Wasserstein autoregressive models for density time series

Data consisting of time-indexed distributions of cross-sectional or intraday returns have been extensively studied in finance, and provide one example in which the data atoms consist of serially dependent probability distributions. Motivated by such data, we propose an autoregressive model for density time series by exploiting the tangent space structure on the space of distributions that is induced by the Wasserstein metric. The densities themselves are not assumed to have any specific parametric form, leading to flexible forecasting of future unobserved densities. The main estimation targets in the order-p Wasserstein autoregressive model are Wasserstein autocorrelations and the vector-valued autoregressive parameter. We propose suitable estimators and establish their asymptotic normality, which is verified in a simulation study. The new order-p Wasserstein autoregressive model leads to a prediction algorithm, which includes a data driven order selection procedure. Its performance is compared to existing prediction procedures via application to four financial return data sets, where a variety of metrics are used to quantify forecasting accuracy. For most metrics, the proposed model outperforms existing methods in two of the data sets, while the best empirical performance in the other two data sets is attained by existing methods based on functional transformations of the densities.     

Analysis of accumulated rounding errors in autoregressive processes

In considering the rounding impact of an autoregressive (AR) process, there are two different models available to be considered. The first assumes that the dynamic system follows an underlying AR model and only the observations are rounded up to a certain precision. The second assumes that the updated observation is a rounded version of an autoregression on previous rounded observations. This article considers the second model and examines behaviour of rounding impacts to the statistical inferences. The conditional maximum‐likelihood estimates for the model are proposed and their asymptotic properties are established, including strong consistency and asymptotic normality. Furthermore, both the classical AR model and the ordinary rounded AR model are no longer reliable when dealing with accumulated rounding errors. The three models are also applied to fit the Ocean Wave data. It turns out that the estimates under distinct models are significantly different. Based on our findings, we strongly recommend that models for dealing with rounded data should be in accordance with the actions of rounding errors.     

Tests Based on Simplicial Depth for AR(1) Models With Explosion

We propose outlier a robust and distribution‐free test for the explosive AR(1) model with intercept based on simplicial depth. In this model, simplicial depth reduces to counting the cases where three residuals have alternating signs. The asymptotic distribution of the test statistic is given by a specific Gaussian process. Conditions for the consistency are given, and the power of the test at finite samples is compared with alternative tests. The new test outperforms these tests in the case of skewed errors and outliers. Finally, we apply the method to crack growth data and compare the results with an OLS approach.     

Testing for parameter stability in nonlinear autoregressive models

In this article we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure, we model the regression function by a single layer feedforward neural network. We show that CUSUM‐type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives, it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change‐point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log‐returns is given.     

Central limit theorems for nonparametric estimators with real‐time random variables

In this article, asymptotic theories for nonparametric methods are studied when they are applied to real‐time data. In particular, we derive central limit theorems for nonparametric density and regression estimators. For this we formally introduce a sequence of real‐time random variables indexed by a parameter related to fine gridding of time domain (or fine discretization). Our results show that the impact of fine gridding is greater in the density estimation case in the sense that strong dependence due to fine gridding severely affects the major strength of nonparametric density estimator (or its data‐adaptive property). In addition, we discuss some issues about nonparametric regression model with fine gridding of time domain.     

Robust Wilcoxon‐Type Estimation of Change‐Point Location Under Short‐Range Dependence

We introduce a robust estimator of the location parameter for the change‐point in the mean based on Wilcoxon statistic and establish its consistency for L₁ near‐epoch dependent processes. It is shown that the consistency rate depends on the magnitude of the change. A simulation study is performed to evaluate the finite sample properties of the Wilcoxon‐type estimator under Gaussianity as well as under heavy‐tailed distributions and disturbances by outliers, and to compare it with a CUSUM‐type estimator. It shows that the Wilcoxon‐type estimator is equivalent to the CUSUM‐type estimator under Gaussianity but outperforms it in the presence of heavy tails or outliers in the data.     

Assessing Time‐Reversibility Under Minimal Assumptions

Abstract. This article considers a simple procedure for assessing whether a weakly dependent univariate stochastic process is time‐reversible. Our approach is based on a simple index of the deviation from zero of the median of the one‐dimensional marginal law of differenced data. An attractive feature of the method is that it requires no moment assumptions. Instead of relying on Gaussian asymptotic approximations, we consider using subsampling and resampling methods to construct confidence intervals for the time‐reversibility parameter, and show that such inference procedures are asymptotically valid under a mild mixing condition. The small‐sample properties of the proposed procedures are examined by means of Monte Carlo experiments and an application to real‐world data is also presented.     

Least tail‐trimmed squares for infinite variance autoregressions

We develop a robust least squares estimator for autoregressions with possibly heavy tailed errors. Robustness to heavy tails is ensured by negligibly trimming the squared error according to extreme values of the error and regressors. Tail‐trimming ensures asymptotic normality and super‐‐convergence with a rate comparable to the highest achieved amongst M‐estimators for stationary data. Moreover, tail‐trimming ensures robustness to heavy tails in both small and large samples. By comparison, existing robust estimators are not as robust in small samples, have a slower rate of convergence when the variance is infinite, or are not asymptotically normal. We present a consistent estimator of the covariance matrix and treat classic inference without knowledge of the rate of convergence. A simulation study demonstrates the sharpness and approximate normality of the estimator, and we apply the estimator to financial returns data. Finally, tail‐trimming can be easily extended beyond least squares estimation for a linear stationary AR model. We discuss extensions to quasi‐maximum likelihood for GARCH, weighted least squares for a possibly non‐stationary random coefficient autoregression, and empirical likelihood for robust confidence region estimation, in each case for models with possibly heavy tailed errors.     

SIGNIFICANT VARIABLE SELECTION AND AUTOREGRESSIVE ORDER DETERMINATION FOR TIME‐SERIES PARTIALLY LINEAR MODELS

This paper is concerned with the regression coefficient and autoregressive order shrinkage and selection via the smoothly clipped absolute deviation (SCAD) penalty for a partially linear model with time‐series errors. By combining the profile semi‐parametric least squares method and SCAD penalty technique, a new penalized estimation for the regression and autoregressive parameters in the model is proposed. We show that the asymptotic property of the resultant estimator is the same as if the order of autoregressive error structure and non‐zero regression coefficients are known in advance, thus achieving the oracle property in the sense of Fan and Li (2001). In addition, based on a prewhitening technique, we construct a two‐stage local linear estimator (TSLLE) for the non‐parametric component. It is shown that the TSLLE is more asymtotically efficient than the one that ignores the autoregressive time‐series error structure. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedure. An example of application on electricity usage data is also illustrated. Copyright © 2014 Wiley Publishing Ltd     

Periodic autoregressive conditional duration

We propose an autoregressive conditional duration (ACD) model with periodic time-varying parameters and multiplicative error form. We name this model periodic autoregressive conditional duration (PACD). First, we study the stability properties and the moment structures of it. Second, we estimate the model parameters, using (profile and two-stage) Gamma quasi-maximum likelihood estimates (QMLEs), the asymptotic properties of which are examined under general regularity conditions. Our estimation method encompasses the exponential QMLE, as a particular case. The proposed methodology is illustrated with simulated data and two empirical applications on forecasting Bitcoin trading volume and realized volatility. We found that the PACD produces better in-sample and out-of-sample forecasts than the standard ACD.     

On an independent and identically distributed mixture bilinear time‐series model

A class of nonlinear time‐series models in which the underlying process follows a finite mixture of bilinear representations is proposed. The mixture feature appears in the conditional distribution of the process which is given as a finite mixture of distributions evaluated at the normed innovations of diagonal bilinear specifications. This class is aimed at capturing special characteristics exhibited by many observed time series such as tail heaviness, multimodality, asymmetry and change in regime. Some probabilistic properties of the proposed model, namely strict and second‐order stationarity, geometric ergodicity, covariance structure, existence of higher order moments, tail behaviour and invertibility, are first studied. Parameter estimation is then performed through the EM algorithm, performance of which is shown via simulation experiments. Applications to some real‐time‐series data are proposed and through which it is shown how neglecting the mixture framework in a bilinear representation results in a loss in adequacy.     

Change‐point detection in panel data

We consider N panels and each panel is based on T observations. We are interested to test if the means of the panels remain the same during the observation period against the alternative that the means change at an unknown time. We provide tests which are derived from a likelihood argument and they are based on the adaptation of the CUSUM method to panel data. Asymptotic distributions are derived under the no change null hypothesis and the consistency of the tests are proven under the alternative. The asymptotic results are shown to work in case of small and moderate sample sizes via Monte Carlo simulations.     

Wavelet‐Based Tests for Comparing Two Time Series with Unequal Lengths

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.     

On a Mixture GARCH Time‐Series Model

Abstract. Recently, there has been a lot of interest in modelling real data with a heavy‐tailed distribution. A popular candidate is the so‐called generalized autoregressive conditional heteroscedastic (GARCH) model. Unfortunately, the tails of GARCH models are not thick enough in some applications. In this paper, we propose a mixture generalized autoregressive conditional heteroscedastic (MGARCH) model. The stationarity conditions and the tail behaviour of the MGARCH model are studied. It is shown that MGARCH models have tails thicker than those of the associated GARCH models. Therefore, the MGARCH models are more capable of capturing the heavy‐tailed features in real data. Some real examples illustrate the results.     

Tests for the Equality of Two Processes' Spectral Densities with Unequal Lengths Using Wavelet Methods

Testing procedures for assessing whether two stationary and independent linear processes with unequal lengths have the same spectral densities or same auto‐covariance functions are investigated. New test statistics are proposed based on the difference of the two wavelet‐based estimates of the two spectral densities. The asymptotic normal distributions of the empirical wavelet coefficients are derived based on Bartlett type approximation of a quadratic form with dependent variables by the corresponding quadratic form with independent and identically distributed (i.i.d.) random variables. The limit distributions of the proposed test statistics are derived from those asymptotic results, and they asymptotically follow known chi‐square distributions. The advantage of those new procedures is that those test statistics are constructed very simply and can be used for two time series with arbitrary lengths. The performance of those new tests is compared with some recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed tests are very comparable to the current tests.     

Robust estimators under semi‐parametric partly linear autoregression: Asymptotic behaviour and bandwidth selection

Abstract. In this article, under a semi‐parametric partly linear autoregression model, a family of robust estimators for the autoregression parameter and the autoregression function is studied. The proposed estimators are based on a three‐step procedure, in which robust regression estimators and robust smoothing techniques are combined. Asymptotic results on the autoregression estimators are derived. Besides combining robust procedures with M‐smoothers, predicted values for the series and detection residuals, which allow to detect anomalous data, are introduced. Robust cross‐validation methods to select the smoothing parameter are presented as an alternative to the classical ones, which are sensitive to outlying observations. A Monte Carlo study is conducted to compare the performance of the proposed criteria. Finally, the asymptotic distribution of the autoregression parameter estimator is stated uniformly over the smoothing parameter.     

Functional Generalized Autoregressive Conditional Heteroskedasticity

Heteroskedasticity is a common feature of financial time series and is commonly addressed in the model building process through the use of autoregressive conditional heteroskedastic and generalized autoregressive conditional heteroskedastic (GARCH) processes. More recently, multivariate variants of these processes have been the focus of research with attention given to methods seeking an efficient and economic estimation of a large number of model parameters. Because of the need for estimation of many parameters, however, these models may not be suitable for modelling now prevalent high‐frequency volatility data. One potentially useful way to bypass these issues is to take a functional approach. In this article, theory is developed for a new functional version of the GARCH process, termed fGARCH. The main results are concerned with the structure of the fGARCH(1,1) process, providing criteria for the existence of strictly stationary solutions both in the space of square‐integrable and continuous functions. An estimation procedure is introduced, and its consistency and asymptotic normality are verified. A small empirical study highlights potential applications to intraday volatility estimation.     

Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models

The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Thm 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.     

Consistent estimation of the memory parameter for nonlinear time series

Abstract. For linear processes, semiparametric estimation of the memory parameter, based on the log‐periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ξ_t and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite‐sample performance of the estimator is investigated in a small Monte Carlo study.