ESTIMATION OF AND INFERENCE ABOUT THE EXPECTED SHORTFALL FOR TIME SERIES WITH INFINITE VARIANCE

2013 ◽  
Vol 29 (4) ◽  
pp. 771-807 ◽  
Author(s):  
Oliver Linton ◽  
Zhijie Xiao
Keyword(s):  
Time Series ◽  
Emerging Market ◽  
Expected Shortfall ◽  
Infinite Variance ◽  
Monte Carlo Experiment ◽  
Dependence Structure ◽  
Market Exchange ◽  
Finite Sample ◽  
Inference Procedure ◽  
Thickness Parameter

We study estimation and inference of the expected shortfall for time series with infinite variance. Both the smoothed and nonsmoothed estimators are investigated. The rate of convergence is determined by the tail thickness parameter, and the limiting distribution is in the stable class with parameters depending on the tail thickness parameter of the time series and on the dependence structure, which makes inference complicated. A subsampling procedure is proposed to carry out statistical inference. We also analyze a nonparametric estimator of the conditional expected shortfall. A Monte Carlo experiment is conducted to evaluate the finite sample performance of the proposed inference procedure, and an empirical application to emerging market exchange rates (from October 1997 to October 2008) is conducted to highlight the proposed study.

scholarly journals Optimal heavy tail estimation – Part 1: Order selection

2017 ◽  
Vol 24 (4) ◽  
pp. 737-744 ◽  
Author(s):  
Manfred Mudelsee ◽  
Miguel A. Bermejo
Keyword(s):  
Time Series ◽  
Heavy Tails ◽  
Characteristic Exponent ◽  
Tail Probability ◽  
Heavy Tail ◽  
Infinite Variance ◽  
Accurate Estimation ◽  
Monte Carlo Experiment ◽  
River Elbe ◽  
Tail Estimation

Abstract. The tail probability, P, of the distribution of a variable is important for risk analysis of extremes. Many variables in complex geophysical systems show heavy tails, where P decreases with the value, x, of a variable as a power law with a characteristic exponent, α. Accurate estimation of α on the basis of data is currently hindered by the problem of the selection of the order, that is, the number of largest x values to utilize for the estimation. This paper presents a new, widely applicable, data-adaptive order selector, which is based on computer simulations and brute force search. It is the first in a set of papers on optimal heavy tail estimation. The new selector outperforms competitors in a Monte Carlo experiment, where simulated data are generated from stable distributions and AR(1) serial dependence. We calculate error bars for the estimated α by means of simulations. We illustrate the method on an artificial time series. We apply it to an observed, hydrological time series from the River Elbe and find an estimated characteristic exponent of 1.48 ± 0.13. This result indicates finite mean but infinite variance of the statistical distribution of river runoff.

scholarly journals TRUNCATED SUM OF SQUARES ESTIMATION OF FRACTIONAL TIME SERIES MODELS WITH DETERMINISTIC TRENDS

2019 ◽  
Vol 36 (4) ◽  
pp. 751-772 ◽  
Author(s):  
Javier Hualde ◽  
Morten Ørregaard Nielsen
Keyword(s):  
Time Series ◽  
Asymptotic Normality ◽  
Relative Strength ◽  
Sum Of Squares ◽  
Monte Carlo Experiment ◽  
Time Series Models ◽  
Finite Sample ◽  
Generalized Polynomial ◽  
Finite Sample Properties ◽  
Deterministic Components

We consider truncated (or conditional) sum of squares estimation of a parametric model composed of a fractional time series and an additive generalized polynomial trend. Both the memory parameter, which characterizes the behavior of the stochastic component of the model, and the exponent parameter, which drives the shape of the deterministic component, are considered not only unknown real numbers but also lying in arbitrarily large (but finite) intervals. Thus, our model captures different forms of nonstationarity and noninvertibility. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to nonuniform convergence of the objective function over a large admissible parameter space, but, in addition, our framework is substantially more involved due to the competition between stochastic and deterministic components. We establish consistency and asymptotic normality under quite general circumstances, finding that results differ crucially depending on the relative strength of the deterministic and stochastic components. Finite-sample properties are illustrated by means of a Monte Carlo experiment.

scholarly journals Optimal heavy tail estimation, Part I: Order selection

2017 ◽  
Author(s):  
Manfred Mudelsee ◽  
Miguel A. Bermejo
Keyword(s):  
Time Series ◽  
Heavy Tails ◽  
Characteristic Exponent ◽  
Tail Probability ◽  
Heavy Tail ◽  
Infinite Variance ◽  
Accurate Estimation ◽  
Monte Carlo Experiment ◽  
River Elbe ◽  
Tail Estimation

Abstract. The tail probability, P, of the distribution of a variable is important for risk analysis of extremes. Many variables in complex geophysical systems show heavy tails, where P decreases with the value, x, of a variable as a power law with characteristic exponent, α. Accurate estimation of α on the basis of data is currently hindered by the problem of the selection of the order, that is, the number of largest x-values to utilize for the estimation. This paper presents a new, widely applicable, data-adaptive order selector, which is based on computer simulations and brute force search. It is the first in a set of papers on optimal heavy tail estimation. The new selector outperforms competitors in a Monte Carlo experiment, where simulated data are generated from stable distributions and AR(1) serial dependence. We calculate error bars for the estimated α by means of simulations. We illustrate the method on an artificial time series. We apply it to an observed, hydrological time series from the river Elbe and find an estimated characteristic exponent of 1.48 ± 0.13. This result indicates finite mean but infinite variance of the statistical distribution of river runoff.

scholarly journals Optimal Reconciliation of Seasonally Adjusted Disaggregates Taking Into Account the Difference Between Direct and Indirect Adjustment of the Aggregate

2021 ◽  
Vol 37 (1) ◽  
pp. 31-51
Author(s):  
Francisco Corona ◽  
Victor M. Guerrero ◽  
Jesús López-Peréz
Keyword(s):  
Time Series ◽  
Economic Indicator ◽  
Monte Carlo Experiment ◽  
Seasonal Adjustment ◽  
Economic Time Series ◽  
Finite Sample ◽  
Vector Autoregressive ◽  
Adjustment Procedure ◽  
Economic Time ◽  
The Difference

Abstract This article presents a new method to reconcile direct and indirect deseasonalized economic time series. The proposed technique uses a Combining Rule to merge, in an optimal manner, the directly deseasonalized aggregated series with its indirectly deseasonalized counterpart. The lastmentioned series is obtained by aggregating the seasonally adjusted disaggregates that compose the aggregated series. This procedure leads to adjusted disaggregates that verify Denton’s movement preservation principle relative to the originally deseasonalized disaggregates. First, we use as preliminary estimates the directly deseasonalized economic time series obtained with the X-13ARIMA-SEATS program applied to all the disaggregation levels. Second, we contemporaneously reconcile the aforementioned seasonally adjusted disaggregates with its seasonally adjusted aggregate, using Vector Autoregressive models. Then, we evaluate the finite sample performance of our solution via a Monte Carlo experiment that considers six Data Generating Processes that may occur in practice, when users apply seasonal adjustment techniques. Finally, we present an empirical application to the Mexican Global Economic Indicator and its components. The results allow us to conclude that the suggested technique is appropriate to indirectly deseasonalize economic time series, mainly because we impose the movement preservation condition to the preliminary estimates produced by a reliable seasonal adjustment procedure.

scholarly journals QUANTILE DOUBLE AUTOREGRESSION

2021 ◽  
pp. 1-47
Author(s):  
Qianqian Zhu ◽  
Guodong Li
Keyword(s):  
Time Series ◽  
Financial Time Series ◽  
Simulation Studies ◽  
Finite Sample ◽  
Conditional Quantile ◽  
New Model ◽  
Conditional Quantiles ◽  
Financial Time ◽  
Portmanteau Tests ◽  
Strict Stationarity

Many financial time series have varying structures at different quantile levels, and also exhibit the phenomenon of conditional heteroskedasticity at the same time. However, there is presently no time series model that accommodates both of these features. This paper fills the gap by proposing a novel conditional heteroskedastic model called “quantile double autoregression”. The strict stationarity of the new model is derived, and self-weighted conditional quantile estimation is suggested. Two promising properties of the original double autoregressive model are shown to be preserved. Based on the quantile autocorrelation function and self-weighting concept, three portmanteau tests are constructed to check the adequacy of the fitted conditional quantiles. The finite sample performance of the proposed inferential tools is examined by simulation studies, and the need for use of the new model is further demonstrated by analyzing the S&P500 Index.

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