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

# Question

Over 40 expanding-window origins (2015-10-01 to 2025-07-01; first_origin, last_origin, n_origins), does a time-series model beat the seasonal-naive forecast of Canadian quarterly CPI inflation (400·Δln of the quarterly-average, not seasonally adjusted CPI) at horizons 1, 2 and 4? And at h = 4, is the mean squared-error loss differential different from zero once we use a HAC (Diebold–Mariano-type) standard error that allows for the serial correlation created by overlapping 4-step errors?

# Data and exclusions

• data/cpi_quarterly.csv has 127 rows and data/cpi_monthly.csv has 381 (n_input_rows). Both SHA-256 hashes match data/README.md. The quarterly averages, recomputed from the monthly file, match to 3e-7. Inflation recomputed from cpi_avg matches to 3e-6, which is just rounding (data_issues).
• Exclusions (out/exclusions.csv): 1995Q1 is dropped because inflation cannot be computed without a previous quarter (127 → 126). No outliers were treated: the 2020 COVID drop and the 2021–22 surge are kept.
• The last estimation sample (origin 2025-07-01) covers 122 quarterly inflation observations, 1995Q2–2025Q3 (n_analysed). The direct regressions at that origin use 121, 120 and 118 rows for h = 1, 2, 4 (n_regression_rows_last_origin).
• The full audit is in out/state/data_audit.md.

# Method

• Seasonal-naive baseline: forecast for quarter t+h = y_{t+h−4}.
• Model dirSAR_bic (model_name, model_description): a direct h-step OLS regression of y_{s+h} on four target-quarter dummies plus p lags y_s, …, y_{s−p+1}. The lag count p ∈ {0,…,4} is chosen by BIC on a common sample, separately at each origin and horizon. The model is then re-estimated on all usable rows, using only quarters ≤ origin. Model selection is part of the expanding window.
• Which lags BIC chose (selected_p_counts):
  • h = 1: p = 0 at 25 origins, p = 1 at 15.
  • h = 2: p = 0 at 17 origins, p = 1 at 23.
  • h = 4: p = 0 at all 40 origins. At this horizon the model is therefore just the expanding-window mean of each quarter.
• Errors: e = actual − forecast, computed on the data as given. No real-time validity is claimed.
• h = 4 test:
  • The loss differential is d_t = e²_snaive − e²_model.
  • The standard error of d̄ uses a Newey–West long-run variance with a Bartlett kernel and truncation lag 4 (hac_kernel_h4, hac_lags_h4). The lag was fixed before seeing results and is ≥ h−1 = 3.
  • The p-value is two-sided from the standard normal.
• Sensitivity checks (sensitivity_dm_h4): Bartlett with lags 3 and 8, rectangular with lag 3, and Student-t(39) p-values. The analysis uses no randomness (random_seed = null).

# Results

All errors are in percentage points of annualised inflation. Forecasts are in out/forecasts.csv (240 rows).

At all three horizons the model's point RMSE and MAE are lower than those of the seasonal-naive forecast.

At h = 4:

• Mean loss differential d̄ = 2.770 squared pp (loss_diff_mean_h4), with HAC SE 1.907 (loss_diff_se_h4).
• DM statistic 1.45 (dm_stat_h4), two-sided p = 0.146 (dm_pvalue_h4).

At conventional levels, the h = 4 improvement cannot be distinguished from zero.

# Uncertainty

• The h = 4 conclusion depends on the HAC choice (sensitivity_dm_h4):

  HAC choice	p-value
  Bartlett, lag 3	0.197
  Rectangular, lag 3	0.114
  Bartlett, lag 4 (primary)	0.146 (t(39): 0.154)
  Bartlett, lag 8	0.024

  The lag-8 result means the estimated long-run variance shrinks when longer lags are included, i.e. there are negative autocorrelations at longer lags. I keep the pre-specified lag 4 as the main result and do not choose a lag after seeing the results.

• h = 1 and h = 2 are also not significant. These supplementary DM tests (Bartlett, lag = h; dm_tests_h1_h2_bartlett_L_h) give p = 0.105 at h = 1 and 0.269 at h = 2. So the point improvements at those horizons are not statistically established either.

• The sample is small. It has 40 overlapping errors and a few extreme quarters (2020–2022), which dominate squared losses. The normal approximation for the DM statistic is therefore rough.

# Limitations and open issues

• There is a single evaluation window of 40 origins, dominated by the COVID and 2021–22 inflation episodes. The ranking could differ in calmer periods.
• The comparison is pseudo-out-of-sample on the current vintage, not real-time.
• The model is chosen by me. Alternatives (seasonal ARIMA, iterated AR) were not evaluated, and no search over models was done on the evaluation sample.
• At h = 4 the selected model is just expanding-window quarterly means. Its edge over seasonal-naive comes from averaging out noise in the single year-ago value.
• DM-test significance at h = 4 is sensitive to the HAC truncation lag, as reported above.
• Everything required was produced (feasibility = "complete").