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Comparing Density Forecsts via Weighted Likelihood Ratio Tests

  • Gianni Amisano
  • Raffaella Giacomini

We propose a test for comparing the out-of-sample accuracy of competing density forecasts of a variable. The test is valid under general conditions: the data can be heterogeneous and the forecasts can be based on (nested or non-nested) parametric models or produced by semi- parametric, non-parametric or Bayesian estimation techniques. The evaluation is based on scoring rules, which are loss functions de?ned over the density forecast and the realizations of the variable. We restrict attention to the logarithmic scoring rule and propose an out-of-sample weighted likelihood ratio test that compares weighted averages of the scores for the competing forecasts. The user-defined weights are a way to focus attention on di¤erent regions of the distribution of the variable. For a uniform weight function, the test can be interpreted as an extension of Vuong (1989)'s likelihood ratio test to time series data and to an out-of-sample testing framework. We apply the tests to evaluate density forecasts of US inflation produced by linear and Markov Switching Phillips curve models estimated by either maximum likelihood or Bayesian methods. We conclude that a Markov Switching Phillips curve estimated by maximum likelihood produces the best density forecasts of inflation.

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Paper provided by University of Brescia, Department of Economics in its series Working Papers with number ubs0504.

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Date of creation: 2005
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Handle: RePEc:ubs:wpaper:ubs0504
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  1. Raffaella Giacomini & Halbert White, 2003. "Tests of Conditional Predictive Ability," Econometrics 0308001, EconWPA.
  2. Francis X. Diebold & Todd A. Gunther & Anthony S. Tay, 1997. "Evaluating density forecasts," Working Papers 97-6, Federal Reserve Bank of Philadelphia.
  3. Söderlind, Paul & Svensson, Lars E.O., 1997. "New Techniques to Extract Market Expectations from Financial Instruments," Seminar Papers 621, Stockholm University, Institute for International Economic Studies.
  4. Lopez, Jose A, 2001. "Evaluating the Predictive Accuracy of Volatility Models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 20(2), pages 87-109, March.
  5. 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.
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  7. Clements, M.P. & Smith J., 1998. "Evaluating The Forecast of Densities of Linear and Non-Linear Models: Applications to Output Growth and Unemployment," The Warwick Economics Research Paper Series (TWERPS) 509, University of Warwick, Department of Economics.
  8. Corradi, Valentina & Swanson, Norman R., 2006. "Bootstrap conditional distribution tests in the presence of dynamic misspecification," Journal of Econometrics, Elsevier, vol. 133(2), pages 779-806, August.
  9. Peter F. Christoffersen & Francis X. Diebold, 1997. "Optimal prediction under asymmetric loss," Working Papers 97-11, Federal Reserve Bank of Philadelphia.
  10. Stock, James H. & Watson, Mark W., 1999. "Forecasting inflation," Journal of Monetary Economics, Elsevier, vol. 44(2), pages 293-335, October.
  11. Weiss, Andrew A, 1996. "Estimating Time Series Models Using the Relevant Cost Function," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(5), pages 539-60, Sept.-Oct.
  12. Donald W.K. Andrews, 1988. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Cowles Foundation Discussion Papers 877R, Cowles Foundation for Research in Economics, Yale University, revised Jul 1989.
  13. Valentina Corradi & Norman Swanson, 2004. "Predective Density and Conditional Confidence Interval Accuracy Tests," Departmental Working Papers 200423, Rutgers University, Department of Economics.
  14. Anthony Tay & Kenneth F. Wallis, 2000. "Density Forecasting: A Survey," Econometric Society World Congress 2000 Contributed Papers 0370, Econometric Society.
  15. Andrew Patton, 2004. "Modelling Asymmetric Exchange Rate Dependence," Working Papers wp04-04, Warwick Business School, Finance Group.
  16. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-33, March.
  17. Diebold, Francis X & Gunther, Todd A & Tay, Anthony S, 1998. "Evaluating Density Forecasts with Applications to Financial Risk Management," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 863-83, November.
  18. Valentina Corradi & Norman R. Swanson, 2003. "A Test for Comparing Multiple Misspecified Conditional Distributions," Departmental Working Papers 200314, Rutgers University, Department of Economics.
  19. Francis X. Diebold & Anthony S. Tay & Kenneth F. Wallis, 1997. "Evaluating Density Forecasts of Inflation: The Survey of Professional Forecasters," NBER Working Papers 6228, National Bureau of Economic Research, Inc.
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  21. repec:cup:etheor:v:13:y:1997:i:6:p:808-17 is not listed on IDEAS
  22. Chang-Jin Kim & Charles R. Nelson, 1999. "State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262112388, June.
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