1# Okun's law in the United States, 1960Q1–2019Q4 (F6-conf-03)23## Question4How much does the quarterly change in the US unemployment rate respond to quarterly real GDP growth, and is that response stable? Estimand: β in Δu_t = α + β·g_t + ε_t, where u_t is the quarterly mean of monthly UNRATE (pp) and g_t = 100·Δln GDPC1_t (% q/q, not annualised). Sample: 1960Q1–2019Q4. Stability is checked with subperiods 1960Q1–1983Q4 and 1984Q1–2019Q4 and a classical Chow test.56## Data and exclusions7- `GDPC1.csv`: 319 rows, 1947Q1–2026Q3, no gaps, no missing values. `UNRATE.csv`: 945 rows, 1948-01 to 2026-09, no month gaps. One value is missing (2025-10-01, empty cell). It falls outside the window, so it has no effect.8- u_t is the mean of the three monthly values. Only complete quarters are used, and all quarters in the window are complete. The regression uses 240 quarters: 96 in subperiod 1 and 144 in subperiod 2. The 1959Q4 levels are used only to form the 1960Q1 differences.9- Everything after 2019Q4 is excluded, as the task requires. That includes the 2020 COVID quarters, which are extreme.10- GDPC1 is the current (2026) vintage. It is revised data, not real-time data. Log growth rates do not depend on the base year.11- Two in-sample quarters have |z| > 4: 1975Q1 and 2009Q1. Both are genuine recession quarters, so I kept them. Their influence is examined below.1213## Method14- Baseline: predict Δu by its sample mean. Model: OLS of Δu on a constant and g. Performance is measured as in-sample RMSE.15- Full-sample inference uses Newey–West HAC standard errors with a Bartlett kernel. The lag length is L = floor(4·(T/100)^(2/9)) = 4 for T = 240, the usual Newey–West rule. The 95% CI is β ± 1.96·SE (normal approximation). Subperiod slope SEs in `table1.csv` are also HAC with L = 4.16- Stability test: classical Chow F = [(SSR_pooled − SSR_1 − SSR_2)/2] / [(SSR_1 + SSR_2)/(T − 4)], with F(2, 236) degrees of freedom.17- Extra diagnostics (in `out/sensitivity.json`, also written by `analysis.py`): HAC SEs at lags 0, 2, 4, 8 and 12; β with the two |z| > 4 quarters dropped; and a HAC-robust Wald version of the break test from an interaction regression with dummies for 1984 onward.1819## Results20| sample | n | α | β | HAC SE(β) |21|---|---|---|---|---|22| full 1960Q1–2019Q4 | 240 | 0.184 | −0.256 | 0.036 |23| 1960Q1–1983Q4 | 96 | 0.254 | −0.259 | 0.039 |24| 1984Q1–2019Q4 | 144 | 0.152 | −0.273 | 0.068 |2526- Full sample: β = −0.256 and α = 0.184. Each extra 1% of quarterly real GDP growth goes with an unemployment rate about 0.26 pp lower than otherwise. The growth rate at which unemployment stays flat is −α/β ≈ 0.72% per quarter.27- Fit: RMSE falls from 0.325 pp (mean-only baseline) to 0.250 pp (OLS). R² is 0.41.28- Stability: the Chow F is 6.09 (p = 0.0026), so the classical test **rejects** equality of (α, β) at 5%. The `stability_conclusion` is therefore `unstable`.29- The two slopes are almost identical (−0.259 vs −0.273). The rejection comes from a shift in the intercept/slope combination, not from a change in β alone:30 - The flat-unemployment growth rate falls from about 0.98% to about 0.55% per quarter. This fits slower trend growth after 1984.31 - In the HAC interaction regression, the slope difference alone is −0.014 (p = 0.85) and the intercept difference alone is −0.10 (p = 0.25). Their joint HAC Wald test still rejects (F = 5.46, p = 0.005). The individual estimates are imprecise because they are highly correlated.32 - So the Okun *slope* looks stable. The *relationship as a whole* (α, β) does not.33- Figure: `figure1.png`. Table: `table1.csv`.3435## Uncertainty36- HAC SE(β) = 0.0358 with L = 4. 95% CI: [−0.326, −0.186]. The interval lies entirely below zero, so `conclusion = negative_effect`.37- Sensitivity of SE(β) to the lag choice:38 - iid OLS: 0.02039 - L = 0: 0.02940 - L = 2: 0.03441 - L = 8: 0.03742 - L = 12: 0.03643 44 The sign conclusion holds for every lag choice. Positive autocorrelation makes the iid SE much too small.45- Dropping 1975Q1 and 2009Q1 moves β to −0.229. The estimate depends somewhat on a few deep-recession quarters but stays clearly negative.46- The classical Chow test assumes iid homoskedastic errors, which the HAC diagnostics show is false. Its p-value is therefore optimistic. The HAC-robust joint test gives the same answer (p ≈ 0.005), which makes the rejection more credible.4748## Limitations and open issues49- **What would invalidate the result:**50 - β is a reduced-form correlation, not a causal effect. Δu and g are determined together, and the static difference form leaves out lags of g. Okun responses are known to be spread over several quarters, so the contemporaneous β probably understates the cumulative response.51 - If the errors have stronger long-memory dependence than a Bartlett kernel with 4 lags can capture, the CI is too narrow. The lag sensitivity above suggests this effect is modest.52- **Break date:** the 1984Q1 split is imposed by the task (start of the Great Moderation), not estimated. I did not run Quandt–Andrews or Bai–Perron tests. Rejection at one fixed date does not show that 1984 is the true break date, and it does not rule out several breaks or gradual drift.53- **The Chow test answers a joint question.** Its rejection should not be read as "Okun's coefficient changed". In these data the slopes are statistically indistinguishable, and the instability is in the intercept, i.e. in trend productivity and labour-force growth.54- **Data vintage:** current-vintage GDP includes decades of revisions and benchmark changes. Results from real-time data, or from GDI instead of GDP, could differ, especially for the noisy quarterly growth figures. I checked neither.55- **Not checked:**56 - asymmetry between recessions and expansions57 - level / gap versions of Okun's law58 - HP- or CBO-gap specifications59 - nonlinearity60 - robustness to excluding entire recessions61- **Outliers:** two quarters have notable leverage (1975Q1, 2009Q1). Robust regression was not used.62