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

# Question

How much does the quarterly change in the US unemployment rate respond to real GDP growth, and is the response stable? Estimand: β in Δu_t = α + β·g_t + ε_t, with Δu_t the change in the quarterly-average UNRATE (pp) and g_t = 100·Δln GDPC1_t (% q/q, not annualised), 1960Q1–2019Q4 (240 quarters). Stability: separate fits for 1960Q1–1983Q4 and 1984Q1–2019Q4, plus a classical Chow F-test of joint equality of (α, β).

# Data and exclusions

• data/GDPC1.csv: 319 rows (1947Q1–2026Q3), no missing values, no duplicate dates, regular quarterly spacing. data/UNRATE.csv: 945 rows (1948-01 to 2026-09); one blank value (2025-10-01) that lies outside the sample window and is never used. SHA-256 hashes match data/README.md.
• Quarterly u_t = mean of the three monthly values (only complete quarters formed). The window 1959Q4–2019Q4 was cut before any computation; nothing after 2019Q4 enters an estimate. 1959Q4 levels are used only to form the 1960Q1 differences.
• The GDP series is the 2026 vintage (revised data, chained 2017 dollars), so this is an ex-post analysis, not a real-time one.
• Samples: full n = 240; subperiod 1 n = 96; subperiod 2 n = 144.

# Method

• Baseline: mean-only prediction of Δu (RMSE reported as baseline_metric).
• Model: OLS of Δu_t on a constant and g_t. Inference for β uses Newey–West HAC standard errors (Bartlett kernel) with 4 lags, chosen by the rule floor(4·(n/100)^(2/9)) = 4 for n = 240; 95% CI = β ± 1.96·SE (normal approximation).
• Subperiod OLS fits use the same specification (HAC SEs with 4 lags reported in table1.csv).
• Chow test: F = [(SSR_pooled − SSR_1 − SSR_2)/2] / [(SSR_1 + SSR_2)/(n − 4)] with (2, 236) df, as required.
• Supplementary checks (out/hac_sensitivity.csv, out/diagnostics.json): SE with 0–12 lags, iid and HC1 SEs; HAC-robust Wald version of the break test using interaction dummies; Ljung–Box on residuals.

# Results

(Table 1 = out/table1.csv; Figure 1 = out/figure1.png, a scatter of Δu against g with the two subperiod fitted lines.)

• Full-sample Okun coefficient β = −0.256: one extra percentage point of quarterly (non-annualised) real growth goes with about a 0.26 pp fall in unemployment in that quarter. α = 0.184 pp, so the implied growth rate that keeps unemployment unchanged is about 0.72 % per quarter (≈2.9 % annualised).
• Fit: residual RMSE 0.250 pp against 0.325 pp for the mean-only baseline (R² = 0.41).
• Stability: the Chow test gives F = 6.09, p = 0.0026, so equality is rejected at 5% → unstable. However, the slopes are almost the same (−0.259 vs −0.273). The rejection comes mainly from the intercept: the implied break-even growth rate falls from about 0.98 % to 0.55 % per quarter, which fits slower trend growth after 1984. In a HAC-robust interaction regression, the joint test also rejects (p = 0.005), but the slope difference on its own clearly does not (p = 0.85) and the intercept difference on its own does not either (p = 0.26). Intercept and slope shifts are hard to separate one at a time, but together they are significant. Conclusion: the responsiveness β looks stable, while the relationship as a whole (α, β) is not.
• Conclusion code: negative_effect, since the 95% CI for β lies entirely below zero.

# Uncertainty

• HAC (4 lags) SE(β) = 0.036. 95% CI [−0.326, −0.185].
• Sensitivity of SE(β) to the lag length: 0.029 (0 lags, = HC1), 0.032 (1), 0.034 (2), 0.036 (4), 0.037 (8), 0.037 (12). The iid OLS SE is 0.020. At every lag the interval stays well below zero, so the sign conclusion does not depend on the bandwidth choice. Residuals are strongly autocorrelated (Ljung–Box at lag 4, p ≈ 6e-13), so the iid SE would understate the uncertainty, and HAC is needed.
• Subperiod 2 has a much wider SE (0.069) than subperiod 1 (0.040), even though it has more observations. This reflects the smaller spread of growth during the Great Moderation and the dominant influence of a few recession quarters (2008–09).

# Limitations and open issues

• The Chow test's validity is doubtful. The classical Chow F assumes homoskedastic, serially uncorrelated errors with a common variance across regimes. Residual variance falls from 0.079 to 0.048 between subperiods, and residuals are strongly autocorrelated. So the classical p-value (0.0026) is not reliable on its own terms. The HAC Wald analogue still rejects (p = 0.005), which is reassuring, but I did not run a Wald test that allows different variances in each regime, or a bootstrap.
• The break date is fixed in advance (1984Q1). No unknown-break test (Quandt–Andrews / Bai–Perron) or rolling-window analysis was done. Instability elsewhere (e.g., jobless recoveries after 1990, 2001, 2009) could be missed or wrongly assigned to 1984.
• The model is static, contemporaneous and bivariate. There are no lags of g or Δu, even though the residual autocorrelation points to dynamics (labour hoarding, delayed adjustment). Then β is the impact response, not the long-run response, and it is a correlation, not a causal effect. Simultaneity between output and unemployment is not addressed.
• Data vintage. Revised 2026-vintage GDP is used. Real-time estimates of β, or estimates on other vintages or chain bases, could differ. Measurement changes in UNRATE (e.g., the 1994 CPS redesign, population-control adjustments) were not corrected for.
• Outliers and influence. Not checked formally (e.g., 1980 credit controls, 2008Q4–2009Q1). Subperiod 2's slope may depend heavily on a handful of recession quarters.
• What would overturn the conclusions: the sign conclusion is robust across every SE choice tried. The "unstable" verdict could be reversed by a test that is robust to heteroskedasticity across regimes, or by a different break date. The finding that "the slope is stable and only the intercept shifts" rests on separate tests that are individually not significant, so it is suggestive rather than established.
• Supplementary files hac_sensitivity.csv and diagnostics.json are produced by analysis.py but are not part of the required outputs.