Lesson M04.L02: t-Tests for Individual Coefficients

Field	Detail
Module	Module 04 — Statistical Inference in OLS
Level	Graduate (ECMT5001, S1 2027)
Duration	35 min
Learning Objective	Construct and interpret t-tests for OLS regression coefficients; distinguish two-sided and one-sided tests; separate statistical significance from economic significance
Provenance	Wooldridge, Introductory Econometrics, 7e, Chapter 4.2 (pp. 124–140)

Explanation

Setting Up the Null Hypothesis

The most common null hypothesis in regression inference is:

\[H_0: \beta_j = 0\]

This tests whether regressor $x_j$ has any partial effect on $y$ (holding the other regressors constant). But more generally, we can test any specific value:

\[H_0: \beta_j = a_j\]

for a hypothesised value $a_j \in \mathbb{R}$. Choosing $a_j = 0$ is by far the most common case in applied work.

The t-Statistic

Under $H_0: \beta_j = a_j$ and the CLM assumptions (MLR.1–MLR.6):

\[t_{\hat{\beta}_j} = \frac{\hat{\beta}_j - a_j}{se(\hat{\beta}_j)} \sim t_{n-k-1}\]

For the standard null $H_0: \beta_j = 0$:

\[t_{\hat{\beta}_j} = \frac{\hat{\beta}_j}{se(\hat{\beta}_j)}\]

This is what regression software reports in the "t-stat" or "t-value" column. The formula says: how many standard errors is $\hat{\beta}_j$ away from zero?

Decision Rules

Two-Sided Test

\[H_0: \beta_j = 0 \qquad H_1: \beta_j \neq 0\]

Reject $H_0$ at significance level $\alpha$ if:

\[|t_{\hat{\beta}_j}| > t_{n-k-1,\, \alpha/2}\]

where $t_{n-k-1,\, \alpha/2}$ is the upper $\alpha/2$ critical value of the $t_{n-k-1}$ distribution. Common values:

Significance level $\alpha$	$t_{n-k-1,\, \alpha/2}$ (large $n$)	Stars convention
10%	1.645	*
5%	1.960	**
1%	2.576	***

One-Sided Test (Upper Tail)

\[H_0: \beta_j \leq 0 \qquad H_1: \beta_j > 0\]

Reject $H_0$ at level $\alpha$ if:

\[t_{\hat{\beta}_j} > t_{n-k-1,\, \alpha}\]

(Note: upper-tail critical value, not $\alpha/2$, since rejection is in one direction only.)

One-Sided Test (Lower Tail)

\[H_0: \beta_j \geq 0 \qquad H_1: \beta_j < 0\]

Reject $H_0$ at level $\alpha$ if:

\[t_{\hat{\beta}_j} < -t_{n-k-1,\, \alpha}\]

Important: Always state your alternative hypothesis before looking at the data; choosing the direction of the test after seeing the sign of $\hat{\beta}_j$ is data snooping and inflates Type I error.

p-Values

The p-value is the probability, under $H_0$, of observing a t-statistic at least as extreme as the one computed:

\[p\text{-value (two-sided)} = 2 \cdot P(T > |t_{\hat{\beta}_j}|) \quad \text{where } T \sim t_{n-k-1}\]

Interpretation: the p-value is the smallest significance level $\alpha$ at which we would reject $H_0$ given the data.

$p < 0.01$: strong evidence against $H_0$ (*** in most software)
$0.01 \leq p < 0.05$: moderate evidence against $H_0$ (**)
$0.05 \leq p < 0.10$: weak evidence against $H_0$ (*)
$p \geq 0.10$: insufficient evidence to reject $H_0$

Reading Regression Output

A typical software output table looks like:

           Coeff    SE       t      p-value
(Intercept) 5.400   0.631   8.56   <0.001 ***
educ        0.082   0.010   8.20   <0.001 ***
exper       0.004   0.002   2.10    0.037  **
female     -0.296   0.036  -8.22   <0.001 ***

To recover the t-statistic manually: divide coefficient by SE. For example: $$t_{\text{educ}} = \frac{0.082}{0.010} = 8.20 \checkmark$$

Economic Significance vs. Statistical Significance

A crucial distinction for applied work:

	Statistically significant	Not statistically significant
Large effect	Ideal: precise and meaningful	Plausible effect, but underpowered — need more data
Small effect	Precisely estimated but trivially small — may not matter in practice	Neither significant nor meaningful

Example. Suppose we have $n = 50{,}000$ observations and estimate that a marketing campaign increases revenue by $0.50 on average ($t = 12.3$, $p < 0.001$). The coefficient is highly statistically significant, but if the campaign costs $200 per customer, the effect is economically negligible.

Conversely, a small study ($n = 40$) might estimate a large wage effect of education but with $t = 1.5$ ($p = 0.14$) — not statistically significant, but the effect may be economically real; we simply lack precision.

Rule for exam write-ups: Always comment on both statistical significance (t-stat, p-value, stars) AND the magnitude and economic interpretation of the coefficient.

Worked Example

Setup. A researcher regresses log hourly wages on three regressors using $n = 526$ observations from the CPS:

\[\log(\text{wage}_i) = \beta_0 + \beta_1\, \text{educ}_i + \beta_2\, \text{exper}_i + \beta_3\, \text{female}_i + u_i\]

The OLS estimation yields the following output ($df = 526 - 3 - 1 = 522$):

Variable	$\hat{\beta}_j$	$se(\hat{\beta}_j)$	$t$-stat	$p$-value
Intercept	0.284	0.104	2.73	0.007
educ	0.092	0.007	13.14	<0.001
exper	0.004	0.001	4.00	<0.001
female	−0.297	0.036	−8.25	<0.001

For large $df \approx 522$, we use $t_{522, 0.025} \approx 1.965$ (5% two-sided) and $t_{522, 0.005} \approx 2.585$ (1% two-sided).

Test 1: Is education statistically significant? ($H_0: \beta_1 = 0$, two-sided, $\alpha = 0.05$)

Step 1 — Hypotheses: $$H_0: \beta_1 = 0 \qquad H_1: \beta_1 \neq 0$$

Step 2 — Test statistic: $$t_{\hat{\beta}_1} = \frac{\hat{\beta}_1}{se(\hat{\beta}_1)} = \frac{0.092}{0.007} = 13.14$$

Step 3 — Distribution under $H_0$: $t_{522}$

Step 4 — Decision: $|13.14| = 13.14 > 1.965$. Reject $H_0$.

Step 5 — Interpretation: Education has a statistically significant positive effect on log wages. Each additional year of schooling is associated with approximately a $9.2\%$ increase in hourly wages, holding experience and gender constant.

Test 2: Is the gender gap statistically significant? ($H_0: \beta_3 = 0$, two-sided, $\alpha = 0.01$)

Step 1 — Hypotheses: $$H_0: \beta_3 = 0 \qquad H_1: \beta_3 \neq 0$$

Step 2 — Test statistic: $$t_{\hat{\beta}_3} = \frac{-0.297}{0.036} = -8.25$$

Step 3 — Distribution under $H_0$: $t_{522}$

Step 4 — Decision: $|-8.25| = 8.25 > 2.585$. Reject $H_0$ at the 1% level.

Step 5 — Interpretation: Women earn statistically and economically significantly less than men: approximately $29.7\%$ lower log wages, holding education and experience constant. The estimate is precise (small SE relative to coefficient) and economically large.

Test 3: Does experience have a positive effect? ($H_0: \beta_2 \leq 0$, one-sided upper, $\alpha = 0.05$)

Step 1 — Hypotheses: $$H_0: \beta_2 \leq 0 \qquad H_1: \beta_2 > 0$$

Step 2 — Test statistic: $$t_{\hat{\beta}_2} = \frac{0.004}{0.001} = 4.00$$

Step 3 — Distribution under $H_0$: $t_{522}$

Step 4 — Decision: $4.00 > t_{522, 0.05} \approx 1.645$. Reject $H_0$.

Step 5 — Interpretation: Experience has a statistically significant positive effect on wages. However, the magnitude is small: each additional year of experience raises log wages by only $0.4\%$ — statistically detectable but economically modest compared to education.

Common Misconception

"A non-significant coefficient means the variable has no effect on Y."

This is incorrect. Failing to reject $H_0: \beta_j = 0$ means we cannot detect a statistically meaningful effect with this sample. The reasons could be: 1. The true effect is genuinely zero. 2. The sample is too small (low power) to detect a real but moderate effect. 3. There is high multicollinearity, inflating $se(\hat{\beta}_j)$. 4. The data have high variance, making all estimates imprecise.

The correct language: "We find no statistically significant evidence of an effect at the 5% level" — not "there is no effect."

Illustration. If $\hat{\beta}_j = 5.0$ with $se(\hat{\beta}_j) = 4.0$, then $t = 1.25$ and $p \approx 0.21$. We fail to reject $H_0$, but the estimated effect is large. A 95% CI would be roughly $5.0 \pm 2 \times 4.0 = (-3, 13)$ — very wide. We simply don't have enough precision to rule out zero, not evidence that the effect is zero.

Practice Prompts

Q1. A model of house prices (thousands of dollars) with $n = 88$ and $k = 4$ regressors yields: $\hat{\beta}_{\text{sqft}} = 0.045$, $se = 0.012$. Test $H_0: \beta_{\text{sqft}} = 0$ against $H_1: \beta_{\text{sqft}} > 0$ at the 5% significance level. State all steps and interpret the result in context.

Answer

**Step 1:** $H_0: \beta_{\text{sqft}} \leq 0$, $H_1: \beta_{\text{sqft}} > 0$ (one-sided upper). **Step 2:** $t = 0.045 / 0.012 = 3.75$. **Step 3:** Under $H_0$, $t \sim t_{n-k-1} = t_{88-4-1} = t_{83}$. **Step 4:** $t_{83, 0.05} \approx 1.663$. Since $3.75 > 1.663$, **reject $H_0$**. **Step 5:** Each additional square foot is associated with a statistically significant \$45 increase in house price (holding other variables constant), significant at the 5% level.

Q2. From the same output: $\hat{\beta}_{\text{pool}} = 4.5$ (thousands of dollars) with $se = 8.2$. A colleague concludes "a pool adds $4,500 to house value." Critically evaluate this claim.

Answer

$t = 4.5 / 8.2 = 0.549$. With $df = 83$, the two-sided critical value at 5% is $\approx 1.989$. Since $0.549 < 1.989$, we **fail to reject** $H_0: \beta_{\text{pool}} = 0$. The estimate suggests a positive effect, but the standard error is nearly twice the coefficient. We cannot detect a statistically significant pool premium with this sample. The colleague should not make a causal claim from a statistically insignificant estimate. The true effect could plausibly be zero (or even negative) — the data are too imprecise to tell.

Q3. A researcher tests $H_0: \beta_j = 1$ (rather than the standard $H_0: \beta_j = 0$) for a coefficient $\hat{\beta}_j = 1.38$ with $se = 0.14$, $n = 200$, $k = 3$. Compute the t-statistic and state whether you reject at the 5% two-sided level.

Answer

**Step 1:** $H_0: \beta_j = 1$, $H_1: \beta_j \neq 1$. **Step 2:** $t = (\hat{\beta}_j - a_j)/se = (1.38 - 1.00) / 0.14 = 0.38 / 0.14 = 2.71$. **Step 3:** $df = 200 - 3 - 1 = 196$. $t_{196, 0.025} \approx 1.972$. **Step 4:** $|2.71| > 1.972$, so **reject $H_0$** at the 5% level. **Conclusion:** The data provide statistically significant evidence that $\beta_j \neq 1$ at the 5% level.

Further Resources

Wooldridge (2019), §4.2: "Testing Hypotheses about a Single Population Parameter: The t Test"
Wooldridge (2019), §4.3: "Confidence Intervals"
Angrist & Pischke (2009), Mostly Harmless Econometrics, §3.1: "Regression Fundamentals"
McCloskey & Ziliak (1996): "The Standard Error of Regressions" — a famous critique of over-reliance on statistical significance in economics