or large and insignificant in a small one. The t ratio measures precision, not importance.
Reading “insignificant” as “zero”. It means the data cannot rule zero out. The interval shows the range of values the data cannot rule out, which is often wide.

Interactive

Reading regression output

Every package prints the same things: the coefficients with their standard errors, t ratios and p-values, then R², adjusted R², and an analysis-of-variance table with the overall F. Here is the survey regression laid out that way. In Explain a figure mode, tap any number to see where it comes from. In Work it out mode one figure is covered and you compute it from the others, as an exam question would ask.

Dependent variableMonthly spending (₹ thousand)

Observations, n600

R²

Adjusted R²

Residual df

s (std. error of regression)

Coefficients
Variable	Coefficient	Std. error	t	p	95% interval
Intercept					to
Income					to
Household size					to
Town					to

Analysis of variance
Source	df	Mean square
Regression	3	2,798.35
Residual	596	6.574
Total	599

Click or tap any figure above to see where it comes from.

Survey regression of monthly household spending on income, household size and a town dummy. 600 households.

Four relations cover almost every missing-figure question: t = coefficient ÷ standard error; the interval is coefficient ± critical value × standard error; R² = regression SS ÷ total SS; and F = (regression SS ÷ k) ÷ (residual SS ÷ (n − k − 1)). Keep them on a card for the exam.

Joint testing

All the slopes at once: the overall F

The t test looks at one coefficient. The overall F asks whether the regression explains anything at all: H₀: β₁ = β₂ = … = β_k = 0, every slope zero at once. The intercept is not in the hypothesis (JNU 2021: the null is the slopes, not “α = 0 and the slopes”).

The overall F statistic

F = \frac{S S R / k}{S S E / ( n - k - 1 )} = \frac{R ^{2} / k}{( 1 - R ^{2} ) / ( n - k - 1 )} \sim F_{k, n - k - 1}

Numerator degrees of freedom k, the number of slopes; denominator n − k − 1. Rejecting means at least one slope is not zero, not that all are.

For the survey regression, F = 425.7 on (3, 596) degrees of freedom. The 5% critical value is 2.62, so the null of “nothing explains spending” is rejected overwhelmingly. The two forms agree: R² = 0.682, so (R² ÷ 3) ÷ ((1 − R²) ÷ 596) = 425.7.

Reading the result carefully matters. A rejected overall F says at least one slope is non-zero (CUET 2020). It does not say every slope is, and it does not say the model is good: with a big enough sample, a regression with a tiny R² is overwhelmingly significant.

Pause and predict

A regression with an intercept and k slopes uses n observations. The numerator and denominator degrees of freedom of the overall F are

Joint testing

Testing restrictions

The overall F is one special case of a general idea. State a null hypothesis that imposes q linear restrictions on the coefficients. Estimate the model twice: once freely (the unrestricted regression) and once with the restrictions built in (the restricted regression). Imposing a restriction can only make the fit worse, never better. The question is whether it makes it much worse.

F for q restrictions

F = \frac{( S S E _{R} - S S E _{U R} ) / q}{S S E _{U R} / ( n - k - 1 )} = \frac{( R _{U R}^{2} - R _{R}^{2} ) / q}{( 1 - R _{U R}^{2} ) / ( n - k - 1 )} \sim F_{q, n - k - 1}

Both regressions must use the same dependent variable and the same observations. The R² form requires that too.

Here k is the number of slopes in the unrestricted model, and q is the number of independent restrictions: count the equals signs. A large F means the restrictions cost too much fit, so reject them. Worked on the survey: does anything beyond income matter for spending? H₀: β_size = β_town = 0, so q = 2. Dropping both gives the restricted regression, whose sum of squared residuals is 4,852.3, against 3,918.2 unrestricted:

Ashanagar survey · two restrictions

Test H₀: the coefficients on household size and the town dummy are both zero, against the regression of spending on income, size and town.

Count the restrictions and the degrees of freedom.
q = 2. The unrestricted model has 3 slopes and 600 observations, so n − k − 1 = 596.
Sums of squares form.
F = ((SSE_R − SSE_UR) ÷ 2) ÷ (SSE_UR ÷ 596) = ((4,852.3 − 3,918.2) ÷ 2) ÷ (3,918.2 ÷ 596) = 71.
R² form, the same F.
R² is 0.6818 unrestricted and 0.6059 restricted, so F = ((R²_UR − R²_R) ÷ 2) ÷ ((1 − R²_UR) ÷ 596) = 71.
Compare with the critical value.
The 5% critical value of F(2, 596) is 3.01, and the p-value is below 0.001.

Answer

F = 71 is far above 3.01, so reject: household size and town matter for spending beyond income. The two forms give identical answers, so use whichever the question’s data suit.

Not every restriction is an exclusion. A null like β₁ + β₂ = 1 is still one linear restriction. To build the restricted regression, substitute: β₁ = 1 − β₂ turns y = α + β₁x₁ + β₂x₂ + β₃x₃ + u into $(y - x_{1}) = α + β_{2} (x_{2} - x_{1}) + β_{3} x_{3} + u$ , a regression of a new dependent variable on new regressors (JNU 2021). The same idea tests constant returns to scale in a Cobb-Douglas function ln Y = a + α ln L + β ln K: the null α + β = 1 is a single linear restriction, so it can be tested with an F, or with a t on the combination, and the two agree (IIT JAM 2025).

Null hypothesis	q	Restricted model	Can an F test do it?
β₁ = β₂ = 0	2	drop x₁ and x₂	Yes
β₀ = 0	1	drop the intercept	Yes (F = t²)
β₁ + β₂ = 1	1	substitute β₁ = 1 − β₂	Yes
β₁ = β₂	1	use x₁ + x₂ as one regressor	Yes
β₁ ÷ β₂ = 0, or β₁β₂ = 0	n/a	not linear in the coefficients	No (the F test handles linear restrictions only, IIT JAM 2022)

Watch out for

The mistakes examiners are fishing for

Using the R² form when the dependent variables differ. The R² of a model for ln y and a model for y measure different things and cannot be subtracted. Use the sums of squares, and only compare models with the same dependent variable.
Putting ESS in the denominator. The denominator of F is the unrestricted residual sum of squares over its degrees of freedom (UGC NET 2023 shows a version with ESS there, and it is the wrong one).
Counting q wrongly. q is the number of independent restrictions. “β₁ = 0, β₂ = 0 and β₃ + β₄ = 1” is q = 3.

Derive it

Why F is t squared for one restriction

When q = 1 the F test and the t test are the same test. That is not a coincidence, and the proof uses the partialling-out result from lesson 1.

Derive it

For one restriction, F is t squared

We want to show

For the single restriction β_j = 0, the F statistic from the restricted and unrestricted regressions equals the square of the t statistic on β_j.

What dropping one regressor costs.
Dropping x_j leaves the restricted regression of y on the other regressors Z. Its residual is My, where M removes Z. The previous derivation showed that My = b_jx̃_j + e, where x̃_j = Mx_j is x_j with Z taken out and e is the unrestricted residual. Since e is orthogonal to x̃_j, squaring and summing leaves no cross term:
$S S E_{R} = \sum (M y)^{2} = b_{j}^{2} \sum x ~_{ij}^{2} + S S E_{U R} ⟹ S S E_{R} - S S E_{U R} = b_{j}^{2} \sum x ~_{ij}^{2}$
The standard error of b_j.
From the variance formula of lesson 2, the variance of b_j is s² ÷ Σx̃_ij², with s² = SSE_UR ÷ (n − k). (The matching diagonal entry of (X′X)⁻¹ is exactly 1 ÷ Σx̃².) So t² = b_j² Σx̃² ÷ s².
Put them together.
$F = \frac{( S S E _{R} - S S E _{U R} ) /1}{S S E _{U R} / ( n - k )} = \frac{b _{j}^{2} \sum x ~ _{ij}^{2}}{s ^{2}} = \frac{b _{j}^{2}}{s ^{2} / \sum x ~ _{ij}^{2}} = \frac{b _{j}^{2}}{Var ( b _{j} )} = t^{2}$
In the survey, dropping household size from the spending regression raises the sum of squared residuals from 3,918.24 to 4,850.91, so F = (4,850.91 − 3,918.24) ÷ (3,918.24 ÷ 596) = 141.868. The t statistic on household size is 11.911, and 11.911² = 141.868. The sum of squared deviations of the cleaned-up size variable is 1,141.086.

Step 1 of 3

Interactive

Why two t tests are not one F

Run a t test on each of two coefficients and you have asked two separate questions. The F test asks one joint question: could both coefficients be zero at the same time? The answers can differ, and the picture shows why. The plane below is the pair of estimated coefficients. The gold box is the two separate 95% intervals. The blue curve is the joint 95% region. When the two regressors move together, the estimates move oppositely (if the data cannot tell x₁ from x₂, a high estimate for one comes with a low estimate for the other), and the joint region becomes a long thin ellipse along the diagonal.

Correlation between x1 and x20.95How closely the two regressors move together.Sample size, n30

t for x11.23

t for x20.26

5% critical t (27 df)2.05

F for both slopes zero7.11

p of that F0.003

Horizontal: the estimate of the coefficient on x₁ (b₁). Vertical: the estimate of the coefficient on x₂ (b₂).

Box of the two separate 95% intervalsJoint 95% region (the ellipse) and the estimateTrue values (0.8, 0.8)○ Origin: both coefficients zero

What it shows

Neither t statistic reaches 2.05, so each coefficient looks no different from zero on its own. Yet F = 7.1 rejects "both are zero". The origin sits inside the gold box but outside the blue ellipse. The two regressors carry nearly the same information, so the data cannot tell which one does the work, but they can tell that one of them does. At these settings (correlation 0.95, 30 observations) this happens in roughly two samples in three; with uncorrelated regressors it almost never does.

Set the correlation near 0.95 and draw a few samples. You will often see both t statistics below the critical value, with the origin inside the gold box, and yet the origin outside the ellipse, so that F rejects. Each variable alone looks dispensable, because the other can stand in for it. Together they are not. This is the classic symptom of multicollinearity, and the Delhi School and UGC NET both ask about it (lesson 8). The reverse can also happen: the origin can lie outside the box but inside the ellipse, so that a t test rejects and a careful joint test does not.

The rule

Test a joint hypothesis with an F, not with a string of t tests. Separate t tests ignore the correlation between the estimates and, at 5% each, do not control the overall error rate at 5%.

Solving the opening question

The opening question, solved

GATE 2021 · moderate

Model (1) has five regressors, model (2) has two. n = 350, R² is 0.3521 and 0.2314. Test H₀: β₃ = β₄ = β₅ = 0.

Name the two models.
Model (1) is unrestricted (five slopes). Model (2) is the restricted model: it is model (1) with β₃ = β₄ = β₅ = 0 imposed, which leaves x₁ and x₂. Same dependent variable, so the R² form is valid.
Count q and the degrees of freedom.
Three restrictions, so q = 3. The unrestricted model has k = 5 slopes, so the denominator has n − k − 1 = 350 − 5 − 1 = 344.
Compute.
F = ((0.3521 − 0.2314) ÷ 3) ÷ ((1 − 0.3521) ÷ 344) = (0.1207 ÷ 3) ÷ (0.6479 ÷ 344) = 0.040233 ÷ 0.0018834 = 21.362.
Decide.
The 5% critical value of F(3, 344) is 2.63. F is far above it, so the three variables are jointly significant.

Answer

21.362. The common mistakes are using n − k as the denominator degrees of freedom, dividing by the wrong q, or using the R² of model (1) alone in the overall-F formula.

Revise

The lesson on one screen

t test: t = b ÷ SE on n − k − 1 df. Interval b ± t_crit × SE. To test β = c use (b − c) ÷ SE. Above about 2 is significant at 5% in a large sample.
Output relations: t = coefficient ÷ SE. R² = SSR ÷ SST. F = (SSR ÷ k) ÷ (SSE ÷ (n − k − 1)). Adjusted R² = 1 − (1 − R²)(n − 1) ÷ (n − k − 1).
Overall F: H₀: every slope is zero. df (k, n − k − 1). Rejection means at least one slope is not zero.
F for restrictions: F = ((SSE_R − SSE_UR) ÷ q) ÷ (SSE_UR ÷ (n − k − 1)), or from R² when the dependent variable is the same. q is the number of restrictions.
Building the restricted model: Exclusion: drop the variables. β₁ + β₂ = 1: substitute and regress (y − x₁) on (x₂ − x₁). β₁ = β₂: use x₁ + x₂ as one regressor.
What F can test: Linear restrictions only. Not ratios or products of coefficients.
One restriction: F = t², and the p-values are identical.
Joint against separate: Two t tests ignore the correlation between estimates. When regressors are highly correlated, each can be insignificant while F rejects.

Practice

Check yourself

0 of 13 correct so far

Compute

A t statistic

A coefficient is 0.42 with standard error 0.15. Find the t statistic for H₀: β = 0.

Degrees of freedom

A regression with an intercept and four slopes uses 120 observations. How many degrees of freedom does a t test on one coefficient have?

An interval

A coefficient is 2.5 with standard error 0.8, and the critical t value is 2.0. Find the upper end of the 95% interval.

Upper limit

The overall F from R²

A regression with an intercept and four slopes, on 51 observations, has R² = 0.4. Find the overall F.

F from sums of squares

The unrestricted model has 4 coefficients including the intercept, n = 64 and SSE = 100. Two restrictions are imposed, and the restricted SSE is 120. Find F.

F from R² for one restriction

R² is 0.55 unrestricted and 0.50 restricted, with q = 1 and 48 residual degrees of freedom. Find F.

F and t

A single added regressor has t = 2.5. Find the F statistic for adding it.

Count the restrictions

H₀: β₁ = 0, β₂ = 0 and β₃ + β₄ = 1. How many restrictions q does an F test have?

Quick questions

1In a multiple regression the overall F test is rejected. We can conclude that

2Which hypothesis can NOT be tested with an F test in a linear regression?

3For testing a single added regressor, the F statistic equals

4The overall F in a regression with k slopes and an intercept, estimated on n observations, has degrees of freedom

5Two coefficients are each insignificant but a joint F test rejects. The most likely cause is

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.

3 questionsPractise →

Standard

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

10 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 13 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.

Two of these questions carry numeric answers. Type the number, rounded as the question says. The Delhi School’s questions on this material (a standardised effect, a nested F test, single significance tests in a home-sales table) are in the hard-problems lesson at the end of the module. See also inference in simple regression, which proves F = t² for the one-regressor case.

← Lesson 2OLS in matrix form and the Gauss-Markov theorem Lesson 4 →R², adjusted R² and comparing models

t tests, F tests and reading regression output

One coefficient: the t test and its interval

The mistakes examiners are fishing for

Reading regression output

All the slopes at once: the overall F

Testing restrictions

The mistakes examiners are fishing for

Why F is t squared for one restriction

For one restriction, F is t squared

Why two t tests are not one F

The opening question, solved

The lesson on one screen

Check yourself

Compute

Quick questions

Past papers, by level

Foundation

Standard

Delhi School level

Read next

t tests, F tests and reading regression output

One coefficient: the t test and its interval

The mistakes examiners are fishing for

Reading regression output

All the slopes at once: the overall F

Testing restrictions

The mistakes examiners are fishing for

Why F is t squared for one restriction

For one restriction, F is t squared

Why two t tests are not one F

The opening question, solved

The lesson on one screen

Check yourself

Compute

Quick questions

Past papers, by level

Foundation

Standard

Delhi School level

Read next