F2-conf-02 — Is the memo's slope really significant?
Question
The memo estimates y_t = α + β x_t + u_t by OLS on 400 monthly observations and, using conventional (iid) standard errors, declares β significant at 5%. The disturbances are serially correlated, so we re-estimate β with a HAC (Newey–West, Bartlett kernel) standard error and assess the claim.
Data and exclusions
data/series.csv: 400 rows read, 1990-01 to 2023-04. Checks: no missing values, no duplicate months, no gaps in the monthly sequence. No exclusions; n_analysed = 400. The data are simulated (synthetic), not observed. No data issues were found.
Method
- OLS of y on x with an intercept (statsmodels).
- Conventional iid SE reproduced as the memo's baseline.
- Residual first-order autocorrelation ρ̂ = Σ û_t û_{t-1} / Σ û_t².
- HAC SE: Newey–West with Bartlett kernel, small-sample df correction applied. Bandwidth choice: Andrews (1991) AR(1) plug-in for the Bartlett kernel, applied to the slope score v_t = (x_t − x̄) û_t: α̂ = 4ρ_v²/((1−ρ_v)²(1+ρ_v)²), S = 1.1447(α̂T)^{1/3}. This gives S = 8.18, rounded up to L = 9 lags. I preferred this data-dependent rule over the fixed Newey–West (1994) rule of thumb, floor(4(T/100)^{2/9}) = 5, because the errors are known to be persistent. The sensitivity table below shows that the conclusion does not depend on this choice.
- Inference uses the asymptotic normal distribution: two-sided p-value and 95% CI = β̂ ± 1.96·SE_HAC.
Results
| quantity | value |
|---|---|
| α̂ (intercept) | 2.0925 |
| β̂ (slope) | 0.1017 |
| SE iid (memo baseline) | 0.0448 (t = 2.27, p = 0.024; this reproduces the memo exactly) |
| residual ρ̂₁ | 0.609 |
| SE HAC (Bartlett, 9 lags) | 0.0654 |
| t_HAC | 1.56 |
| p_HAC (two-sided) | 0.120 |
| 95% CI (HAC) | [−0.026, 0.230] |
I reproduced the memo's point estimates and its iid SE. After allowing for serial correlation, however, the SE is about 46% larger, the t-statistic falls to 1.56 and the 95% CI includes zero. The memo's claim that the slope is significant at 5% is not supported.
Uncertainty
Sensitivity of the HAC SE to the bandwidth. All values come from out/diagnostics.json.
| lags | SE | t | p (normal) | p (t, 398 df) |
|---|---|---|---|---|
| 0 (White, heterosk. only) | 0.0490 | 2.08 | 0.038 | 0.039 |
| 5 (NW rule of thumb) | 0.0645 | 1.58 | 0.115 | 0.116 |
| 8 | 0.0653 | 1.56 | 0.119 | 0.120 |
| 9 (chosen) | 0.0654 | 1.56 | 0.120 | 0.121 |
| 12 | 0.0657 | 1.55 | 0.121 | 0.122 |
| 18 | 0.0663 | 1.54 | 0.125 | 0.125 |
| 24 | 0.0660 | 1.54 | 0.123 | 0.124 |
The slope is significant only if serial correlation is ignored (0 lags). For every bandwidth from 5 to 24 lags, p is between 0.115 and 0.125.
Limitations and open issues
- Newey–West tests with Bartlett kernels tend to over-reject in finite samples when the errors are persistent. Fixed-b or other more conservative inference would probably push p even higher, so it would strengthen the conclusion rather than reverse it.
- The Andrews plug-in assumes an AR(1) approximation for the score.
- A non-rejection here does not show that β = 0. The CI is wide and includes economically non-trivial positive values.
- β is a linear-projection coefficient, not a causal effect.