Autoregressive Conditional Heteroscedastic Process Research Articles

A multivariate heavy-tailed integer-valued GARCH process with EM algorithm-based inference

A new multivariate integer-valued Generalized AutoRegressive Conditional Heteroscedastic (GARCH) process based on a multivariate Poisson generalized inverse Gaussian distribution is proposed. The estimation of parameters of the proposed multivariate heavy-tailed count time series model via maximum likelihood method is challenging since the likelihood function involves a Bessel function that depends on the multivariate counts and its dimension. As a consequence, numerical instability is often experienced in optimization procedures. To overcome this computational problem, two feasible variants of the expectation-maximization (EM) algorithm are proposed for estimating the parameters of our model under low and high-dimensional settings. These EM algorithm variants provide computational benefits and help avoid the difficult direct optimization of the likelihood function from the proposed process. Our model and proposed estimation procedures can handle multiple features such as modeling of multivariate counts, heavy-tailedness, overdispersion, accommodation of outliers, allowances for both positive and negative autocorrelations, estimation of cross/contemporaneous-correlation, and the efficient estimation of parameters from both statistical and computational points of view. Extensive Monte Carlo simulation studies are presented to assess the performance of the proposed EM algorithms. Two empirical applications of our approach are provided. The first application concerns modeling bivariate count time series data on cannabis possession-related offenses in Australia, while the second one involves modeling intraday high-frequency financial transactions data from multiple holdings in the U.S. financial market.

Dynamic spatiotemporal ARCH models

ABSTRACT Geo-referenced data are characterised by an inherent spatial dependence due to geographical proximity. In this paper, we introduce a dynamic spatiotemporal autoregressive conditional heteroscedasticity (ARCH) process to describe the effects of (i) the log-squared time-lagged outcome variable, the temporal effect, (ii) the spatial lag of the log-squared outcome variable, the spatial effect, and (iii) the spatiotemporal effect on the volatility of an outcome variable. We derive a generalised method of moments (GMM) estimator based on the linear and quadratic moment conditions. We show the consistency and asymptotic normality of the GMM estimator. After studying the finite-sample performance in simulations, the model is demonstrated by analysing monthly log-returns of condominium prices in Berlin from 1995 to 2015, for which we found significant volatility spillovers.

Nearly unstable integer‐valued ARCH process and unit root testing

AbstractThis paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom.

Change detection from high-resolution airborne laser scans using penalized composite likelihood screening

The study of elevation change using pre- and post-event raster grids obtained by airborne laser scans is central to many scientific areas, including geomorphology, landscape science and environmental science. Separating real elevation change from noise is a nontrivial task due to complex sources of measurement uncertainty causing errors to be spatially variable. For high-resolution grids, an additional hurdle is the excessive computational effort required by common statistical models for the spatial error. This paper introduces a screening methodology for change detection in high-resolution elevation difference grids using a composite likelihood regularization framework. We propose to model the spatial error by a sparse spatial autoregressive conditional heteroscedasticity process and obtain simultaneous detection of change location and change size estimation by maximizing a penalized composite likelihood objective. The methodology is implemented through a fast coordinate-wise maximization algorithm with computational complexity growing linearly with the grid size. Numerical studies on simulated and real data show satisfactory accuracy for both change detection and parameter estimation.

Convergence rate bounds for iterative random functions using one-shot coupling

One-shot coupling is a method of bounding the convergence rate between two copies of a Markov chain in total variation distance, which was first introduced in Roberts and Rosenthal (Process Appl 99:195–208, 2002) and generalized in Madras and Sezer (Bernoulli 16:882–908, 2010). The method is divided into two parts: the contraction phase, when the chains converge in expected distance and the coalescing phase, which occurs at the last iteration, when there is an attempt to couple. One-shot coupling does not require the use of any exogenous variables like a drift function or a minorization constant. In this paper, we summarize the one-shot coupling method into the One-Shot Coupling Theorem. We then apply the theorem to two families of Markov chains: the random functional autoregressive process and the autoregressive conditional heteroscedastic process. We provide multiple examples of how the theorem can be used on various models including ones in high dimensions. These examples illustrate how the theorem’s conditions can be verified in a straightforward way. The one-shot coupling method appears to generate tight geometric convergence rate bounds.

Noise content assessment in GNSS coordinate time-series with autoregressive and heteroscedastic random errors

SUMMARYProper representation of the stochastic process of the Global Navigation Satellite System (GNSS) coordinate time-series lays a foundation for realistic velocities estimates and their uncertainties, as well as identifications of functional effects, generally called periodic/offset signals. Various noise models (e.g. autoregressive process) have been used to describe the stochastic process of GNSS data. A realistic stochastic model prefers proper consideration of individual error characteristics of observations. The usual autoregressive (AR) noise model assumes its white-noise components sharing the same constant variances, which might degrade the modelling accuracy of stochastic process. Through using a generalized autoregressive conditional heteroscedasticity (GARCH) process to quantify variance variation of the white-noise components, our current investigation has constituted an AR-GARCH noise model to realize the stochastic model and subsequent noise content assessment. This noise framework and its adjustment algorithm are further considered in the identifications of functional effects. For evaluating the performance of our current algorithm, 500 emulated real time-series were used, they consist of a linear trend, seasonal periodic signals, offsets, gaps (up to 10 per cent) and an AR-GARCH noise process. The algorithm's identification power for functional effects was investigated within a case study in the absence of periodic/offset signals. Furthermore, the algorithm's results were compared with the current state-of-the-art noise models (e.g. white plus flicker noise) using 15 real GNSS coordinate time-series. The results demonstrated that the presented stochastic model has been identified as a preferred noise model. Its model's algorithm can offer reliable noise content although sporadic unidentified periodic/offset signals are still masked in time-series. Moreover, the error volatility of white-noise components was identified via an autoregressive conditional heteroscedasticity Lagrange multiplier (ARCH LM) test and confirmed in the time-series.

Directional spatial autoregressive dependence in the conditional first- and second-order moments

In contrast to classical econometric approaches which are based on prespecified isotropic weighting schemes, we suggest that the spatial weighting matrix in the presence of directional dependencies should be estimated. We identify this direction based on different candidate neighbourhood sets. In this paper, we consider two different types of processes – namely spatial autoregressive and spatial autoregressive conditional heteroscedastic processes – and derive the consistency of the corresponding maximum likelihood estimates in the presence of directional dependencies. Moreover, Monte Carlo simulation results indicate that the model’s performance improves with sample size and with smaller neighbourhood subset sizes. Finally, we apply this approach to aerosol observations over the North Atlantic region and show that their spatial dependence matches the direction of the trade winds in this area.

Longevity swaps for longevity risk management in life insurance products

PurposeThe life insurance industry has been exposed to high levels of longevity risk born from the mismatch between realized mortality trends and anticipated forecast. Annuity providers are exposed to extended periods of annuity payments. There are no immediate instruments in the market to counter the risk directly. This paper aims to develop appropriate instruments for hedging longevity risk and providing an insight on how existing products can be tailor-made to effectively immunize portfolios consisting of life insurance using a cointegration vector error correction model with regime-switching (RS-VECM), which enables both short-term fluctuations, through the autoregressive structure [AR(1)] and long-run equilibria using a cointegration relationship. The authors also develop synthetic products that can be used to effectively hedge longevity risk faced by life insurance and annuity providers who actively hold portfolios of life insurance products. Models are derived using South African data. The authors also derive closed-form expressions for hedge ratios associated with synthetic products written on life insurance contracts as this will provide a natural way of immunizing the associated portfolios. The authors further show how to address the current liquidity challenges in the longevity market by devising longevity swaps and develop pricing and hedging algorithms for longevity-linked securities. The use of a cointergrating relationship improves the model fitting process, as all the VECMs and RS-VECMs yield greater criteria values than their vector autoregressive model (VAR) and regime-switching vector autoregressive model (RS-VAR) counterpart’s, even though there are accruing parameters involved.Design/methodology/approachThe market model adopted from Ngai and Sherris (2011) is a cointegration RS-VECM for this enables both short-term fluctuations, through the AR(1) and long-run equilibria using a cointegration relationship (Johansen, 1988, 1995a, 1995b), with a heteroskedasticity through the use of regime-switching. The RS-VECM is seen to have the best fit for Australian data under various model selection criteria by Sherris and Zhang (2009). Harris (1997) (Sajjad et al., 2008) also fits a regime-switching VAR model using Australian (UK and US) data to four key macroeconomic variables (market stock indices), showing that regime-switching is a significant improvement over autoregressive conditional heteroscedasticity (ARCH) and generalised autoregressive conditional heteroscedasticity (GARCH) processes in the account for volatility, evidence similar to that of Sherris and Zhang (2009) in the case of Exponential Regressive Conditional Heteroscedasticity (ERCH). Ngai and Sherris (2011) and Sherris and Zhang (2009) also fit a VAR model to Australian data with simultaneous regime-switching across many economic and financial series.FindingsThe authors develop a longevity swap using nighttime data instead of usual income measures as it yields statistically accurate results. The authors also develop longevity derivatives and annuities including variable annuities with guaranteed lifetime withdrawal benefit (GLWB) and inflation-indexed annuities. Improved market and mortality models are developed and estimated using South African data to model the underlying risks. Macroeconomic variables dependence is modeled using a cointegrating VECM as used in Ngai and Sherris (2011), which enables both short-run dependence and long-run equilibrium. Longevity swaps provide protection against longevity risk and benefit the most from hedging longevity risk. Longevity bonds are also effective as a hedging instrument in life annuities. The cost of hedging, as reflected in the price of longevity risk, has a statistically significant effect on the effectiveness of hedging options.Research limitations/implicationsThis study relied on secondary data partly reported by independent institutions and the government, which may be biased because of smoothening, interpolation or extrapolation processes.Practical implicationsAn examination of South Africa’s mortality based on industry experience in comparison to population mortality would demand confirmation of the analysis in this paper based on Belgian data as well as other less developed economies. This study shows that to provide inflation-indexed life annuities, there is a need for an active market for hedging inflation in South Africa. This would demand the South African Government through the help of Actuarial Society of South Africa (ASSA) to issue inflation-indexed securities which will help annuities and insurance providers immunize their portfolios from longevity risk.Social implicationsIn South Africa, there is an infant market for inflation hedging and no market for longevity swaps. The effect of not being able to hedge inflation is guaranteed, and longevity swaps in annuity products is revealed to be useful and significant, particularly using developing or emerging economies as a laboratory. This study has shown that government issuance or allowing issuance, of longevity swaps, can enable insurers to manage longevity risk. If the South African Government, through ASSA, is to develop a projected mortality reference index for South Africa, this would allow the development of mortality-linked securities and longevity swaps which ultimately maximize the social welfare of life assurance policy holders.Originality/valueThe paper proposes longevity swaps and static hedging because they are simple, less costly and practical with feasible applications to the South African market, an economy of over 50 million people. As the market for MLS develops further, dynamic hedging should become possible.

Non‐linear Kalman filters comparison for generalised autoregressive conditional heteroscedastic clutter parameter estimation

In this work, the authors analyse the estimation of the generalised autoregressive conditional heteroscedastic (GARCH) process conditional variance based on three non-linear filtering approaches: extended Kalman filter (EKF), unscented Kalman filter and cubature Kalman filter. The authors present a state model for a GARCH process and derive an EKF including second-order non-linear terms for simultaneous estimation of state and parameters. Using synthetic data, the authors evaluate the consistency and the correlation of the innovations for the three filters, by means of numerical simulations. The authors also study the performance of smoothed versions of the non-linear Kalman filters using real clutter data in comparison with a conventional quasi-maximum likelihood estimation method for the GARCH process coefficients. The authors show that with all methods the process coefficients estimates are of the same order and the resulting conditional variances are commensurable. However, the non-linear Kalman filters greatly reduce the computational load. These kind of filters could be used for the radar detector based on a GARCH clutter model that uses an adaptive threshold that demands the conditional variance at each decision instant.

Fourth Moment Structure of Markov Switching Multivariate GARCH Models

Abstract We derive sufficient conditions for the existence of second and fourth moments of Markov switching multivariate generalized autoregressive conditional heteroscedastic processes in the general vector specification. We provide matrix expressions in closed form for such moments, which are obtained by using a Markov switching vector autoregressive moving-average representation of the initial process. These expressions are shown to be readily programmable in addition of greatly reducing the computational cost. As theoretical applications of the results, we derive the spectral density matrix of the squares and cross products, propose a new definition of multivariate kurtosis measure to recognize heavy-tailed features in financial real data, and provide a matrix expression in closed form of the impulse-response function for the volatility. An empirical example illustrates the results.

A STUDY ON THE RETURN VOLATILITY OF THE COFFEE BEANS PRICE USING ARCH MODELS

In the case of primary commodities, price volatility would arise mainly due to disturbances in supply, whereas for industrial raw materials, it would be the result of disturbances in demand. In the analysis of commodity markets can be seen that information, hedging, speculation and physical availability are factors that can influence their volatility. Moreover, increased volatility in commodity markets can justify the use of information-based processes for modeling the pattern of return volatility of these commodities. Since the relevance of autoregressive conditional heteroscedasticity (ARCH) family models in the solution of problems in economic and financial areas due to their applicability and interpretation (the relation between return and volatility) have been provided, the aim of this work was to compare the Bayesian estimates for the parameters of ARCH processes with normal and Student’s t distributions for the conditional distribution of the return series of coffee beans price. In addition, informative prior distributions were suggested and posterior summaries were obtained by Monte Carlo Markov Chain simulation methods. Results showed that the proposed Bayesian approach provides satisfactory estimates and that the ARCH process with Student’s t distribution adjusts better to the data.

Exchange market pressure in South Africa and Kenya: An analysis using parametric and non-parametric extreme value theory

Orientation: Exchange market pressure (EMP) is the selling pressure of domestic currency or excess demand needed for foreign currency. Research purpose: The purpose of this study was to analyse EMP using extreme value theory (EVT) and to compare two commonly used EVT methods. Motivation for the study: To determine whether the EMP of two African countries can be modelled with EVT, and if so, which method would be best suited. To determine periods of extreme pressure or currency crisis by using these methods. Lastly, to study the individual components of the EMP index during these periods of stress of crises. Research design/approach and method: The monthly data of the three components of the EMP index for two African countries (Republic of South Africa [RSA] and Kenya) were studied for a period of 19 years (1999–2017). The data were modelled using the generalised Pareto distribution (GPD) with the peak over threshold (POT) method using maximum likelihood estimation. Moreover, the data were also modelled using the non-parametric Hill estimate. Appropriate estimated and empirical quantiles are reported in order to determine periods of extreme pressure or crises. The components of the EMP RSA data set were modelled with appropriate autoregressive moving averages and/or autoregressive conditional heteroscedasticity/generalised autoregressive conditional heteroscedasticity (ARMA/ARCH/GARCH) processes to ensure independent and identically distributed variables. Main findings: Reliable and accurate estimates were obtained for the scale and shape parameter using the GPD POT method for both countries. Positive shape parameters confirmed generalised extreme value distributions with heavy tails of the Frechet type. Similarly, accurate and reliable shape estimates were computed using the non-parametric Hill estimator for both countries. Practical/managerial implications: The GPD POT method more closely reflected estimates to the empirical qauntiles compared to the Hill method. It is feasible to model the EMP data of two African countries with EVT using both POT estimation methods. However, the GPD method using maximum likelihood estimation was more accurate compared to the non-parametric Hill estimate. Contribution/value-add: It is feasible to model the EMP data of two African countries using EVT using both POT methods. As a consequence periods of extreme pressure could be identified. Therefore the individual components of the EMP at those periods of extreme pressure could be studied more closely.

A discrete-time risk model with Poisson ARCH claim-number process

In this paper, we propose a discrete-time risk model with the claim number following an integer-valued autoregressive conditional heteroscedasticity (ARCH) process with Poisson deviates. In this model, the current claim number depends on the previous observations. Within this framework, the equation for finding the adjustment coefficient is derived. Numerical studies are also carried out to examine the impact of the Poisson ARCH dependence structure on the ruin probability.

Moving Average Market Timing in European Energy Markets: Production Versus Emissions

This paper searches for stochastic trends and returns predictability in key energy asset markets in Europe over the last decade. The financial assets include Intercontinental Exchange Futures Europe (ICE-ECX) carbon emission allowances (the main driver of interest), European Energy Exchange (EEX) Coal ARA futures and ICE Brent oil futures (reflecting the two largest energy sources in Europe), Stoxx600 Europe Oil and Gas Index (the main energy stock index in Europe), EEX Power Futures (representing electricity), and Stoxx600 Europe Renewable Energy index (representing the sunrise energy industry). This paper finds that the Moving Average (MA) technique beats random timing for carbon emission allowances, coal, and renewable energy. In these asset markets, there seems to be significant returns predictability of stochastic trends in prices. The results are mixed for Brent oil, and there are no predictable trends for the Oil and Gas index. Stochastic trends are also missing in the electricity market as there is an ARFIMA-FIGARCH process in the day-ahead power prices. The empirical results are interesting for several reasons. We identified the data generating process in EU electricity prices as fractionally integrated (0.5), with a fractionally integrated Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) process in the residual. This is a novel finding. The order of integration of order 0.5 implies that the process is not stationary but less non-stationary than the non-stationary I(1) process, and that the process has long memory. This is probably because electricity cannot be stored. Returns predictability with MA rules requires stochastic trends in price series, indicating that the asset prices should obey the I(1) process, that is, to facilitate long run returns predictability. However, all the other price series tested in the paper are I(1)-processes, so that their returns series are stationary. The empirical results are important because they give a simple answer to the following question: When are MA rules useful? The answer is that, if significant stochastic trends develop in prices, long run returns are predictable, and market timing performs better than does random timing.

Commodity price volatility and U.S. monetary policy: Commodity price overshooting revisited

AbstractCommodity price volatility has created concerns for central bank policy‐makers. Recent commodity prices peaked in the consequences of the financial crisis of 2007, and they have remained relatively volatile since. As they are often seen as being connected in a cause and effect relationship with inflation and real output, the driving forces behind commodity price volatility are necessary for the conduct of monetary policy. Using an autoregressive moving average with an exponential generalized autoregressive conditional heteroscedastic (ARMA‐EGARCH) process, we extract the conditional variance series to find volatility spillover between monetary policy and commodity price index. The findings show that the volatility of agricultural commodity price index and other commodities price indices overshoot the long‐run equilibrium price in response to an impulse in monetary policy.

Test for tail index constancy of GARCH innovations based on conditional volatility

This study considers the problem of testing whether the tail index of the GARCH innovations undergoes a change according to the values of conditional volatilities. Special attention is paid to power-transformed and threshold generalized autoregressive conditional heteroscedasticity processes that can accommodate the GARCH family. We show that the proposed test asymptotically follows a functional of a standard Brownian motion under some regularity conditions. To evaluate our method, we carry out a simulation study and real data analysis using the return series of the Google stock price and DowJones index.

Exploiting tail shape biases to discriminate between stable and student t alternatives

SummaryThe nonnormal stable laws and Student t distributions are used to model the unconditional distribution of financial asset returns, as both models display heavy tails. The relevance of the two models is subject to debate because empirical estimates of the tail shape conditional on either model give conflicting signals. This stems from opposing bias terms. We exploit the biases to discriminate between the two distributions. A sign estimator for the second‐order scale parameter strengthens our results. Tail estimates based on asset return data match the bias induced by finite‐variance unconditional Student t data and the generalized autoregressive conditional heteroscedasticity process.

Comparison of ARCH / GARCH model and Elman Recurrent Neural Network on data return of closing price stock

Time series data can be modeled using the Autoregressive Intergrated Moving Average (ARIMA) model. ARIMA model requires the assumption of homoscedasticity. But on financial data, like stock data, has a high fluctuation so that the variant is heteroscedastic. To solve this problem, Autoregressive Conditional Heteroscedasticity (ARCH) or Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model is used. The ARCH process introduced in Engle allows the conditional variance to change over time as a function of past errors leaving the unconditional variance constant. The ARCH model was generalized to the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and introduced in Bollerslev allowing for a much more flexible lag structure.Another model that can be used is the Elman Recurrent Neural Network model (ERNN) which is a nonlinear model. The purpose of this research is to determine the best model between ARCH / GARCH model and ERNN model based on MSE value. The data used is the return of daily closing price of BRI stocks period of January 1st, 2015 to 17th February 2017. In this study, the best model obtained is GARCH (1,1) which has the smallest MSE value.

Self-weighted recursive estimation of GARCH models

ABSTRACTThe generalized autoregressive conditional heteroscedasticity (GARCH) processes are frequently used to investigate and model financial returns. They are routinely estimated by computationally complex off-line estimation methods, for example, by the conditional maximum likelihood procedure. However, in many empirical applications (especially in the context of high-frequency financial data), it seems necessary to apply numerically more effective techniques to calibrate and monitor such models. The aims of this contribution are: (i) to review the previously introduced recursive estimation algorithms and to derive self-weighted alternatives applying general recursive identification instruments, and (ii) to examine these methods by means of simulations and an empirical application.

Commodity Price Volatility and U.S. Monetary Policy: Commodity Price Overshooting Revisited

Commodity price volatility can create concern for central bank policy-makers. Recent commodity prices peaked in the aftermath of the financial crisis of 2007, and they have remained relatively volatile since. As they are often seen as being connected in a cause and effect relationship with inflation and real output, the driving forces behind commodity price volatility are crucial for the conduct of monetary policy (Svensson, 2005). Using an autoregressive moving average with an exponential generalized autoregressive conditional heteroscedastic (ARMA-EGARCH) process, we extract the conditional variance series to identify volatility spillovers between monetary policies and commodity price index. The findings show that the volatility of agricultural commodity price index and other commodities price indices overshoot their long-run equilibrium in response to an impulse in monetary policy.