Solving Canada's Inflation 'Price Puzzle': Why a Classical Model Beat Deep Learning at Forecasting CPI

Canada's Consumer Price Index — the number behind interest-rate decisions, wage negotiations, and government benefit indexing — has spent the past three decades on a mostly one-way climb. A new academic study set out to answer which forecasting method actually predicts it best, testing more than 30 years of monthly data going back to January 1995.

The contest pitted a classical statistical technique — a seasonal ARIMA model with exogenous variables (SARIMAX) — against three more modern machine-learning approaches: an LSTM neural network, a GRU network, and a weighted ensemble of automated models. The classical model won convincingly on every accuracy measure that matters for forecasting, even though the machine-learning models scored marginally higher on a pure statistical fit metric.

Along the way, the study also tackled a decades-old riddle in monetary economics known as the 'price puzzle': simple statistical models often show that when a central bank raises interest rates, prices rise afterward rather than fall — the opposite of what basic economic theory predicts. Using a technique that accounts for the Bank of Canada's own forward-looking behaviour — it often raises rates in anticipation of inflation, not purely in reaction to it — the study shows the puzzle dissolves. Once that anticipation is properly modelled, higher interest rates behave exactly as classical theory expects: they cool inflation over time, and Canada's central bank turns out to be reacting to expected inflation more aggressively than textbook rules typically assume.

Why Canada's CPI number carries so much weight

Statistics Canada compiles the CPI monthly against a base year of 2002 (indexed to 100), tracking prices across 491 elementary product and service aggregates chosen to represent what urban and rural households actually buy. Shelter carries the largest weight in the basket at just over 29%, followed by food (nearly 17%), transportation (just under 17%), and household operations (around 13%), with the remainder split across recreation, clothing, health, and other categories. The index is compiled through a rigorous three-stage sampling process spanning geographic areas, retail outlets, and individual products, and it directly feeds two of the most consequential processes in the Canadian economy: the Bank of Canada's interest-rate decisions, and the statutory indexing of pensions and other public transfer payments.

A classical model beats three AI challengers

The study built four separate forecasting models on the same Canadian CPI and interest-rate data, training each on data through December 2023 and testing its forecasts against 2024–2025 data the model hadn't seen. On every error-based accuracy measure — Root Mean Square Error, Mean Absolute Error, and Mean Absolute Percentage Error — the SARIMAX model came out on top, while the LSTM, GRU, and ensemble models scored progressively worse. The one metric where the pattern flipped was R-squared, a measure of how much of the historical data's variation a model's fit explains: there, the more complex models scored marginally higher. The study reads that combination — weaker forecasts from models with stronger in-sample fit — as a textbook signature of overfitting: the machine-learning models were learning the noise in the training data rather than the CPI's genuine underlying structure, which then hurt them on data they hadn't seen before.

The gap is not small. SARIMAX's forecast error (RMSE of 0.91) is roughly a third of LSTM's (1.40), and less than half of GRU's or the weighted ensemble's — a meaningful difference given the CPI index itself moves across a range of roughly 90 to 170 points over the sample period.

Untangling a 30-year-old monetary-policy puzzle

Before building its final forecasting model, the study ran a simpler baseline regression of CPI on lagged interest rates, and got a result that has puzzled economists since the early 1990s: a positive, statistically significant relationship in which higher interest rates appear to precede higher, not lower, inflation. Economists call this the 'price puzzle,' and the usual explanation is not that monetary policy is genuinely counterproductive, but that a naive regression can't distinguish a central bank's own forecast-driven behaviour from the effect of its policy. If the Bank of Canada raises rates today because it expects inflation next quarter, a simple model will see 'rates up, then inflation up' and misread the sequence as causation running the wrong way.

The study's fix uses a Two-Stage Least Squares (2SLS) instrumental-variables approach: it first models the interest rate itself as a function of the forward-looking information a central bank would actually use to set it, producing a 'purged' interest-rate series stripped of that anticipatory component. Feeding this purged rate into the full model instead of the raw policy rate flips the relationship from perverse to expected, and the resulting coefficient implies the Bank of Canada raises interest rates by roughly 4.79 percentage points for every 1 percentage point rise in anticipated inflation — comfortably satisfying the 'Taylor Principle,' the standard benchmark (a feedback ratio above 1) economists use to judge whether a central bank's policy rule is stabilizing rather than destabilizing.

The EconoLens View

What this analysis is really doing is untangling two very different economic stories that look identical in a simple chart: interest rates went up, then prices went up. The first story is reverse causation — the Bank of Canada raises rates because it already expects inflation to rise, so rates and prices move together for reasons that have nothing to do with rates causing inflation. The second story is the real causal chain the paper isolates: once that anticipatory signal is stripped out, a rate increase causes a cooling of domestic demand through the output gap channel, which in turn causes both demand-driven and imported cost-push inflation to decay rather than compound. That decay — a long-run multiplier of 0.157 — is the paper's central cause-and-effect claim: a rate hike today causes measurably lower inflation several months later, not the other way around. The practical upshot is a split verdict: monetary policy is shown to be doing its job on the inflation rate, cooling it toward a stable near 2.6–2.7%, but it has no causal channel back to the price level itself — so the cost-of-living burden from past shocks remains, regardless of how well future rate hikes work.

What comes after 2027, on the model's own numbers

Running the winning SARIMAX model forward, the study projects Canada's CPI climbing from a January 2026 level of about 165.5 to roughly 169.5 by December 2026, then continuing upward to about 174.1 by December 2027. Translated into a year-over-year inflation rate rather than the raw index, that path is far less dramatic: the model expects headline inflation to average around 2.6% in 2026 and to settle near 2.7% in 2027 — essentially flat rather than accelerating. The study is explicit, however, that a stable inflation rate is not the same as a falling price level — on its own numbers, the absolute cost of the same basket of goods keeps climbing permanently higher, even once the rate of increase has stabilized.

Data and the forecasting tournament setup

The study uses monthly Canadian CPI data (Statistics Canada, base year 2002 = 100), the 3-month average interbank interest rate (sourced via the Federal Reserve Bank of St. Louis's FRED database), a Total Commodity Price Index (Bank of Canada's BCPI, base year 1972 = 100), and the Bank of Canada's own output-gap estimate, spanning January 1995 through December 2025. Before any modelling, the authors ran three standard diagnostic checks on the CPI series. A Shapiro-Wilk test rejected normality (p well below 0.01: the series does not follow a bell-curve distribution). A Ljung-Box test rejected the null hypothesis that the series is white noise (p effectively zero), confirming genuine, statistically significant autocorrelation rather than randomness. An Augmented Dickey-Fuller test for stationarity returned a p-value of 0.99 — far too high to reject non-stationarity — confirming the raw CPI series is non-stationary, meaning its statistical properties shift over time rather than staying constant, which most forecasting models assume they won't. To correct this, the authors applied both a first-order non-seasonal difference and a seasonal difference (12 months apart), after which the transformed series tested as stationary. A box plot of the transformed series found no significant outliers.

Four models were then trained on data through December 2023 and evaluated on 2024–2025 data held back for testing: the SARIMAX(1,1,1)(1,1,1)[12] specification described below, an LSTM recurrent neural network, a GRU network, and a weighted ensemble combining multiple automated (AutoML) models. All four were scored on the same four metrics — RMSE, MAE, MAPE, and R-squared.

The instrumental-variables fix for the price puzzle

Before settling on its final specification, the study first estimated a baseline model using the raw, un-instrumented interest rate as a regressor, which produced the classic price-puzzle result: a positive coefficient (β_baseline = 0.4205) implying an estimated price-stickiness duration of 2.38 months — the inverse of the coefficient — before a rate change would show up in prices, a result that, taken at face value, implies higher rates raise rather than lower inflation. Following the kind of approach used in the literature to address this sort of model misspecification, the authors instead constructed a forward-looking monetary-policy reaction function via Two-Stage Least Squares: in the first stage, they model the interest rate as a function of the information available to a forward-looking central bank; in the second stage, they use the fitted, 'purged' interest-rate series — stripped of that anticipatory, endogenous component — as the exogenous regressor inside the full SARIMAX model.

Reading the SARIMAX(1,1,1)(1,1,1)[12] parameter table

Four results stand out. First, the Output Gap coefficient (β = 0.38891, p<0.001) is large, positive, and highly significant — a textbook New Keynesian Phillips Curve result, pointing to domestic economic slack, not global cost pressure, as the dominant, statistically reliable driver of Canadian retail inflation in this specification. Second, the purged interest-rate coefficient (β_Purged_IR = 0.20857, p=0.0025) is positive and significant even after instrumenting — the study reads it through the policy-feedback relationship it implies rather than as a simple pass-through effect: inverting it (1 / 0.2086) gives approximately 4.79, a figure the study interprets as the Bank of Canada's policy-feedback ratio, comfortably above the 1.0 threshold the Taylor Principle requires of a stabilizing monetary-policy rule. Third, the commodity-price regressor (CommodityFisher, β=0.00353, p=0.2132) is statistically insignificant — read not as evidence that Canada is insulated from global commodity shocks, but as evidence that the Bank of Canada's aggressive policy-feedback loop squeezes domestic demand (via the output gap) hard enough to neutralize commodity pass-through before it reaches headline retail prices. Fourth, the seasonal moving-average term (ma.S.L12 = −0.80274, p<0.0001) is large, negative, and highly significant, pointing to a strong annual self-correcting mechanism in Canadian retail pricing — consistent with recurring seasonal patterns such as winter energy-price spikes or harvest-linked food-price swings reversing within about a year.

Because this long-run multiplier sits below 1, the study concludes that an initial interest-rate or cost-push shock does not compound indefinitely — it decays over time, which the authors read as evidence of a monetarily stable system in which the Bank of Canada's policy actions are ultimately self-correcting rather than destabilizing.

How the model was checked for overfitting

A model that fits its training data far better than its test data has memorized rather than learned — the classic overfitting signature the study says explains the machine-learning models' weaker forecast performance despite their higher R-squared. By that standard, the winning SARIMAX specification checks out: its accuracy metrics stay close between training and testing (RMSE moves from 0.944 to 0.595, R-squared from 0.997 to 0.924) rather than collapsing on unseen data the way an overfit model's would. A Ljung-Box test on the model's residuals returned a p-value of 0.9126 — far above the 0.05 threshold — failing to reject the null hypothesis that the residuals are white noise, meaning the model has captured essentially all of the systematic structure in the data, leaving only random, unpredictable noise behind. Visual inspection of the residual autocorrelation (ACF) and partial autocorrelation (PACF) functions, both falling within the 95% confidence bounds at every lag, supports the same conclusion.

Out-of-sample validation against real Statistics Canada data

As a further check, the study compared its own forecasts for the first seven months of 2026 against the actual, officially released Statistics Canada figures for those same months. The model's predictions (165.5, 166.4, 167.3, 167.8, 168.6, 168.9, and 169.3 for January through July) tracked closely against the real released values (165.0, 165.9, 167.4, 168.0, 169.6, 169.0, and 169.9), with every actual reading falling inside the model's 95% forecast interval. The single largest gap between prediction and actual came in May 2026, which the study attributes to a real-world shock outside the model's training data: an escalation in Middle East geopolitical tensions during that period drove up global crude-oil and gasoline prices, which fed through Canada's supply chain into higher transport and grocery costs — exactly the kind of unanticipated external event a model trained purely on historical patterns through 2023 has no way of foreseeing.

The 2026–2027 outlook, month by month

The projected path is a steady, parallel climb in both years rather than a sudden acceleration: the 2026 monthly index rises from 165.5 in January to 169.5 in December, and the 2027 path shifts the whole curve upward by roughly the same shape, from 170.0 in January to 174.1 in December. Both years share a consistent seasonal fingerprint: prices climb fastest from spring into midsummer, ease slightly through August and September, tick back up in October–November, and fall in a sharp seasonal trough each December.

Translating the index into an inflation rate — and a discrepancy worth flagging

Per the source study's own figure, the annual average implied by these monthly figures is 2.60% (SD 0.20) for 2026 and 2.69% (SD 0.02) for 2027. Its concluding narrative text, however, separately states rounder figures of roughly 2.5% for 2026 flattening to roughly 2.3% for 2027 — a discrepancy with its own table that this draft flags rather than silently resolves. The table figures above, being the more granular, directly computed values, are treated as the primary reference here; both years point to a flattening, near-target inflation rate either way, so the discrepancy affects the precise decimal reading of the forecast, not its overall conclusion.

Limitations

The forecast is a statistical projection built from historical trend, seasonal, and estimated policy-feedback patterns; it is not a structural macroeconomic model and cannot anticipate a genuine shock the way a model with explicit supply-and-demand channels might. The paper's own out-of-sample check illustrates this directly: its May 2026 forecast showed the largest deviation from the actual released figure, which the authors attribute to an unanticipated Middle East-linked spike in global crude and gasoline prices during that period — exactly the kind of exogenous event a purely time-series model, however well-fitted to the past, has no mechanism to see coming. The comparison is also specific to one country's inflation series over one historical window and one specific set of preprocessing choices (first-order non-seasonal and seasonal differencing, a training/testing split ending December 2023); a different window or preprocessing choice could plausibly favour a different model.

What EconoLens sees in this result

This is the third installment of a running tournament by overlapping members of this author team, each pitting classical time-series models against machine-learning and deep-learning challengers on a different country's consumer price index: a classical exponential-smoothing model (ETS) won on US CPI-U, a deep-learning LSTM network won on India's CPI-IW, and now a classical SARIMAX model wins on Canada's CPI. Read across all three, the pattern is not 'simple models always beat complex ones' — LSTM's win on India's series rules that reading out — but something narrower and, on the evidence so far, more durable: no single architecture dominates across countries, and each central bank's own inflation series has to be tested on its own terms rather than assumed to inherit whichever model won elsewhere. For an outlet covering economic forecasting across multiple countries, that is arguably the more transferable finding than any single model's win in any single country.