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

# Question

The earlier memo (data/memo.md) regressed monthly y on x by OLS (with intercept, 400 months) and said the slope is significant at 5%, using conventional iid standard errors. The task description says the disturbances follow an AR(1) with autocorrelation of about 0.7 and that x is persistent. Under those conditions iid standard errors are too small. I re-estimated β in y_t = α + β x_t + u_t and tested H0: β = 0 with a Newey–West (Bartlett) HAC standard error.

# Data and exclusions

data/series.csv has 400 rows covering 1990-01 to 2023-04 (n_input_rows). There are no gaps, no duplicate months and no missing values, so I removed nothing (out/exclusions.csv), n_analysed = 400 and data_issues is empty. The x and y series have lag-1 autocorrelations of 0.66 and 0.62 (out/state/data_audit.md). The data are synthetic.

# Method

• OLS of y on a constant and x. The conventional iid standard error (se_iid) is the memo's measure and serves as the baseline.
• rho_resid is the lag-1 autocorrelation of the OLS residuals.
• HAC standard error: Newey–West with the Bartlett kernel, through statsmodels cov_type="HAC", including the small-sample correction n/(n−k).
• Bandwidth: I used the Andrews (1991) AR(1) plug-in for the Bartlett kernel, fitted to the slope score v_t = (x_t − x̄)·û_t. This targets the persistence that actually drives the slope's variance. The score's lag-1 autocorrelation is 0.401, which gives S = 8.18, so hac_lags = 8. I also set a floor at the Newey–West rule of thumb, floor(4(T/100)^{2/9}) = 5. Details are under bandwidth_details.
• Inference: t = β̂/se_hac, a two-sided normal p-value, and a 95% CI of β̂ ± 1.96·se_hac.
• No randomness is involved (random_seed = null).

# Results

• alpha_hat = 2.0925 and beta_hat = 0.1017. se_iid = 0.0448, t_iid = 2.27 and p_iid = 0.0238. These match the memo to its stated rounding (memo_reproduced), so the memo's arithmetic is correct.
• rho_resid = 0.609. The residuals are strongly autocorrelated, so the iid assumption fails.
• se_hac = 0.0653 with hac_lags = 8. That is 1.46 times the iid standard error (se_ratio_hac_over_iid).
• t_hac = 1.56 and p_hac = 0.119. The 95% CI is [beta_ci_low, beta_ci_high] = [−0.026, 0.230], which includes 0.
• Verdict: memo_claim_supported = false. Once the standard error allows for serial correlation, the slope is not significant at the 5% level.

# Uncertainty

The verdict holds for every bandwidth tried, from 4 to 24 lags: hac_sensitivity gives p between 0.114 and 0.123, and any_sensitivity_p_lt_005 = false. The point estimate is positive and of the same size as the memo's. The data are consistent with a modest positive association but do not reject β = 0 at 5%.

# Limitations and open issues

• Newey–West tests over-reject in finite samples when persistence is high. This makes the HAC p-value, if anything, too small, which strengthens the "not significant" verdict rather than weakening it. I did not pursue fixed-b or other refinements because the task does not require them.
• rho_resid (0.61) is a little below the stated value of about 0.7. This fits the known downward bias of residual autocorrelation estimates and does not change the conclusion.
• β is a linear-projection coefficient. It is not a causal effect.