Asymptotic spectral theory for nonlinear time series

We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditions, in our asymptotic spectral theory we impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear time series.Comment: Published in at http://dx.doi.org/10.1214/009053606000001479 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Some Univariate Time Series Properties of Output

This paper deals with the size of the random walk property of Colombia's output in two periods 1925-1994 and 1950-1994. GDP and GDPPC were both found to be integrated of order one a result which is very well known. The sequences are highly persistent, specially in the period 1950-1994. The forecast error when an innovation of 1 percent enters into the economy is about 1.5 percent in the very long run, when GDP is considered. The response is about 1.3 percent in the case of GDPPC, which seems to give support to the idea that population growth is a source of nonstationarity in some macroeconomic aggregates. For the larger sample(1925-1994) persistence is less. This result could cast some doubt on the method of estimation of GDP for the period 1925-1950. Finally, evidence of nonlinearity is found only in Hodrick- Prescott filtered variables dated between 1925 and 1994. This leaves open the question about whether the HP filter introduces nonlinearity in the hight frequency variable that it generates.Unit roots, persistence, nonlinearities, logistic function, ESTAR and, LSTAR. models.

Testing for Non-Linear Dependence in Univariate Time Series: An Empirical Investigation of the Austrian Unemployment Rate

The modelling of univariate time series is a subject of great importance in a variety of fields, in regional science and economics, and beyond. Time series modelling involves three major stages:model identification, model%0D estimation and diagnostic checking. This current paper focuses its attention on the model identification stage in general and on the issue of testing for non-linear dependence in particular. If the null hypothesis of independence is rejected, then the alternative hypothesis implies the existence of linear or non-linear dependence. The test of this hypothesis is of crucial importance. If the data are linearly dependent, the linear time series models have to be specified (generally within the SARIMA methodology). If the data are non-linearly dependent, then non-linear time series modelling (such as ARCH, GARCH and autoregressive neural network models) must be employed. Several tests have recently been developed for this purpose. In this paper we make a modest attempt to investigate the power of five competing tests (McLeod-Li-test, Hsieh-test, BDS-test, Terävirta''''s neural network test) in a real world application domain of unemployment rate prediction in order to determine what kind of non-linear specification they have good power against, and which not. The results obtained indicate that that all the tests reject the hypothesis of mere linear dependence in our application. But if interest is focused on predicting the conditional mean of the series, the neural network test is most informative for model identification and its use is therefore highly%0D recommended.

Asymptotic properties of weighted least squares estimation in weak parma models

The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Theorem 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.Weak periodic autoregressive moving average models; Seasonality; Weighted least squares; Asymptotic normality; Strong consistency; Weak periodic white noise; Strong mixing.

Higher order moments of bilinear time series processes with symmetrically distributed errors

Statistical Methods

Weakly dependent functional data

Functional data often arise from measurements on fine time grids and are obtained by separating an almost continuous time record into natural consecutive intervals, for example, days. The functions thus obtained form a functional time series, and the central issue in the analysis of such data consists in taking into account the temporal dependence of these functional observations. Examples include daily curves of financial transaction data and daily patterns of geophysical and environmental data. For scalar and vector valued stochastic processes, a large number of dependence notions have been proposed, mostly involving mixing type distances between $\sigma$-algebras. In time series analysis, measures of dependence based on moments have proven most useful (autocovariances and cumulants). We introduce a moment-based notion of dependence for functional time series which involves $m$-dependence. We show that it is applicable to linear as well as nonlinear functional time series. Then we investigate the impact of dependence thus quantified on several important statistical procedures for functional data. We study the estimation of the functional principal components, the long-run covariance matrix, change point detection and the functional linear model. We explain when temporal dependence affects the results obtained for i.i.d. functional observations and when these results are robust to weak dependence.Comment: Published in at http://dx.doi.org/10.1214/09-AOS768 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

LAM modelling of East European economies: Methodology, EU accession and privatisation

The paper presents a new approach to modelling economies in transition, where the adjustment processes are often nonlinear and data series are short. The model presented in the paper, the LAM-3 model, is the latest development in a series of long-run adjustment models, used for simulation and forecasting of several East European Economies. In particular the model contains short-run equations with bilinear error correction derived from a structural vector autoregressive model. The paper also gives derivations of two long-run relationships of the model, those for full-capacity output (reformable and non-reformable labour) and the relationship linking prices, money, incomes and exchange rates. The short-run part evolves around the specification of price and wage dynamics according to the NAIRU principle. Due to the fact that series of data for Eastern European countries are short, the parameters are evaluated with the use of global optimization technique (repetitive stochastic guestimation) rather than by a traditional econometric method. The model was estimated and applied for Czech Republic, Estonia, Latvia, Lithuania, Poland, Slovak Republic, Romania and Ukraine. For each country it consists of 3 long-run and 21 short-run relationships. Examples of simulations presented here evaluate European Union accessibility through inflation correlation measures and Aghion-Blanchard optimal speed of privatisation

Generalized and Subset Integrated Autoregressive Moving Average Bilinear Time Series Models

Generalized integrated autoregressive moving average bilinear model which is capable of achieving stationary for all non linear series is proposed and compared with subset generalized integrated autoregressive moving average bilinear model using the residual variance to see which perform better. The parameters of the proposed models are estimated using Newton-Raphson iterative method and Marquardt algorithm and the statistical properties of the derived estimates were investigated. An algorithm was proposed to eliminate redundant parameters from the full order generalized integrated autoregressive moving average bilinear models. To determine the order of the models, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) were adopted. Generalized integrated autoregressive moving average bilinear models are fitted to Wolfer sunspot numbers and stationary conditions are satisfied. Generalized integrated autoregressive moving average bilinear model performed better than subset generalized integrated autoregressive bilinear model. Keywords: Stationary, Newton-Raphson, Residual Variance, Marquardt Algorithm and Parameters