1# Okun's law in U.S. quarterly data, 1960Q1–2019Q4 (F2-conf-04)23## Question4How strongly does the change in the U.S. unemployment rate co-move with quarterly real GDP growth? We estimate the slope β of the linear projection Δu_t = α + β g_t + e_t, where u_t is the quarterly mean 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), and ask whether β is negative and significantly different from zero at 5% using autocorrelation-robust (Newey–West) inference.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`. Current vintage, used as given.8- One data issue (`data_issues`): UNRATE for 2025-10 is empty. It lies outside the window used and is dropped without affecting any analysed quarter. Every quarter 1959Q4–2019Q4 has all three monthly unemployment observations and a GDP value (asserted in the script).9- Exclusions (`out/exclusions.csv`, quarterly units): 319 quarters in the union of both series → 240 after restricting to 1960Q1–2019Q4 (79 removed; 1959Q4 levels used only to form the 1960Q1 differences) → 240 after dropping missing differences (0 removed). Sample: `first_quarter` = 1960Q1, `last_quarter` = 2019Q4, `n_analysed` = 240.1011## Method12- OLS of Δu_t on a constant and g_t (statsmodels). The conventional homoskedastic/independent SE (`se_iid`) is the naïve baseline.13- Newey–West HAC SE with Bartlett kernel (`se_hac`), no small-sample degrees-of-freedom correction. Bandwidth `hac_lags` = 4 from the Newey–West (1994) rule of thumb for the Bartlett kernel, L = floor(4·(T/100)^{2/9}) with T = 240 (= floor(4.86)); the alternative common rule 0.75·T^{1/3} ≈ 4.66 gives the same order. The choice is supported by the residual autocorrelations, which are sizeable at lags 1–2 and small by lag 4 (`resid_acf_lags1to4`).14- 95% CI = β̂ ± 1.96·`se_hac` (normal critical value, `z_crit_95`); two-sided test of H0: β = 0 with the HAC t-statistic.15- Deterministic; no random numbers (`random_seed` = null). Results were verified by an independent numpy re-implementation (agreement < 1e-14) and by a byte-identical re-run (`out/state/checks.md`).1617## Results18- Okun coefficient `beta_hat` = −0.256: a 1 percentage point higher quarterly (non-annualised) real GDP growth is associated with a 0.256 pp larger fall in the unemployment rate in the same quarter.19- Intercept `alpha_hat` = 0.184 pp; `r_squared` = 0.408. Mean quarterly growth `mean_growth` = 0.754%, so the implied growth rate consistent with unchanged unemployment is about −α̂/β̂ ≈ 0.72% per quarter (derived from `alpha_hat` and `beta_hat`).20- `okun_negative_5pct` = true: β̂ < 0 and the HAC t-statistic is −7.14 (`t_stat_hac`, p ≈ 1e-12, `p_value_hac`).21- Figure: `out/figure1.png` (scatter of Δu_t against g_t with the fitted OLS line).2223## Uncertainty24- `se_iid` = 0.0200 (baseline) vs `se_hac` = 0.0358 (L = 4): the HAC SE is about 1.8 times the naïve SE, consistent with positively autocorrelated residuals (lag-1 autocorrelation 0.39, `resid_acf_lags1to4`).25- 95% HAC CI: [`beta_ci_low`, `beta_ci_high`] = [−0.326, −0.186]. The naïve interval would be [−0.295, −0.217] (`baseline_ci_iid_low`, `baseline_ci_iid_high`), i.e. too narrow.26- Bandwidth sensitivity (`se_hac_by_lag`): SE ranges from 0.029 (L = 0, heteroskedasticity-robust only) to 0.037 (L = 8); with L = 12 it is 0.036. At the largest of these (≈0.037) |t| ≈ 6.9, so the significance conclusion does not depend on the bandwidth.2728## Limitations and open issues29- β is a reduced-form linear projection coefficient, not a causal effect; it is a full-sample average and ignores possible structural change over 1960–2019 and asymmetries between expansions and recessions.30- Data are the current vintage, not real-time; GDP revisions mean results describe today's view of history.31- Newey–West inference is asymptotic; with strong persistence, fixed-b critical values would widen the CI somewhat, but given |t| ≈ 7 the conclusion would not change. This was not computed.32- The bandwidth is a rule of thumb rather than a data-dependent (Andrews/Newey–West automatic) selection; sensitivity over 0–12 lags is reported instead.33- No other limitations: the task was completed as specified (`feasibility` = "complete").34