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Hypothesis Testing with the t Statistic

The t test evaluates a hypothesis about a single coefficient using t=β^j−βj,0se(β^j)t=\dfrac{\hat{\beta}_j-\beta_{j,0}}{\mathrm{se}(\hat{\beta}_j)}. The most common case tests H0:βj=0H_0:\beta_j=0 against a one- or two-sided alternative, comparing ∣t∣|t| to a critical value from the t n−k−1t_{\,n-k-1} distribution. A coefficient can be statistically significant yet economically trivial, so significance and magnitude are different questions.

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.

-4-3-2-101234+t*−t*t = 3.0000.1430.757β̂ = 0.45
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

The t ratio asks how many standard errors the estimate sits from the value under the null. Far from zero means the data are hard to reconcile with no effect. But a tiny effect can clear the significance bar in a huge sample, so always read the size of the coefficient too.

Formulas

t statistic for a single coefficient
t=β^j−βj,0se(β^j)t=\frac{\hat{\beta}_j-\beta_{j,0}}{\mathrm{se}(\hat{\beta}_j)}
Most often βj,0=0\beta_{j,0}=0; reject H0H_0 when ∣t∣|t| exceeds the critical value.
Default Stata t ratio
t=β^jse(β^j)t=\frac{\hat{\beta}_j}{\mathrm{se}(\hat{\beta}_j)}
This is the t value Stata prints when testing βj=0\beta_j=0.

Worked examples

Scenario

In `regress lwage educ exper`, the educ coefficient is 0.092 with a standard error of 0.007.

Solution

The t statistic, 0.092 divided by 0.007, is about 13.1, far beyond the two-sided 5 percent critical value near 1.96, so you reject H0:βeduc=0H_0:\beta_{\mathrm{educ}}=0. Stata’s `test educ` and the printed p-value confirm strong evidence of a schooling effect.

NoteA one-sided test of H0:βeduc≤0H_0:\beta_{\mathrm{educ}}\le 0 uses a single-tail critical value near 1.65.

Common mistakes

  • ✗Reading statistical significance as economic importance. A precisely estimated but tiny coefficient can be significant yet matter little.
  • ✗Thinking a large p-value proves βj=0\beta_j=0. Failing to reject means insufficient evidence, not confirmation of the null.
  • ✗Using a two-sided critical value for a directional hypothesis. One-sided alternatives use a one-tail critical value, which is smaller in magnitude.
  • ✗Believing the t test stays exact under any conditions. Exactness needs the CLM; otherwise it is a large-sample approximation.

Revision bullets

  • •t statistic: t=(β^j−βj,0)/se(β^j)t=(\hat{\beta}_j-\beta_{j,0})/\mathrm{se}(\hat{\beta}_j), usually with βj,0=0\beta_{j,0}=0.
  • •Compare ∣t∣|t| to a t n−k−1t_{\,n-k-1} critical value; one-sided tests use a one-tail value.
  • •Statistical significance is distinct from economic significance.
  • •A large p-value means insufficient evidence, not proof of no effect.
  • •Exact t inference relies on the CLM; otherwise it is asymptotic.

Quick check

The t statistic for testing H0:βj=0H_0:\beta_j=0 is computed as:

A coefficient is statistically significant at the 1 percent level but implies a 0.01 percent change in yy. The best reading is:

Compared with a two-sided test at 5 percent, a one-sided alternative uses a critical value that is:

Connected topics

Gauss-MarkovAsymptoticsCLM normalityConf. intervalsF testUnits & Scale

Sources

  1. Wooldridge, Introductory Econometrics, Ch. 4
    Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Sec. 4.2
  2. McCloskey & Ziliak (1996)
    McCloskey & Ziliak (1996), The Standard Error of Regressions, Journal of Economic Literature 34(1), 97-114
How to cite this page
Dr. Phil's Quant Lab. (2026). Hypothesis Testing with the t Statistic. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-t-test
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