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

# Question

Across 40 rolling quarterly origins, 2015-10-01 to 2025-07-01 (n_origins, first_origin, last_origin), does a time-series model beat the seasonal-naive forecast of annualised, not seasonally adjusted, quarterly CPI inflation at h = 1, 2 and 4? And at h = 4, is the mean squared-error loss differential distinguishable from zero once we use a standard error that allows for the serial correlation caused by overlapping forecasts?

# Data and exclusions

Input: data/cpi_quarterly.csv, 127 rows (n_input_rows). data/cpi_monthly.csv (381 rows) was used only to cross-check the file. Both SHA-256 hashes match data/README.md. The quarterly file agrees with the monthly file: quarterly averages match to within 3.3e-7, and recomputing inflation reproduces infl_annualised to within 3.3e-6. There are no duplicates and no gaps (details in out/state/data_audit.md). The one data issue (data_issues) is that inflation is undefined for 1995Q1. Exclusions are listed in out/exclusions.csv: 127 → 126 (drop 1995Q1) → 122 (quarters after the last origin are not used for estimation) → 118 (first 4 quarters used to build lags). So the last estimation sample has 118 observations (n_analysed). Quarters after an origin are used only as realised targets for that origin.

# Method

• Baseline (snaive): the forecast for quarter t+h is the inflation value of quarter t+h−4.
• Model (sar_aic, model_description): OLS regression of inflation on four quarterly seasonal intercepts plus p own lags. At every origin, p ∈ {0,…,4} is chosen by AIC on a common sample. The estimation window expands from 1995Q2 to the origin and uses only quarters ≤ origin. Multi-step forecasts are made by iterating the model forward. Across origins AIC chose p = 1 fifteen times and p = 2 twenty-five times (supp_selected_p_counts).
• Evaluation: errors are actual − forecast. RMSE and MAE are computed over the 40 origins for each horizon.
• Diebold–Mariano-type test at h = 4: d_t = e²_snaive − e²_model. Its long-run variance uses a Newey–West (Bartlett) estimator with truncation lag 3 = h − 1 (hac_lags_h4, hac_kernel_h4). The test uses a two-sided normal p-value.
• The script is deterministic and draws no random numbers (random_seed = null).

# Results

The seasonal AR has lower RMSE and lower MAE than seasonal-naive at all three horizons in this sample. At h = 4, the mean loss differential is 2.83 squared pp (loss_diff_mean_h4), with a HAC SE of 2.08 (loss_diff_se_h4). This gives DM = 1.36 (dm_stat_h4) and a two-sided p = 0.174 (dm_pvalue_h4).

# Uncertainty

At the pre-specified lag of 3 with a Bartlett kernel, the h = 4 improvement is not statistically distinguishable from zero at conventional levels (dm_pvalue_h4). The Harvey–Leybourne–Newbold small-sample-corrected p-value is 0.22 (supp_hln_pvalue_h4). The SE depends heavily on the HAC choice (supp_dm_se_sensitivity_h4): it is 1.85 at Bartlett lag 4, 1.37 at Bartlett lag 6, and 1.70 with a rectangular kernel at lag 3. All of these are smaller than the lag-3 Bartlett SE because the loss differential has negative autocorrelation at higher orders. Lag 3 was fixed before looking at results and is the value reported. The sensitivity means the conclusion is fragile, not robust, and choosing a lag after seeing the results would be specification searching. No significance tests were run at h = 1 or h = 2, so the improvements there are point estimates only.

# Limitations and open issues

• Only 40 origins, and the period includes the 2020 pandemic shock and the 2021–22 inflation surge. A few large errors can dominate squared-error loss.
• With T = 40, the asymptotic normal approximation of the DM test is rough.
• The errors use the current data vintage. CPI is essentially unrevised, but no claim of real-time validity is made. A quarter is treated as known at the end of that quarter, whereas publication actually comes about three weeks later.
• Only one model was tried, so there is no multiple-model search to adjust for.
• supp_* keys are supplementary and not required by the task.