Hierarchical Forecast Reconciliation (MinT)

Summary

Collections of time series with aggregation constraints (product or geographic hierarchies, or crossed “grouped” structures) need coherent forecasts: aggregates must equal the sum of their components. Reconciliation forecasts every series at every level independently (the base forecasts $\hat{y}$), then linearly projects them onto the coherent subspace, $\tilde{y}=SP\hat{y}$. Wickramasuriya, Athanasopoulos & Hyndman show (i) the covariance needed by the earlier GLS approach of Hyndman et al. (2011) is not identifiable; (ii) the reconciled error covariance is $SP{W}_h{P}^{\top }{S}^{\top }$ where $W_h$ is the covariance of base forecast errors (Lemma 1); (iii) minimising its trace subject to unbiasedness gives the closed form $P=(S^{\top }{W}_h^{-1}S{)}^{-1}{S}^{\top }{W}_h^{-1}$ (Theorem 1), with a cheaper equivalent form; and (iv) the result is never worse than the base forecasts in $W_h^{-1}$-weighted loss. With a shrinkage estimate of $W_h$ — MinT(Shrink) — it dominates bottom-up, OLS and WLS in simulations and on 555 Australian tourism series.

Overview

A retailer forecasts total sales, sales by country, region, store, and also by product group; “the cross-product of these two hierarchies often results in a very large collection, comprising millions of individual time series” (Sec. 1). Three traditional strategies exist:

• Bottom-up: forecast only the most disaggregated series and add up. It “ignores relationships between series, and performs particularly poorly on highly disaggregated data which tend to have a low signal-to-noise ratio.”
• Top-down: forecast the total and split by proportions. Hyndman et al. (2011) showed any top-down method is biased even when base forecasts are unbiased.
• Independent forecasts at all levels: uses the best model per level, but the results do not add up.

Reconciliation keeps the third strategy and repairs its incoherence — and, because each level sees a different signal-to-noise trade-off, the repair is also a forecast combination that improves accuracy at all levels.

Main Content

Summing matrix, base and reconciled forecasts (Sec. 2.1) ^def-summing-matrix

Let $y_t\in {\mathbb{R}}^m$ stack all series and $b_t\in {\mathbb{R}}^n$ the bottom-level series. Then

$$y_t=S{b}_t,$$

with $S\in {\mathbb{R}}^{m\times n}$ the summing matrix, one row per series. For a two-level tree (Total → A, B; A → AA, AB, AC; B → BA, BB), $m=8$, $n=5$ and

$$S=\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 1& 1& 0& 0\\ 0& 0& 0& 1& 1\\ & & I_5& & \end{array}\right].$$

Let ${\hat{y}}_T(h)$ be $h$-step base forecasts from any method. Every linear reconciliation method has the form

$${\tilde{y}}_T(h)=SP{\hat{y}}_T(h),P\in {\mathbb{R}}^{n\times m}:$$

$P$ maps base forecasts to bottom-level forecasts, and $S$ sums them up. Bottom-up is $P=[0_{n\times (m-n)}{I}_n]$; top-down is $P=[p{0}_{n\times (m-1)}]$ for proportions $p$.

Unbiasedness condition ^thm-unbiased

If the base forecasts are unbiased, $\mathbb{E}[{\hat{y}}_T(h)\mid {\mathcal{I}}_T]=S{\beta }_T(h)$, the reconciled forecasts are unbiased iff $SPS=S$, equivalently $PS=I_n$ (Sec. 2.1). Top-down violates this.

Why the 2011 GLS approach fails (Sec. 2.2). Hyndman et al. modelled ${\hat{y}}_T(h)=S{\beta }_T(h)+{\epsilon }_h$ with coherency-error covariance ${\Sigma }_h$, giving $P=(S^{\top }{\Sigma }_h^{†}S{)}^{-1}{S}^{\top }{\Sigma }_h^{†}$. But the residuals are $(I_m-SP)\hat{y}$, and $I_m-SP$ is idempotent with rank $m-n<m$, so ${\Sigma }_h$ cannot be identified. They fell back to OLS, $P=(S^{\top }S{)}^{-1}{S}^{\top }$.

Lemma 1 — covariance of reconciled forecast errors ^thm-mint-lemma1

For any $P$ with $SPS=S$,

$$\mathrm{Var}[y_{t+h}-{\tilde{y}}_t(h)\mid {\mathcal{I}}_t]=SP{W}_h{P}^{\top }{S}^{\top },W_h=\mathbb{E}[{\hat{e}}_t(h){\hat{e}}_t(h{)}^{\top }\mid {\mathcal{I}}_t],$$

where ${\hat{e}}_t(h)=y_{t+h}-{\hat{y}}_t(h)$ are the base forecast errors. Proof sketch (App. A.1): $\tilde{e}=SP\hat{e}+(I-SP)y_{t+h}$ and $(I-SP)S=0$. Under Gaussian errors this yields prediction intervals for any unbiased reconciliation method.

Theorem 1 — MinT reconciliation ^thm-mint

Let $W_h$ be positive definite. The $P$ that minimises $\mathrm{tr}[SP{W}_h{P}^{\top }{S}^{\top }]$ subject to $SPS=S$ is

$$P=(S^{\top }{W}_h^{-1}S{)}^{-1}{S}^{\top }{W}_h^{-1},$$

or equivalently

$$P=J-J{W}_{h}U(U^{\top }{W}_{h}U{)}^{-1}{U}^{\top },$$

where $S^{\top }=[C^{\top }{I}_n]$, $J=[0_{n\times m^{\ast }}{I}_n]$, $U^{\top }=[I_{m^{\ast }}-C]$ and $m^{\ast }=m-n$ is the number of aggregate series. The second form inverts a single $m^{\ast }\times m^{\ast }$ matrix rather than an $n\times n$ and an $m\times m$ one, which is what makes MinT scale.

The reconciled forecasts are therefore the GLS projection of $\hat{y}$ onto the coherent subspace $\{\check{y}:U^{\top }\check{y}=0\}$ in the metric $W_h^{-1}$:

$${\tilde{y}}_T(h)=\arg \underset{\check{y}:U^{\top }\check{y}=0}{\min }[{\hat{y}}_T(h)-\check{y}{]}^{\top }{W}_h^{-1}[{\hat{y}}_T(h)-\check{y}]=\hat{y}-W_{h}U(U^{\top }{W}_{h}U{)}^{-1}{U}^{\top }\hat{y}.$$

Reconciliation never hurts (Sec. 2.3) ^thm-mint-pythagoras

By the generalised Pythagorean inequality for this projection, for every coherent realisation $y_{T+h}$,

$$[y_{T+h}-\hat{y}{]}^{\top }{W}_h^{-1}[y_{T+h}-\hat{y}]\ge [y_{T+h}-\tilde{y}{]}^{\top }{W}_h^{-1}[y_{T+h}-\tilde{y}].$$

MinT forecasts are “at least as good as the incoherent base forecasts.” The authors conclude that “applying forecast reconciliation should always be preferred, and approaches that use limited information such as bottom-up or top-down should be avoided” (Sec. 5).

Under the “error-additivity” assumption $\hat{e}=S\hat{a}$ (so $W_h=S{V}_h{S}^{\top }$ is singular), every unbiased $P$ attains the same variance; the Moore–Penrose inverse in the first form gives OLS and in the second gives bottom-up — explaining why both earlier methods appeared “optimal.”

Estimating $W_h$ (Sec. 2.4)

Label	$W_h\propto$	Assumption / use
OLS	$I$	Uncorrelated, equal-variance errors at all levels — impossible in a hierarchy
WLS${}_v$	$\mathrm{diag}({\hat{W}}_1)$	Variance scaling by in-sample one-step residual variances
WLS${}_s$	$\Lambda =\mathrm{diag}(S1)$	Structural scaling: variance ∝ number of bottom series aggregated; needs no residuals (e.g. judgmental forecasts)
MinT(Sample)	${\hat{W}}_1=\frac{1}{T}{\sum }_t{\hat{e}}_t(1){\hat{e}}_t(1{)}^{\top }$	Full sample covariance; poor or singular when $m\gtrsim T$
MinT(Shrink)	${\lambda }_D{\hat{W}}_{1,D}+(1-{\lambda }_D){\hat{W}}_1$	Schäfer–Strimmer shrinkage of off-diagonals toward zero, ${\hat{\lambda }}_D={\sum }_{i\ne j}\widehat{\mathrm{Var}}({\hat{r}}_{ij})/{\sum }_{i\ne j}{\hat{r}}_{ij}^2$

All use one-step in-sample residuals and assume $W_h=k_h{W}_1$; the constant $k_h$ cancels in point forecasts but matters for intervals (left to later work).

Evidence

• Simulations (Sec. 3). With two bottom series and error correlation $-0.8$, MinT(Shrink) cuts top-level one-step RMSE by ≈30% versus base. Across designs (correlated ARIMA hierarchies forecast with deliberately misspecified ETS; seasonal series; a 2,047-series five-level hierarchy) “MinT(Shrink) consistently shows the largest improvements”; gains are largest under model misspecification, and bottom-up often fails to improve on base forecasts.
• Australian domestic tourism (Sec. 4, Table 8). 555 monthly series (a geographic hierarchy of 7 states → 27 zones → 76 regions, crossed with 4 purposes of travel), 1998-2016. Rolling-window evaluation: 96-month training window, 1-12-step forecasts, rolled forward one month at a time (132 one-step … 121 twelve-step forecasts per series). With ARIMA base forecasts, average RMSE over $h=1$-$12$ changes relative to base by:

Level	BU	OLS	WLS${}_v$	MinT(Shrink)
Australia (total)	+22.2%	−1.3%	+3.1%	+0.5%
States	+7.0%	−4.1%	−4.6%	−6.4%
Zones	+1.8%	−3.5%	−5.6%	−7.0%
Regions	+0.5%	−2.1%	−4.7%	−5.6%
Australia by purpose	+7.2%	−4.2%	−8.0%	−12.4%

Bottom-up is worst because fewer than 50% of bottom-level models even detect seasonality; reconciliation “bring[s] informative signals from the higher levels of aggregation to the lower levels and vice versa.” It also “implicitly models spatial autocorrelations.”

Examples

Three-series hierarchy, by hand. Total $=A+B$, so $S=\left[\begin{array}{cc}1& 1\\ 1& 0\\ 0& 1\end{array}\right]$, $U^{\top }=[1,-1,-1]$. Base forecasts $\hat{y}=(100,60,50)$ are incoherent: $U^{\top }\hat{y}=-10$.

• OLS ($W=I$): $\tilde{y}=\hat{y}+\frac{10}{3}(1,-1,-1)=(103.3,56.7,46.7)$ — the discrepancy is split equally.
• WLS with $W=\mathrm{diag}(4,1,5)$: $WU=(4,-1,-5)$, $U^{\top }WU=10$, so $\tilde{y}=\hat{y}+(4,-1,-5)=(104,59,45)$ — the noisiest forecast ($B$) absorbs most of the adjustment, the most precise ($A$) barely moves. By Lemma 1 the total error variance falls from $\mathrm{tr}W=10$ to $\mathrm{tr}[SPW{P}^{\top }{S}^{\top }]=5.8$.
• Full MinT with positive covariance between Total and $B$ errors, $W=\left[\begin{array}{ccc}4& 1& 2\\ 1& 1& 0\\ 2& 0& 5\end{array}\right]$: $\tilde{y}=(102.5,60,42.5)$; trace $7.5$ vs $10$.

import numpy as np
def mint(S, yhat, W):
    Wi = np.linalg.inv(W)
    P = np.linalg.solve(S.T @ Wi @ S, S.T @ Wi)   # (S'W^-1 S)^-1 S'W^-1
    return S @ P @ yhat, S @ P @ W @ P.T @ S.T     # reconciled mean, error covariance
def shrink_cov(E):                                 # E: (T, m) one-step residuals
    W = E.T @ E / len(E); D = np.diag(np.diag(W))
    Z = E / E.std(0); R = Z.T @ Z / len(E)
    varR = ((Z[:, :, None] * Z[:, None, :] - R) ** 2).sum(0) / (len(E) * (len(E) - 1))
    off = ~np.eye(len(W), dtype=bool)
    lam = np.clip(varR[off].sum() / (R[off] ** 2).sum(), 0, 1)
    return lam * D + (1 - lam) * W

Marketing use. Geo-level sales forecasts (DMA → region → national) used for budgeting or as counterfactual baselines should agree with the national forecast finance already uses; MinT delivers that agreement while improving the noisy geo-level forecasts with national-level trend and seasonal signal.

Connections

• Probabilistic Forecasting - Overview — coherence as the structural requirement of forecasting at scale.
• Local vs Global Forecasting Models — reconciliation is a post-hoc way to share information across series, complementary to sharing parameters.
• Forecast Evaluation and Backtesting — the rolling-window design of Sec. 4 is the template for evaluating reconciliation.
• Hierarchical Models — a different sense of “hierarchical”: partial pooling of parameters through a prior, vs MinT’s pooling of forecasts through linear aggregation constraints. Both shrink noisy low-level estimates toward better-determined aggregates.
• Empirical Bayes Interpretation of Shrinkage — the shrinkage covariance estimator that makes MinT feasible when $m>T$.
• Multivariate Persistence and Cointegration — modelling cross-series dynamics directly (VAR); MinT instead exploits cross-series error correlation after univariate forecasting.

See Also

• Proper Scoring Rules (CRPS, Log Score, Pinball Loss) — the energy score evaluates joint coherent probabilistic forecasts; Lemma 1 supplies the Gaussian covariance.
• Single Marketing Time Series — the ARIMA/ETS base forecasters.
• DeepAR and Global Autoregressive Neural Forecasters — the DeepAR paper cites Hyndman et al. (2011) as the way to leverage hierarchical structure; base forecasts from any global model can be reconciled.
• Geo-Experiment Methodology - Overview — geo panels are natural hierarchies.