# Re-examination of the results memo: is the slope of y on x significant?

## Question
The memo (`data/memo.md`) ran OLS of y on x with an intercept on 400 monthly observations. It used conventional (iid) standard errors and concluded that the slope is significant at the 5% level. The disturbances are serially correlated and x is persistent, so the iid standard error is not valid. I re-estimate the slope β in y_t = α + β x_t + u_t and test H0: β = 0 with a Newey–West HAC standard error. Then I check whether the memo's claim still holds.

## Data and exclusions
- `data/series.csv` has 400 rows (`n_input_rows`), running monthly from 1990-01 to 2023-04. There are no gaps, no duplicates and no missing values (`out/state/data_audit.md`). Its SHA-256 matches the bundle manifest.
- I excluded no observations, so `n_analysed` = 400 (`out/exclusions.csv`: 400 − 0 = 400). `data_issues` is an empty list.
- I used the memo only as something to check. None of its numbers feed into the computation.

## Method
- **Point estimate:** OLS with an intercept on the full sample (`beta_hat`, `alpha_hat`).
- **Baseline (the memo's measure):** the conventional iid OLS standard error (`se_iid`, `baseline_t_iid`, `baseline_p_iid`).
- **Residual persistence:** `rho_resid` = Σû_t û_{t-1} / Σû_t².
- **HAC standard error (`se_hac`):** Newey–West with a Bartlett kernel (weights 1 − j/(L+1)) and the small-sample factor T/(T−2). I computed it with statsmodels and checked it against my own numpy implementation (`out/state/checks.md`).
- **Bandwidth (`hac_lags`):** a data-dependent rule:
  - First, the Andrews (1991) AR(1) plug-in for the Bartlett kernel, applied to the slope score v_t = (x_t − x̄)û_t. Its formula is S = 1.1447·(α̂(1)·T)^{1/3}, with α̂(1) = 4ρ_v²/((1−ρ_v)²(1+ρ_v)²). The score's estimated autocorrelation is ρ_v = `rho_score` = 0.401, which gives S = `andrews_bandwidth_raw` ≈ 8.2.
  - I rounded S up to 9 and kept the larger of that and the Newey–West (1994) rule of thumb floor(4(T/100)^{2/9}) = 5 (`nw1994_rule_lags`). That gives L = 9 months.
  - Why a plug-in rather than a fixed rule: with error autocorrelation near 0.7 and a persistent regressor, a fixed rule like 5 lags is likely to cut off autocorrelation that is still present.
- **Test and interval:** t_hac = β̂/se_hac, a two-sided p-value from the standard normal (`p_hac`), and a 95% interval β̂ ± 1.96·se_hac.
- **Decision rule:** `memo_claim_supported` = (p_hac < 0.05).
- No random numbers are used (`random_seed` = null).

## Results
| quantity | value | key |
|---|---|---|
| intercept | 2.0925 | `alpha_hat` |
| slope | 0.1017 | `beta_hat` |
| iid SE (memo) | 0.0448 | `se_iid` |
| iid t / p | 2.27 / 0.0238 | `baseline_t_iid`, `baseline_p_iid` |
| residual AR(1) coefficient | 0.609 | `rho_resid` |
| HAC lags | 9 | `hac_lags` |
| HAC SE | 0.0654 | `se_hac` |
| HAC SE / iid SE | 1.46 | `se_ratio_hac_over_iid` |
| HAC t | 1.56 | `t_hac` |
| HAC p (two-sided) | 0.120 | `p_hac` |
| 95% CI (HAC) | [−0.026, 0.230] | `beta_ci_low`, `beta_ci_high` |
| memo claim supported | false | `memo_claim_supported` |

- My recomputation reproduces the memo's point estimates, SE, t and p to rounding. The memo's arithmetic is correct.
- The residuals are strongly autocorrelated (`rho_resid` = 0.61).
- The HAC standard error is 46% larger than the iid one (`se_ratio_hac_over_iid`). With it, the t-statistic falls to 1.56 (`t_hac`) and the p-value rises to 0.12 (`p_hac`). The 95% interval contains zero.
- **Verdict:** the memo's claim that the slope is significant at the 5% level is **not supported** (`memo_claim_supported` = false).

## Uncertainty
- **Bandwidth:** the verdict does not depend on the bandwidth choice. Across 5, 8, 9, 12, 18 and 24 lags, the HAC SE ranges from 0.0645 to 0.0663 and the p-value from 0.115 to 0.125 (`hac_sensitivity_by_lag`). Even the smallest bandwidth (the 1994 rule of thumb, 5 lags) gives p = 0.115.
- **Reference distribution:** using a Student-t instead of the normal would only make the p-value larger.
- **Point estimate:** β̂ = 0.10 is positive. The data cannot distinguish it from zero, or from values up to about 0.23, at the 95% level.

## Limitations and open issues
- **Known size distortion:** Newey–West tests are known to over-reject in finite samples when regressors and errors are persistent. If anything, the true p-value is probably higher than 0.12, which reinforces the conclusion. I did not compute fixed-b (Kiefer–Vogelsang) critical values or a parametric GLS/Cochrane–Orcutt alternative because the task does not require them.
- **What the estimand means:** β is a linear-projection coefficient, i.e. an association. The memo's wording that "x is a significant predictor" is also loose. The regression is contemporaneous, not out-of-sample prediction, and nothing here supports a causal reading.
- **Bandwidth convention:** Andrews' S is the Bartlett truncation parameter (weight 1 − j/S), while statsmodels uses weights 1 − j/(L+1). Rounding S up therefore adds roughly one lag of extra conservatism. The sensitivity table shows this does not matter.
- **Simulated data:** the data are synthetic, as stated in TASK.md, so the conclusions apply to this dataset only.