ARFIMA Model Research Articles

Estimation of Value-at-Risk Adjusted under the Capital Asset Pricing Model Based on ARMAX-GARCH Approach

Investors having an understanding of investment statistics are important. Especially quantitative tools related to investment risk measurement. Value-at-Risk Adjusted is one of the investment risk measurement tools, which assumes that returns are not normally distributed.This paper intends to measure investment risk based onValue-at-Risk Adjustedor called Modified Value-at-Risk under the Capital Asset Pricing Model. It is assumed that the return of the market index has a non-constant average and there is a long memory effect. The average of the return of the market index is estimated using ARFIMA models.It is also assumed that the stock risk premium correlates with market risk premiums, and stock risk premiums some time before. The correlation will be analyzed using the ARMAX-GARCH model approach. The Modified Value-at-Risk was then formulated based on the Capital asset Pricing Model with the ARMAX-GARCH model approach.To measure the performance of Modified Value-at-Risk that has been formulated is done with back testing. Back testing is carried out based on the Lopez II method. As a case study, analyzed some data on 10 stocks traded on the capital market in Indonesia.The results of the analysis show that the market index return risk premium significantly follows the ARFIMA model, and the 10 share risk premium significantly follows the ARMAX-GARCH model. Based on the results of back testing calculations indicate that the Value-at-Risk Adjustedor Modified Value-at-Risk is very suitable to be used to measure investment risk in the 10 stocks analyzed.

Forecasting the realized volatility of CSI 300

According to the characteristics of realized volatility existing in Shanghai and Shenzhen 300 index (China Securities Index 300, CSI 300), the GARCH family model is introduced to describe the ARCH effect of error sequences in HAR, ARFIMA, and ARFIMAX models. Then, the HAR-GARCH family, the ARFIMA-GARCH family, and the ARFIMAX-GARCH family models set is proposed, which contains 33 kinds of models. Using the quasi maximum likelihood method, the parameters of the all models are estimated with normal (N) and skewed student t (SKST) distributions. By rolling window technology, one-step-ahead rolling prediction of realized volatility for CSI 300 is conducted. The results from prediction accuracy by model confidence set (MCS) test show that the realized volatility prediction models with skewed student t distribution possess higher precision than those of normal distribution in general. The ARFIMAX and other six models in ARFIMAX-GARCH family form the optimal prediction models set, which can forecast the realized volatility of CSI 300 with better accuracy.

On History of Mathematical Economics: Application of Fractional Calculus

Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed.

Gold price modeling in Indonesia using ARFIMA method

Gold investment is the best choice to control finance. Gold is easy to resell if there is a financial need at the unpredictable moment. The data of gold price in Indonesia is a long-term memory data series or a time series data that has a long-term dependency. ARFIMA model is an appropriate model for such long-term memory data series since ARFIMA model has a different parameter value (d) of integer while the value of d in ARFIMA model is non-integer value due to the long-term memory or the long-term dependency. This research aims to obtain the best ARFIMA model of gold price data in Indonesia. It is obtained the value of ARFIMA model (1,d,[3]) with d=1,05716 as the best model.

Forecasting realised volatility using ARFIMA and HAR models

Recent literature provides mixed empirical evidence with respect to the forecasting performance of ARFIMA and HAR models. This paper compares the forecasting performance of both models using high frequency data of 100 stocks representing 10 business sectors for the period 2000-2010. We allow for different sectors, changing market conditions, variation in the sampling frequency and forecasting horizons. For the overall sample and using the 300 sec sampling frequency, the forecasting performance of both models is indistinguishable. However, differences arise under different market regimes, forecasting horizons and sampling frequencies. ARFIMA models are superior for the crisis and pre-crisis sub-samples. HAR forecasts are less sensitive to regime change and to longer forecasting horizons. Variations in forecasting performance could also be explained using differences in the levels of persistence underlying each model.

Forecasting Realized Volatility of Russian stocks using Google Trends and Implied Volatility

This work proposes to forecast the Realized Volatility (RV) and the Value-at-Risk (VaR) of the most liquid Russian stocks using GARCH, ARFIMA and HAR models, including both the implied volatility computed from options prices and Google Trends data. The in-sample analysis showed that only the implied volatility had a significant effect on the realized volatility across most stocks and estimated models, whereas Google Trends did not have any significant effect. The outof-sample analysis highlighted that models including the implied volatility improved their forecasting performances, whereas models including internet search activity worsened their performances in several cases. Moreover, simple HAR and ARFIMA models without additional regressors often reported the best forecasts for the daily realized volatility and for the daily Value-at-Risk at the 1 % probability level, thus showing that efficiency gains more than compensate any possible model misspecifications and parameters biases. Our empirical evidence shows that, in the case of Russian stocks, Google Trends does not capture any additional information already included in the implied volatility.

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.

Analysis of complex time series based on EMD energy entropy plane

Empirical mode decomposition (EMD) is a self-adaptive signal processing method that can be applied to nonlinear and non-stationary processes perfectly. In view of this good ability of EMD, in this paper, we propose a new method—EMD energy entropy plane—which combines two different tools—EMD energy entropy and complexity-entropy causality plane—to analyze time series. Firstly, we apply EMD energy entropy plane to synthetic data, such as logistic map, Henon map, ARFIMA model and so on, finding that the EMD energy entropy plane presents different trends and distributions when the map is in periodic cycles and chaos. Then we demonstrate the application of EMD energy entropy plane in stock markets. Results show that it is an effective tool of distinguishing two kinds of financial markets. In addition, the introduction of multi-scale reveals the variation law of EMD energy entropy plane at different scales.

FORECASTING FRESH WATER AND MARINE FISH PRODUCTION IN MALAYSIA USING ARIMA AND ARFIMA MODELS

Malaysia is surrounded by sea, rivers and lakes which provide natural sources of fish for human consumption. Hence, fish is one source of protein supply to the country and fishery is a sub-sector that contribute to the national gross domestic product. Since fish forecasting is crucial in fisheries management for managers and scientists, time series modelling can be one useful tool. Time series modelling have been used in many fields of studies including the fields of fisheries. In a previous research, the ARIMA and ARFIMA models were used to model marine fish production in Malaysia and the ARFIMA model emerged to be a better forecast model. In this study, we consider fitting the ARIMA and ARFIMA to both the marine and freshwater fish production in Malaysia. The process of model fitting was done using the “ITSM 2000, version 7.0” software. The performance of the models were evaluated using the mean absolute error, root mean square error and mean absolute percentage error. It was found in this study that the selection of the best fit model depends on the forecast accuracy measures used.

Evaluation of Long Memory on the Malaysia Exchange Rate Market

This research evaluates the presence of long memory or long-term dependence on the Malaysian exchange rate. Daily, weekly and monthly data are evaluated against the US dollar (USD) covering from January 2005 to March 2018. Evaluation of long memory is based on the Geweke and Porter-Hudak estimation and the Maximum Likelihood Estimation. The result suggests the presence of long memory on all the daily, weekly and monthly data. Results show that shock on the Malaysian exchange rate persist longer than expected. The forecast capability also concludes that addition of the long memory presence from ARIMA model to ARFIMA model could improve the model forecast.

Long range dependence in the Bitcoin market: A study based on high-frequency data

Using the high-frequency data of Bitcoin, this paper investigates the long memory characteristics of the unconditional and conditional volatilities of Bitcoin at different time scales using the local Whittle (LW) estimator, the exact local Whittle (ELW) estimator and the ARMA–FIAPARCHmodel. The results show that the long memory parameter is significant and quite stable for both unconditional and conditional volatility measures across different time scales. This paper also examines the long memory characteristics of the unconditional and conditional “realized” volatilities of Bitcoin at different time scales using the local Whittle (LW) estimator, exact local Whittle (ELW) estimator and the ARFIMA model. Long memory is found to be significant and stable also in case of unconditional and conditional “realized” volatilities. The study also undertakes quarterly non-overlapping rolling window analysis to develop deeper insights into the evolution of long memory parameter, d, over the period. The results indicate high persistence in the Bitcoin market. This study has useful implications for different investors and market participants having varying exposures in the Bitcoin market depending on their trading horizons. The findings can help them in forecasting the expected volatility in the Bitcoin market and thereby in developing and implementing trading strategies.

Prediction intervals in the ARFIMA model using bootstrap G

This paper presents a bootstrap resampling scheme to build pre-diction intervals for future values in fractionally autoregressive movingaverage (ARFIMA) models. Standard techniques to calculate forecastintervals rely on the assumption of normality of the data and do nottake into account the uncertainty associated with parameter estima-tion. Bootstrap procedures, as nonparametric methods, can overcomethese diculties. In this paper, we test two bootstrap prediction in-tervals based on the nonparametric bootstrap in the residuals of theARFIMA model. In this paper, two bootstrap prediction intervals areproposed based on the nonparametric bootstrap in the residuals ofthe ARFIMA model. The rst one is the well known percentile boot-strap, (Thombs and Schucany, 1990; Pascual et al., 2004), never usedfor ARFIMA models to the knowlegde of the authors. For the secondapproach, the intervals are calculated using the quantiles of the empir-ical distribution of the bootstrap prediction errors (Masarotto, 1990;Bisaglia e Grigoletto, 2001). The intervals are compared, througha Monte Carlo experiment, to the asymptotic interval, under Gaus-sian and non-Gaussian error distributions. The results show that thebootstrap intervals present coverage rates closer to the nominal levelassumed, when compared to the asymptotic standard method. An ap-plication to real data of temperature in New York city is also presentedto illustrate the procedures.

Autoregressive Times Series Methods for Time Domain Astronomy

Celestial objects exhibit a wide range of variability in brightness at different wavebands. Surprisingly, the most common methods for characterizing time series in statistics -- parametric autoregressive modeling -- is rarely used to interpret astronomical light curves. We review standard ARMA, ARIMA and ARFIMA (autoregressive moving average fractionally integrated) models that treat short-memory autocorrelation, long-memory $1/f^\alpha$ `red noise', and nonstationary trends. Though designed for evenly spaced time series, moderately irregular cadences can be treated as evenly-spaced time series with missing data. Fitting algorithms are efficient and software implementations are widely available. We apply ARIMA models to light curves of four variable stars, discussing their effectiveness for different temporal characteristics. A variety of extensions to ARIMA are outlined, with emphasis on recently developed continuous-time models like CARMA and CARFIMA designed for irregularly spaced time series. Strengths and weakness of ARIMA-type modeling for astronomical data analysis and astrophysical insights are reviewed.

Q-SampEnAve: an adaptive measurement to recognize the patterns for short-term financial time series

Adaptive algorithms are increasingly discussed because of few parameter constraints, which can respond to different inputs and make the results more accurate. In this paper, we propose a self-tuning complexity measurement for short time series, called q-sample entropy average (q-SampEnAve), which is a data-driven approach. This approach avoids the impact of tolerance r in traditional sample entropy (SampEn) by considering all r candidates. Besides, q-sample entropy (q-SampEn), as an intermediate stage, could serve as a good discriminative tool as well. By discussing parameter q, we find that the q plays a decisive role in the results that large positive q filters out the large $$p_i$$ in a probability distribution, and large negative q filter out the rare events. These two factors make our new method more accurate and complete than existing methods. In simulated experiments, ARFIMA model, uniform distribution and logistic map are employed. It reveals that sequences generated two-component ARFIMA models are distinctly separated from uniform distribution in (0, 1) and logistic mapping ( $$a = 4$$ ). Besides, even for the sequences generated by ARFIMA model with different parameters, four states (mean, std, kurtosis and skewness) show a little different. Most importantly, uniform distribution in (0, 1) and logistic mapping ( $$a =4$$ ) are relatively disordered. On the contrary, the system generated by the ARFIMA model are relatively regular. For stock indices, the performance of SSE Composite Index (000001.SS) and SZSE Composite Index (399001.SZ) are identical, while Hang Seng Index ( $$\hat{}$$ HSI) behaves closer to Dow Jones Industrial Average ( $$\hat{}$$ DJI) and Nasdaq Composite ( $$\hat{}$$ IXIC) than indices in mainland China (000001.SS and 399001.SZ), which may be related to the economic policy and the market environment. It also illustrates that the US stock market is relatively mature and more suitable for investment. Finally, we give a brief discussion about the application in random bit sequence, which illustrates that random bit sequences generated from different parameters can be recognized.

Analysis of financial stock markets through the multiscale cross-distribution entropy based on the Tsallis entropy

In this paper, we propose a multiscale cross-distribution entropy (MCDE) method based on the Tsallis entropy to analyze financial stock markets. In order to evaluate the effectiveness of this method, we employ it into ARFIMA models. Then, applying the proposed method to analyze financial time series, we conclude that it provides us more exact, detailed and clearer information about the relationships between pairs of financial stock markets in comparison with the multiscale cross-sample entropy. Furthermore, the results show that the embedding dimension m has little influence on MCDE. The stability of financial time series could be affected by the order q. Moreover, the MCDE results of $$q > 0$$ tend to be more stable than the results obtained by $$q < 0$$ , and it may arise from producing amounts of entropy as $$q < 0$$ . The larger the entropy is, the more active the financial time series is.

On Attempts to Use Models Incorporating Long-Range Dependence in Long-Term Volatility Forecasting

ARFIMA models, as advocated by Jiang and Tian for use in long-term volatility forecasting, are found in a follow-up empirical study to be dominated by a certain simple historical predictor of stock price volatility at a five-year horizon. (This particular historical predictor is not recommended over more conventional methods, such as fifteen-year trailing historical volatility, due to bias-related concerns.) A relationship is observed between the estimated fractional-differencing parameter and the predictability of volatility. For companies with estimated values of d around 0.3, volatility forecast-errors (using several forecast methods) are significantly smaller than for those with estimated d in the range of about (0.4, 0.5). Negative coefficients on ARFIMA forecasts, after controlling for long-run historical volatility within certain multivariate volatility prediction-models, is suggestive of a relationship between ARFIMA prediction-results and phenomena like structural breaks, which are not captured by the ARFIMA approach.

Recent Developments in Macro-Econometric Modeling: Theory and Applications

Developments in macro-econometrics have been evolving since the aftermath of the Second World War.[...]

A new measurement of financial time irreversibility based on information measures method

In this paper, we propose a new measurement of time irreversibility based on information measures method to analyze the financial stock markets. In order to examine the effectiveness of this method, we employ it into ARFIMA models. Applying the new method to quantifying time irreversibility of 33 financial indices evolving over the period 2002–2016, we conclude that the stock daily prices of the companies are indeed time irreversible and the degree of irreversibility varies with time for each company. According to the values of irreversibility, we could rank the companies. Also we obtain that the values of annualized irreversibility may have little effect on the coefficient of variation. Moreover, in order to find patterns arising among different periods, we use the principal component analysis (PCA) and hierarchical clustering, the results obtained by these two standard techniques in data mining are in agreement.

Quantifying the range of cross-correlated fluctuations using a [formula omitted]–[formula omitted] dependent AHXA coefficient

Recently, based on analogous height cross-correlation analysis (AHXA), a cross-correlation coefficient ρ×(L) has been proposed to quantify the levels of cross-correlation on different temporal scales for bivariate series. A limitation of this coefficient is that it cannot capture the full information of cross-correlations on amplitude of fluctuations. In fact, it only detects the cross-correlation at a specific order fluctuation, which might neglect some important information inherited from other order fluctuations. To overcome this disadvantage, in this work, based on the scaling of the qth order covariance and time delay L, we define a two-parameter dependent cross-correlation coefficient ρq(L) to detect and quantify the range and level of cross-correlations. This new version of ρq(L) coefficient leads to the formation of a ρq(L) surface, which not only is able to quantify the level of cross-correlations, but also allows us to identify the range of fluctuation amplitudes that are correlated in two given signals. Applications to the classical ARFIMA models and the binomial multifractal series illustrate the feasibility of this new coefficient ρq(L). In addition, a statistical test is proposed to quantify the existence of cross-correlations between two given series. Applying our method to the real life empirical data from the 1999–2000 California electricity market, we find that the California power crisis in 2000 destroys the cross-correlation between the price and the load series but does not affect the correlation of the load series during and before the crisis.

COMPARATIVE STUDY ON RETAIL SALES FORECASTING BETWEEN SINGLE AND COMBINATION METHODS

In today’s competitive global economy, businesses must adjust themselves constantly to ever-changing markets. Therefore, predicting future events in the marketplace is crucial to the maintenance of successful business activities. In this study, sales forecasts for a global furniture retailer operating in Turkey were made using state space models, ARIMA and ARFIMA models, neural networks, and Adaptive Network-based Fuzzy Inference System (ANFIS). Also, the forecasting performances of some widely used combining methods were evaluated by comparison with the weekly sales data for ten products. According to the best of our knowledge, this study is the first time that the recently developed state space models, also called ETS (Error-Trend-Seasonal) models, and the ANFIS model have been tested within combining methods for forecasting retail sales. Analysis of the results of the single models in isolation indicated that none of them outperformed all the others across all the time series investigated. However, the empirical results suggested that most of the combined forecasts examined could achieve statistically significant increases in forecasting accuracy compared with individual models and with the forecasts generated by the company’s current system.