Inference

• Statistical inference uses the sampling distribution of estimators to make statements about population parameters.
• Two main tools: hypothesis tests and confidence intervals.

• Null hypothesis (\(H_0\)): the claim held to be true unless the data provide sufficient evidence against it.
• Alternative hypothesis (\(H_1\)): the claim we are trying to establish.

• The econometrician carries the burden of proof — must show the data provide enough evidence to reject \(H_0\) in favor of \(H_1\).

• Type I error: wrongful conviction — rejecting \(H_0\) when it is true.
• Type II error: letting a guilty person go free — failing to reject \(H_0\) when \(H_1\) is true.

• But lowering \(\alpha\) comes at a cost: requiring stronger evidence also makes it harder to reject when \(H_1\) is true (Type II error increases).
• By convention, \(\alpha\) is set at 5% or 1%.

To carry out step 3, we need to know the distribution of the test statistic under \(H_0\).

• This requires one more assumption.

• Justification: \(U\) is the sum of many unobserved factors. By the CLT, such sums are approximately normal.

• Limitations:
  • How many factors? Are they independent? Is the combination additive?
  • Normality may be poor in skewed or heavy-tailed settings (e.g., wages, which are bounded below).
  • For binary or count outcomes, the linear model itself is often problematic for broader reasons.

• With large samples, we can relax this assumption using asymptotic approximations (Lecture 3b).

• Denote \(\text{sd}(\hat\beta_j) = \sqrt{\text{Var}(\hat\beta_j \mid X)} = \sigma / \sqrt{\text{SST}_j(1 - R_j^2)}\) for the slope coefficients (\(j = 1, \ldots, k\)). Standardizing:

\[ \frac{\hat\beta_j - \beta_j}{\text{sd}(\hat\beta_j)} \sim N(0, 1) \]

• If \(\sigma^2\) were known, we could use:

\[ Z = \frac{\hat\beta_j - \beta_{j,0}}{\color{red}{\text{sd}(\hat\beta_j)}} \sim N(0, 1) \quad \text{under } H_0 \]

• Reject \(H_0\) if \(|Z| > z_{1-\alpha/2}\) (the critical value): the value chosen so that \(P(|Z| > z_{1-\alpha/2}) = \alpha\).

• But this substitution has a cost: \(\hat\sigma\) is itself estimated from data, introducing extra randomness. The resulting statistic is not \(N(0,1)\).
• Gosset (1908), working with small samples at the Guinness brewery under the pseudonym “Student,” showed that it follows a distribution with heavier tails — the \(t\)-distribution.

• Decision rule: reject \(H_0\) if \(|t| > t_{1-\alpha/2, \, n-k-1}\) (the critical value).
• This is an exact finite-sample result under MLR.1–MLR.6. In Lecture 3b, we develop large-sample alternatives that do not require normality.

\[ t = \frac{\hat\beta_j - \beta_{j,0}}{\text{se}(\hat\beta_j)} = \underbrace{\frac{\hat\beta_j - \beta_j}{\text{se}(\hat\beta_j)}}_{\sim \, t_{n-k-1}} + \underbrace{\frac{\beta_j - \beta_{j,0}}{\text{se}(\hat\beta_j)}}_{\text{nonzero shift}} \]

• The \(t\)-statistic is shifted away from zero — the larger the shift, the more likely we reject.
• The probability of correctly rejecting \(H_0\) when \(H_1\) is true is called power.

• Power is higher when:
  • The effect is larger: \(|\beta_j - \beta_{j,0}|\) is big
  • The estimation is more precise: \(\text{se}(\hat\beta_j)\) is small

• The \(p\)-value is the smallest significance level at which \(H_0\) would be rejected.

\[ p\text{-value} = P(|T| > |t|) \quad \text{where } T \sim t_{n-k-1} \]

• Decision rule: reject \(H_0\) if \(p\text{-value} < \alpha\).
• The \(p\)-value summarizes the strength of evidence against \(H_0\) — smaller means stronger evidence.

• If \(H_0: \beta_j = 0\) is rejected: \(X_j\) is statistically significant.
• If not rejected: \(X_j\) is statistically insignificant.

• Caution: failing to reject does not mean \(H_0\) is true — it may mean we lack the precision to detect the effect (low power).

• \(t = -0.08311 / 0.02600 = -3.197\)
• Critical value at 5% with 137 df: \(\approx 1.98\)
• \(|t| = 3.197 > 1.98\): reject \(H_0\). Skipping classes is statistically significant.

• Reject \(H_0\) if \(t\) falls in the shaded rejection region.
• One-sided tests have more power to detect effects in the hypothesized direction.

• Important: the direction must be chosen before seeing the data. Choosing after seeing \(\hat\beta_j\) inflates the Type I error.

• \(H_0: \beta_{\text{exper}} \leq 0\) vs. \(H_1: \beta_{\text{exper}} > 0\)
• \(t = 0.0041 / 0.0017 = 2.41\)
• Critical value at 5% (522 df): \(\approx 1.65\)
• \(t = 2.41 > 1.65\): reject \(H_0\).

In the college GPA example, test \(H_0: \beta_{\text{skipped}} = -0.1\) against a two-sided alternative.

\[ t = \frac{-0.08311 - (-0.1)}{0.026} = \frac{0.01689}{0.026} \approx 0.65 \]

• \(|t| = 0.65 < 1.98\): fail to reject. The data are consistent with each skipped class reducing GPA by 0.1 points.

Substituting \(t = (\hat\beta_j - \beta_j) / \text{se}(\hat\beta_j)\) and rearranging:

\[ P\!\left( \hat\beta_j - t_{1-\alpha/2, \, n-k-1} \cdot \text{se}(\hat\beta_j) \leq \beta_j \leq \hat\beta_j + t_{1-\alpha/2, \, n-k-1} \cdot \text{se}(\hat\beta_j) \right) = 1 - \alpha \]

The \(100(1-\alpha)\%\) confidence interval for \(\beta_j\):

\[ \hat\beta_j \pm t_{1-\alpha/2, \, n-k-1} \cdot \text{se}(\hat\beta_j) \]

What is the relationship between a CI and a two-sided test?

• Fail to reject \(H_0: \beta_j = \beta_{j,0}\) at level \(\alpha\) \(\iff\) \(\beta_{j,0}\) lies inside the \(100(1-\alpha)\%\) CI.
• The CI is the set of all values that would not be rejected by a two-sided test.

• Zero is outside this interval — consistent with rejecting \(H_0: \beta_{\text{skipped}} = 0\).
• \(-0.1\) is inside this interval — consistent with failing to reject \(H_0: \beta_{\text{skipped}} = -0.1\).

Example: 401(k) participation and firm size

\[ \text{prate} = \beta_0 + \beta_1 \, \text{mrate} + \beta_2 \, \text{age} + \beta_3 \, \text{totemp} + U \]

where prate = participation rate (%), mrate = employer match rate, age = plan age (years), totemp = total employees.

             Estimate Std. Error t value Pr(>|t|)
totemp      -1.291e-04  3.666e-05  -3.521 0.000443 ***

• \(\hat\beta_3\) is highly statistically significant (\(p < 0.001\)).
• But a 10,000-employee increase reduces participation by only \(10{,}000 \times 0.00013 = 1.3\) percentage points.
• Is that economically meaningful? That depends on context — statistical significance alone doesn’t answer the question.

Example: Is one year at a junior college worth as much as one year at a university?

\[ \log(\text{wage}) = \beta_0 + \beta_1 \, \text{jc} + \beta_2 \, \text{univ} + \beta_3 \, \text{exper} + U \]

\[ H_0: \beta_1 = \beta_2 \qquad \text{vs.} \qquad H_1: \beta_1 < \beta_2 \]

• The \(t\)-statistic is:

\[ t = \frac{\hat\beta_1 - \hat\beta_2}{\text{se}(\hat\beta_1 - \hat\beta_2)} \]

• But \(\text{se}(\hat\beta_1 - \hat\beta_2)\) requires \(\text{Cov}(\hat\beta_1, \hat\beta_2)\), which is not always reported.
• In practice, software handles this directly (linearHypothesis in R, test in Stata). But the idea behind it is instructive.

Substitute \(\beta_1 = \theta + \beta_2\) into the regression:

\[ \begin{aligned} \log(\text{wage}) &= \beta_0 + (\theta + \beta_2) \, \text{jc} + \beta_2 \, \text{univ} + \beta_3 \, \text{exper} + U \\ &= \beta_0 + \theta \, \text{jc} + \beta_2 (\text{jc} + \text{univ}) + \beta_3 \, \text{exper} + U \end{aligned} \]

• Run this transformed regression. The coefficient on jc is \(\hat\theta\), and its standard error is \(\text{se}(\hat\theta)\).
• Test \(H_0: \theta = 0\) with the usual \(t\)-test — no need for covariance.

Example: Do performance statistics matter for baseball players’ salaries?

\[ \begin{aligned} \log(\text{salary}) = \beta_0 &+ \beta_1 \, \text{years} + \beta_2 \, \text{gamesyr} \\ &+ \beta_3 \, \text{bavg} + \beta_4 \, \text{hrunsyr} + \beta_5 \, \text{rbisyr} + U \end{aligned} \]

where bavg = batting average, hrunsyr = home runs/year, rbisyr = RBIs/year.

\[ H_0: \beta_3 = \beta_4 = \beta_5 = 0 \qquad \text{vs.} \qquad H_1: \text{at least one} \neq 0 \]

• This is not a valid size-\(\alpha\) test. Even if the tests were independent:

\[ P(\text{reject at least one} \mid H_0) = 1 - (1 - \alpha)^q \]

• With \(q = 3\) tests at \(\alpha = 0.05\): \(1 - 0.95^3 \approx 0.143\) — nearly three times the nominal level.

• We need a test designed for joint hypotheses.

• If \(H_0\) is true, the excluded variables don’t help explain \(Y\) — dropping them should barely worsen the fit.
• If dropping them worsens the fit substantially, that’s evidence against \(H_0\).

• How do we measure “worsening the fit”? Compare the sum of squared residuals: \(\text{SSR}_r\) vs. \(\text{SSR}_{ur}\).

• \(\text{SSR}_r \geq \text{SSR}_{ur}\) always, so \(F \geq 0\).
• Under \(H_0\): excluded variables have no effect, so \(\text{SSR}_r \approx \text{SSR}_{ur}\) and \(F \approx 0\).
• Under \(H_1\): dropping them worsens fit substantially, so \(F\) is large.

\[ F = \frac{(198.31 - 183.19)/3}{183.19/347} = \frac{5.04}{0.528} = 9.55 \]

• Critical value at 5% with \((3, 347)\) df: \(2.63\)
• \(F = 9.55 > 2.63\): reject \(H_0\). Performance statistics are jointly significant.

• Useful when software reports \(R^2\) but not SSR.
• Requires the same dependent variable in both models (otherwise SST does not cancel).

• Why? Multicollinearity: correlated regressors make individual effects hard to isolate, but their joint contribution can still be large.
• This is precisely when the \(F\)-test is most valuable.

• The restricted model is just the intercept (\(R^2_r = 0\)), so:

\[ F = \frac{R^2 / k}{(1 - R^2) / (n - k - 1)} \]

• Most software reports this \(F\)-statistic automatically.

• For a single restriction, the two are equivalent: \(t^2 = F\) and \(t^2_{n-k-1} \sim F_{1,\, n-k-1}\).