A Bayesian approach for predicting food and beverage sales in staff canteens and restaurants

Accurate demand forecasting is one of the key aspects for successfully managing restaurants and staff canteens. In particular, properly predicting future sales of menu items allows for a precise ordering of food stock. From an environmental point of view, this ensures a low level of pre-consumer food waste, while from the managerial point of view, this is critical to the profitability of the restaurant. Hence, we are interested in predicting future values of the daily sold quantities of given menu items. The corresponding time series show multiple strong seasonalities, trend changes, data gaps, and outliers. We propose a forecasting approach that is solely based on the data retrieved from point-of-sale systems and allows for a straightforward human interpretation. Therefore, we propose two generalized additive models for predicting future sales. In an extensive evaluation, we consider two data sets consisting of multiple time series collected at a casual restaurant and a large staff canteen and covering a period of 20 months. We show that the proposed models fit the features of the considered restaurant data. Moreover, we compare the predictive performance of our method against the performance of other well-established forecasting approaches.     

Euro area real-time density forecasting with financial or labor market frictions

We compare real-time density forecasts for the euro area using three DSGE models. The benchmark is the Smets and Wouters model, and its forecasts of real GDP growth and inflation are compared with those from two extensions. The first adds financial frictions and expands the observables to include a measure of the external finance premium. The second allows for the extensive labor-market margin and adds the unemployment rate to the observables. The main question that we address is whether these extensions improve the density forecasts of real GDP and inflation and their joint forecasts up to an eight-quarter horizon. We find that adding financial frictions leads to a deterioration in the forecasts, with the exception of longer-term inflation forecasts and the period around the Great Recession. The labor market extension improves the medium- to longer-term real GDP growth and shorter- to medium-term inflation forecasts weakly compared with the benchmark model.     

Forecasting the UK economy with a medium-scale Bayesian VAR

We estimate a Bayesian VAR (BVAR) for the UK economy and assess its performance in forecasting GDP growth and CPI inflation in real time relative to forecasts from COMPASS, the Bank of England’s DSGE model, and other benchmarks. We find that the BVAR outperformed COMPASS when forecasting both GDP and its expenditure components. In contrast, their performances when forecasting CPI were similar. We also find that the BVAR density forecasts outperformed those of COMPASS, despite under-predicting inflation at most forecast horizons. Both models over-predicted GDP growth at all forecast horizons, but the issue was less pronounced in the BVAR. The BVAR’s point and density forecast performances are also comparable to those of a Bank of England in-house statistical suite for both GDP and CPI inflation, as well as to the official Inflation Report projections. Our results are broadly consistent with the findings of similar studies for other advanced economies.     

Bayesian forecasting and portfolio decisions using dynamic dependent sparse factor models

We extend the recently introduced latent threshold dynamic models to include dependencies among the dynamic latent factors which underlie multivariate volatility. With an ability to induce time-varying sparsity in factor loadings, these models now also allow time-varying correlations among factors, which may be exploited in order to improve volatility forecasts. We couple multi-period, out-of-sample forecasting with portfolio analysis using standard and novel benchmark neutral portfolios. Detailed studies of stock index and FX time series include: multi-period, out-of-sample forecasting, statistical model comparisons, and portfolio performance testing using raw returns, risk-adjusted returns and portfolio volatility. We find uniform improvements on all measures relative to standard dynamic factor models. This is due to the parsimony of latent threshold models and their ability to exploit between-factor correlations so as to improve the characterization and prediction of volatility. These advances will be of interest to financial analysts, investors and practitioners, as well as to modeling researchers.     

Bayesian median autoregression for robust time series forecasting

We develop a Bayesian median autoregressive (BayesMAR) model for time series forecasting. The proposed method utilizes time-varying quantile regression at the median, favorably inheriting the robustness of median regression in contrast to the widely used mean-based methods. Motivated by a working Laplace likelihood approach in Bayesian quantile regression, BayesMAR adopts a parametric model bearing the same structure as autoregressive models by altering the Gaussian error to Laplace, leading to a simple, robust, and interpretable modeling strategy for time series forecasting. We estimate model parameters by Markov chain Monte Carlo. Bayesian model averaging is used to account for model uncertainty, including the uncertainty in the autoregressive order, in addition to a Bayesian model selection approach. The proposed methods are illustrated using simulations and real data applications. An application to U.S. macroeconomic data forecasting shows that BayesMAR leads to favorable and often superior predictive performance compared to the selected mean-based alternatives under various loss functions that encompass both point and probabilistic forecasts. The proposed methods are generic and can be used to complement a rich class of methods that build on autoregressive models.     

Optimal probabilistic forecasts: When do they work?

Proper scoring rules are used to assess the out-of-sample accuracy of probabilistic forecasts, with different scoring rules rewarding distinct aspects of forecast performance. Herein, we re-investigate the practice of using proper scoring rules to produce probabilistic forecasts that are ‘optimal’ according to a given score and assess when their out-of-sample accuracy is superior to alternative forecasts, according to that score. Particular attention is paid to relative predictive performance under misspecification of the predictive model. Using numerical illustrations, we document several novel findings within this paradigm that highlight the important interplay between the true data generating process, the assumed predictive model and the scoring rule. Notably, we show that only when a predictive model is sufficiently compatible with the true process to allow a particular score criterion to reward what it is designed to reward, will this approach to forecasting reap benefits. Subject to this compatibility, however, the superiority of the optimal forecast will be greater, the greater is the degree of misspecification. We explore these issues under a range of different scenarios and using both artificially simulated and empirical data.     

Open economy forecasting with a DSGE-VAR: Head to head with the RBNZ published forecasts

We construct a DSGE-VAR model for competing head to head with the long history of published forecasts of the Reserve Bank of New Zealand. We also construct a Bayesian VAR model with a Minnesota prior for forecast comparison. The DSGE-VAR model combines a structural DSGE model with a statistical VAR model based on the in-sample fit over the majority of New Zealand’s inflation-targeting period. We evaluate the real-time out-of-sample forecasting performance of the DSGE-VAR model, and show that the forecasts from the DSGE-VAR are competitive with the Reserve Bank of New Zealand’s published, judgmentally-adjusted forecasts. The Bayesian VAR model with a Minnesota prior also provides a competitive forecasting performance, and generally, with a few exceptions, out-performs both the DSGE-VAR and the Reserve Bank’s own forecasts.     

False dichotomy alert: Improving subjective-probability estimates vs. raising awareness of systemic risk

Taleb et al. (2022) portray the superforecasting research program as a masquerade that purports to build “survival functions for tail assessments via sports-like tournaments.” But that never was the goal. The program was designed to help intelligence analysts make better probability judgments, which required posing rapidly resolvable questions. From a signal detection theory perspective, the superforecasting and Taleb et al. programs are complementary, not contradictory (a point Taleb and Tetlock (2013) recognized). The superforecasting program aims at achieving high hit rates at low cost in false-positives, whereas Taleb et al. prioritize alerting us to systemic risk, even if that entails a high false-positive rate. Proponents of each program should, however, acknowledge weaknesses in their cases. It is unclear: (a) how Taleb et al. (2022) can justify extreme error-avoidance trade-offs, without tacit probability judgments of rare, high-impact events; (b) how much superforecasting interventions can improve probability judgments of such events.     

Bayesian modification indices for IRT models

Bayesian modification indices are presented that provide information for the process of model evaluation and model modification. These indices can be used to investigate the improvement in a model if fixed parameters are re-specified as free parameters. The indices can be seen as a Bayesian analogue to the modification indices commonly used in a frequentist framework. The aim is to provide diagnostic information for multi-parameter models where the number of possible model violations and the related number of alternative models is too large to render estimation of each alternative practical. As an example, the method is applied to an item response theory (IRT) model, that is, to the two-parameter model. The method is used to investigate differential item functioning and violations of the assumption of local independence.     

贝叶斯非线性分层模型在多元索赔准备金评估中的应用

本文将贝叶斯非线性分层模型应用于基于不同业务线的多元索赔准备金评估中,设计了一种合适的模型结构,将非线性分层模型与贝叶斯方法结合起来,应用WinBUGS软件对精算实务中经典流量三角形数据进行建模分析,并使用MCMC方法得到了索赔准备金完整的预测分布。这种方法扩展并超越了已有多元评估方法中最佳估计和预测均方误差估计的研究范畴。在贝叶斯框架下结合后验分布实施推断对非寿险公司偿付能力监管和行业决策具有重要作用。     

A direct Monte Carlo approach for Bayesian analysis of the seemingly unrelated regression model

Computationally efficient methods for Bayesian analysis of seemingly unrelated regression (SUR) models are described and applied that involve the use of a direct Monte Carlo (DMC) approach to calculate Bayesian estimation and prediction results using diffuse or informative priors. This DMC approach is employed to compute Bayesian marginal posterior densities, moments, intervals and other quantities, using data simulated from known models and also using data from an empirical example involving firms’ sales. The results obtained by the DMC approach are compared to those yielded by the use of a Markov Chain Monte Carlo (MCMC) approach. It is concluded from these comparisons that the DMC approach is worthwhile and applicable to many SUR and other problems.     

Macroeconomic forecasting with large Bayesian VARs: Global-local priors and the illusion of sparsity

A class of global-local hierarchical shrinkage priors for estimating large Bayesian vector autoregressions (BVARs) has recently been proposed. We question whether three such priors: Dirichlet-Laplace, Horseshoe, and Normal-Gamma, can systematically improve the forecast accuracy of two commonly used benchmarks (the hierarchical Minnesota prior and the stochastic search variable selection (SSVS) prior), when predicting key macroeconomic variables. Using small and large data sets, both point and density forecasts suggest that the answer is no. Instead, our results indicate that a hierarchical Minnesota prior remains a solid practical choice when forecasting macroeconomic variables. In light of existing optimality results, a possible explanation for our finding is that macroeconomic data is not sparse, but instead dense.     

On the forecasting performance of a small-scale DSGE model

Dynamic stochastic general equilibrium (DSGE) models have recently become standard tools for policy analysis. Nevertheless, their forecasting properties have still barely been explored. In this article, we address this problem by examining the quality of forecasts of the key U.S. economic variables: the three-month Treasury bill yield, the GDP growth rate and GDP price index inflation, from a small-size DSGE model, trivariate vector autoregression (VAR) models and the Philadelphia Fed Survey of Professional Forecasters (SPF). The ex post forecast errors are evaluated on the basis of the data from the period 1994–2006. We apply the Philadelphia Fed “Real-Time Data Set for Macroeconomists” to ensure that the data used in estimating the DSGE and VAR models was comparable to the information available to the SPF.Overall, the results are mixed. When comparing the root mean squared errors for some forecast horizons, it appears that the DSGE model outperforms the other methods in forecasting the GDP growth rate. However, this characteristic turned out to be statistically insignificant. Most of the SPF's forecasts of GDP price index inflation and the short-term interest rate are better than those from the DSGE and VAR models.     

Validation and forecasting accuracy in models of climate change

Forecasting researchers, with few exceptions, have ignored the current major forecasting controversy: global warming and the role of climate modelling in resolving this challenging topic. In this paper, we take a forecaster’s perspective in reviewing established principles for validating the atmospheric-ocean general circulation models (AOGCMs) used in most climate forecasting, and in particular by the Intergovernmental Panel on Climate Change (IPCC). Such models should reproduce the behaviours characterising key model outputs, such as global and regional temperature changes. We develop various time series models and compare them with forecasts based on one well-established AOGCM from the UK Hadley Centre. Time series models perform strongly, and structural deficiencies in the AOGCM forecasts are identified using encompassing tests. Regional forecasts from various GCMs had even more deficiencies. We conclude that combining standard time series methods with the structure of AOGCMs may result in a higher forecasting accuracy. The methodology described here has implications for improving AOGCMs and for the effectiveness of environmental control policies which are focussed on carbon dioxide emissions alone. Critically, the forecast accuracy in decadal prediction has important consequences for environmental planning, so its improvement through this multiple modelling approach should be a priority.     

Inversion copulas from nonlinear state space models with an application to inflation forecasting

We propose the construction of copulas through the inversion of nonlinear state space models. These copulas allow for new time series models that have the same serial dependence structure as a state space model, but with an arbitrary marginal distribution, and flexible density forecasts. We examine the time series properties of the copulas, outline serial dependence measures, and estimate the models using likelihood-based methods. Copulas constructed from three example state space models are considered: a stochastic volatility model with an unobserved component, a Markov switching autoregression, and a Gaussian linear unobserved component model. We show that all three inversion copulas with flexible margins improve the fit and density forecasts of quarterly U.S. broad inflation and electricity inflation.     

Pair trading based on quantile forecasting of smooth transition GARCH models

Pair trading is a statistical arbitrage strategy used on similar assets with dissimilar valuations. We utilize smooth transition heteroskedastic models with a second-order logistic function to generate trading entry and exit signals and suggest two pair trading strategies: the first uses the upper and lower threshold values in the proposed model as trading entry and exit signals, while the second strategy instead takes one-step-ahead quantile forecasts obtained from the same model. We employ Bayesian Markov chain Monte Carlo sampling methods for updating the estimates and quantile forecasts. As an illustration, we conduct a simulation study and empirical analysis of the daily stock returns of 36 stocks from U.S. stock markets. We use the minimum square distance method to select ten stock pairs, choose additional five pairs consisting of two companies in the same industrial sector, and then finally consider pair trading profits for two out-of-sample periods in 2014 within a six-month time frame as well as for the entire year. The proposed strategies yield average annualized returns of at least 35.5% without a transaction cost and at least 18.4% with a transaction cost.