Ebook A Test for Density Forecast Comparison with Applications to Risk Management
Forecasting density has been at the core of finance and economics research agenda. For instance, most of the classical finance theories such as asset pricing, portfolio selection, and option valuation aim to model the surrounding uncertainty via a parametric distribution function. Extracting information about market participants’ expectations from option prices can be considered another form of density forecasting exercise (Soderling and Svensson, 1997; Jackwerth and Rubinstein, 1996). Besides, there have been increasing interest in density forecasts of inflation, unemployment and output (Clements and Smith, 2000; Clements, 2002). Some popular risk measurement tools such as the Value-at-Risk (VaR) and expected shortfalls (ES), which aim to model the tails of portfolio return distributions, can also be evaluated from the perspective of density forecasting.
Consequently, in time series econometrics, there has been extensive literature on evaluating density forecast models: Diebold et al. (1998), Diebold et al. (1999), Clements and Smith (2000), Berkowitz (2001), Hong (2002), among others. In particular, Diebold et al. (1998), Granger (1999a, 1999b), Granger and Pesaran (2000a, 2000b), Pesaran and Skouras (2001), and Patton and Timmermann (2003) discussed the issue of forecast optimality.
While the research on evaluating each density forecast model has been very versatile since the seminal paper of Diebold et al. (1998), there has been much less effort in comparing alternative density forecast models. Considering the recent empirical evidence on volatility clustering and asymmetry and fat-tailedness in financial return series, we believe that a formal test of relative optimality of a given model compared with alternative distribution and volatility specifications in the context of density forecasts will contribute to the existing literature.
Deciding on which distribution and/or volatility specification to use for a particular asset is a common task even for finance practitioners and risk professionals. For instance, in spite of the massive literature on volatility forecasting, a clear consensus on which model to use has not yet been reached. As argued in Poon and Granger (2003), most of the (volatility) forecasting studies do not produce very conclusive results because only a subset of alternative models are compared, with a potential bias towards the method developed by the authors. It is further claimed that lack of a uniform forecast evaluation technique makes volatility forecasting a difficult task. Being able to choose the most suitable volatility and distribution specifications is a more demanding task. However, in this paper we demonstrate that this gap can be filled by a rigorous density forecast comparison methodology.
The main aim of this paper is to propose a test for comparing various density forecast models, all of which can be possibly misspecified. The proposed test for density forecast comparison enables us to assess which volatility and/or distribution are statistically more appropriate to mimic the time series behavior of a return series. Since every particular density forecast model may be misspecified, comparison of a set of candidate models should be based on the “distances” of these models to the true, unknown model. Our starting point is to utilize Kullback and Leibler’s (1951) information criterion (KLIC) and then relate it to the likelihood ratio (LR) statistics of Berkowitz (2001) for density forecast evaluation. In particular, we define the distance of a postulated density forecast model to the true model as the minimum KLIC divergence measure between the two distributions, as suggested by Vuong (1989). This distance can be estimated by the sample likelihood ratio, which is equivalent to the sample likelihood ratio of the density of the inverse normal transform of the probability integral transforms (PIT) of the process with respect to the models’s density forecast to a standard normal density. This equivalence enables us to use the LR statistic developed in Berkowitz (2001) to estimate conveniently the minimum KLIC. One immediate extension of the proposed test is to compare the predictive abilities of alternative density forecast models in the tails within the context of VaR and ES. For this purpose, a tail minimum KLIC discrepancy measure based on the censored likelihoods is used. Another innovation of this paper is to use our distance measure as a forecast loss function in the framework of White’s (2000) reality check.
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