Density Forecasting: A Survey

A density forecast of the realization of a random variable at some future time is an estimate of the probability distribution of the possible future values of that variable. This article presents a selective survey of applications of density forecasting in macroeconomics and finance, and discusses some issues concerning the production, presentation and evaluation of density forecasts.

A Multi-Step Forecast Density

This paper makes two contribution to the literature on density forecasts. First, we propose a novel bootstrap approach to estimate forecasting densities based on nonparametric techniques. The method is based on the Markov Bootstrap that is suitable to resample dependent data. The combination of nonparametric and bootstrap methods delivers density forecasts that are flexible in capturing markovian dependence (linear and nonlinear) occurring in any moment of the distribution. Second, we improve the testing approach to evaluate density forecasts by considering a set of tests for dynamical misspecification such as autocorrelation, heteroskedasticity and neglected nonlinearity. The approach is useful because rejections of the tests give insights into ways to improve the forecasting model. By Monte Carlo simulations we show that the proposed evaluation strategy has much higher power to detect misspecification of the density forecasts compared to previous analysis. The proposed nonparametric-bootstrap forecasting method exhibits the ability to capture correctly the dynamics of linear and nonlinear time series models. We also investigate the performance at higher orders and propose methods to deal with the \u201ccurse of dimensionality\u201d. Finally, we empirically investigate the relevance of the method in out-of-sample forecasting the density of 3 business cycles variables for the US: real GDP, the Coincident Indicator and Industrial Production. The results indicate that the method gives reliable density forecasts for all variables and performs better compared to parametric forecasting methods.

Using conditional kernel density estimation for wind power density forecasting

Of the various renewable energy resources, wind power is widely recognized as one of the most promising. The management of wind farms and electricity systems can benefit greatly from the availability of estimates of the probability distribution of wind power generation. However, most research has focused on point forecasting of wind power. In this paper, we develop an approach to producing density forecasts for the wind power generated at individual wind farms. Our interest is in intraday data and prediction from 1 to 72 hours ahead. We model wind power in terms of wind speed and wind direction. In this framework, there are two key uncertainties. First, there is the inherent uncertainty in wind speed and direction, and we model this using a bivariate VARMA-GARCH (vector autoregressive moving average-generalized autoregressive conditional heteroscedastic) model, with a Student t distribution, in the Cartesian space of wind speed and direction. Second, there is the stochastic nature of the relationship of wind power to wind speed (described by the power curve), and to wind direction. We model this using conditional kernel density (CKD) estimation, which enables a nonparametric modeling of the conditional density of wind power. Using Monte Carlo simulation of the VARMA-GARCH model and CKD estimation, density forecasts of wind speed and direction are converted to wind power density forecasts. Our work is novel in several respects: previous wind power studies have not modeled a stochastic power curve; to accommodate time evolution in the power curve, we incorporate a time decay factor within the CKD method; and the CKD method is conditional on a density, rather than a single value. The new approach is evaluated using datasets from four Greek wind farms

The ACD Model: Predictability of the Time Between Concecutive Trades

Forecasting ability of several parameterizations of ACD models are compared to benchmark linear autoregressions for inter-trade durations. The estimation of parametric ACD models requires both the choice of a conditional density for durations and the specification of a functional form for the conditional mean duration. Our results provide guidance for choosing among different parameterizations and for developing better forecasting models to predict one-step-ahead, multi-step-ahead, and the whole density of time durations. For evaluating density forecasts, we propose a new constructive test, which is based on the series of probability integral transforms. The choice of the conditional distribution for inter-trade durations does not seem to affect the out-of sample performances of the ACD at short, as well as longer, horizons. Yet, this choice becomes critical when forecasting the density.

Forecasting the density of asset returns

In this paper we introduce a transformation of the Edgeworth-Sargan series expansion of the Gaussian distribution, that we call Positive Edgeworth-Sargan (PES). The main advantage of this new density is that it is well defined for all values in the parameter space, as well as it integrates up to one. We include an illustrative empirical application to compare its performance with other distributions, including the Gaussian and the Student's t, to forecast the full density of daily exchange-rate returns by using graphical procedures. Our results show that the proposed function outperforms the other two models for density forecasting, then providing more reliable value-at-risk forecasts.Density forecasting, Edgeworth-Sargan distribution, probability integral transformations, P-value plots, VaR

A comparison of in-sample forecasting methods

In-sample forecasting is a recent continuous modification of well-known forecasting methods based on aggregated data. These aggregated methods are known as age-cohort methods in demography, economics, epidemiology and sociology and as chain ladder in non-life insurance. Data is organized in a two-way table with age and cohort as indices, but without measures of exposure. It has recently been established that such structured forecasting methods based on aggregated data can be interpreted as structured histogram estimators. Continuous in-sample forecasting transfers these classical forecasting models into a modern statistical world including smoothing methodology that is more efficient than smoothing via histograms. All in-sample forecasting estimators are collected and their performance is compared via a finite sample simulation study. All methods are extended via multiplicative bias correction. Asymptotic theory is being developed for the histogram-type method of sieves and for the multiplicatively corrected estimators. The multiplicative bias corrected estimators improve all other known in-sample forecasters in the simulation study. The density projection approach seems to have the best performance with forecasting based on survival densities being the runner-up

Evaluating density forecasts

The authors propose methods for evaluating and improving density forecasts. They focus primarily on methods that are applicable regardless of the particular user's loss function, though they take explicit account of the relationships between density forecasts, action choices, and the corresponding expected loss throughout. They illustrate the methods with a detailed series of examples, and they discuss extensions to improving and combining suboptimal density forecasts, multistep-ahead density forecast evaluation, multivariate density forecast evaluation, monitoring for structural change and its relationship to density forecasting, and density forecast evaluation with known loss function.Forecasting

Evaluating Density Forecasts

We propose methods for evaluating density forecasts. We focus primarily on methods that are applicable regardless of the particular user's loss function. We illustrate the methods with a detailed simulation example, and then we present an application to density forecasting of daily stock market returns. We discuss extensions for improving suboptimal density forecasts, multi-step-ahead density forecast evaluation, multivariate density forecast evaluation, monitoring for structural change and its relationship to density forecasting, and density forecast evaluation with known loss function.