Comparing the forecasting performances of linear models for electricity prices with high RES penetration

We compare alternative univariate versus multivariate models and frequentist versus Bayesian autoregressive and vector autoregressive specifications for hourly day-ahead electricity prices, both with and without renewable energy sources. The accuracy of point and density forecasts is inspected in four main European markets (Germany, Denmark, Italy, and Spain) characterized by different levels of renewable energy power generation. Our results show that the Bayesian vector autoregressive specifications with exogenous variables dominate other multivariate and univariate specifications in terms of both point forecasting and density forecasting.     

Comparing density forecasts in a risk management context

We compare multivariate and univariate approaches to assessing the accuracy of competing density forecasts of a portfolio return in the downside part of the support. We argue that the common practice of performing multivariate forecast comparisons can be problematic in the context of assessing portfolio risk, since better multivariate forecasts do not necessarily correspond to better aggregate portfolio return forecasts. This is illustrated by examples that involve (skew) elliptical distributions and an application to daily returns of a number of US stock prices. In addition, time-varying test statistics and Value-at-Risk forecasts provide empirical evidence of regime changes over the last decades.     

Theory-coherent forecasting

We consider a method for producing multivariate density forecasts that satisfy moment restrictions implied by economic theory, such as Euler conditions. The method starts from a base forecast that might not satisfy the theoretical restrictions and forces it to satisfy the moment conditions using exponential tilting. Although exponential tilting has been considered before in a Bayesian context (Robertson et al. 2005), our main contributions are: (1) to adapt the method to a classical inferential context with out-of-sample evaluation objectives and parameter estimation uncertainty; and (2) to formally discuss the conditions under which the method delivers improvements in forecast accuracy. An empirical illustration which incorporates Euler conditions into forecasts produced by Bayesian vector autoregressions shows that the improvements in accuracy can be sizable and significant.     

Evaluating DSGE model forecasts of comovements

This paper develops and applies tools to assess multivariate aspects of Bayesian Dynamic Stochastic General Equilibrium (DSGE) model forecasts and their ability to predict comovements among key macroeconomic variables. We construct posterior predictive checks to evaluate conditional and unconditional density forecasts, in addition to checks for root-mean-squared errors and event probabilities associated with these forecasts. The checks are implemented on a three-equation DSGE model as well as the Smets and Wouters (2007) model using real-time data. We find that the additional features incorporated into the Smets–Wouters model do not lead to a uniform improvement in the quality of density forecasts and prediction of comovements of output, inflation, and interest rates.     

Forecasting with Global Vector Autoregressive Models: a Bayesian Approach

This paper develops a Bayesian variant of global vector autoregressive (B‐GVAR) models to forecast an international set of macroeconomic and financial variables. We propose a set of hierarchical priors and compare the predictive performance of B‐GVAR models in terms of point and density forecasts for one‐quarter‐ahead and four‐quarter‐ahead forecast horizons. We find that forecasts can be improved by employing a global framework and hierarchical priors which induce country‐specific degrees of shrinkage on the coefficients of the GVAR model. Forecasts from various B‐GVAR specifications tend to outperform forecasts from a naive univariate model, a global model without shrinkage on the parameters and country‐specific vector autoregressions. Copyright © 2016 John Wiley & Sons, Ltd.     

Likelihood-based scoring rules for comparing density forecasts in tails

We propose new scoring rules based on conditional and censored likelihood for assessing the predictive accuracy of competing density forecasts over a specific region of interest, such as the left tail in financial risk management. These scoring rules can be interpreted in terms of Kullback-Leibler divergence between weighted versions of the density forecast and the true density. Existing scoring rules based on weighted likelihood favor density forecasts with more probability mass in the given region, rendering predictive accuracy tests biased toward such densities. Using our novel likelihood-based scoring rules avoids this problem.     

Marginalized Predictive Likelihood Comparisons of Linear Gaussian State‐Space Models with Applications to DSGE,DSGE‐VAR,and VAR Models

The predictive likelihood is useful for ranking models in forecast comparison exercises using Bayesian inference. We discuss how it can be estimated, by means of marzginalization, for any subset of the observables in linear Gaussian state‐space models. We compare macroeconomic density forecasts for the euro area of a DSGE model to those of a DSGE‐VAR, a BVAR and a multivariate random walk over 1999:Q1–2011:Q4. While the BVAR generally provides superior forecasts, its performance deteriorates substantially with the onset of the Great Recession. This is particularly notable for longer‐horizon real GDP forecasts, where the DSGE and DSGE‐VAR models perform better. Copyright © 2016 John Wiley & Sons, Ltd.     

Macroeconomic forecasting with a vector arima model: A case study of the finnish economy

The vector ARIMA (VARIMA) model is a multivariate generalization of the univariate ARIMA model. VARIMA can accomodate assumptions on exogeneity and on contemporaneous relationships. Exogeneous forecasts and non-zero future shocks make it possible to generate alternative forecasts. In a case study VARIMA well describes developments in the 1970's and successfully competes with judgemental methods and ARIMA in providing a general outlook of the early 1980's.     

A bootstrap-based non-parametric forecast density

Interest in density forecasts (as opposed to solely modeling the conditional mean) arises from the possibility of dynamics in higher moments of a time series, as well as in forecasting the probability of future events in some applications. By combining the idea of Markov bootstrapping with that of kernel density estimation, this paper presents a simple non-parametric method for estimating out-of-sample multi-step density forecasts. The paper also considers a host of evaluation tests for examining the dynamic misspecification of estimated density forecasts by targeting autocorrelation, heteroskedasticity and neglected non-linearity. These tests are useful, as a rejection of the tests gives insight into ways to improve a particular forecasting model. In an extensive Monte Carlo analysis involving a range of commonly used linear and non-linear time series processes, the non-parametric method is shown to work reasonably well across the simulated models for a suitable choice of the bandwidth (smoothing parameter). Furthermore, an application of the method to the U.S. Industrial Production series provides multi-step density forecasts that show no sign of dynamic misspecification.     

Efficient estimation of Bayesian VARMAs with time‐varying coefficients

Empirical work in macroeconometrics has been mostly restricted to using vector autoregressions (VARs), even though there are strong theoretical reasons to consider general vector autoregressive moving averages (VARMAs). A number of articles in the last two decades have conjectured that this is because estimation of VARMAs is perceived to be challenging and proposed various ways to simplify it. Nevertheless, VARMAs continue to be largely dominated by VARs, particularly in terms of developing useful extensions. We address these computational challenges with a Bayesian approach. Specifically, we develop a Gibbs sampler for the basic VARMA, and demonstrate how it can be extended to models with time‐varying vector moving average (VMA) coefficients and stochastic volatility. We illustrate the methodology through a macroeconomic forecasting exercise. We show that in a class of models with stochastic volatility, VARMAs produce better density forecasts than VARs, particularly for short forecast horizons.     

Bayesian semiparametric multivariate GARCH modeling

This paper proposes a Bayesian nonparametric modeling approach for the return distribution in multivariate GARCH models. In contrast to the parametric literature the return distribution can display general forms of asymmetry and thick tails. An infinite mixture of multivariate normals is given a flexible Dirichlet process prior. The GARCH functional form enters into each of the components of this mixture. We discuss conjugate methods that allow for scale mixtures and nonconjugate methods which provide mixing over both the location and scale of the normal components. MCMC methods are introduced for posterior simulation and computation of the predictive density. Bayes factors and density forecasts with comparisons to GARCH models with Student-t$t$ innovations demonstrate the gains from our flexible modeling approach.     

Conditional value-at-risk forecasts of an optimal foreign currency portfolio

This study provides daily conditional value-at-risk (C-VaR) forecasts for a foreign currency portfolio comprising the USD/EUR, USD/JPY, and USD/BRL currencies. To do so, we estimate multivariate stochastic volatility models with time-varying conditional correlations using a Bayesian Markov chain Monte Carlo algorithm. Then, given the model-specific currency return density forecasts, we make the optimal portfolio choice by minimizing the C-VaR through numerical optimization. According to out-of-sample experiment, including emerging markets into the currency basket is essential for downside risk management, and considering model uncertainty as well as the parameter uncertainty can improve the portfolio performance.     

Forecasting performance of multivariate time series models with full and reduced rank: an empirical examination

Since Quenouille's influential work on multiple time series, much progress has been made towards the goal of parameter reduction and model fit. Relatively less attention has been paid to the systematic evaluation of out-of-sample forecast performance of multivariate time series models. In this paper, we update the hog data set studied by Quenouille (and other researchers who followed him). We re-estimate his model with extended observations (1867–1966), and generate recursive one- to four-steps-ahead forecasts for the period of 1967 through 2000. These forecasts are compared to forecasts from an unrestricted vector autoregression, a reduced rank regression model, an index model and a cointegration-based error correction model. The error correction model that takes into account both nonstationarity of the data and rank reduction performs best at all four forecasting horizons. However, differences among competing models are statistically insignificant in most cases. No model consistently encompasses the others at all four horizons.     

Evaluating Point and Density Forecasts of DSGE Models

This paper investigates the accuracy of forecasts from four dynamic stochastic general equilibrium (DSGE) models for inflation, output growth and the federal funds rate using a real‐time dataset synchronized with the Fed's Greenbook projections. Conditioning the model forecasts on the Greenbook nowcasts leads to forecasts that are as accurate as the Greenbook projections for output growth and the federal funds rate. Only for inflation are the model forecasts dominated by the Greenbook projections. A comparison with forecasts from Bayesian vector autoregressions shows that the economic structure of the DSGE models which is useful for the interpretation of forecasts does not lower the accuracy of forecasts. Combining forecasts of several DSGE models increases precision in comparison to individual model forecasts. Comparing density forecasts with the actual distribution of observations shows that DSGE models overestimate uncertainty around point forecasts. Copyright © 2013 John Wiley & Sons, Ltd.     

Local rank tests in a multivariate nonparametric relationship

Consider a multivariate nonparametric model where the unknown vector of functions depends on two sets of explanatory variables. For a fixed level of one set of explanatory variables, we provide consistent statistical tests, called local rank tests, to determine whether the multivariate relationship can be explained by a smaller number of functions. We also provide estimators for the smallest number of functions, called local rank, explaining the relationship. The local rank tests and the estimators of local rank are defined in terms of the eigenvalues of a kernel-based estimator of some matrix. The asymptotics of the eigenvalues is established by using the so-called Fujikoshi expansion along with some techniques of the theory of U-statistics. We present a simulation study which examines the small sample properties of local rank tests. We also apply the local rank tests and the local rank estimators to a demand system given by a newly constructed data set. This work can be viewed as a “local” extension of the tests for a number of factors in a nonparametric relationship introduced by Stephen Donald.     

Does judgment improve macroeconomic density forecasts?

This paper presents empirical evidence on how judgmental adjustments affect the accuracy of macroeconomic density forecasts. Judgment is defined as the difference between professional forecasters’ densities and the forecast densities from statistical models. Using entropic tilting, we evaluate whether judgments about the mean, variance and skew improve the accuracy of density forecasts for UK output growth and inflation. We find that not all judgmental adjustments help. Judgments about point forecasts tend to improve density forecast accuracy at short horizons and at times of heightened macroeconomic uncertainty. Judgments about the variance hinder at short horizons, but can improve tail risk forecasts at longer horizons. Judgments about skew in general take value away, with gains seen only for longer horizon output growth forecasts when statistical models took longer to learn that downside risks had reduced with the end of the Great Recession. Overall, density forecasts from statistical models prove hard to beat.     

Forecasting cryptocurrencies under model and parameter instability

This paper studies the predictability of cryptocurrency time series. We compare several alternative univariate and multivariate models for point and density forecasting of four of the most capitalized series: Bitcoin, Litecoin, Ripple and Ethereum. We apply a set of crypto-predictors and rely on dynamic model averaging to combine a large set of univariate dynamic linear models and several multivariate vector autoregressive models with different forms of time variation. We find statistically significant improvements in point forecasting when using combinations of univariate models, and in density forecasting when relying on the selection of multivariate models. Both schemes deliver sizable directional predictability.     

Conditional predictive density evaluation in the presence of instabilities

We propose new methods for evaluating predictive densities. The methods include Kolmogorov–Smirnov and Cramér–von Mises-type tests for the correct specification of predictive densities robust to dynamic mis-specification. The novelty is that the tests can detect mis-specification in the predictive densities even if it appears only over a fraction of the sample, due to the presence of instabilities. Our results indicate that our tests are well sized and have good power in detecting mis-specification in predictive densities, even when it is time-varying. An application to density forecasts of the Survey of Professional Forecasters demonstrates the usefulness of the proposed methodologies.     

Evaluating quantile-bounded and expectile-bounded interval forecasts

In many different contexts, decision-making is improved by the availability of probabilistic predictions. The accuracy of probabilistic forecasting methods can be compared using scoring functions and insight provided by calibration tests. These tests evaluate the consistency of predictions with the observations. Our main agenda in this paper is interval forecasts and their evaluation. Such forecasts are usually bounded by two quantile forecasts. However, a limitation of quantiles is that they convey no information regarding the size of potential exceedances. By contrast, the location of an expectile is dictated by the whole distribution. This prompts us to propose expectile-bounded intervals. We provide interpretation, a consistent scoring function and a calibration test. Before doing this, we reflect on the evaluation of forecasts of quantile-bounded intervals and expectiles, and suggest extensions of previously proposed calibration tests in order to guard against strategic forecasting. We illustrate ideas using day-ahead electricity price forecasting.     

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.