# F2-conf-02 — Is the slope of y on x really significant?

## Question
The memo (`data/memo.md`) says the OLS slope of y on x (0.1017, conventional SE 0.0448, p = 0.024) is significant at the 5% level. The task is to re-estimate β in y_t = α + β x_t + u_t by full-sample OLS and to test it with a Newey–West (Bartlett-kernel HAC) standard error, which stays valid when the disturbances are serially correlated.

## Data and exclusions
`data/series.csv` has 400 rows covering 1990-01 to 2023-04. I checked for missing values, duplicate months and gaps in the monthly sequence and found none. No observations were excluded, so n = 400 and `data_issues` is empty. The data are synthetic.

## Method
- OLS of y on a constant and x.
- **Conventional SE (iid errors).** This is the memo's measure and the naïve baseline.
- **HAC SE.** Newey–West with the Bartlett kernel, computed with statsmodels `cov_type="HAC"` with the small-sample correction (T/(T−k)).
- **Bandwidth choice.** I used the Andrews (1991) AR(1) plug-in rule for the Bartlett kernel: L = floor(1.1447·(α̂·T)^{1/3}), with α̂ = 4ρ²/((1−ρ)²(1+ρ)²).
  - ρ is the lag-1 autocorrelation of the slope score v_t = (x_t − x̄)·û_t. This is the series whose long-run variance enters the slope's variance.
  - The estimate is ρ = 0.40, which gives **L = 8 months**.
  - For comparison, the Newey–West (1994) rule of thumb floor(4(T/100)^{2/9}) gives 5.
- **Inference.** The two-sided p-value and the 95% CI use the asymptotic normal reference. The p-value from a t(398) reference is also reported and is almost the same.

## Results
| quantity | value |
|---|---|
| α̂ (intercept) | 2.0925 |
| β̂ (slope) | 0.1017 |
| conventional SE (iid) | 0.0448 (t = 2.27, p = 0.024) |
| lag-1 autocorrelation of residuals | 0.61 |
| HAC SE (Bartlett, 8 lags) | 0.0653 |
| HAC t-statistic | 1.56 |
| HAC p-value (two-sided) | 0.119 |
| 95% CI (HAC) | [−0.026, 0.230] |

I reproduce the memo's point estimates, conventional SE, t-statistic and p-value exactly.

**Verdict.** The OLS residuals are strongly autocorrelated (ρ̂ = 0.61), and x is also persistent (lag-1 autocorrelation 0.66). Under those conditions the iid standard error understates the uncertainty. The HAC standard error is about 46% larger. With it, the slope is **not statistically significant at the 5% level**: p ≈ 0.12, and the 95% CI includes zero. The memo's claim is **not supported** (`memo_claim_supported = false`).

## Uncertainty
The conclusion holds across bandwidths:

| lags | HAC SE | p-value |
|---|---|---|
| 5 | 0.0645 | 0.115 |
| 8 | 0.0653 | 0.119 |
| 12 | 0.0657 | 0.121 |
| 24 | 0.0660 | 0.123 |
| 36 | 0.0613 | 0.097 |

Only the heteroskedasticity-only SE (0 lags, SE 0.0490, p = 0.038) crosses 5%. That SE ignores autocorrelation, so it is not appropriate here. Full details are in `hac_sensitivity_by_lags` in `results.json`.

## Limitations and open issues
- With persistent errors and a persistent regressor, Newey–West tests are known to over-reject somewhat in samples of this size. The true p-value is therefore, if anything, larger, which only strengthens the conclusion that significance is not established. Fixed-b or other alternative inference methods were not run.
- The bandwidth rule assumes the score is roughly AR(1).
- "Not significant" means the data cannot distinguish β from zero at 5%. It is not evidence that β = 0.
- β is a linear-projection coefficient: it describes an association, not a causal effect, and it is not a forecasting relationship.
- No random numbers were used (`random_seed = null`).