Semiparametric Estimation of Fractional Cointegrating Subspaces

We consider a common components model for multivariate fractional cointegration, in which the s>=1 components have different memory parameters. The cointegrating rank is allowed to exceed 1. The true cointegrating vectors can be decomposed into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters, denoted by d_k for k=1,...,s. We estimate each cointegrating subsspace separately using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered, differenced observations. The averaging uses the first m Fourier frequencies, with m fixed. We will show that any vector in the k'th estimated coingetraging subspace is, with high probability, close to the k'th true cointegrating subspace, in the sense that the angle between the estimated cointegrating vector and the true cointegrating subspace converges in probability to zero. The angle is O_p(n^{- \alpha_k}), where n is the sample size and \alpha_k is the shortest distance between the memory parameters corresponding to the given and adjacent subspaces. We show that the cointegrating residuals corresponding to an estimated cointegrating vector can be used to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to \infty more slowly than n. We also show how these memory parameter estimates can be used to test for fractional cointegration and to consistently identify the cointegrating subspaces.Fractional Cointegration; Long Memory; Tapering; Periodogram

Is exposure to secondhand smoke associated with cognitive parameters of children and adolescents?—a systematic literature review

PURPOSE: Despite the known association of second hand smoke (SHS) with increased risk of ill health and mortality, the effects of SHS exposure on cognitive functioning in children and adolescents are unclear. Through a critical review of the literature we sought to determine whether a relationship exists between these variables. METHODS: The authors systematically reviewed articles (dated 1989–2012) that investigated the association between SHS exposure (including in utero due to SHS exposure by pregnant women) and performance on neurocognitive and academic tests. Eligible studies were identified from searches of Web of Knowledge, MEDLINE, Science Direct, Google Scholar, CINAHL, EMBASE, Zetoc, and Clinicaltrials.gov. RESULTS: Fifteen articles were identified, of which 12 showed inverse relationships between SHS and cognitive parameters. Prenatal SHS exposure was inversely associated with neurodevelopmental outcomes in young children, whereas postnatal SHS exposure was associated with poor academic achievement and neurocognitive performance in older children and adolescents. Furthermore, SHS exposure was associated with an increased risk of neurodevelopmental delay. CONCLUSIONS: Recommendations should be made to the public to avoid sources of SHS and future research should investigate interactions between SHS exposure and other risk factors for delayed neurodevelopment and poor cognitive performance

Rank and factor loadings estimation in time series tensor factor model by pre-averaging

The idiosyncratic components of a tensor time series factor model can exhibit serial correlations, (e.g., finance or economic data), ruling out many state-of-the-art methods that assume white/independent idiosyncratic components. While the traditional higher order orthogonal iteration (HOOI) is proved to be convergent to a set of factor loading matrices, the closeness of them to the true underlying factor loading matrices are in general not established, or only under i.i.d. Gaussian noises. Under the presence of serial and cross-correlations in the idiosyncratic components and time series variables with only bounded fourth-order moments, for tensor time series data with tensor order two or above, we propose a pre-averaging procedure that can be considered a random projection method. The estimated directions corresponding to the strongest factors are then used for projecting the data for a potentially improved re-estimation of the factor loading spaces themselves, with theoretical guarantees and rate of convergence spelt out when not all factors are pervasive. We also propose a new rank estimation method, which utilizes correlation information from the projected data. Extensive simulations are performed and compared to other state-of-the-art or traditional alternatives. A set of tensor-valued NYC taxi data is also analyzed

Estimating fractional cointegration in the presence of polynomial trends

We propose and derive the asymptotic distribution of a tapered narrow-band least squares estimator (NBLSE) of the cointegration parameter ÃÂò in the framework of fractional cointegration. This tapered estimator is invariant to deterministic polynomial trends. In particular, we allow for arbitrary linear time trends that often occur in practice. Our simulations show that, in the case of no deterministic trends, the estimator is superior to ordinary least squares (OLS) and the nontapered NBLSE proposed by P.M. Robinson when the levels have a unit root and the cointegrating relationship between the series is weak. In terms of rate of convergence, our estimator converges faster under certain circumstances, and never slower, than either OLS or the nontapered NBLSE. In a data analysis of interest rates, we find stronger evidence of cointegration if the tapered NBLSE is used for the cointegration parameter than if OLS is used.Statistics Working Papers Serie

Semiparametric Estimation of Fractional Cointegrating Subspaces

We consider a common components model for multivariate fractional cointegration, in which the s ¸ 1 components have different memory parameters. The cointegrating rank is allowed to exceed 1. The true cointegrating vectors can be decomposed into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters, denoted by dk, for k = 1; : : : ; s. We estimate each cointegrating subspace separately using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered, differenced observations. The averaging uses the first m Fourier frequencies, with m fixed. We will show that any vector in the k’th estimated cointegrating subspace is, with high probability, close to the k’th true cointegrating subspace, in the sense that the angle between the estimated cointegrating vector and the true cointegrating subspace converges in probability to zero. This angle is Op(n¡®k ), where n is the sample size and ®k is the shortest distance between the memory parameters corresponding to the given and adjacent subspaces. We show that the cointegrating residuals corresponding to an estimated cointegrating vector can be used to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to 1 more slowly than n. We also show how these memory parameter estimates can be used to test for fractional cointegration and to consistently identify the cointegrating subspaces.Statistics Working Papers Serie

On the Correlation Matrix of the Discrete Fourier Transform and the Fast Solution of Large Toeplitz Systems For Long-Memory Time Series

For long-memory time series, we show that the Toeplitz system §n(f)x = b can be solved in O(n log5=2 n) operations using a well-known version of the preconditioned conjugate gradient method, where §n(f) is the n£n covariance matrix, f is the spectral density and b is a known vector. Solutions of such systems are needed for optimal linear prediction and interpolation. We establish connections between this preconditioning method and the frequency domain analysis of time series. Indeed, the running time of the algorithm is determined by rate of increase of the condition number of the correlation matrix of the discrete Fourier transform vector, as the sample size tends to 1. We derive an upper bound for this condition number. The bound is of interest in its own right, as it sheds some light on the widely-used but heuristic approximation that the standardized DFT coefficients are uncorrelated with equal variances. We present applications of the preconditioning methodology to the forecasting and smoothing of volatility in a long memory stochastic volatility model, and to the evaluation of the Gaussian likelihood function of a long-memory model.Statistics Working Papers Serie

Impact of Exponential Smoothing on Inventory Costs in Supply Chains

It is common for firms to forecast stationary demand using simple exponential smoothing due to the ease of computation and understanding of the methodology. In this paper we show that the use of this methodology can be extremely costly in the context of inventory in a two-stage supply chain when the retailer faces AR(1) demand. We show that under the myopic order-up-to level policy, a retailer using exponential smoothing may have expected inventory-related costs more than ten times higher than when compared to using the optimal forecast. We demonstrate that when the AR(1) coefficient is less than the exponential smoothing parameter, the supplier’s expected inventory-related cost is less when the retailer uses optimal forecasting as opposed to exponential smoothing. We show there exists an additional set of cases where the sum of the expected inventory-related costs of the retailer and the supplier is less when the retailer uses optimal forecasting as opposed to exponential smoothing even though the supplier’s expected cost is higher. In this paper, we study the impact on the naive retailer, the sophisticated supplier, and the two-stage chain as a whole of the supplier sharing its forecasting expertise with the retailer. We provide explicit formulas for the supplier’s demand and the mean squared forecast errors for both players under various scenarios.College of Business Administration, Department of Information Systems and Supply Chain Management, Rider University; Sy Syms School of Business, Yeshiva University; Department of Information, Operations, and Management Science, Leonard N. Stern School of Business, New York UniversityOperations Management Working Papers Serie

Adaptation to Walking Direction in Biological Motion

The direction that we see another person walking provides us with an important cue to their intentions, but little is known about how the brain encodes walking direction across a neuronal population. The current study used an adaptation technique to investigate the sensory coding of perceived walking direction. We measured perceived walking direction of point-light stimuli before and after adaptation, and found that adaptation to a specific walking direction resulted in repulsive perceptual aftereffects. The magnitude of these aftereffects was tuned to the walking direction of the adaptor relative to the test, with local repulsion of perceived walking direction for test stimuli oriented on either side of the adapted walking direction. The specific tuning profiles that we observed are well explained by a population-coding model, in which perceived walking direction is coded in terms of the relative activity across a bank of sensory channels with peak tuning distributed across the full 360° range of walking directions. Further experiments showed specificity in how horizontal (azimuth) walking direction is coded when moving away from the observer compared to when moving toward the observer. Moreover, there was clear specificity in these perceptual aftereffects for walking direction compared to a nonbiological form of 3D motion (a rotating sphere). These results indicate the existence of neural mechanisms in the human visual system tuned to specific walking directions, provide insight into the number of sensory channels and how their responses are combined to encode walking direction, and demonstrate the specificity of adaptation to biological motion

Asymptotics for Duration-Driven Long Range Dependent Processes

We consider processes with second order long range dependence resulting from heavy tailed durations. We refer to this phenomenon as duration-driven long range dependence (DDLRD), as opposed to the more widely studied linear long range dependence based on fractional differencing of an $iid$ process. We consider in detail two specific processes having DDLRD, originally presented in Taqqu and Levy (1986), and Parke (1999). For these processes, we obtain the limiting distribution of suitably standardized discrete Fourier transforms (DFTs) and sample autocovariances. At low frequencies, the standardized DFTs converge to a stable law, as do the standardized sample autocovariances at fixed lags. Finite collections of standardized sample autocovariances at a fixed set of lags converge to a degenerate distribution. The standardized DFTs at high frequencies converge to a Gaussian law. Our asymptotic results are strikingly similar for the two DDLRD processes studied. We calibrate our asymptotic results with a simulation study which also investigates the properties of the semiparametric log periodogram regression estimator of the memory parameter