Forecasting and Explaining the Phillips Curve: A SHAP-Based Comparison of Machine Learning and Traditional Time-Series Models for Canadian Unemployment and Inflation

Louis Agyekum

REVIEW 4 major objections 6 minor 2 references

This paper claims that the best model for Canadian inflation depends on the forecast horizon: ARIMA wins at one month, while Random Forest and XGBoost cut errors by 30-75 percent at six and twelve months.

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2026-08-01 04:40 UTC pith:C7433MWD

load-bearing objection The horizon-dependent crossover is likely a direct-versus-iterated forecasting artifact, not evidence about ARIMA vs. tree ensembles. the 4 major comments →

arxiv 2607.22453 v1 pith:C7433MWD submitted 2026-07-24 stat.AP

Forecasting and Explaining the Phillips Curve: A SHAP-Based Comparison of Machine Learning and Traditional Time-Series Models for Canadian Unemployment and Inflation

Louis Agyekum This is my paper

classification stat.AP MSC 62M1062M2062P20

keywords Phillips curveinflation forecastingmachine learningXGBoostRandom ForestARIMASHAPwalk-forward validation

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish a horizon-dependent crossover in inflation forecasting: a simple linear ARIMA model is statistically the most accurate one month ahead, but tree-based ensembles become substantially more accurate at six- and twelve-month horizons, with RMSE reductions of roughly 30-75 percent versus ARIMA and VAR. It also tries to show that the nonlinear unemployment-inflation relationship implied by the winning tree model is flat across most of the observed range and steepens sharply only in the extreme high-unemployment tail associated with the 2020 pandemic shock. If both claims hold, forecasters should choose models by horizon and regime rather than relying on a single global champion, and the Phillips curve debate should pay attention to tail nonlinearity at high unemployment, not only in tight labor markets.

Core claim

The central empirical discovery is a statistically significant reversal of forecasting superiority. In a walk-forward expanding-window evaluation on monthly Canadian data, ARIMA beats every machine-learning and deep-learning contestant at the one-month horizon, a result confirmed by a formal test of equal predictive accuracy. At six and twelve months, Random Forest and XGBoost beat ARIMA and VAR by large margins, while LSTM and GRU deteriorate badly. On the interpretation side, the paper uses SHAP on the best XGBoost model to recover an implied Phillips curve: recent inflation lags dominate the forecast, lagged unemployment contributes a smaller but meaningful channel, and the marginal effec

What carries the argument

The central mechanism is a horizon-conditional walk-forward comparison: six model classes are re-estimated on an expanding window and scored on the same out-of-sample period, with each pairwise gap tested by a formal equal-predictive-accuracy statistic. On top of that, SHAP, a game-theoretic attribution method that decomposes each prediction into additive feature contributions, is applied to the final XGBoost model, and the smoothed dependence between lagged unemployment and its SHAP contribution is read as a data-driven Phillips curve.

Load-bearing premise

The entire evaluation window starts after the COVID shock, so the claimed horizon crossover is measured only in pandemic and post-pandemic conditions; if the pre-2020 tranquil period had been included, the ranking might look different.

What would settle it

Re-run the walk-forward with an out-of-sample period that includes 2015-2019 calm months, or split the evaluation into pre-pandemic and post-pandemic blocks; if ARIMA also matches or beats the tree ensembles at six and twelve months in the calm block, the claimed crossover does not generalize beyond the volatile period.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

At one-month horizons, forecasters can reasonably keep parsimonious linear ARIMA models; the machine-learning overhead is not paying off in this sample.
At six- and twelve-month horizons, tree ensembles offer large, statistically significant error reductions, so longer-range inflation forecasts should not default to linear benchmarks.
Recurrent neural networks are not a reliable choice on short macroeconomic samples: they were competitive only at one month and deteriorated fastest at longer horizons.
No single model dominates across all macroeconomic regimes, so reported 'best model' rankings should be accompanied by the horizon and regime in which they were produced.
The implied Phillips curve from the winning model is flat over most of the observed unemployment range, which is consistent with a fairly flat curve; the steep negative tail is a pandemic-driven feature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Because the out-of-sample window begins only in September 2020, the crossover is measured under post-pandemic volatility; the natural next test is whether the same crossover appears when evaluating on tranquil pre-2020 periods.
If the horizon dependence is real, a forecast-combination scheme that switches between ARIMA at short horizons and tree ensembles at long horizons could beat any single model; the paper does not test this, but it follows directly.
The reverse-direction result hints that inflation lags predict unemployment more strongly than unemployment predicts inflation; this editorial inference suggests the economic 'Phillips curve channel' in forecasting may run more from inflation to unemployment than the reverse.
The extreme-tail steepening rests on very few observations, so the paper's own transparency suggests treating it as an episode-specific pattern; a cross-country replication with more high-unemployment episodes would show whether it is a real economic regularity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The horizon-dependent crossover is likely a direct-versus-iterated forecasting artifact, not evidence about ARIMA vs. tree ensembles.

read the letter

Short version: the central result—ARIMA wins at h=1, tree ensembles at h=6/12—is supported by the reported RMSEs and Diebold-Mariano tests, but I think the comparison is contaminated by an unstated forecasting-strategy choice. The ML models are trained with 12 lags of the predictors as features and the h-step-ahead target as the response, which is direct forecasting. ARIMA and VAR are standard one-step models, and their h-step forecasts are almost certainly produced by iterating the one-step recursion. So the crossover may just be direct vs. iterated, not nonlinearity or model class. The paper never acknowledges this, and it is not in the limitations section. That makes the headline claim fragile.

Credit where due: the walk-forward design is proper, the DM tests use the Harvey-Leybourne-Newbold correction, and the paper is honest about the undefined DM statistic at h=12. The SHAP analysis is also carefully caveated: the authors explicitly say the steep high-unemployment tail rests on a handful of pandemic observations and is not a structural causal estimate. The regime-conditional results and the Random Forest robustness check are useful. This is a competent implementation of a standard pipeline.

The other soft spots are real but secondary. The OOS window is entirely post-August 2020, so the rankings aren't tested in a tranquil regime—the paper does flag this. The VAR(11) chosen by AIC is overfitted and likely explains the VAR's terrible long-horizon errors. No code or data are shared. All of these are fixable.

My bottom line: the paper is worth engaging, but the direct/iterated issue needs to be addressed before the crossover can be taken at face value. I'd send it to peer review, but I'd want the authors to either explicitly compare direct and iterated versions of both model classes or reframe the claim as 'direct ML beats iterated ARIMA at long horizons.' As is, it is a useful cautionary example for what can go wrong in forecast horse races.

Referee Report

4 major / 6 minor

Summary. The paper compares six forecasting models (ARIMA, VAR, Random Forest, XGBoost, LSTM, GRU) for monthly Canadian CPI inflation over January 2012–April 2026, using expanding-window walk-forward validation at horizons 1, 3, 6, and 12 months. It reports a horizon-dependent crossover: ARIMA is statistically most accurate at h=1, while Random Forest and XGBoost are more accurate at h=6 and h=12, with Diebold–Mariano tests for most cells. It then uses SHAP on a full-sample XGBoost model to describe an implied unemployment–inflation relationship, finding a relatively flat relationship over most of the unemployment range and a steep negative slope at very high (pandemic-era) unemployment. The paper emphasizes the horizon- and regime-conditional nature of model rankings rather than a single global winner.

Significance. If the forecasting crossover is valid, the paper contributes to the ongoing debate about when machine learning methods add value in macroeconomic forecasting, complementing recent US-focused studies with Canadian evidence. The walk-forward design, transparent reporting of the undefined DM statistic at h=12, and explicit robustness checks (e.g., RF-based SHAP, reversed prediction direction) are strengths. The SHAP side illustrates both the potential and the fragility of using post hoc interpretability to recover economic relationships. However, the central crossover claim is clouded by a forecasting-mechanism confound and by an out-of-sample window that is entirely post-COVID, so the empirical regularity as stated is not yet firmly established.

major comments (4)

[Sections 4.5–4.6] The paper never specifies whether forecasts are direct or iterated. The ML models are defined with features dated at t and targets at t+h (direct multi-step), while ARIMA and VAR are conventional one-step models whose h-step forecasts are normally iterated. Thus the headline comparison is direct-ML versus iterated-linear, and the h=6/h=12 gains of tree ensembles may be a direct-forecasting artifact rather than evidence about model class. This confound is load-bearing for the central claim. The authors should either estimate direct versions of ARIMA/VAR (single-equation projections on t-dated lags) or iterated versions of RF/XGBoost, and report both; at minimum, the discussion must acknowledge the confound and temper the attribution of the crossover to nonlinearity/ML flexibility.
[Section 4.6; Table 4] The out-of-sample evaluation window begins in September 2020, with no pre-2020 evaluation points; Section 5.6 explicitly states that no OOS points fall in the pre-COVID regime. Consequently, all reported RMSE rankings and DM tests are estimated only under post-pandemic and disinflation conditions. The abstract and conclusion present ARIMA-best-at-h=1 and trees-best-at-h=6/12 as a general statement about model classes, but the design cannot support that generalization. The authors should reframe the claim as specific to the 2020–2026 turbulent period, or extend the sample/backtest to include tranquil pre-2020 OOS observations.
[Table 6; Abstract] At h=12, the DM statistic for XGBoost versus ARIMA is undefined (NaN), yet the abstract states that 'Random Forest and XGBoost become notably superior at six and twelve months, reducing RMSE by approximately 30–75 percent.' Only Random Forest has a significant DM test at h=12; XGBoost's advantage is descriptive only. This overstates the statistical support. The paper should either compute a valid test (e.g., using a different long-run variance estimator or fixed-smoothing inference) or explicitly state that the XGBoost 12-month result is not formally tested.
[Sections 5.8 and 7] The SHAP analysis is presented as recovering an 'implied, nonlinear Phillips curve,' but SHAP values are computed from a predictive model fit to the full sample and are not identified as structural causal parameters. The paper itself acknowledges this in Section 7, yet the title and contribution claim 'Explaining the Phillips Curve' and the abstract describes the SHAP result in relatively unhedged language. Additionally, the extreme-tail steepening rests on a small number of pandemic-era observations, as disclosed in Section 5.8. The interpretive language should be consistently hedged so that readers do not mistake a post hoc description of a predictive model for a structural economic finding.

minor comments (6)

[Section 4.6] Please specify explicitly for each model whether forecasts at h>1 are produced directly or iteratively; this is essential for interpreting Table 4.
[Table 4] Bold the lowest RMSE in each horizon or add a footnote to make the 'winner' immediately visible, given the discussion references it.
[Section 5.4] The selected ARIMA(4,1,4) has fairly high orders for n=172; please comment on parameter redundancy and perhaps compare with a more parsimonious model (e.g., ARIMA(1,1,1)) as a robustness check.
[Equation (10)] MAPE with y_t in the denominator can be unstable when inflation is near or below zero (the sample minimum is -0.37); consider reporting symmetric MAPE or another scale-free measure.
[Data Availability] The link 'https://www.statscan.ca' appears to be a typo; the correct domain is 'statcan.gc.ca'.
[Author Contributions] The text says 'All authors declare to have contributed equally' but the paper lists a single author; please correct this template artifact.

Circularity Check

0 steps flagged

No significant circularity: the forecasting comparison is genuinely out-of-sample and the SHAP interpretation is explicitly descriptive.

full rationale

The central claim (a horizon-dependent crossover in forecast accuracy) is produced by the expanding-window walk-forward design in Section 4.6, where each model is re-estimated using only data available through t and evaluated on t+h; the RMSE and Diebold-Mariano results in Tables 4 and 6 are empirical outputs of that design, not identities. The SHAP analysis in Sections 4.8 and 5.8 is explicitly post hoc: the paper states that the SHAP-based interpretation 'is not a structural causal estimate' and that the in-sample R^2 'should be interpreted only as a check ... not as evidence of genuine forecasting accuracy.' The extreme-tail steepening of the implied Phillips curve is also disclosed as resting on a small number of pandemic-era observations. The only author-overlapping citation (Nortey et al. 2025) appears as related work and as a point of comparison, not as evidence for any load-bearing empirical claim; under the hard rules it is therefore not circularity. Concerns about direct-versus-iterated forecasting and the entirely post-COVID evaluation window are validity threats, not instances of a result being equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The paper's central claims rest on a modest number of modeling choices: hand-selected ML hyperparameters, AIC-selected model orders, and the interpretive leap that SHAP dependence plots from a predictive model represent an economic Phillips curve. The forecasting comparison itself is data-driven and out-of-sample, but the structural interpretation adds a non-empirical assumption.

free parameters (6)

ARIMA order (p,d,q) for inflation = (4,1,4)
Selected by AIC grid search over p,q in {0..4}, d in {0,1}; not clear whether re-selected in each expanding window, so model order may use full-sample information.
VAR lag order = 11 (AIC; BIC/HQ choose 1)
AIC-selected; overfitting concern flagged by authors.
Random Forest hyperparameters = 300 trees, max_depth=6, min_samples_leaf=3
Chosen by hand; not tuned on validation data.
XGBoost hyperparameters = 300 rounds, depth=4, learning_rate=0.03, row/col subsample=0.8
Chosen by hand to mitigate overfitting.
LSTM/GRU architecture = 32 units, dropout=0.2, window=12
Chosen by hand; early stopping on training loss.
Regime boundaries = pre-COVID: 2012-01 to 2020-02; COVID: 2020-03 to 2020-12; surge: 2021-01 to 2023-06; disinflation: 2023-07 onward
Author-defined a priori; partially validated by PELT (Feb 2021 break), but Jan 2024 extra break not modeled.

axioms (6)

domain assumption The Phillips curve relationship is a meaningful bivariate object; unemployment and inflation are jointly determined and cointegrated.
Central to interpretation; Johansen test supports cointegration but does not establish structural relationship.
domain assumption Forecast loss is symmetric quadratic (RMSE) and DM test with squared-error loss is the relevant metric.
Standard in forecasting literature but not justified in economic terms.
ad hoc to paper SHAP dependence plots from a predictive model can be interpreted as an implied structural Phillips curve.
This is the key interpretive leap in the paper; the authors acknowledge it is not a structural causal estimate.
ad hoc to paper The sample's regime boundaries are correct and the Jan 2024 break is ignored.
PELT detects Jan 2024 break in both series, not incorporated in the four-regime analysis.
domain assumption The expanding-window walk-forward forecasts mimic real-time forecasting.
But data vintages are not real-time; paper acknowledges revised data limitation.
standard math Standard regularity conditions for DM test and unit root tests hold.
Unproved background econometric assumptions.

pith-pipeline@v1.3.0-alltime-deepseek · 16861 in / 15448 out tokens · 148148 ms · 2026-08-01T04:40:43.794958+00:00 · methodology

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read the original abstract

This study evaluates the out-of-sample forecasting ability of six model types: ARIMA, VAR, Random Forest, XGBoost, LSTM, and GRU, for monthly Canadian inflation from January 2012 to April 2026 (n = 172). The evaluation employs expanding-window walk-forward validation across 1-, 3-, 6-, and 12-month horizons. Results reveal a horizon-dependent shift: ARIMA significantly outperforms all machine learning and deep learning models at the one-month horizon (Diebold-Mariano p < 0.05). However, Random Forest and XGBoost become notably superior at six and twelve months, reducing RMSE by approximately 30-75 percent compared to ARIMA and VAR. LSTM and GRU perform well only at the shortest horizon, likely due to overfitting given the limited data. Analyzing four macroeconomic sub-periods shows that no single model consistently dominates. SHAP analysis of the top-performing XGBoost model indicates that lagged inflation is more influential than unemployment, which only becomes significantly impactful during the pandemic tail. The findings clarify when machine learning methods can surpass traditional benchmarks.

Figures

Figures reproduced from arXiv: 2607.22453 by Louis Agyekum.

**Figure 1.** Figure 1: Canadian unemployment and inflation rates, January 2012–April 2026, with shaded regime windows (pre-COVID, COVID shock, inflation surge, disinflation) and the Bank of Canada's 2% inflation target shown as a reference line [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗

**Figure 2.** Figure 2: The empirical Phillips curve: unemployment versus inflation, colored by macroeconomic regime (left panel), and a comparison of linear (OLS) and nonparametric (LOWESS) fits over the full sample (right panel) [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Figure 3: Distributional and autocorrelation diagnostics for the unemployment rate and inflation rate: histograms with kernel density overlay (left column), autocorrelation functions (middle column), and partial autocorrelation functions (right column), 24 lags. 5.1 Stationarity and Diagnostic Tests [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

**Figure 4.** Figure 4: PELT structural break detection for the unemployment rate (top panel; detected breakpoints May 2017 and January 2024) and the inflation rate (bottom panel; detected breakpoints February 2021 and January 2024). The PELT algorithm identifies two breakpoints in each series. For the unemployment rate, breaks occur in May 2017 and January 2024; for the inflation rate, breaks occur in February 2021 and January 2… view at source ↗

**Figure 5.** Figure 5: CUSUM test for structural stability based on recursive residuals from a first-order autoregressive benchmark. Unemployment rate (left panel, CUSUM p = 0.667); inflation rate (right panel, CUSUM p = 0.275). In contrast to the PELT results, the CUSUM test fails to reject the null hypothesis of parameter stability for either series over the full sample (unemployment: p = 0.667; inflation: p = 0.275). This app… view at source ↗

**Figure 6.** Figure 6: Forecast accuracy (RMSE) across all six models and four forecast horizons: heatmap (left panel) and grouped bar chart (right panel) [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗

**Figure 7.** Figure 7: Actual versus predicted inflation rate, ARIMA model, one-month-ahead forecasts, September 2020–April 2026. 5.6 Regime-Conditional Forecast Accuracy [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

**Figure 8.** Figure 8: Regime-conditional RMSE by model, separately for each of the four forecast horizons [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗

**Figure 9.** Figure 9: Best-performing model (lowest RMSE) by macroeconomic regime and forecast horizon. Cells marked “insufficient data” correspond to regime×horizon combinations with fewer than three out-of-sample evaluation points. The regime-conditional results in [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗

**Figure 10.** Figure 10: Diebold-Mariano test statistics for each machine learning and deep learning challenger versus ARIMA, by forecast horizon. Asterisks denote significance at the 5% level [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗

**Figure 13.** Figure 13: SHAP dependence plot for the one-month lag of the unemployment rate (left panel, colored by the interacting one-month lag of inflation) and a LOWESS-smoothed implied marginal effect (right panel), pooled across the full sample [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗

**Figure 14.** Figure 14: SHAP-implied marginal effect of lagged unemployment on predicted inflation, plotted separately by macroeconomic regime: pre-COVID (n = 96), inflation surge (n = 36), and disinflation (n = 27). The COVID shock regime is omitted from this panel because it has fewer than the minimum required number of observations (eight) for a regime-specific LOWESS fit [PITH_FULL_IMAGE:figures/full_fig_p039_14.png] view at source ↗

**Figure 16.** Figure 16: SHAP-implied dependence between lagged unemployment and predicted inflation, Random Forest model (left panel), and a normalized shape comparison between Random Forest and XGBoost (right panel) [PITH_FULL_IMAGE:figures/full_fig_p042_16.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

Bai, J., & Perron, P. (2003). Computation and analysis of multiple structural change models. Journal of Applied Econometrics, 18(1), 1–22. https://doi.org/10.1002/jae.659 Coulombe, P. G., Leroux, M., Stevanovic, D., & Surprenant, S. (2022). How is machine learning useful for macroeconomic forecasting? Journal of Applied Econometrics, 37(5), 920–964. Diebo...

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https://doi.org/10.3934/DSFE.2025007 Smith, S. C., Timmermann, A., & Wright, J. H. (2023). Breaks in the Phillips curve: Evidence from panel data (NBER Working Paper No. 31773). National Bureau of Economic Research. Statistics Canada. (2026). Labour force characteristics by province, monthly, seasonally unadjusted (Table 14-10-0374-01). https://www150.sta...

work page doi:10.3934/dsfe.2025007 2023

[1] [1]

Bai, J., & Perron, P. (2003). Computation and analysis of multiple structural change models. Journal of Applied Econometrics, 18(1), 1–22. https://doi.org/10.1002/jae.659 Coulombe, P. G., Leroux, M., Stevanovic, D., & Surprenant, S. (2022). How is machine learning useful for macroeconomic forecasting? Journal of Applied Econometrics, 37(5), 920–964. Diebo...

arXiv 2003

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C., Timmermann, A., & Wright, J

https://doi.org/10.3934/DSFE.2025007 Smith, S. C., Timmermann, A., & Wright, J. H. (2023). Breaks in the Phillips curve: Evidence from panel data (NBER Working Paper No. 31773). National Bureau of Economic Research. Statistics Canada. (2026). Labour force characteristics by province, monthly, seasonally unadjusted (Table 14-10-0374-01). https://www150.sta...

work page doi:10.3934/dsfe.2025007 2023