Hypothesis Testing

ECON 3209 · Week 9, Lecture 1 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

  1. Define null and alternative hypotheses in regression analysis.
  2. Understand test statistics, critical values, and p-values.
  3. Distinguish Type I and Type II errors.
  4. Apply coefficient significance testing in Python.
  5. Interpret regression tests in economically meaningful language.

Why inference matters

  • OLS gives point estimates, but point estimates alone are not enough.
  • We need to know whether an estimated effect could be due to sampling noise.
  • Inference helps us decide whether evidence is strong or weak.
  • Policy conclusions should depend on both effect size and uncertainty.
  • Hypothesis testing is one of the main tools for statistical inference.

Null and alternative hypotheses

  • The null hypothesis is the benchmark claim we test.
  • The alternative hypothesis is what we consider if the null is rejected.
  • In simple regression, a common test is \[H_0: \beta_1 = 0 \quad \text{vs} \quad H_1: \beta_1 \neq 0\]
  • This asks whether the regressor has a statistically detectable linear effect.
  • The null is not “proved true”; it is either rejected or not rejected.

Five components of a hypothesis test

  1. Null hypothesis (\(H_0\)): the benchmark claim.
  2. Alternative hypothesis (\(H_1\)): the competing claim.
  3. Test statistic: a rule such as the \(t\) statistic.
  4. Rejection region: the values that lead us to reject \(H_0\).
  5. Conclusion: the decision stated in economic language.
  • Good hypothesis testing is a sequence of decisions, not just a p-value.

Test statistic for a coefficient

\[t = \frac{\hat{\beta}_1 - \beta_{1,0}}{se(\hat{\beta}_1)}\]

  • \(\beta_{1,0}\) is the value under the null hypothesis.
  • Usually \(\beta_{1,0} = 0\).
  • A large absolute \(t\) value suggests evidence against the null.
  • The p-value translates the test statistic into probability language.

Type I and Type II errors

  • Type I error: rejecting a true null hypothesis.
  • Type II error: failing to reject a false null hypothesis.
  • The significance level \(\alpha\) controls the probability of Type I error.
  • Lower \(\alpha\) reduces false positives but can increase false negatives.
  • There is always a trade-off between caution and power.

Understanding the p-value

  • The p-value is the probability of observing a test statistic at least as extreme as the one obtained, assuming the null is true.
  • A small p-value means the data would be unusual under the null.
  • Common cut-offs are 10%, 5%, and 1%.
  • A p-value is not the probability that the null is true.
  • Statistical significance is not the same as economic importance.

Fitting a model and reading p-values

Manual t-statistic calculation

Decision rule

  • Choose a significance level, such as \(\alpha = 0.05\).
  • If p-value < 0.05, reject the null at the 5% level.
  • If p-value \ge 0.05, do not reject the null.
  • Always report the estimated effect as well as significance.
  • Practical significance may matter even when p-values are moderate.

Rejection rules for the t-test

  • Right-tail test: reject \(H_0\) if \(t > t_{\alpha, df}\).
  • Two-tail test: reject \(H_0\) if \(|t| > t_{\alpha/2, df}\).
  • P-value rule: reject \(H_0\) whenever \(p \leq \alpha\).
  • These rules are equivalent ways to express the same decision.

Kerala MPC example

  • Suppose the Kerala finance department studies whether household income raises household expenditure.
  • This is the marginal propensity to consume (MPC) question.
  • Test \[H_0: \beta_1 = 0 \qquad \text{vs} \qquad H_1: \beta_1 > 0\]
  • If \(\hat{\beta}_1 = 0.62\) and \(SE(\hat{\beta}_1) = 0.11\), then \[t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)} = \frac{0.62}{0.11} \approx 5.64\]
  • Since this exceeds a typical right-tail critical value such as \(t_{0.05,30} = 1.697\), we reject \(H_0\): the MPC is positive and significant, so a minimum-wage increase is expected to raise aggregate demand.

Statistical versus economic significance

  • A tiny effect can be statistically significant in a very large sample.
  • A large effect can be statistically insignificant in a very noisy or small sample.
  • Economists care about magnitude, sign, and context.
  • Example: a 0.02 increase in repayment rate may be statistically significant but economically trivial.
  • Never interpret p-values without the coefficient itself.

Comparing stronger and weaker effects

Two-sided and one-sided tests

Common mistakes in hypothesis testing

  • Treating p < 0.05 as proof of a large effect.
  • Ignoring the sign and size of the coefficient.
  • Confusing failure to reject with proof that the null is true.
  • Running many tests without clear hypotheses.
  • Forgetting that bad data can make “significant” results meaningless.

Exercise

Estimate a regression of yield_ on fertilizer using simulated data. Print the slope, its standard error, the t-statistic, and the p-value. Then state whether you reject \(H_0: \beta_1 = 0\) at the 5% level.

Summary

  • ✅ Hypothesis testing helps decide whether an observed effect is statistically credible.
  • ✅ The null and alternative hypotheses formalise the question.
  • ✅ The t-statistic compares the estimate with its standard error.
  • ✅ Type I and Type II errors describe different testing mistakes.
  • ✅ P-values guide decisions but do not measure economic importance.
  • ✅ Good inference combines significance, magnitude, and context.

Next Lecture

  • We will study confidence intervals for regression coefficients.
  • These intervals give a range of plausible values for the true parameter.