Density nowcasts of euro area real GDP growth: does pooling matter?

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1 Density nowcasts of euro area real GDP growth: does pooling matter? Marta Bańbura 1 and Lorena Saiz 2 Abstract In this paper we review different strategies to combine and evaluate individual density forecasts from a suite of models typically used for nowcasting the euro area real GDP growth. The individual density now/forecasts are obtained by using Bayesian estimation combined with a simulation smoother. The densities are then combined using different approaches, ranging from linear and logarithmic pooling to a general non-linear pooling (i.e. beta-transform linear pooling), and with different weighting schemes: dynamic and static weights (e.g. optimal weights, equal weights, based on recursive log-score and continuous ranked probability score (CRPS)). Apart from relative accuracy of different combinations we also assess their calibration using the popular probability integral transform tests. Keywords: Nowcasting, probability forecasts, forecast combination. JEL classification: E32, E37, C53. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the European Central Bank. 1 European Central Bank; 2 European Central Bank ;

2 Extended abstract 1 Introduction In contrast to point forecasts, density forecasts provide estimates of the entire probability distribution of future outcomes. This can be useful to assess the degree of uncertainty surrounding the central forecast, probability of tail events, probability of entering into a recession, balance of risks associated with alternative forecast, etc. As in the point forecast case the rationale for combining predictive forecasts from several models could include insurance against structural breaks, model misspecification or uncertainty in model selection (Geweke and Amisano, 2011). In addition, combinations of normal densities give as a result a more flexible distribution, which can accommodate non-gaussian features (e.g. skewed distribution or fat tails). However, the combination of density forecasts does not necessarily lead to better accuracy compared to individual densities. This is because the resulting combined density tends to have a higher dispersion (Hall and Mitchell, 2009), in particular in the linear pooling case (Ranjan and Gneiting, 2010, 2013). In this paper we study this problem for a particular case of forecasting problem, namely the nowcasting. Nowcasting is forecasting with very short (occasionally negative) forecast horizons and requires modelling frameworks that are able to handle data at mixed frequency and released non-synchronously (see e.g. Bańbura, Giannone, and Reichlin, 2011; Banbura, Giannone, Modugno, and Reichlin, 2013, for a survey). Nowcasts are relevant as they are one of few systematically informative forecasts in macro (see e.g. Giannone, Reichlin, and Small, 2008). We focus on nowcasts for euro area real GDP growth. Previous work on predictive forecasts in the context of nowcasting includes e.g. Aastveit, Gerdrup, Jore, and Thorsrud (2014), Mazzi, Mitchell, and Montana (2014) or Proietti, Marczak, and Mazzi (2015). In this paper we consider somewhat different composition of models and more importantly we focus on the role of different pooling and weighting approaches and different evaluation methods. 2 Individual models We consider two classes of models: Mixed frequency dynamic factor models These are similar to the factor models for the euro area in Bańbura and Rünstler (2011). To account for parameter uncertainty we consider different approaches: re-sampling (Wall and Stoffer, 2002) and Bayesian estimation (Bai and Wang, 2015). Bridge equations In this type of model, the nowcasts are obtained for a regression of (quarterly) GDP on some (higher frequency) predictors aggregated to quarterly frequency. To handle the ragged edge problem auxiliary models, Bayesian AR or VAR or factor models are used to forecast the predictors to close the target period of interest. For the latter, given the draws of the parameters the forecasts are simulated using the simulation smoother of Durbin and Koopman (2002). For the (V)ARs we use the standard Minnesota priors (see e.g. Kadiyala and Karlsson, 1997; Bańbura, Giannone, and Reichlin, 2010). The bridge equations are estimated using the standard Bayesian approach. 2

3 3 Combination and evaluation of individual densities 3.1 Pooling approaches Various methods have been proposed for combining individual forecast densities. The most frequently used method is the linear opinion pooling which is weighted linear combination of individual probability forecasts (Stone, 1961). The resulting distribution is a mixture of individual distributions. Consequently, even if the individual distributions are Gaussian, this method can accommodate very well situations with departures of gaussianity and multi-modal distributions. However, dispersion tends to increase under linear aggregation, even when the individual densities are properly calibrated. The logarithmic opinion pool Winkler (1968) is typically uni-modal and less dispersed than the linear opinion pool. The Betatransformed linear pool (Ranjan and Gneiting, 2010) combines the traditional linear opinion pool with a beta transform to non-linearly recalibrate the linear pool. 3.2 Weighting schemes The weighting scheme is also important for the characteristics of the combined density. The weights are typically based on past forecast performance computed recursively and, therefore, time-varying. The typical measures of forecast performance are the average log score, continuous ranked probability score (CRPS), and quadratic probability score (QPS). In contrast to the point forecast case, using equal weights does not always deliver the most accurate results in density forecasting. One alternative that has proved to give acceptable results in terms of forecast accuracy is to use optimal weights. This is, to choose the weights that maximize the past forecast performance (e.g. weights that maximize the likelihood (Geweke and Amisano, 2011) or the log score of the combined distribution (Hall and Mitchell, 2007)). 3.3 Evaluation To compare the accuracy of competing density forecasts (e.g. based on different models or on different sets of information) scoring rules are typically used. They are numerical measures assessing the calibration and sharpness (or dispersion) of the predictive distribution. The most popular scoring rules are: i) quadratic probability score (QPS) (Brier, 1950); ii) logarithmic score (logs) (Good, 1952); iii) continuous ranked probability score (CRPS) (Matheson and Winkler, 1976). Logarithmic score is commonly used and easy to compute despite its high sensitiveness to the presence of outliers. The logarithmic score is the logarithm of the predictive density evaluated at the observed outcome. The higher probability the estimated density gives to the outcome, the higher the log score will be. Therefore, over the forecast cycle the log score is expected to increase as more information is available and those densities with higher log scores at each forecast horizon will be preferred. The density nowcasts will be accurate when they provide a precise description of the reality, i.e. the distribution of the observed data. The standard way of evaluating the correct specification of density forecasts is using the probability integral transform (PIT) or the cumulative distribution function evaluated at the observed outcome (Diebold, Gunther, and Tay, 1998). A density forecast is well calibrated if the PITs are uniformly, independently (for forecast horizon equal to 1) and identically distributed. An alternative approach is to work with the inverse PITs and check that they are normally, independently and identically distributed. The evaluation of the calibration is therefore tested by performing standard statistical tests of uniformity, normality, independence, and identical distribution (see more details in Rossi and Sekhposyan, 3

4 2014). The PIT informs in which percentiles of the nowcast densities the actual observation falls. If the densities are well calibrated, we should observe that these PITs are uniformly distributed U(0,1). The PITs histogram is a good tool for visual assessment of the uniformity. Furthermore, statistical tests such as Kolmogorov-Smirnov or Anderson-Darling are a good instrument to evaluate the departures from uniformity. Alternatively, Berkowitz (2001) showed that if the PITs are U(0,1), the inverse standard normal transformation of the PITs is N(0,1). Two traditional tests of normality are Berkowitz (2001) and Doornik and Hansen (2008). 4 Probabilistic nowcasts of euro area GDP The nowcasts are evaluated over on a real time data base. The data set contains around 30 monthly indicators covering hard data (notably industrial production but also retail trade, unemployment rate, trade or orders) and soft indicators (notably European Commission and Purchasing Managers surveys) as well as several financial variables. For each quarter 12 updates are considered: 4 forecasts, 6 nowcasts and 2 backcasts. The nowcasts are evaluated over on a real time data base. 4

5 References Aastveit, K. A., K. R. Gerdrup, A. S. Jore, and L. A. Thorsrud (2014): Nowcasting GDP in real-time: A density combination approach, Journal of Business and Economic Statistics, 32(1), Bai, J., and P. Wang (2015): Identification and Bayesian Estimation of Dynamic Factor Models, Journal of Business & Economic Statistics, 33(2), Banbura, M., D. Giannone, M. Modugno, and L. Reichlin (2013): Now-casting and the real-time data flow, in Handbook of Economic Forecasting, ed. by A. T. Graham Elliott, pp North Holland. Bańbura, M., D. Giannone, and L. Reichlin (2010): Large Bayesian VARs, Journal of Applied Econometrics, 25(1), (2011): Nowcasting, in The Oxford Handbook of Economic Forecasting, ed. by D. Hendry, and M. Clements. Oxford University Press. Bańbura, M., and G. Rünstler (2011): A look into the factor model black box. Publication lags and the role of hard and soft data in forecasting GDP., International Journal of Forecasting, 27, Berkowitz, J. (2001): Testing Density Forecasts, with Applications to Risk Management, Journal of Business & Economic Statistics, 19(4), Brier, G. W. (1950): Verification of forecasts expressed in terms of probability, Monthly Weather Review, 78, 13. Diebold, F. X., T. A. Gunther, and A. S. Tay (1998): Evaluating Density Forecasts with Applications to Financial Risk Management, International Economic Review, 39(4), Doornik, J. A., and H. Hansen (2008): An Omnibus Test for Univariate and Multivariate Normality, Oxford Bulletin of Economics and Statistics, 70(s1), Durbin, J., and S. Koopman (2002): A simple and efficient simulation smoother for state space time series analysis, Biometrika, 89(3), Geweke, J., and G. Amisano (2011): Optimal prediction pools, Journal of Econometrics, 164(1), Giannone, D., L. Reichlin, and D. Small (2008): Nowcasting: The real-time informational content of macroeconomic data, Journal of Monetary Economics, 55(4), Good, I. J. (1952): Rational Decisions, Journal of the Royal Statistical Society, 14, Hall, S., and J. Mitchell (2009): Recent developments in density forecasting, in Applied econometrics, ed. by T. Mills, and K. Patterson, vol. 2 of Palgrave Handbook of Econometrics. MacMillan. Hall, S. G., and J. Mitchell (2007): Combining density forecasts, International Journal of Forecasting, 23(1), Kadiyala, K. R., and S. Karlsson (1997): Numerical Methods for Estimation and Inference in Bayesian VAR-Models, Journal of Applied Econometrics, 12(2), Matheson, J. E., and R. L. Winkler (1976): Scoring rules for continuous probability distributions, Management Science, 22,

6 Mazzi, G. L., J. Mitchell, and G. Montana (2014): Density Nowcasts and Model Combination: Nowcasting Euro-Area GDP Growth over the Recession, Oxford Bulletin of Economics and Statistics, 76(2), Proietti, T., M. Marczak, and G. Mazzi (2015): EuroMInd-D: A density estimate of monthly gross domestic product for the euro area, Hohenheim Discussion Papers in Business, Economics and Social Sciences , University of Hohenheim, Faculty of Business, Economics and Social Sciences. Ranjan, R., and T. Gneiting (2010): Combining probability forecasts, Journal of the Royal Statistical Society Series B, 72(1), (2013): Combining predictive distributions, Electronic Journal of Statistics, 7, Rossi, B., and T. Sekhposyan (2014): Evaluating predictive densities of US output growth and inflation in a large macroeconomic data set, International Journal of Forecasting, 30(3), Wall, K. D., and D. S. Stoffer (2002): A State Space approach to bootstrapping conditional forecasts in ARMA models, Journal of Time Series Analysis, 23, Winkler, R. L. (1968): The Consensus of Subjective Probability Distributions, Management Science, 15, B61 B75. 6

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