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

# Question

The memo (data/memo.md) reports an OLS slope of y on x of 0.1017 with a conventional SE of 0.0448 (t = 2.27, p = 0.0238) and concludes significance at 5%. We re-estimate β in y_t = α + β x_t + u_t on the full sample with a Newey–West (Bartlett) HAC standard error, because the disturbances are serially correlated, and judge whether the claim holds.

# Data and exclusions

data/series.csv: 400 monthly rows, 1990-01 to 2023-04, consecutive, no missing values or duplicates (n_input_rows, n_analysed = 400). No exclusions (out/exclusions.csv: 400 → 400); data_issues is empty. Full audit in out/state/data_audit.md. The data are simulated.

# Method

OLS of y on a constant and x. Naïve baseline: the conventional iid SE (se_iid), the memo's measure. Main inference: Newey–West HAC covariance with Bartlett kernel, no small-sample df correction, normal reference distribution. Bandwidth: Andrews (1991) AR(1) plug-in for the Bartlett kernel, applied to the slope score v_t = (x_t − x̄)·û_t: S_T = 1.1447·(α(1)·T)^{1/3}, α(1) = 4r²/((1−r)²(1+r)²), where r is the lag-1 autocorrelation of v_t (score_rho = 0.401). This gives andrews_bandwidth_raw = 8.18, which we floor to hac_lags = 8 months. This rule is data-dependent and suited to the persistence of the score. The Newey–West (1994) rule of thumb, floor(4(T/100)^{2/9}) = 5 (nw_rule_of_thumb_lags), and other lag choices are reported as sensitivity checks (hac_sensitivity). The script is deterministic and uses no random numbers (random_seed = null).

# Results

• beta_hat = 0.1017, alpha_hat = 2.0925. These match the memo exactly, as do the iid SE se_iid = 0.0448, t (t_iid) = 2.27 and p (p_iid) = 0.0238.
• Lag-1 autocorrelation of the OLS residuals: rho_resid = 0.61. The errors are strongly serially correlated, and x is persistent too, so the iid SE understates the uncertainty.
• HAC SE (se_hac) = 0.0651, about 1.45× the iid SE. t_hac = 1.56, p_hac = 0.118.
• 95% CI (beta_ci_low, beta_ci_high) = [−0.026, 0.229], which contains zero.
• Verdict: memo_claim_supported = false. Once serial correlation is allowed for, the slope is not statistically significant at the 5% level.

# Uncertainty

The verdict holds across bandwidths (hac_sensitivity): 5 lags gives SE 0.0644 (p = 0.114), 12 lags gives 0.0655 (p = 0.120), and 24 lags gives 0.0659 (p = 0.122). Using t(398) critical values instead of normal ones changes p by less than 0.001. Only the heteroskedasticity-only (0-lag) SE, 0.0489, gives p = 0.037 < 0.05, but that SE ignores the serial correlation the task requires us to handle. All checks are listed in out/state/checks.md.

# Limitations and open issues

• HAC tests based on kernels and normal critical values tend to over-reject in finite samples when the regressor and errors are persistent. A more conservative approach (e.g., fixed-b critical values) would make rejection less likely still, so the "not significant" verdict is, if anything, conservative.
• The point estimate (0.10) is unchanged. "Not significant" means the data cannot rule out β = 0; it is not evidence that β = 0.
• Only the Bartlett kernel was used, as the task specifies. The data are synthetic.