Non-Negative Forecast Reconciliation: Optimal Methods and Operational Solutions

Abstract

In many different applications such as retail, energy, and tourism, forecasts for a set of related time series must satisfy both linear and non-negativity constraints, as negative values are meaningless in practice. Standard regression-based reconciliation approaches achieve coherence with linear constraints, but may generate negative forecasts, reducing interpretability and usability. This paper develops and evaluates three alternative strategies for non-negative forecast reconciliation. First, reconciliation is formulated as a non-negative least squares problem and solved with the operator splitting quadratic program, allowing flexible inclusion of additional constraints. Second, we propose an iterative non-negative reconciliation with immutable forecasts, offering a practical optimization-based alternative. Third, we investigate a family of set-negative-to-zero heuristics that achieve efficiency and interpretability at minimal computational cost. Using the Australian Tourism Demand dataset, we compare these approaches in terms of forecast accuracy and computation time. The results show that non-negativity constraints consistently improve accuracy compared to base forecasts. Overall, set-negative-to-zero achieve near-optimal performance with negligible computation time, the block principal pivoting algorithm provides a good accuracy–efficiency compromise, and the operator splitting quadratic program offers flexibility for incorporating additional constraints in large-scale applications.

1. Introduction

Forecasting plays a central role in supporting decision-making across business, policy, healthcare and scientific domains [1]. In many applications, forecasts are required not only for individual time series but also for collections of related series that are connected through linear relationships. For example, forecasts of product demand must sum consistently across categories, forecasts of regional electricity consumption must aggregate to the national total, and forecasts of monthly tourism flows should align with quarterly or annual aggregates. In these contexts, base forecasts are typically produced, either independently for each series or through multivariate models, that focus only on improving forecast accuracy, without considering the underlying linear constraints among the series [2]. As a consequence, the base forecasts are generally incoherent, meaning that they do not satisfy the required linear constraints that link the different levels or components [3]. For this reason, a post-forecasting process—called forecast reconciliation [4]—aimed to adjust a set of incoherent “base” forecasts of a multivariate time series so that they satisfy the constraints, should be applied.

In hierarchical and grouped time series, two well-known and commonly used approaches [5] are bottom-up and top-down. In the bottom-up approach, forecasts are first generated for the most disaggregated series and then aggregated to produce predictions for higher-level series [6,7]. The top-down approach begins with forecasts at the fully aggregated level, which are subsequently allocated to the lower-level series using predetermined proportions [8,9,10]. It is worth noting that traditional reconciliation strategies fail to exploit all available information. Hyndman et al. [3] introduced an optimal reconciliation approach, using a regression model to optimally combine the base forecasts for all series so that the resulting forecasts are coherent.

Since its introduction, the linear regression reconciliation framework has been applied and developed in different contexts. Methodological contributions include the cross-sectional framework, where forecasts are linked through contemporaneous structures [11,12,13,14,15], the temporal framework, where the same variable is available for different frequencies [16,17], and the cross-temporal framework, which integrates both dimensions within a unified setting [18,19,20,21,22]. In addition, forecast reconciliation is applied in various disciplines: retail [23,24,25], energy [24,26,27,28,29,30,31], tourism [10,21,32,33] and economics [19,34], among others. Athanasopoulos et al. [4] provide a comprehensive review that synthesizes and organizes the expanding literature on forecast reconciliation.

Many forecasting applications involve variables that can only take non-negative values, such as sales revenues, product demand, or counts of individuals. In these contexts, it is essential that the reconciliation process preserves non-negativity, since negative forecasts lack practical meaning and may lead to wrong managerial decisions. To address this issue, Wickramasuriya et al. [13] reformulated reconciliation as a non-negative least squares (NNLS) problem [35], including non-negativity directly within the optimal reconciliation framework using the Block Principal Pivoting algorithm [36] and establishing formal properties, such as conditions for existence, uniqueness, and optimality of NNLS reconciliation. Although theoretically rigorous, this approach can be computationally demanding in large-scale applications [4,21].

Building on this theoretical foundation, we focus on a more applied perspective. Our objective is to examine and compare alternative procedures, both optimization-based and heuristic, for enforcing non-negativity in reconciled forecasts, with particular attention to computational feasibility and suitability for different frameworks (cross-sectional, temporal and cross-temporal). In addition, we focus on assessing the empirical performance of different approaches, highlighting the trade-offs between forecast accuracy, robustness, and computational efficiency.

Beyond optimization-based reconciliation, alternative heuristic procedures have also been proposed, designed to achieve non-negativity with reduced computational burden. Kourentzes and Athanasopoulos [37] proposed an iterative correction algorithm to eliminate negative reconciled forecasts in the context of intermittent demand series. Another heuristic, proposed by Di Fonzo and Girolimetto [20], is the set-negative-to-zero procedure, which replaces negative reconciled forecasts with zeros while preserving coherence through bottom-up. This strategy provides a useful baseline against which more elaborate procedures can be compared.

In this paper, we contribute to the development of non-negative forecast reconciliation methods by proposing and evaluating three distinct approaches. First, we formulate the reconciliation problem as a NNLS program, and solve it using the operator splitting quadratic program introduced by Stellato et al. [38]. This approach is flexible and allows the incorporation of additional types of constraints, such as immutable forecasts [39]. Second, we propose an iterative non-negative reconciliation procedure with immutable forecasts, offering an alternative optimization-based strategy inspired by the procedure of Kourentzes and Athanasopoulos [37]. Third, we investigate a family of heuristics, known as set-negative-to-zero, originally introduced by Di Fonzo and Girolimetto [20], which provide accurate reconciled forecasts at minimal computational cost. Finally, we evaluate these approaches in a large-scale empirical study using the Australian Tourism Demand dataset [12,13]. Our analysis highlights both forecasting accuracy and computational efficiency to achieve a balance between theoretical property and practical feasibility. In doing so, we provide methodological insights for researchers and practitioners facing the dual requirements of coherence and non-negativity in forecasting. All procedures examined in this study have been implemented in the R package FoReco [40].

The remainder of the paper is structured as follows. Section 2.1 reviews the general framework for forecast reconciliation. Section 3 introduces and develops the non-negative reconciliation procedures. Section 4 presents the empirical application and reports the results, focusing on accuracy and computational performance across methods. Finally, Section 5 provides the conclusion. The code and data for reproducing the results are available at https://github.com/danigiro/vn525nn, accessed on 23 October 2025.

2. Forecast Reconciliation

2.1. Zero-Constrained and Structural Representation

Let ${\mathit{y}}_t={[y_{1,t},\dots ,y_{i,t},\dots ,y_{n,t}]}^{\prime }$ be an n-variate linearly constrained time series observed at the most temporally disaggregated level, with a seasonality of period m (e.g., $m=12$ for monthly data, $m=4$ for quarterly data). The cross-sectional constraints [12,19] can be expressed as

$${\mathit{C}}_{cs}{\mathit{y}}_t={\mathbf{0}}_{(n_a\times 1)},t=1,\dots ,T$$

(1)

where ${\mathit{C}}_{cs}$ is an $(n_a\times n)$ zero-constraints cross-sectional matrix. Following Girolimetto and Di Fonzo [41], ${\mathit{y}}_t$ can be organized into two sets of variables such that ${\mathit{y}}_t={\left[{\mathit{u}}_t^{\prime }{\mathit{b}}_t^{\prime }\right]}^{\prime }$, where the $n_a$ “constrained” series ${\mathit{u}}_t$ are linked to the $n_b$ “free” series ${\mathit{b}}_t$ through the cross-sectional linear combination matrix ${\mathit{A}}_{cs}$ via ${\mathit{u}}_t={\mathit{A}}_{cs}{\mathit{b}}_t$. In general, Girolimetto and Di Fonzo [41] show that ${\mathit{C}}_{cs}=\left[{\mathit{I}}_{n_a}-{\mathit{A}}_{cs}\right]$. A different, equivalent way of expressing the constraints (1) is the structural representation

$${\mathit{y}}_t={\mathit{S}}_{cs}{\mathit{b}}_t,$$

where the cross-sectional structural matrix

$${\mathit{S}}_{cs}=\left[\begin{array}{c}{\mathit{A}}_{cs}\\ {\mathit{I}}_{n_b}\end{array}\right],$$

characterizes the entire system by representing ${\mathit{y}}_t$ as a linear combination of the free components ${\mathit{b}}_t$.

When we are dealing with a genuine hierachical/grouped time series [3], this two sets are naturally formed by the upper- and the bottom-level time series, respectively. For example, in a simple two-level hierarchy where $y_{T,t}=y_{X,t}+y_{Y,t}$, we have ${\mathit{A}}_{cs}=\left[\begin{array}{cc}1& 1\end{array}\right]$, ${\mathit{S}}_{cs}=\left[\begin{array}{cc}1& 1\\ 1& 0\\ 0& 1\end{array}\right]$, and ${\mathit{C}}_{cs}=\left[\begin{array}{ccc}1& -1& -1\end{array}\right]$.

Looking to the temporal framework [16], let $K=\{k_p,k_{p-1},\dots ,k_2,k_1\}$ be the set of p factors of m, in descending order, with $k_1=1$ and $k_p=m$ (e.g., for quarterly series, $m=4$, $p=3$, and $K=\{4,2,1\}$). Then, $x_{i,j}^{\left[k\right]}={\sum }_{t=(j-1)k+1}^{jk}{y}_{i,t}$ is the temporally aggregated value of the series $i=1,...,n$ for a factor k at time $j=1,...,{\displaystyle \frac{T}{k}}$. For a fixed temporal aggregation order $k\in K$, we stack the observations into a column vector ${\mathit{x}}_{i,\tau }^{\left[k\right]}$ and the complete vector for all temporal aggregation orders for a single series i is ${\mathit{x}}_{i,\tau }={\left[\begin{array}{cccc}{\mathit{x}}_{i,\tau }^{\left[k_p\right]\prime }& {\mathit{x}}_{i,\tau }^{\left[k_{p-1}\right]\prime }& \dots & {\mathit{x}}_{i,\tau }^{\left[1\right]\prime }\end{array}\right]}^{\prime }$. Given

$${\mathit{A}}_{te}=\left[\begin{array}{c}{\mathbf{1}}_{k_p}\\ {\mathit{I}}_{{\displaystyle \frac{m}{k_{p-1}}}}\otimes {\mathbf{1}}_{k_{p-1}}\\ ⋮\\ {\mathit{I}}_{{\displaystyle \frac{m}{k_2}}}\otimes {\mathbf{1}}_{k_2}\end{array}\right],$$

we can construct the temporal structural matrix ${\mathit{S}}_{te}=\left[\begin{array}{c}{\mathit{A}}_{te}\\ {\mathit{I}}_m\end{array}\right]$, which relates ${\mathit{x}}_{i,\tau }$ to the most disaggregated series ${\mathit{x}}_{i,\tau }^{\left[1\right]}$ through ${\mathit{x}}_{i,\tau }={\mathit{S}}_{te}{\mathit{x}}_{i,\tau }^{\left[1\right]}$, and the zero-constraints temporal matrix is ${\mathit{C}}_{te}=\left[I_{k^{*}}-{\mathit{A}}_{te}\right]$, such that ${\mathit{C}}_{te}{\mathit{x}}_{i,\tau }={\mathbf{0}}_{[k^{*}\times (m+k^{*})]}$, where $k^{*}={\sum }_{k\in \mathcal{K}\setminus \left\{1\right\}}{\displaystyle \frac{m}{k}}$ is the number of upper time series in the temporal hierarchy [21].

To unify both frameworks, the series are stacked into an $[n\times (m+k^{*})]$ matrix ${\mathit{X}}_{\tau }=\left[\begin{array}{c}{\mathit{x}}_{1,\tau }\\ ⋮\\ {\mathit{x}}_{n,\tau }\end{array}\right]$ for $\tau =1,\dots ,N$ where the rows represent cross-sectional, and columns temporal, dimensions. The cross-temporal zero-constrained representation for the complete set of observations ${\mathit{x}}_{\tau }=\mathrm{vec}\left({\mathit{X}}_{\tau }^{\prime }\right)$ is given by ${\mathit{C}}_{ct}{\mathit{x}}_{\tau }={\mathbf{0}}_{[(n_{a}m+n{k}^{*})\times 1]}$, where ${\mathit{C}}_{ct}=\left[\begin{array}{c}{\mathit{C}}^{*}\\ {\mathit{I}}_n\otimes {\mathit{C}}_{te}\end{array}\right]$ is the full rank zero-constraints cross-temporal matrix with ${\mathit{C}}^{*}=\left[\begin{array}{cc}{\mathbf{0}}_{(n_{a}m\times n{k}^{*})}& {\mathit{I}}_m\otimes {\mathit{C}}_{cs}\end{array}\right]{\mathit{P}}^{\prime }$ [19], $\mathit{P}$ is the commutation matrix [42] such that $\mathit{P}\mathrm{vec}\left({\mathit{X}}_{\tau }\right)=\mathrm{vec}\left({\mathit{X}}_{\tau }^{\prime }\right)$, and the operator $\mathrm{vec}(·)$ converts a matrix into a vector. Alternatively, the structural representation is ${\mathit{x}}_{\tau }={\mathit{S}}_{ct}{\mathit{b}}_{\tau }^{\left[1\right]}$, where ${\mathit{S}}_{ct}={\mathit{S}}_{cs}\otimes {\mathit{S}}_{te}$ is the cross-temporal summation matrix, and ${\mathit{b}}_{\tau }^{\left[1\right]}=\left[\begin{array}{c}{\mathit{x}}_{1,\tau }^{\left[1\right]}\\ ⋮\\ {\mathit{x}}_{n,\tau }^{\left[1\right]}\end{array}\right]$ contains the most disaggregated ($k_1=1$) constrained time series.

For the most aggregated forecast horizon $h=1,\dots ,H$, let ${\widehat{\mathit{X}}}_h$ be the h-step-ahead base forecasts matrix where the rows represent cross-sectional, and columns temporal, dimensions. These base forecasts are generally incoherent, meaning ${\mathit{C}}_{ct}{\widehat{\mathit{x}}}_h\ne \mathbf{0}$ with ${\widehat{\mathit{x}}}_h=\mathrm{vec}\left({\hat{\mathit{X}}}_h^{\prime }\right)$. Forecast reconciliation adjusts these base forecasts to obtain reconciled forecasts ${\tilde{\mathit{x}}}_h$, which satisfy the linear constraints ${\mathit{C}}_{ct}{\tilde{\mathit{x}}}_h=\mathbf{0}$. To summarise, ${\mathit{X}}_h$ represents the matrix of true coherent values, ${\widehat{\mathit{X}}}_h$ are the matrix of incoherent base forecasts, and ${\tilde{\mathit{X}}}_h$ are the matrix of reconciled forecasts.

2.2. Regression-Based Reconciliation

In this section, we address regression-based reconciliation in a general setting, without reference to a specific cross-sectional, temporal, or cross-temporal framework. In addition,

Table 1. Summary of the main symbols used in Section 2.2, including their descriptions and equivalent notation in the cross-sectional (cs), temporal (te), and cross-temporal (ct) forecasting frameworks. Dimensions and indexing conventions are provided to clarify how the general notation maps to each specific framework. Note that starting from matrices $\mathit{X}$, $\widehat{\mathit{X}}$ and $\tilde{\mathit{X}}$ described in Section 2.1, the cross-sectional framework corresponds to working on a single column j (${\mathit{X}}_{\{·,j\}}$, ${\widehat{\mathit{X}}}_{\{·,j\}}$, ${\tilde{\mathit{X}}}_{\{·,j\}}$), the temporal framework on a single row i (${\mathit{X}}_{\{i,·\}}$, ${\widehat{\mathit{X}}}_{\{i,·\}}$, ${\tilde{\mathit{X}}}_{\{i,·\}}$) and the cross-temporal framework on the vectorised form ($\mathrm{vec}\left(\mathit{X}\right)$, $\mathrm{vec}(\hat{\mathit{X}})$, $\mathrm{vec}(\tilde{\mathit{X}})$). In addition, we omit the subscript h to simplify notation.

		Framework
Symbol	Description	cs	te	ct
$\mathit{x}$	$(n^{*}\times 1)$ vector of true (unknown) coherent vector	${\mathit{X}}_{\{·,j\}}$	${\mathit{X}}_{\{i,·\}}$	$\mathrm{vec}\left(\mathit{X}\right)$
$\widehat{\mathit{x}}$	$(n^{*}\times 1)$ vector of incoherent base forecasts	${\widehat{\mathit{X}}}_{\{·,j\}}$	${\widehat{\mathit{X}}}_{\{i,·\}}$	$\mathrm{vec}\left(\widehat{\mathit{X}}\right)$
$\tilde{\mathit{x}}$	$(n^{*}\times 1)$ vector of reconciled forecasts	${\tilde{\mathit{X}}}_{\{·,j\}}$	${\tilde{\mathit{X}}}_{\{i,·\}}$	$\mathrm{vec}\left(\tilde{\mathit{X}}\right)$
$\mathit{C}$	$(n_a^{*}\times n^{*})$ zero-constraints matrix such that $\mathit{C}\mathit{x}=\mathbf{0}$	${\mathit{C}}_{cs}$	${\mathit{C}}_{te}$	${\mathit{C}}_{ct}$
$\mathit{S}$	$(n^{*}\times n_b^{*})$ structural matrix	${\mathit{S}}_{cs}$	${\mathit{S}}_{te}$	${\mathit{S}}_{ct}$
$n^{*}$	Total number of elements in the vectors	n	$m+k^{*}$	$n(m+k^{*})$
$n_a^{*}$, $n_b^{*}$	Numbers of row and colum in the zero-constraints matrix $\mathit{C}$ and the structural matrix $\mathit{S}$, respectively	$n_a$, $n_b$	$k^{*}$, m	$nm+n_b{k}^{*}$, $n_{b}m$
$\mathit{\beta }$	$(n_b^{*}\times 1)$ vector of target forecasts corrisponding to the free components (cross-sectional)/high-frequency univariate series (temporal)/high-frequency free series (cross-temporal)

summarizes the main symbols used throughout this section, along with their descriptions and the corresponding notation in the different forecasting frameworks: cross-sectional (cs), temporal (te), and cross-temporal (ct). The table provides a unified reference for the vectors of true values, base forecasts, reconciled forecasts, as well as the structural and constraint matrices. Dimensions and indexing conventions are also specified to clarify how the general notation maps to each specific framework.

Starting from a zero-constrained representation, the regression-based approach assumes that the base forecasts $\widehat{y}$ are related to the true (but unobserved) coherent forecasts y by a linear model [12,19,43]

$$\widehat{\mathit{x}}=\mathit{x}+\mathit{\epsilon },$$

(2)

where $\widehat{\mathit{x}}$ is an $(n^{*}\times 1)$ vector of base forecasts, $\mathit{x}$ is an $(n^{*}\times 1)$ vector of target (true) forecasts, and $\mathit{\epsilon }$ is an $(n^{*}\times 1)$ vector of zero-mean errors with a known positive definite covariance matrix $\Omega =E\left[\mathit{\epsilon }{\mathit{\epsilon }}^{\prime }\right]$. The target forecasts $\mathit{x}$ must satisfy a system of linearly independent constraints:

$$\mathit{C}\mathit{x}={\mathbf{0}}_{(n_a^{*}\times 1)}.$$

(3)

The reconciliation process involves finding reconciled forecasts $\tilde{\mathit{x}}$ that are “as close as possible” to the base forecasts $\widehat{\mathit{x}}$ according to a pre-specified metric, while simultaneously satisfying the constraints. This is achieved by minimizing a linearly constrained generalized least squares (GLS) objective function:

$$\tilde{\mathit{x}}=\underset{\mathit{x}}{arg\; min}{\left(\mathit{x}-\widehat{\mathit{x}}\right)}^{\prime }{\Omega }^{-1}\left(\mathit{x}-\widehat{\mathit{x}}\right)\mathrm{s}.\mathrm{t}.\mathit{C}\mathit{x}={\mathbf{0}}_{(n_a^{*}\times 1)}.$$

(4)

The closed-form solution [43,44] for the reconciled forecasts $\tilde{\mathit{x}}$ is given by

$$\tilde{\mathit{x}}=\widehat{\mathit{x}}-\Omega {\mathit{C}}^{\prime }{\left(\mathit{C}\Omega {\mathit{C}}^{\prime }\right)}^{-1}\mathit{C}\widehat{\mathit{x}}=\mathit{M}\widehat{\mathit{x}},$$

(5)

where the reconciliation matrix $\mathit{M}={\mathit{I}}_{n^{*}}-\Omega {\mathit{C}}^{\prime }{\left(\mathit{C}\Omega {\mathit{C}}^{\prime }\right)}^{-1}\mathit{C}$ is a projection matrix. This formula essentially adjusts the base forecasts $\widehat{\mathit{x}}$ by a linear combination of their coherency errors, $\mathit{C}\widehat{\mathit{x}}$.

Equivalently, using the structural representation of a linear constrained time series [3,10,41], the reconciled forecasts can be derived through the linear model

$$\widehat{\mathit{x}}=\mathit{S}\mathit{\beta }+\mathit{\epsilon },$$

(6)

where $\widehat{\mathit{x}}$ is an $(n^{*}\times 1)$ vector of base forecasts, $\mathit{\beta }$ is an $(n_b^{*}\times 1)$ vector of free components’ target forecasts, and $\mathit{\epsilon }$ is an $(n^{*}\times 1)$ vector of zero-mean errors with a known positive definite covariance matrix $\Omega =E\left[\mathit{\epsilon }{\mathit{\epsilon }}^{\prime }\right]$. Minimizing the GLS objective function

$${\left(\widehat{\mathit{x}}-\mathit{S}\mathit{\beta }\right)}^{\prime }{\Omega }^{-1}\left(\widehat{\mathit{x}}-\mathit{S}\mathit{\beta }\right)$$

(7)

results in $\tilde{\mathit{\beta }}={\left({\mathit{S}}^{\prime }{\Omega }^{-1}\mathit{S}\right)}^{-1}{\mathit{S}}^{\prime }{\Omega }^{-1}\widehat{\mathit{x}}$, from which the whole reconciled vector can be computed as [3,19]

$$\tilde{\mathit{x}}=\mathit{S}\tilde{\mathit{\beta }}=\mathit{S}{\left({\mathit{S}}^{\prime }{\Omega }^{-1}\mathit{S}\right)}^{-1}{\mathit{S}}^{\prime }{\Omega }^{-1}\widehat{\mathit{x}}=\mathit{S}\tilde{\mathit{G}}\widehat{\mathit{x}},$$

(8)

where $\tilde{\mathit{G}}={\left({\mathit{S}}^{\prime }{\Omega }^{-1}\mathit{S}\right)}^{-1}{\mathit{S}}^{\prime }{\Omega }^{-1}$ and $\mathit{M}=\mathit{S}\tilde{\mathit{G}}$. If the base forecasts $\widehat{\mathit{x}}$ are unbiased, the reconciled forecasts $\tilde{\mathit{x}}$ are also unbiased and achieve minimum variance, since the weight matrix $\tilde{\mathit{G}}$ minimizes the trace of the reconciled forecasts’ covariance matrix (MinT, [12]):

$$\tilde{\mathit{G}}=\underset{\mathit{G}}{arg\; min}\mathrm{tr}\left(\mathit{S}\mathit{G}\Omega {\mathit{G}}^{\prime }{\mathit{S}}^{\prime }\right)\mathrm{s}.\mathrm{t}.\mathit{G}\mathit{S}={\mathit{I}}_{n_b^{*}},$$

(9)

where $\mathrm{tr}(·)$ denotes the trace of a square matrix.

In practice, the covariance matrix $\Omega$ is generally unknown and its accurate estimation is crucial for effective reconciliation. For the cross-sectional case, several approximations are commonly used. Simple choices include the identity matrix [3], assuming uncorrelated errors with equal variance, and the variance scaling [11], which accounts for heterogeneous error variances but still overlooks correlations. Moreover a shrinkage covariance [12] combining a structured target with the sample covariance can capture the full error dependence and obtain a robust, non-singular estimator. For the temporal case,

For the temporal case, modelling error dependencies across forecast horizons is still an important issue [21]. Simple approaches assume independence or constant variance over time [16], while others methods estimate autocovariances from forecast residuals and impose some struture for different forecast horizons [17,45].

For the cross-temporal case, both cross-sectional and temporal dependencies must be captured at once, which greatly increases dimensionality. As the number of series and forecast horizons grows, the resulting covariance matrix can become extremely large, making direct estimation from limited data unstable or even infeasible. To overcome this, reconciliation methods [19,21] typically use structured forms, shrinkage, or dimension-reduction techniques that exploit the underlying hierarchy and temporal dependence to obtain tractable and reliable covariance estimates.

2.3. Iterative Cross-Temporal Reconciliation

The iterative cross-temporal reconciliation proposed by Di Fonzo and Girolimetto [19] provides a heuristic alternative to the cross-temporal optimal combination method. This method produces reconciled forecasts by alternating reconciliation steps along one dimension (either cross-sectional or temporal) in a cyclic fashion until convergence. In details, the iteration $j\ge 1$ can be described as follows:

Step 1 

compute the temporally reconciled forecasts (${\tilde{\mathit{X}}}_{\mathrm{te}}^{\left(j\right)}$) for each variable $i\in \{1,\dots ,n\}$ of ${\tilde{\mathit{X}}}_{\mathrm{cs}}^{(j-1)}$;

Step 2 

compute the time-by-time cross-sectional reconciled forecasts (${\tilde{\mathit{X}}}_{\mathrm{cs}}^{\left(j\right)}$) for all the temporal aggregation levels of ${\tilde{\mathit{X}}}_{\mathrm{te}}^{\left(j\right)}$.

These two steps are performed iteratively until a convergence criterion is met. Typically, convergence is achieved when the remaining discrepancies (${\mathit{D}}_{\mathrm{te}}={\mathit{C}}_{te}{\tilde{\mathit{X}}}_{\mathrm{cs}}^{\left(j\right)\prime }$) fall below a predefined positive tolerance value $\delta$ (e.g., $\delta ={10}^{-6}$). The matrix ${\tilde{\mathit{X}}}_{\mathrm{cs}}^{\left(j\right)}$ from the final iteration contains the cross-temporal reconciled forecasts $\tilde{\mathit{X}}$. At $j=0$, the starting values are given by ${\tilde{\mathit{X}}}_{\mathrm{cs}}^{\left(0\right)}=\widehat{\mathit{X}}$ (see Section 2.1).

Note that the temporal-then-cross-sectional sequence is presented here, but one may also begin with the cross-sectional step by inverting the order of Step 1 and Step 2. Note, however, that the final reconciled values may depend on the chosen order.

Other heuristic procedures have also been proposed, such as Kourentzes and Athanasopoulos [18] and Yagli et al. [46]. However, these are not discussed in detail here, as they fall outside the scope of this work. In particular, the KA heuristic relies on projection matrices, which are not available for most of the solutions presented in Section 3, while [46], whenever it achieves cross-temporally coherent results, can be considered as a special case of the iterative procedure described above [20].

3. Non-Negative Reconciliation

When applying a “free” reconciliation procedure (i.e., without imposing non-negativity), the resulting forecasts are not guaranteed to remain non-negative. Sometimes, this can be problematic in practice, since negative values are meaningless for variables such as demand, revenue, or counts, and can undermine both interpretability and credibility of the forecasts. A simple remedy is to replace negative reconciled values with zero [47,48]. However, this naive adjustment generally breaks the aggregation constraints, producing forecasts that are no longer coherent with the underlying hierarchical or temporal structure.

To overcome this issue, non-negativity constraints can be explicitly incorporated into the optimization formulations presented in Section 2.2. In particular, the projection (4) and the structural (7) problem can be re-formulated, respectively, as

$${\tilde{\mathit{x}}}_0=\underset{\mathit{x}}{arg\; min}{\left(\mathit{x}-\widehat{\mathit{x}}\right)}^{\prime }{\Omega }^{-1}\left(\mathit{x}-\widehat{\mathit{x}}\right)\mathrm{s}.\mathrm{t}.\mathit{C}\mathit{x}={\mathbf{0}}_{(n_a^{*}\times 1)},\mathit{x}\ge 0,$$

(10)

and

$${\tilde{\mathit{\beta }}}_0=\underset{\mathit{\beta }}{arg\; min}{\left(\widehat{\mathit{x}}-\mathit{S}\mathit{\beta }\right)}^{\prime }{\Omega }^{-1}\left(\widehat{\mathit{x}}-\mathit{S}\mathit{\beta }\right)\mathrm{s}.\mathrm{t}.\mathit{\beta }\ge 0.$$

(11)

Both formulations refer to non-negative least squares (NNLS) problems [35,49], firstly studied in the context of forecast reconciliation by Wickramasuriya et al. [13].

Several algorithms have been proposed for solving NNLS problems. Wickramasuriya et al. [13] show that the Block Principal Pivoting ($bpv$) algorithm [36] achieves strong performance in terms of both solution accuracy and computational efficiency. This solver is available in the FoReco package [40], which also provides an alternative implementation based on the Operator Splitting Quadratic Program ($osqp$) [38,50], a flexible framework that can incorporate additional types of constraints, such as immutable forecasts [39].

In addition to optimization-based solvers, we consider three heuristic approaches that combine ease of use, interpretability, and accuracy:

• The Negative Forecasts Correction Algorithm (nfca), an iterative correction procedure proposed by Kourentzes and Athanasopoulos [37].
• The iterative Non-Negative reconciliation with Immutable Constraints (nnic), where non-negativity is progressively enforced by treating corrected values as immutable in successive reconciliations [39,40].
• The bottom-up and top-down variants of the Set-Negative-To-Zero (sntz) procedure [20], which are very simple, competitive in accuracy, and computationally faster compared to the more intensive nonlinear optimization methods.

3.1. Block Principal Pivoting (bpv)

Since Lawson and Hanson [35], a wide range of approaches for addressing non-negative least squares problems have been proposed. The idea behind the Block Principal Pivoting [36] is to transform an inequality-constrained least squares problem into a sequence of equality-constrained sub-problems. $bpv$ incorporates a subset selection of variables to exchange and a backup rule to guarantee convergence in a finite number of iterations. Further details may be found in Wickramasuriya et al. [13].

3.2. Operator Splitting Quadratic Program (osqp)

The Operator Splitting Quadratic Program [38,50] solver is based on the Alternating Direction Method of Multipliers (ADMM) [51,52] and provides high-accuracy solutions by solving a quasi-definite linear system with a largely constant coefficient matrix across iterations. $osqp$ is highly robust, requiring neither positive definiteness of the objective function nor linear independence of the constraints. Further details may be found in Stellato et al. [38].

3.3. Negative Forecasts Correction Algorithm (nfca)

An alternative approach for ensuring non-negative reconciled forecasts is the Negative Forecasts Correction Algorithm [37]. Unlike optimization-based algorithms, that impose non-negativity constraints directly within the reconciliation problem, this method applies an iterative correction procedure. The central idea of the algorithm is straightforward: rather than discarding coherence by simply truncating negatives to zero, it progressively eliminates them while redistributing the necessary adjustments throughout the hierarchy. Formally, let ${\tilde{\mathit{x}}}^{\left(0\right)}$ denote the $free$ reconciled forecasts. At each iteration $r\ge 1$, a correction vector is constructed as follows. First, consider the $(n^{*}\times 1)$ vector ${\mathit{\delta }}^{\left(r\right)}$, with components

$${\delta }_i^{\left(r\right)}=\left\{\begin{array}{cc}-{\tilde{x}}_i^{(r-1)}\hfill & \mathrm{if}{\tilde{x}}_i^{(r-1)}<0\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.,i=1,\dots ,n^{*}.$$

(12)

This auxiliary vector is linearly transformed into a coherent correction vector using the reconciliation matrix $\mathit{M}$: ${\tilde{\mathit{\delta }}}^{\left(r\right)}=\mathit{M}{\mathit{\delta }}^{\left(r\right)}$, and the reconciled forecasts are updated as ${\tilde{\mathit{x}}}^{\left(r\right)}={\tilde{\mathit{x}}}^{(r-1)}+{\tilde{\mathit{\delta }}}^{\left(r\right)}$. The procedure is repeated until all components of ${\tilde{\mathit{x}}}^{\left(r\right)}$ are non-negative, or until changes fall below a predefined tolerance.

It is worth noting that the nfca algorithm is not an exact solution to the non-negative least squares problem. Rather, it should be considered as a heuristic: it enforces non-negativity by repeated coherent corrections rather than by solving a constrained optimization problem. As a result, while it is simple to code and often works well in small systems where negatives are rare, it may encounter difficulties when applied to larger or more complex hierarchies. In such settings, the number of required iterations may increase substantially, and convergence is not guaranteed. Furthermore, when many forecasts turn negative at once, numerical stability may be an issue.

3.4. Iterative Non-Negative Reconciliation with Immutable Constraints (NNIC)

Building on [53], and inspired by the nfca heuristic [37], the iterative Non-Negative reconciliation with Immutable Constraints provides an effective procedure to enforce non-negativity in reconciled forecasts. This heuristic can be viewed as an analogue of the Block Principal Pivoting algorithm for non-negative least squares. Rather than solving the constrained optimization problem in a single step, $nnic$ adopts an iterative pivoting strategy: whenever a forecast is found to be negative, it is fixed at zero and incorporated into the set of immutable constraints, while the remaining elements are left free to adjust.

Formally, assume that the $free$ reconciled forecasts’ vector, $\tilde{\mathit{x}}$, has $l^{\left(0\right)}\ge 1$ negative entries. Denote ${\tilde{\mathit{x}}}^{\left(0\right)}=\tilde{\mathit{x}}$, and let ${\mathit{C}}^{\left(0\right)}=\mathit{C}$ be the constraints matrix in expression (3). At each iteration $r\ge 1$, the constraint set is updated as

$${\mathit{C}}^{\left(r\right)}=\left[\begin{array}{c}{\mathit{C}}^{(r-1)}\\ {\mathit{E}}^{\left(r\right)}\end{array}\right],$$

(13)

where ${\mathit{E}}^{\left(r\right)}$ is an $[l^{(r-1)}\times n^{*}]$ matrix whose rows are zero everywhere except for a single unit entry in the columns corresponding to the negative elements of ${\tilde{\mathit{x}}}^{(r-1)}$. The next iterate reconciled forecasts, ${\tilde{\mathit{x}}}^{\left(r\right)}$, is obtained by applying the projection approach reconciliation formula (5) (or the structural formulation (8) following the immutable reconciliation framework of [39]) with the updated constraints matrix ${\mathit{C}}^{\left(r\right)}$. The procedure terminates once all components are non-negative or a pre-specified iteration limit is reached.

Although effective in many applications, nnic has some limitations. Its iterative nature can become computationally demanding when a large number of negatives are present, and convergence is not theoretically guaranteed in all settings. Nonetheless, empirical evidence indicates that $nnic$ frequently coincides with the solutions obtained by exact NNLS solvers, while being considerably simpler to implement.

3.5. Set-Negative-to-Zero (sntz): Bottom-Up and Top-Down Variants

In this section, we consider a simple heuristic strategy without any computationally intensive numerical optimization. For simplicity, we focus on a cross-sectional hierarchy with upper- and bottom-level series, assuming that at least one bottom-level free-reconciled forecast is negative.

The original $sntz$ procedure ($snt{z}_{bu}$), as described by Di Fonzo and Girolimetto [20], operates by first setting any negative bottom-level (corresponding, in temporal and cross-temporal frameworks, to the high-frequency and high-frequency bottom-level series, respectively) free-reconciled forecasts to zero, and then reconstructing the full set of reconciled forecasts using a bottom-up aggregation. Then, the non-negative bottom-level forecasts are left unchanged, and the upper-level forecasts are adjusted to restore coherence. Consequently, the reconciled top-level forecast may differ from its free-reconciled counterpart. This bottom-focused approach seems appropriate when the forecaster has greater confidence in the bottom-level base forecasts, and prefers minimal adjustments at this level, accepting potential changes at the upper levels.

However, in several practical settings (i.e., intermittent time series), bottom-level forecasts are more difficult to forecast, while upper-level aggregates may be more stable and accurate. In such cases, it may be more appropriate to preserve the upper-level forecasts. Motivated by this consideration, we propose a top-down variant, denoted $snt{z}_{td}$, to preserve the upper-level forecasts. In this approach, negative bottom-level forecasts are first replaced with zero, which typically creates a discrepancy between the adjusted bottom-level forecasts and the upper-level totals. So, this discrepancy is redistributed across the bottom-level series according to a set of distributional weights. Finally, the adjusted bottom-level forecasts are aggregated upward, ensuring that the hierarchy remains coherent and all series are non-negative, while the upper-level forecasts are left unchanged.

In summary, using $snt{z}_{bu}$ means that the forecaster is confident on the bottom-level free-reconciled forecasts, that are touched as little as possible, and is willing to accept (hopefully small) changes in the forecasts at the upper levels of the hierarchy. On the other hand, $snt{z}_{td}$ should be preferred when the upper level (aggregated) series’ free-reconciled forecasts are deemed to be relatively more accurate than those (disaggregated) of the bottom level variables, and the practitioner would prefer to retain coherency and non-negativity without no further adjustment of the free-reconciled forecasts of the upper level variables in the hierarchy.

• A numerical illustration

To illustrate the two variants, consider a simple hierarchy with one top-level variable a and three bottom-level components $b_1$, $b_2$, and $b_3$, such that

$$a=b_1+b_2+b_3.$$

Suppose the free-reconciled forecasts are $\tilde{a}=40$, ${\tilde{b}}_1=35$, ${\tilde{b}}_2=-5$, and ${\tilde{b}}_3=10$, which are coherent by construction. Under $snt{z}_{bu}$, the negative bottom-level forecast is set to zero, i.e., ${\tilde{b}}_{2,0}=0$. The adjusted top-level forecast is then computed as ${\tilde{a}}_0={\tilde{b}}_1+{\tilde{b}}_3=45$, that is ${\tilde{b}}_{1,0}={\tilde{b}}_1=35$ and ${\tilde{b}}_{3,0}={\tilde{b}}_3=10$.

• General formulation of ${\mathit{sntz}}_{\mathit{td}}$

Let $\tilde{a}>0$ denote the $free$ reconciled forecast of the top-level variable, and ${\{{\tilde{b}}_i\}}_{i=1,\dots ,n_b}$ the corresponding reconciled forecasts of $n_b$ bottom-level (in our example, $n_b=3$) series, such that $\tilde{a}={\sum }_{i=1}^{n_b}{\tilde{b}}_i$. Suppose that at least one ${\tilde{b}}_i$ is negative. Define the index sets

$$I^{+}=\{i:{\tilde{b}}_i>0\},I^{-}=\{i:{\tilde{b}}_i\le 0\},$$

and compute the discrepancy

$$d=\tilde{a}-\sum _{i\in I^{+}}{\tilde{b}}_i=\sum _{i\in I^{-}}{\tilde{b}}_i.$$

As the set $I^{-}$ is not empty, the discrepancy d is negative since $\tilde{a}={\sum }_{i=1}^{n_b}{\tilde{b}}_i=\underset{>0}{\underbrace{{\displaystyle \sum _{i\in I^{+}}}{\tilde{b}}_i}}+\underset{<0}{\underbrace{{\displaystyle \sum _{i\in I^{-}}}{\tilde{b}}_i}}$. Let ${\left\{w_i\right\}}_{i=1}^{n_b}$ be non-negative distribution coefficients satisfying $w_i=0$ for $i\in I^{-}$, $w_i>0$ for $i\in I^{+}$, and ${\sum }_{i\in I^{+}}{w}_i=1$. We consider three alternative specifications of $w_i$, $i\in I^{+}$:

• Proportional distribution, $snt{z}_{tdp}$: ${\displaystyle w_i=\frac{{\tilde{b}}_i}{{\sum }_{j\in I^{+}}{\tilde{b}}_j},i\in I^{+}}$
• Squared proportional distribution, $snt{z}_{tdsp}$: ${\displaystyle w_i=\frac{{\tilde{b}}_i^2}{{\sum }_{j\in I^{+}}{\tilde{b}}_j^2},i\in I^{+}.}$
• Variance-weighted distribution, $snt{z}_{tdvw}$: ${\displaystyle w_i=\frac{{\widehat{\sigma }}_i^2}{{\sum }_{j\in I^{+}}{\widehat{\sigma }}_j^2},i\in I^{+}.}$

The choice of weights depends on the structure of the series and the desired trade-off between simplicity and robustness. When the series have comparable scale and uncertainty, the proportional variant ($snt{z}_{tdp}$) provides a transparent and balanced allocation of the discrepancy. Under sparsity, when many small or zero components are present, or when it is desirable to protect small positive forecasts from being reduced below zero, the squared-proportional variant ($snt{z}_{tdsp}$) is preferable, as it concentrates the adjustment on larger components. In the presence of heteroskedasticity or markedly unequal forecast uncertainty, the variance-weighted scheme ($snt{z}_{tdvw}$) is recommended, since it allocates larger reductions to noisier series while preserving more reliable ones.

Finally, according to the top-down set-negative-to-zero heuristic, we derive the non-negative reconciled forecasts as

$${\tilde{a}}_0=\tilde{a},{\tilde{b}}_{i,0}=\left\{\begin{array}{cc}0\hfill & \mathrm{if}i\in I^{-}\hfill \\ {\tilde{b}}_i+w_{i}d\hfill & \mathrm{if}i\in I^{+}\hfill \end{array}\right.$$

and, more generally, ${\tilde{\mathit{x}}}_0=\mathit{S}{\tilde{\mathit{b}}}_0$ with, in this case, ${\tilde{\mathit{x}}}_0={\left[\begin{array}{cc}{\tilde{a}}_0& {\mathit{b}}_0^{\prime }\end{array}\right]}^{\prime }$ and ${\tilde{\mathit{b}}}_0={\left[\begin{array}{ccc}{\tilde{b}}_{1,0}& \dots & {\tilde{b}}_{n_b,0}\end{array}\right]}^{\prime }$. By construction, the adjusted bottom-level forecasts are non-negative and coherent:

$${\displaystyle {\displaystyle \sum _{i=1}^{n_b}}{\tilde{b}}_{i,0}=\sum _{i\in I^{+}}{\tilde{b}}_{i,0}+\sum _{i\in I^{-}}{\tilde{b}}_{i,0}=\sum _{i\in I^{+}}{\tilde{b}}_i+\sum _{i\in I^{+}}{w}_i\left(\tilde{a}-\sum _{i\in I^{+}}{\tilde{b}}_i\right)=\tilde{a}={\tilde{a}}_0.}$$

where, by definition, ${\displaystyle {\displaystyle \sum _{i=1}^{n_b}}{w}_i=\sum _{i\in I^{+}}{w}_i=1}$.

The proposed procedure always decreases the value of the positive freely reconciled forecasts ${\tilde{b}}_i$, $i\in I^{+}$. For some i it may happen that ${\tilde{b}}_i<\left|w_{i}d\right|$, which would result in a negative reconciled forecast ${\tilde{b}}_{i,0}$. This issue may be easily overcome by iterating the procedure, setting to zero the negative forecasts generated in the previous step, and distributing the new discrepancy as shown before. In practice, this simple iteration converges in finitely many steps and guarantees non-negativity for all components. A negative value can reappear only when $w_i>{\tilde{b}}_i/\left|d\right|$, a situation that was empirically rare in our experiments, no case required more than two passes (see Figure 1).

In conclusion,

Table 2. Reconciled forecasts for the toy example introduced in Section 2. The column $free$ reports the unrestricted reconciled forecasts, which may include negative values, while the columns $snt{z}_{bu}$, $snt{z}_{tdp}$, $snt{z}_{tdsp}$, and $snt{z}_{tdvw}$ show the results obtained with alternative non-negative reconciliation schemes, which enforce coherence and eliminate negative components.

	Type of Forecast
Variable	$\mathit{free}$	${\mathit{sntz}}_{\mathit{bu}}$	${\mathit{sntz}}_{\mathit{tdp}}$	${\mathit{sntz}}_{\mathit{tdsp}}$	${\mathit{sntz}}_{\mathit{tdvw}}$
${\tilde{a}}_{\phantom{1}}$	40	45	40	40	40
${\tilde{b}}_1$	35	35	31.1	30.4	31
${\tilde{b}}_2$	−5	0	0	0	0
${\tilde{b}}_3$	10	10	8.9	9.6	9

illustrates the application of these variants to the toy example above, assuming ${\widehat{\sigma }}_1^2=64$ and ${\widehat{\sigma }}_3^2=16$. The table reports the $free$ reconciled forecasts and the corresponding non-negative reconciled values under $snt{z}_{bu}$, $snt{z}_{tdp}$, $snt{z}_{tdsp}$, and $snt{z}_{tdvw}$, respectively.

4. Australian Tourism Demand Dataset

The empirical analysis builds on the Australian tourism demand dataset originally studied in Wickramasuriya et al. [12,13], extending it to the context of cross-temporal forecast reconciliation under non-negativity constraints [21]. The dataset consists of monthly measures of tourist flows, expressed in visitor nights (VN), collected through the Australian Government’s National Visitor Survey. The data cover 228 monthly observations, from January 1998 to December 2016, capturing both arrivals and the number of nights spent in tourist facilities.

The dataset has a cross-sectional grouped structure, which arises from combining a geographic hierarchy with a classification by purpose of travel. The geographic hierarchy disaggregates the country into seven states, further subdivided into 27 zones and 76 regions, with 111 geographic divisions in total. However, six zones consist of only a single region (South Coast NSW, ACT NSW, West Coast VIC, North WA, South WA, South TAS), which leads to 105 non-redundant nodes rather than the theoretical 111. In this sense, the geographic hierarchy can be considered “unbalanced” [33] (see Figure 2,

Table A1. Geographical divisions of Australia in States, Zones e Regions. Zones formed by a single region have been highlighted in italics.

Serie	Name	Label	Serie	Name	Label
Total			Continue: Regions
1	Australia	Total	49	Gippsland	BCB
States			50	Phillip Island	BCC
2	New South Wales (NSW)	A	51	Central Murray	BDA
3	Victoria (VIC)	B	52	Goulburn	BDB
4	Queensland (QLD)	C	53	High Country	BDC
5	South Australia (SA)	D	54	Melbourne East	BDD
6	Western Australia (WA)	E	55	Upper Yarra	BDE
7	Tasmania (TAS)	F	56	MurrayEast	BDF
8	Northern Territory (NT)	G	57	Mallee	BEA
Zones			58	Wimmera	BEB
9	Metro NSW	AA	59	Western Grampians	BEC
10	Nth Coast NSW	AB	60	Bendigo Loddon	BED
	Sth Coast NSW	AC	61	Macedon	BEE
11	Sth NSW	AD	62	Spa Country	BEF
12	Nth NSW	AE	63	Ballarat	BEG
	ACT	AF	64	Central Highlands	BEG
13	Metro VIC	BA	65	Gold Coast	CAA
	West Coast VIC	BB	66	Brisbane	CAB
14	East Coast VIC	BC	67	Sunshine Coast	CAC
15	Nth East VIC	BD	68	Central Queensland	CBA
16	Nth West VIC	BE	69	Bundaberg	CBB
17	Metro QLD	CA	70	Fraser Coast	CBC
18	Central Coast QLD	CB	71	Mackay	CBD
19	Nth Coast QLD	CC	72	Whitsundays	CCA
20	Inland QLD	CD	73	Northern	CCB
21	Metro SA	DA	74	Tropical North Queensland	CCC
22	Sth Coast SA	DB	75	Darling Downs	CDA
23	Inland SA	DC	76	Outback	CDB
24	West Coast SA	DD	77	Adelaide	DAA
25	West Coast WA	EA	78	Barossa	DAB
	Nth WA	EB	79	Adelaide Hills	DAC
	Sth WA	EC	80	Limestone Coast	DBA
	Sth TAS	FA	81	Fleurieu Peninsula	DBB
26	Nth East TAS	FB	82	Kangaroo Island	DBC
27	Nth West TAS	FC	83	Murraylands	DCA
28	Nth Coast NT	GA	84	Riverland	DCB
29	Central NT	GB	85	Clare Valley	DCC
Regions			86	Flinders Range and Outback	DCD
30	Sydney	AAA	87	Eyre Peninsula	DDA
31	Central Coast	AAB	88	Yorke Peninsula	DDB
32	Hunter	ABA	89	Australia’s Coral Coast	EAA
33	North Coast NSW	ABB	90	Experience Perth	EAB
34	South Coast	ACA	91	Australia’s SouthWest	EAC
35	Snowy Mountains	ADA	92	Australia’s North West	EBA
36	Capital Country	ADB	93	Australia’s Golden Outback	ECA
37	The Murray	ADC	94	Hobart and the South	FAA
38	Riverina	ADD	95	East Coast	FBA
39	Central NSW	AEA	96	Launceston, Tamar and the North	FBB
40	New England North West	AEB	97	North West	FCA
41	Outback NSW	AEC	98	WildernessWest	FCB
42	Blue Mountains	AED	99	Darwin	GAA
43	Canberra	AFA	100	Kakadu Arnhem	GAB
44	Melbourne	BAA	101	Katherine Daly	GAC
45	Peninsula	BAB	102	Barkly	GBA
46	Geelong	BAC	103	Lasseter	GBB
47	Western	BBA	104	Alice Springs	GBC
48	Lakes	BCA	105	MacDonnell	GBD

Source: [12,21].

in Appendix B and [21]).

In addition, tourism demand is disaggregated by purpose of travel (PoT) into four categories: holiday (Hol), visiting friends and relatives (Vis), business (Bus), and other (Oth). This classification generates 24 additional nodes (six single-region zones, each crossed with four PoT categories) that are duplications and therefore excluded. Accounting for these adjustments, we have a total of 525 non-redundant nodes [21,33], rather than the theoretical 555 [12,13]. Within this structure, the most disaggregated level (the “bottom” series) comprises 304 variables. These combine to produce 221 additional aggregate series, yielding 525 distinct time series overall (see

Table 3. Grouped time series for Australian tourism flows.

	Number of Series
	Geographical Division (GD)	Purpose of Travel (PoT)	Total
Australia	1	4	5
States	7	28	35
Zones *	21	84	105
Regions	76	304	380
Total	105	420	525

* 6 Zones with only one Region are included in the Regions.

). This update refines the structure originally presented in Table 7 of Wickramasuriya et al. [12].

From a temporal perspective, the dataset consists of monthly observations ($m=12$), which can be aggregated into 2-, 3-, 4-, 6-, and 12-monthly series, giving $\mathcal{K}=1,2,3,4,6,12$.

An important feature of this dataset is that for a large number of time series at least one of the values observed is zero, as shown in

Table 4. Number (#) and percentage of time series with values equal to 0, divided by temporal aggregation order (k) and cross-sectional level (${\mathcal{L}}_0$ Australia, ${\mathcal{L}}_1$ States, ${\mathcal{L}}_2$ Zones, ${\mathcal{L}}_3$ Regions, ${\mathcal{L}}_4$ PoT, ${\mathcal{L}}_5={\mathcal{L}}_1\times {\mathcal{L}}_4$, ${\mathcal{L}}_6={\mathcal{L}}_2\times {\mathcal{L}}_4$, ${\mathcal{L}}_7={\mathcal{L}}_3\times {\mathcal{L}}_4$).

k	${\mathcal{L}}_0$	${\mathcal{L}}_1$	${\mathcal{L}}_2$	${\mathcal{L}}_3$	${\mathcal{L}}_4$	${\mathcal{L}}_5$	${\mathcal{L}}_6$	${\mathcal{L}}_7$	Tot
# Series	1	7	21	76	4	28	84	304	525
1				13		1	25	200	239
				(17%)		(4%)	(30%)	(66%)	(46%)
2				1		1	13	131	146
				(1%)		(4%)	(15%)	(43%)	(28%)
3							6	98	104
							(7%)	(32%)	(20%)
4							3	76	79
							(4%)	(25%)	(15%)
6								54	54
								(18%)	(10%)
12								16	16
								(5%)	(3%)

. Approximately 46% (239 out of 525) of the monthly series contain at least one zero. This proportion decreases as the temporal aggregation order increases, but even among the annual series, 16 (around 3% of the total) contain at least one zero. At the regional level, 13 of the 76 monthly series also display at least one zero. In summary, Table 4 provides an overview of the sparsity pattern within the Australian Tourism Demand dataset, revealing the extent of zero-inflation across both temporal and hierarchical dimensions. Such information is crucial for interpreting the empirical results, as the presence of structural zeros introduces additional challenges for both model estimation and reconciliation. In particular, it highlights the practical relevance of enforcing non-negativity: when many series contain zeros or near-zero values, unconstrained reconciliation may easily generate negative forecasts, undermining interpretability and coherence. Consequently, this feature of the dataset further motivates the use of non-negative reconciliation procedures capable of handling sparse (or intermittent) time series effectively.

4.1. Forecasting Experiment

The forecasting experiment adopts a recursive evaluation design with an expanding training window. The initial training sample spans January 1998 to December 2008 (10 years, 120 months), and forecasts are generated for the subsequent year (2009). The training window is then progressively expanded by one month (e.g., January 1998–January 2009 to forecast February 2009–January 2010), continuing until the final training sample, January 1998–December 2015. This results in 85 distinct forecast origins and, correspondingly, 85 replications of the forecasting experiment.

Forecast horizons vary depending on the temporal aggregation level: up to six steps ahead for bimonthly series, four steps for quarterly, three for four-monthly, two for semiannual, and one for annual aggregates. Base forecasts are generated using ARIMA and ETS [54] models, estimated by minimizing the corrected Akaike Information Criterion (AICc). Both modeling approaches employ the default implementations of the R package forecast [55], following the procedures described in Hyndman and Khandakar [56]. In addition, models are fitted to log-transformed series, with forecasts back-transformed as in Wickramasuriya et al. [13].

Among the four sets of base forecasts ($arima$, $arima+log$, $ets$, $ets+log$), only $ets+log$ forecasts were carried forward to reconciliation. This choice is justified by two considerations: first, $ets+log$ forecasts are guaranteed to be nonnegative; and second, they consistently achieve the lowest average relative mean squared error (AvgRelMSE; see Section 4.2) across most cases (see Figure A1 in Appendix B). Base forecasts were subsequently reconciled using the R package FoReco [40] with six different reconciliation procedures:

$cs\left(shr\right)$

   Optimal cross-sectional approach with shrinkage covariance matrix [12];

$te\left(str\right)$

    Optimal temporal approach with diagonal covariance matrix based on the temporal structural matrix ${\mathit{S}}_{te}$ [16];

$te\left(wlsv\right)$

    Optimal temporal approach with diagonal covariance matrix based on in-sample residuals [16];

$ite(wlsv,shr)$

  Iterative approach with temporal diagonal covariance matrix [16] and cross-sectional shrinkage covariance matrix [12];

$ct\left(str\right)$

    Optimal cross-temporal approach with diagonal covariance matrix based on the temporal structural matrix ${\mathit{S}}_{ct}$ [16];

$ct\left(wlsv\right)$

    Optimal cross-temporal approach with diagonal covariance matrix based on in-sample residuals [19];

$ct\left(bdshr\right)$

     Optimal cross-temporal approach with block-diagonal covariance matrix based on in-sample residuals [19].

The first three procedures generate reconciled forecasts that are coherent along only one dimension (either cross-sectional or temporal).

4.2. Performance Measures for Multiple Comparisons

Forecast accuracy is evaluated using mean squared error (MSE) and its relative counterpart. Let ${\widehat{e}}_{i,j,t}^{\left[k\right],h}$ denote the forecast error for series $i=1,\dots ,n$, method $j=0,\dots ,J$, temporal aggregation level $k\in \mathcal{K}$, forecast origin $t=1,\dots ,q$, and forecast horizon $h=1,\dots ,h_k$:

$${\widehat{e}}_{i,j,t}^{\left[k\right],h}=y_{i,t+h}^{\left[k\right]}-{\widehat{y}}_{i,j,t}^{\left[k\right],h}.$$

The MSE is defined as the average of squared errors across forecast origins:

$${\mathrm{MSE}}_{i,j}^{\left[k\right],h}={\displaystyle \frac{1}{q}}{\displaystyle \sum _{t=1}^q}{\left({\widehat{e}}_{i,j,t}^{\left[k\right],h}\right)}^2.$$

Relative mean squared error (rMSE) compares each method j against a benchmark (method 0):

$${\mathrm{rMSE}}_{i,j}^{\left[k\right],h}={\displaystyle \frac{{\mathrm{MSE}}_{i,j}^{\left[k\right],h}}{{\mathrm{MSE}}_{i,0}^{\left[k\right],h}}}.$$

The average relative MSE (AvgRelMSE) is then defined as the geometric mean of rMSE values across series, temporal aggregation levels, and horizons:

$${\mathrm{AvgRelMSE}}_j={\left({\displaystyle \prod _{i=1}^n}\prod _{k\in \mathcal{K}}{\displaystyle \prod _{h=1}^{h_k}}{\mathrm{rMSE}}_{i,j}^{\left[k\right],h}\right)}^{{\displaystyle \frac{1}{n\left|\mathcal{K}\right|m}}}.$$

This index can be reported for specific groups of variables, across multiple forecast horizons, and at different levels of temporal aggregation. To assess whether performance differences are statistically significant, we employ the non-parametric Friedman test and the post hoc Multiple Comparison with the Best (MCB) Nemenyi procedure [18,57,58,59].

4.3. Non-Negative Reconciled Forecasts

Although the $ets+log$ forecasts are nonnegative by construction, reconciliation does not guarantee that this property is preserved. Indeed, negative reconciled forecasts occur under all reconciliation techniques considered (see

Table 5. Summary statistics of the negative reconciled forecasts using ETS base forecasts (with the log-transformation): replication’s number with at least one negative value (# rep); series’ number (# series) with at least one negative value in one replication (min and max); min and max of negative values in all the replications (values).

Label	# rep	# series		Values		# rep	# series		Values
		min	max	min	max		min	max	min	max
	Monthly forecasts ($k=1$)					Two-monthly forecasts ($k=2$)
$cs\left(shr\right)$	85	2	12	−58.35	−0.00031	85	1	8	−83.01	−0.00064
$te\left(str\right)$	41	1	4	−4.81	−0.01459	7	1	1	−1.27	−0.06806
$te\left(wlsv\right)$	43	1	5	−5.29	−0.00514	5	1	1	−1.13	−0.18561
$ite(wlsv,shr)$	85	2	8	−18.89	−0.00125	73	1	4	−21.21	−0.00102
$ct\left(str\right)$	85	10	32	−45.33	−0.00004	85	1	17	−71.76	−0.00510
$ct\left(wlsv\right)$	85	1	6	−29.97	−0.00558	60	1	4	−30.81	−0.01798
$ct\left(bdshr\right)$	85	2	13	−21.12	−0.00064	84	1	8	−20.15	−0.00558
	Quarterly forecasts ($k=3$)					Four-monthly forecasts ($k=4$)
$cs\left(shr\right)$	79	1	6	−71.64	−0.00051	69	1	4	−53.33	−0.00108
$te\left(str\right)$	–	–	–	–	–	–	–	–	–	1 –
$te\left(wlsv\right)$	–	–	–	–	–	–	–	–	–	1 –
$ite(wlsv,shr)$	49	1	3	−20.60	−0.00064	42	1	3	−23.73	−0.00851
$ct\left(str\right)$	84	1	10	−84.84	−0.00765	79	1	8	−100.25	−0.00223
$ct\left(wlsv\right)$	31	1	2	−19.00	−0.01225	25	1	3	−21.76	−0.06677
$ct\left(bdshr\right)$	82	1	6	−26.81	−0.00231	71	1	5	−30.29	−0.00017
	Semi-annual forecasts ($k=6$)					Annual forecasts ($k=12$)
$cs\left(shr\right)$	42	1	2	−26.78	−0.00734	5	1	1	−23.60	−0.60212
$te\left(str\right)$	–	–	–	–	–	–	–	–	–	–
$te\left(wlsv\right)$	–	–	–	–	–	–	–	–	–	–
$ite(wlsv,shr)$	21	1	3	−23.07	−0.01458	5	1	1	−16.51	−1.89363
$ct\left(str\right)$	69	1	7	−130.78	−0.05021	43	1	2	−131.82	−0.19975
$ct\left(wlsv\right)$	11	1	2	−19.76	−0.39761	4	1	1	−12.83	−1.74552
$ct\left(bdshr\right)$	56	1	3	−35.96	−0.01885	9	1	1	−33.97	−1.08399

). To address this, we evaluate seven alternative strategies for producing nonnegative reconciled forecasts (Section 3): block principal pivoting ($bpv$) [13], operator splitting quadratic programming ($osqp$) [38], the negative forecasts correction algorithm (nfca) [60], iterative nonnegative reconciliation with immutable constraints ($nnic$), and four variants of the set-negative-to-zero heuristic, original bottom-up ($snt{z}_{bu}$) [20], top-down proportional ($snt{z}_{tdp}$), top-down squared proportional ($snt{z}_{tdsp}$), and top-down variance-weighted ($snt{z}_{tdvw}$).

In Section 4, we evaluate the performance of alternative procedures for reconciling forecasts that may produce negative values, focusing on two key dimensions of assessment: forecasting accuracy and computational efficiency. By comparing both optimization-based and heuristic approaches, we want to show that computationally lighter procedures can achieve performance comparable to, or even superior to, that of more demanding optimal procedures. In doing so, we provide implementation insights on how non-negativity constraints can be applied in a large-scale setting without compromising either coherence or accuracy.

4.4. Results

Table 6. Average relative MSE (AvgRelMSE) across different temporal aggregation levels (monthly 1, two-monthly 2, quarterly 3, four-monthly 4, semi-annual 6, annual 12 and all), reconciliation approaches ($cs\left(shr\right)$, $te\left(str\right)$, $te\left(wlsv\right)$, $ite(wlsv,shr)$, $ct\left(str\right)$, $ct\left(wlsv\right)$, $ct\left(bdshr\right)$) and non-negative ($bpv$, $osqp$, nfca, $nnic$, $snt{z}_{bu}$, $snt{z}_{tdp}$, $snt{z}_{tdsp}$, $snt{z}_{tdvw}$); $free$ indicates the unrestricted reconciled forecasts, which may include negative values. Bold values indicate the best performing approach within each block (reconciliation approach) and, blue underlined and green italics values denote the overall best and the second-best performance, respectively.

	Temporal Aggregation Level
Label	1	2	3	4	6	12	all	1	2	3	4	6	12	all
	$cs\left(shr\right)$							$te\left(str\right)$
$free$	0.979279	0.977149	0.968516	0.959679	0.950190	0.942428	0.971733	0.978197	0.981807	0.979275	0.970618	0.955839	0.948815	0.975631
$\left.\begin{array}{c}bpv\\ osqp\\ nnic\end{array}\right\}$	0.977990	0.976276	0.968187	0.959392	0.950029	0.942396	0.970907	0.978193	0.981805	0.979276	0.970617	0.955841	0.948814	0.975628
$nfca$	0.977953	0.976012	0.968163	0.959401	0.950000	0.942398	0.970831	0.978193	0.981805	0.979276	0.970617	0.955841	0.948814	0.975628
$snt{z}_{bu}$	0.978009	0.976293	0.968147	0.959379	0.949981	0.942395	0.970908	0.978194	0.981804	0.979276	0.970616	0.955843	0.948813	0.975629
$snt{z}_{tdp}$	0.978054	0.976361	0.968155	0.959399	0.949989	0.942387	0.970946	0.978192	0.981803	0.979274	0.970615	0.955841	0.948815	0.975627
$snt{z}_{tdsp}$	0.978035	0.976378	0.968159	0.959398	0.949988	0.942387	0.970942	0.978192	0.981803	0.979274	0.970615	0.955840	0.948815	0.975627
$snt{z}_{tdvw}$	0.978067	0.976383	0.968155	0.959401	0.949988	0.942385	0.970956	0.978194	0.981804	0.979276	0.970617	0.955843	0.948815	0.975628
	$te\left(wlsv\right)$							$ite(wlsv,shr)$
$free$	0.977797	0.981215	0.978636	0.969898	0.955040	0.948397	0.975091	0.965780	0.965329	0.958496	0.946306	0.925542	0.906115	0.957432
$\left.\begin{array}{c}bpv\\ osqp\\ nnic\end{array}\right\}$	0.977791	0.981213	0.978637	0.969897	0.955041	0.948396	0.975088	0.965324	0.965132	0.958274	0.946106	0.925353	0.905947	0.957122
$nfca$	0.977791	0.981213	0.978637	0.969897	0.955041	0.948396	0.975088	0.965304	0.965104	0.958249	0.946084	0.925330	0.905920	0.957099
$snt{z}_{bu}$	0.977793	0.981213	0.978638	0.969896	0.955044	0.948396	0.975089	0.965339	0.965113	0.958252	0.946079	0.925340	0.905960	0.957118
$snt{z}_{tdp}$	0.977791	0.981211	0.978636	0.969895	0.955041	0.948397	0.975087	0.965258	0.965062	0.958206	0.946044	0.925314	0.905983	0.957062
$snt{z}_{tdsp}$	0.977789	0.981211	0.978635	0.969895	0.955040	0.948397	0.975087	0.965227	0.965033	0.958178	0.946018	0.925291	0.905981	0.957034
$snt{z}_{tdvw}$	0.977793	0.981213	0.978638	0.969896	0.955044	0.948397	0.975089	0.965305	0.965105	0.958246	0.946075	0.925341	0.905980	0.957102
	$ct\left(str\right)$							$ct\left(wlsv\right)$
$free$	0.984083	0.986868	0.981204	0.970437	0.949994	0.933418	0.978475	0.971016	0.972262	0.967195	0.956613	0.938178	0.922365	0.965031
$\left.\begin{array}{c}bpv\\ osqp\\ nnic\end{array}\right\}$	0.981578	0.984560	0.978940	0.968185	0.948245	0.931837	0.976164	0.970660	0.972115	0.967051	0.956487	0.938074	0.922239	0.964802
$nfca$	0.981578	0.984570	0.978937	0.968178	0.948243	0.931810	0.976163	0.970659	0.972109	0.967043	0.956480	0.938067	0.922232	0.964797
$snt{z}_{bu}$	0.981914	0.984635	0.978871	0.968004	0.947865	0.930293	0.976208	0.970723	0.972069	0.966998	0.956421	0.938029	0.922214	0.964800
$snt{z}_{tdp}$	0.982056	0.984944	0.979322	0.968553	0.948586	0.931386	0.976551	0.970735	0.972092	0.967031	0.956461	0.938081	0.922292	0.964825
$snt{z}_{tdsp}$	0.981960	0.984816	0.979192	0.968394	0.948371	0.931052	0.976419	0.970731	0.972086	0.967025	0.956454	0.938069	0.922274	0.964819
$snt{z}_{tdvw}$	0.982074	0.984986	0.979396	0.968688	0.948834	0.931990	0.976634	0.970738	0.972093	0.967030	0.956458	0.938076	0.922280	0.964826
	$ct\left(bdshr\right)$
$free$	0.964932	0.964332	0.957661	0.945062	0.924328	0.906373	0.956526
$\left.\begin{array}{c}bpv\\ osqp\\ nnic\end{array}\right\}$	0.964320	0.963973	0.957260	0.944642	0.923871	0.905896	0.956035
$nfca$	0.964284	0.963898	0.957185	0.944566	0.923809	0.905858	0.955979
$snt{z}_{bu}$	0.964377	0.963854	0.957090	0.944448	0.923693	0.905891	0.955975
$snt{z}_{tdp}$	0.964389	0.963877	0.957124	0.944490	0.923749	0.905977	0.956002
$snt{z}_{tdsp}$	0.964387	0.963874	0.957123	0.944489	0.923742	0.905964	0.955999
$snt{z}_{tdvw}$	0.964392	0.963880	0.957123	0.944486	0.923739	0.905952	0.956002

summarizes the Average Relative MSE (AvgRelMSE) across temporal aggregation levels, reconciliation approaches, and non-negative procedure. Overall, the results indicate improvements in forecast performance for all series and temporal aggregation levels when using non-negative reconciliation strategies, compared to both the $base$ forecasts (all the AvgRelMSE values $<1$) and the $free$ reconciliations. The Multiple Comparison with the Best (MCB) tests (Figure A2 and Figure A3 in Appendix B) show that all non-negative approaches consistently outperform the base forecasts, while improvements over $free$ reconciliation are not always statistically significant.

When considering only the non-negative procedures, forecast accuracy does not provide a conclusive ranking. In this application, $osqp$ and $bpv$ converge to the same results, as expected. It’s worth noting that the iterative non-negative reconciliation approach $nnic$ converges at the same outcomes, although this consistency may not be generalized (see Appendix A). Relative to the optimal results guaranteed by $osqp$ and $bpv$, heuristic approaches perform competitively and, in some cases, outperform the optimal methods. Notably, the $snt{z}_{bu}$ and $snt{z}_{tdsp}$ heuristics show particularly strong performance across different temporal levels.

Analyzing the results separately for high-frequency bottom-level series and the remaining time series provides additional insights (Figure A2 and Figure A3 in Appendix B). For high-frequency bottom-level series, imposing non-negativity constraints significantly improve over both $free$ and $base$ reconciliations, as indicated by MCB tests in Figure A2. However, for the temporal approaches ($te\left(str\right)$ and $te\left(wlsv\right)$), the differences are not statistically significant, because the $free$ reconciliation produces few negative values (see Table 5). While improvements for $snt{z}_{bu}$ are expected, almost all the top-down $sntz$ heuristics also provide statistically significant improvements for bottom-level series. Considering all the remaining time series including both bottom-level series at temporal aggregation levels $k>1$ and upper-level series at any temporal aggregation level (Figure A2), the $free$ reconciliation often is competitive with several reconciliation approaches, such as $ct\left(bdshr\right)$. However, heuristics like $snt{z}_{bu}$, and $snt{z}_{tdsp}$ are consistently statistically superior or at least not worse than the $free$ reconciliation approach.

In addition to forecast accuracy, we evaluate the computational performance of the different non-negative procedures. Figure 3 presents boxplots of the reconciliation times across replications of the forecasting experiment, while a more detailed analysis, including comparisons between structural and projection formulations, is provided in Figure A4 of Appendix B. The boxplots of Figure 3 show that the numerical optimization procedures (nfca, $nnic$, $bpv$, and $osqp$) generally require considerably more computation time than set-negative-to-zero. In particular, $osqp$ is highly sensitive to user-defined convergence parameters, resulting in longer computation times for larger structure, such as cross-temporal cases ($ct\left(str\right)$ and $ct\left(wlsv\right)$), while for smaller hierarchies (e.g., temporal $te\left(str\right)$ and $te\left(wlsv\right)$) the time is similar to $sntz$. Further information on the solver hyperparameter settings and computational environment used in the experiments is provided in Appendix C.

Among the optimization-based approaches, $bpv$ offers a good balance between efficiency and accuracy, while $nnic$ performs well with diagonal covariance structures but loses efficiency when $free$ reconciliation times increase, as observed in $ct\left(bdshr\right)$. A further consideration concerns nfca, which is the most computationally demanding procedure, typically requiring more resources than both optimization-based and heuristic alternatives. Achieving convergence was also more challenging: whereas $bpv$ and $osqp$ converged with tolerances of ${10}^{-6}$ (and $nnic$ with ${10}^{-5}$), for nfca we had to relax the tolerance to ${10}^{-3}$. These factors substantially limit the practicality of this procedure in large-scale or time-sensitive settings. At the same time, all set-negative-to-zero heuristics showed computation times very close to $free$, confirming them as the fastest non-negative options.

In summary, the results show that non-negative reconciliation strategies consistently improve forecast accuracy while preserving coherence. Among these, the set-negative-to-zero heuristics, particularly $snt{z}_{bu}$ and $snt{z}_{tdsp}$, stand out for achieving near-optimal accuracy with computational costs comparable to the $free$ reconciliation. Figure 4, which plots median computation time against AvgRelMSE, confirms that these heuristics lie close to the efficiency frontier, whereas optimization-based methods deliver limited or no additional accuracy gains at substantially higher costs.Given that accuracy differences among methods are relatively small on average, computational efficiency becomes the decisive criterion for practical deployment. In this regard, $snt{z}_{bu}$ and $snt{z}_{tdsp}$ offer the best balance between accuracy and scalability, making them particularly well suited for large-scale, online, or real-time forecasting applications where faster reconciliation is essential.

The online appendix, provided as Supplementary Material, reports results using the mean absolute error as the accuracy metric, as well as the complete set of results for all alternative base forecasts ($arima$, $ets$, and $arima+log$). These additional analyses are consistent with the findings reported for $ets+log$.

5. Conclusions

In this paper, we examined non-negative reconciliation methods for multiple time series subject to linear aggregation constraints. Our results show that imposing non-negativity consistently improves the accuracy and interpretability of forecasts compared to both base forecasts and $free$ reconciliations, while preserving coherence across all levels of aggregation. These gains are particularly evident for disaggregated series, where negative forecasts are most likely to arise, but we also observed improvements at more aggregated levels, indicating that the benefits propagate throughout the system.

A key contribution of our study is the systematic comparison of optimization-based and heuristic reconciliation procedures. While algorithms such as block principal pivoting and operator splitting quadratic programming deliver optimal reconciled solutions, they require substantial computation time and can be sensitive to parameter settings. Moreover, the iterative method $nnic$ shows competitive results but with reduced efficiency in some settings, while the heuristic nfca is even more computationally demanding and particularly sensitive to convergence settings compared to the other procedures. Instead, we found that simple set-negative-to-zero heuristics, and especially the bottom-up and squared-proportional top-down variants, achieve near-optimal forecast at almost no additional computational cost. In several cases, these heuristics even outperformed the optimization-based methods, demonstrating their robustness, scalability, and suitability for large-scale forecasting problems.

In conclusion, we show that addressing non-negativity in reconciled forecasts is not only a methodological refinement, but a practical necessity for high-dimensional, real-time, and resource-constrained forecasting environments. Our results suggest that practitioners can confidently adopt heuristic reconciliation procedures to obtain accurate and coherent forecasts while avoiding the computational burden of more complex optimization methods. The empirical evidence presented in this study further indicates that the choice among non-negative reconciliation approaches should be guided by the intended application and computational constraints. The set-negative-to-zero ($sntz$) heuristics deliver near-optimal accuracy at negligible computational cost, making them particularly suitable for large-scale, online, or real-time forecasting settings. The block principal pivoting ($bpv$) algorithm provides a balanced trade-off between accuracy and computational efficiency, representing a robust default among optimization-based methods. Finally, the operator splitting quadratic program ($osqp$) offers the highest degree of flexibility, allowing for the inclusion of additional constraints, with acceptable tuning time but some sensitivity to tolerance settings in large cross–temporal hierarchies. Future research could extend this analysis by considering alternative loss functions or accuracy metrics that explicitly account for non-negativity constraints, thereby better reflecting the practical implications of sign violations in strictly non-negative forecasting applications.