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

## Question
The previous memo (`data/memo.md`) regressed monthly `y` on `x` by OLS with an intercept and used conventional (iid) standard errors to call the slope significant at the 5% level. The task states that the disturbances are AR(1) with autocorrelation around 0.7 and that x is persistent. Under those conditions iid standard errors are too small. This report re-estimates the slope β in y_t = α + β x_t + u_t with a Newey–West (HAC) standard error and checks whether the memo's claim holds.

## Data and exclusions
`data/series.csv` has 400 rows (`n_input_rows`), covering 1990-01 to 2023-04 with consecutive months. It has no missing values, no duplicate months and no gaps. No rows were removed (`out/exclusions.csv`), so `n_analysed` = 400 and `data_issues` is empty. File hashes and summary statistics are in `out/state/data_audit.md`. The memo was used only as a claim to check: its numbers were recomputed, not copied.

## Method
- I fit OLS of y on x with an intercept on the full sample, giving `beta_hat` and `alpha_hat`. The naïve baseline is the conventional standard error `se_iid`, which is the memo's measure.
- `rho_resid` is the lag-1 autocorrelation of the OLS residuals.
- HAC standard error: Newey–West with a Bartlett kernel, through statsmodels with the small-sample factor n/(n−k). I also checked it against a hand-coded implementation (`out/state/checks.md`).
- Bandwidth: Andrews (1991) AR(1) plug-in rule for the Bartlett kernel. I applied it to the slope's score v_t = (x_t − x̄)û_t and used S = 1.1447·(α̂(1)·T)^{1/3} with α̂(1) = 4r²/((1−r)²(1+r)²). The score's AR(1) coefficient is r = `rho_score` = 0.40, which gives S = `andrews_bandwidth` = 8.18, so `hac_lags` = 8. I chose a data-dependent rule because both x and u are persistent. The fixed Newey–West (1994) rule of thumb gives 5 lags (`nw_rule_of_thumb_lags`), and I report it as a sensitivity check.
- Inference uses the normal distribution: `t_hac` = `beta_hat`/`se_hac`, `p_hac` is two-sided, and the 95% CI is `beta_hat` ± 1.96·`se_hac`. No randomness is involved (`random_seed` = null).

## Results
- `beta_hat` = 0.1017 and `alpha_hat` = 2.0925. Together with `se_iid` = 0.0448, `t_iid` = 2.27 and `p_iid` = 0.0238, these reproduce the memo's numbers exactly.
- `rho_resid` = 0.61, so the residuals are strongly positively autocorrelated.
- `se_hac` = 0.0653 with `hac_lags` = 8. That is 1.46 times the iid standard error (`se_ratio_hac_to_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 contains zero.
- **Verdict: the memo's claim is not supported** (`memo_claim_supported` = false). Once serial correlation is accounted for, the slope is not significant at 5%.

## Uncertainty
- The verdict does not depend on the bandwidth. Over lags 5, 8, 12, 16 and 24, the HAC standard error ranges from 0.0645 to 0.0660 and p from 0.115 to 0.123 (`hac_sensitivity`). Every one of these p-values is above 0.05 (`sensitivity_all_insignificant` = true).
- The point estimate is unchanged. Only its precision changes: the effective sample size is much smaller than 400 because both series are persistent.

## Limitations and open issues
- Newey–West tests with persistent data tend to over-reject in finite samples. If anything, the true p-value is likely larger, which would make the "not significant" verdict more conservative, not less. I did not compute fixed-b or other refinements because the task does not require them.
- The residual autocorrelation (0.61) is below the stated value of about 0.7. That fits the known downward bias of the residual autocorrelation estimate in finite samples. It is not a data problem.
- "Not significant at 5%" does not show that β = 0. The CI still allows sizeable positive slopes.
- β is a linear-projection coefficient, not a causal effect.