# Okun's law in the United States, 1960Q1–2019Q4 (F6-conf-03)

# Question

How strongly do quarterly changes in the US unemployment rate respond to real output growth, and has that response been stable? Estimand: β in Δu_t = α + β·g_t + ε_t. Here Δu_t is the change in the quarterly-average UNRATE (pp) and g_t = 100·Δln GDPC1_t (% q/q, not annualised). The sample is 1960Q1–2019Q4. Stability is assessed with separate fits for 1960Q1–1983Q4 and 1984Q1–2019Q4 and a classical Chow test.

# Data and exclusions

• data/GDPC1.csv has 319 quarterly rows (1947Q1–2026Q3) and data/UNRATE.csv has 945 monthly rows (1948-01 – 2026-09) (n_input_rows). The SHA-256 hashes match the provenance README (see out/state/data_audit.md).
• Data issues (data_issues): UNRATE has one missing value, at 2025-10-01. It lies outside the window and is removed by the 2019Q4 cutoff, so no imputation was needed. Both series extend past 2019Q4, and those rows are dropped before any computation (information rule).
• Quarterly u_t is the simple mean of the 3 monthly values. A check confirms that every quarter in the window has all 3 months present.
• Exclusion ledger (out/exclusions.csv), on the quarterly grid: 319 → drop the 51 quarters before 1959Q4 → 268 → drop the 27 quarters after 2019Q4 → 241 → drop 1959Q4, which serves only as the lag level → 240. The regression uses n_analysed = 240 quarters, from sample_start = 1960-01-01 to sample_end = 2019-10-01. Subperiods have n_sub1 = 96 and n_sub2 = 144 quarters.

# Method

• Baseline: a mean-only prediction of Δu. Its RMSE is baseline_metric.
• Main model: full-sample OLS of Δu_t on a constant and g_t. Inference uses Newey–West HAC standard errors (Bartlett kernel, small-sample correction). The lag length comes from the Newey–West rule of thumb floor(4·(n/100)^(2/9)), which gives hac_lags = 4 for n = 240. The 95% CI is β ± 1.96·effect_se (normal approximation). The conclusion code is set from the sign of this interval.
• Stability: separate OLS fits on each subperiod (beta_sub1, beta_sub2). The classical Chow F-test uses F = [(SSR_pooled − SSR_1 − SSR_2)/2] / [(SSR_1 + SSR_2)/(n − 4)] with df (2, 236) (chow_f_stat, chow_p_value).
• Supplementary diagnostic (not required): a HAC Wald test of the break, using a full-sample regression with a post-1984 dummy and a dummy×g interaction (same 4 lags). It is run both jointly and for the slope alone (hac_wald_break_*). This checks whether the Chow result survives heteroskedasticity and autocorrelation.
• No randomness is involved (random_seed = null). out/analysis.py regenerates results.json, table1.csv and figure1.png deterministically.

# Results

• Okun coefficient (full sample): effect_estimate = −0.256 pp per 1% q/q growth, with intercept = 0.184 pp. So 1 pp of extra quarterly real growth goes with an unemployment rate about a quarter of a point lower that quarter. The model has r_squared = 0.41.
• The model fits better than the baseline: residual RMSE model_metric = 0.250 pp versus mean-only RMSE baseline_metric = 0.325 pp.
• Subperiod slopes are almost identical: beta_sub1 = −0.259 and beta_sub2 = −0.273. The intercepts differ: intercept_sub1 = 0.254 and intercept_sub2 = 0.152.
• Chow test: chow_f_stat = 6.09, chow_p_value = 0.0026. Hence stability_conclusion = unstable: joint equality of (α, β) is rejected at 5%.
• The rejection comes from the intercept, not the slope. The HAC Wald test of a joint break also rejects (hac_wald_break_joint_p = 0.0046). The HAC test of a slope-only break does not (hac_wald_break_slope_p = 0.85). In other words, the growth rate at which unemployment stays flat has fallen since 1984 (consistent with slower trend growth), while the Okun slope itself shows no detectable change.
• Table 1 (out/table1.csv) lists n, intercept, slope and HAC slope SE for the full sample and each subperiod. Figure 1 (out/figure1.png) is a scatter of Δu against g with the two subperiod fitted lines.
• conclusion = negative_effect: the 95% CI lies entirely below zero.

# Uncertainty

• The HAC SE is effect_se = 0.036, giving a 95% CI for β of [effect_ci_low, effect_ci_high] = [−0.326, −0.185].
• The HAC SE is about 1.8 times the iid OLS SE (ols_se_beta_iid = 0.020), which points to serial correlation and/or heteroskedasticity in the residuals.
• Subperiod HAC SEs: se_beta_sub1 = 0.039 (3 lags) and se_beta_sub2 = 0.069 (4 lags). The difference between the subperiod slopes (≈0.014) is far smaller than these SEs.
• Residual dispersion differs across subperiods (sigma_resid_sub1 = 0.277, sigma_resid_sub2 = 0.218). The classical Chow test assumes equal error variance and no serial correlation, so its p-value is only approximate. The HAC Wald diagnostic reaches the same joint conclusion.
• The results are sensitive to the HAC lag choice, but only one lag rule was reported. Alternative lags were not tabulated.

# Limitations and open issues

• What would invalidate the result: (i) If the break date is not 1984Q1. It is imposed by the task, and an endogenous break search (e.g. Quandt–Andrews) was not run. Testing at a date the data suggested would distort the size of the test. (ii) If g_t is endogenous: β is a reduced-form projection, not a causal effect. Unemployment and output are jointly determined, and measurement error in GDP growth biases |β| towards zero (attenuation). (iii) Nonlinearity or asymmetry: unemployment may rise faster in recessions than it falls in expansions. A linear model averages over this, and the large positive Δu outliers in recessions (Figure 1) have high leverage.
• Data vintage: GDPC1 is the 2026 vintage, so revised GDP is used. Coefficients estimated on real-time data would differ. That does not matter for this ex post question, but it does matter for any forecasting use.
• Not checked: sensitivity to the HAC lag length or to kernel/prewhitening choices; Andrews or fixed-b HAC inference; dynamic specifications (lags of g_t, the gap version of Okun's law); robustness to dropping recession quarters or outliers; trend breaks inside each subperiod (e.g. jobless recoveries after 1990, 2001 and 2009).
• Main threats: The Chow test assumes homoskedastic, serially independent errors, and both assumptions look violated (see Uncertainty). The qualitative "unstable" verdict is backed by the HAC Wald test, but the exact p-value should not be over-interpreted. The "unstable" label refers to (α, β) jointly. Readers should not take it to mean that the Okun slope changed: the data do not support that.