
on Econometric Time Series 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  New methods are developed for identifying, estimating and performing inference with nonstationary time series that have autoregressive roots near unity. The approach subsumes unit root (UR), local unit root (LUR), mildly integrated (MI) and mildly explosive (ME) speciï¬ cations in the new model formulation. It is shown how a new parameterization involving a localizing rate sequence that characterizes departures from unity can be consistently estimated in all cases. Simple pivotal limit distributions that enable valid inference about the form and degree of nonstationarity apply for MI and ME speciï¬ cations and new limit theory holds in UR and LUR cases. Normalizing and variance stabilizing properties of the new parameterization are explored. Simulations are reported that reveal some of the advantages of this alternative formulation of nonstationary time series. A housing market application of the methods is conducted that distinguishes the diï¬€ering forms of house price behavior in Australian state capital cities over the past decade. 
Keywords:  Cauchy limit distribution, Local to unity, Localizing rate sequence, Mild integration, Mildly explosive process, Unit root 
JEL:  C22 
Date:  2021–10 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:2304&r= 
By:  Yanbo Liu (School of Economics, Shandong University); Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  This paper explores predictive regression models with stochastic unit root (STUR) components and robust inference procedures that encompass a wide class of persistent and timevarying stochastically nonstationary regressors. The paper extends the mechanism of endogenously generated instrumentation known as IVX, showing that these methods remain valid for short and longhorizon predictive regressions in which the predictors have STUR and local STUR (LSTUR) generating mechanisms. Both mean regression and quantile regression methods are considered. The asymptotic distributions of the IVX estimators are new and require some new methods in their derivation. The distributions are compared to previous results and, as in earlier work, lead to pivotal limit distributions for Wald testing procedures that remain robust for both single and multiple regressors with various degrees of persistence and stochastic and ï¬ xed local departures from unit roots. Numerical experiments corroborate the asymptotic theory, and IVX testing shows good power and size control. The IVX methods are illustrated in an empirical application to evaluate the predictive capability of economic fundamentals in forecasting S\&P 500 excess returns. 
Keywords:  IVX, Long horizon, LSTUR, Predictability, Quantile regression, Robustness, Short horizon, STUR 
JEL:  C12 C22 G01 
Date:  2021–10 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:2305&r= 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University); Igor Kheifets (HSE University) 
Abstract:  A semiparametric triangular systems approach shows how multicointegration can occur naturally in an I(1) cointegrated regression model. The framework reveals the source of multicointegration as singularity of the long run error covariance matrix in an I(1) system, a feature noted but little explored in earlier work. Under such singularity, cointegrated I(1) systems embody a multicointegrated structure and may be analyzed and estimated without appealing to the associated I(2) system but with consequential asymptotic properties that can introduce asymptotic bias into conventional methods of cointegrating regression. The present paper shows how estimation of such systems may be accomplished under multicointegration without losing the nice properties that hold under simple cointegration, including mixed normality and pivotal inference. The approach uses an extended version of highdimensional trend IV estimation with deterministic orthonormal instruments that leads to mixed normal limit theory and pivotal inference in singular multicointegrated systems in addition to standard cointegrated I(1) systems. Wald tests of general linear restrictions are constructed using a ï¬ xedb long run variance estimator that leads to robust pivotal HAR inference in both cointegrated and multicointegrated cases. Simulations show the properties of the estimation and inferential procedures in ï¬ nite samples, contrasting the cointegration and multicointegration cases. An empirical illustration to housing stocks, starts and completions is provided. 
Keywords:  Cointegration, HAR inference, Integration, Long run variance matrix, Multicointegration, Singularity, Trend IV estimation 
JEL:  C12 C13 C22 
Date:  2021–10 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:2306&r= 
By:  Tommaso Proietti (CEIS & DEF, Università di Roma "Tor Vergata"); Federico Maddanu (Università di Roma "Tor Vergata") 
Abstract:  The paper proposes a novel model for time series displaying persistent stationary cycles, the fractional sinusoidal waveform process. The underlying idea is to allow the parameters that regulate the amplitude and phase to evolve according to fractional noise processes. Its advantages with respect to popular alternative specifications, such as the Gegenbauer process, are twofold: the autocovariance function is available in closed form, which opens the way to exact maximum likelihood estimation; secondly the model encompasses deterministic cycles, so that discrete spectra arise as a limiting case. A generalization of the process, featuring multiple components, an additive `red noise' component and exogenous variables, provides a model for climate time series with mixed spectra. Our illustrations deal with the change in amplitude and phase of the intraannual component of carbon dioxide concentrations in Mauna Loa, and with the estimation and the quantification of the contribution of orbital cycles to the variability of paleoclimate time series. 
Keywords:  Mixed Spectrum. Cyclical long memory. Paleoclimatic data 
Date:  2021–10–19 
URL:  http://d.repec.org/n?u=RePEc:rtv:ceisrp:518&r= 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d â‰¥ 1 2: Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d 
Keywords:  Discrete Fourier transform, Fractional Brownian motion, Fractional integration, Log periodogram regression, Nonstationarity, Operator decomposition, Semiparametric estimation, Whittle likelihood 
JEL:  C22 
Date:  2021–10 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:2303&r= 
By:  Abhishek K. Umrawal; Joshua C. C. Chan 
Abstract:  We propose a new quadraticprogrammingbased method of approximating a nonstandard density using a multivariate Gaussian density. Such nonstandard densities usually arise while developing posterior samplers for unobserved components models involving inequality constraints on the parameters. For instance, Chat et al. (2016) propose a new model of trend inflation with linear inequality constraints on the stochastic trend. We implement the proposed new method for this model and compare it to the existing approximation. We observe that the proposed new method works as good as the existing approximation in terms of the final trend estimates while achieving greater gains in terms of sample efficiency. 
Date:  2021–10 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2110.12149&r= 
By:  Alessandra Luati (University of Bologna); Francesca Papagni (Free University of Bozen); Tommaso Proietti (CEIS & DEF, Università di Roma "Tor Vergata") 
Abstract:  This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalized autocovariance function of a stationary random process. The generalized autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. The general result on the asymptotic efficiency is then applied to the class of Gaussian stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a YuleWalker system of equations in the generalized autocovariance estimator. 
Keywords:  CramérRao lower bound; Frequency Domain; Minimum Contrast Estimation; Periodogram 
Date:  2021–10–14 
URL:  http://d.repec.org/n?u=RePEc:rtv:ceisrp:515&r= 