Exact Local Whittle Estimation of Fractionally Cointegrated Systems

Semiparametric estimation of a bivariate fractionally cointegrated system is considered. The new estimator employs the exact local Whittle approach developed by Shimotsu and Phillips (2003a) and estimates the two memory parameters jointly with the cointegrating vector. It permits both (asymptotically) stationary and nonstationary stochastic trends and/or equilibrium errors without relying on differencing or data tapering. Indeed, the asymptotic properties of the estimator depend only on the difference of the two memory parameters. The estimator of the memory parameters is shown to be consistent and asymptotically normally distributed in both stationary and nonstationary cases.

Exact Local Whittle Estimation of Fractional Integration with Unknown Mean and Time Trend

Recently, Shimotsu and Phillips (2005) developed a new semiparametric estimator, the exact local Whittle (ELW) estimator, of the memory parameter (d) in fractionally integrated processes. The ELW estimator has been shown to be consistent and have the same N(0,1/4) limit distribution for all values of d if the optimization covers an interval of width less than 9/2 and the mean of the process is known. With the intent to provide an efficient semiparametric estimator suitable for economic data, we extend the ELW estimator so that it accommodates an unknown mean and a polynomial time trend. We show that the resulting feasible ELW estimator is consistent and has a N(0,1/4) limit distribution for -1/2

Gaussian semiparametric estimation of multivariate fractionally integrated processes

This paper analyzes the semiparametric estimation of multivariate long-range dependent processes. The class of spectral densities considered includes multivariate fractionally integrated processes, which are not covered by the existing literature. This paper also establishes the consistency of the multivariate Gaussian semiparametric estimator, which has not been shown in the other works. Asymptotic normality of the multivariate Gaussian semiparametric estimator is also established, and the proposed estimator is shown to have a smaller limiting variance than the two-step Gaussian semiparametric estimator studied by Lobato (1999). Gaussianity is not assumed in the asymptotic theory.

Simple (but effective) tests of long memory versus structural breaks

This paper proposes two simple tests that are based on certain time domain properties of I(d) processes. First, if a time series follows an I(d) process, then each subsample of the time series also follows an I(d) process with the same value of d. Second, if a time series follows an I(d) process, then its dth differenced series follows an I(0) process. Simple as they may sound, these properties provide useful tools to distinguish between true and spurious I(d) processes. In the first test, we split the sample into b subsamples, estimate d for each subsample, and compare them with the estimate of d from the full sample. In the second test, we estimate d, use the estimate to take the dth difference of the sample, and apply the KPSS test and Phillips-Perron test to the differenced data and its partial sum. Both tests are applicable to both stationary and nonstationary I(d) processes. Simulations show that the proposed tests have good power against the spurious long memory models considered in the literature. The tests are applied to the daily realized volatility of the S&P 500 index.long memory, fractional integration, structural breaks, realized volatility

Exact Local Whittle Estimation of Fractional Integration with Unknown Mean and Time Trend

Recently Shimotsu and Phillips (2002, Essex Discussion Paper 535) developed a new semiparametric estimator, the exact local Whittle (ELW) estimator, of the memory parameter (d) in fractionally integrated processes. The ELW estimator has been shown to be consistent and have the same N(0, 1/4 ) limit distribution for all values of d. With economic applications in mind, we extend the ELW estimator so that it accommodates an unknown mean and a linear time trend. We show the resulting feasible ELW estimator is consistent for d > -1/2 and has a N(0, 1/4 ) limit distribution for d in (-1/2, 2) (d in (-1/2, 7/4) when the data have a linear trend) except for a few negligible intervals. A simulation study shows that the feasible ELW estimator inherits the desirable properties of the ELW estimator also in small samples.

Gaussian Semiparametric Estimation of Multivariate Fractionally Integrated Processes

This paper analyzes the semiparametric estimation of multivariate long-range dependent processes. The class of spectral densities considered is motivated by and includes those of multivariate fractionally integrated processes. The paper establishes the consistency of the multivariate Gaussian semiparametric estimator (GSE), which has not been shown in other work, and the asymptotic normality of the GSE estimator. The proposed GSE estimator is shown to have a smaller limiting variance than the two-step GSE estimator studied by Lobato (1999). Gaussianity is not assumed in the asymptotic theory. Some simulations confirm the relevance of the asymptotic results in samples of the size used in practical work.fractional integration, long memory, semiparametric estimation

Exact Local Whittle Estimation of Fractionally Cointegrated Systems

Semiparametric estimation of a bivariate fractionally cointegrated system is considered. We propose a two-step procedure that accommodates both (asymptotically) stationary (d =1/2) stochastic trend and/or equilibrium error. A tapered version of the local Whittle estimator of Robinson (2008) is used as the first-stage estimator, and the second-stage estimator employs the exact local Whittle approach of Shimotsu and Phillips (2005). The consistency and asymptotic distribution of the two-step estimator are derived. The estimator of the memory parameters has the same Gaussian asymptotic distribution in both the stationary and nonstationary case. The convergence rate and the asymptotic distribution of the estimator of the cointegrating vector are affected by the difference between the memory parameters. Further, the estimator has a Gaussian asymptotic distribution when the difference between the memory parameters is less than 1/2.discrete Fourier transform, fractional cointegration, long memory, nonstationarity, semiparametric estimation, Whittle likelihood