# F2-conf-02 — Is the slope of y on x really significant?

# Question

The memo (data/memo.md) says the OLS slope of y on x (0.1017, conventional SE 0.0448, p = 0.024) is significant at the 5% level. The task is to re-estimate β in y_t = α + β x_t + u_t by full-sample OLS and to test it with a Newey–West (Bartlett-kernel HAC) standard error, which stays valid when the disturbances are serially correlated.

# Data and exclusions

data/series.csv has 400 rows covering 1990-01 to 2023-04. I checked for missing values, duplicate months and gaps in the monthly sequence and found none. No observations were excluded, so n = 400 and data_issues is empty. The data are synthetic.

# Method

• OLS of y on a constant and x.
• Conventional SE (iid errors). This is the memo's measure and the naïve baseline.
• HAC SE. Newey–West with the Bartlett kernel, computed with statsmodels cov_type="HAC" with the small-sample correction (T/(T−k)).
• Bandwidth choice. I used the Andrews (1991) AR(1) plug-in rule for the Bartlett kernel: L = floor(1.1447·(α̂·T)^{1/3}), with α̂ = 4ρ²/((1−ρ)²(1+ρ)²).
  • ρ is the lag-1 autocorrelation of the slope score v_t = (x_t − x̄)·û_t. This is the series whose long-run variance enters the slope's variance.
  • The estimate is ρ = 0.40, which gives L = 8 months.
  • For comparison, the Newey–West (1994) rule of thumb floor(4(T/100)^{2/9}) gives 5.
• Inference. The two-sided p-value and the 95% CI use the asymptotic normal reference. The p-value from a t(398) reference is also reported and is almost the same.

# Results

I reproduce the memo's point estimates, conventional SE, t-statistic and p-value exactly.

Verdict. The OLS residuals are strongly autocorrelated (ρ̂ = 0.61), and x is also persistent (lag-1 autocorrelation 0.66). Under those conditions the iid standard error understates the uncertainty. The HAC standard error is about 46% larger. With it, the slope is not statistically significant at the 5% level: p ≈ 0.12, and the 95% CI includes zero. The memo's claim is not supported (memo_claim_supported = false).

# Uncertainty

The conclusion holds across bandwidths:

Only the heteroskedasticity-only SE (0 lags, SE 0.0490, p = 0.038) crosses 5%. That SE ignores autocorrelation, so it is not appropriate here. Full details are in hac_sensitivity_by_lags in results.json.

# Limitations and open issues

• With persistent errors and a persistent regressor, Newey–West tests are known to over-reject somewhat in samples of this size. The true p-value is therefore, if anything, larger, which only strengthens the conclusion that significance is not established. Fixed-b or other alternative inference methods were not run.
• The bandwidth rule assumes the score is roughly AR(1).
• "Not significant" means the data cannot distinguish β from zero at 5%. It is not evidence that β = 0.
• β is a linear-projection coefficient: it describes an association, not a causal effect, and it is not a forecasting relationship.
• No random numbers were used (random_seed = null).