# Okun's law in U.S. quarterly data, 1960Q1–2019Q4 (F2-conf-04)

## Question
How large is the Okun coefficient β in the linear projection Δu_t = α + β g_t + e_t, where u_t is the quarterly average of monthly UNRATE (pp) and g_t = 100·ln(GDPC1_t/GDPC1_{t−1}) is quarter-on-quarter real GDP growth (percent, not annualised)? Is it negative and significant at the 5% level when the uncertainty allows for serially correlated errors? β is a descriptive projection coefficient. It is not a causal effect.

## Data and exclusions
Inputs are FRED current-vintage files: `UNRATE.csv` with 945 data rows and `GDPC1.csv` with 319 data rows (`n_input_rows`). Their SHA-256 values match `data/README.md`. The full audit is in `out/state/data_audit.md`. UNRATE has one empty value, at 2025-10-01 (`data_issues`). That quarter lies outside the window used, so the gap has no effect. Unemployment is averaged by quarter only when all 3 months are present.

Exclusion ledger (`out/exclusions.csv`), counted in GDPC1 quarters (319):
1. Keep only quarters that have a complete UNRATE quarterly average: 319 → 314. This removes 1947Q1–Q4, before UNRATE starts, and 2025Q4.
2. Drop quarters with no preceding calendar quarter, where the first difference is undefined: 314 → 312. This removes 1948Q1 and 2026Q1.
3. Restrict to the 1960Q1–2019Q4 window: 312 → 240.

The result is n = 240 quarters (`n_analysed`), running from `first_quarter` = 1960Q1 to `last_quarter` = 2019Q4. The 1959Q4 levels are used only to form the 1960Q1 differences.

## Method
- **Estimator.** OLS of Δu_t on a constant and g_t.
- **Baseline.** The conventional homoskedastic, independent-errors standard error (`se_iid`).
- **Main inference.** A Newey–West HAC standard error with Bartlett kernel and no small-sample degrees-of-freedom correction (`se_hac`). It is implemented by hand and cross-checked against statsmodels.
- **Bandwidth.** The Newey–West/Stock–Watson rule of thumb L = floor(4·(T/100)^{2/9}) = floor(4.86) = 4 quarters (`hac_lags`). I chose it before looking at the results. It is a standard automatic choice for T = 240, and four quarters covers one year of dependence, which suits quarterly macro data.
- **CI and test.** The 95% CI is β̂ ± 1.96·se_hac, using a normal critical value (`critical_value`). The test is a two-sided z-test of H0: β = 0.
- **Randomness.** None is used (`random_seed` = null).
- **Figure.** `out/figure1.png` plots Δu_t against g_t with the OLS line.

## Results
- β̂ = −0.256 (`beta_hat`). A quarter with 1 percentage point higher q/q real GDP growth goes with an unemployment rate about 0.26 pp lower than it would otherwise be in that quarter.
- α̂ = 0.184 pp (`alpha_hat`). R² = 0.408 (`r_squared`).
- Mean quarterly growth in the sample is 0.754% (`mean_growth`). The implied growth rate that keeps unemployment unchanged is −α̂/β̂ ≈ 0.72% per quarter, calculated from `alpha_hat` and `beta_hat`.
- The HAC standard error is 0.0358 with 4 lags (`se_hac`, `hac_lags`). The 95% CI is [−0.326, −0.186] (`beta_ci_low`, `beta_ci_high`), with t = −7.14 (`t_stat_hac`) and p ≈ 9.5e−13 (`p_value_hac`).
- The Okun coefficient is negative and significantly different from zero at 5%: `okun_negative_5pct` = true.

## Uncertainty
The baseline iid standard error is 0.0200 (`se_iid`), giving a CI of [−0.295, −0.217] (`baseline_beta_ci_low_iid`, `baseline_beta_ci_high_iid`). The residuals are positively autocorrelated, with first-order autocorrelation 0.39 (`resid_lag1_autocorr`). As a result the HAC standard error is about 1.8 times the iid one, so the iid interval overstates precision.

The HAC standard error changes little with the bandwidth: from 0.032 to 0.037 across lags 1–12 (`se_hac_by_lags`). With 0 lags (heteroskedasticity-robust only) it is 0.029. The significance conclusion is the same at every bandwidth tried.

## Limitations and open issues
- The data are the current vintage, not real-time. Revisions to GDP would change the numbers somewhat.
- β is a reduced-form projection. It is not a structural or causal effect, and it may vary over time, which this analysis does not test.
- The HAC inference is asymptotic, uses a normal critical value and a fixed rule-of-thumb bandwidth, and has no small-sample correction. Fixed-b or other refinements would widen the interval slightly. With |t| ≈ 7 they would not change the conclusion.
- Contemporaneous Δu uses quarterly averages of monthly data, which itself induces some serial correlation. The HAC standard error is designed to absorb this.
- No part of the task was infeasible (`feasibility` = "complete").