1# Okun's law in U.S. quarterly data, 1960Q1–2019Q4 (F2-conf-04)23## Question4Estimate β in the linear projection Δu_t = α + β g_t + e_t, where u_t is the quarterly average of monthly UNRATE (pp), Δu_t = u_t − u_{t−1}, and g_t = 100·ln(GDPC1_t/GDPC1_{t−1}) (percent, quarter-on-quarter, not annualised), over 1960Q1–2019Q4, with a Newey–West (Bartlett) HAC standard error, and decide whether β is negative and significantly different from zero at 5% (two-sided).56## Data and exclusions7- `data/UNRATE.csv`: 945 monthly rows, `data/GDPC1.csv`: 319 quarterly rows (`n_input_rows`). Both SHA-256 checksums match `data/README.md` (see `out/state/data_audit.md`).8- One data issue: UNRATE for 2025-10 is an empty field; it lies outside the 1959Q4–2019Q4 levels window and has no effect (`data_issues`).9- Exclusions (`out/exclusions.csv`, unit = quarters of the merged quarterly panel, 319): step 1 keeps 1959Q4–2019Q4 (78 removed, information rule); step 2 drops 1959Q4, which serves only as the differencing base (1 removed). 319 − 79 = 240 = `n_analysed`; sample runs `first_quarter` = 1960Q1 to `last_quarter` = 2019Q4. Every quarter in the window has all 3 monthly unemployment observations.1011## Method12- Quarterly u_t = mean of the three monthly SA rates; Δu_t and g_t as defined in the task (log growth ×100, not annualised).13- OLS of Δu_t on a constant and g_t (statsmodels). Baseline uncertainty: conventional homoskedastic, serially independent OLS SE (`se_iid`).14- HAC: Newey–West Bartlett kernel, weights 1 − j/(L+1), no small-sample degrees-of-freedom correction, normal critical value 1.96. Bandwidth L = `hac_lags` = 4, from the Newey–West (1994) plug-in rule floor(4·(T/100)^{2/9}) with T = 240 (`hac_lags_rule_nw1994`); the textbook rule floor(0.75·T^{1/3}) gives the same value (`hac_lags_rule_sw`). The statsmodels HAC SE was verified against a hand-coded Bartlett estimator.15- No randomness (`random_seed` = null). `out/analysis.py` regenerates `out/results.json`, `out/figure1.png`, `out/hac_sensitivity.csv`, `out/exclusions.csv`.1617## Results18- Okun coefficient `beta_hat` = −0.256: a 1 percentage-point higher quarterly real GDP growth is associated with a 0.256 pp larger fall in the unemployment rate in that quarter.19- Intercept `alpha_hat` = 0.184 pp; `r_squared` = 0.408.20- Mean quarterly growth `mean_growth` = 0.754% (about 3.0% annualised). The growth rate at which predicted Δu = 0 is −α̂/β̂ ≈ 0.72% per quarter (derived from `alpha_hat` and `beta_hat`).21- Figure: `out/figure1.png` (scatter of Δu_t against g_t with the fitted line).22- `okun_negative_5pct` = true: β̂ < 0 and H0: β = 0 is rejected at 5% with the HAC SE (`t_stat_hac` = −7.14, `p_value_hac` ≈ 9.5e−13).2324## Uncertainty25- `se_iid` = 0.0200 (naïve baseline; 95% CI [`baseline_ci_iid_low`, `baseline_ci_iid_high`] = [−0.295, −0.217]).26- `se_hac` = 0.0358 with `hac_lags` = 4; 95% HAC CI [`beta_ci_low`, `beta_ci_high`] = [−0.326, −0.186].27- The HAC SE is about 1.8 times the iid SE (ratio of `se_hac` to `se_iid`), consistent with positively autocorrelated OLS residuals (`resid_autocorr_lag1_4`: 0.39, 0.29, 0.14, 0.07 at lags 1–4).28- Bandwidth sensitivity (`out/hac_sensitivity.csv`): over L = 0…12 the HAC SE never exceeds `se_hac_max_over_lags_0_12` = 0.0371 and the largest p-value is `p_value_hac_max_over_lags_0_12` ≈ 5.7e−12, so the significance conclusion does not depend on the bandwidth. A small-sample df correction (n/(n−k)) changes the SE by under 0.5% (same CSV).2930## Limitations and open issues31- β is a descriptive linear-projection (reduced-form) coefficient, not a causal effect. It is also contemporaneous only (no lags of g_t), as the task specifies.32- Current-vintage data are used, not real-time data; GDP revisions and the 2017 chain base are taken as given.33- The coefficient is assumed constant over 1960–2019; parameter stability (for example across decades or recessions) was not tested because the task does not ask for it.34- The HAC inference is asymptotic (normal critical values, Bartlett kernel). With T = 240 and |t| ≈ 7, the conclusion is far from the margin, so fixed-b or other small-sample refinements are unlikely to change it. They were not computed.35- No part of the task was infeasible (`feasibility` = "complete").36