# F3-conf-04 — Multi-horizon forecasts of Canadian quarterly CPI inflation

## Question
Over 40 expanding-window origins (`first_origin` = 2015-10-01 to `last_origin` = 2025-07-01, `n_origins` = 40), does a seasonal AR model forecast annualised, non-seasonally-adjusted quarterly CPI inflation better than the seasonal-naive forecast at h = 1, 2, 4, and is the h = 4 squared-error loss differential significantly different from zero when its serial correlation (from overlapping 4-step errors) is accounted for?

## Data and exclusions
Input `data/cpi_quarterly.csv` (127 rows; `n_input_rows`) — SHA-256 matches the README; `data/cpi_monthly.csv` (381 rows) was used only for an audit (quarterly averages and inflation reproduced to rounding precision, see `out/state/data_audit.md`). Issues (`data_issues`): 1995Q1 inflation is undefined and dropped; `infl_annualised` used as given; the 2020 COVID and 2021–23 inflation-surge observations are kept. Exclusion ledger (`out/exclusions.csv`): 127 → 126 (1995Q1) → 118 observations in the last estimation sample (`n_analysed` = 118; 4 initial lags and the 4 post-origin quarters 2025Q4–2026Q3, which serve only as targets).

## Method
- Baseline `snaive`: forecast of quarter t+h = inflation of quarter t+h−4.
- Model `ar_seasdum` (`model_name`, `model_description`): OLS regression of inflation on four quarterly seasonal intercepts plus p own lags, p ∈ {0,…,4} chosen by AIC at each origin on a common sample; estimated on all quarters ≤ origin (expanding window from 1995Q2); multi-step forecasts are iterated. AIC picked p = 1 at 15 origins and p = 2 at 25 (`selected_ar_order_counts`). No randomness (`random_seed` = null).
- Evaluation: RMSE and MAE over the 40 origins per horizon; relative improvement 1 − RMSE_model/RMSE_snaive.
- h = 4 test: d_t = e²_snaive,t − e²_model,t; standard error of d̄ from a Newey–West (Bartlett kernel, `hac_kernel_h4`) long-run variance with truncation lag `hac_lags_h4` = 3 (= h − 1, fixed in advance); DM statistic compared with N(0,1), two-sided.

## Results
| h | RMSE snaive | RMSE model | MAE snaive | MAE model | 1 − RMSE ratio |
|---|---|---|---|---|---|
| 1 | 2.849 (`rmse_snaive_h1`) | 2.192 (`rmse_model_h1`) | 2.137 (`mae_snaive_h1`) | 1.519 (`mae_model_h1`) | 0.231 (`rel_improvement_rmse_h1`) |
| 2 | 2.847 (`rmse_snaive_h2`) | 2.460 (`rmse_model_h2`) | 2.126 (`mae_snaive_h2`) | 1.550 (`mae_model_h2`) | 0.136 (`rel_improvement_rmse_h2`) |
| 4 | 2.882 (`rmse_snaive_h4`) | 2.340 (`rmse_model_h4`) | 2.181 (`mae_snaive_h4`) | 1.594 (`mae_model_h4`) | 0.188 (`rel_improvement_rmse_h4`) |

Units: percentage points of annualised inflation. The seasonal AR has lower RMSE and MAE than the seasonal-naive forecast at all three horizons (RMSE gains of 14–23 %).

At h = 4: d̄ = 2.834 squared pp (`loss_diff_mean_h4`), HAC SE = 2.082 (`loss_diff_se_h4`), DM = 1.36 (`dm_stat_h4`), two-sided p = 0.174 (`dm_pvalue_h4`). With the pre-specified test, the h = 4 gain is **not** statistically distinguishable from zero at conventional levels. The model had the smaller squared error at 29 of 40 origins (`sensitivity_h4.n_origins_model_better_sqerr_h4`).

## Uncertainty
- Sensitivity of the h = 4 test (reported in `sensitivity_h4`, not the primary result): Harvey–Leybourne–Newbold small-sample correction with t(39) gives p = 0.222 (`hln_pvalue_t39`). The sample autocorrelations of d_t at lags 3 and 4 are negative (−0.28, −0.41; `d_autocorr_lag1_to_4`), so longer Bartlett truncations *shrink* the SE: p = 0.125 at lag 4, 0.039 at lag 6, 0.018 at lag 8 (`bartlett_L4_pvalue`, `bartlett_L6_pvalue`, `bartlett_L8_pvalue`). The verdict is therefore sensitive to the bandwidth; the primary lag 3 was fixed from the h − 1 rule before results were seen, and we do not switch to a lag that delivers significance. With 40 overlapping observations the long-run variance is poorly estimated whichever lag is used.
- The relative-improvement figures at h = 1 and h = 2 have no formal test here (not required); they are point estimates from 40 forecasts each.

## Limitations and open issues
- Errors are computed on the current vintage; no real-time validity is claimed.
- The evaluation window contains the 2020 pandemic shock and the 2021–23 inflation surge; d_t is unlikely to be covariance-stationary, which the DM asymptotics assume, and a few large errors dominate squared-error loss.
- Only one model class was tried (no search across models on the evaluation sample), which avoids data snooping but means the result is specific to this specification.
- The DM test treats forecasts as given (Diebold–Mariano), not as a test of population model superiority with estimation uncertainty (West/Giacomini–White).