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Full-Text Articles in Economics
Robust Testing For Explosive Behavior With Strongly Dependent Errors, Yiu Lim Lui, Peter C. B. Phillips, Jun Yu
Robust Testing For Explosive Behavior With Strongly Dependent Errors, Yiu Lim Lui, Peter C. B. Phillips, Jun Yu
Cowles Foundation Discussion Papers
A heteroskedasticity-autocorrelation robust (HAR) test statistic is proposed to test for the presence of explosive roots in financial or real asset prices when the equation errors are strongly dependent. Limit theory for the test statistic is developed and extended to heteroskedastic models. The new test has stable size properties unlike conventional test statistics that typically lead to size distortion and inconsistency in the presence of strongly dependent equation errors. The new procedure can be used to consistently time-stamp the origination and termination of an explosive episode under similar conditions of long memory errors. Simulations are conducted to assess the finite …
Weak Identification Of Long Memory With Implications For Inference, Peter C. B. Phillips
Weak Identification Of Long Memory With Implications For Inference, Peter C. B. Phillips
Cowles Foundation Discussion Papers
This paper explores weak identification issues arising in commonly used models of
economic and financial time series. Two highly popular configurations are shown to
be asymptotically observationally equivalent: one with long memory and weak autoregressive dynamics, the other with antipersistent shocks and a near-unit autoregressive
root. We develop a data-driven semiparametric and identification-robust approach to
inference that reveals such ambiguities and documents the prevalence of weak identification in many realized volatility and trading volume series. The identification-robust empirical evidence generally favors long memory dynamics in volatility and volume, a conclusion that is corroborated using social-media news flow data.
Asymptotic Theory For Near Integrated Process Driven By Tempered Linear Process, Farzad Sabzikar, Qiying Wang, Peter C.B. Phillips
Asymptotic Theory For Near Integrated Process Driven By Tempered Linear Process, Farzad Sabzikar, Qiying Wang, Peter C.B. Phillips
Cowles Foundation Discussion Papers
This paper develops an asymptotic theory for near-integrated random processes and some associated regressions when the errors are tempered linear processes. Tempered processes are stationary time series that have a semi-long memory property in the sense that the autocovariogram of the process resembles that of a long memory model for moderate lags but eventually diminishes exponentially fast according to the presence of a decay factor governed by a tempering parameter. When the tempering parameter is sample size dependent, the resulting class of processes admits a wide range of behavior that includes both long memory, semi-long memory, and short memory processes. …
Weak Convergence To Stochastic Integrals For Econometric Applications, Hanying Liang, Peter C.B. Phillips, Hanchao Wang, Qiying Wang
Weak Convergence To Stochastic Integrals For Econometric Applications, Hanying Liang, Peter C.B. Phillips, Hanchao Wang, Qiying Wang
Cowles Foundation Discussion Papers
Limit theory involving stochastic integrals is now widespread in time series econometrics and relies on a few key results on function space weak convergence. In establishing weak convergence of sample covariances to stochastic integrals, the literature commonly uses martingale and semimartingale structures. While these structures have wide relevance, many applications in econometrics involve a cointegration framework where endogeneity and nonlinearity play a major role and lead to complications in the limit theory. This paper explores weak convergence limit theory to stochastic integral functionals in such settings. We use a novel decomposition of sample covariances of functions of I(1) and I(0) …
Long Memory And Long Run Variation, Peter C.B. Phillips
Long Memory And Long Run Variation, Peter C.B. Phillips
Cowles Foundation Discussion Papers
A commonly used defining property of long memory time series is the power law decay of the autocovariance function. Some alternative methods of deriving this property are considered working from the alternate definition in terms of a fractional pole in the spectrum at the origin. The methods considered involve the use of (i) Fourier transforms of generalized functions, (ii) asymptotic expansions of Fourier integrals with singularities, (iii) direct evaluation using hypergeometric function algebra, and (iv) conversion to a simple gamma integral. The paper is largely pedagogical but some novel methods and results involving complete asymptotic series representations are presented. The …
Nonlinearity And Temporal Dependence, Xiaohong Chen, Lars P. Hansen, Marine Carrasco
Nonlinearity And Temporal Dependence, Xiaohong Chen, Lars P. Hansen, Marine Carrasco
Cowles Foundation Discussion Papers
Nonlinearities in the drift and diffusion coefficients influence temporal dependence in scalar diffusion models. We study this link using two notions of temporal dependence: beta-mixing and rho-mixing. We show that beta-mixing and rho-mixing with exponential decay are essentially equivalent concepts for scalar diffusions. For stationary diffusions that fail to be rho-mixing, we show that they are still beta-mixing except that the decay rates are slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. Some have spectral densities that diverge at frequency zero in a manner …
Nonlinearity And Temporal Dependence, Xiaohong Chen, Lars P. Hansen, Marine Carrasco
Nonlinearity And Temporal Dependence, Xiaohong Chen, Lars P. Hansen, Marine Carrasco
Cowles Foundation Discussion Papers
Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: rho-mixing, beta-mixing and alpha-mixing. Stationary diffusions that are rho-mixing have mixing coefficients that decay exponentially to zero. When they fail to be rho-mixing, they are still beta-mixing and alpha-mixing; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state-dependent, Poisson sampling …
Long Run Covariance Matrices For Fractionally Integrated Processes, Peter C.B. Phillips, Chang Sik Kim
Long Run Covariance Matrices For Fractionally Integrated Processes, Peter C.B. Phillips, Chang Sik Kim
Cowles Foundation Discussion Papers
An asymptotic expansion is given for the autocovariance matrix of a vector of stationary long-memory processes with memory parameters d satisfying 0 < d < 1/2. The theory is then applied to deliver formulae for the long run covariance matrices of multivariate time series with long memory.
A Complete Asymptotic Series For The Autocovariance Function Of A Long Memory Process, Offer Lieberman, Peter C.B. Phillips
A Complete Asymptotic Series For The Autocovariance Function Of A Long Memory Process, Offer Lieberman, Peter C.B. Phillips
Cowles Foundation Discussion Papers
An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d in (-1/2,1/2). The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA(p,d,q) model. The leading term of the expansion is of the order O (1/ k 1-2 d ), where k is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O(1/ k 3-2d ). The derivation uses Erdélyi’s (1956) expansion for Fourier-type integrals when there are critical points …
Optimal Estimation Of Cointegrated Systems With Irrelevant Instruments, Peter C.B. Phillips
Optimal Estimation Of Cointegrated Systems With Irrelevant Instruments, Peter C.B. Phillips
Cowles Foundation Discussion Papers
It has been know since Phillips and Hansen (1990) that cointegrated systems can be consistently estimated using stochastic trend instruments that are independent of the system variables. A similar phenomenon occurs with deterministically trending instruments. The present work shows that such “irrelevant” deterministic trend instruments may be systematically used to produce asymptotically efficient estimates of a cointegrated system. The approach is convenient in practice, involves only linear instrumental variables estimation, and is a straightforward one step procedure with no loss of degrees of freedom in estimation. Simulations reveal that the procedure works well in practice, having little finite sample bias …
Adaptive Local Polynomial Whittle Estimation Of Long-Range Dependence, Donald W.K. Andrews, Yixiao Sun
Adaptive Local Polynomial Whittle Estimation Of Long-Range Dependence, Donald W.K. Andrews, Yixiao Sun
Cowles Foundation Discussion Papers
The local Whittle (or Gaussian semiparametric) estimator of long range dependence, proposed by Künsch (1987) and analyzed by Robinson (1995a), has a relatively slow rate of convergence and a finite sample bias that can be large. In this paper, we generalize the local Whittle estimator to circumvent these problems. Instead of approximating the short-run component of the spectrum, φ(λ), by a constant in a shrinking neighborhood of frequency zero, we approximate its logarithm by a polynomial. This leads to a “local polynomial Whittle” (LPW) estimator. We specify a data-dependent adaptive procedure that adjusts the degree of the polynomial to the …
Higher-Order Improvements Of The Parametric Bootstrap For Long-Memory Gaussian Processes, Donald W.K. Andrews, Offer Lieberman
Higher-Order Improvements Of The Parametric Bootstrap For Long-Memory Gaussian Processes, Donald W.K. Andrews, Offer Lieberman
Cowles Foundation Discussion Papers
This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter d 0 are included. The results establish that the bootstrap provides higher-order improvements over the delta method. Analogous results are given for tests. The CIs and tests are based on one or other of two approximate maximum likelihood estimators. The first estimator solves the first-order conditions with respect to the covariance parameters of a “plug-in” log-likelihood function that has the unknown mean replaced by the sample mean. The second …
Error Bounds And Asymptotic Expansions For Toeplitz Product Functionals Of Unbounded Spectra, Offer Lieberman, Peter C.B. Phillips
Error Bounds And Asymptotic Expansions For Toeplitz Product Functionals Of Unbounded Spectra, Offer Lieberman, Peter C.B. Phillips
Cowles Foundation Discussion Papers
This paper establishes error orders for integral limit approximations to traces of powers to the p th order) of products of Toeplitz matrices. Such products arise frequently in the analysis of stationary time series and in the development of asymptotic expansions. The elements of the matrices are Fourier transforms of functions which we allow to be bounded, unbounded, or even to vanish on [- π,π ], thereby including important cases such as the spectral functions of fractional processes. Error rates are also given in the case in which the matrix product involves inverse matrices. The rates are sharp up to …
Local Whittle Estimation Of Fractional Integration, Katsumi Shimotsu, Peter C.B. Phillips
Local Whittle Estimation Of Fractional Integration, Katsumi Shimotsu, Peter C.B. Phillips
Cowles Foundation Discussion Papers
An exact form of the local Whittle likelihood is studied with the intent of developing a general purpose estimation procedure for the memory parameter ( d ) that does not rely on tapering or differencing prefilters. The resulting exact local Whittle estimator is shown to be consistent and to 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 initial value of the process is known.
Local Polynomial Whittle Estimation, Donald W.K. Andrews, Yixiao Sun
Local Polynomial Whittle Estimation, Donald W.K. Andrews, Yixiao Sun
Cowles Foundation Discussion Papers
The local Whittle (or Gaussian semiparametric) estimator of long range dependence, proposed by Künsch (1987) and analyzed by Robinson (1995a), has a relatively slow rate of convergence and a finite sample bias that can be large. In this paper, we generalize the local Whittle estimator to circumvent those problems. Instead of approximating the short-run component of the spectrum, φ(λ), by a constant in a shrinking neighborhood of frequency zero, we approximate its logarithm by a polynomial. This leads to a “local polynomial Whittle” (LPW) estimator. Following the work of Robinson (1995a), we establish the asymptotic bias, variance, mean-squared error (MSE), …
Modified Local Whittle Estimation Of The Memory Parameter In The Nonstationary Case, Katsumi Shimotsu, Peter C.B. Phillips
Modified Local Whittle Estimation Of The Memory Parameter In The Nonstationary Case, Katsumi Shimotsu, Peter C.B. Phillips
Cowles Foundation Discussion Papers
Semiparametric estimation of the memory parameter is studied in models of fractional integration in the nonstationary case, and some new representation theory for the discrete Fourier transform of a fractional process is used to assist in the analysis. A limit theory is developed for an estimator of the memory parameter that covers a range of values of d commonly encountered in applied work with economic data. The new estimator is called the modified local Whittle estimator and employs a version of the Whittle likelihood based on frequencies adjacent to the origin and modified to take into account the form of …
Local Whittle Estimation In Nonstationary And Unit Root Cases, Katsumi Shimotsu, Peter C.B. Phillips
Local Whittle Estimation In Nonstationary And Unit Root Cases, Katsumi Shimotsu, Peter C.B. Phillips
Cowles Foundation Discussion Papers
Asymptotic properties of the local Whittle estimator in the nonstationary case (d > 1/2) are explored. For 1/2 < d < 1, the estimator is shown to be consistent, and its limit distribution and the rate of convergence depend on the value of d . For d = 1, the limit distribution is mixed normal. For d > 1 and when the process has a linear trend, the estimator is shown to be inconsistent and to converge in probability to unity.