Long memory or shifting means in geophysical time series?

Macroeconomic Dynamics, 2010

Structural breaks and switching processes are known to induce apparent long memory in a time series. Here we show that any significant time variation in the mean renders the sample correlogram (and related spectral estimates) inconsistent. In particular, smooth time variation in the mean-i.e., even a weak trend, either stochastic or deterministic-induces apparent long memory. This apparent long memory can be eliminated by either high-pass filtering or by detrending. Here we demonstrate the effectiveness in this regard of nonlinear detrending via penalized-spline nonparametric regression. A time-varying mean can be of economic interest in its own right. This suggests that isolating out and separately examining both a local mean (i.e., a nonlinear trend or the realization of a stochastic trend) and deviations from it is preferable as a modeling strategy to simply estimating a fractionally integrated model. We illustrate the superiority of this strategy using stock return volatility data.

Hydrology and Earth System Sciences, 2013

A short memory process that encounters occasional structural breaks in mean can show a slower rate of decay in the autocorrelation function and other properties of fractional integrated I (d) processes. In this paper we employed a procedure for estimating the fractional differencing parameter in semiparametric contexts proposed by Geweke and Porter-Hudak (1983) to analyse nine daily rainfall data sets across Malaysia. The results indicate that all the data sets exhibit long memory. Furthermore, an empirical fluctuation process using the ordinary least square (OLS)-based cumulative sum (CUSUM) test for the break date was applied. Break dates were detected in all data sets. The data sets were partitioned according to their respective break date, and a further test for long memory was applied for all subseries. Results show that all subseries follows the same pattern as the original series. The estimate of the fractional parameters d 1 and d 2 on the subseries obtained by splitting the original series at the break date confirms that there is a long memory in the data generating process (DGP). Therefore this evidence shows a true long memory not due to structural break.

Journal of Time Series Analysis, 2001

We consider a fractional exponential, or FEXP estimator of the memory parameter of a stationary Gaussian long‐memory time series. The estimator is constructed by fitting a FEXP model of slowly increasing dimension to the log periodogram at all Fourier frequencies by ordinary least squares, and retaining the corresponding estimated memory parameter. We do not assume that the data were necessarily generated by a FEXP model, or by any other finite‐parameter model. We do, however, impose a global differentiability assumption on the spectral density except at the origin. Because of this, and its use of all Fourier frequencies, we refer to the FEXP estimator as a broadband semiparametric estimator. We demonstrate the consistency of the FEXP estimator, and obtain expressions for its asymptotic bias and variance. If the true spectral density is sufficiently smooth, the FEXP estimator can strongly outperform existing semiparametric estimators, such as the Geweke–Porter‐Hudak (GPH) and Gaussi...

arXiv preprint arXiv:0901.0762, 2009

We present the results of a simulation study into the properties of 12 different estimators of the Hurst parameter, H, or the fractional integration parameter, d, in long memory time series. We compare and contrast their performance on simulated Fractional Gaussian Noises and fractionally integrated series with lengths between 100 and 10,000 data points and H values between 0.55 and 0.90 or d values between 0.05 and 0.40. We apply all 12 estimators to the Campito Mountain data and estimate the accuracy of their estimates using the Beran goodness of fit test for long memory time series.

2016

International Journal of Forecasting, 1988

An important test of the adequacy of a stochastic model is its ability to forecast accurately. In hydrology as in many other disciplines, the performance of the model in producing one step ahead forecasts is of particular interest. The ability of several stationary nonseasonal time series models to produce accurate forecasts is examined in this paper. Statistical tests are employed to determine if the forecasts generated by a particular model are better than the forecasts produced by an alternative procedure. The results of the study indicate that for the data sets examined, there is no significant difference in forecast performance between the nonseasonal autoregressive moving average model and a nonparametric regression model. ARMA, Fractional Gaussian noise, Fractional differencing, Fractional ARMA, Forecasting.

Water Resources Research, 1969

A variety of geophysical records are examined to determine the dependence upon the lag s of a quantity called 'rescaled range,' denoted by R(t, s)/S(t, s). If there had been no appreciable dependence between two values of the record at very distant points in time, the ratio R/S would have been proportional to s ø'5. But, in fact, as first noted by Edwin I•Iurst, the R/S ratio of hydrological and other geophysical records is proportional to s a with H % 0.5. I-Iurst's original claims must be tightened and hedged, and his estimates of H must be discarded, but his general idea will be shown to be correct. We have shown elsewhere that this behavior of R/S means that the strength of long-range statistical dependence in geophysical records is considerable.

RePEc: Research Papers in Economics, 2014

Fractionally integrated processes have become a standard class of models to describe the long memory features of economic and …nancial time series data. However, it has been demonstrated in numerous studies that structural break processes and non-linear features can often be confused as being long memory. The question naturally arises whether it is possible empirically to determine the source of long memory as being genuinely long memory in the form of a fractionally integrated process or whether the long range dependence is of a di¤erent nature. In this paper we suggest a testing procedure that helps discriminating between such processes. The idea is based on the feature that nonlinear transformations of stationary fractionally integrated Gaussian processes decrease the order of memory in a speci…c way which is determined by the Hermite rank of the transformation. In principle, a non-linear transformation of the series can make the series short memory I(0). We suggest using the Wald test of Shimotsu (2007) to test the null hypothesis that a vector time series of properly transformed variables is I(0). Our testing procedure is designed such that even non-stationary fractionally integrated processes are permitted under the null hypothesis. The test is shown to have good size and to be robust against certain types of deviations from Gaussianity. The test is also shown to be consistent against a broad class of processes that are non-fractional but still exhibit (spurious) long memory. In particular, the test is shown to have excellent power against a class of stationary and non-stationary random level shift models as well as Markov switching GARCH processes where the break and transition probabilities are allowed to be time varying.

Hydrology and Earth System Sciences Discussions

A short memory process that encounters occasional structural breaks in mean can show a slower rate of decay in the autocorrelation function and other properties of fractional integrated I(d) processes. In this paper we employed a procedure for estimating the fractional differencing parameter in semi parametric contexts proposed by Geweke and Porter-Hudak to analyze nine daily rainfall data sets across Malaysia. The results indicate that all the data sets exhibit long memory. Furthermore, an empirical fluctuation process using the Ordinary Least Square (OLS) based cumulative sum (CUSUM) test with F-statistic for the break date were applied, break dates were detected in all data sets. The data sets were partitioned according to their respective break date and further test for long memory was applied for all subseries. Results show that all subseries follows the same pattern with the original series. The estimate of the fractional parameters d 1 and d 2 on the subseries obtained by splitting the original series at the break-date, confirms that there is a long memory in the DGP. Therefore this evidence shows a true long memory not due to structural break.