Chapter 17 - Quantile Prediction
Introduction
In a general prediction or forecasting problem, the objective of the exercise is to “find an approximation of an unobserved variable that is a function of the observations” (Gourieroux and Monfort, 1995). In this chapter, the object of interest is the conditional -quantile of a scalar variable , , where the probability is given and denotes an information set generated by some covariates and/or lags of .1
The motivation for studying quantiles is twofold. In some situations, predicting a particular quantile is required by, say, an outside regulator. Which particular quantile is to be predicted depends on the context. In the banking sector, for example, the 1% or 5% quantile describing the left tail of the profit and loss account’s distribution (the so-called Value-at-Risk) is of interest to risk managers (as per the recommendations of the Basel Committee on Banking Supervision, 1996, Basel Committee on Banking Supervision, 2011). Systemic real risk (defined as the 5% quantile of quarterly growth in real GDP) and systemic financial risk (defined as the 5% quantile of a system-wide financial risk indicator) are of concern to policymakers (see, e.g., De Nicolò and Lucchetta, 2010). Lower and upper quantiles may be of interest in the studies of unemployment duration (see, e.g., Koenker and Xiao, 2002) or wage inequalities (see, e.g., Machado and Mata, 2005). More often, however, one wishes to obtain a collection of conditional quantiles that can characterize the entire conditional distribution.2 For instance, predicting several conditional quantiles of future inflation gives some idea on the entire distribution of the latter, which can be vital in assessing the risk to inflation stability due to macroeconomic shocks (see, e.g., Manzana and Zeromb, 2010).
In many other cases, however, quantiles are not given as primitive. Here, the objective is to provide an optimal point forecast of and optimality can be closely tied to the decision maker’s loss function. To elaborate more, consider a forecaster (e.g., policy maker, firm, government, Central Bank, or international organization) whose loss function depends on the realization of the target variable (e.g., inflation rate, GDP growth, budget deficit) and on its forecast . For example, the reputation (or reward) of professional forecasters is likely to depend on the accuracy with which they forecast the variable of interest. The optimal point forecast of is the value of that given the forecaster’s information minimizes the expected loss (see, e.g., Elliott and Timmermann, 2008).
In realistic applications, it is natural to assume that the loss is a smooth function that is everywhere positive except when in which case . This guarantees that the loss is minimum when the forecaster has perfect foresight. For imperfect forecasts, , and the resulting loss is positive . The embarrassment costs to the forecaster resulting from underpredicting the target (i.e., ) as compared to overpredicting the target (i.e., ) by the same magnitude are likely to differ. This can be captured by letting the forecaster’s loss be an asymmetric function for which it is possible that . So let be parameterized as:where denotes the so-called “tick” or “check” function defined for any scalar asand where denotes the usual indicator function, and is any strictly increasing real function.3 The parameter describes the degree of asymmetry in the forecaster’s loss function: values less than one half indicate that overpredicting induces greater loss to the forecaster than underpredicting by the same magnitude. In the symmetric case, equals one half and the forecaster’s embarrassment costs due to over- and underpredictions are the same.
If the forecaster’s loss is of the form given in Eq. (1), then the optimal forecast of is its conditional quantile Komunjer, 2005, Komunjer and Vuong, 2010b, Gneiting, 2011. Interestingly, the real function , which enters the forecaster’s loss need not be known. This means that quantiles are optimal forecasts for a large class of decision makers obtained by letting vary in the set of monotone increasing functions. In the simple case where is identity, one obtains , which is a well-known loss function in the literature on quantile estimation (see, e.g., the seminal paper by Koenker and Bassett (1978)).
Not only do the loss functions in (1) yield quantiles as optimal predictors of , but they are the only ones to do so. In other words, whenever the forecast corresponds to the conditional quantile of , the forecaster’s loss is necessarily of the form in Eq. (1). This result establishes a strong link between the quantiles as optimal predictors of and the underlying forecast loss functions for a given probability level . In certain situations, however, may be unknown and the question is whether it is possible to uncover it from the properties of the optimal forecast of . Elliott et al. (2005) show that the answer is yes provided one focuses on the so-called “lin-lin” losses in (1) obtained when is identity.
There is a variety of applications in economics and finance where quantile prediction has been of interest. In finance, for example, the question of Value-at-Risk (VaR) measurement and prediction has been the leading motivation for quantile prediction in the last 20 years.4 An excellent survey of such applications can be found in Manganelli and Engle, 2004, Kuester et al., 2007. Two books by Christoffersen, 2003, McNeil et al., 2005 give overviews of quantile prediction for risk management in general.
Quantile prediction has also been used in the construction of interval forecasts and the closely related density forecasts. An early example is Granger et al. (1989), who construct interval forecasts based on estimated time-series models for ARCH processes using quantile regression. Numerous applications have been proposed since; a survey can be found in Chatfield (1993). Among more recent work, Hansen (2006) proposes a two-step quantile estimation method that focuses on incorporating parameter estimation uncertainty in interval forecasts. Examples of quantile use in density forecasting can be found in the survey by Tay and Wallis (2000). Applications are in a variety of fields, ranging from the construction of the so-called “fan charts”5 in macroeconomics (see, e.g., Manzana and Zeromb, 2010), predicting densities of financial return series (see, e.g., Cenesizoglu and Timmermann, 2008, Kim et al., 2010), to the paired comparison model of sports ratings (see, e.g., Koenker and Bassett, 2010).
Interestingly, conditional quantile forecasts have also been used to generate conditional volatility predictions that are of great importance in many area of finance. The quantile approach to volatility forecasting exploits an interesting result: that, for a variety of probability distributions, there is a surprising constancy of the ratio of the standard deviation to the interval between symmetric tail quantiles. This means that the conditional volatility can be approximated by a simple time-invariant function of the interval between symmetric conditional quantiles, even though the conditional volatility and distribution of financial returns may vary over time. Taylor (2005) uses this approach to construct volatility forecasts from quantile forecasts produced by a variety of VaR methods.
Finally, in the context of forecast evaluation, Patton and Timmermann (2007) proposed using quantiles as a way to test for forecast efficiency under more general loss functions than quadratic loss.
Depending on the type of data that they use, the above applications can be classified into two broad categories: cross-sectional and dynamic. An example of a cross-sectional application would be the one in which quantiles of the distribution of wages across a population of individuals are predicted given a variety of individuals’ socioeconomic characteristics, such as their age, education, or experience. An example of a dynamic application would be the one in which one tries to predict the quantiles of various financial asset or portfolio returns given the information set generated by lagged returns. To focus the scope of this chapter, we shall primarily turn our attention to the quantile prediction methods that use dynamic (time-series) data. In particular, this means that we shall mostly omit from this review the existing work on conditional quantiles that only applies to independent and identically distributed (iid) data. That work is more geared towards cross-sectional applications, which for the most part shall remain outside of the scope of the chapter. To emphasize the dynamic nature of the problem, we shall hereafter denote the conditional quantile of interest as where where is the information set generated by lagged observations of as well as those of the covariates available up to time .
In this chapter, we shall try to address two key issues. Our first question of interest is that of prediction (forecasting): how to predict given the information available up to time ? This question is closely related to the question of measuring even when the time observations have become available. As in many forecasting situations, the object of interest itself is not observable. Thus, we cannot directly compute the prediction (or forecasting) errors , which complicates the prediction problem.
Our second question of interest is that of forecast evaluation: how to assess the accuracy of the predictor of ? As with the prediction problem, this question is closely related to the question of evaluating the accuracy of the measurement of . Understanding the underpinnings of forecast evaluation is particularly important if one is interested in comparing alternative quantile predictors.
The chapter is organized as follows: in Section 2 we review the three families of prediction approaches based on fully parametric, semi-parametric, and non-parametric methods. Section 3 discusses the conditional quantile forecast evaluation techniques. In Section 4, some of the issues specific to quantiles, such as near-extreme quantiles and multivariate quantiles, are considered. Section 5 concludes and discusses possible venues for future research. Throughout, we shall use the terms “predictor” (resp. “prediction”) and “forecast” (resp. “forecasting”) interchangeably.
Section snippets
Prediction
Depending on the strength of the model assumptions that they employ, existing quantile prediction methods can be categorized into three broad categories: fully parametric, semi-parametric, and non-parametric approaches. Manganelli and Engle, 2004, Kuester et al., 2007 offer similar classifications in their respective surveys on VaR prediction. In this section, we first set up the quantile forecasting problem, then offer a review of the existing prediction approaches.
Evaluation
Given the range of approaches available for producing conditional quantile predictions, it is necessary to have adequate tools for their evaluation. There are several dimensions according to which it is possible to classify the existing evaluation techniques. For example, one could separately consider evaluation criteria based on in-sample performance of conditional quantile predictions versus out-of-sample. Another classification is into absolute versus relative evaluation approaches. Absolute
Specific Issues
We now address some of the issues specific to the quantiles, such as quantiles whose probabilities are far in the tails ( close to zero or one), the problem of quantile crossings, which can occur if several quantiles of the same variable are measured or forecast at the same time, as well as the issue of multivariate quantiles.
Conclusion and Directions for Future Research
The goal of this chapter was to review the existing methods for quantile prediction and predictive ability testing. Existing prediction approaches were classified according to the strength of the model assumptions they rely on. Thus, we separately considered fully parametric, semi-parametric, and non-parametric methods. Existing forecast evaluation tests were grouped into two categories: absolute and relative. Relative merits of various forecasting models and evaluation methods were reviewed in
Acknowledgments
The author would like to thank the Editors, Allan Timmermann and Graham Elliott, as well as two anonymous referees for providing excellent suggestions and comments. She would also like to thank the participants of the 2011 St. Louis Fed Handbook of Forecasting Conference for their valuable feedback. Financial support from the National Science Foundation SES-0962473 is gratefully acknowledged.
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