Random Level Shifts Research Articles

Pricing VIX futures: A framework with random level shifts

We propose a DRLS framework to price VIX futures by modeling the logVIX series dynamics using the ARFIMA and HAR models that introduce the random level shifts component. Compared with other traditional time series models, our model allows the change of the theoretical mean value of the VIX index by time, which is more reasonable since there are different volatility states under different market environments. Using the Kalman filter, we can derive the explicit formula of the VIX futures price without calculating numerical integration that is different from models without random level shifts. The empirical results show that the DRLS framework performs better in both in-sample estimating and out-of-sample forecasting than directly pricing models without random level shifts and is much simpler.

Forecasting U.S. Yield Curve Using the Dynamic Nelson–Siegel Model with Random Level Shift Parameters

In this paper, we develop a new model based on the classical dynamic Nelson-Siegel model by introducing random level shift (RLS) parameters. The built-in RLS can capture cyclical fluctuations in interest rates and structural breaks induced by technological progress, financial crisis, major monetary policy interventions, etc. In addition, the model can be used to forecast future structural breaks. We apply the model to fit and forecast daily U.S. Treasury yield curves and the model outperforms other widely used models. The empirical results show that the model not only has a better in-sample fit with residuals exhibiting less persistence but also has superior out-of-sample performance. Moreover, the model performs very well especially for short-term and long-term bonds, and the performance improves as the forecasting horizon increases.

Modeling Volatility of Asset and Volume of Trade Returns in the Nigerian Stock Market in the Presence of Random Level Shifts

This study investigated the impact of volatility shock persistence on the conditional variance in the Nigerian stock returns using symmetric and asymmetric higher order GARCH family models in the presence of random level shifts and non-Gaussian errors. The study utilised Bai and Perron methodology to detect structural breakpoints in the conditional variance of daily stock and volume of trade returns in the Nigerian stock market from 2nd January, 1998 to 22nd March, 2017. The study employed symmetric GARCH (3,2) and GARCH (2,1)-M models to estimate volatility of asset returns, symmetric GARCH (2,2) and GARCH (2,1)-M to model volatility of volume of trade returns and asymmetric EGARCH (2,2), TGARCH (3,2) and PGARCH (2,3) models to measure the volatility of asset returns as well as asymmetric EGARCH (2,1), TGARCH (1,3) and PGARCH (3,2) models to estimate volatility of volume of trade returns. These models were optimally selected using information criteria and log likelihood as the best fitting symmetric and asymmetric GARCH models to estimate the conditional volatility of asset and volume of trade returns in the Nigerian stock market with and without structural breaks. Results revealed that when random level shifts were ignored in volatility models, the shocks persistence were very high with long memory and variance explosion. But when the random level shifts were incorporated into the GARCH models, there was a significant reduction in the volatility shocks persistence and long memory. Moreover, volatility half-lives also declined drastically while accounting for these sudden level shifts in variance. The study found asymmetry without leverage effects as well as a positive risk-return tradeoff for both asset and volume of trade returns in the Nigerian stock market. The Nigeria banking reform of 2004, the Global Financial and Economic Crises, as well as other local events in Nigeria, were found to have negative and significant impacts on the Nigerian stock market. The study provided some policy recommendations.

An empirical note about estimation and forecasting Latin American Forex returns volatility: the role of long memory and random level shifts components

A set of RLS-type models with ARMA and ARFIMA dynamics is estimated and compared in a forecasting exercise with ARFIMA, GARCH and FIGARCH models. It is an extension of Rodriguez (N Am J Econ Financ 42:393–420, 2017) but using more countries and working with squared returns in the forecasting exercise. The estimation results show: (i) existence of RLS; (ii) measurement errors except in Chile and Colombia. Regarding the fractional parameter, the estimates are quite small indicating the possible absence of long memory with possible exceptions of Chile and Colombia. The forecast exercise using the 10% MCS of Hansen et al. (Econometrica 79:453–497, 2011) and the ratios of MSFE indicate absence of the RLS-ARFIMA models while RLS-ARMA models are selected. In general, the results of the estimations and forecasts indicate probable absence of long memory or its small magnitude, which would makes this characteristic not only unnecessary but also irrelevant to capture the variations of the low frequencies of the series.

A multivariate test against spurious long memory

This paper provides a multivariate score-type test against spurious long memory. We prove the consistency of the test against the alternatives of random level shifts and smooth trends. The test statistic is based on the weighted sum of the partial derivatives of the multivariate local Whittle likelihood function. To apply the test to fractionally cointegrated series, the test statistic is calculated for the linearly transformed system after estimating the cointegrating matrix. We derive the limiting distribution and show consistency of this procedure. The test is applied to log-absolute returns and log-realized volatilities of the S&P 500, DAX, FTSE, and NIKKEI.

Modeling Latin-American stock and Forex markets volatility: Empirical application of a model with random level shifts and genuine long memory

Following Varneskov and Perron (2017a,b), I apply the RLS-ARFIMA(p,d,q) models to the daily stock and Forex market returns volatility of Argentina, Brazil, Mexico and Peru. Further, two sets of high-frequency data are also used. The model is a parametric state-space model with an estimation framework that combines long memory and level shifts by decomposing the underlying process into a mixture model and ARFIMA dynamics. The results of the estimates are not conclusive as those obtained in Varneskov and Perron (2017a,b). In fact, the very small magnitudes of the fractional parameter estimates suggest that only the high-frequency series could be modeled as RLS-ARFIMA models. The other (daily) series would be modeled as RLS-ARMA models with measurement errors except in the case of the Forex market of Brazil where there is no evidence of measurement errors. Another possibility is to accept the small magnitudes of the estimates of the fractional parameter as evidence of genuine long memory and, in that case, a larger group of series can be modeled as RLS-ARFIMA models. The forecasts are evaluated from two perspectives: one using 10% of the Model Confidence Set of Hansen, Lunde, and Nason (2011) and the other using a recent statistic proposed by Knüppel (2015) to evaluate density forecasts. The results favor the RLS-ARMA and RLS-ARFIMA models although we found some differences between the two approaches for the cases of Brazil and Mexico (stocks) and Argentina (Forex). Finally, forecasts are used to calculate the VaR at 1%, 5% and 10%. The results support broadly the RLS-ARFIMA models with one or two exceptions.

Combining long memory and level shifts in modelling and forecasting the volatility of asset returns

We propose a parametric state space model of asset return volatility with an accompanying estimation and forecasting framework that allows for ARFIMA dynamics, random level shifts and measurement errors. The Kalman filter is used to construct the state-augmented likelihood function and subsequently to generate forecasts, which are mean and path-corrected. We apply our model to eight daily volatility series constructed from both high-frequency and daily returns. Full sample parameter estimates reveal that random level shifts are present in all series. Genuine long memory is present in most high-frequency measures of volatility, whereas there is little remaining dynamics in the volatility measures constructed using daily returns. From extensive forecast evaluations, we find that our ARFIMA model with random level shifts consistently belongs to the 10% Model Confidence Set across a variety of forecast horizons, asset classes and volatility measures. The gains in forecast accuracy can be very pronounced, especially at longer horizons.

Medium band least squares estimation of fractional cointegration in the presence of low-frequency contamination

This paper introduces a new estimator of the fractional cointegrating vector between stationary long memory processes that is robust to low-frequency contamination such as random level shifts, outliers, Markov switching means, and certain deterministic trends. In particular, the proposed medium band least squares (MBLS) estimator uses sample-size-dependent trimming of frequencies in the vicinity of the origin to account for such contamination. Consistency and asymptotic normality of the MBLS estimator are established, a feasible inference procedure is proposed, and rigorous tools for assessing the cointegration strength and testing MBLS against the existing narrow band least squares estimator are developed. Finally, the asymptotic framework for the MBLS estimator is used to provide new perspectives on volatility factors in an empirical application to long-span realized variance series for S&P 500 equities.

Modelling the volatility of commodities prices using a stochastic volatility model with random level shifts

We use the approach of Qu and Perron (Econom J 16(3):309–339, 2013) for the modeling and inference of volatility of a set of commodity prices in the presence of random level shifts of unknown timing, magnitude and frequency. Our approach contributes to the study of commodities in several aspects. First, we test for the presence of a genuine long-memory process in the volatility of commodities. Second, we determine that the random level shifts are certainly the main source of variation in the commodity price volatility. Finally, we estimate the volatility and its components as latent variables, thereby making it possible to evaluate their level of correlation with macroeconomic variables in small open economies such as Latin-American countries where the dependence on commodity price volatility is high. We use six commodity series: agriculture, livestock, gold, oil, industrial metals and a general commodity index. All series cover the period from January 1983 until December 2013 in daily frequency. The results show that although the occurrence of a level shift is rare, (about once every 1.5 or 1.8 years), this component clearly contributes most to the variation in the volatility. Furthermore, isolating the level shift component from the overall volatility indicates a strong relationship of this component with a set of business cycle indicators of several Latin American countries.

Modelling exchange rate volatility with random level shifts

ABSTRACTRecent literature has shown that the volatility of exchange rate returns displays long memory features. It has also been shown that if a short memory process is contaminated by level shifts, the estimate of the long memory parameter tends to be upward biased. In this article, we directly estimate a random level shift model to the logarithm of the absolute returns of five exchange rates series, in order to assess whether random level shifts (RLSs) can explain this long memory property. Our results show that there are few level shifts for the five series, but once they are taken into account the long memory property of the series disappears. We also provide out-of-sample forecasting comparisons, which show that, in most cases, the RLS model outperforms popular models in forecasting volatility. We further support our results using a variety of robustness checks.

Modeling Latin-American stock markets volatility: Varying probabilities and mean reversion in a random level shift model

Following Xu and Perron (2014), I applied the extended RLS model to the daily stock market returns of Argentina, Brazil, Chile, Mexico and Peru. This model replaces the constant probability of level shifts for the entire sample with varying probabilities that record periods with extremely negative returns. Furthermore, it incorporates a mean reversion mechanism with which the magnitude and the sign of the level shift component vary in accordance with past level shifts that deviate from the long-term mean. Therefore, four RLS models are estimated: the Basic RLS, the RLS with varying probabilities, the RLS with mean reversion, and a combined RLS model with mean reversion and varying probabilities. The results show that the estimated parameters are highly significant, especially that of the mean reversion model. An analysis of ARFIMA and GARCH models is also performed in the presence of level shifts, which shows that once these shifts are taken into account in the modeling, the long memory characteristics and GARCH effects disappear. Also, I find that the performance prediction of the RLS models is superior to the classic models involving long memory as the ARFIMA(p,d,q) models, the GARCH and the FIGARCH models. The evidence indicates that except in rare exceptions, the RLS models (in all its variants) are showing the best performance or belong to the 10% of the Model Confidence Set (MCS). On rare occasions the GARCH and the ARFIMA models appear to dominate but they are rare exceptions. When the volatility is measured by the squared returns, the great exception is Argentina where a dominance of GARCH and FIGARCH models is appreciated.

An application of a random level shifts model to the volatility of Peruvian stock and exchange rate returns

The literature has shown that the volatility of stock and forex rate market returns shows the characteristic of long memory. Another fact that is shown in the literature is that this feature may be spurious and volatility actually consists of a short memory process contaminated with random level shifts (RLS). In this paper, we follow recent econometric approaches estimating an RLS model to the logarithm of the absolute value of stock and forex returns. The model consists of the sum of a short-term memory component and a component of level shifts. The second component is specified as the cumulative sum of a process that is zero with probability ‘1-alpha’ and is a random variable with probability ‘alpha’. The results show that there are level shifts that are rare, but once they are taken into account, the characteristic or property of long memory disappears. Also, the presence of General Autoregressive Conditional Heteroscedasticity (GARCH) effects is eliminated when included or deducted level shifts. An exercise of out-of-sample forecasting shows that the RLS model has better performance than traditional models for modelling long memory such as the models ARFIMA (p,d,q).

Modified local Whittle estimator for long memory processes in the presence of low frequency (and other) contaminations

We propose a modified local-Whittle estimator of the memory parameter of a long memory time series process which has good properties under an almost complete collection of contamination processes that have been discussed in the literature, mostly separately. These contaminations include processes whose spectral density functions dominate at low frequencies such as random level shifts, deterministic level shifts and deterministic trends. We show that our modified estimator has the usual asymptotic distribution applicable for the standard local Whittle estimator in the absence of such contaminations. We also show how the estimator can be modified to further account for additive noise and that our modification for low frequency contamination reduces the bias due to short-memory dynamics. Through extensive simulations, we show that the proposed estimator provides substantial efficiency gains compared to existing semiparametric estimators in the presence of contaminations, with little loss of efficiency when these are absent.

A detailed evaluation of format-compliant encryption methods for JPEG XR-compressed images

JPEG XR is the most recent still image coding standard, and custom security features for this format are required for fast adoption of the standard. Format-compliant encryption schemes are important for many application scenarios but need to be highly customised to a specific recent format like JPEG XR. This paper proposes, discusses, and evaluates a set of format-compliant encryption methods for the JPEG XR standard: coefficient scan order permutation, sign bit encryption, transform-based encryption, random level shift encryption, index-based VLC encryption, and encrypting entire frequency bands are considered. All algorithms are thoroughly evaluated by discussing possible compression impact, by assessing visual security and cryptographic security, and by discussing applicability in real-world scenarios. Most techniques are found to be insecure and, in a cryptographic sense, have a limited range of applicability and cannot be applied to JPEG XR bitstreams in an efficient manner. Encrypting entire frequency bands is identified to be a good solution in case a weaker form of format compliance can be accepted.

Forecasting return volatility: Level shifts with varying jump probability and mean reversion

We extend the random level shift (RLS) model of Lu and Perron (2010) to the volatility of asset prices, which consists of a short memory process and a random level shift component. Motivated by empirical features, (a) we specify a time-varying probability of shifts as a function of large negative lagged returns; and (b) we incorporate a mean reverting mechanism so that the sign and magnitude of the jump component change according to the deviations of past jumps from their long run mean. This allows the possibility of forecasting the sign and magnitude of the jumps. We estimate the model using twelve different series, and compare its forecasting performance with those of a variety of competing models at various horizons. A striking feature is that the modified RLS model has the smallest mean square forecast errors in 64 of the 72 cases, while it is a close second for the other 8 cases. The improvement in forecast accuracy is often substantial, especially for medium- to long-horizon forecasts. This is strong evidence that our modified RLS model offers important gains in forecasting performance.

Modeling gene expression evolution with an extended Ornstein-Uhlenbeck process accounting for within-species variation.

Much of the phenotypic variation observed between even closely related species may be driven by differences in gene expression levels. The current availability of reliable techniques like RNA-Seq, which can quantify expression levels across species, has enabled comparative studies. Ornstein-Uhlenbeck (OU) processes have been proposed to model gene expression evolution as they model both random drift and stabilizing selection and can be extended to model changes in selection regimes. The OU models provide a statistical framework that allows comparisons of specific hypotheses of selective regimes, including random drift, constrained drift, and expression level shifts. In this way, inferences may be made about the mode of selection acting on the expression level of a gene. We augment this model to include within-species expression variance, allowing for modeling of nonevolutionary expression variance that could be caused by individual genetic, environmental, or technical variation. Through simulations, we explore the reliability of parameter estimates and the extent to which different selective regimes can be distinguished using phylogenies of varying size using both the typical OU model and our extended model. We find that if individual variation is not accounted for, nonevolutionary expression variation is often mistaken for strong stabilizing selection. The methods presented in this article are increasingly relevant as comparative expression data becomes more available and researchers turn to expression as a primary evolving phenotype.

A stochastic volatility model with random level shifts and its applications to S&P 500 and NASDAQ return indices

This paper proposes a framework for the modelling, inference and forecasting of volatility in the presence of level shifts of unknown timing, magnitude and frequency. First, we consider a stochastic volatility model comprising both a level shift and a short‐memory component, with the former modelled as a compound binomial process and the latter as an AR(1). Next, we adopt a Bayesian approach for inference and develop algorithms to obtain posterior distributions of the parameters and the two latent components. Then, we apply the model to daily S&P 500 and NASDAQ returns over the period 1980.1–2010.12. The results show that although the occurrence of a level shift is rare, about once every 2 years, this component clearly contributes most to the variation in the volatility. The half‐life of a typical shock from the AR(1) component is short, on average between 9 and 15 days. Interestingly, isolating the level shift component from the overall volatility reveals a stronger relationship between volatility and business cycle movements. Although the paper focuses on daily index returns, the methods developed can potentially be used to study the low‐frequency variation in realized volatility or the volatility of other financial or macroeconomic variables.

Memory Parameter Estimation in the Presence of Level Shifts and Deterministic Trends

We propose estimators of the memory parameter of a time series that are robust to a wide variety of random level shift processes, deterministic level shifts and deterministic time trends. The estimators are simple trimmed versions of the popular log-periodogram regression estimator that employ certain sample size-dependent and, in some cases, data-dependent trimmings which discard low-frequency components. We also show that a previously developed trimmed local Whittle estimator is robust to the same forms of data contamination. Regardless of whether the underlying long/shortmemory process is contaminated by level shifts or deterministic trends, the estimators are consistent and asymptotically normal with the same limiting variance as their standard untrimmed counterparts. Simulations show that the trimmed estimators perform their intended purpose quite well, substantially decreasing both finite sample bias and root mean-squared error in the presence of these contaminating components. Furthermore, we assess the tradeoffs involved with their use when such components are not present but the underlying process exhibits strong short-memory dynamics or is contaminated by noise. To balance the potential finite sample biases involved in estimating the memory parameter, we recommend a particular adaptive version of the trimmed log-periodogram estimator that performs well in a wide variety of circumstances. We apply the estimators to stock market volatility data to find that various time series typically thought to be long-memory processes actually appear to be short or very weak long-memory processes contaminated by level shifts or deterministic trends.

MEMORY PARAMETER ESTIMATION IN THE PRESENCE OF LEVEL SHIFTS AND DETERMINISTIC TRENDS

We propose estimators of the memory parameter of a time series that are robust to a wide variety of random level shift processes, deterministic level shifts, and deterministic time trends. The estimators are simple trimmed versions of the popular log-periodogram regression estimator that employ certain sample-size-dependent and, in some cases, data-dependent trimmings that discard low-frequency components. We also show that a previously developed trimmed local Whittle estimator is robust to the same forms of data contamination. Regardless of whether the underlying long- or short-memory process is contaminated by level shifts or deterministic trends, the estimators are consistent and asymptotically normal with the same limiting variance as their standard untrimmed counterparts. Simulations show that the trimmed estimators perform their intended purpose quite well, substantially decreasing both finite-sample bias and root mean-squared error in the presence of these contaminating components. Furthermore, we assess the trade-offs involved with their use when such components are not present but the underlying process exhibits strong short-memory dynamics or is contaminated by noise. To balance the potential finite-sample biases involved in estimating the memory parameter, we recommend a particular adaptive version of the trimmed log-periodogram estimator that performs well in a wide variety of circumstances. We apply the estimators to stock market volatility data to find that various time series typically thought to be long-memory processes actually appear to be short- or very weak long-memory processes contaminated by level shifts or deterministic trends.

When long memory meets the Kalman filter: A comparative study

The finite sample properties of the state space methods applied to long memory time series are analyzed through Monte Carlo simulations. The state space setup allows to introduce a novel modeling approach in the long memory framework, which directly tackles measurement errors and random level shifts. Missing values and several alternative sources of misspecification are also considered. It emerges that the state space methodology provides a valuable alternative for the estimation of the long memory models, under different data generating processes, which are common in financial and economic series. Two empirical applications highlight the practical usefulness of the proposed state space methods.