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

## Question
The memo says the OLS slope of `y` on `x` (monthly, 1990-01 to 2023-04) is significant at the 5% level. It reached that conclusion with conventional i.i.d. standard errors. The task says the regression errors are serially correlated (AR(1), roughly 0.7), so I re-estimate the slope β in y_t = α + β x_t + u_t. I then test H0: β = 0 using a HAC (Newey–West, Bartlett kernel) standard error.

## Data and exclusions
`data/series.csv` has 400 rows (`n_input_rows`) covering 400 consecutive months. There are no missing values, no duplicate months and no gaps. No rows were excluded (`out/exclusions.csv`), so `n_analysed` = 400. `data_issues` is empty. Checksums and the full audit are in `out/state/data_audit.md`. I used the memo only for comparison: its numbers are recomputed, not taken as given.

## Method
- OLS of y on a constant and x over the full sample. This gives `alpha_hat` and `beta_hat`. The conventional standard error `se_iid` is the baseline.
- `rho_resid` is the first-order autocorrelation of the OLS residuals.
- HAC standard error: Newey–West with a Bartlett kernel and the small-sample correction (statsmodels `cov_type="HAC"`). **Bandwidth choice:** Andrews (1991) AR(1) plug-in rule for the Bartlett kernel, applied to the slope score v_t = (x_t − x̄)·û_t. The score's AR(1) coefficient is `rho_score`. The raw bandwidth is `andrews_bandwidth_raw` = 8.18, rounded up to `hac_lags` = 9 months. I preferred this data-dependent rule because both the errors and x are persistent. The fixed Newey–West rule of thumb, floor(4(T/100)^{2/9}) = 5 (`nw_rule_of_thumb_lags`), adapts less to persistence. The script also reports a sensitivity grid of lag choices (`hac_sensitivity`).
- Inference uses `t_hac` = `beta_hat`/`se_hac` with a standard normal reference distribution, a two-sided `p_hac`, and a 95% CI of `beta_hat` ± 1.96·`se_hac`.
- The script uses no randomness (`random_seed` = null).

## Results
- **The memo's numbers reproduce:** `alpha_hat` = 2.0925, `beta_hat` = 0.1017, `se_iid` = 0.0448, `t_iid` = 2.27, `p_iid` = 0.0238.
- **Residuals are strongly autocorrelated:** `rho_resid` = 0.609.
- **With the HAC standard error:** `se_hac` = 0.0654 with `hac_lags` = 9, which is about 1.46 times `se_iid`. This gives `t_hac` = 1.556 and `p_hac` = 0.120. The 95% CI is [`beta_ci_low`, `beta_ci_high`] = [−0.026, 0.230], which includes 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. Only the uncertainty was understated.

## Uncertainty
The conclusion does not depend on the bandwidth (`hac_sensitivity`). For every lag choice from 5 to 36 months, p is between 0.097 and 0.123. The only setting that gives p < 0.05 is lag 0 (p = 0.038), which is a heteroskedasticity-only correction with no allowance for autocorrelation. At 5% the slope is not significant; at 10% it is borderline at best.

## Limitations and open issues
- HAC tests with a normal reference distribution are known to over-reject in small samples when the series are highly persistent. That would only weaken significance further, so it reinforces the verdict. I did not compute fixed-b or other refinements because the task does not require them.
- The 95% CI is a large-sample approximation.
- Not rejecting H0 does not show that β = 0. The data are simply not precise enough to establish significance at the 5% level.
- The data are synthetic, as stated in TASK.md.