EVALUATING REAL‐TIME VAR FORECASTS WITH AN INFORMATIVE DEMOCRATIC PRIOR

This paper proposes Bayesian forecasting in a vector autoregression using a democratic prior. This prior is chosen to match the predictions of survey respondents. In particular, the unconditional mean for each series in the vector autoregression is centered around long‐horizon survey forecasts. Heavy shrinkage toward the democratic prior is found to give good real‐time predictions of a range of macroeconomic variables, as these survey projections are good at quickly capturing endpoint shifts. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Bayesian Model Averaging and exchange rate forecasts

Exchange rate forecasting is hard and the seminal result of Meese and Rogoff [Meese, R., Rogoff, K., 1983. Empirical exchange rate models of the seventies: Do they fit out of sample? Journal of International Economics 14, 3–24] that the exchange rate is well approximated by a driftless random walk, at least for prediction purposes, still stands despite much effort at constructing other forecasting models. However, in several other macro and financial forecasting applications, researchers in recent years have considered methods for forecasting that effectively combine the information in a large number of time series. In this paper, I apply one such method for pooling forecasts from several different models, Bayesian Model Averaging, to the problem of pseudo out-of-sample exchange rate predictions. For most currency–horizon pairs, the Bayesian Model Averaging forecasts using a sufficiently high degree of shrinkage, give slightly smaller out-of-sample mean square prediction error than the random walk benchmark. The forecasts generated by this model averaging methodology are however very close to, but not identical to, those from the random walk forecast.     

Reconciled boosted models for GEFCom2017 hierarchical probabilistic load forecasting

When forecasting time series in a hierarchical configuration, it is necessary to ensure that the forecasts reconcile at all levels. The 2017 Global Energy Forecasting Competition (GEFCom2017) focused on addressing this topic. Quantile forecasts for eight zones and two aggregated zones in New England were required for every hour of a future month. This paper presents a new methodology for forecasting quantiles in a hierarchy which outperforms a commonly-used benchmark model. A simulation-based approach was used to generate demand forecasts. Adjustments were made to each of the demand simulations to ensure that all zonal forecasts reconciled appropriately, and a weighted reconciliation approach was implemented to ensure that the bottom-level zonal forecasts summed correctly to the aggregated zonal forecasts. We show that reconciling in this manner improves the forecast accuracy. A discussion of the results and modelling performances is presented, and brief reviews of hierarchical time series forecasting and gradient boosting are also included.     

Forecasting crude oil prices with DSGE models

In this study, we conducted an oil prices forecasting competition among a set of structural models, including vector autoregression and dynamic stochastic general equilibrium (DSGE) models. Our results highlight two principles. First, forecasts should exploit the fact that real oil prices are mean reverting over long horizons. Second, models should not replicate the high volatility of the oil prices observed in samples. By following these principles, we show that an oil sector DSGE model performs much better at real oil price forecasting than random walk or vector autoregression.     

Modelling and Forecasting Fiscal Variables for the Euro Area*

In this paper, we assess the possibility of producing unbiased forecasts for fiscal variables in the Euro area by comparing a set of procedures that rely on different information sets and econometric techniques. In particular, we consider autoregressive moving average models, Vector autoregressions, small‐scale semistructural models at the national and Euro area level, institutional forecasts (Organization for Economic Co‐operation and Development), and pooling. Our small‐scale models are characterized by the joint modelling of fiscal and monetary policy using simple rules, combined with equations for the evolution of all the relevant fundamentals for the Maastricht Treaty and the Stability and Growth Pact. We rank models on the basis of their forecasting performance using the mean square and mean absolute error criteria at different horizons. Overall, simple time‐series methods and pooling work well and are able to deliver unbiased forecasts, or slightly upward‐biased forecast for the debt–GDP dynamics. This result is mostly due to the short sample available, the robustness of simple methods to structural breaks, and to the difficulty of modelling the joint behaviour of several variables in a period of substantial institutional and economic changes. A bootstrap experiment highlights that, even when the data are generated using the estimated small‐scale multi‐country model, simple time‐series models can produce more accurate forecasts, because of their parsimonious specification.     

Content horizons for univariate time-series forecasts

This paper investigates the maximum horizon at which conditioning information can be shown to have value for univariate time series forecasts. In particular, we consider the problem of determining the horizon beyond which forecasts from univariate time series models of stationary processes add nothing to the forecast implicit in the unconditional mean. We refer to this as the content horizon for forecasts, and provide a formal definition of the corresponding forecast content function at horizons s=1,… S. This function depends upon parameter estimation uncertainty as well as on autocorrelation structure of the process. We show that for autoregressive processes it is possible to give an asymptotic expression for the forecast content function, and show by simulation that the expression gives a good approximation even at modest sample sizes. The results are applied to the growth rate of GDP and to inflation, using US and Canadian data.     

FRED-SD: A real-time database for state-level data with forecasting applications

We construct a real-time dataset (FRED-SD) with vintage data for the U.S. states that can be used to forecast both state-level and national-level variables. Our dataset includes approximately 28 variables per state, including labor-market, production, and housing variables. We conduct two sets of real-time forecasting exercises. The first forecasts state-level labor-market variables using five different models and different levels of industrially disaggregated data. The second forecasts a national-level variable exploiting the cross-section of state data. The state-forecasting experiments suggest that large models with industrially disaggregated data tend to have higher predictive ability for industrially diversified states. For national-level data, we find that forecasting and aggregating state-level data can outperform a random walk but not an autoregression. We compare these real-time data experiments with forecasting experiments using final-vintage data and find very different results. Because these final-vintage results are obtained with revised data that would not have been available at the time the forecasts would have been made, we conclude that the use of real-time data is essential for drawing proper conclusions about state-level forecasting models.     

Forecasting large datasets with Bayesian reduced rank multivariate models

The paper addresses the issue of forecasting a large set of variables using multivariate models. In particular, we propose three alternative reduced rank forecasting models and compare their predictive performance for US time series with the most promising existing alternatives, namely, factor models, large‐scale Bayesian VARs, and multivariate boosting. Specifically, we focus on classical reduced rank regression, a two‐step procedure that applies, in turn, shrinkage and reduced rank restrictions, and the reduced rank Bayesian VAR of Geweke ( 1996 ). We find that using shrinkage and rank reduction in combination rather than separately improves substantially the accuracy of forecasts, both when the whole set of variables is to be forecast and for key variables such as industrial production growth, inflation, and the federal funds rate. The robustness of this finding is confirmed by a Monte Carlo experiment based on bootstrapped data. We also provide a consistency result for the reduced rank regression valid when the dimension of the system tends to infinity, which opens the way to using large‐scale reduced rank models for empirical analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

REAL‐TIME FORECASTING OF INFLATION AND OUTPUT GROWTH WITH AUTOREGRESSIVE MODELS IN THE PRESENCE OF DATA REVISIONS

We examine how the accuracy of real‐time forecasts from models that include autoregressive terms can be improved by estimating the models on ‘lightly revised’ data instead of using data from the latest‐available vintage. The benefits of estimating autoregressive models on lightly revised data are related to the nature of the data revision process and the underlying process for the true values. Empirically, we find improvements in root mean square forecasting error of 2–4% when forecasting output growth and inflation with univariate models, and of 8% with multivariate models. We show that multiple‐vintage models, which explicitly model data revisions, require large estimation samples to deliver competitive forecasts. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

A Dynamic Panel Data Approach to the Forecasting of the GDP of German Länder

Abstract

In this paper, we make multi-step forecasts of the annual growth rates of the real GDP for each of the 16 German Länder simultaneously. We apply dynamic panel models accounting for spatial dependence between regional GDP. We find that both pooling and accounting for spatial effects help to improve the forecast performance substantially. We demonstrate that the effect of accounting for spatial dependence is more pronounced for longer forecasting horizons (the forecast accuracy gain is about 9% for a 1-year horizon and exceeds 40% for a 5-year horizon). We recommend incorporating a spatial dependence structure into regional forecasting models, especially when long-term forecasts are made.     

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.     

Macroeconomic Forecasting Performance under Alternative Specifications of Time‐Varying Volatility

This paper compares alternative models of time‐varying volatility on the basis of the accuracy of real‐time point and density forecasts of key macroeconomic time series for the USA. We consider Bayesian autoregressive and vector autoregressive models that incorporate some form of time‐varying volatility, precisely random walk stochastic volatility, stochastic volatility following a stationary AR process, stochastic volatility coupled with fat tails, GARCH and mixture of innovation models. The results show that the AR and VAR specifications with conventional stochastic volatility dominate other volatility specifications, in terms of point forecasting to some degree and density forecasting to a greater degree. Copyright © 2014 John Wiley & Sons, Ltd.     

DeepAR: Probabilistic forecasting with autoregressive recurrent networks

Probabilistic forecasting, i.e., estimating a time series’ future probability distribution given its past, is a key enabler for optimizing business processes. In retail businesses, for example, probabilistic demand forecasts are crucial for having the right inventory available at the right time and in the right place. This paper proposes DeepAR, a methodology for producing accurate probabilistic forecasts, based on training an autoregressive recurrent neural network model on a large number of related time series. We demonstrate how the application of deep learning techniques to forecasting can overcome many of the challenges that are faced by widely-used classical approaches to the problem. By means of extensive empirical evaluations on several real-world forecasting datasets, we show that our methodology produces more accurate forecasts than other state-of-the-art methods, while requiring minimal manual work.     

Forecasting using supervised factor models

This paper examines the theoretical and empirical properties of a supervised factor model based on combining forecasts using principal components (CFPC), in comparison with two other supervised factor models (partial least squares regression, PLS, and principal covariate regression, PCovR) and with the unsupervised principal component regression, PCR. The supervision refers to training the predictors for a variable to forecast. We compare the performance of the three supervised factor models and the unsupervised factor model in forecasting of U.S. CPI inflation. The main finding is that the predictive ability of the supervised factor models is much better than the unsupervised factor model. The computation of the factors can be doubly supervised together with variable selection, which can further improve the forecasting performance of the supervised factor models. Among the three supervised factor models, the CFPC best performs and is also most stable. While PCovR also performs well and is stable, the performance of PLS is less stable over different out-of-sample forecasting periods. The effect of supervision gets even larger as forecast horizon increases. Supervision helps to reduce the number of factors and lags needed in modelling economic structure, achieving more parsimony.     

Order-invariant tests for proper calibration of multivariate density forecasts

Established tests for proper calibration of multivariate density forecasts based on Rosenblatt probability integral transforms can be manipulated by changing the order of variables in the forecasting model. We derive order-invariant tests. The new tests are applicable to densities of arbitrary dimensions and can deal with parameter estimation uncertainty and dynamic misspecification. Monte Carlo simulations show that they often have superior power relative to established approaches. We use the tests to evaluate generalized autoregressive conditional heteroskedasticity-based multivariate density forecasts for a vector of stock market returns and macroeconomic forecasts from a Bayesian vector autoregression with time-varying parameters.     

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.     

Online adaptive lasso estimation in vector autoregressive models for high dimensional wind power forecasting

Wind power forecasts with lead times of up to a few hours are essential to the optimal and economical operation of power systems and markets. Vector autoregression (VAR) is a framework that has been shown to be well suited to predicting for several wind farms simultaneously by considering the spatio-temporal dependencies in their time series. Lasso penalisation yields sparse models and can avoid overfitting the large numbers of coefficients in higher dimensional settings. However, estimation in VAR models usually does not account for changes in the spatio-temporal wind power dynamics that are related to factors such as seasons or wind farm setup changes, for example. This paper tackles this problem by proposing a time-adaptive lasso estimator and an efficient coordinate descent algorithm for updating the VAR model parameters recursively online. The approach shows good abilities to track changes in the multivariate time series dynamics on simulated data. Furthermore, in two case studies it shows clearly better predictive performances than the non-adaptive lasso VAR and univariate autoregression.     

Macroeconomic forecasting and structural change

The aim of this paper is to assess whether modeling structural change can help improving the accuracy of macroeconomic forecasts. We conduct a simulated real‐time out‐of‐sample exercise using a time‐varying coefficients vector autoregression (VAR) with stochastic volatility to predict the inflation rate, unemployment rate and interest rate in the USA. The model generates accurate predictions for the three variables. In particular, the forecasts of inflation are much more accurate than those obtained with any other competing model, including fixed coefficients VARs, time‐varying autoregressions and the naïve random walk model. The results hold true also after the mid 1980s, a period in which forecasting inflation was particularly hard. Copyright © 2011 John Wiley & Sons, Ltd.