Combining Density Forecasts using Focused Scoring Rules

We investigate the added value of combining density forecasts focused on a specific region of support. We develop forecast combination schemes that assign weights to individual predictive densities based on the censored likelihood scoring rule and the continuous ranked probability scoring rule (CRPS) and compare these to weighting schemes based on the log score and the equally weighted scheme. We apply this approach in the context of measuring downside risk in equity markets using recently developed volatility models, including HEAVY, realized GARCH and GAS models, applied to daily returns on the S&P 500, DJIA, FTSE and Nikkei indexes from 2000 until 2013. The results show that combined density forecasts based on optimizing the censored likelihood scoring rule significantly outperform pooling based on equal weights, optimizing the CRPS or log scoring rule. In addition, 99% Value-at-Risk estimates improve when weights are based on the censored likelihood scoring rule.

[1]  P. Hansen,et al.  Realized GARCH: A Joint Model of Returns and Realized Measures of Volatility , 2010 .

[2]  Daniel B. Nelson.  CONDITIONAL HETEROSKEDASTICITY IN ASSET RETURNS: A NEW APPROACH , 1991 .

[3]  F. Diebold,et al.  Comparing Predictive Accuracy , 1994, Business Cycles.

[4]  Vu,et al.  Time-Varying Combinations of Predictive Densities Using Nonlinear Filtering , 2012 .

[5]  Leif Anders Thorsrud,et al.  Nowcasting GDP in Real Time: A Density Combination Approach , 2014 .

[6]  J. Stock,et al.  Combination forecasts of output growth in a seven-country data set , 2004 .

[7]  Shaun P. Vahey,et al.  Real-Time Inflation Forecast Densities from Ensemble Phillips Curves , 2010 .

[8]  Neil Shephard,et al.  Realising the future: forecasting with high frequency based volatility (HEAVY) models , 2010 .

[9]  Francesco Ravazzolo,et al.  Combined Density Nowcasting in an Uncertain Economic Environment , 2014 .

[10]  J. Geweke,et al.  Optimal Prediction Pools , 2008 .

[11]  Peter Reinhard Hansen,et al.  The Model Confidence Set , 2010 .

[12]  R. Engle,et al.  CAViaR , 1999 .

[13]  P. Hansen,et al.  A Forecast Comparison of Volatility Models: Does Anything Beat a Garch(1,1)? , 2004 .

[14]  F. Ravazzolo,et al.  Optimal Portfolio Choice under Decision-Based Model Combinations , 2015 .

[15]  T. Gneiting,et al.  Comparing Density Forecasts Using Threshold- and Quantile-Weighted Scoring Rules , 2011 .

[16]  Model risk of risk models , 2016 .

[17]  S. Hall,et al.  Combining density forecasts , 2007 .

[18]  Guofu Zhou,et al.  Out-of-Sample Equity Premium Prediction: Combination Forecasts and Links to the Real Economy , 2009 .

[19]  Francis X. Diebold,et al.  Modeling and Forecasting Realized Volatility , 2001 .

[20]  Peter F. Christoffersen.  Evaluating Interval Forecasts , 1998 .

[21]  Michel Terraza,et al.  Forecasts of the seasonal fractional integrated series , 2004 .

[22]  Piotr Cofta,et al.  The Model of Confidence , 2007 .

[23]  T. Gneiting,et al.  Combining Predictive Distributions , 2011, 1106.1638.

[24]  Gianni Amisano,et al.  Comparing Density Forecasts via Weighted Likelihood Ratio Tests , 2007 .

[25]  C. De Mol,et al.  Optimal Combination of Survey Forecasts , 2012 .

[26]  Drew D. Creal,et al.  Generalized autoregressive score models with applications ∗ , 2010 .

[27]  A. Timmermann.  Forecast Combinations , 2005 .

[28]  Combining Density Forecasts using Focused Scoring Rules , 2015 .

[29]  A. Timmermann.  Chapter 4 Forecast Combinations , 2006 .

[30]  Daniel F. Waggoner,et al.  Confronting Model Misspecification in Macroeconomics , 2012 .

[31]  B. Hansen.  Autoregressive Conditional Density Estimation , 1994 .

[32]  Francesco Ravazzolo,et al.  Optimal Portfolio Choice Under Decision-Based Model Combinations: DECISION-BASED MODEL COMBINATIONS , 2016 .

[33]  J. M. Bates,et al.  The Combination of Forecasts , 1969 .

[34]  C. Granger,et al.  Economic and Statistical Measures of Forecast Accuracy , 1999 .

[35]  Marco Del Negro,et al.  Dynamic Prediction Pools: An Investigation of Financial Frictions and Forecasting Performance , 2014 .

[36]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[37]  James Mitchell,et al.  Evaluating, Comparing and Combining Density Forecasts Using the Klic with an Application to the Bank of England and Niesr 'Fan' Charts of Inflation , 2005 .

[38]  Halbert White,et al.  Tests of Conditional Predictive Ability , 2003 .

[39]  Yong Bao,et al.  Comparing Density Forecast Models , 2007 .

[40]  A. Raftery,et al.  Strictly Proper Scoring Rules, Prediction, and Estimation , 2007 .

[41]  N. Shephard,et al.  Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise , 2006 .

[42]  Shaun P. Vahey,et al.  Combining forecast densities from VARs with uncertain instabilities , 2010 .

[43]  Ivana Komunjer,et al.  Evaluation and Combination of Conditional Quantile Forecasts , 2002 .

[44]  George Kapetanios,et al.  Generalised Density Forecast Combinations , 2014 .

[45]  Christophe Boucher,et al.  Risk models–at–risk , 2014 .

[46]  Anthony S. Tay,et al.  Evaluating Density Forecasts with Applications to Financial Risk Management , 1998 .

[47]  Dick van Dijk,et al.  Likelihood-based scoring rules for comparing density forecasts in tails , 2011 .