Density Forecasts Research Articles

Deep-Based Conditional Probability Density Function Forecasting of Residential Loads

This paper proposes a direct model for conditional probability density forecasting of residential loads, based on a deep mixture network. Probabilistic residential load forecasting can provide comprehensive information about future uncertainties in demand. An end-to-end composite model comprising convolution neural networks (CNNs) and gated recurrent unit (GRU) is designed for probabilistic residential load forecasting. Then, the designed deep model is merged into a mixture density network (MDN) to directly predict probability density functions (PDFs). In addition, several techniques, including adversarial training, are presented to formulate a new loss function in the direct probabilistic residential load forecasting (PRLF) model. Several state-of-the-art deep and shallow forecasting models are also presented in order to compare the results. Furthermore, the effectiveness of the proposed deep mixture model in characterizing predicted PDFs is demonstrated through comparison with kernel density estimation, Monte Carlo dropout, a combined probabilistic load forecasting method and the proposed MDN without adversarial training.

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.

Online Estimation of DSGE Models

Thiis paper illustrates the usefulness of sequential Monte Carlo (SMC) methods in approximating DSGE model posterior distributions. We show how the tempering schedule can be chosen adaptively, document the accuracy and runtime benefits o fgeneralized data tempering for “online” estimation (that is, re-estimating a model asnew data become available), and provide examples of multimodal posteriors that are well captured by SMC methods. We then use the online estimation of the DSGE model to compute pseudo-out-of-sample density forecasts and study the sensitivity ofthe predictive performance to changes in the prior distribution. We find that making priors less informative (compared to the benchmark priors used in the literature) by increasing the prior variance does not lead to a deterioration of forecast accuracy.

Predicting international equity returns: Evidence from time-varying parameter vector autoregressive models

In this paper, we forecast monthly stock returns of eight advanced economies using a time-varying parameter vector autoregressive model (TVP-VAR) with mixture innovations. Compared to standard TVP-VARs, our proposed model automatically detects whether time-variation in the parameters is needed through the introduction of a latent process. This framework is capable of dynamically detecting whether a given regression coefficient is constant or time-varying during distinct time periods. We moreover compare the performance of this model with a wide range of nested alternative time-varying and constant parameter VAR models. Our results indicate that our proposed framework outperforms its competitors in terms of point and density forecasts. A portfolio allocation exercise confirms the superiority of our proposed model. In addition, a copula-based analysis shows that it pays off to adopt a multivariate modeling framework during periods of stress, like the recent financial crisis.

Forecasting with a Panel Tobit Model

We use a dynamic panel Tobit model with heteroskedasticity to generate point, set, and density forecasts for a large cross-section of short time series of censored observations. Our fully Bayesian approach allows us to flexibly estimate the cross-sectional distribution of heterogeneous coefficients and then implicitly use this distribution as prior to construct Bayes forecasts for the individual time series. We construct set forecasts that explicitly target the average coverage probability for the cross-section. We present a novel application in which we forecast bank-level charge-off rates for credit card and residential real estate loans, comparing various versions of the panel Tobit model.

From Fixed-Event to Fixed-Horizon Density Forecasts: Obtaining Measures of Multi-Horizon Uncertainty From Survey Density Forecasts

Surveys of professional forecasters produce precise and timely point forecasts for key macroeconomic variables. However, the accompanying density forecasts are not as widely utilized, and there is no consensus about their quality. This is partly because such surveys are often conducted for events. For example, in each quarter, panelists are asked to forecast output growth and inflation for the current calendar year and the next, implying that the forecast horizon changes with each survey round. The fixed-event nature limits the usefulness of survey density predictions for policymakers and market participants, who often wish to characterize uncertainty a fixed number of periods ahead (fixed-horizon). Is it possible to obtain fixed-horizon density forecasts using the available fixed-event ones? We propose a density combination approach that weights fixed-event density forecasts according to a uniformity of the probability integral transform criterion, aiming at obtaining a correctly calibrated fixed-horizon density forecast. Using data from the US Survey of Professional Forecasters, we show that our combination method produces competitive density forecasts relative to widely used alternatives based on historical forecast errors or Bayesian VARs. Thus, our proposed fixed-horizon predictive densities are a new and useful tool for researchers and policymakers.

Density Forecasts in Panel Data Models: A Semiparametric Bayesian Perspective

This paper constructs individual-specific density forecasts for a panel of firms or households using a dynamic linear model with common and heterogeneous coefficients and cross-sectional heteroskedasticity. The panel considered in this paper features a large cross-sectional dimension N but short time series T. Due to the short T, traditional methods have difficulty in disentangling the heterogeneous parameters from the shocks, which contaminates the estimates of the heterogeneous parameters. To tackle this problem, I assume that there is an underlying distribution of heterogeneous parameters, model this distribution nonparametrically allowing for correlation between heterogeneous parameters and initial conditions as well as individual-specific regressors, and then estimate this distribution by pooling the information from the whole cross-section together. Theoretically, I prove that both the estimated common parameters and the estimated distribution of the heterogeneous parameters achieve posterior consistency, and that the density forecasts asymptotically converge to the oracle forecast. Methodologically, I develop a simulation-based posterior sampling algorithm specifically addressing the nonparametric density estimation of unobserved heterogeneous parameters. Monte Carlo simulations and an application to young firm dynamics demonstrate improvements in density forecasts relative to alternative approaches.

Weigh(T)Ing the Basket: Aggregate and Component-Based Inflation Forecasts for the Euro Area

We compare direct forecasts of HICP and HICP excluding energy and food in the euro area and five member countries to aggregated forecasts of their main components from large Bayesian VARs with a shared set of predictors. We focus on conditional point and density forecasts, in line with forecasting practices at many policy institutions. Our main findings are that point forecasts perform similarly using both approaches, whereas directly forecasting aggregate indices tends to yield better density forecasts. In the aftermath of the Great Financial Crisis, relative forecasting performance was typically only affected temporarily. Inflation forecasts made by Eurosystem/ECB staff perform similarly or slightly better than those from our models for the euro area. JEL Classification: C11, C32, C53, E37

Density Forecast Combinations: The Real-Time Dimension

Density forecast combinations are examined in real-time using the log score to compare five methods: fixed weights, static and dynamic prediction pools, as well as Bayesian and dynamic model averaging. Since real-time data involves one vintage per time period and are subject to revisions, the chosen actuals for such comparisons typically differ from the information that can be used to compute model weights. The terms observation lag and information lag are introduced to clarify the different time shifts involved for these computations and we discuss how they influence the combination methods. We also introduce upper and lower bounds for the density forecasts, allowing us to benchmark the combination methods. The empirical study employs three DSGE models and two BVARs, where the former are variants of the Smets and Wouters model and the latter are benchmarks. The models are estimated on real-time euro area data and the forecasts cover 2001–2014, focusing on inflation and output growth. We find that some combinations are superior to the individual models for the joint and the output forecasts, mainly due to over-confident forecasts of the BVARs during the Great Recession. Combinations with limited weight variation over time and with positive weights on all models provide better forecasts than those with greater weight variation. For the inflation forecasts, the DSGE models are better overall than the BVARs and the combination methods. JEL Classification: C11, C32, C52, C53, E37

Proper Scoring Rules for Evaluating Asymmetry in Density Forecasting

This paper proposes a novel asymmetric continuous probabilistic score (ACPS) for evaluating and comparing density forecasts. It extends the proposed score and defines a weighted version, which emphasizes regions of interest, such as the tails or the center of a variable’s range. The ACPS is of general use in any situation where the decision maker has asymmetric preferences in the evaluation of the forecasts. In an artificial experiment, the implications of varying the level of asymmetry are illustrated. Then, the proposed score is applied to assess and compare density forecasts of macroeconomic relevant datasets (unemployment rate) and of commodity prices (oil and electricity prices) with a particular focus on the recent COVID crisis period.

Does the Phillips Curve Help to Forecast Euro Area Inflation?

We find that it does, but choosing the right specification is not trivial. We unveil notable model instability, with breaks in the performance of most simple Phillips curves. Euro area inflation was particularly hard to forecast in the run-up to the EMU and after the sovereign debt crisis, when the trend and for the latter period, also the amount of slack, were harder to pin down. Yet, some specifications outperform a univariate benchmark most of the time and are thus a useful element in a forecaster's toolkit. We base these conclusions on an extensive forecast evaluation over 1994 - 2018, an extraordinarily long period by euro area standards. We complement the analysis using real-time data over 2005-2018. As lessons for practitioners, we find that: (i) the key type of time variation to consider is an inflation trend; (ii) a simple filter-based output gap works well overall as a measure of economic slack, but after the Great Recession it is outperformed by endogenously estimated slack or by estimates from international economic institutions; (iii) external variables do not bring forecast gains; (iv) newer generation Phillips curve models with several time-varying features are a promising avenue for forecasting, especially when density forecasts are of interest, and finally, (v) averaging over a wide range of modelling choices offers some hedge against breaks in forecast performance.

Bayesian VAR Forecasts, Survey Information and Structural Change in the Euro Area

We incorporate external information extracted from the European Central Bank’s Survey of Professional Forecasters into the predictions of a Bayesian VAR, using entropic tilting and soft conditioning. The resulting conditional forecasts significantly improve the plain BVAR point and density forecasts. Importantly, we do not restrict the forecasts at a specific quarterly horizon, but their possible paths over several horizons jointly, as the survey information comes in the form of one- and two-year-ahead expectations. Besides improving the accuracy of the variable that we target, the spillover effects on “other-than-targeted” variables are relevant in size and statistically significant. We document that the baseline BVAR exhibits an upward bias for GDP growth after the financial crisis, and our results provide evidence that survey forecasts can help mitigate the effects of structural breaks on the forecasting performance of a popular macroeconometric model. Furthermore, we provide evidence of unstable VAR dynamics, especially during and after the recent Great Recession.

Perceived Uncertainty Shocks, Excess Optimism-Pessimism, and Learning in the Business Cycle

What are the effects of beliefs, sentiment, and uncertainty, over the business cycle? To answer this question, we develop a behavioral New Keynesian macroeconomic model, in which we relax the assumption of rational expectations. Agents are, instead, boundedly rational: they have a finite-planning horizon, and they learn about the economy over time. Moreover, we allow agents to have a potentially asymmetric loss function in forecasting, which creates a direct channel for expected variances to affect the economy. In forming expectations, agents may be subject to shifts in optimism and pessimism (sentiment) and their beliefs may be influenced by their perceptions about future uncertainty. We estimate the behavioral model using Bayesian methods and exploit a large number of subjective expectation series (both point and density forecasts) at different horizons from the Survey of Professional Forecasters. We find that sentiment shocks are the key source of business cycle fluctuations. Shifts in perceived uncertainty can also affect real activity and inflation through a confidence channel, as they play an important role in belief formation. Overall, the results shed light on the importance of behavioral forces over the business cycles, and on the contribution and interaction of first-moment - sentiment - shocks versus second-moment - perceived uncertainty - shocks.

No-Arbitrage Priors, Drifting Volatilities, and the Term Structure of Interest Rates

We propose a method to produce density forecasts of the term structure of government bond yields that accounts for (i) the possible mispecification of an underlying Gaussian Affine Term Structure Model (GATSM) and (ii) the time varying volatility of interest rates. For this, we derive a Bayesian prior from a GATSM and use it to estimate the coefficients of a BVAR for the term structure, specifying a common, multiplicative, time varying volatility for the VAR disturbances. Results based on U.S. data show that this method significantly improves the precision of point and density forecasts of the term structure. While this paper focuses on term structure modelling, the proposed method can be applied for a wide range of alternative models, including DSGE models, and is a generalization of the method of Del Negro and Schorfheide (2004) to VARs featuring drifting volatilities. The method also generalizes the model of Giannone et al. (2012), by specifying hierarchically not only the prior variance but also the prior mean of the VAR coefficients. Our results show that both time variation in volatilities, and a hierarchical specification for the prior means, improve model fit and forecasting performance.

Time-Varying Trend Models for Forecasting Inflation in Australia

We investigate whether a class of trend models with various error term structures can improve upon the forecast performance of commonly used time series models when forecasting CPI inflation in Australia. The main result is that trend models tend to provide more accurate point and density forecasts compared to conventional autoregressive and Phillips curve models. The best short-term forecasts come from a trend model with stochastic volatility in the transitory component, while medium to long-run forecasts are better made by specifying a moving average component. We also find that trend models can capture various dynamics in periods of significance which conventional models cannot. This includes the dramatic reduction in inflation when the RBA adopted inflation targeting, the one-off 10 per cent Goods and Services Tax inflationary episode in 2000, and the gradual decline in inflation since 2014.

Forecasting with Bayesian Grouped Random Effects in Panel Data

In this paper, we estimate and leverage latent constant group structure to generate the point, set, and density forecasts for short dynamic panel data. We implement a nonparametric Bayesian approach to simultaneously identify coefficients and group membership in the random effects which are heterogeneous across groups but fixed within a group. This method allows us to flexibly incorporate subjective prior knowledge on the group structure that potentially improves the predictive accuracy. In Monte Carlo experiments, we demonstrate that our Bayesian grouped random effects (BGRE) estimators produce accurate estimates and score predictive gains over standard panel data estimators. With a data-driven group structure, the BGRE estimators exhibit comparable accuracy of clustering with the Kmeans algorithm and outperform a two-step Bayesian grouped estimator whose group structure relies on Kmeans. In the empirical analysis, we apply our method to forecast the investment rate across a broad range of firms and illustrate that the estimated latent group structure facilitates forecasts relative to standard panel data estimators.

Where does the tail begin? An approach based on scoring rules

Learning about the tail shape of time series is important in, e.g., economics, finance, and risk management. However, it is well known that estimates of the tail index can be very sensitive to the choice of the number k of tail observations used for estimation. We propose a procedure that determines where the tail begins by choosing k in a data-driven fashion using scoring rules. So far, scoring rules have mainly been used to compare density forecasts. We also demonstrate how our proposal can be used in multivariate applications in the system risk literature. The advantages of our choice of k are illustrated in simulations and an empirical application to Value-at-Risk forecasts for five U.S. blue-chip stocks.

Forecasting Hourly Power Load Considering Time Division: A Hybrid Model Based on K-means Clustering and Probability Density Forecasting Techniques

In comparison with traditional point forecasting method, probability density forecasting can reflect the load fluctuation more effectively and provides more information. This paper proposes a hybrid hourly power load forecasting model, which integrates K-means clustering algorithm, Salp Swarm Algorithm (SSA), Least Square Support Vector Machine (LSSVM), and kernel density estimation (KDE) method. Firstly, the loads at 24 times a day are grouped into three categories according to the K-means clustering algorithm, which correspond to the valley period, flat period, and peak period of the load, respectively. Secondly, the load point forecasting value is obtained by LSSVM method optimized by SSA algorithm. Furthermore, the kernel density estimation method is employed to fit the forecasting error of SSA-LSSVM in different time periods, and the probability density function of the error distribution is obtained. The final load probability density forecasting result is obtained by combining the point forecasting value and the error fitting result, and then the upper and lower limits of the confidence interval under the given confidence level are solved. In this paper, the performance of the model is evaluated by two indicators named interval coverage and interval average width. Meanwhile, in comparison with several other models, it can be concluded that the proposed model can effectively improve the forecasting effect.

Model instability in predictive exchange rate regressions.

In this paper we aim to improve existing empirical exchange rate models by accounting for uncertainty with respect to the underlying structural representation. Within a flexible Bayesian framework, our modeling approach assumes that different regimes are characterized by commonly used structural exchange rate models, with transitions across regimes being driven by a Markov process. We assume a time‐varying transition probability matrix with transition probabilities depending on a measure of the monetary policy stance of the central bank at home and in the USA. We apply this model to a set of eight exchange rates against the US dollar. In a forecasting exercise, we show that model evidence varies over time, and a model approach that takes this empirical evidence seriously yields more accurate density forecasts for most currency pairs considered.

Oil price shocks and economic growth: The volatility link

This paper shows that oil shocks impact economic growth primarily through the conditional variance of growth. Our comparison of models focuses on density forecasts. Over a range of dynamic models, oil shock measures and data, we find a robust link between oil shocks and the volatility of economic growth. We then develop a new measure of oil shocks and show that it is superior to existing measures; it indicates that the conditional variance of growth increases in response to an indicator of the local maximum oil price exceedance. The empirical results uncover a large pronounced asymmetric response of the growth volatility to oil price changes. The uncertainty about future growth is considerably lower than with a benchmark AR(1) model when no oil shocks are present.