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

## Question
The memo (data/memo.md) reports an OLS slope of y on x of 0.1017 with a conventional SE of 0.0448 (t = 2.27, p = 0.0238) and concludes significance at 5%. We re-estimate β in y_t = α + β x_t + u_t on the full sample with a Newey–West (Bartlett) HAC standard error, because the disturbances are serially correlated, and judge whether the claim holds.

## Data and exclusions
`data/series.csv`: 400 monthly rows, 1990-01 to 2023-04, consecutive, no missing values or duplicates (`n_input_rows`, `n_analysed` = 400). No exclusions (out/exclusions.csv: 400 → 400); `data_issues` is empty. Full audit in out/state/data_audit.md. The data are simulated.

## Method
OLS of y on a constant and x. Naïve baseline: the conventional iid SE (`se_iid`), the memo's measure. Main inference: Newey–West HAC covariance with Bartlett kernel, no small-sample df correction, normal reference distribution. Bandwidth: Andrews (1991) AR(1) plug-in for the Bartlett kernel, applied to the slope score v_t = (x_t − x̄)·û_t: S_T = 1.1447·(α(1)·T)^{1/3}, α(1) = 4r²/((1−r)²(1+r)²), where r is the lag-1 autocorrelation of v_t (`score_rho` = 0.401). This gives `andrews_bandwidth_raw` = 8.18, which we floor to `hac_lags` = 8 months. This rule is data-dependent and suited to the persistence of the score. The Newey–West (1994) rule of thumb, floor(4(T/100)^{2/9}) = 5 (`nw_rule_of_thumb_lags`), and other lag choices are reported as sensitivity checks (`hac_sensitivity`). The script is deterministic and uses no random numbers (`random_seed` = null).

## Results
- `beta_hat` = 0.1017, `alpha_hat` = 2.0925. These match the memo exactly, as do the iid SE `se_iid` = 0.0448, t (`t_iid`) = 2.27 and p (`p_iid`) = 0.0238.
- Lag-1 autocorrelation of the OLS residuals: `rho_resid` = 0.61. The errors are strongly serially correlated, and x is persistent too, so the iid SE understates the uncertainty.
- HAC SE (`se_hac`) = 0.0651, about 1.45× 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: `memo_claim_supported` = false. Once serial correlation is allowed for, the slope is **not** statistically significant at the 5% level.

## Uncertainty
The verdict holds across bandwidths (`hac_sensitivity`): 5 lags gives SE 0.0644 (p = 0.114), 12 lags gives 0.0655 (p = 0.120), and 24 lags gives 0.0659 (p = 0.122). Using t(398) critical values instead of normal ones changes p by less than 0.001. Only the heteroskedasticity-only (0-lag) SE, 0.0489, gives p = 0.037 < 0.05, but that SE ignores the serial correlation the task requires us to handle. All checks are listed in out/state/checks.md.

## Limitations and open issues
- HAC tests based on kernels and normal critical values tend to over-reject in finite samples when the regressor and errors are persistent. A more conservative approach (e.g., fixed-b critical values) would make rejection less likely still, so the "not significant" verdict is, if anything, conservative.
- The point estimate (0.10) is unchanged. "Not significant" means the data cannot rule out β = 0; it is not evidence that β = 0.
- Only the Bartlett kernel was used, as the task specifies. The data are synthetic.