Interest in the interface of nonstationarity and nonlinearity has been increasing in the econometric literature. The motivation for this development maybe be traced to the perceived possibility that processes following nonlinear models maybe mistakenly taken to be unit root or long-memory nonstationary. This paper considers the possibility that processes may exhibit both long memory and nonlinearity. We test against the possibility that the process u_t in the model (1-L)^d y_t = u_t is nonlinear. We do not assume a particular parametric form for the nonlinear process but construct a pure significance test. Clearly, such a test could be straightforwardly constructed if d were known. Unfortunately, if a linear model is assumed while estimating d the power of the test will be reduced. We propose new more powerful tests for this problem. We present Monte Carlo evidence on the performance of the new tests and apply them to Yen real exchange rates.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID358364_code021204670.pdf?abstractid=358364&mirid=1&type=2

Testing for Neglected Nonlinearity in Long Memory Models

This article constructs tests for the presence of nonlinearity of unknown form in addition to a fractionally integrated, long-memory component in a time series process The tests are based on artificial neural network approximations and do not restrict the parametric form of the nonlinearity Some theoretical results for the new tests are obtained, and detailed simulation evidence on the power of the tests is presented The new methodology is then applied to a wide variety of economic and financial time series

https://www.econstor.eu/bitstream/10419/62906/1/377023701.pdf

Testing for Neglected Nonlinearity in Long-Memory Models

Many authors have investigated the possibility of long-range dependence, or long memory, in asset returns, but very little evidence has been found for long memory in either stock or exchange rate returns. This paper examines the presence of long-range dependence in a sample of 34 stock index returns using the semiparametric methods suggested by Geweke and Porter-Hudak (1983) and Robinson (1995). The results provide significant and robust evidence of fractional dynamics in most major and small stock markets over the sample periods. Depending on the test used, statistically significant long memory is detected in approximately 65% of the series. It appears that most stock returns are long-term dependent, suggesting stock market inefficiency and a high degree of predictability.

Long range dependence in stock market returns

Summary

This paper is concerned with the estimation and construction of confidence intervals for the impulse response function (IRF) from strongly persistent time series. A non-parametric, time domain estimator, based on an autoregressive (AR) approximation is shown to have good theoretical and small sample properties for the estimation of the IRF. An alternative procedure of using a semi-parametric local Whittle (LW) estimator of the long-memory parameter and then obtaining estimates of the short run parameters and IRF is also considered. The second part of the paper investigates the most appropriate methods for estimating the variability and the construction of confidence intervals for the estimated IRF. Particular attention is given to a generic semi-parametric sieve bootstrap based on an autoregressive approximation of the unknown data generating mechanism. The validity of bootstrap inference on the IRF, based on the autoregressive approximation, is proven under mild assumptions. The findings in this paper indicate that a good strategy for analysing IRF is to estimate by semi-parametric AR approximations, and to use the sieve bootstrap for estimating confidence intervals. Simulation evidence indicates this approach appears to be a very good strategy for processes with either short or long memory. An empirical example concerning the persistence of real exchange rate series is included.

Estimation and inference for impulse response functions from univariate strongly persistent processes

Several studies have tested for long-range dependence in macroeconomic and financial time series but very few have assessed the usefulness of long-memory models as forecast-generating mechanisms. This study tests for fractional differencing in the US monetary indices (simple sum and divisia) and compares the out-of-sample fractional forecasts to benchmark forecasts. The long-memory parameter is estimated using Robinson's Gaussian semi-parametric and multivariate log-periodogram methods. The evidence amply suggests that the monetary series possess a fractional order between one and two. Fractional out-of-sample forecasts are consistently more accurate (with the exception of the M3 series) than benchmark autoregressive forecasts but the forecasting gains are not generally statistically significant. In terms of forecast encompassing, the fractional model encompasses the autoregressive model for the divisia series but neither model encompasses the other for the simple sum series.  Copyright © 2006 John Wiley & Sons, Ltd.

/pdf/long-memory-forecasting-of-us-monetary-indices-273a2k2rrt.pdf

Long-memory forecasting of US monetary indices

. Two-stage estimators have been proposed in fractional autoregressive integrated moving-average (ARIMA) systems which first estimate the long-run features of the system semi-parametrically and then estimate the short-run features by usual methods in a second stage. Although asymptotic theory is available for the estimates in the first stage of such a procedure, we are aware of no results concerning the estimates in the second stage. In this paper we provide a stochastic order of magnitude associated with an estimator in this class and discuss this result.

Stochastic orders of magnitude associated with two‐stage estimators of fractional arima systems

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Stochastic orders of magnitude associated with two‐stage estimators of fractional arima systems