1# F2-conf-02 — Re-examining the memo: is the slope really significant?23## Question4The memo says that the OLS slope of `y` on `x` (400 monthly observations, 1990-01 to 2023-04) is statistically significant at the 5% level. That conclusion relies on conventional i.i.d. standard errors. We re-estimate β in y_t = α + β x_t + u_t, compute a standard error that remains valid when the disturbances are serially correlated (Newey–West HAC), and check whether the claim still holds.56## Data and exclusions7`data/series.csv` (SHA-256 c29dde22…bc514) has 400 rows (`n_input_rows`). The months run consecutively with no gaps, and there are no missing values or duplicates. No rows were excluded (`out/exclusions.csv`), so `n_analysed` = 400. No data problems were found (`data_issues` = []). The memo was used only as a set of claims to check. The data are simulated.89## Method10- OLS of y on a constant and x on the full sample. This gives `alpha_hat` and `beta_hat`.11- Baseline, which is the memo's measure: the conventional OLS standard error, `se_iid`, with t and p from Student-t(398) (`baseline_t_iid`, `baseline_p_iid`).12- Residual first-order autocorrelation: `rho_resid` = Σ û_t û_{t−1} / Σ û_t².13- HAC standard error: Newey–West with a Bartlett kernel and the small-sample correction n/(n−k) (statsmodels `cov_type="HAC"`).14- Bandwidth: Andrews (1991) AR(1) plug-in for the Bartlett kernel, applied to the slope score v_t = (x_t − x̄)û_t. The formula is S = 1.1447·(a₁T)^{1/3}, with a₁ = 4ρ_v²/((1−ρ_v)²(1+ρ_v)²), rounded up. This rule is driven by the data. It responds to the persistence of both x and u, which matters because the task describes both as persistent. The fixed Newey–West (1994) rule of thumb, 4(T/100)^{2/9}, ignores that persistence. The estimated AR(1) coefficient of the score is `rho_score`, the raw bandwidth is `andrews_bandwidth_raw`, and the bandwidth used is `hac_lags`.15- Inference: t_hac = β̂/se_hac, a two-sided p-value from the standard normal, and a 95% CI of β̂ ± 1.96·se_hac.16- Decision rule: the memo's claim counts as supported if and only if `p_hac` < 0.05.17- The analysis is fully deterministic. No randomness is used (`random_seed` = null).1819## Results20- Memo reproduced exactly: `alpha_hat` = 2.0925, `beta_hat` = 0.1017, `se_iid` = 0.0448, `baseline_t_iid` = 2.27, `baseline_p_iid` = 0.0238.21- The residuals are strongly autocorrelated: `rho_resid` = 0.609.22- The slope score has `rho_score` = 0.401. This gives `andrews_bandwidth_raw` = 8.18, so `hac_lags` = 9 months.23- HAC standard error: `se_hac` = 0.0654. That is 1.46 times the i.i.d. standard error (`se_ratio_hac_to_iid`).24- `t_hac` = 1.56 and `p_hac` = 0.120.25- 95% CI for β: [`beta_ci_low`, `beta_ci_high`] = [−0.026, 0.230]. The interval contains zero.26- **Verdict: `memo_claim_supported` = false.** Once serial correlation is accounted for, the slope is not significant at the 5% level. The point estimate itself is unchanged.2728## Uncertainty29The verdict does not depend on the bandwidth (`hac_sensitivity`). With 5 lags (the NW rule of thumb, `nw_rule_of_thumb_lags`), `p_hac` = 0.115. With 6, 8, 12, 18 and 24 lags, p ranges from 0.117 to 0.125. In every case se_hac falls between 0.0645 and 0.0663, and the t-statistic never exceeds 1.58.3031## Limitations and open issues32- Bartlett-kernel HAC tests are known to over-reject in samples of moderate size when x and u are both persistent. Fixed-b critical values or a longer bandwidth would make the test more conservative. Our test does not reject even with the normal reference, so this would only strengthen the conclusion.33- The Andrews plug-in assumes an AR(1) approximation for the score.34- "Not significant" means the data cannot exclude β = 0. It does not show that β = 0. The point estimate is 0.10 and the CI is wide.35- The data are simulated. The slope is a linear-projection coefficient and should not be read as a causal effect.36