COMMON SEASONAL FEATURES: GLOBAL UNEMPLOYMENT

Seasonal patterns in economic time series are generally examined from a univariate point of view. Using extensions of the unit root literature, important classes of seasonal processes are deterministic, stationary stochastic or mean reverting, and unit root stochastic. Time series tests have been developed for each of these. This paper examines seasonality in a multivariate context. Systems of economic variables can have trends, cycles and unit roots as well as the various types of seasonality. Restrictions such as cointegration and common cycles are here applied also to examine multivariate seasonal behaviour of economic variables. If each of a collection of series has a certain type of seasonality but a linear combination of these series can be found without seasonality, then the seasonal is said to be ‘common’. New tests are developed to determine if seasonal characteristics are common to a set of time series. These tests can be employed in the presence of various other time series structures. The analysis is applied to OECD data on unemployment for the period 1975.1 to 1993.4, and it is found that four diverse countries (Australia, Canada, Japan and USA) not only have common trends in their unemployment, but also have common deterministic seasonal features and a common cycle/stochastic seasonal feature. Such a collection of characteristics were not found in other groups of OECD countries.     

Can the random walk model be beaten in out-of-sample density forecasts? Evidence from intraday foreign exchange rates

It has been documented that random walk outperforms most economic structural and time series models in out-of-sample forecasts of the conditional mean dynamics of exchange rates. In this paper, we study whether random walk has similar dominance in out-of-sample forecasts of the conditional probability density of exchange rates given that the probability density forecasts are often needed in many applications in economics and finance. We first develop a nonparametric portmanteau test for optimal density forecasts of univariate time series models in an out-of-sample setting and provide simulation evidence on its finite sample performance. Then we conduct a comprehensive empirical analysis on the out-of-sample performances of a wide variety of nonlinear time series models in forecasting the intraday probability densities of two major exchange rates—Euro/Dollar and Yen/Dollar. It is found that some sophisticated time series models that capture time-varying higher order conditional moments, such as Markov regime-switching models, have better density forecasts for exchange rates than random walk or modified random walk with GARCH and Student-t innovations. This finding dramatically differs from that on mean forecasts and suggests that sophisticated time series models could be useful in out-of-sample applications involving the probability density.     

Nonlinear Time Series Modelling: An Introduction

Recent developments in nonlinear time series modelling are reviewed. Three main types of nonlinear model are discussed: Markov Switching, Threshold Autoregression and Smooth Transition Autoregression. Classical and Bayesian estimation techniques are described for each model. Parametric tests for nonlinearity are reviewed with examples from the three types of model. Finally forecasting and impulse response analysis is developed.     

Selecting a Nonlinear Time Series Model using Weighted Tests of Equal Forecast Accuracy*

Nonlinear time series models have become fashionable tools to describe and forecast a variety of economic time series. A closer look at reported empirical studies, however, reveals that these models apparently fit well in‐sample, but rarely show a substantial improvement in out‐of‐sample forecasts, at least over linear models. One of the many possible reasons for this finding is the use of inappropriate model selection criteria and forecast evaluation criteria. In this paper we therefore propose a novel criterion, which we believe does more justice to the very nature of nonlinear models. Simulations show that this criterion outperforms those criteria currently in use, in the sense that the true nonlinear model is more often found to perform better in out‐of‐sample forecasting than a benchmark linear model. An empirical illustration for US GDP emphasizes its relevance.     

On a double-threshold autoregressive heteroscedastic time series model

Tong's threshold models have been found useful in modelling nonlinearities in the conditional mean of a time series. The threshold model is extended to the so-called double-threshold ARCH(DTARCH) model, which can handle the situation where both the conditional mean and the conditional variance specifications are piecewise linear given previous information. Potential applications of such models include financial data with different (asymmetric) behaviour in a rising versus a falling market and business cycle modelling. Model identification, estimation and diagnostic checking techniques are developed. Maximum likelihood estimation can be achieved via an easy-to-use iteratively weighted least squares algorithm. Portmanteau-type statistics are also derived for checking model adequacy. An illustrative example demonstrates that asymmetric behaviour in the mean and the variance could be present in financial series and that the DTARCH model is capable of capturing these phenomena.     

A unified approach to nonlinearity,structural change,and outliers

This paper demonstrates that the class of conditionally linear and Gaussian state-space models offers a general and convenient framework for simultaneously handling nonlinearity, structural change and outliers in time series. Many popular nonlinear time series models, including threshold, smooth transition and Markov-switching models, can be written in state-space form. It is then straightforward to add components that capture parameter instability and intervention effects. We advocate a Bayesian approach to estimation and inference, using an efficient implementation of Markov Chain Monte Carlo sampling schemes for such linear dynamic mixture models. The general modelling framework and the Bayesian methodology are illustrated by means of several examples. An application to quarterly industrial production growth rates for the G7 countries demonstrates the empirical usefulness of the approach.     

A nonparametric measure of convergence towards purchasing power parity

It has been claimed that the deviations from purchasing power parity are highly persistent and have quite long half‐lives under the assumption of a linear adjustment of real exchange rates. However, inspired by trade cost models, nonlinear adjustment has been widely employed in recent empirical studies. This paper proposes a simple nonparametric procedure for evaluating the speed of adjustment in the presence of nonlinearity, using the largest Lyapunov exponent of the time series. The empirical result suggests that the speed of convergence to a long‐run price level is indeed faster than what was found in previous studies with linear restrictions. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability results for nonlinear error correction models

This paper studies analogs of Granger's representation theorem in the context of a general nonlinear vector autoregressive error correction model. The model allows for nonlinear autoregressive conditional heteroskedasticity and the conditional distribution involved can be a mixture distribution of a rather general type. Mixture models of this kind can be thought of as generalizations of threshold models and they have attracted attention in the recent time series and econometrics literature. The paper develops a useful transformation which shows how the nonlinear error correction model can be transformed to a nonlinear vector autoregressive model so that available results on the stationarity or nonstationarity of the latter can be used for the former. The most satisfactory results are obtained in a model in which a specific structural relation between the nonlinearity and equilibrium correction prevails. Without this structural relation only a lower bound for the number of long-run equilibrium relations can explicitly be determined because the exact number depends on properties of the first and second moments of a nonlinear stationary component of the process.     

Forecasting aggregates using panels of nonlinear time series

Macroeconomic time series such as total unemployment or total industrial production concern data which are aggregated across regions, sectors, or age categories. In this paper we examine whether forecasts for these aggregates can be improved by considering panel models for the disaggregate series. As many macroeconomic variables have nonlinear properties, we specifically focus on panels of nonlinear time series. We discuss the representation of such models, parameter estimation and a method for generating forecasts. We illustrate the usefulness of our approach for simulated data and for the US coincident index, making use of state-specific component series.     

On the efficient estimation methods for the macro-economic models nonlinear in variables

All the macro-economic models have the nonlinearity in variables within their simultaneous equations systems. I propose a full information estimation method for such models. The method is (i) asymptotically efficient, (ii) feasible in the contemporary computer technology as it consists of calculations very much like the nonlinear multipliers, and (iii) hopefully applicable to the undersized sample case which prevails in the macro-economic model building. Though two other methods are also investigated, one is found to be asymptotically inefficient, and another turns out to be inapplicable to the undersized sample case.     

Nonstationary nonlinear heteroskedasticity

In this paper, we consider time series with the conditional heteroskedasticities that are given by nonlinear functions of integrated processes. Such time series are said to have nonlinear nonstationary heteroskedasticity (NNH), and the functions generating conditional heterogeneity are called heterogeneity generating functions (HGF's). Various statistical properties of time series with NNH are investigated for a wide class of HGF's. For NNH models with a variety of HGF's, volatility clustering and leptokurtosis, which are common features of ARCH type models, are manifest. In particular, it is shown that the sample autocorrelations of their squared processes vanish only very slowly, or do not even vanish at all, in the limit. Volatility clustering is therefore well expected. The NNH models with certain types of HGF's indeed have sample characteristics that are very similar to those of ARCH type models. Moreover, the sample kurtosis of the NNH model either diverges or has a stable limiting distribution with support truncated on the left by the kurtosis of the innovations. This would well explain the presence of leptokurtosis in many observed time series data. To illustrate the empirical relevancy of our model, we analyze the spreads between the forward and spot rates of USD/DM exchange rates. It is found that the conditional variances of the spreads can be well modelled as a nonlinear function of the levels of the spot rates.     

几类非线性协整模型研究综述

协整分析方法经过20多年的发展成为计量经济学界的一个前沿工具,在经济与金融领域得到了广泛的应用。线性协整分析已经成熟,而非线性协整的理论与方法仍在持续研究中。本文回顾了最近20年非线性协整的发展历史,其中包括结构变化、门限非线性、马尔可夫转换和平滑转换等几类非线性协整模型,强调了这些非线性机制的本质区别,总结了已取得的一些重要研究成果,最后对该问题的最新发展动向加以概括。     

Nonlinear IV unit root tests in panels with cross-sectional dependency

We propose a unit root test for panels with cross-sectional dependency. We allow general dependency structure among the innovations that generate data for each of the cross-sectional units. Each unit may have different sample size, and therefore unbalanced panels are also permitted in our framework. Yet, the test is asymptotically normal, and does not require any tabulation of the critical values. Our test is based on nonlinear IV estimation of the usual augmented Dickey–Fuller type regression for each cross-sectional unit, using as instruments nonlinear transformations of the lagged levels. The actual test statistic is simply defined as a standardized sum of individual IV t-ratios. We show in the paper that such a standardized sum of individual IV t-ratios has limit normal distribution as long as the panels have large individual time series observations and are asymptotically balanced in a very weak sense. We may have the number of cross-sectional units arbitrarily small or large. In particular, the usual sequential asymptotics, upon which most of the available asymptotic theories for panel unit root models heavily rely, are not required. Finite sample performance of our test is examined via a set of simulations, and compared with those of other commonly used panel unit root tests. Our test generally performs better than the existing tests in terms of both finite sample sizes and powers. We apply our nonlinear IV method to test for the purchasing power parity hypothesis in panels.     

Detecting nonlinearity in time series by model selection criteria

This article analyzes the use of model selection criteria for detecting nonlinearity in the residuals of a linear model. Model selection criteria are applied for finding the order of the best autoregressive model fitted to the squared residuals of the linear model. If the order selected is not zero, this is considered as an indication of nonlinear behavior. The BIC and AIC criteria are compared to some popular nonlinearity tests in three Monte Carlo experiments. We conclude that the BIC model selection criterion seems to offer a promising tool for detecting nonlinearity in time series. An example is shown to illustrate the performance of the tests considered and the relationship between nonlinearity and structural changes in time series.     

NONLINEAR BUSINESS CYCLE MODELLING

Abstract. Literature which employs nonlinearities to explain economic fluctuations, commonly called business cycles, is surveyed. Relaxation of the linearity assumption significantly increases the range of possible dynamic solution paths and introduces the possibility that business cycles are endogenously determined. The dominant post-war modelling strategy has been the Frisch (1933) (and Slutsky, 1937) inspired one of developing essentially (log) linear economic models which produce damped cycles (or monotonic damping) to propagate the energy provided by repeated random (or autocorrelated) shocks. The cycle is exogenously driven, since it would die out in the absence of shocks. Deterministic (nonstochastic) nonlinear models can produce a wide range of endogenous fluctuations, including: stable limit cycles; growth cycles; and chaotic output, which have the appearance of random fluctuations. Further, the same model can produce qualitatively different outputs according to starting and parameter values. If the possibility of shocks to parameters is admitted, then behaviour can change abruptly following shocks. Evidence on the existence of nonlinearities and chaos in macroeconomic time series is assessed and alternative approaches to modelling dynamic economic development, related to the work of Keynes, Marx, Schumpeter and Shackle, are discussed. Their ideas have not proved readily amenable to mathematical modelling, but attempts to encapsulate some of them are reviewed.     

A flexible approach to parametric inference in nonlinear and time varying time series models

Many structural break and regime-switching models have been used with macroeconomic and financial data. In this paper, we develop an extremely flexible modeling approach which can accommodate virtually any of these specifications. We build on earlier work showing the relationship between flexible functional forms and random variation in parameters. Our contribution is based around the use of priors on the time variation that is developed from considering a hypothetical reordering of the data and distance between neighboring (reordered) observations. The range of priors produced in this way can accommodate a wide variety of nonlinear time series models, including those with regime-switching and structural breaks. By allowing the amount of random variation in parameters to depend on the distance between (reordered) observations, the parameters can evolve in a wide variety of ways, allowing for everything from models exhibiting abrupt change (e.g. threshold autoregressive models or standard structural break models) to those which allow for a gradual evolution of parameters (e.g. smooth transition autoregressive models or time varying parameter models). Bayesian econometric methods for inference are developed for estimating the distance function and types of hypothetical reordering. Conditional on a hypothetical reordering and distance function, a simple reordering of the actual data allows us to estimate our models with standard state space methods by a simple adjustment to the measurement equation. We use artificial data to show the advantages of our approach, before providing two empirical illustrations involving the modeling of real GDP growth.     

An empirical analysis of neural network memory structures for basin water quality forecasting

This research investigates the cumulative multi-period forecast accuracy of a diverse set of potential forecasting models for basin water quality management. The models are characterized by their short-term (memory by delay or memory by feedback) and long-term (linear or nonlinear) memory structures. The experiments are conducted as a series of forecast cycles, with a rolling origin of a constant fit size. The models are recalibrated with each cycle, and out-of-sample forecasts are generated for a five-period forecast horizon. The results confirm that the JENN and GMNN neural network models are generally more accurate than competitors for cumulative multi-period basin water quality prediction. For example, the JENN and GMNN models reduce the cumulative five-period forecast errors by as much as 50%, relative to exponential smoothing and ARIMA models. These findings are significant in view of the increasing social and economic consequences of basin water quality management, and have the potential for extention to other scientific, medical, and business applications where multi-period predictions of nonlinear time series are critical.     

A nonlinear panel data model of cross-sectional dependence

This paper proposes a nonlinear panel data model which can endogenously generate both ‘weak’ and ‘strong’ cross-sectional dependence. The model’s distinguishing characteristic is that a given agent’s behaviour is influenced by an aggregation of the views or actions of those around them. The model allows for considerable flexibility in terms of the genesis of this herding or clustering type behaviour. At an econometric level, the model is shown to nest various extant dynamic panel data models. These include panel AR models, spatial models, which accommodate weak dependence only, and panel models where cross-sectional averages or factors exogenously generate strong, but not weak, cross sectional dependence. An important implication is that the appropriate model for the aggregate series becomes intrinsically nonlinear, due to the clustering behaviour, and thus requires the disaggregates to be simultaneously considered with the aggregate. We provide the associated asymptotic theory for estimation and inference. This is supplemented with Monte Carlo studies and two empirical applications which indicate the utility of our proposed model as a vehicle to model different types of cross-sectional dependence.     

Sequential numerical integration in nonlinear state space models for microeconometric panel data

This paper discusses the estimation of a class of nonlinear state space models including nonlinear panel data models with autoregressive error components. A health economics example illustrates the usefulness of such models. For the approximation of the likelihood function, nonlinear filtering algorithms developed in the time‐series literature are considered. Because of the relatively simple structure of these models, a straightforward algorithm based on sequential Gaussian quadrature is suggested. It performs very well both in the empirical application and a Monte Carlo study for ordered logit and binary probit models with an AR(1) error component. Copyright © 2008 John Wiley & Sons, Ltd.     

SPATIAL LIMITED DEPENDENT VARIABLE MODELS: A REVIEW FOCUSED ON SPECIFICATION,ESTIMATION, AND HEALTH ECONOMICS APPLICATIONS

Modeling individual choices is one of the main aim in microeconometrics. Discrete choice models have been widely used to describe economic agents' utility functions and most of them play a paramount role in applied health economics. On the other hand, spatial econometrics collects a series of econometric tools, which are particularly useful when we deal with spatially distributed data sets. Accounting for spatial dependence can avoid inconsistency problems of the commonly used statistical estimators. However, the complex structure of spatial dependence in most of the nonlinear models still precludes a large diffusion of these spatial techniques. The purpose of this paper is then twofold. The former is to review the main methodological problems and their different solutions in spatial nonlinear modeling. The latter is to review their applications to health issues, especially those appeared in the last few years, by highlighting the main reasons why spatial discrete neighboring effects should be considered and suggesting possible future lines of development in this emerging field. Particular attention has been paid to cross‐sectional spatial discrete choice modeling. However, discussions on the main methodological advancements in other spatial limited dependent variable models and spatial panel data models are also included.