# Okun's law in U.S. quarterly data, 1960Q1–2019Q4

# Question

Estimate β in 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}) (quarter-on-quarter, not annualised). The sample is the 240 quarters 1960Q1–2019Q4. β is estimated by OLS, its standard error is Newey–West (Bartlett) HAC, and we test whether β is negative and significant at the 5% level.

# Data and exclusions

• data/UNRATE.csv: 945 monthly rows read; data/GDPC1.csv: 319 quarterly rows read. SHA-256 checksums match data/README.md.
• Monthly unemployment was averaged to calendar quarters. The script checks that every quarter from 1959Q4 to 2019Q4 has all three months, and that GDPC1 has no gaps in that range. Only levels from 1959Q4 to 2019Q4 are used; 1959Q4 serves only to form the 1960Q1 differences.
• Data issue: UNRATE is missing for 2025-10 (empty field). That month is outside the window, so it was not used and does not affect any result.
• No other exclusions were made. n = 240.

# Method

• OLS of Δu_t on a constant and g_t (statsmodels).
• Baseline: the conventional i.i.d./homoskedastic OLS standard error.
• HAC: Newey–West with the Bartlett kernel and bandwidth L = floor(4·(T/100)^(2/9)) = floor(4·2.4^(2/9)) = 4 lags for T = 240. This is the Newey–West (1994) rule of thumb, and 4 quarters equals one year of dependence. The OLS residuals support this choice: their autocorrelations at lags 1–4 are 0.39, 0.29, 0.14 and 0.07, so dependence dies out after about one year. No small-sample degrees-of-freedom correction was applied. The 95% CI is β̂ ± 1.96·se_hac, and the test uses the normal distribution, which is the standard asymptotic HAC inference.
• The script out/analysis.py is deterministic and uses no randomness. It writes out/results.json, out/figure1.png, and out/diagnostics.json (sensitivity output, not a required deliverable).

# Results

When quarterly real GDP growth is 1 percentage point higher, the quarterly change in the unemployment rate is about 0.26 pp lower on average. The fitted line implies unemployment is stable (Δu = 0) at g = α̂/(−β̂) ≈ 0.72% per quarter, roughly 2.9% annualised. That is close to the sample mean growth of 0.754% per quarter. Figure: out/figure1.png.

# Uncertainty

• 95% HAC CI: [−0.326, −0.186]. The HAC t-statistic is −7.14 (p ≈ 1e-12).

• The Okun coefficient is negative and significantly different from zero at the 5% level (okun_negative_5pct = true).

• The HAC standard error is about 1.8 times the naïve one, because the residuals are positively autocorrelated. The naïve standard error overstates precision.

• Bandwidth sensitivity of the HAC SE:

  lags	0 (White)	2	4	6	8	12
  se(β̂)	0.0288	0.0342	0.0358	0.0369	0.0371	0.0364

  A degrees-of-freedom correction changes the SE by less than 0.5%. The conclusion does not depend on the bandwidth.

# Limitations and open issues

• The data are the current vintage, not real-time, so GDP revisions are included.
• The result is a linear projection over the full sample, not a structural or causal estimate. The relation may be unstable over time: we did not test for breaks (for example, post-1984 or the 2008 recession), and we did not test for asymmetry.
• The specification is static. It omits lags of g_t, so part of the dynamic Okun relation ends up in the serially correlated residuals, which the HAC SE accounts for.
• HAC inference uses the asymptotic normal distribution. Fixed-b or other small-sample refinements were not applied. Given t ≈ −7, they would not change the conclusion.