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

## Question
The memo says the OLS slope of y on x is significant at the 5% level. It reached that conclusion using conventional (iid) standard errors. The task description says the regression errors are AR(1) with autocorrelation of about 0.7 and that x is persistent. Under those conditions iid standard errors understate the uncertainty. I re-estimate β in y_t = α + β x_t + u_t with a HAC (Newey–West, Bartlett) standard error and check whether the memo's claim holds.

## Data and exclusions
`data/series.csv` contains 400 monthly rows from 1990-01 to 2023-04 (`n_input_rows`). There are no gaps, no missing values and no duplicate months. No rows were removed (`out/exclusions.csv`), so `n_analysed` = 400. No data issues were found (`data_issues` = []). The checksums and summary statistics are in `out/state/data_audit.md`.

## Method
- I ran OLS with an intercept on the full sample. The baseline is the conventional iid standard error (`se_iid`).
- The headline standard error is a Newey–West HAC standard error with a Bartlett kernel, without a small-sample degrees-of-freedom correction (`se_hac`).
- Bandwidth (`hac_lags`): I used the Andrews (1991) AR(1) plug-in bandwidth for the Bartlett kernel, 1.1447·(α(1)·T)^{1/3}. It is computed on the series whose long-run variance drives the slope's variance, the score v_t = (x_t − x̄)û_t. The score's AR(1) coefficient is `rho_score` = 0.401. The raw bandwidth (`andrews_bandwidth_raw` = 8.18) is rounded up to 9 lags. For comparison, the Newey–West (1994) rule of thumb gives 5 lags (`nw1994_rule_lags`), and I also report a grid of 0–36 lags (`out/sensitivity.csv`).
- Inference uses the normal reference distribution: t_hac = β̂/se_hac, a two-sided p-value, and a 95% CI of β̂ ± 1.96·se_hac.
- The method uses no randomness (`random_seed` = null).

## Results
- `alpha_hat` = 2.0925 and `beta_hat` = 0.1017.
- Using iid errors, `se_iid` = 0.0448, `t_iid` = 2.27 and `p_iid` = 0.0238. These match the memo exactly, so I reproduced its numbers rather than taking them as given.
- The first-order autocorrelation of the residuals is `rho_resid` = 0.609. That is strongly positive, in line with the persistent errors described in the task.
- Using HAC with 9 lags, `se_hac` = 0.0652, about 1.46 times `se_iid`. This gives `t_hac` = 1.56 and `p_hac` = 0.119.
- The 95% HAC CI runs from `beta_ci_low` = −0.026 to `beta_ci_high` = 0.230. It includes zero.
- **Verdict:** `memo_claim_supported` = false. Once serial correlation is accounted for, the slope is not significant at the 5% level. The memo's significance comes from understating the standard error.

## Uncertainty
The verdict does not depend on the bandwidth chosen:
- With the NW1994 rule of 5 lags, the standard error is 0.0644 (`se_hac_nw1994_rule`).
- For every bandwidth from 5 to 36 lags, the HAC p-value stays between 0.096 and 0.125 (`sensitivity_min_p`, `sensitivity_max_p`).
- Only 0 or 1 lags give p < 0.05, and those bandwidths are clearly too short given the residual autocorrelation.

## Limitations and open issues
- HAC inference is asymptotic. With persistent x and errors, Newey–West tests with short bandwidths tend to over-reject in finite samples. If that applies here, the true p-value is probably even larger, which would only strengthen the verdict.
- I did not try fixed-b or other alternative inference methods, because the task does not require them.
- The sample residual autocorrelation (0.61) is below the roughly 0.7 stated in the task. That is consistent with the downward bias of autocorrelation estimates from OLS residuals and does not change the conclusion.
- The data are simulated.
- "Not significant" does not mean the effect is zero. The CI is wide and still includes economically meaningful positive slopes.