Economics Commons^™

Robust Testing For Explosive Behavior With Strongly Dependent Errors, Yui Lim Lui, Peter C. B. Phillips, Jun Yu

Research Collection School Of Economics

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 …

Volatility Puzzle: Long Memory Or Anti-Persistency, Shuping Shi, Jun Yu

Research Collection School Of Economics

The log realized volatility (RV) is often modeled as an autoregressive fractionally integrated moving average model ARFIMA(1,d,01,d,0). Two conflicting empirical results have been found in the literature. One stream shows that log RV has a long memory (i.e., the fractional parameter d > 0). The other stream suggests that the autoregressive coefficient α is near unity with antipersistent errors (i.e., d α close to 0 and d close to 0.5) from Model 2Model 2 (ARFIMA(1,d,01,d,0) with α close to unity and d close to –0.5). An intuitive explanation is given. For the 10 financial assets considered, despite that no definitive conclusions …

Asymptotic Theory For Explosive Fractional Ornstein–Uhlenbeck Processes, Hui Jiang, Yajuan Pan, Weilin Liao, Qingshan Yang, Jun Yu

Research Collection School Of Economics

This paper proposes estimators for the parameters of an explosive fractional Ornstein-Uhlenbeck process. The asymptotic properties for the diffusion estimators are developed under the in-fill asymptotic scheme, while the asymptotic properties for the drift estimators are developed under the double asymptotic scheme for the full range of the Hurst parameter. Simulation results demonstrate the effectiveness of the proposed estimators, and the asymptotic distributions provide a good approximation in finite samples. Empirical applications are presented to demonstrate the model’s usefulness and the practical value of the asymptotic theory.

Modeling And Forecasting Realized Volatility With The Fractional Ornstein-Uhlenbeck Process, Xiaohu Wang, Weilin Xiao, Jun Yu

Research Collection School Of Economics

This paper proposes to model and forecast realized volatility (RV) using the fractional Ornstein-Uhlenbeck (fO-U) process with a general Hurst parameter, H. A two-stage method is introduced for estimating parameters in the fO-U process based on discrete-sampled observations. In the first stage, H is estimated based on the ratio of two second-order differences of observations from different frequencies. In the second stage, with the estimated , the other parameters of the model are estimated by the method of moments. All estimators have closed-form expressions and are easy to implement. A large sample theory of the proposed estimators is derived. Extensive …

Volatility Puzzle: Long Memory Or Antipersistency, Shuping Shi, Jun Yu

Research Collection School Of Economics

The log realized volatility (RV) is often modeled as an autoregressive fractionally integrated moving average model ARFIMA(1, d, 0). Two conflicting empirical results have been found in the literature. One stream shows that log RV has a long memory (i.e., the fractional parameter d > 0). The other stream suggests that the autoregressive coefficient α is near unity with antipersistent errors (i.e., d

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 …

Robust Testing For Explosive Behavior With Strongly Dependent Errors, Yiu Lim Lui, Peter C. B. Phillips, Jun Yu

Research Collection School Of Economics

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 …

Essays On Long Memory Time Series And Panel Models, Shuyao Ke

Dissertations and Theses Collection (Open Access)

This dissertation studies different long memory models. The first chapter considers a time series regression model where both the regressors and error term are locally stationary long memory processes with time-varying memory parameters, and the regression coefficients are also allowed to be time-varying. We consider a frequency-domain least squares estimator with kernelized discrete Fourier transform and derive its pointwise asymptotic normality and uniform consistency. A specification test on the constancy of coefficients is provided. The second chapter studies a linear regression panel data model with interactive fixed effects where the regressors, factors and idiosyncratic error terms are all stationary but …

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.

Weak Identification Of Long Memory With Implications For Inference, Jia Li, Peter C. B. Phillips, Shuping Shi, Jun Yu

Research Collection School Of Economics

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.

Different Strokes For Different Folks: Long Memory And Roughness, Shuping Shi, Jun Yu

SMU Economics and Statistics Working Paper Series

The log realized volatility of financial assets is often modeled as an autoregressive fractionally integrated moving average model (ARFIMA) process, denoted by ARFIMA(p, d, q), with p = 1 and q = 0. Two conflicting results have been found in the literature regarding the dynamics. One stream shows that the data series has a long memory (i.e., the fractional parameter d > 0) with strong mean reversion (i.e., the autoregressive coefficient |α1| ≈ 0). The other stream suggests that the volatil-ity is rough (i.e., d < 0) with highly persistent dynamic (i.e., α1 → 1). To consolidate the findings, this paper first examines the finite sample properties of alternative estimation methods employed in the literature for the ARFIMA(1, d, 0) model and then applies the outperforming techniques to a wide range of financial assets. The candidate methods include two parametric maximum likeli-hood (ML) methods (the maximum time-domain modified profile likelihood (MPL) and maximum frequency-domain likelihood) and two semiparametric methods (the local Whittle method and log periodogram estimation method). The two parametric methods work well across all parameter set-tings, with the MPL method outperforming. In contrast, the two semiparametric methods have a very large upward bias for d and an equally large downward bias for α1 when α1 is close to unity. The poor performance of the semiparametric methods in the presence of a highly persistent dynamic might lead to a false conclusion of long memory. In the empirical applications, we find that the log realized volatilities of exchange rate futures over the past decade have a long memory, where the point estimate of d is between 0.4 and 0.5 and the estimate of α1 is near zero. For other finan-cial assets considered (including stock indices and industry indices), we find that they have rough volatility, with the point estimate of d being negative and the point estimates of α1 close to unity.

Persistent And Rough Volatility, Xiaobin Liu, Shuping Shi, Jun Yu

Research Collection School Of Economics

This paper contributes to an ongoing debate on volatility dynamics. We introduce a discrete-time fractional stochastic volatility (FSV) model based on the fractional Gaussian noise. The new model has the same limit as the fractional integrated stochastic volatility (FISV) model under the in-fill asymptotic scheme. We study the theoretical properties of both models and introduce a memory signature plot for a model-free initial assessment. A simulated maximum likelihood (SML) method, which maximizes the time-domain log-likelihoods obtained by the importance sampling technique, is employed to estimate the model parameters. Simulation studies suggest that the SML method can accurately estimate both models. …

Three Essays On Nonstationary Time Series Econometrics, Yiu Lim Lui

Dissertations and Theses Collection (Open Access)

This dissertation comprises three papers that separately study different nonstationary time series models.

The first paper, titled as "The Grid Bootstrap for Continuous Time Models", is a joint work with Professor Jun Yu and Professor Weilin Xiao. It considers the grid bootstrap for constructing confidence intervals for the persistence parameter in a class of continuous-time models driven by a Lévy process. Its asymptotic validity is discussed under the assumption that the sampling interval (h) shrinks to zero, the time span (N) goes to infinity or both. Its improvement over the in-fill asymptotic theory is achieved by expanding the coefficient-based statistic …

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

Research Collection School Of Economics

Limit theory involving stochastic integrals is now widespread in time series econometrics and relies on a few key results on functional weak convergence. In establishing such convergence, the literature commonly uses martingale and semimartingale structures. While these structures have wide relevance, many applications involve a cointegration framework where endogeneity and nonlinearity play major roles and complicate 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) time series that simplifies the asymptotics and our limit results for …

Estimating The Volatility Occupation Time Via Regularized Laplace Inversion, Jia Li, Viktor Todorov, Tauchen

Research Collection School Of Economics

We propose a consistent functional estimator for the occupation time of the spot variance of an asset price observed at discrete times on a finite interval with the mesh of the observation grid shrinking to zero. The asset price is modeled nonparametrically as a continuous-time Itô semimartingale with nonvanishing diffusion coefficient. The estimation procedure contains two steps. In the first step we estimate the Laplace transform of the volatility occupation time and, in the second step, we conduct a regularized Laplace inversion. Monte Carlo evidence suggests that the proposed estimator has good small-sample performance and in particular it is far …

Gdp Per Capita In Africa Before The Global Financial Crisis: Persistence, Mean Reversion And Long Memory Features, Luis A. Gil-Alana, Olaoluwa S. Yaya, Olanrewaju I. Shittu

CBN Journal of Applied Statistics (JAS)

This paper examined the long memory features of GDP per capita data before the global financial crisis, using a sample of 26 African countries. The study employed fractional integration and tested the stability of the differencing parameter across the sample period for each country. The results indicated that most of the countries’ GDP series were I(1) or higher. Evidence of mean reversion was observed in 10 countries where the disturbances were autocorrelated. There was strong evidence against mean reversion in the remaining 16 countries. The results also indicated that the fractional differencing parameter was stable in 17 countries, while the …

Forecasting Nigerian Stock Market Returns Using Arima And Artificial Neural Network Models, Godknows M. Isenah, Olusanya E. Olubusoye

CBN Journal of Applied Statistics (JAS)

The study reports empirical evidence that artificial neural network based models are applicable to forecasting of stock market returns. The Nigerian stock market logarithmic returns time series was tested for the presence of memory using the Hurst coefficient before the models were trained. The test showed that the logarithmic returns process is not a random walk and that the Nigerian stock market is not efficient. Two artificial neural network based models were developed in the study. These networks are TECH (4-3-1) and TECH (3-3-1)whose out-of-sample forecast performance was compared with a baseline ARIMA (3,0,1) model. The results obtained in the …

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 Regressors And Predictive Regressions: A Two-Stage Rebalancing Approach, Alex Maynard, Aaron Smallwood, Mark E. Wohar

Economics Faculty Publications

Predictability tests with long memory regressors may entail both size distortion and incompatibility between the orders of integration of the dependent and independent variables. Addressing both problems simultaneously, this paper proposes a two-step procedure that rebalances the predictive regression by fractionally differencing the predictor based on a first-stage estimation of the memory parameter. Extensive simulations indicate that our procedure has good size, is robust to estimation error in the first stage, and can yield improved power over cases in which an integer order is assumed for the regressor. We also extend our approach beyond the standard predictive regression context to …

On Fractionally Integrated Logistic Smooth Transitions In Time Series, Olanrewaju I. Shittu, Yaya S. Olaolua

CBN Journal of Applied Statistics (JAS)

Long memory and nonlinearity are two key features of some macroeconomic time series which are characterized by persistent shocks that seem to rise faster during recession than it falls during expansion. A variant of nonlinear time series model together with long memory are used to examine these features in inflation series for three economies. The results which compares favourably with that of van Dijk et al. (2002) elicit some interesting attributes of inflation in the developed and developing economies.

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

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

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 …

Are Australia's Savings And Investment Fractionally Cointegrated?, Arusha V. Cooray, B. Felmingham

Faculty of Commerce - Papers (Archive)

This paper uses an Autoregressive Fractionally Integrated Moving Average (ARFIMA) process to determine if Australia’s savings and investment are fractionally cointegrated. The study finds the two series to be fractionally cointegrated implying that deviations from equilibrium are persistent.

Essays On Models For Financial Volatility, Mihaela Oana Craioveanu

LSU Doctoral Dissertations

This research is focused on models for volatility. After the introduction of realized volatility as a consistent estimator for daily volatility, time series models without latent variables have been used to model and forecast volatility. The first part of this research provides a critical review of some of the commonly used realized volatility models and addresses the problem of stationarity and lag selection. In the empirical part we apply our methodology to thirty Dow Jones Industrial Average stocks from the NYSE TAQ dataset. We address the lag selection problem for each of the stocks considered. We find that models based …

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

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

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

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 …