
on Econometric Time Series 
By:  Costantini, Mauro (Department of Economics, University of Vienna, Vienna, Austria); Lupi, Claudio (Department SEGeS, Faculty of Economics, University of Molise, Campobasso, Italy) 
Abstract:  In this paper we propose a simple extension to the panel case of the covariateaugmented Dickey Fuller (CADF) test for unit roots developed in Hansen (1995). The extension we propose is based on a pvalues combination approach that takes into account crosssection dependence. We show that the test is easy to compute, has good size properties and gives power gains with respect to other popular panel approaches. A procedure to compute the asymptotic pvalues of Hansen’s CADF test is also a sidecontribution of the paper. We also complement Hansen (1995) and Caporale and Pittis (1999) with some new theoretical results. Two empirical applications are carried out for illustration purposes on international data to test the PPP hypothesis and the presence of a unit root in international industrial production indices. 
Keywords:  Unit root, panel data, approximate pvalues, Monte Carlo 
JEL:  C22 C23 F31 
Date:  2011–01 
URL:  http://d.repec.org/n?u=RePEc:ihs:ihsesp:261&r=ets 
By:  Antonis Demos (www.aueb.gr/users/demos); Dimitra Kyriakopoulou 
Abstract:  n this paper we derive the bias approximations of the Maximum Likelihood (ML) and QuasiMaximum Likelihood (QML) Estimators of the EGARCH(1,1) parameters and we check our theoretical results through simulations. With the approximate bias expressions up to O(1/T), we are then able to correct the bias of all estimators. To this end, a Monte Carlo exercise is conducted and the results are presented and discussed. We conclude that, for given sets of parameters values, the bias correction works satisfactory for all parameters. The results for the bias expressions can be used in order to formulate the approximate Edgeworth distribution of the estimators. 
Date:  2010–06–10 
URL:  http://d.repec.org/n?u=RePEc:aue:wpaper:1108&r=ets 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University); Tassos Magdalinos (University of Nottingham) 
Abstract:  Nielsen (2009) shows that vector autoregression is inconsistent when there are common explosive roots with geometric multiplicity greater than unity. This paper discusses that result, provides a coexplosive system extension and an illustrative example that helps to explain the finding, gives a consistent instrumental variable procedure, and reports some simulations. Some exact limit distribution theory is derived and a useful new reverse martingale central limit theorem is proved. 
Keywords:  Coexplosive behavior, Common roots, Endogeneity, Forward instrumentation, Geometric multiplicity, Reverse martingale 
JEL:  C22 
Date:  2011–01 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1777&r=ets 
By:  Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  The delta method and continuous mapping theorem are among the most extensively used tools in asymptotic derivations in econometrics. Extensions of these methods are provided for sequences of functions, which are commonly encountered in applications, and where the usual methods sometimes fail. Important examples of failure arise in the use of simulation based estimation methods such as indirect inference. The paper explores the application of these methods to the indirect inference estimator (IIE) in first order autoregressive estimation. The IIE uses a binding function that is sample size dependent. Its limit theory relies on a sequencebased delta method in the stationary case and a sequencebased implicit continuous mapping theorem in unit root and local to unity cases. The new limit theory shows that the IIE achieves much more than bias correction. It changes the limit theory of the maximum likelihood estimator (MLE) when the autoregressive coefficient is in the locality of unity, reducing the bias and the variance of the MLE without affecting the limit theory of the MLE in the stationary case. Thus, in spite of the fact that the IIE is a continuously differentiable function of the MLE, the limit distribution of the IIE is not simply a scale multiple of the MLE but depends implicitly on the full binding function mapping. The unit root case therefore represents an important example of the failure of the delta method and shows the need for an implicit mapping extension of the continuous mapping theorem. 
Keywords:  Binding function, Delta method, Exact bias, Implicit continuous maps, Indirect inference, Maximum likelihood 
JEL:  C23 
Date:  2011–01 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1781&r=ets 
By:  Xiaohu Wang (Singapore Management University); Peter C.B. Phillips (Cowles Foundation, Yale University); Jun Yu (Singapore Management University) 
Abstract:  Multivariate continuous time models are now widely used in economics and finance. Empirical applications typically rely on some process of discretization so that the system may be estimated with discrete data. This paper introduces a framework for discretizing linear multivariate continuous time systems that includes the commonly used Euler and trapezoidal approximations as special cases and leads to a general class of estimators for the mean reversion matrix. Asymptotic distributions and bias formulae are obtained for estimates of the mean reversion parameter. Explicit expressions are given for the discretization bias and its relationship to estimation bias in both multivariate and in univariate settings. In the univariate context, we compare the performance of the two approximation methods relative to exact maximum likelihood (ML) in terms of bias and variance for the Vasicek process. The bias and the variance of the Euler method are found to be smaller than the trapezoidal method, which are in turn smaller than those of exact ML. Simulations suggest that when the mean reversion is slow the approximation methods work better than ML, the bias formulae are accurate, and for scalar models the estimates obtained from the two approximate methods have smaller bias and variance than exact ML. For the square root process, the Euler method outperforms the Nowman method in terms of both bias and variance. Simulation evidence indicates that the Euler method has smaller bias and variance than exact ML, Nowman's method and the Milstein method. 
Keywords:  Bias, Diffusion, Euler approximation, Trapezoidal approximation, Milstein approximation 
JEL:  C15 G12 
Date:  2011–01 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1778&r=ets 
By:  Qiying Wang (University of Sydney); Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  We provide a limit theory for a general class of kernel smoothed U statistics that may be used for specification testing in time series regression with nonstationary data. The framework allows for linear and nonlinear models of cointegration and regressors that have autoregressive unit roots or near unit roots. The limit theory for the specification test depends on the self intersection local time of a Gaussian process. A new weak convergence result is developed for certain partial sums of functions involving nonstationary time series that converges to the intersection local time process. This result is of independent interest and useful in other applications. 
Keywords:  Intersection local time, Kernel regression, Nonlinear nonparametric model, OrnsteinUhlenbeck process, Specification tests, Weak convergence 
JEL:  C14 C22 
Date:  2011–01 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1779&r=ets 
By:  Chirok Han (Korea University); Peter C.B. Phillips (Cowles Foundation, Yale University) 
Abstract:  First difference maximum likelihood (FDML) seems an attractive estimation methodology in dynamic panel data modeling because differencing eliminates fixed effects and, in the case of a unit root, differencing transforms the data to stationarity, thereby addressing both incidental parameter problems and the possible effects of nonstationarity. This paper draws attention to certain pathologies that arise in the use of FDML that have gone unnoticed in the literature and that affect both finite sample peformance and asymptotics. FDML uses the Gaussian likelihood function for first differenced data and parameter estimation is based on the whole domain over which the loglikelihood is defined. However, extending the domain of the likelihood beyond the stationary region has certain consequences that have a major effect on finite sample and asymptotic performance. First, the extended likelihood is not the true likelihood even in the Gaussian case and it has a finite upper bound of definition. Second, it is often bimodal, and one of its peaks can be so peculiar that numerical maximization of the extended likelihood frequently fails to locate the global maximum. As a result of these pathologies, the FDML estimator is a restricted estimator, numerical implementation is not straightforward and asymptotics are hard to derive in cases where the peculiarity occurs with nonnegligible probabilities. We investigate these problems, provide a convenient new expression for the likelihood and a new algorithm to maximize it. The peculiarities in the likelihood are found to be particularly marked in time series with a unit root. In this case, the asymptotic distribution of the FDMLE has bounded support and its density is infinite at the upper bound when the time series sample size T approaching infinity. As the panel width n approaching infinity the pathology is removed and the limit theory is normal. This result applies even for T fixed and we present an expression for the asymptotic distribution which does not depend on the time dimension. When n,T approaching infinity, the FDMLE has smaller asymptotic variance than that of the bias corrected MLE, an outcome that is explained by the restricted nature of the FDMLE. 
Keywords:  Asymptote, Bounded support, Dynamic panel, Efficiency, First difference MLE, Likelihood, Quartic equation, Restricted extremum estimator 
JEL:  C22 C23 
Date:  2011–01 
URL:  http://d.repec.org/n?u=RePEc:cwl:cwldpp:1780&r=ets 
By:  Cristina Amado (Universidade do Minho  NIPE); Timo Teräsvirta (CREATES, School of Economics and Management, Aarhus University) 
Abstract:  In this paper, we propose two parametric alternatives to the standard GARCH model. They allow the variance of the model to have a smooth timevarying structure of either additive or multiplicative type. The suggested parameterisations describe both nonlinearity and structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. The main focus is on the multiplicative decom position that decomposes the variance into an unconditional and conditional component. A modelling strategy for the timevarying GARCH model based on the multiplicative decomposition of the variance is developed. It is heavily dependent on Lagrange multiplier type misspeci.cation tests. Finitesample properties of the strategy and tests are examined by simulation. An empirical application to daily stock returns and another one to daily exchange rate returns illustrate the functioning and properties of our modelling strategy in practice. The results show that the long memory type behaviour of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance. 
Keywords:  Conditional heteroskedasticity; Structural change; Lagrange multiplier test; Misspeci.cation test; Nonlinear time series; Timevarying parameter model. 
JEL:  C12 C22 C51 C52 
Date:  2011 
URL:  http://d.repec.org/n?u=RePEc:nip:nipewp:01/2011&r=ets 