How to compare interpretatively different models for the conditional variance function. (English) Zbl 1511.62422
Summary: This study considers regression-type models with heteroscedastic Gaussian errors. The conditional variance is assumed to depend on the explanatory variables via a parametric or non-parametric variance function. The variance function has usually been selected on the basis of the log-likelihoods of fitted models. However, log-likelihood is a difficult quantity to interpret – the practical importance of differences in log-likelihoods has been difficult to assess. This study overcomes these difficulties by transforming the difference in log-likelihood to easily interpretative difference in the error of predicted deviation. In addition, methods for testing the statistical significance of the observed difference in test data log-likelihood are proposed.
MSC:
| 62P20 | Applications of statistics to economics |
Keywords:
conditional variance; variance function; predictive likelihood; log-scoring rule; predictive density; out-of-sample testing; model performance measureReferences:
| [1] | Abramowitz, M. and Stegun, I. 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10, Edited by: Abramowitz, M. and Stegun, I. New York: Dover. · Zbl 0543.33001 |
| [2] | Aitkin, M. 1987. Modelling variance heterogeneity in normal regression using GLIM. Appl. Stat., 36: 332-339. · doi:10.2307/2347792 |
| [3] | Amisano, G. and Giacomini, R. 2007. Comparing density forecasts via weighted likelihood ratio tests. J. Bus. Econ. Stat., 25: 177-190. |
| [4] | Asuncion, A. and Newman, D. 2008. UCI machine learning repository, Available at http://www.ics.uci.edu/ mlearn/MLRepository.html |
| [5] | Bao, Y., Lee, T. H. and Saltoglu, B. 2007. Comparing density forecast models. J. Forecasting, 26: 203-225. · doi:10.1002/for.1023 |
| [6] | Cawley, G. and Talbot, N. 2005. Constructing Bayesian formulations of sparse kernel learning methods. Neural Netw., 18: 674-683. · Zbl 1077.68758 · doi:10.1016/j.neunet.2005.06.002 |
| [7] | Cawley, G., Talbot, N., Foxall, R., Dorling, S. and Mandic, D. 2004. Heteroscedastic kernel ridge regression. Neurocomputing, 57: 105-124. · doi:10.1016/j.neucom.2004.01.005 |
| [8] | Corradi, V. and Swanson, N. 2005. A test for comparing multiple misspecified conditional distributions. Econom. Theory, 21: 991-1016. · Zbl 1081.62011 · doi:10.1017/S0266466605050498 |
| [9] | Corradi, V. and Swanson, N. 2006. Predictive density and conditional confidence interval accuracy tests. J. Econom., 135: 187-228. · Zbl 1418.62438 · doi:10.1016/j.jeconom.2005.07.026 |
| [10] | Diebold, F., Gunther, T. and Tay, A. 1998. Evaluating density forecasts with applications to financial risk management. Int. Econ. Rev., 39: 863-883. · doi:10.2307/2527342 |
| [11] | Dorling, S., Foxall, R., Mandic, D. and Cawley, G. 2003. Maximum-likelihood cost functions for neural network models of air quality data. Atmos. Environ., 37: 3435-3443. · doi:10.1016/S1352-2310(03)00323-6 |
| [12] | Feller, W. 1971. An Introduction to Probability Theory and Its Applications, Vol. 2, New York: Wiley. · Zbl 0219.60003 |
| [13] | Fleming, J., Kirby, C. and Ostdiek, B. 2001. The economic value of volatility timing. J. Finance, 56: 329-352. · doi:10.1111/0022-1082.00327 |
| [14] | Geweke, J. and Amisano, G. 2008. Evaluating the predictive distributions of Bayesian models of asset returns. Available at SSRN: http://ssrn.com/abstract=1154496 |
| [15] | Giacomini, R. and White, H. 2006. Tests of conditional predictive ability. Econometrica, 74: 1545-1578. · Zbl 1187.91151 · doi:10.1111/j.1468-0262.2006.00718.x |
| [16] | Gneiting, T., Balabdaoui, F. and Raftery, A. 2007. Probabilistic forecasts, calibration and sharpness. J. R. Stat. Soc.: B, 69: 243-268. · Zbl 1120.62074 · doi:10.1111/j.1467-9868.2007.00587.x |
| [17] | Gneiting, T. and Raftery, A. 2007. Strictly proper scoring rules, prediction, and estimation. J. Am. Stat. Assoc., 102: 359-378. · Zbl 1284.62093 |
| [18] | Jørgensen, B. 1997. The Theory of Dispersion Models, London: Chapman and Hall. · Zbl 0928.62052 |
| [19] | Juutilainen, I. and Röning, J. 2006. Planning of strength margins using joint modelling of mean and dispersion. Mater. Manufact. Process., 21: 367-373. |
| [20] | Mitchell, J. and Hall, S. 2005. Evaluating, comparing and combining density forecasts using the KLIC with an application to the Bank of England and NIESR Fan charts of inflation. Oxford Bull. Econ. Stat., 67: 995-1033. · doi:10.1111/j.1468-0084.2005.00149.x |
| [21] | Rigby, R. and Stasinopoulos, D. 2000. Construction of reference centiles using mean and dispersion additive models. Statistician, 49: 41-50. |
| [22] | Ronchetti, E. 1997. Robust linear model selection by cross-validation. J. Am. Stat. Assoc., 92: 1017-1023. · Zbl 1067.62551 |
| [23] | Smyth, G., Huele, A. and Verbyla, A. 2001. Exact and approximate REML for heteroscedastic regression. Stat. Model., 1: 161-175. · Zbl 1104.62080 · doi:10.1191/147108201128140 |
| [24] | Yeh, I. C. 1998. Modeling of strength of high performance concrete using artificial neural networks. Cement Concrete Res., 28: 1797-1808. · doi:10.1016/S0008-8846(98)00165-3 |
| [25] | Yu, K. and Jones, M. 2004. Likelihood-based local linear estimation of the conditional variance function. J. Am. Stat. Assoc., 99: 139-144. · Zbl 1089.62507 |
| [26] | Zele, M. and Juricic, D. 2005. Estimation of the confidence limits for the quadratic forms in normal variables using a simple Gaussian distribution approximation. Comput. Stat., 20: 137-150. · Zbl 1091.62004 · doi:10.1007/BF02736127 |
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