Long Memory Estimation Research Articles

Improving the daily pan evaporation estimation of long short-term memory and support vector regression models by using the Wild Horse Optimizer algorithm

ABSTRACT Evaporation is a basic element in the hydrological cycle that plays a vital role in a region's water balance. In this paper, the Wild Horse Optimizer (WHO) algorithm was used to optimize long short-term memory (LSTM) and support vector regression (SVR) to estimate daily pan evaporation (Ep). Primary meteorological variables including minimum temperature (Tmin), maximum temperature (Tmax), sunshine hours (SSH), relative humidity (RH), and wind speed (WS) were collected from two synoptic meteorological stations with different climates which are situated in Fars province, Iran. One of the stations is located in Larestan city with a hot desert climate and the other is in Abadeh city with a cold dry climate. The partial mutual information (PMI) algorithm was utilized to identify the efficient input variables (EIVs) on Ep. The results of the PMI algorithm proved that the Tmax, Tmin, and RH for Larestan station and also the Tmax, Tmin, and SSH for Abadeh station are the EIVs on Ep. The results showed the LSTM–WHO hybrid model for both stations can ameliorate the daily Ep estimation and it can also reduce the estimation error. Therefore, the LSTM–WHO hybrid model was proposed as a powerful model compared to standalone models in estimating daily Ep.

Modeling Long Memory and Regime Switching with an MRS-FIEGARCH Model: A Simulation Study

Recent research suggests that long memory can be caused by regime switching and is easily confused with it. However, if the causes of confusion were properly controlled, they could be distinguished. Motivated by this idea, our study aims to distinguish between the long memory and regime switching of financial volatility. We firstly modeled the long memory and regime switching of volatility using the Fractionally Integrated Exponential GARCH (FIEGARCH) and Markov Regime-Switching EGARCH (MRS-EGARCH) frameworks, respectively, and performed a simulation study on their finite-sample properties when innovations followed a non-normal distribution. Subsequently, we demonstrated the confusion between the FIEGARCH and MRS-EGARCH processes using simulations. A recent study theoretically proved that the time-varying smoothing probability series can induce the presence of significant long memory in the regime-switching process. To control for its effect, the two-stage two-state FIEGARCH and MRS-FIEGARCH frameworks are proposed. The Monte Carlo studies showed that both frameworks can effectively distinguish between the pure FIEGARCH and pure MRS-EGARCH processes. When the MRS-FIEGARCH model was further employed to fit series generated with the MRS-FIEGARCH process, it outperformed the ordinary FIEGARCH model. Finally, an empirical study of NASDAQ index return was conducted to demonstrate that our MRS-FIEGARCH model can provide potentially more reliable long-memory estimates, identify the volatility states and outperform both the FIEGARCH and MRS-EGARCH models.

Asymptotics of bivariate local Whittle estimators with applications to fractal connectivity

Several methodological and numerical issues behind the local Whittle estimation of long and short memory in bivariate stationary time series with possible fractional cointegration are reexamined. These issues include the asymptotic normality for all model parameters, local Whittle plots for phase parameter and fractal connectivity, and others. For fractal connectivity, in particular, it is advocated to work with a model parametrization for which the model parameters associated with this phenomenon are identifiable and could be tested naturally within the local Whittle estimation framework. A simulation study and data applications are also considered.

Application of Wavelet-Based Maximum Likelihood Estimator in Measuring Market Risk for Fossil Fuel

Energy commodity prices are inherently volatile, since they are determined by the volatile global demand and supply of fossil fuel extractions, which in the long-run will affect the observed climate patterns. Measuring the risk associated with energy price changes, therefore, ultimately provides us with an important tool to study the economic drivers of climate changes. This study examines the potential use of long-memory estimation methods in capturing such risk. In particular, we are interested in investigating the energy markets’ efficiency at the aggregated level, using a novel wavelet-based maximum likelihood estimator (waveMLE). We first compare the performance of various conventional estimators with this new method. Our simulated results show that waveMLE in general outperforms these previously well-established estimators. Additionally, we document that while energy returns realizations follow a white-noise and are generally independent, volatility processes exhibits a certain degree of long-range dependence.

Market efficiency of Baltic stock markets: A fractional integration approach

We investigate financial market efficiency in the time series of four daily Baltic stock market indices, namely: Baltic Benchmark Gross Index (OMXBBGI), all share index of Tallin-Lithuanian (OMXT), all share index of Riga (OMXR) and all share index of Vilnius (OMXV), based on historical data from 1 January, 2000 to 22 January 2016. We use fractional integration methods to test the hypothesis of market efficiency. Realizing that long-memory estimation could be spurious in the presence of structural breaks, we identify bull and bear market phases from each of the time series. Applying the fractional integration approach, we find that the random walk hypothesis of market efficiency is generally rejected in the overall, and at two bull and one bear sub-samples of the four Baltic stock indices. The volatility at the bear markets of these stocks persists more than the volatility at the bull markets. Our results therefore provide evidence for weak form of market efficiency in the Baltic stock markets, with some exceptions. As a way of policy, the results are relevant to portfolio managers and policy makers in a number of ways.

Wavelets and estimation of long memory in nonstationary models: Does anything beat the exact local whittle estimator?

ABSTRACTIn this article, we analyze the performance of five estimation methods for the long memory parameter d. The goal of our article is to construct a wavelet estimate for the fractional differencing parameter in nonstationary long memory processes that dominate the well-known estimate of Shimotsu and Phillips (2005). The simulation results show that the wavelet estimation method of Lee (2005) with several tapering techniques performs better under most cases in nonstationary long memory. The comparison is based on the empirical root mean squared error of each estimate.

A wavelet lifting approach to long-memory estimation

Reliable estimation of long-range dependence parameters is vital in time series. For example, in environmental and climate science such estimation is often key to understanding climate dynamics, variability and often prediction. The challenge of data collection in such disciplines means that, in practice, the sampling pattern is either irregular or blighted by missing observations. Unfortunately, virtually all existing Hurst parameter estimation methods assume regularly sampled time series and require modification to cope with irregularity or missing data. However, such interventions come at the price of inducing higher estimator bias and variation, often worryingly ignored. This article proposes a new Hurst exponent estimation method which naturally copes with data sampling irregularity. The new method is based on a multiscale lifting transform exploiting its ability to produce wavelet-like coefficients on irregular data and, simultaneously, to effect a necessary powerful decorrelation of those coefficients. Simulations show that our method is accurate and effective, performing well against competitors even in regular data settings. Armed with this evidence our method sheds new light on long-memory intensity results in environmental and climate science applications, sometimes suggesting that different scientific conclusions may need to be drawn.

Spectral bandwidth selection for long memory

Long-memory parameter estimation using log-periodogram regression relies largely on the frequency bandwidth and the order of estimation. Literature shows that a data-dependent plug-in method for the bandwidth significantly increases the MSE’s. In a long memory time series with mild short range effect, a simple approach to determine the bandwidth size is suggested based on the spectral analysis. Monte Carlo simulation results and empirical applications show that the proposed bandwidth selection performs satisfactorily.

Long memory and ARFIMA modelling: The case of CPI inflation rate in Ghana

In macroeconomic theory and economic policy, changes in the general price level or the rate of inflation plays an essential role. Hence, for example, one of the motives behind the adoption of Inflation Targeting policy (IT) by Ghana and the treaty espoused by the European Monetary Union, known as the Maastricht Treaty, was the convergence of inflation rates. On the other hand there is a controversy about which is the order of integration in the inflation rates, some authors arguing that this variable is stationary I(0) Whittle others saying it is nonstationary I(1). In this study we examine the CPI inflation rates of Ghana from a different perspective allowing for fractional degrees of differentiation. Thus, the methodology is based on long memory or long-range dependence processes, using fractional integration and employing techniques based on Whittle parametric and semiparametric methods and autoregressive fractionally integrated moving average (ARFIMA) models. Standard I(0)/I(1) methods were also employed. Our findings indicate that long memory exists in the CPI inflation rate of Ghana. After processing fractional differencing and determining the short memory components, the following two models, ARFIMA(3,0.427,1) and ARFIMA(2,0.499,1) were respectively specified to describe the pre and post introduction of IT policy in May 2007. Consequently, the CPI inflation rate of Ghana is fractionally integrated and mean reverting. Long memory in financial time series has important implications for the critical explanation of financial time series behaviour, as it could provide an opportunity to earn speculative profits in financial markets and cast disbelief on the correctness of the EMH. For instance, when price changes exhibit long memory or long-range dependence, asset pricing models based on the Efficient Market Hypothesis (EMH) may overestimate or underestimate investment risk. Furthermore, the presence of long memory in inflation rates can provide vital information about the likely impact of shocks (e.g. demand/supply) on the economy with respect to time. The results obtained in this study would be very useful in setting up monetary policies or consolidating previous policies such as IT in order to enhance economic growth. Moreover, estimation of long memory in inflation rates can serve as an evaluation tool to assess the performance of monetary policy under different dispensations. Lastly, the presence of long memory can assist in identifying inflationary pressures in the economy.

Estimation of Long Memory in Integrated Variance

A stylized fact is that realized variance has long memory. We show that, when the instantaneous volatility is a long memory process of order d, the integrated variance is characterized by the same long-range dependence. We prove that the spectral density of realized variance is given by the sum of the spectral density of the integrated variance plus that of a measurement error, due to the sparse sampling and market microstructure noise. Hence, the realized volatility has the same degree of long memory as the integrated variance. The additional term in the spectral density induces a finite-sample bias in the semiparametric estimates of the long memory. A Monte Carlo simulation provides evidence that the corrected local Whittle estimator of Hurvich et al. (2005) is much less biased than the standard local Whittle estimator and the empirical application shows that it is robust to the choice of the sampling frequency used to compute the realized variance. Finally, the empirical results suggest that the volatility series are more likely to be generated by a nonstationary fractional process.

Wavelets and Estimation of Long Memory in Log Volatility and Time Series Perturbed by Noise

Percival and Walden (2002) present a wavelet methodology of the least squares estimation of the long memory parameter for fractionally differenced processes. We suggest that the general idea of using wavelets for estimating long memory could be used for the estimation of long memory in time series perturbed by noise. One prominent example thereof is the time series of log-Garman-Klass estimates of log volatility of financial markets. The estimator of Percival and Walden (2002) is biased if the long memory time series is perturbed by noise. We propose a new estimator of the long memory parameter which combines (in its construction) the frequency-domain approach of Sun & Phillips (2003) and the approach of Percival & Walden (2002). We illustrate the properties of the proposed estimator via Monte Carlo simulations. The results show that the estimator may be useful for the estimation of the long memory in volatility.

What explains long memory in futures price volatility?

Long memory in futures price volatility is a well-documented stylized fact with implications for market efficiency, risk management, forecasting and option pricing bias. The implications of long-memory differ, however, based on whether it is of a ‘fractional’ or of a ‘stochastic’ type. The aims of this article are to determine, in the case of agricultural commodity futures data, which type better describes price volatility and also to evaluate several competing explanations for findings of long memory. The evidence presented here finds little support for three out of four potential explanations, namely, excessive noise in the volatility measure, bias in the long-memory estimator and understated SEs of the long-memory parameter. For the data considered, price volatility appears to be most likely generated by a nonfractional long-memory process such as a stochastic break or stochastic unit root.

Long memory in REIT volatility revisited: genuine or spurious, and self-similar?

This paper revisits the Real Estate Investment Trust (REIT) long-memory literature and addresses two important research questions: one, whether the observed long memory in REIT volatility is genuine or spurious (that is, caused by structural changes); and, two, a related one – whether the long memory is self-similar. Regarding the first question, we find strong evidence for the coexistence of pure long memory and structural breaks in all developed countries under study when daily data are used. But for the emerging markets under study some show coexistence while others show only pure long memory. Such a finding is also shared by both developed and emerging markets when it comes to using lower frequency data (weekly and monthly). As for the second question, we find support for self-similarity when we compare the daily and weekly long-memory estimates for the developed markets, implying that long memory is an intrinsic feature of the data. However, the support is not strong enough to completely rule out the possibility of structural breaks. Moreover, the support is found reduced when we consider the emerging markets and the monthly estimates from the developed markets. This is possibly due to the small sample size in both cases. Overall our findings have important implications for volatility modeling and forecasting.

Estimation of Long Memory in Integrated Variance

A stylized fact is that realized variance has long memory. We show that, when the instantaneous volatility is driven by a fractional Brownian motion, the integrated variance is characterized by long-range dependence. As a consequence, the realized variance inherits this property when prices are observed continuously and without microstructure noise, and the spectral densities of integrated and realized variance coincide. However, prices are not observed continuously, so that the realized variance is affected by a measurement error. Discrete sampling and market microstructure noise induce a finite-sample bias in the fractionally integration semiparametric estimates. A Monte Carlo simulation analysis provides evidence of such a bias for common sampling frequencies.

Semiparametric estimation for seasonal long-memory time series using generalized exponential models

We consider a generalized exponential (GEXP) model in the frequency domain for modeling seasonal long-memory time series. This model generalizes the fractional exponential (FEXP) model [Beran, J., 1993. Fitting long-memory models by generalized linear regression. Biometrika 80, 817–822] to allow the singularity in the spectral density occurring at an arbitrary frequency for modeling persistent seasonality and business cycles. Moreover, the short-memory structure of this model is characterized by the Bloomfield [1973. An exponential model for the spectrum of a scalar time series. Biometrika 60, 217–226] model, which has a fairly flexible semiparametric form. The proposed model includes fractionally integrated processes, Bloomfield models, FEXP models as well as GARMA models [Gray, H.L., Zhang, N.-F., Woodward, W.A., 1989. On generalized fractional processes. J. Time Ser. Anal. 10, 233–257] as special cases. We develop a simple regression method for estimating the seasonal long-memory parameter. The asymptotic bias and variance of the corresponding long-memory estimator are derived. Our methodology is applied to a sunspot data set and an Internet traffic data set for illustration.

Long-memory in high-frequency exchange rate volatility under temporal aggregation

This paper applies log-periodogram estimators of the fractional difference parameter to the volatility of the US dollar exchange rate returns of 11 European currencies, and under temporal aggregation from an underlying half-hourly intra-day frequency. Particular attention is paid to the sequencing of the nonlinear transformation of returns and their temporal aggregation. The results reported confirm that long-memory in absolute returns constitutes an intrinsic and empirically significant characteristic of the exchange rates considered. At the practical level, our results lend support to the proposal that nonlinear transformation prior to temporal aggregation can return meaningful long-memory parameter estimates. Our findings also illustrate the advantages of long-memory parameter estimation based on the smoothed periodogram applied to absolute returns in controlling for noise induced by temporal aggregation in the processing of high-frequency data.

Long-memory analysis of time series with missing values.

The estimation of long memory is often restricted by missing data. We examine the effects on the estimation of long memory of three simple gap-filling techniques: interpolation, random, and mean filling. Numerical simulations show that the gap-filling techniques introduce significant deviations from the expected scaling behavior for both persistent and antipersistent time series. For persistent time series the interpolation method provides a reliable estimation of long memory for scales longer than the largest likely gap.

Robust Automatic Bandwidth for Long Memory

The choice of bandwidth, or number of harmonic frequencies, is crucial to semiparametric estimation of long memory in a covariance stationary time series as it determines the rate of convergence of the estimate, and a suitable choice can insure robustness to some non‐standard error specifications, such as (possibly long‐memory) conditional heteroscedasticity. This paper considers mean squared error minimizing bandwidths proposed in the literature for the local Whittle, the averaged periodogram and the log periodogram estimates of long memory. Robustness of these optimal bandwidth formulae to conditional heteroscedasticity of general form in the errors is considered. Feasible approximations to the optimal bandwidths are assessed in an extensive Monte Carlo study that provides a good basis for comparison of the above‐mentioned estimates with automatic bandwidth selection.

Estimation spectrale avec mémoire longue et hétéroscédasticité conditionnelle

Semiparametric spectral methods are valuable in that they yield inference on time series under very broad assumptions. This contribution analyzes the averaged periodogram statistic in the framework of a generalized linear process with (possibly long memory) conditional heteroscedasticity in the innovations. It is shown that the averaged periodogram statistic is appropriate for asymptotically normal estimation of the spectrum of a weakly dependent process at frequency zero and for consistent estimation of long memory and stationary cointegration in strongly dependent processes.

An Approximate Wavelet MLE of Short- and Long-Memory Parameters

By design a wavelet's strength rests in its ability to localize a process simultaneously in time-scale space. The wavelet's ability to localize a time series in time-scale space directly leads to the computational efficiency of the wavelet representation of a N£ N matrix operator by allowing the N largest elements of the wavelet represented operator to represent the matrix operator (Devore, et al. (1992a) and (1992b)). This property allows many dense matrices to have sparse representation when transformed by wavelets. In this paper we generalize the long-memory parameter estimator of McCoy and Walden (1996) to estimate simultaneously the short and long-memory parameters. Using the sparse wavelet representation of a matrix operator, we are able to approximate an ARFIMA model's likelihood function with the series' wavelet coefficients and their variances. Maximization of this approximate likelihood function over the short and long-memory parameter space results in the approximate wavelet maximum likelihood estimates of the ARFIMA model. By simultaneously maximizing the likelihood function over both the short and long-memory parameters and using only the wavelet coefficient's variances, the approximate wavelet MLE provides a fast alternative to the frequency-domain MLE. Furthermore, the simulation studies found herein reveal the approximate wavelet MLE to be robust over the invertible parameter region of the ARFIMA model's moving average parameter, whereas the frequency-domain MLE dramatically deteriorates as the moving average parameter approaches the boundaries of invertibility.