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

# Question

The memo (data/memo.md) regressed monthly y on x by OLS with an intercept and, using conventional (iid) standard errors, concluded the slope is significant at 5%. We re-estimate β in y_t = α + β x_t + u_t on the full sample with a HAC (Newey–West, Bartlett) standard error valid under serially correlated disturbances, and judge whether the memo's claim holds.

# Data and exclusions

data/series.csv (simulated): 400 consecutive months 1990-01 to 2023-04 (n_input_rows), no missing values, no duplicate months, no gaps (data_issues = []). No rows excluded (out/exclusions.csv); n_analysed = 400. Audit details and SHA-256 hashes: out/state/data_audit.md.

# Method

1. OLS of y on x with intercept (alpha_hat, beta_hat), conventional SE (se_iid) = the memo's baseline.
2. Residual first-order autocorrelation rho_resid = Σu_t u_{t-1}/Σu_t².
3. HAC Newey–West SE with Bartlett kernel (weights 1 − j/(L+1)), no small-sample degrees-of-freedom correction, standard-normal reference distribution. Bandwidth: Andrews (1991) AR(1) plug-in rule for the Bartlett kernel, 1.1447·(α̂(1)·T)^{1/3}, with α̂(1)=4ρ̂²/((1−ρ̂)²(1+ρ̂)²), where ρ̂ is the AR(1) coefficient of the slope score (x_t − x̄)û_t (hac_bandwidth_details.rho_score = 0.40). This gives 8.18 and we use floor → hac_lags = 8. This data-dependent rule was preferred to the fixed Newey–West (1994) rule of thumb floor(4(T/100)^{2/9}) = 5 (hac_bandwidth_details.nw_rule_of_thumb_lags) because the task describes highly persistent errors and a persistent regressor; sensitivity to L is reported. The HAC SE was checked against a hand-coded sandwich computation.
4. Test of H0: β = 0: t = β̂/se_hac, two-sided normal p-value, 95% CI β̂ ± 1.96·se_hac. Decision rule: the memo's claim counts as supported iff p_hac < 0.05. No randomness (random_seed = null).

# Results

• Point estimates reproduce the memo: alpha_hat = 2.0925, beta_hat = 0.1017; the baseline se_iid = 0.0448 gives t = 2.27 (t_iid) and p = 0.0238 (p_iid), matching the memo's figures.
• The OLS residuals are strongly autocorrelated: rho_resid = 0.609.
• HAC: se_hac = 0.0651 with hac_lags = 8, i.e. about 1.45 times 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: the memo's claim that the slope is significant at 5% is not supported (memo_claim_supported = false). The point estimate is unchanged; what changes is the uncertainty. The iid SE understates it because both the errors and x are persistent.

# Uncertainty

Sensitivity to bandwidth (hac_lag_sensitivity): for L = 5, 8, 12, 16, 24 the HAC SE ranges over 0.0644–0.0659 and p over 0.114–0.122 (hac_lag_sensitivity_min_p_excl_0 = 0.114), so the verdict holds for every bandwidth tried. Only L = 0, the heteroskedasticity-robust SE that ignores autocorrelation, gives p = 0.037. The ratio se_hac/se_iid ≈ 1.45 matches the approximate AR(1)-style inflation factor sqrt((1+ρ_xρ_u)/(1−ρ_xρ_u)) ≈ 1.53, computed from the lag-1 autocorrelations of x (0.66) and of the residuals (0.61) (out/state/checks.md).

# Limitations and open issues

• Newey–West inference with Bartlett weights and a normal reference distribution can over-reject in finite samples when errors are this persistent (ρ ≈ 0.6–0.7). Fixed-b or other small-sample refinements would usually widen the interval further, so they would strengthen the verdict rather than reverse it. We did not compute them.
• The bandwidth is a choice. The verdict does not depend on it in the range we checked (5–24 lags), but we did not search every possible procedure.
• Non-significance at 5% does not show that β = 0. The data are consistent with a positive slope of the estimated size, and also with no association at all.
• The data are simulated, as TASK.md states.