No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
A Brief Survey on the Choice of Parameters for: “Kernel density estimation for time series data”
Abstract
This paper presents an overview of the empirical performance of some of the common methods for parameter selection in the area of enhanced dynamic kernel density and distribution estimation with exponentially declining weights. It is shown that exponential weighting delivers accurate nonparametric density and quantile evaluations, without common corrections for scale and/or location in most of the financial time series considered, provided that parameters are chosen appropriately with computationally heavy Least-Squares routines. For more time-efficient numerical optimisations and/or simple kernel adaptive estimation strategies, Least-Squares routines may be re-written with exponentially weighted binned kernel estimators. This insures equally effective parameters evaluation under the different choices of kernel functional forms, though binning strategy becomes an important component of estimations. On the other hand, it is also highlighted that if the estimations target is to mine time-varying nonparametric quantiles, kernel functional forms and bandwidths may not be necessary for these evaluations. Combining exponential weights with empirical distribution estimator provides a very similar quantile performance to the kernel enhanced estimator, while parametric specifications may provide a better extreme quantiles outlook.
... This can lead to significant discrepancies between the fitted parametric model and the actual data distribution [27]. In contrast, the nonparametric method does not necessitate prior knowledge of the sample model, allowing for the extraction of distribution characteristics directly from the provided data [28]. Therefore, this paper employs the nonparametric kernel density estimation technique to fit the actual data, yielding kernel density expressions for wind and light outputs after conducting goodness-of-fit and accuracy assessments. ...
... (2) Environmental Protection Costs of Microgrids The operation of distributed microgrid power generation, as well as that of the main grid, can lead to environmental pollution issues. The environmental protection costs are specifically defined as follows: (28) where h signifies the level of pollutant emissions; H represents the total emissions, including CO 2 , SO 2 , and NO X ; u ih and u gridh are the emission coefficients for DG and the main grid, respectively; C h denotes the treatment cost associated with emissions of the h-th pollutant; and P grid refers to the power output from the main grid. ...
In recent years, the global electric vehicle (EV) sector has experienced rapid growth, resulting in major load variations in microgrids due to uncontrolled charging behaviors. Simultaneously, the unpredictable nature of distributed energy output complicates effective integration, leading to frequent limitations on wind and solar energy utilization. The combined integration of distributed energy sources with electric vehicles introduces both opportunities and challenges for microgrid scheduling; however, relevant research to inform practical applications is currently insufficient. This paper tackles these issues by first introducing a method for generating typical wind–solar output scenarios through kernel density estimation and a combination strategy using Frank copula functions, accounting for the complementary traits and uncertainties of wind and solar energy. Building on these typical scenarios, a two-level optimization model for a microgrid is created, integrating demand response and vehicle-to-grid (V2G) interactions of electric vehicles. The model’s upper level aims to minimize operational and environmental costs, while the lower level seeks to reduce the total energy expenses of electric vehicles. Simulation results demonstrate that this optimization model improves the economic efficiency of the microgrid system, fosters regulated EV electricity consumption, and mitigates load variations, thus ensuring stable microgrid operation.
... Our work is not the first one to stress the limitations of the maximum likelihood approach in the selection of the two free parameters. We can indeed cite an article following a least-square approach [40] and another one maximizing a uniformity criterion for the PITs by the mean of an artificial neural network [48]. Our method differs from the latter as it also takes into account the independence of the PITs and it requires only standard statistical tools compared to artificial neural networks. ...
In dynamic operating environment, equipment degradation exhibits significant complexity and variability, which can adversely impact the accuracy of predicting the remaining useful life (RUL). A time-varying non-parametric kernel density estimation (KDE) strategy, integrated with an adaptive kernel auxiliary particle filter (AKAPF) is presented, which is used for monitoring the degraded status of systems and enhance RUL prediction. Firstly, an exponential weighted time-varying RUL prediction model is developed by employing non-parametric KDE in combination with a forgetting factor. Subsequently, the relative density between samples is computed using the k-nearest neighbor (KNN) approach, which facilitates the determination of sparsity and density, enabling the adaptive selection of kernel bandwidth. Next, AKAPF is employed to update the degradation state of the dynamic system and determine the forgetting factor ω, thereby achieving optimal adaptive time-varying weights. Through iterative updates and loops, this real-time prediction system effectively identifies the degradation state and assesses the RUL. Finally, numerical experiments, along with data from the NASA dataset and a high-temperature furnace case study are conducted to validate the robustness and practicality of the proposed model.
Since the financial crisis, risk management has been of growing interest to investors and the approach of Value-at-Risk has gained wide acceptance. Investing in Cryptocurrencies brings not only huge rewards but also huge risks. For this purpose, this paper investigates whether Cryptocurrencies investors’ decisions can rely on the pragmatic and parsimonious approaches for Value-at-Risk forecasting. Specifically, we suggest a parsimonious reflected gamma specification under the GAS framework, consider other GAS special cases and the Exponential Weights driven nonparametric methods, which fall into the same modelling category as the well-known and widely recognised original RiskMetrics™ approach. We focus on the returns for BTC, LTC and ETH and find that progress upon RiskMetricks™ may provide valuable gains in exposure modelling of Cryptocurrencies under the rough and primary backtesting conditions, though not all of the considered approaches demonstrate consistency at the selected risk confidence levels. In our setting, Laplace GAS specification, which controls for time-variation both in scale (volatility) and skewness (asymmetric responses to positive and negative volatility) parameters, performs the best at the most of the levels. We also find that controlling for time-variation in the degrees of freedom (tails) of the Student's t may be a worthwhile consideration, though such approach may still yield more conservative investors’ strategies than its Laplace asymmetric alternative. Reflected gamma and Extreme Value Theory linked Double Pareto specifications also demonstrate a modest performance, but likely suffer from the lack of asymmetry in their parameters, as our Reflected Gamma parametrisation accounts for time-variation in the tails, unlike Pareto specifications and does not outperform asymmetric Laplace specification. Data- driven nonparametric methods seem to struggle the most in approximating downside tail risks due to the sharp corrections in Cryptocurrencies’ value.
We test the presence of regime changes in the GARCH volatility dynamics of Bitcoin log–returns using Markov–switching GARCH (MSGARCH) models. We also compare MSGARCH to traditional single–regime GARCH specifications in predicting one–day ahead Value–at–Risk (VaR). The Bayesian approach is used to estimate the model parameters and to compute the VaR forecasts. We find strong evidence of regime changes in the GARCH process and show that MSGARCH models outperform single–regime specifications when predicting the VaR.
Time Dependent Kernel Density Estimation (TDKDE) used in modelling time varying phenomenon requires two input parameters known as bandwidth and discount to perform. A Maximum Likelihood Estimation (MLE) procedure is commonly used to estimate these parameters in a set of data but this method has a weakness; it may not produce stable kernel estimates. In this article, a novel estimation procedure is developed using Artificial Neural Networks which eliminates this inherent issue. Moreover, evaluating the performance of the kernel estimation in terms of the uniformity of Probability Integral Transform (PIT) shows a significant improvement using the proposed method. A real-life application of TDKDE parameter estimation on NASDQ stock returns validates the flawless performance of the new technique.
We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
The recent advent of smart meters has led to large micro-level datasets. For the first time, the electricity consumption at individual sites is available on a near real-time basis. Efficient management of energy resources, electric utilities, and transmission grids, can be greatly facilitated by harnessing the potential of this data. The aim of this study is to generate probability density estimates for consumption recorded by individual smart meters. Such estimates can assist decision making by helping consumers identify and minimize their excess electricity usage, especially during peak times. For suppliers, these estimates can be used to devise innovative time-of-use pricing strategies aimed at their target consumers. We consider methods based on conditional kernel density (CKD) estimation with the incorporation of a decay parameter. The methods capture the seasonality in consumption, and enable a nonparametric estimation of its conditional density. Using eight months of half-hourly data for one thousand meters, we evaluate point and density forecasts, for lead times ranging from one half-hour up to a week ahead. We find that the kernel-based methods outperform a simple benchmark method that does not account for seasonality, and compare well with an exponential smoothing method that we use as a sophisticated benchmark. To gauge the financial impact, we use density estimates of consumption to derive prediction intervals of electricity cost for different time-of-use tariffs. We show that a simple strategy of switching between different tariffs, based on a comparison of cost densities, delivers significant cost savings for the great majority of consumers.
Continuous data are often measured or used in binned or rounded form. In this paper we follow up on Hall's work analyzing the effect of using equally-spaced binned data in a kernel density estimator. It is shown that a surprisingly large amount of binning does not adversely affect the integrated mean squared error of a kernel estimate.
The Rcpp package simplifies integrating C++ code with R. It provides a consistent C++ class hierarchy that maps various types of R objects (vectors, matrices, functions, environments, . . . ) to dedicated C++ classes. Object interchange between R and C++ is managed by simple, flexible and extensible concepts which include broad support for C++ Standard Template Library idioms. C++ code can both be compiled, linked and loaded on the fly, or added via packages. Flexible error and exception code handling is provided. Rcpp substantially lowers the barrier for programmers wanting to combine C++ code with R.
Variance targeting estimation (VTE) is a technique used to alleviate the numerical difficulties encountered in the quasi-maximum likelihood estimation (QMLE) of GARCH models. It relies on a reparameterization of the model and a first-step estimation of the unconditional variance. The remaining parameters are estimated by quasi maximum likelihood (QML) in a second step. This paper establishes the asymptotic distribution of the estimators obtained by this method in univariate GARCH models. Comparisons with the standard QML are provided and the merits of the variance targeting method are discussed. In particular, it is shown that when the model is misspecified, the VTE can be superior to the QMLE for long-term prediction or value-at-risk calculation. An empirical application based on stock market indices is proposed. Copyright The Author 2011. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com., Oxford University Press.
A copula defines the probability that observations from two time series lie below given quantiles. It is proposed that stationarity
tests constructed from indicator variables be used to test against the hypothesis that the copula is changing over time. Tests
associated with different quantiles may point to changes in different parts of the copula. The tests are still effective when
prefiltering is carried out to correct for persistent changes in volatility. However, a “median quadrant association test”
that is robust to changing volatility provides a good overall test against a time-varying copula. An empirical illustration
on financial contagion is provided. (JEL: C12, G32)
A copula models the relationships between variables independently of their marginal distributions. When the variables are time series, the copula may change over time. Recursive procedures based on indicator variables are proposed for tracking these changes over time. Estimation of the unknown parameters is by maximum likelihood. When the marginal distributions change, pre-filtering is necessary before constructing the indicator variables on which the recursions are based. This entails estimating time-varying quantiles and a simple method based on time-varying histograms is proposed. The techniques are applied to the Hong Kong and Korean stock market indices. Some interesting and unexpected movements are detected, particularly after the attack on the Hong Kong dollar in 1997.
Leave-one-out cross-validation is a popular and readily implemented heuristic for bandwidth selection in nonparametric smoothing problems. In this note we elucidate the role of leave-one-out selection criteria by discussing a criterion introduced by Sarda (J. Statist. Plann. Inference35 (1993) 65–75) for bandwidth selection for kernel distribution function estimators (KDFEs). We show that for this problem, use of the leave-one-out KDFE in the selection procedure is asymptotically equivalent to leaving none out. This contrasts with kernel density estimation, where use of the leave-one-out density estimator in the selection procedure is critical.Unfortunately, simulations show that neither method works in practice, even for samples of size as large as 1000. In fact, we show that for any fixed bandwidth, the expected value of the derivative of the leave-none-out criterion is asymptotically positive. This result and our simulations suggest that the criteria are increasing and that for sufficiently large samples (e.g., n = 100), the smallest available bandwidth will always be selected, thus contradicting the optimality result of Sarda for this estimator.As an alternative to minimizing a selection criterion, we propose a plug-in estimator of the asymptotically optimal bandwidth. Simulations suggest that the plug-in is a good estimator of the asymptotically optimal bandwidth even for samples as small as 10 observations and is not too far from the finite sample bandwidth.
Application of nonparametric density estimators generally requires the specification of a "smoothing parameter." The kernel estimator, for example, is not fully defined until a window width, or scaling, for the kernels has been chosen. Many "data-driven" techniques have been suggested for the practical choice of smoothing parameter. Of these, the most widely studied is the method of cross-validation. Our own simulations, as well as those of many other investigators, indicate that cross-validated smoothing can be an extremely effective practical solution. However, many of the most basic properties of cross-validated estimators are unknown. Indeed, recent results show that cross-validated estimators can fail even to be consistent for seemingly well-behaved problems. In this paper we will review the application of cross-validation to the smoothing problem, and establish consistency for certain cross-validated kernels and histograms.
This paper introduces the Dynamic Additive Quantile (DAQ) model that ensures the monotonicity of conditional quantile estimates. The DAQ model is easily estimable and can be used for computation and updating of the Value-at-Risk. An asymptotically efficient estimator of the DAQ is obtained by maximizing an objective function based on the inverse KLIC measure. An alternative estimator proposed in the paper is the Method of L-Moments estimator (MLM). The MLM estimator is consistent, but generally not fully efficient. Goodness-of-fit tests and diagnostic tools for the assessment of the model are also provided. For illustration, the DAQ model is estimated from a series of returns on the Toronto Stock Exchange (TSX) market index.
We present a new method for data‐based selection of the bandwidth in kernel density estimation which has excellent properties. It improves on a recent procedure of Park and Marron (which itself is a good method) in various ways. First, the new method has superior theoretical performance; second, it also has a computational advantage; third, the new method has reliably good performance for smooth densities in simulations, performance that is second to none in the existing literature. These methods are based on choosing the bandwidth to (approximately) minimize good quality estimates of the mean integrated squared error. The key to the success of the current procedure is the reintroduction of a non‐stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance.
The interest in forecasting the Value at Risk (VaR) has been growing over the last two decades, due to the practical relevance of this risk measure for financial and insurance institutions. Furthermore, VaR forecasts are often used as a testing ground when fitting alternative models for representing the dynamic evolution of time series of financial returns. There are vast numbers of alternative methods for constructing and evaluating VaR forecasts. In this paper, we survey the new benchmarks proposed in the recent literature.
We present a simple methodology for modeling the time variation in volatilities and other higher-order moments using a recursive updating scheme that is similar to the familiar RiskMetrics™ approach. The parameters are updated using the score of the forecasting distribution, which allows the parameter dynamics to adapt automatically to any non-normal data features, and increases the robustness of the subsequent estimates. The new approach nests several of the earlier extensions to the exponentially weighted moving average (EWMA) scheme. In addition, it can be extended easily to higher dimensions and alternative forecasting distributions. The method is applied to Value-at-Risk forecasting with (skewed) Student’s distributions and a time-varying degrees of freedom and/or skewness parameter. We show that the new method is as good as or better than earlier methods for forecasting the volatility of individual stock returns and exchange rate returns.
One criterion proposed in the literature for selecting the smoothing parameter(s) in RosenblattParzen nonparametric constant kernel estimators of a probability density function is a leave-out-one-at-a-time nonparametric maximum likelihood method. Empirical work with this estimator in the univariate case showed that it worked quite well for short tailed distributions. However, it drastically oversmoothed for long tailed distributions. In this paper it is shown that this nonparametric maximum likelihood method will not select consistent estimates of the density for long tailed distributions such as the double exponential and Cauchy distributions. A remedy which was found for estimating long tailed distributions was to apply the nonparametric maximum likelihood procedure to a variable kernel class of estimators. This paper considers one data set, which is a pseudo-random sample of size 100 from a Cauchy distribution, to illustrate the problem with the leave-out-one-at-a-time nonparametric maximum likelihood method and to illustrate a remedy to this problem via a variable kernel class of estimators.
We derive the asymptotic properties of nonparametric estimators of the drift and diffusion functions, and the local time, of a discretely sampled diffusion process that is possibly nonstationary. We provide complete two-dimensional asymptotics in both the time span and the sampling interval, allowing for a precise characterization of their distribution, as well as the determination of optimal bandwidths for these estimators.
Copula-based models provide a great deal of flexibility in modeling multivariate distributions, allowing the researcher to specify the models for the marginal distributions separately from the dependence structure (copula) that links them to form a joint distribution. In addition to flexibility, this often also facilitates estimation of the model in stages, reducing the computational burden. This chapter reviews the growing literature on copula-based models for economic and financial time series data, and discusses in detail methods for estimation, inference, goodness-of-fit testing, and model selection that are useful when working with these models. A representative data set of two daily equity index returns is used to illustrate all of the main results.
The efficient management of wind farms and electricity systems benefit greatly from accurate wind power quantile forecasts. For example, when a wind power producer offers power to the market for a future period, the optimal bid is a quantile of the wind power density. An approach based on conditional kernel density (CKD) estimation has previously been used to produce wind power density forecasts. The approach is appealing because: it makes no distributional assumption for wind power; it captures the uncertainty in forecasts of wind velocity; it imposes no assumption for the relationship between wind power and wind velocity; and it allows more weight to be put on more recent observations. In this paper, we adapt this approach. As we do not require an estimate of the entire wind power density, our new proposal is to optimise the CKD-based approach specifically towards estimation of the desired quantile, using the quantile regression objective function. Using data from three European wind farms, we obtained encouraging results for this new approach. We also achieved good results with a previously proposed method of constructing a wind power quantile as the sum of a point forecast and a forecast error quantile estimated using quantile regression.
New tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas are presented. To obtain asymptotic properties, we derive a new limit result for the sequential empirical tail copula process. Moreover, consistency of both the tests and a break-point estimator are proven. We analyze the finite sample behavior of the tests by Monte Carlo simulations. Finally, and crucial from a risk management perspective, we apply the new findings to datasets from energy and financial markets.
The experience from the global financial crisis has raised serious concerns about the accuracy of standard risk measures as tools for the quantification of extreme downward risks. A key reason for this is that risk measures are subject to a model risk due, e.g. to specification and estimation uncertainty. While regulators have proposed that financial institutions assess the model risk, there is no accepted approach for computing such a risk. We propose a remedy for this by a general framework for the computation of risk measures robust to model risk by empirically adjusting the imperfect risk forecasts by outcomes from backtesting frameworks, considering the desirable quality of VaR models such as the frequency, independence and magnitude of violations. We also provide a fair comparison between the main risk models using the same metric that corresponds to model risk required corrections.
This paper reports the application of a new kernel density estimation model based on the Nadaraya-Watson estimator, for the problem of wind power uncertainty forecasting. The new model is described, including the use of kernels specific to the wind power problem. A novel time-adaptive approach is presented. The quality of the new model is benchmarked against a splines quantile regression model currently in use in the industry. The case studies refer to two distinct wind farms in the United States and show that the new model produces better results, evaluated with suitable quality metrics such as calibration, sharpness, and skill score.
A time-varying probability density function, or the corresponding cumulative distribution function, may be estimated nonparametrically by using a kernel and weighting the observations using schemes derived from time series modelling. The parameters, including the bandwidth, may be estimated by maximum likelihood or cross-validation. Diagnostic checks may be carried out directly on residuals given by the predictive cumulative distribution function. Since tracking the distribution is only viable if it changes relatively slowly, the technique may need to be combined with a filter for scale and/or location. The methods are applied to data on the NASDAQ index and the Hong Kong and Korean stock market indices.
We review different approaches to nonparametric density and regression estimation. Kernel estimators are motivated from local averaging and solving ill-posed problems. Kernel estimators are compared to k-NN estimators, orthogonal series and splines. Pointwise and uniform confidence bands are described, and the choice of smoothing parameter is discussed. Finally, the method is applied to nonparametric prediction of time series and to semiparametric estimation.
In this article we study nonparametric regression quantile estimation by kernel weighted local linear fitting. Two such estimators are considered. One is based on localizing the characterization of a regression quantile as the minimizer of E{pp(Y — a)|X = x}, where ρp is the appropriate “check” function. The other follows by inverting a local linear conditional distribution estimator and involves two smoothing parameters, rather than one. Our aim is to present fully operational versions of both approaches and to show that each works quite well; although either might be used in practice, we have a particular preference for the second. Our automatic smoothing parameter selection method is novel; the main regression quantile smoothing parameters are chosen by rule-of-thumb adaptations of state-of-the-art methods for smoothing parameter selection for regression mean estimation. The techniques are illustrated by application to two datasets and compared in simulations.
Several approaches can be made to the choice of bandwidth in the kernel smoothing of distribution functions. Recent proposals by Sarda (1993) and by Altman & Leger (1995) are analogues of the ‘leave-one-out’ and ‘plug-in’ methods which have been widely used in density estimation. In contrast, a method of crossvalidation appropriate to the smoothing of distribution functions is proposed. Selection of the bandwidth parameter is based on unbiased estimation of a mean integrated squared error curve whose minimising value defines an optimal smoothing parameter. This procedure is shown to lead to asymptotically optimal bandwidth choice, not just in the usual first-order sense but also in the second-order sense in which kernel methods improve on the standard empirical distribution function. Some general theory on the performance of optimal, data-based methods of bandwidth choice is also provided, leading to results which do not have analogues in the context of density estimation. The numerical performances of all the methods discussed in the paper are compared. A bandwidth based on a simple reference distribution is also included. Simulations suggest that the crossvalidatory proposal works well, although the simple reference bandwidth is also quite effective.
A class of density estimates using a superposition of kernels where the kernel parameter can depend on the nearest neighbor distances is studied by the use of simulated data. Their performance using several measures of error is superior to that of the usual Parzen estimators. A tentative solution is given to the problem of calibrating the kernel peakedness when faced with a finite sample set.
The use of a kernel estimator as a smooth estimator for a distribution function has been suggested by many authors An expression for the bandwidth that minimizes the mean integrated square error asymptotically has been available for some time. However, few practical data based methods ior estimating this bandwidth have been investigated. In this paper we propose multisstage plug-in type estimater for this optimal bandwith and derive its asymptotic properties. In particular we show that two stages are required for good asymptotic properties. This behavior is verified for finite samples using a simulation study.
All “continuous” data is rounded to some extent, and this rounding introduces errors into the estimation of the parent density and its derivatives. In this paper we examine the influence of rounding errors on the two most popular types of nonparametric density estimator—those based on kernels and those based on generalized Fourier series. Our results lead directly to several important conclusions which are very relevant to the practical problems of density estimation.
The test is based on the maximum difference between an empirical and a hypothetical cumulative distribution. Percentage points are tabled, and a lower bound to the power function is charted. Confidence limits for a cumulative distribution are described. Examples are given. Indications that the test is superior to the chi-square test are cited.
The exponentially weighted moving average (EWMA) estimator is widely used to forecast the conditional volatility of short horizon asset returns. The EWMA estimator is appropriate when returns are drawn from a normal conditional distribution, but when the conditional distribution of returns is fat-tailed - as is commonly found in practice - the EWMA estimator will be inefficient in the sense that it will attach too much weight to extreme returns. In this paper, we propose a simple generalisation of the EWMA estimator that nests both the standard EWMA estimator, and other EWMA estimators that are robust to leptokurtosis in the conditional distribution of returns. We illustrate the new estimator by forecasting the value at risk of aggregate equity portfolios for the US, the UK and Japan using historical simulation. Backtesting results show that a robust EWMA estimator based on the absolute value of returns rather than their squares offers an improvement over the standard EWMA estimator.
We propose a new modified (biased) cross-validation method for adaptively determining the bandwidth in a nonparametric density estimation setup. It is shown that the method provides consistent minimizers. Some simulation results are reported on which compare the small sample behavior of the new and the classical cross-validation selectors.
Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series Xr We consider estimators of the joint probability density of Xt at different t-values, of conditional probability densities, and of the conditional expectation of functionals of Xv given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.
a b s t r a c t This survey reviews the large and growing literature on copula-based models for economic and financial time series. Copula-based multivariate models allow the researcher to specify the models for the marginal distributions separately from the dependence structure that links these distributions to form a joint distribution. This allows for a much greater degree of flexibility in specifying and estimating the model, freeing the researcher from considering only existing multivariate distributions. The author surveys estimation and inference methods and goodness-of-fit tests for such models, as well as empirical applications of these copulas for economic and financial time series.
We propose a class of observation-driven time series models referred to as generalized autoregressive score (GAS) models. The mechanism to update the parameters over time is the scaled score of the likelihood function. This new approach provides a unified and consistent framework for introducing time-varying parameters in a wide class of nonlinear models. The GAS model encompasses other well-known models such as the generalized autoregressive conditional heteroskedasticity, autoregressive conditional duration, autoregressive conditional intensity, and Poisson count models with time-varying mean. In addition, our approach can lead to new formulations of observation-driven models. We illustrate our framework by introducing new model specifications for time-varying copula functions and for multivariate point processes with time-varying parameters. We study the models in detail and provide simulation and empirical evidence. Copyright © 2012 John Wiley & Sons, Ltd.
This paper considers the problem of selecting optimal bandwidths for variable (sample-point adaptive) kernel density estimation. A data-driven variable bandwidth selector is proposed, based on the idea of approximating the log-bandwidth function by a cubic spline. This cubic spline is optimized with respect to a cross-validation criterion. The proposed method can be interpreted as a selector for either integrated squared error (ISE) or mean integrated squared error (MISE) optimal bandwidths. This leads to reflection upon some of the differences between ISE and MISE as error criteria for variable kernel estimation. Results from simulation studies indicate that the proposed method outperforms a fixed kernel estimator (in terms of ISE) when the target density has a combination of sharp modes and regions of smooth undulation. Moreover, some detailed data analyses suggest that the gains in ISE may understate the improvements in visual appeal obtained using the proposed variable kernel estimator. These numerical studies also show that the proposed estimator outperforms existing variable kernel density estimators implemented using piecewise constant bandwidth functions.
This paper utilizes the most flexible skewed generalized t (SGT) distribution for describing petroleum and metal volatilities that are characterized by leptokurtosis and skewness in order to provide better approximations of the reality. The empirical results indicate that the forecasted Value-at-Risk (VaR) obtained using the SGT distribution provides the most accurate out-of-sample forecasts for both the petroleum and metal markets. With regard to the unconditional and conditional coverage tests, the SGT distribution produces the most appropriate VaR estimates in terms of the total number of rejections; this is followed by the nonparametric distribution, generalized error distribution (GED), and finally the normal distribution. Similarly, in the dynamic quantile test, the VaR estimates generated by the SGT and nonparametric distributions perform better than that generated by other distributions. Finally, in the superior predictive test, the SGT distribution has significantly lower capital requirements than the nonparametric distribution for most commodities.
Extreme value theory is used to derive asymptotically motivated models for unusual or rare events, e.g. the upper or lower tails of a distribution. A new flexible extreme value mixture model is proposed combining a non-parametric kernel density estimator for the bulk of the distribution with an appropriate tail model. The complex uncertainties associated with threshold choice are accounted for and new insights into the impact of threshold choice on density and quantile estimates are obtained. Bayesian inference is used to account for all uncertainties and enables inclusion of expert prior information, potentially overcoming the inherent sparsity of extremal data. A simulation study and empirical application for determining normal ranges for physiological measurements for pre-term infants is used to demonstrate the performance of the proposed mixture model. The potential of the proposed model for overcoming the lack of consistency of likelihood based kernel bandwidth estimators when faced with heavy tailed distributions is also demonstrated.
In this paper we study both the level of Value-at-Risk (VaR) disclosure and the accuracy of the disclosed VaR figures for a sample of US and international commercial banks. To measure the level of VaR disclosures, we develop a VaR Disclosure Index that captures many different facets of market risk disclosure. Using panel data over the period 1996–2005, we find an overall upward trend in the quantity of information released to the public. We also find that Historical Simulation is by far the most popular VaR method. We assess the accuracy of VaR figures by studying the number of VaR exceedances and whether actual daily VaRs contain information about the volatility of subsequent trading revenues. Unlike the level of VaR disclosure, the quality of VaR disclosure shows no sign of improvement over time. We find that VaR computed using Historical Simulation contains very little information about future volatility.
We propose exponentially weighted quantile regression (EWQR) for estimating time-varying quantiles. The EWQR cost function can be used as the basis for estimating the time-varying expected shortfall associated with the EWQR quantile forecast. We express EWQR in a kernel estimation framework, and then modify it by adapting a previously proposed double kernel estimator in order to provide greater accuracy for tail quantiles that are changing relatively quickly over time. We introduce double kernel quantile regression, which extends the double kernel idea to the modeling of quantiles in terms of regressors. In our empirical study of 10 stock returns series, the versions of the new methods that do not accommodate the leverage effect were able to outperform GARCH-based methods and CAViaR models. Copyright The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oupjournals.org, Oxford University Press.
"Discrimination information," or Kullback-Leibler loss, is an appropriate measure of distance in problems of discrimination. We examine it in the context of nonparametric kernel density estimation and show that its asymptotic properties are profoundly influenced by tail properties of the kernel and of the unknown density. We suggest ways of choosing the kernel so as to reduce loss, and describe the extent to which likelihood cross-validation asymptotically minimises loss. Likelihood cross-validation generally leads to selection of a window width of the correct order of magnitude, but not necessarily to a window with the correct first-order properties. However, if the kernel is chosen appropriately, then likelihood cross-validation does result in asymptotic minimisation of Kullback-Leibler loss.
We propose a new nonparametric conditional cumulative distribution function kernel estimator that admits a mix of discrete and categorical data along with an associated nonparametric conditional quantile estimator. Bandwidth selection for kernel quantile regression remains an open topic of research. We employ a conditional probability density function-based bandwidth selector proposed by Hall, Racine, and Li that can automatically remove irrelevant variables and has impressive performance in this setting. We provide theoretical underpinnings including rates of convergence and limiting distributions. Simulations demonstrate that this approach performs quite well relative to its peers; two illustrative examples serve to underscore its value in applied settings.
It is often proposed (R. P. W. Duin, I.E.E.E. Trans. Comput.25 (1976), 1175-1179; J. D. F. Habbema, J. Hermans, and J. Remme, "Compstat 1978" (Corsten and Hermans, Eds.), pp. 178-185, "Compstat 1974" (G. Bruckman, Ed.), pp. 101-110;[16] and [17]) that Kullback-Leibler loss or likelihood cross-validation be used to select the window size when a kernel density estimate is constructed for purposes of discrimination. Some numerical work (E. F. Schuster and G. G. Gregory, "Fifteenth Annual Symposium on the Interface of Computer Science and Statistics" (W. F. Eddy, Ed.), pp. 295-298, Springer-Verlag, New York, 1981) argues against this proposal, but a major theoretical contribution (Y. S. Chow, S. Geman, and L. D. Wu, Ann. Statist.11 (1983), 25-38) demonstrates consistency in the important case of compactly supported kernels. In the present paper we argue that in the context of Kullback-Leibler loss and likelihood cross-validation, compactly supported kernels are an unwise choice. They can result in unnecessarily large loss, and can lead to infinite loss when likelihood cross-validation is used to select window size. Compactly supported kernels often dictate that window size be chosen by trading off one part of the variance component of loss against the other, with scant regard for bias; compare the classical theory, where minimum loss is achieved by trading off variance against bias.
The article considers nonparametric estimation of value-at-risk (VaR) and associated standard error estimation for dependent financial returns. Theoretical properties of the kernel VaR estimator are investigated in the context of dependence. The presence of dependence affects the variance of the VaR estimates and has to be taken into consideration in order to obtain adequate assessment of their variation. An estimation procedure of the standard errors is proposed based on kernel estimation of the spectral density of a derived series. The performance of the VaR estimators and the proposed standard error estimation procedure are evaluated by theoretical investigation, simulation of commonly used models for financial returns, and empirical studies on real financial return series. Copyright 2005, Oxford University Press.
Density forecasting is increasingly more important and commonplace, for example in financial risk management, yet little attention has been given to the evaluation of density forecasts. The authors develop a simple and operational framework for density forecast evaluation. They illustrate the framework with a detailed application to density forecasting of asset returns in environments with time-varying volatility. Finally, the authors discuss several extensions. Copyright 1998 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
A complete theory for evaluating interval forecasts has not been worked out to date. Most of the literature implicitly assumes homoskedastic errors even when this is clearly violated and proceed by merely testing for correct unconditional coverage. Consequently, the author sets out to build a consistent framework for conditional interval forecast evaluation, which is crucial when higher-order moment dynamics are present. The new methodology is demonstrated in an application to the exchange rate forecasting procedures advocated in risk management. Copyright 1998 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.