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

## Question
The earlier memo (`data/memo.md`) regressed monthly `y` on `x` by OLS (with intercept, 400 months) and said the slope is significant at 5%, using conventional iid standard errors. The task description says the disturbances follow an AR(1) with autocorrelation of about 0.7 and that `x` is persistent. Under those conditions iid standard errors are too small. I re-estimated β in y_t = α + β x_t + u_t and tested H0: β = 0 with a Newey–West (Bartlett) HAC standard error.

## Data and exclusions
`data/series.csv` has 400 rows covering 1990-01 to 2023-04 (`n_input_rows`). There are no gaps, no duplicate months and no missing values, so I removed nothing (`out/exclusions.csv`), `n_analysed` = 400 and `data_issues` is empty. The x and y series have lag-1 autocorrelations of 0.66 and 0.62 (`out/state/data_audit.md`). The data are synthetic.

## Method
- OLS of y on a constant and x. The conventional iid standard error (`se_iid`) is the memo's measure and serves as the baseline.
- `rho_resid` is the lag-1 autocorrelation of the OLS residuals.
- HAC standard error: Newey–West with the Bartlett kernel, through statsmodels `cov_type="HAC"`, including the small-sample correction n/(n−k).
- Bandwidth: I used the Andrews (1991) AR(1) plug-in for the Bartlett kernel, fitted to the slope score v_t = (x_t − x̄)·û_t. This targets the persistence that actually drives the slope's variance. The score's lag-1 autocorrelation is 0.401, which gives S = 8.18, so `hac_lags` = 8. I also set a floor at the Newey–West rule of thumb, floor(4(T/100)^{2/9}) = 5. Details are under `bandwidth_details`.
- Inference: t = β̂/se_hac, a two-sided normal p-value, and a 95% CI of β̂ ± 1.96·se_hac.
- No randomness is involved (`random_seed` = null).

## Results
- `alpha_hat` = 2.0925 and `beta_hat` = 0.1017. `se_iid` = 0.0448, `t_iid` = 2.27 and `p_iid` = 0.0238. These match the memo to its stated rounding (`memo_reproduced`), so the memo's arithmetic is correct.
- `rho_resid` = 0.609. The residuals are strongly autocorrelated, so the iid assumption fails.
- `se_hac` = 0.0653 with `hac_lags` = 8. That is 1.46 times the iid standard error (`se_ratio_hac_over_iid`).
- `t_hac` = 1.56 and `p_hac` = 0.119. The 95% CI is [`beta_ci_low`, `beta_ci_high`] = [−0.026, 0.230], which includes 0.
- **Verdict: `memo_claim_supported` = false.** Once the standard error allows for serial correlation, the slope is not significant at the 5% level.

## Uncertainty
The verdict holds for every bandwidth tried, from 4 to 24 lags: `hac_sensitivity` gives p between 0.114 and 0.123, and `any_sensitivity_p_lt_005` = false. The point estimate is positive and of the same size as the memo's. The data are consistent with a modest positive association but do not reject β = 0 at 5%.

## Limitations and open issues
- Newey–West tests over-reject in finite samples when persistence is high. This makes the HAC p-value, if anything, too small, which strengthens the "not significant" verdict rather than weakening it. I did not pursue fixed-b or other refinements because the task does not require them.
- `rho_resid` (0.61) is a little below the stated value of about 0.7. This fits the known downward bias of residual autocorrelation estimates and does not change the conclusion.
- β is a linear-projection coefficient. It is not a causal effect.