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Comparing Density Forecasts via Weighted Likelihood Ratio Tests: Asymptotic and Bootstrap Methods

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  • Giacomini, Raffaella

Abstract

This paper proposes and analyzes tests that can be used to compare the accuracy of alternative conditional density forecasts of a variable. The tests are also valid in the broader context of model selection based on out-of-sample predictive ability. We restrict attention to the case of density forecasts derived from non-nested parametric models, with known or estimated parameters. The evaluation makes use of scoring rules, which are loss functions defined over the density forecast and the realizations of the variable. In particular, we consider the logarithmic scoring rule, which leads to the development of asymptotic and bootstrap 'weighted likelihood ratio' tests. The name comes from the fact that the tests compare weighted averages of the scores over the available sample, as a way to focus attention on different regions of the distribution of the variable. For a uniform weight function, the asymptotic test can be interpreted as an extension of Vuong (1989)' s likelihood ratio test for non-nested hypotheses to time series data and to an out-of-sample testing framework. A Monte Carlo simulation explores the size and power properties of this last test in finite samples. An application using S&P500 daily returns shows how the tests can be used to compare the performance of density forecasts obtained from GARCH models with different distributional assumptions.

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Bibliographic Info

Paper provided by Department of Economics, UC San Diego in its series University of California at San Diego, Economics Working Paper Series with number qt59s2g5j5.

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Date of creation: 01 Jun 2002
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Handle: RePEc:cdl:ucsdec:qt59s2g5j5

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Keywords: density forecasting; scoring rules; predictive ability; forecast comparison;

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  1. West, Kenneth D, 1996. "Asymptotic Inference about Predictive Ability," Econometrica, Econometric Society, vol. 64(5), pages 1067-84, September.
  2. Peter F. Christoffersen & Francis X. Diebold, 1997. "Optimal prediction under asymmetric loss," Working Papers 97-11, Federal Reserve Bank of Philadelphia.
  3. Francis X. Diebold & Jose A. Lopez, 1996. "Forecast Evaluation and Combination," NBER Technical Working Papers 0192, National Bureau of Economic Research, Inc.
  4. Newey, Whitney & West, Kenneth, 2014. "A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 33(1), pages 125-132.
  5. Jose A. Lopez, 1995. "Evaluating the predictive accuracy of volatility models," Research Paper 9524, Federal Reserve Bank of New York.
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Cited by:
  1. Raffaella Giacomini & Halbert White, 2003. "Tests of conditional predictive ability," Boston College Working Papers in Economics 572, Boston College Department of Economics.
  2. Isao Ishida, 2005. "Scanning Multivariate Conditional Densities with Probability Integral Transforms," CIRJE F-Series CIRJE-F-369, CIRJE, Faculty of Economics, University of Tokyo.
  3. Valentina Corradi & Norman R. Swanson, 2003. "A Test for Comparing Multiple Misspecified Conditional Distributions," Departmental Working Papers 200314, Rutgers University, Department of Economics.
  4. Michael P. Clements & Philip Hans Franses & Norman R. Swanson, 2003. "Forecasting economic and financial time-series with non-linear models," Departmental Working Papers 200309, Rutgers University, Department of Economics.
  5. Valentina Corradi & Norman Swanson, 2003. "The Block Bootstrap for Parameter Estimation Error In Recursive Estimation Schemes, With Applications to Predictive Evaluation," Departmental Working Papers 200313, Rutgers University, Department of Economics.
  6. Corradi, Valentina & Swanson, Norman R., 2004. "A test for the distributional comparison of simulated and historical data," Economics Letters, Elsevier, vol. 85(2), pages 185-193, November.
  7. Stefania D'Amico, 2004. "Density Estimation and Combination under Model Ambiguity," Computing in Economics and Finance 2004 273, Society for Computational Economics.
  8. Corradi, Valentina & Swanson, Norman R., 2006. "Predictive density and conditional confidence interval accuracy tests," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 187-228.
  9. Stefania D'Amico, 2005. "Density selection and combination under model ambiguity: an application to stock returns," Finance and Economics Discussion Series 2005-09, Board of Governors of the Federal Reserve System (U.S.).
  10. Hall, Stephen G. & Mitchell, James, 2007. "Combining density forecasts," International Journal of Forecasting, Elsevier, vol. 23(1), pages 1-13.

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