# F2-conf-02 — Re-examining the memo's slope significance claim

## Question
Is the OLS slope β in y_t = α + β x_t + u_t (monthly, 1990-01 to 2023-04) significantly different from zero at the 5% level once the standard error allows for serially correlated disturbances? The previous memo said yes, but it used conventional i.i.d. OLS standard errors.

## Data and exclusions
`data/series.csv` (simulated data): 400 rows, 400 consecutive months with no gaps, no duplicates and no missing values. I made no exclusions, so n = 400. I found no data issues.

## Method
- OLS of y on x with an intercept on the full sample (statsmodels).
- Naïve baseline: the conventional OLS standard error, as in the memo.
- Residual diagnostic: the lag-1 autocorrelation of the OLS residuals.
- HAC standard error: Newey–West with a Bartlett kernel and the small-sample correction n/(n−k).
- **Bandwidth choice.** I used Andrews' (1991) AR(1) plug-in for the Bartlett kernel, S = 1.1447·(α̂(1)·n)^{1/3} with α̂(1) = 4ρ²/((1−ρ)²(1+ρ)²). Here ρ is the lag-1 autocorrelation of the score series v_t = (x_t − x̄)·û_t, which is the series whose long-run variance the HAC estimator targets. With ρ ≈ 0.40 this gives **8 lags**. The Newey–West (1994) rule of thumb, floor(4(n/100)^{2/9}), gives 5 lags. I report a sensitivity check over 5, 8, 12, 18 and 24 lags.
- Inference: t = β̂/se_hac, a two-sided normal p-value, and a 95% CI of β̂ ± 1.96·se_hac.

## Results
| quantity | value |
|---|---|
| α̂ (intercept) | 2.0925 |
| β̂ (slope) | 0.1017 |
| se, i.i.d. OLS | 0.0448 (t = 2.27, p = 0.024) — matches the memo |
| residual lag-1 autocorrelation | 0.609 |
| se, HAC (Bartlett, 8 lags) | 0.0653 |
| t (HAC) | 1.56 |
| p (HAC, two-sided) | 0.119 |
| 95% CI (HAC) | [−0.026, 0.230] |

I reproduced the memo's point estimates and its conventional standard error exactly. The residuals are strongly autocorrelated (0.61), and the regressor is also persistent (lag-1 autocorrelation ≈ 0.66). Together these make the i.i.d. standard error too small. The HAC standard error is about 46% larger, and the slope is **not** significant at 5%: p ≈ 0.12, and the 95% CI includes zero.

**Verdict: the memo's claim that "the slope is statistically significant at the 5% level" is not supported.**

## Uncertainty
The conclusion does not depend on the bandwidth:

| lags | se_hac | t | p |
|---|---|---|---|
| 5 (NW rule of thumb) | 0.0645 | 1.58 | 0.115 |
| 8 (Andrews plug-in, main) | 0.0653 | 1.56 | 0.119 |
| 12 | 0.0657 | 1.55 | 0.121 |
| 18 | 0.0663 | 1.54 | 0.125 |
| 24 | 0.0660 | 1.54 | 0.123 |

At every bandwidth p is above 0.11. The point estimate is positive, so the data do not show that the slope is zero. They also do not give evidence at the 5% level that it differs from zero.

## Limitations and open issues
- Newey–West HAC inference with persistent errors and a persistent regressor can over-reject in samples of this size, because the HAC standard error tends to be biased downward. Fixed-b or other better-sized methods (e.g., Kiefer–Vogelsang critical values, or larger bandwidths) would, if anything, give larger p-values. This strengthens the "not significant" verdict rather than weakening it.
- The p-values use the asymptotic normal reference distribution. A t(398) reference gives practically the same values.
- The estimand is a linear projection coefficient. Nothing here implies a causal effect or that x predicts y out of sample.
- The data are simulated. I did not use a random seed because nothing in the analysis is stochastic.