
on Econometric Time Series 
By:  Oscar Martinez (Universitat Rovira i Virgili, Spain); Jose Olmo (Department of Economics, City University, London) 
Abstract:  One of the main implications of the effcient market hypothesis (EMH) is that expected future returns on financial assets are not predictable if investors are risk neutral. In this paper we argue that financial time series offer more information than that this hypothesis seems to supply. In particular we postulate that runs of very large returns can be predictable for small time periods. In order to prove this we propose a TAR(3,1)GARCH(1,1) model that is able to describe two different types of extreme events: a first type generated by large uncertainty regimes where runs of extremes are not predictable and a second type where extremes come from isolated dread/joy events. This model is new in the literature in nonlinear processes. Its novelty resides on two features of the model that make it different from previous TAR methodologies. The regimes are motivated by the occurrence of extreme values and the threshold variable is defined by the shock affecting the process in the preceding period. In this way this model is able to uncover dependence and clustering of extremes in high as well as in low volatility periods. This model is tested with data from General Motors stock prices corresponding to two crises that had a substantial impact in fnancial markets worldwide; the Black Monday of October 1987 and September 11th, 2001. By analyzing the periods around these crises we find evidence of statistical significance of our model and thereby of predictability of extremes for September 11th but not for Black Monday. These findings support the hypotheses of a big negative event producing runs of negative returns in the first case, and of the burst of a worldwide stock market bubble in the second example. 
Keywords:  Asymmetries, crises; Extreme values; Hypothesis testing; Leverage effect; Nonlinearities; Threshold models 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cty:dpaper:0808&r=ets 
By:  Xiaohong Chen (Cowles Foundation, Yale University); Lars P. Hansen (Dept. of Economics and Statistics, University of Chicago); Marine Carrasco (Dept. of Economics, University of Montreal) 
Abstract:  Nonlinearities in the drift and diffusion coefficients influence temporal dependence in scalar diffusion models. We study this link using two notions of temporal dependence: betamixing and rhomixing. We show that betamixing and rhomixing with exponential decay are essentially equivalent concepts for scalar diffusions. For stationary diffusions that fail to be rhomixing, we show that they are still betamixing 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 similar to that of stochastic processes with long memory. Finally we show how nonlinear, statedependent, Poisson sampling alters the unconditional distribution as well as the temporal dependence. 
Keywords:  Mixing, Diffusion, Strong dependence, Long memory, Poisson sampling 
JEL:  C12 C13 C22 C50 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1652&r=ets 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  Some limit properties for information based model selection criteria are given in the context of unit root evaluation and various assumptions about initial conditions. Allowing for a nonparametric short memory component, standard information criteria are shown to be weakly consistent for a unit root provided the penalty coefficient C_n > infinity and C_n/n > 0 as n > infinity. Strong consistency holds when C_n/(loglog n)^3 > infinity under conventional assumptions on initial conditions and under a slightly stronger condition when initial conditions are infinitely distant in the unit root model. The limit distribution of the AIC criterion is obtained. 
Keywords:  AIC, Consistency, Model selection, Nonparametric, Unit root 
JEL:  C22 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1653&r=ets 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  A local limit theorem is proved for sample covariances of nonstationary time series and integrable functions of such time series that involve a bandwidth sequence. The resulting theory enables an asymptotic development of nonparametric regression with integrated or fractionally integrated processes that includes the important practical case of spurious regressions. Some local regression diagnostics are suggested for forensic analysis of such regresssions, including a local R² and a local Durbin Watson (DW) ratio, and their asymptotic behavior is investigated. The most immediate findings extend the earlier work on linear spurious regression (Phillips, 1986), showing that the key behavioral characteristics of statistical significance, low DW ratios and moderate to high R^2 continue to apply locally in nonparametric spurious regression. Some further applications of the limit theory to models of nonlinear functional relations and cointegrating regressions are given. The methods are also shown to be applicable in partial linear semiparametric nonstationary regression. 
Keywords:  Brownian motion, Kernel method, Local R^2, Local DurbinWatson ratio, Local time, Integrated process, Nonparametric regression, Spurious regression 
JEL:  C23 C25 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1654&r=ets 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University); Tassos Magdalinos (University of Nottingham, UK) 
Abstract:  It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient and a limit normal distribution for the t ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continues to hold in spite of the degeneracy in the limit theory and is therefore robust to initial conditions that extend to the infinite past. 
Keywords:  Cauchy limit distribution, Cointegration, Distant past initialization, Infinite past initialization, Random orthonormalization, Singular limit theory 
JEL:  C22 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1655&r=ets 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  May 2008 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 formulae are useful in many ways including the calculation of long run variation matrices for multivariate time series with long memory and the econometric estimation of such models. 
Keywords:  Asymptotic expansion, Autocovariance function, Fractional pole, Fourier integral, Generalized function, Long memory, Long range dependence, Singularity 
JEL:  C22 C32 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1656&r=ets 
By:  ; Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel estimates, and frequently leads to illposed inverse problems. In functional cointegrating regressions where the regressor is an integrated time series, it is shown here that inverse and illposed inverse problems do not arise. Remarkably, nonparametric kernel estimation of a structural nonparametric cointegrating regression is consistent and the limit distribution theory is mixed normal, giving simple useable asymptotics in practical work. The results provide a convenient basis for inference in structural nonparametric regression with nonstationary time series. The methods may be applied to a wide range of empirical models where functional estimation of cointegrating relations is required. 
Keywords:  Brownian Local time, Cointegration, Functional regression, Gaussian process, Integrated process, Kernel estimate, Nonlinear functional, Nonparametric regression, Structural estimation, Unit root 
JEL:  C14 C22 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1657&r=ets 
By:  Xu Cheng (Dept. of Economics, Yale University); Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  Some convenient limit properties of usual information criteria are given for cointegrating rank selection. Allowing for a nonparametric short memory component and using a reduced rank regression with only a single lag, standard information criteria are shown to be weakly consistent in the choice of cointegrating rank provided the penalty coefficient C_n > infinity and C_n/n > 0 as n > infinity. The limit distribution of the AIC criterion, which is inconsistent, is also obtained. The analysis provides a general limit theory for semiparametric reduced rank regression under weakly dependent errors. The method does not require the specification of a full model, is convenient for practical implementation in empirical work, and is sympathetic with semiparametric estimation approaches to cointegration analysis. Some simulations results on finite sample performance of the criterion are reported. 
Keywords:  Cointegrating rank, Consistency, Information criteria, Model selection, Nonparametric, Short memory, Unit roots 
JEL:  C22 C32 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1658&r=ets 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University); Tassos Magdalinos (University of Nottingham, UK); Liudas Giraitis (Queen Mary, University of London, UK) 
Abstract:  A limit theory is established for autoregressive time series that smooths the transition between local and moderate deviations from unity and provides a transitional form that links conventional unit root distributions and the standard normal. Edgeworth expansions of the limit theory are given. These expansions show that the limit theory that holds for values of the autoregressive coefficient that are closer to stationarity than local (i.e., deviations of the form =1 + (c/n), where n is the sample size and c < 0) holds up to the second order. Similar expansions around the limiting Cauchy density are provided for the mildly explosive case. 
Keywords:  Edgeworth expansion, Local to unity, Moderate deviations, Unit root distribution 
JEL:  C22 
Date:  2008–05 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1659&r=ets 
By:  Mohamed Boutahar (GREQAM  Groupement de Recherche en Économie Quantitative d'AixMarseille  Université de la Méditerranée  AixMarseille II  Université Paul Cézanne  AixMarseille III  Ecole des Hautes Etudes en Sciences Sociales  CNRS : UMR6579); Gilles Dufrénot (GREQAM  Groupement de Recherche en Économie Quantitative d'AixMarseille  Université de la Méditerranée  AixMarseille II  Université Paul Cézanne  AixMarseille III  Ecole des Hautes Etudes en Sciences Sociales  CNRS : UMR6579); Anne PeguinFeissolle (GREQAM  Groupement de Recherche en Économie Quantitative d'AixMarseille  Université de la Méditerranée  AixMarseille II  Université Paul Cézanne  AixMarseille III  Ecole des Hautes Etudes en Sciences Sociales  CNRS : UMR6579) 
Abstract:  This paper generalizes the standard long memory modeling by assuming that the long memory parameter d is stochastic and time varying: we introduce a STAR process on this parameter characterized by a logistic function. We propose an estimation method of this model. Some simulation experiments are conducted. The empirical results suggest that this new model offers an interesting alternative competing framework to describe the persistent dynamics in modelling some financial series. 
Keywords:  Longmemory, Logistic function, STAR 
Date:  2008–04–23 
URL:  http://d.repec.org/n?u=RePEc:hal:papers:halshs00275254_v2&r=ets 
By:  Silja Kinnebrock; Mark Podolskij 
Abstract:  This paper introduces a new estimator to measure the expost covariation between highfrequency financial time series under market microstructure noise. We provide an asymptotic limit theory (including feasible central limit theorems) for standard methods such as regression, correlation analysis and covariance, for which we obtain the optimal rate of convergence. We demonstrate some positive semidefinite estimators of the covariation and construct a positive semidefinite estimator of the conditional covariance matrix in the central limit theorem. Furthermore, we indicate how the assumptions on the noise process can be relaxed and how our method can be applied to nonsynchronous observations. We also present an empirical study of how highfrequency correlations, regressions and covariances change through time. 
Keywords:  Central Limit Theorem; Diffusion Models; Market Microstructure Noise; Nonsynchronous Trading; HighFrequency Data; Semimartingale Theory; 
Date:  2008 
URL:  http://d.repec.org/n?u=RePEc:sbs:wpsefe:2008fe25&r=ets 
By:  Andrea Vaona (Istituto Ricerche Economiche, Faculty of Economic Sciences, University of Lugano, Switzerland.) 
Abstract:  We show that some nonparametric specification tests can be robust to disturbance autocorrelation. This robustness can be affected by the specification of the true model and by the sample size. Once applied to the prediction of changes in the Euro Repo rate by means of an index based on ECB wording, we find that the least sensitive nonparametric tests can have a comparable performance to a RESET test with robust standard errors. 
Keywords:  nonparametric misspecification tests, serial correlation, central bank communication. 
JEL:  C14 C15 E5 
Date:  2008–04–11 
URL:  http://d.repec.org/n?u=RePEc:lug:wpaper:0803&r=ets 