Methods for forecasting time series adhering to linear constraints have seen notable development in recent years, especially with the advent of forecast reconciliation. This paper extends forecast reconciliation to the open question of non-linearly constrained time series. Non-linear constraints can emerge with variables that are formed as ratios such as mortality rates and unemployment rates. On the methodological side, Non-linearly Constrained Reconciliation (NLCR) is proposed. This algorithm adjusts forecasts that fail to meet non-linear constraints, in a way that ensures the new forecasts meet the constraints. The NLCR method is a projection onto a non-linear surface, formulated as a constrained optimisation problem. On the theoretical side, optimisation methods are again used, this time to derive sufficient conditions for when the NLCR methodology is guaranteed to improve forecast accuracy. Finally on the empirical side, NLCR is applied to two datasets from demography and economics and shown to significantly improve forecast accuracy relative to relevant benchmarks.

Forecast reconciliation has attracted significant research interest in recent years, with most studies taking the hierarchy of time series as given. We extend existing work that uses time series clustering to construct hierarchies, with the goal of improving forecast accuracy, in three ways. First, we investigate multiple approaches to clustering, including not only different clustering algorithms, but also the way time series are represented and how distance between time series is defined. We find that cluster-based hierarchies lead to improvements in forecast accuracy relative to two-level hierarchies. Second, we devise an approach based on random permutation of hierarchies, keeping the structure of the hierarchy fixed, while time series are randomly allocated to clusters. In doing so, we find that improvements in forecast accuracy that accrue from using clustering do not arise from grouping together similar series but from the structure of the hierarchy. Third, we propose an approach based on averaging forecasts across hierarchies constructed using different clustering methods, that is shown to outperform any single clustering method. All analysis is carried out on two benchmark datasets and a simulated dataset. Our findings provide new insights into the role of hierarchy construction in forecast reconciliation and offer valuable guidance on forecasting practice.

This paper extends the technique of gradient boosting with a focus on using domain-specific models instead of trees. The domain of mortality forecasting is considered as an application. The two novel contributions are to use well-known stochastic mortality models as weak learners in gradient boosting rather than trees, and to include a penalty that shrinks mortality forecasts in adjacent age groups and nearby geographical regions closer together. The proposed method demonstrates superior forecasting performance based on US male mortality data from 1969 to 2019. The proposed approach also enables us to interpret and visualize the results. The boosted model with age-based shrinkage yields the most accurate national-level mortality forecast. For state-level forecasts, spatial shrinkage provides further improvement in accuracy in addition to the benefits of age-based shrinkage. This improvement can be attributed to data sharing across states with large and small populations in adjacent regions and states with common risk factors.

Variational inference (VI) is a popular method to estimate statistical and econometric models. The key to VI is the selection of a tractable density to approximate the Bayesian posterior. For large and complex models a common choice is to assume independence between multivariate blocks in a partition of the parameter space. While this simplifies the problem it can reduce accuracy. This paper proposes using vector copulas to capture dependence between the blocks parsimoniously. Tailored multivariate marginals are constructed using learnable cyclically monotone transformations. We call the resulting joint distribution a “dependent block posterior” approximation. Vector copula models are suggested that make tractable and flexible variational approximations. They allow for differing marginals, numbers of blocks, block sizes and forms of between block dependence. They also allow for solution of the variational optimization using fast and efficient stochastic gradient methods. The efficacy and versatility of the approach is demonstrated using four different statistical models and 16 datasets which have posteriors that are challenging to approximate. In all cases, our method produces more accurate posterior approximations than benchmark VI methods that either assume block independence or factor-based dependence, at limited additional computational cost.

A novel forecast linear augmented projection (FLAP) method is introduced, which reduces the forecast error variance of any unbiased multivariate forecast without introducing bias. The method first constructs new component series which are linear combinations of the original series. Forecasts are then generated for both the original and component series. Finally, the full vector of forecasts is projected onto a linear subspace where the constraints implied by the combination weights hold. It is proven that the trace of the forecast error variance is non-increasing with the number of components, and mild conditions are established for which it is strictly decreasing. It is also shown that the proposed method achieves maximum forecast error variance reduction among linear projection methods. The theoretical results are validated through simulations and two empirical applications based on Australian tourism and FRED-MD data. Notably, using FLAP with Principal Component Analysis (PCA) to construct the new series leads to substantial forecast error variance reduction.

PURPOSE:Cancer treatment remains a significant and escalating healthcare expense worldwide. Although annual reports on total cancer care costs are available, the potential impact of evolving treatment guidelines and the introduction of new drugs on future budgeting remains largely uncertain. The aim of this study was to examine the trends in the use of Pharmaceutical Benefits Scheme (PBS)-subsidised cancer drugs in Australia over the past decade. METHODS:PBS codes for all PBS-subsidised cancer drugs that were listed in government-endorsed treatment protocols were obtained and used to retrieve usage data. Their patterns of use, represented by the number of prescriptions (services) processed by Services Australia, were analysed for the period between 2012 and 2022. RESULTS:The overall prescribing of cancer drugs is outpacing Australia's population growth, primarily due to an ageing population and the accelerated rise in cancer diagnoses observed over the past decade. From 846 eviQ protocols, 141 cancer drugs were available on the PBS, of which kinase inhibitor (39 drugs) and monoclonal antibody drugs (24 drugs) had the highest increase in use during the study period; 16% and 23% respectively. Of the 8 drug classes, hormonal agents (20 drugs) were the most prescribed. CONCLUSION:The utilisation of PBS-listed cancer drugs is increasing faster than population growth, especially for high-cost monoclonal antibody and kinase inhibitor drugs, indicating continued pressure on government spending.

We present two novel approaches to overcome the limitations of data-sparse local house price indexes and combine them into a single model pipeline that is simple, computationally efficient, and interpretable. The first contribution is a new spatio-temporal regularization of least squares dummy variable models, such as the repeat sales regression used here. This regularization encodes prior knowledge of the proximity of houses in space and their sales in time. It handles missing values in a natural way. The second is nonlocal regularization using truncated principal component analysis (PCA) applied to the resulting national collection of local price indexes. The PCA loadings show that there are important underlying socioeconomic factors that can be leveraged in the construction of Australian market indexes. This PCA reveals important socioeconomic factors, showing that many local markets can be described by a few broad aspects of the national market, consisting of a general trend that contrasts regions influenced by the mining industry with Sydney and Melbourne, and another trend that highlights lifestyle.

The Bayesian statistical paradigm provides a principled and coherent approach to probabilistic forecasting. Uncertainty about all unknowns that characterize any forecasting problem – model, parameters, latent states – is able to be quantified explicitly and factored into the forecast distribution via the process of integration or averaging. Allied with the elegance of the method, Bayesian forecasting is now underpinned by the burgeoning field of Bayesian computation, which enables Bayesian forecasts to be produced for virtually any problem, no matter how large or complex. The current state of play in Bayesian forecasting in economics and finance is the subject of this review. The aim is to provide the reader with an overview of modern approaches to the field, set in some historical context, with sufficient computational detail given to assist the reader with implementation.

Manifold learning obtains a low-dimensional representation of an underlying Riemannian manifold supporting high-dimensional data. Kernel density estimates of the low-dimensional embedding with a fixed bandwidth fail to account for the way manifold learning algorithms distort the geometry of the Riemannian manifold. We propose a novel distortion-corrected kernel density estimator (DC-KDE) for any manifold learning embedding, with a bandwidth that depends on the estimated Riemannian metric at each data point. Exploiting the geometric information of the manifold leads to more accurate density estimation, which subsequently could be used for anomaly detection. To compare our proposed estimator with a fixed-bandwidth kernel density estimator, we run two simulations including one with data lying in a 100 dimensional ambient space. We demonstrate that the proposed DC-KDE improves the density estimates as long as the manifold learning embedding is of sufficient quality, and has higher rank correlations with the true manifold density. Further simulation results are provided via a supplementary R shiny app. The proposed method is applied to density estimation in statistical manifolds of electricity usage with the Irish smart meter data.

We explore the application of Mixture Density Networks (MDN) to house price sales data for the Australian market and construct price indexes for geographically smaller regions without using house characteristics. This is achieved by tracking the price distributions in time, where a deep neural network based on an embedding architecture calculates the parameters of a Gaussian mixture to approximate this cross-sectional density. Out-of-sample tests comparing our model to several established price index approaches in econometrics, including repeat sales and dynamic factor models, demonstrate that such networks can achieve similar results to established linear econometric baselines in data-poor settings. The MDN demonstrates practical performance under limited data conditions and provides full distributional outputs for each region and time point. Our exploration also shows the usefulness of this type of model for studying time-dependent quantile-based aspects of the market, and some example applications to the Sydney housing market are discussed.

Collections of time series formed via aggregation are prevalent in many fields. These are commonly referred to as hierarchical time series and may be constructed cross-sectionally across different variables, temporally by aggregating a single series at different frequencies, or even generalised beyond aggregation as time series that respect linear constraints. When forecasting such time series, a desirable condition is for forecasts to be coherent: to respect the constraints. The past decades have seen substantial growth in this field with the development of reconciliation methods that ensure coherent forecasts and improve forecast accuracy. This paper serves as a comprehensive review of forecast reconciliation and an entry point for researchers and practitioners dealing with hierarchical time series. The scope of the article includes perspectives on forecast reconciliation from machine learning, Bayesian statistics and probabilistic forecasting, as well as applications in economics, energy, tourism, retail demand and demography.

This paper presents a formal framework and proposes algorithms to extend forecast reconciliation to discrete-valued data to extend forecast reconciliation to discrete-valued data, including low counts. A novel method is introduced based on recasting the optimisation of scoring rules as an assignment problem, which is solved using quadratic programming. The proposed framework produces coherent joint probabilistic forecasts for count hierarchical time series. Two discrete reconciliation algorithms are also proposed and compared against generalisations of the top-down and bottom-up approaches for count data. Two simulation experiments and two empirical examples are conducted to validate that the proposed reconciliation algorithms improve forecast accuracy. The empirical applications are forecasting criminal offences in Washington D.C. and product unit sales in the M5 dataset. Compared to benchmarks, the proposed framework shows superior performance in both simulations and empirical studies.

A new methodology is proposed to approximate the time-dependent house price distribution at a fine regional scale using Gaussian mixtures. The means, variances and weights of the mixture components are related to time, location and dwelling type through a non linear function trained by a deep functional approximator. Price indices are derived as means, medians, quantiles or other functions of the estimated distributions. Price densities for larger regions, such as a city, are calculated via a weighted sum of the component density functions. The method is applied to a data set covering all of Australia at a fine spatial and temporal resolution. In addition to enabling a detailed exploration of the data, the proposed index yields lower prediction errors in the practical task of individual dwelling price projection from previous sales values within the three major Australian cities. The estimated quantiles are also found to be well calibrated empirically, capturing the complexity of house price distributions.

The practical importance of coherent forecasts in hierarchical forecasting has inspired many studies on forecast reconciliation. Under this approach, so-called base forecasts are produced for every series in the hierarchy and are subsequently adjusted to be coherent in a second reconciliation step. Reconciliation methods have been shown to improve forecast accuracy, but will, in general, adjust the base forecast of every series. However, in an operational context, it is sometimes necessary or beneficial to keep forecasts of some variables unchanged after forecast reconciliation. In this paper, we formulate reconciliation methodology that keeps forecasts of a pre-specified subset of variables unchanged or "immutable". In contrast to existing approaches, these immutable forecasts need not all come from the same level of a hierarchy, and our method can also be applied to grouped hierarchies. We prove that our approach preserves unbiasedness in base forecasts. Our method can also account for correlations between base forecasting errors and ensure non-negativity of forecasts. We also perform empirical experiments, including an application to sales of a large scale online retailer, to assess the impacts of our proposed methodology.

Standard selection criteria for forecasting models focus on information that is calculated for each series independently, disregarding the general tendencies and performances of the candidate models. In this paper, we propose a new way to statistical model selection and model combination that incorporates the base-rates of the candidate forecasting models, which are then revised so that the per-series information is taken into account. We examine two schemes that are based on the precision and sensitivity information from the contingency table of the base rates. We apply our approach on pools of exponential smoothing models and a large number of real time series and we show that our schemes work better than standard statistical benchmarks. We discuss the connection of our approach to other cross-learning approaches and offer insights regarding implications for theory and practice.

We develop a framework for forecasting multivariate data that follow known linear constraints. This is particularly common in forecasting where some variables are aggregates of others, commonly referred to as hierarchical time series, but also arises in other prediction settings. For point forecasting, an in-creasingly popular technique is reconciliation, whereby forecasts are made for all series (so-called base forecasts) and subsequently adjusted to cohere with the constraints. We extend reconciliation from point forecasting to probabilistic forecasting. A novel definition of reconciliation is developed and used to con-struct densities and draw samples from a reconciled probabilistic forecast. In the elliptical case, we prove that true predictive distributions can be recovered using reconciliation even when the location and scale of base predictions are chosen arbitrarily. Reconciliation weights are estimated to optimise energy or variogram score. The log score is not considered since it is improper when comparing unreconciled to reconciled forecasts, a result also proved in this paper. Due to randomness in the objective function, optimisation uses stochastic gradient descent. This method improves upon base forecasts in simulated and empirical data, particularly when the base forecasting models are severely misspecified. For milder misspecification, extending popular reconciliation methods for point forecasting results in similar perfor-mance to score optimisation.(c) 2022 Elsevier B.V. All rights reserved.

Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases.

Variational Bayesian (VB) methods produce posterior inference in a time frame considerably smaller than traditional Markov Chain Monte Carlo approaches. Although the VB posterior is an approximation, it has been shown to produce good parameter estimates and predicted values when a rich classes of approximating distributions are considered. In this paper, we propose the use of recursive algorithms to update a sequence of VB posterior approximations in an online, time series setting, with the computation of each posterior update requiring only the data observed since the previous update. We show how importance sampling can be incorporated into online variational inference allowing the user to trade accuracy for a substantial increase in computational speed. The proposed methods and their properties are detailed in two separate simulation studies. Additionally, two empirical illustrations are provided, including one where a Dirichlet Process Mixture model with a novel posterior dependence structure is repeatedly updated in the context of predicting the future behaviour of vehicles on a stretch of the US Highway 101.

A geometric interpretation is developed for so-called reconciliation methodologies used to forecast time series that adhere to known linear constraints. In particular, a general framework is established that nests many existing popular reconciliation methods within the class of projections. This interpretation facilitates the derivation of novel theoretical results. First, reconciliation via projection is guaranteed to improve forecast accuracy with respect to a class of loss functions based on a generalised distance metric. Second, the Minimum Trace (MinT) method minimises expected loss for this same class of loss functions. Third, the geometric interpretation provides a new proof that forecast reconciliation using projections results in unbiased forecasts, provided that the initial base forecasts are also unbiased. Approaches for dealing with biased base forecasts are proposed. An extensive empirical study of Australian tourism flows demonstrates the theoretical results of the paper and shows that bias correction prior to reconciliation outperforms alternatives that only bias-correct or only reconcile forecasts.