# Report — F2-conf-02: is the memo's slope really significant?

## Question
The memo (`data/memo.md`) concludes that the OLS slope of y on x is significant at the 5% level. The standard errors it uses assume independent errors. TASK.md, however, says the disturbances follow an AR(1) process with autocorrelation of about 0.7 and that x is also persistent. I re-estimated β in y_t = α + β x_t + u_t on the full sample with a HAC (Newey–West, Bartlett-kernel) standard error and checked whether the claim still holds.

## Data and exclusions
`data/series.csv` contains 400 monthly rows (`n_input_rows`), 1990-01 to 2023-04. It has no missing values, no duplicate months and no gaps. No rows were excluded (`out/exclusions.csv`), so `n_analysed` = 400. No data issues were found (`data_issues` = []). Hashes and summary statistics are in `out/state/data_audit.md`.

## Method
- OLS of y on a constant and x.
- **Baseline:** the conventional i.i.d. standard error (`se_iid`). This reproduces the memo's figures: `alpha_hat` = 2.0925, `beta_hat` = 0.1017, `se_iid` = 0.0448, `t_iid` = 2.27, `p_iid` = 0.0238.
- **Robust inference:** Newey–West HAC covariance with a Bartlett kernel and a small-sample factor n/(n−k), computed with statsmodels.
- **Bandwidth choice:** the Andrews (1991) AR(1) plug-in for the Bartlett kernel, applied to the slope score (x_t − x̄)û_t. The score's estimated lag-1 autocorrelation is `bandwidth_details.rho_score` ≈ 0.40, which gives a raw bandwidth of 8.18 (`bandwidth_details.andrews_bw_raw`). Rounding up gives `hac_lags` = 9 months.
  - I preferred this data-driven rule to the Newey–West (1994) rule of thumb (5 lags, `bandwidth_details.nw1994_rule_of_thumb_lags`) because the errors are strongly persistent.
- t-statistics, p-values and the 95% CI use the standard normal distribution. No randomness is involved (`random_seed` = null).

## Results
- `beta_hat` = 0.1017 and `alpha_hat` = 2.0925.
- The first-order autocorrelation of the residuals is `rho_resid` = 0.61. This confirms strong serial correlation.
- The HAC standard error is `se_hac` = 0.0654 with 9 lags, about 1.46 times `se_iid`.
- This gives `t_hac` = 1.56 and `p_hac` = 0.120. The 95% CI is [`beta_ci_low`, `beta_ci_high`] = [−0.026, 0.230], which includes 0.
- **Verdict:** `memo_claim_supported` = false. Once the serial correlation is accounted for, the slope is not significant at the 5% level. The point estimate itself matches the memo; only the inference changes.

## Uncertainty
- **Sensitivity to the bandwidth** (`hac_sensitivity`, also in `out/hac_sensitivity.csv`): HAC p-values range from 0.079 (1 lag) to 0.125 (20 lags). Every lag from 1 to 24 gives p > 0.05 (`sensitivity_all_p_above_005_for_lags_ge_1` = true).
- Only 0 lags gives p < 0.05 (0.038). That is White's heteroskedasticity-only standard error, which ignores the autocorrelation and is therefore not valid here.
- **Small-sample bias:** with persistent regressors and errors, Bartlett HAC standard errors tend to be biased downward in finite samples, so these HAC p-values, if anything, overstate significance. This strengthens the verdict.

## Limitations and open issues
- The bandwidth rule is one reasonable choice among several. The conclusion holds for every choice tested from 1 to 24 lags.
- I used normal critical values. Fixed-b or Student-t critical values would be more conservative and would only reinforce the verdict.
- β is a linear-projection coefficient, not a causal effect.
- The data are synthetic, as TASK.md states.