Confidence Intervals

ECON 3209 · Week 9, Lecture 2 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

Explain what a confidence interval is and what it is not.
Construct a 95% confidence interval for an OLS coefficient.
Interpret interval width using sample size and variability.
Compute coefficient confidence intervals in Python.
Relate confidence intervals to hypothesis testing.

From point estimate to interval estimate

A point estimate gives one best guess for a parameter.
A confidence interval gives a range of plausible values.
Wider intervals indicate more uncertainty.
Narrower intervals indicate greater precision.
Intervals are often more informative than a yes/no test result.

The 95% confidence interval formula

\[CI_{0.95} = [\hat{\beta}_i - 1.96 \times SE(\hat{\beta}_i),\; \hat{\beta}_i + 1.96 \times SE(\hat{\beta}_i)]\]

\[\hat{\beta}_1 \pm t_{\alpha/2,\,df} \times se(\hat{\beta}_1)\]

The centre is the estimated coefficient.
The margin of error depends on the critical value and standard error.
For large samples, 1.96 is the common 95% normal critical value.
Larger standard errors create wider intervals.
Confidence intervals come from the sampling distribution of the estimator.

Correct interpretation

A 95% confidence interval does not mean there is a 95% probability that the fixed true parameter lies inside this specific interval.
It means that in 95% of all random samples, the interval construction method will contain the true \(\beta\).
In practice, we treat the interval as a plausible range for the coefficient.
The interval combines effect size and uncertainty in one object.
This is why intervals are widely reported in empirical studies.

Estimating a model and extracting intervals

Manual 95% CI calculation

Kerala example: confidence interval for rubber-price response

Suppose we regress the Kerala rubber price on national rubber production and obtain \[\hat{\beta}_{production} = -238.89, \qquad SE(\hat{\beta}_{production}) = 53.59\]
A manual 95% confidence interval is \[-238.89 \pm 1.96(53.59) = [-343.93,\,-133.85]\]
Because the interval is entirely negative, higher national production is associated with lower Kerala rubber prices in this example.
The interval also shows uncertainty: the effect may be moderately negative or strongly negative, but the sign is consistently below zero.

What affects interval width?

Larger samples reduce standard errors and narrow intervals.
Greater noise in the data widens intervals.
Less variation in the regressor widens intervals.
Higher confidence levels produce wider intervals.
Precision is earned through better data and better design.

Comparing small and large samples

Plotting a confidence interval

Link with hypothesis testing

If the 95% confidence interval excludes zero, we reject \(H_0: \beta_1 = 0\) at the 5% level.
If the interval includes zero, we do not reject at the 5% level.
So confidence intervals and two-sided hypothesis tests are closely related.
Intervals often give more intuition because they show magnitude and uncertainty together.
Researchers should report both estimates and intervals.

Reporting intervals clearly

Report the coefficient and the confidence interval together.
State the confidence level explicitly.
Use economic units, not only statistical symbols.
Example: “A one-unit increase in fertilizer raises yield by 0.28 tonnes/ha (95% CI: 0.22 to 0.34).”
Clear reporting improves interpretation and transparency.

Using confidence intervals responsibly

A narrow interval around zero may still imply little economic importance.
A wide positive interval indicates uncertainty even if the point estimate is large.
Confidence intervals do not solve bias problems from bad identification.
Good intervals require good models and good data.
Statistical tools cannot replace economic reasoning.

Exercise

Simulate a regression of repayment_rate on income. Estimate the slope and calculate a 95% confidence interval both with model.conf_int() and manually using the t critical value.

Summary

✅ Confidence intervals provide a range of plausible parameter values.
✅ A 95% interval equals estimate ± critical value × standard error.
✅ Larger samples and less noise produce narrower intervals.
✅ Confidence intervals and two-sided hypothesis tests are closely linked.
✅ Reporting intervals improves interpretation beyond p-values alone.
✅ Precision must always be discussed together with economic meaning.

Next Lecture

We will apply t and F tests in Python.
The goal is to test single coefficients and groups of coefficients using statsmodels output.