Memory Parameter Research Articles

Time varying market efficiency of the GCC stock markets

This paper investigates the time-varying levels of weak-form market efficiency for the GCC stock markets over the period spanning from May 2005 to September 2013. We use two empirical approaches: (1) the generalized autoregressive conditional heteroscedasticity in mean (GARCH-M) model with state space time varying parameter (Kalman filter), and (2) a rolling technique sample test of the fractional long memory parameter d. As long memory estimation methods, we use the detrended fluctuation analysis (DFA) technique, the modified R/S statistic, the exact local whittle (ELW) and the feasible Exact Local Whittle (FELW) methods. Moreover, we use the Bai and Perron (1998, 2003) multiple structural breaks technique to test and date the time varying behavior of stock market efficiency. Empirical results show that GCC markets have different degrees of time-varying efficiency, and also have experiencing periods of efficiency improvement. Results also show evidence of structural breaks in all GCC markets. Moreover, we observe that the recent financial shocks such as Arab spring and subprime crises have a significant impact on the time path evolution of market efficiency.

Long memory volatility of gold price returns: How strong is the evidence from distinct economic cycles?

This paper examines the long memory behavior in the volatility of gold returns using daily data for the period 1985–2009. We divided the whole sample into eight sub-samples in order to analyze the robustness and consistency of our results during different crisis periods. This constitutes our main contribution. We cover four major world crises, namely, (i) the US stock market crash of 1987; (ii) the Asian financial crisis of 1997; (iii) the World Trade Center terrorist attack of 2001 and finally, (iv) the sub-prime crisis of 2007, in order to investigate how the fractional integrated parameter of the FIGARCH(1,d,1) model evolves over time. Our findings are twofold: (i) there is evidence of long memory in the conditional variance over the whole sample period; (ii) when we consider the sub-sample analysis, the results show mixed evidence. Thus, for the 1985–2003 period the long memory parameter is positive and statistically significant in the pre-crisis sub-samples, and there is no evidence of long memory in the crisis sub-sample periods; however the reverse pattern occurs for the 2005–2009 period. This highlights the unique characteristics of the 2007 sub-prime crisis.

Bayesian Inference on the Memory Parameter for Gamma-Modulated Regression Models

In this work, we propose a Bayesian methodology to make inferences for the memory parameter and other characteristics under non-standard assumptions for a class of stochastic processes. This class generalizes the Gamma-modulated process, with trajectories that exhibit long memory behavior, as well as decreasing variability as time increases. Different values of the memory parameter influence the speed of this decrease, making this heteroscedastic model very flexible. Its properties are used to implement an approximate Bayesian computation and MCMC scheme to obtain posterior estimates. We test and validate our method through simulations and real data from the big earthquake that occurred in 2010 in Chile.

A modified test against spurious long memory

This paper considers the situation where a time series is composed of a fractionally integrated component with long memory parameter d∈(−1/2;1/2) and some contamination in the form of level shifts or trends. The test against spurious long memory by Qu (2011) is consistent in this case as the standard local Whittle estimator for unknown d is upward biased. As demonstrated in this work, the power can be improved by removing the fractional component from the series prior to application of the test. This task can be accomplished by using the modified local Whittle approach by Hou and Perron (2014). This estimator is robust against contaminations and yields nearly unbiased point estimates of d, irrespective of whether contaminations are present or not. The suggested testing procedure has similar size properties as the original test and is often more powerful.

A dangerous use of the fading memory term for non linear kinematic models at variable temperature

Abstract The purpose of this note is to shed the light on a specific problem linked to the use of a variable fading memory parameter in a nonlinear kinematic rule. A strong variation of this parameter versus temperature may generate cumbersome behaviours even with very simple temperature histories. A better solution is to use several non linear kinematic variables with a constant fading memory term.

A goodness-of-fit test for marginal distribution of linear random fields with long memory

This paper addresses the problem of fitting a known distribution function to the marginal distribution of a stationary long memory moving average random field observed on increasing \(\nu \)-dimensional “cubic” domains when its mean \(\mu \) and scale \(\sigma \) are known or unknown. Using two suitable estimators of \(\mu \) and a classical estimate of \(\sigma \), a modification of the Kolmogorov–Smirnov statistic is defined based on the residual empirical process and having a Cauchy-type limit distribution, independent of \(\mu ,\sigma \) and the long memory parameter d. Based on this result, a simple goodness-of-fit test for the marginal distribution is constructed, which does not require the estimation of d or any other underlying nuisance parameters. The result is new even for the case of time series, i.e., when \(\nu =1\). Findings of a simulation study investigating the finite sample behavior of size and power of the proposed test is also included in this paper.

Monte Carlo simulation for comparison of different estimators of long memory parameter: An application of ARFIMA model for forecasting commodity price

Time series with long memory or long-range dependence occurs frequently in agricultural commodity prices. For describing long memory, fractional integration is considered. The autoregressive fractionally integrated moving-average (ARFIMA) model along with its different estimation procedures is inve stigated. For the present investigation, the daily spot prices of mustard in Mumbai market are used. Autocorrelation (ACF) and partial autocorrelation (PACF) functions showed a slow hyperbolic decay indicating the presence of long memory. On the basis of minimum AIC values, the best model is identified for each series. Evaluation of forecasting is carried out with root mean squares prediction error (RMSPE), mean absolute prediction error (MAPE) and relative mean absolute prediction error (RMAPE). The residuals of the fitted models were used for diagnostic checking. Long memory parameter of ARFIMA model is computed by Geweke and Porter-Hudak (GPH), Gaussian semiparametric and wavelet method by using Maximal overlap discrete wavelet transform (MODWT). To this end, a comparison in the performance of different estimation procedures is carried out by Monte Carlo simulation technique. The R software package has been used for data analysis.

The Universality of Homogeneous Polynomial Forms and Critical Limits

Nourdin et al. [9] established the following universality result: if a sequence of off-diagonal homogeneous polynomial forms in i.i.d. standard normal random variables converges in distribution to a normal, then the convergence also holds if one replaces these i.i.d. standard normal random variables in the polynomial forms by any independent standardized random variables with uniformly bounded third absolute moment. The result, which was stated for polynomial forms with a finite number of terms, can be extended to allow an infinite number of terms in the polynomial forms. Based on a contraction criterion derived from this extended universality result, we prove a central limit theorem for a strongly dependent nonlinear processes, whose memory parameter lies at the boundary between short and long memory.

Power transformations of absolute returns and long memory estimation

Abstract Different power transformations of absolute returns of various financial assets have been found to display different magnitudes of sample autocorrelations, a property referred to as the Taylor effect. In this paper, we consider the long memory stochastic volatility model for the returns, under which, the asymptotic rate of decay of the autocorrelations of powers of absolute returns is governed by their long memory parameter. Although the true long memory parameter of powers of absolute returns is the same across different powers, we show that the local Whittle estimator of the long memory parameter has finite-sample bias that differs across the power transformations chosen. A Monte-Carlo experiment provides evidence in support of our theoretical finding that the reported variation of the estimates of the long memory parameter for power transformations of returns could be due to finite-sample bias of the estimator. The local Whittle estimates of powers of absolute returns for the S&P500 index and the DM/USD exchange rate are also examined.

Higher-order improvements of the sieve bootstrap for fractionally integrated processes

This paper investigates the accuracy of bootstrap-based inference in the case of long memory fractionally integrated processes. The re-sampling method is based on the semi-parametric sieve, whereby the dynamics in the process used to produce the bootstrap draws are captured by an autoregressive approximation. Application of the sieve method to data pre-filtered by a semi-parametric estimate of the long memory parameter is also explored. Higher-order improvements yielded by both forms of re-sampling are demonstrated using Edgeworth expansions for a broad class of statistics that includes first- and second-order moments, the discrete Fourier transform and regression coefficients. The methods are then applied to the problem of estimating the sampling distributions of the sample mean and of selected sample autocorrelation coefficients, in experimental settings. In the case of the sample mean, the pre-filtered version of the bootstrap is shown to avoid the distinct underestimation of the sampling variance of the mean which the raw sieve method demonstrates in finite samples, higher-order accuracy of the latter notwithstanding. Pre-filtering also produces gains in terms of the accuracy with which the sampling distributions of the sample autocorrelations are reproduced, most notably in the part of the parameter space in which asymptotic normality does not obtain.

A bootstrap approximation for the distribution of the Local Whittle estimator

The asymptotic properties of the Local Whittle estimator of the memory parameter d have been widely analysed and its consistency and asymptotic distribution have been obtained for values of d∈(−1/2,1] in a wide range of situations. However, the asymptotic distribution may be a poor approximation of the exact one in several cases, e.g. with small sample sizes or even with larger samples when d>0.75. In other situations the asymptotic distribution is unknown, as for example in a noninvertible context or in some nonlinear transformations of long memory processes, where only consistency is obtained. For all these cases a bootstrap strategy based on resampling a (perhaps locally) standardised periodogram is proposed. A Monte Carlo analysis shows that this strategy leads to a good approximation of the exact distribution of the Local Whittle estimator in those situations where the asymptotic distribution is not reliable.

History dependent quantum walk on the cycle with an unbalanced coin

Recently, a new model of quantum walk, utilizing recycled coins, was introduced; however little is yet known about its properties. In this paper, we study its behavior on the cycle graph. In particular, we will consider its time averaged distribution and how it is affected by the walk’s “memory parameter”—a real parameter, between zero and eight, which affects the walk’s coin flip operator. Despite an infinite number of different parameters, our analysis provides evidence that only a few produce non-uniform behavior. Our analysis also shows that the initial state, and cycle size modulo four all affect the behavior of this walk. We also prove an interesting relationship between the recycled coin model and a different memory-based quantum walk recently proposed.

A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector

In the present investigation, three mathematical models on a common single strain mosquito-transmitted diseases are considered. The first one is based on ordinary differential equations, and other two models are based on fractional order differential equations. The proposed models are validated using published monthly dengue incidence data from two provinces of Venezuela during the period 1999–2002. We estimate several parameters of these models like the order of the fractional derivatives (in case of two fractional order systems), the biting rate of mosquito, two probabilities of infection, mosquito recruitment and mortality rates, etc., from the data. The basic reproduction number, R0, for the ODE system is estimated using the data. For two fractional order systems, an upper bound for, R0, is derived and its value is obtained using the published data. The force of infection, and the effective reproduction number, R(t), for the three models are estimated using the data. Sensitivity analysis of the mosquito memory parameter with some important responses is worked out. We use Akaike Information Criterion (AIC) to identify the best model among the three proposed models. It is observed that the model with memory in both the host, and the vector population provides a better agreement with epidemic data. Finally, we provide a control strategy for the vector-borne disease, dengue, using the memory of the host, and the vector.

Modelling and forecasting of retail price of arhar dal in Karnal, Haryana

Forecasting retail price of agricultural commodities is of utmost importance for planning in advance to resist any abnormalities. In this regard, the autoregressive fractionally integrated moving-average (ARFIMA) model is employed. ARFIMA model searches for a non-integer differencing parameter d to difference the data to capture long memory. The model is applied for modelling and forecasting of daily retail price of pigeonpea (Cajanas cajan) in Karnal during January, 2011 to July, 2013. Augmented Dickey-Fuller (ADF) test and Philips Peron (PP) test are used for testing the stationarity of the series. Autocorrelation (ACF) and partial autocorrelation (PACF) functions showed a slow hyperbolic decay indicating the presence of long memory. In the present price series, long memory parameter is found to be significant. On the basis of minimum AIC values, the best model is identified. To this end, evaluation of forecasting is carried out with root mean squares prediction error (RMSPE), mean absolute prediction error (MAPE) and relative mean absolute prediction error (RMAPE). The residuals of the fitted models were used for diagnostic checking. It is found that ARFIMA model has been able to capture the long memory present in the data set. The R software package has been used for data analysis.

A Test of the Long Memory Hypothesis Based on Self-Similarity

AbstractThis paper develops a new test of true versus spurious long memory, based on log-periodogram estimation of the long memory parameter using skip-sampled data. A correction factor is derived to overcome the bias in this estimator due to aliasing. The procedure is designed to be used in the context of a conventional test of significance of the long memory parameter, and a composite test procedure is described that has the properties of known asymptotic size and consistency. The test is implemented using the bootstrap, with the distribution under the null hypothesis being approximated using a dependent-sample bootstrap technique to approximate short-run dependence following fractional differencing. The properties of the test are investigated in a set of Monte Carlo experiments. The procedure is illustrated by applications to exchange rate volatility and dividend growth series.

Long range dependency and forecasting of housing price index and mortgage market rate: evidence of subprime crisis

Article history: Received January 20, 2015 Received in revised format 6 February 2015 Accepted 25 March 2015 Available online March 25 2015 In this paper, we examine and forecast the House Price Index (HPI) and mortgage market rate in terms of the description of the subprime crisis. We use a semi-parametric local polynomial Whittle estimator proposed by Shimotsu et al. (2005) [Shimotsu, K., & Phillips, P.C.B. (2005), Exact local Whittle estimation of fractional integration. The Annals of Statistics, 33(4), 18901933.] in a long memory parameter time series. Empirical investigation of HPI and mortgage market rate shows that these variables are more persistent when the d estimates are found on the Shimotsu method than on the one of Kunsch (1987) [Kunsch, H.R. (1987). Statistical aspects of self-similar processes. In Y. Prokhorov and V.V. Sazanov (eds.), Proceedings of the First World Congress of the Bernoulli Society, VNU Science Press, Utrecht, 67-74.]. The estimating forecast values are more realistic and they strongly reflect the present US economy actuality in the two series as indicated by the forecast evaluation topics. Growing Science Ltd. All rights reserved. 5 © 201

Fractal Structure of the Stock Markets of Leading Asian Countries

(ProQuest: ... denotes formulae omitted.)I. IntroductionIn the last two decades, borrowing and firm value issues receded into the background due to the structured financial instruments and the exotic options coming into prominence. During this period, some models have been improved by financial engineers in order to evaluate these instruments and to model the path of financial time series. However, these models are generally based on the assumptions of conventional financial theory, such as normal distribution and random walk, which are deeply criticized in different studies (Ramasamy and Helmi, 2011).As the existing conventional theory oversimplifies the complicated structure of financial markets and explains the problems under ideal and normal conditions, we need accurate and credible models in order to build up an effective finance theory (Velasquez, 2009). Conventional theory based on the Bachelier's (1900) study reduced the tail probabilities to an insignificant level using normal distribution. However, this insignificance forms the basis of models such as Markowitz (1952), Sharpe (1964), Fama (1965, 1970) and Black and Scholes (1973), which are the basic studies of conventional theory. The scandals and crises of the last 50 years, including Enron (2001), Parmalat (2003), Black Monday (1987), the collapse of Long Term Capital Management (LTCM) (1998) and the mortgage crisis (2008), demonstrated that the random walk assumption of the Efficient Market Hypothesis is not valid. This is because, if the distribution of price changes were normal, five standard deviation observations would be seen every 7,000 years. However, in this case changes are seen almost every 4 to 5 years. As stated by Mandelbrot and Hudson (2004), the chain is only as strong as its weakest link and the weakest links in conventional theory are the Efficient Market Hypothesis, random walk and normal distribution assumptions.The first and the most important criticism of conventional theory was made by Mandelbrot regarding the ignoring of self-similarity and long memory properties of the financial time series. It should be noted that while the long memory issue was associated with Mandelbrot, the first person who adressed this topic was Hurst (1951). That is why Mandelbrot (1972) used the H notation as the long memory parameter in honor of Hurst1. In that period, Hurst showed that there were significant correlations between the floods. According to his studies, a good flood year had a tendency to be followed by another good flood year, and similarly a bad flood year was followed by another bad flood year. That is, the movements of good and bad years are not in a random process (Grabbe, 2001). Hurst explained this correlation via the power law. One of the most crucial observations of Hurst can be explained as follows. Natural events generally appear to follow a normal distribution but only when the occurrence order of the events is not taken into account (Mandelbrot and Hudson, 2004). The statistic introduced by Hurst measures the persistency between the events and it is known as the Rescaled Range (R/S). The Rescaled Range is the range of the partial sums of the deviations from the mean of any time series (Chaitip et al., 2011). Inspired by the Bible, Mandelbrot and Wallis (1968) named the persistency situations the Joseph Effect and adapted them into the financial time series theory as the long memory issue (Grabbe, 2001).2 Mandelbrot (1963, 1983) also contributed to the emergence of self-similarity and the fractal dimension as new topics in finance theory. In fact, all of these models paved the way for the evolution of conventional finance theory.According to conventional finance theory based on the random walk assumption of Fama (1965), there is no correlation between asset returns, and therefore asset prices develop with the geometric Brownian motion. Almost all models of conventional finance theory are formed under this assumption and probability distributions of the returns are accepted as normal. …

INFERENCE ON NONSTATIONARY TIME SERIES WITH MOVING MEAN

A semiparametric model is proposed in which a parametric filtering of a nonstationary time series, incorporating fractionally differencing with short memory correction, removes correlation but leaves a nonparametric deterministic trend. Estimates of the memory parameter and other dependence parameters are proposed, and shown to be consistent and asymptotically normally distributed with parametric rate. Tests with standard asymptotics for I(1) and other hypotheses are thereby justified. Estimation of the trend function is also considered. We include a Monte Carlo study of finite-sample performance.

Limited-memory LDL⊤ factorization of symmetric quasi-definite matrices with application to constrained optimization

We propose a generalization of the limited-memory Cholesky factorization of Lin and More (SIAM J. Sci. Comput. 21(1), 24---45, 1999) to the symmetric indefinite case with special interest in symmetric quasi-definite matrices. We use this incomplete factorization to precondition two formulations of linear systems arising from regularized interior-point methods for quadratic optimization. An advantage of the limited-memory approach is predictable memory requirements. We establish existence of incomplete factors when the input matrix is an H-matrix but our numerical results illustrate that the factorization succeeds more generally. An appropriate diagonal shift is applied whenever the input matrix is not quasi definite. As the memory parameter increases an efficiency measure of the preconditioner suggested by Scott and T?ma (2013) improves. The combination of the 3×3 block formulation analyzed by Greif, Moulding, and Orban (SIAM J. Optim. 24(1), 49---83, 2014), the SYMAMD ordering, and a moderate memory parameter results in encouraging performance.

Long Memory Volatility Persistence in High Frequency Precious Metals Returns

Using high frequency data, this paper examines the long memory property in the conditional volatility of the precious metals return series at different time frequencies using FIGARCH models. Very significant long memory characteristics have been detected in absolute returns by using Semiparametric local Whittle estimation of the long memory parameter. Estimation of the long memory parameter across many different data sampling frequencies gives consistent estimates of the long memory parameter, indicating that the series are exactly to show some degree of self-similarity. Results indicate that the long memory property remains quite consistent across different time frequencies for both unconditional and conditional volatility measures. This study is useful for investors and traders (with different trading horizons) and it can be used in predicting expected future volatility and in designing and implementing trading strategies at different time frequencies.