1# Okun's law in the United States, 1960–2019 (F6-conf-03)23## Question4How strongly does the quarterly change in the US unemployment rate (Δu_t, pp) respond to quarterly real GDP growth (g_t = 100·Δln GDPC1, % not annualised), in Δu_t = α + β·g_t + ε_t over 1960Q1–2019Q4? Was the relation stable between 1960Q1–1983Q4 and 1984Q1–2019Q4? β is a reduced-form (descriptive) association, not a causal effect.56## Data and exclusions7Inputs: `data/GDPC1.csv` (319 quarterly rows) and `data/UNRATE.csv` (945 monthly rows) (`n_input_rows`); SHA-256 hashes match the README (see `out/state/data_audit.md`). Quarterly u_t is the mean of the three monthly UNRATE values. Exclusions (`out/exclusions.csv`, counted in GDP quarters): 51 quarters before 1959Q4 dropped, 27 quarters after 2019Q4 dropped (information rule), 1959Q4 used only as the lag level → 240 quarters analysed (`n_analysed`), from `sample_start` = 1960-01-01 to `sample_end` = 2019-10-01; subperiods have `n_sub1` = 96 and `n_sub2` = 144 quarters. Data issues (`data_issues`): one blank UNRATE value (2025-10), outside the sample, so it has no effect; both files extend past 2019Q4 and were truncated; GDPC1 is the current revised vintage.89## Method10- Baseline: mean-only prediction of Δu; RMSE `baseline_metric`.11- Main estimate: full-sample OLS of Δu on a constant and g. Standard errors are Newey–West HAC (Bartlett kernel, no small-sample correction). The lag length is `hac_lags` = 4, from the rule floor(4·(n/100)^(2/9)) with n = 240. The 95% CI is β ± 1.96·SE.12- Stability: separate OLS on each subperiod. The classical Chow test compares the pooled sum of squared residuals with the sum from the two subperiod fits, F = ((SSR_p − SSR_u)/2)/(SSR_u/(n−4)), with df (2, 236).13- Supplementary diagnostics (not required): HAC SE at other lag lengths (`hac_se_sensitivity`), and a HAC t-test of a break in the slope alone, run as a dummy-interaction regression (`slope_break_hac_pvalue`).14- The analysis is deterministic and uses no random numbers (`random_seed` = null). Outputs: `out/table1.csv` (slope_se = HAC SE, 4 lags, in each row) and `out/figure1.png`.1516## Results17- Full sample: β = −0.256 (`effect_estimate`) and α = 0.184 pp (`intercept`), so 1 pp higher quarterly real growth goes with about a 0.26 pp larger fall in the unemployment rate that quarter. R² = 0.41 (`r_squared_full`).18- Fit: residual RMSE is 0.250 pp (`model_metric`), against 0.325 pp for the mean-only baseline (`baseline_metric`).19- The 95% HAC interval lies entirely below zero, so `conclusion` = `negative_effect`.20- Subperiod slopes are nearly equal: −0.259 in 1960–83 (`beta_sub1`) and −0.273 in 1984–2019 (`beta_sub2`).21- The joint Chow test of (α, β) rejects equality: F = 6.09 (`chow_f_stat`), p = 0.0026 (`chow_p_value`), so `stability_conclusion` = `unstable`.22- The rejection comes from the intercept, not the slope. The intercept falls from 0.254 (`intercept_sub1`) to 0.152 (`intercept_sub2`), while the slope-only HAC break test gives p = 0.85 (`slope_break_hac_pvalue`). One reading is that the growth rate needed to keep unemployment stable (−α/β) fell after 1984, but this is an interpretation that was not tested formally.2324Table 1 (`out/table1.csv`) gives n, intercept, slope and HAC SE per sample. Figure 1 (`out/figure1.png`) shows the scatter with the two subperiod fitted lines.2526## Uncertainty27- HAC SE of β = 0.0358 (`effect_se`). 95% CI: [−0.326, −0.186] (`effect_ci_low`, `effect_ci_high`).28- The classical OLS SE is 0.0200 (`ols_classical_se_full`), so correcting for autocorrelation and heteroskedasticity widens the SE by about 80%.29- Lag sensitivity (`hac_se_sensitivity`): SE is 0.029 at 0 lags, 0.034 at 2, 0.037 at 8 and 0.036 at 12. The interval excludes zero at every lag length tried.30- The Chow p-value assumes homoskedastic, serially uncorrelated errors. The HAC/classical SE ratio shows these assumptions fail here, so that p-value is probably too small (over-rejects). The stability verdict is required to rest on the classical test and is reported as such, but should be read with this caveat.3132## Limitations and open issues33- **What would invalidate the result:**34 - The Chow test's i.i.d.-error assumption is violated (serial correlation is evident). A HAC-robust joint Wald test might not reject, so the `unstable` verdict is fragile. It was not computed, because the task specifies the classical test.35 - The break date (1984Q1) is imposed, not estimated. Unknown-break tests (sup-F/Quandt–Andrews) were not run.36 - The sample-split result depends on that choice.37- **Not checked:**38 - Endogeneity / simultaneity. β is a projection coefficient, not a structural parameter.39 - Dynamic specifications with lagged growth. The contemporaneous-only model may misattribute timing.40 - Real-time vs revised GDP. The current GDPC1 vintage is used, and earlier vintages might give different values.41 - Outliers (e.g. the 1980–82 recessions, 2008–09) and their influence.42 - Asymmetry between expansions and recessions.43 - Changes in UNRATE methodology (e.g. the 1994 CPS redesign).44- **Main threats:**45 - The Chow rejection is driven by an intercept shift, probably reflecting slower trend growth and labour-force changes. This means "unstable" here does not mean the Okun slope changed: the slope-only test does not reject (`slope_break_hac_pvalue`).46 - The HAC lag choice changes the SE modestly but not the sign conclusion.47