# 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):

Keep only quarters that have a complete UNRATE quarterly average: 319 → 314. This removes 1947Q1–Q4, before UNRATE starts, and 2025Q4.
Drop quarters with no preceding calendar quarter, where the first difference is undefined: 314 → 312. This removes 1948Q1 and 2026Q1.
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").

spb/wp29_uqo