# Report — F2-conf-02: is the memo's slope really significant?

# Question

The memo (data/memo.md) concludes that the OLS slope of y on x is significant at the 5% level. The standard errors it uses assume independent errors. TASK.md, however, says the disturbances follow an AR(1) process with autocorrelation of about 0.7 and that x is also persistent. I re-estimated β in y_t = α + β x_t + u_t on the full sample with a HAC (Newey–West, Bartlett-kernel) standard error and checked whether the claim still holds.

# Data and exclusions

data/series.csv contains 400 monthly rows (n_input_rows), 1990-01 to 2023-04. It has no missing values, no duplicate months and no gaps. No rows were excluded (out/exclusions.csv), so n_analysed = 400. No data issues were found (data_issues = []). Hashes and summary statistics are in out/state/data_audit.md.

# Method

• OLS of y on a constant and x.
• Baseline: the conventional i.i.d. standard error (se_iid). This reproduces the memo's figures: alpha_hat = 2.0925, beta_hat = 0.1017, se_iid = 0.0448, t_iid = 2.27, p_iid = 0.0238.
• Robust inference: Newey–West HAC covariance with a Bartlett kernel and a small-sample factor n/(n−k), computed with statsmodels.
• Bandwidth choice: the Andrews (1991) AR(1) plug-in for the Bartlett kernel, applied to the slope score (x_t − x̄)û_t. The score's estimated lag-1 autocorrelation is bandwidth_details.rho_score ≈ 0.40, which gives a raw bandwidth of 8.18 (bandwidth_details.andrews_bw_raw). Rounding up gives hac_lags = 9 months.
  • I preferred this data-driven rule to the Newey–West (1994) rule of thumb (5 lags, bandwidth_details.nw1994_rule_of_thumb_lags) because the errors are strongly persistent.
• t-statistics, p-values and the 95% CI use the standard normal distribution. No randomness is involved (random_seed = null).

# Results

• beta_hat = 0.1017 and alpha_hat = 2.0925.
• The first-order autocorrelation of the residuals is rho_resid = 0.61. This confirms strong serial correlation.
• The HAC standard error is se_hac = 0.0654 with 9 lags, about 1.46 times se_iid.
• This gives t_hac = 1.56 and p_hac = 0.120. The 95% CI is [beta_ci_low, beta_ci_high] = [−0.026, 0.230], which includes 0.
• Verdict: memo_claim_supported = false. Once the serial correlation is accounted for, the slope is not significant at the 5% level. The point estimate itself matches the memo; only the inference changes.

# Uncertainty

• Sensitivity to the bandwidth (hac_sensitivity, also in out/hac_sensitivity.csv): HAC p-values range from 0.079 (1 lag) to 0.125 (20 lags). Every lag from 1 to 24 gives p > 0.05 (sensitivity_all_p_above_005_for_lags_ge_1 = true).
• Only 0 lags gives p < 0.05 (0.038). That is White's heteroskedasticity-only standard error, which ignores the autocorrelation and is therefore not valid here.
• Small-sample bias: with persistent regressors and errors, Bartlett HAC standard errors tend to be biased downward in finite samples, so these HAC p-values, if anything, overstate significance. This strengthens the verdict.

# Limitations and open issues

• The bandwidth rule is one reasonable choice among several. The conclusion holds for every choice tested from 1 to 24 lags.
• I used normal critical values. Fixed-b or Student-t critical values would be more conservative and would only reinforce the verdict.
• β is a linear-projection coefficient, not a causal effect.
• The data are synthetic, as TASK.md states.