# F2-conf-02 — Re-examining the memo: is the slope really significant?

# Question

The memo says that the OLS slope of y on x (400 monthly observations, 1990-01 to 2023-04) is statistically significant at the 5% level. That conclusion relies on conventional i.i.d. standard errors. We re-estimate β in y_t = α + β x_t + u_t, compute a standard error that remains valid when the disturbances are serially correlated (Newey–West HAC), and check whether the claim still holds.

# Data and exclusions

data/series.csv (SHA-256 c29dde22…bc514) has 400 rows (n_input_rows). The months run consecutively with no gaps, and there are no missing values or duplicates. No rows were excluded (out/exclusions.csv), so n_analysed = 400. No data problems were found (data_issues = []). The memo was used only as a set of claims to check. The data are simulated.

# Method

• OLS of y on a constant and x on the full sample. This gives alpha_hat and beta_hat.
• Baseline, which is the memo's measure: the conventional OLS standard error, se_iid, with t and p from Student-t(398) (baseline_t_iid, baseline_p_iid).
• Residual first-order autocorrelation: rho_resid = Σ û_t û_{t−1} / Σ û_t².
• HAC standard error: Newey–West with a Bartlett kernel and the small-sample correction n/(n−k) (statsmodels cov_type="HAC").
• Bandwidth: Andrews (1991) AR(1) plug-in for the Bartlett kernel, applied to the slope score v_t = (x_t − x̄)û_t. The formula is S = 1.1447·(a₁T)^{1/3}, with a₁ = 4ρ_v²/((1−ρ_v)²(1+ρ_v)²), rounded up. This rule is driven by the data. It responds to the persistence of both x and u, which matters because the task describes both as persistent. The fixed Newey–West (1994) rule of thumb, 4(T/100)^{2/9}, ignores that persistence. The estimated AR(1) coefficient of the score is rho_score, the raw bandwidth is andrews_bandwidth_raw, and the bandwidth used is hac_lags.
• Inference: t_hac = β̂/se_hac, a two-sided p-value from the standard normal, and a 95% CI of β̂ ± 1.96·se_hac.
• Decision rule: the memo's claim counts as supported if and only if p_hac < 0.05.
• The analysis is fully deterministic. No randomness is used (random_seed = null).

# Results

• Memo reproduced exactly: alpha_hat = 2.0925, beta_hat = 0.1017, se_iid = 0.0448, baseline_t_iid = 2.27, baseline_p_iid = 0.0238.
• The residuals are strongly autocorrelated: rho_resid = 0.609.
• The slope score has rho_score = 0.401. This gives andrews_bandwidth_raw = 8.18, so hac_lags = 9 months.
• HAC standard error: se_hac = 0.0654. That is 1.46 times the i.i.d. standard error (se_ratio_hac_to_iid).
• t_hac = 1.56 and p_hac = 0.120.
• 95% CI for β: [beta_ci_low, beta_ci_high] = [−0.026, 0.230]. The interval contains zero.
• Verdict: memo_claim_supported = false. Once serial correlation is accounted for, the slope is not significant at the 5% level. The point estimate itself is unchanged.

# Uncertainty

The verdict does not depend on the bandwidth (hac_sensitivity). With 5 lags (the NW rule of thumb, nw_rule_of_thumb_lags), p_hac = 0.115. With 6, 8, 12, 18 and 24 lags, p ranges from 0.117 to 0.125. In every case se_hac falls between 0.0645 and 0.0663, and the t-statistic never exceeds 1.58.

# Limitations and open issues

• Bartlett-kernel HAC tests are known to over-reject in samples of moderate size when x and u are both persistent. Fixed-b critical values or a longer bandwidth would make the test more conservative. Our test does not reject even with the normal reference, so this would only strengthen the conclusion.
• The Andrews plug-in assumes an AR(1) approximation for the score.
• "Not significant" means the data cannot exclude β = 0. It does not show that β = 0. The point estimate is 0.10 and the CI is wide.
• The data are simulated. The slope is a linear-projection coefficient and should not be read as a causal effect.