# F2-conf-04 — Okun's law, U.S. quarterly data, 1960Q1–2019Q4

# Question

Estimate β in Δu_t = α + β g_t + e_t. Here u_t is the quarterly average of the monthly unemployment rate (UNRATE), and g_t = 100·ln(GDPC1_t/GDPC1_{t−1}) is real GDP growth in percent quarter-on-quarter, not annualised. The sample is 1960Q1–2019Q4 (240 quarters). The standard error must stay valid when the errors are serially correlated (Newey–West HAC). The question is whether β is negative and significant at the 5% level.

# Data and exclusions

• data/UNRATE.csv: 945 monthly rows, 1948-01 to 2026-09. data/GDPC1.csv: 319 quarterly rows, 1947Q1 to 2026Q3. Both are the current FRED vintage and were used as given. Dates are unique, monotonic and regularly spaced.
• Quarterly u_t is the mean of the 3 months in the quarter. A quarter counts only if all 3 months are present.
• Data issue: UNRATE has an empty value for 2025-10. This is outside the window needed (1959Q4–2019Q4), so it has no effect. GDPC1 has no missing values.
• The 1959Q4 levels are used only to form the 1960Q1 differences. No observation after 2019Q4 is used. All 240 quarters in the window have complete data, so none were excluded.

# Method

• OLS of Δu_t on a constant and g_t, using statsmodels.
• Naïve SE: the conventional homoskedastic, independent-errors OLS standard error.
• HAC SE: Newey–West with the Bartlett kernel. Bandwidth L = floor(4·(T/100)^{2/9}) = floor(4·2.4^{0.222}) = 4 quarters. This is the standard Newey–West (1994) rule-of-thumb plug-in for the Bartlett kernel. Four quarters is also one year, which is a natural horizon for quarterly macro data. statsmodels' small-sample correction (factor T/(T−k)) is applied.
• 95% CI: β̂ ± 1.96·se_hac, using the normal approximation, which is the standard asymptotic justification for HAC inference. Two-sided test of H0: β = 0 with the HAC SE.
• The whole analysis is deterministic and uses no random numbers. Running out/analysis.py regenerates results.json, figure1.png and a supplementary file, diagnostics.json.

# Results

A 1 percentage point increase in quarterly real GDP growth goes with a 0.26 pp fall in the unemployment rate that quarter. The implied growth rate at which unemployment stays constant is −α̂/β̂ ≈ 0.72% per quarter, close to the sample mean growth of 0.75%. The Okun coefficient is negative and significantly different from zero at the 5% level (okun_negative_5pct = true). Figure: out/figure1.png.

# Uncertainty

• The OLS residuals are serially correlated. Their autocorrelations at lags 1–4 are 0.39, 0.29, 0.14 and 0.07, and the Durbin–Watson statistic is 1.22. A HAC correction is therefore warranted. The HAC SE is about 1.8 times the naïve SE.
• The SE is not very sensitive to the bandwidth. HAC SE by lag: 0 → 0.0290 (heteroskedasticity only, already larger than the iid 0.0200), 1 → 0.0325, 2 → 0.0344, 4 → 0.0360, 6 → 0.0370, 8 → 0.0373, 12 → 0.0366, 16 → 0.0343. Even the largest of these SEs gives |t| > 6, so the significance conclusion holds for every bandwidth tried.
• Using a t(238) critical value (1.970) instead of 1.96 would change the CI only in the fourth decimal place.

# Limitations and open issues

• β is a linear-projection coefficient, not a structural parameter. The relation is contemporaneous only (no lags of g), and it ignores possible instability over time. For example, the Okun coefficient is widely thought to vary over time and over the business cycle, and none of that is tested here.
• The data are the current vintage. Revised GDP differs from what was known in real time.
• The bandwidth comes from a rule of thumb rather than a data-driven choice, but the sensitivity check above shows the conclusion does not depend on it.
• Newey–West inference is asymptotic. With T = 240 and moderate persistence the approximation should be adequate, but fixed-b or other refinements were not computed.