# F2-conf-02 — Is the memo's slope really significant?

## Question
The memo (`data/memo.md`) regressed monthly `y` on `x` by OLS with an intercept and, using conventional (iid) standard errors, concluded the slope is significant at 5%. We re-estimate β in y_t = α + β x_t + u_t on the full sample with a HAC (Newey–West, Bartlett) standard error valid under serially correlated disturbances, and judge whether the memo's claim holds.

## Data and exclusions
`data/series.csv` (simulated): 400 consecutive months 1990-01 to 2023-04 (`n_input_rows`), no missing values, no duplicate months, no gaps (`data_issues` = []). No rows excluded (`out/exclusions.csv`); `n_analysed` = 400. Audit details and SHA-256 hashes: `out/state/data_audit.md`.

## Method
1. OLS of y on x with intercept (`alpha_hat`, `beta_hat`), conventional SE (`se_iid`) = the memo's baseline.
2. Residual first-order autocorrelation `rho_resid` = Σu_t u_{t-1}/Σu_t².
3. HAC Newey–West SE with Bartlett kernel (weights 1 − j/(L+1)), no small-sample degrees-of-freedom correction, standard-normal reference distribution. Bandwidth: Andrews (1991) AR(1) plug-in rule for the Bartlett kernel, 1.1447·(α̂(1)·T)^{1/3}, with α̂(1)=4ρ̂²/((1−ρ̂)²(1+ρ̂)²), where ρ̂ is the AR(1) coefficient of the slope score (x_t − x̄)û_t (`hac_bandwidth_details.rho_score` = 0.40). This gives 8.18 and we use floor → `hac_lags` = 8. This data-dependent rule was preferred to the fixed Newey–West (1994) rule of thumb floor(4(T/100)^{2/9}) = 5 (`hac_bandwidth_details.nw_rule_of_thumb_lags`) because the task describes highly persistent errors and a persistent regressor; sensitivity to L is reported. The HAC SE was checked against a hand-coded sandwich computation.
4. Test of H0: β = 0: t = β̂/se_hac, two-sided normal p-value, 95% CI β̂ ± 1.96·se_hac. Decision rule: the memo's claim counts as supported iff `p_hac` < 0.05. No randomness (`random_seed` = null).

## Results
- Point estimates reproduce the memo: `alpha_hat` = 2.0925, `beta_hat` = 0.1017; the baseline `se_iid` = 0.0448 gives t = 2.27 (`t_iid`) and p = 0.0238 (`p_iid`), matching the memo's figures.
- The OLS residuals are strongly autocorrelated: `rho_resid` = 0.609.
- HAC: `se_hac` = 0.0651 with `hac_lags` = 8, i.e. about 1.45 times the iid SE. `t_hac` = 1.56, `p_hac` = 0.118; 95% CI [`beta_ci_low`, `beta_ci_high`] = [−0.026, 0.229], which contains zero.
- **Verdict: the memo's claim that the slope is significant at 5% is not supported** (`memo_claim_supported` = false). The point estimate is unchanged; what changes is the uncertainty. The iid SE understates it because both the errors and x are persistent.

## Uncertainty
Sensitivity to bandwidth (`hac_lag_sensitivity`): for L = 5, 8, 12, 16, 24 the HAC SE ranges over 0.0644–0.0659 and p over 0.114–0.122 (`hac_lag_sensitivity_min_p_excl_0` = 0.114), so the verdict holds for every bandwidth tried. Only L = 0, the heteroskedasticity-robust SE that ignores autocorrelation, gives p = 0.037. The ratio se_hac/se_iid ≈ 1.45 matches the approximate AR(1)-style inflation factor sqrt((1+ρ_xρ_u)/(1−ρ_xρ_u)) ≈ 1.53, computed from the lag-1 autocorrelations of x (0.66) and of the residuals (0.61) (`out/state/checks.md`).

## Limitations and open issues
- Newey–West inference with Bartlett weights and a normal reference distribution can over-reject in finite samples when errors are this persistent (ρ ≈ 0.6–0.7). Fixed-b or other small-sample refinements would usually widen the interval further, so they would strengthen the verdict rather than reverse it. We did not compute them.
- The bandwidth is a choice. The verdict does not depend on it in the range we checked (5–24 lags), but we did not search every possible procedure.
- Non-significance at 5% does not show that β = 0. The data are consistent with a positive slope of the estimated size, and also with no association at all.
- The data are simulated, as `TASK.md` states.