Multiple Regression & Inferenceintermediate

Confidence Intervals for Coefficients

A confidence interval (CI) for a coefficient is $\hat{\beta}_j \pm c\cdot\mathrm{se}(\hat{\beta}_j)$ , where $c$ is the $t_{\,n-k-1}$ critical value for the chosen confidence level. A 95 percent CI collects the values of $\beta_j$ not rejected by a two-sided test, so it is the dual of the t test. If the interval excludes zero, the coefficient is significant at that level.

Try it yourself

Significance ⇄ confidence interval

Significance and the confidence interval say the same thing two ways — and “significant” is not the same as “large”.

Take one regression slope β̂ with standard error se. The t-statistic t = β̂ / se lands on the t-density: if it falls in a red rejection tail we reject H₀: β = 0. The same evidence is the interval β̂ ± t*·se below. The two always agree: the interval excludes 0 ⇔ the test rejects.

95% CI [0.143, 0.757]contains 0? no

Reject H₀: β = 0 at α = 0.05

The CI excludes 0, so the test rejects. Same conclusion.

t = β̂ / se3.000

p-value (two-tail)0.006

critical t*2.048

df = n − k28

CI low0.143

CI high0.757

Magnitude ≠ significance: t depends on β̂ relative tose. A tiny β̂ with an even smaller se is “significant” yet economically trivial. Always read the CI width, not just the verdict.

Coefficient β̂0.45

Std. error se0.150

Sample size n30

Regressors k (incl. intercept)2

Significance α0.05

Try this

Discussion.Load “Significant but tiny”: the estimate clears every significance test yet the whole interval sits between 0.01 and 0.03. Would you call that effect important? Now argue the opposite case from “Big but imprecise”. What does each one tell you that a lone p-value hides?

Two-sided test of H₀: β = 0 with t ~ Student-t(df = n − k). The p-value and t* come from the Student-t distribution (regularized incomplete beta), exact to textbook precision. The CI is β̂ ± t*·se; it excludes 0 exactly when p < α.

Why it matters

Instead of a single number, a CI reports a plausible range for the true effect given the sampling noise. Wider intervals signal less precise estimates. Reading whether zero or some economically meaningful value sits inside the interval is often more informative than a bare reject or fail-to-reject.

Formulas

Confidence interval for a coefficient

\hat{\beta}_j \pm c\cdot\mathrm{se}(\hat{\beta}_j)

c

is the two-sided

t_{\,n-k-1}

critical value, near 1.96 for 95 percent in large samples.

Test-CI duality

\text{reject } H_0:\beta_j=a \iff a \notin \text{CI}

Values outside the CI are exactly the nulls a two-sided test rejects.

Worked examples

Scenario

Stata reports educ = 0.092 with a 95 percent interval of [0.078, 0.106] from `regress lwage educ exper`.

Solution

Because the whole interval lies above zero, you reject $H_0:\beta_{\mathrm{educ}}=0$ at 5 percent, matching the t test. The interval also says the data are consistent with returns to schooling between about 7.8 and 10.6 percent per year.

NoteStata prints this interval automatically; widen it with `regress ..., level(99)`.

Common mistakes

✗Saying there is a 95 percent probability the true $\beta_j$ lies in one computed interval. The parameter is fixed; the 95 percent refers to the procedure across repeated samples.
✗Thinking a CI and a two-sided t test can disagree at the same level. They are mathematically equivalent for the same null and level.
✗Believing a wider interval means a larger effect. Width reflects precision, driven by the standard error, not the size of the coefficient.
✗Assuming a CI accounts for specification error. It captures sampling uncertainty only, not bias from omitted variables or a wrong model.

Revision bullets

•CI: $\hat{\beta}_j \pm c\cdot\mathrm{se}(\hat{\beta}_j)$ with $c$ from the $t_{\,n-k-1}$ distribution.
•A CI is the dual of the two-sided t test; excluding zero means significant.
•Confidence refers to the procedure over repeated samples, not one interval.
•Width measures precision, not the magnitude of the effect.
•A CI reflects sampling noise only, not specification bias.

Quick check

A 95 percent confidence interval for $\beta_j$ is best interpreted as:

If a 95 percent confidence interval for $\beta_j$ excludes zero, then a two-sided t test of $H_0:\beta_j=0$ at 5 percent will:

Connected topics

Gauss-Markov Asymptotics CLM normality t test F test Prediction

Sources

Wooldridge, Introductory Econometrics, Ch. 4
Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Sec. 4.3
Hill, Griffiths & Lim, Ch. 3
Hill, Griffiths & Lim (2018), Principles of Econometrics, 5th ed., Ch. 3

How to cite this page

Dr. Phil's Quant Lab. (2026). Confidence Intervals for Coefficients. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-confidence-intervals

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