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

# Question

The previous memo (data/memo.md) regressed monthly y on x by OLS with an intercept and used conventional (iid) standard errors to call the slope significant at the 5% level. The task states that the disturbances are AR(1) with autocorrelation around 0.7 and that x is persistent. Under those conditions iid standard errors are too small. This report re-estimates the slope β in y_t = α + β x_t + u_t with a Newey–West (HAC) standard error and checks whether the memo's claim holds.

# Data and exclusions

data/series.csv has 400 rows (n_input_rows), covering 1990-01 to 2023-04 with consecutive months. It has no missing values, no duplicate months and no gaps. No rows were removed (out/exclusions.csv), so n_analysed = 400 and data_issues is empty. File hashes and summary statistics are in out/state/data_audit.md. The memo was used only as a claim to check: its numbers were recomputed, not copied.

# Method

• I fit OLS of y on x with an intercept on the full sample, giving beta_hat and alpha_hat. The naïve baseline is the conventional standard error se_iid, which is the memo's measure.
• rho_resid is the lag-1 autocorrelation of the OLS residuals.
• HAC standard error: Newey–West with a Bartlett kernel, through statsmodels with the small-sample factor n/(n−k). I also checked it against a hand-coded implementation (out/state/checks.md).
• Bandwidth: Andrews (1991) AR(1) plug-in rule for the Bartlett kernel. I applied it to the slope's score v_t = (x_t − x̄)û_t and used S = 1.1447·(α̂(1)·T)^{1/3} with α̂(1) = 4r²/((1−r)²(1+r)²). The score's AR(1) coefficient is r = rho_score = 0.40, which gives S = andrews_bandwidth = 8.18, so hac_lags = 8. I chose a data-dependent rule because both x and u are persistent. The fixed Newey–West (1994) rule of thumb gives 5 lags (nw_rule_of_thumb_lags), and I report it as a sensitivity check.
• Inference uses the normal distribution: t_hac = beta_hat/se_hac, p_hac is two-sided, and the 95% CI is beta_hat ± 1.96·se_hac. No randomness is involved (random_seed = null).

# Results

• beta_hat = 0.1017 and alpha_hat = 2.0925. Together with se_iid = 0.0448, t_iid = 2.27 and p_iid = 0.0238, these reproduce the memo's numbers exactly.
• rho_resid = 0.61, so the residuals are strongly positively autocorrelated.
• se_hac = 0.0653 with hac_lags = 8. That is 1.46 times the iid standard error (se_ratio_hac_to_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 contains zero.
• Verdict: the memo's claim is not supported (memo_claim_supported = false). Once serial correlation is accounted for, the slope is not significant at 5%.

# Uncertainty

• The verdict does not depend on the bandwidth. Over lags 5, 8, 12, 16 and 24, the HAC standard error ranges from 0.0645 to 0.0660 and p from 0.115 to 0.123 (hac_sensitivity). Every one of these p-values is above 0.05 (sensitivity_all_insignificant = true).
• The point estimate is unchanged. Only its precision changes: the effective sample size is much smaller than 400 because both series are persistent.

# Limitations and open issues

• Newey–West tests with persistent data tend to over-reject in finite samples. If anything, the true p-value is likely larger, which would make the "not significant" verdict more conservative, not less. I did not compute fixed-b or other refinements because the task does not require them.
• The residual autocorrelation (0.61) is below the stated value of about 0.7. That fits the known downward bias of the residual autocorrelation estimate in finite samples. It is not a data problem.
• "Not significant at 5%" does not show that β = 0. The CI still allows sizeable positive slopes.
• β is a linear-projection coefficient, not a causal effect.