Nonlinear Deterministic Trend Research Articles

Unit root testing with slowly varying trends

A unit root test is proposed for time series with a general nonlinear deterministic trend component. It is shown that asymptotically the pooled OLS estimator of overlapping blocks filters out any trend component that satisfies some Lipschitz condition. Under both fixed‐b and small‐b block asymptotics, the limiting distribution of the t‐statistic for the unit root hypothesis is derived. Nuisance parameter corrections provide heteroskedasticity‐robust tests, and serial correlation is accounted for by pre‐whitening. A Monte Carlo study that considers slowly varying trends yields both good size and improved power results for the proposed tests when compared to conventional unit root tests.

Persistence and efficiency of OECD stock markets: linear and nonlinear fractional integration approaches

The analysis of the dynamics of the share price series using the fractional integration technique that accommodates nonlinear deterministic trend based on the Chebyshev polynomials in time is considered in this study. It is important to account for the nonlinear behavior and autoregressive structure of the error disturbance term of the share price indices if correct estimates of the integration order are to be obtained. The findings of this study show that it is important to put the autoregressive order into consideration following the significant effect it imposes on the fractional integration estimates, and there are evidences of nonlinearities in nine countries. The nonlinear effect significantly imparts on the degree of persistence of the nine countries by either inducing more or less persistence. However, it is more profound for Belgium, Japan and Hungary whose share market statuses change from efficient to inefficient, as their d estimates under the linearity case, respectively, fall from 0.9715, 0.9998 and 0.9344 to 0.3230, 0.3112 and 0.8579 under the nonlinearity case. For the sake of policy, these findings have important implications for portfolio management and diversification, investment choices, and policy making and forecast.

A parametric stationarity test with smooth breaks

Abstract We propose a test to investigate the stationarity null against the unit-root alternative where a Fourier component is employed to approximate nonlinear deterministic trend of unknown form. A parametric adjustment is also adopted to accommodate possible stationary error. The asymptotic distribution of the test under the null is derived, and the asymptotic critical values are tabulated. We also show that it is a consistent test. Even with small sample sizes often encountered in empirical applications, our parametric stationarity test employing Fourier term has good size and power properties when trend breaks are gradual. The validity of the Fisher hypothesis for 15 OECD countries is investigated to illustrate the usefulness of our test.

Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component

AbstractThis paper proposes a new test for the presence of a nonlinear deterministic trend approximated by a Fourier expansion in a univariate time series for which there is no prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. Our approach builds on the work of Perron and Yabu () and is based on a Feasible Generalized Least Squares procedure that uses a super‐efficient estimator of the sum of the autoregressive coefficients α when α = 1. The resulting Wald test statistic asymptotically follows a chi‐square distribution in both the I(0) and I(1) cases. To improve the finite sample properties of the test, we use a bias‐corrected version of the OLS estimator of α proposed by Roy and Fuller (). We show that our procedure is substantially more powerful than currently available alternatives. We illustrate the usefulness of our method via an application to modelling the trend of global and hemispheric temperatures.

Nonlinear trend stationarity in global and hemispheric temperatures

ABSTRACTThis study investigates the issue of whether our Earth’s surface temperature exhibits a stochastic trend. Using state-of-the-art econometrics, we analyse the latest available temperature anomaly data. Our results indicate that both global and hemispheric temperatures may appear to have a stochastic trend when in fact they are stationary around a nonlinear deterministic trend and structural breaks are responsible. Furthermore, the nonlinearity found in the temperature trend is more complex than what has been reported in previous studies.

Modeling U.S. Historical Time-Series Prices and Inflation Using Various Linear and Nonlinear Long-Memory Approaches

This paper estimates the complete historical US price data by employing a relatively new statistical methodology based on long memory. We consider, in addition to the standard case, the possibility of nonlinearities in the form of nonlinear deterministic trends as well as the possibility that persistence exists at both the zero frequency and a frequencies away from zero. We model the fractional nonlinear case using Chebyshev polynomials and model the fractional cyclical structures as a Gegenbauer process. We find in the latter case that that secular (i.e., long-run) persistence and cyclical persistence matter in the behavior of prices, producing long-memory effects that imply mean reversion at both the long-run and cyclical frequencies.

Fractional integration and nonlinear deterministic trends in the analysis of time series data

ABSTRACTThis article examines the interaction between fractional integration and nonlinear structures by using for the latter the Chebyshev polynomials in time that can be taken as an alternative, less abrupt way of modelling breaks in time series data. A Lagrange multiplier test, developed for testing the order of integration in the context of nonlinear deterministic trends, is implemented in three well-known and previously studied time series data: the Nile river data, the temperatures in the Northern hemisphere and CO2 emissions in the US. The results suggest that the second and especially the third time series display nonlinear behaviour still with fractional degrees of differentiation.

An IV Test for a Unit Root in Generally Trending and Correlated Panels

This paper proposes an IV-based panel unit root test that is general enough to accommodate general error serial and cross-section dependence, and a potentially nonlinear deterministic trend function. These allowances make the new test one of the most general around. It is also very simple to implement. Indeed, the IV statistic is asymptotically invariant to not only to all nuisance parameters characterizing the dependence of the errors and the true trend function, but also the deterministic specification of the fitted test regression.

Towards homoscedastic nonlinear cointegration for structural health monitoring

The paper presents the homoscedastic nonlinear cointegration. The method leads to stable variances in nonlinear cointegration residuals. The adapted Breusch–Pagan test procedure is developed to test for the presence of heteroscedasticity (or homoscedasticity) in the cointegration residuals obtained from the nonlinear cointegration analysis. Three different time series – i.e. one with a nonlinear quadratic deterministic trend, simulated vibration data and experimental wind turbine data – are used to illustrate the application of the proposed method. The proposed approach can be used for effective removal of nonlinear trends from various types of data and for reliable structural damage detection based on data that are corrupted by environmental and/or operational nonlinear trends.

Persistence, mean reversion and non-linearities in the US housing prices over 1830–2013

ABSTRACTThe objective of this study is to provide a direct estimate of the degree of persistence of measures of nominal and real house prices for the US economy, covering the longest possible annual sample of data, namely 1830–2013. The estimation of the degree of persistence accommodates for non-linear (deterministic) trends using Chebyshev polynomials in time. In general, the results show a high degree of persistence in the series along with a component of non-linear behaviour. In general, if we assume uncorrelated errors, non-linearities are observed in both nominal and real prices, but this hypothesis is rejected in favour of linear models for the log-transformation of the data. However, if autocorrelated errors are permitted, non-linearities are observed in all cases, and mean reversion is found in the case of logged prices, though given the wide confidence intervals, the unit root null hypothesis cannot be rejected in these cases.

Testing fractional persistence and non-linearities in the natural gas market: An application of non-linear deterministic terms based on Chebyshev polynomials in time

Studying variations of natural gas prices in relation to consumer prices may give us better indicators for the analysis of economic activity. This paper deals with the analysis of natural gas spot prices using fractional integration techniques in the context of non-linear deterministic trends. We find nonstationarity with mean reverting coefficients (i.e., orders of integration in the range (0.5, 1)) in the daily and monthly series, as well as in their logarithmic transformations. Evidences of non-linearities are only obtained in the monthly series which may be a consequence of the higher degree of volatility associated with this frequency.

Semiparametric generalized long-memory modeling of some mena stock market returns: A wavelet approach

This paper proposes a new class of semiparametric generalized long-memory models with FIAPARCH errors that extends the conventional GARMA model to incorporate nonlinear deterministic trend and allows for time-varying volatility. To estimate the parameters, we implement a wavelet theory. We provide an empirical application to some MENA stock markets and find that the proposed model offers an interesting framework to describe seasonal long-range dependence and nonlinear trend in return as well as persistence to shocks in conditional volatility. The predictive results also indicate that this model outperforms the traditional FARMA-FIAPARCH process.

Testing for long memory in the presence of non-linear deterministic trends with Chebyshev polynomials

AbstractThis paper examines the interaction between non-linear deterministic trends and long run dependence by means of employing Chebyshev time polynomials and assuming that the detrended series displays long memory with the pole or singularity in the spectrum occurring at one or more possibly non-zero frequencies. The combination of the non-linear structure with the long memory framework produces a model which is linear in parameters and therefore it permits the estimation of the deterministic terms by standard OLS-GLS methods. Moreover, the orthogonality property of Chebyshev’s polynomials makes them especially attractive to approximate non-linear structures of data. We present a procedure which allows us to test (possibly fractional) orders of integration at various frequencies in the presence of the Chebyshev trends with no effect on the standard limit distribution of the method. Several Monte Carlo experiments are conducted and the results indicate that the method performs well. An empirical application, using data of real exchange rates is also carried out at the end of the article.

Nonlinear Cointegration Approach for Condition Monitoring of Wind Turbines

Monitoring of trends and removal of undesired trends from operational/process parameters in wind turbines is important for their condition monitoring. This paper presents the homoscedastic nonlinear cointegration for the solution to this problem. The cointegration approach used leads to stable variances in cointegration residuals. The adapted Breusch-Pagan test procedure is developed to test for the presence of heteroscedasticity in cointegration residuals obtained from the nonlinear cointegration analysis. Examples using three different time series data sets—that is, one with a nonlinear quadratic deterministic trend, another with a nonlinear exponential deterministic trend, and experimental data from a wind turbine drivetrain—are used to illustrate the method and demonstrate possible practical applications. The results show that the proposed approach can be used for effective removal of nonlinear trends form various types of data, allowing for possible condition monitoring applications.

Modelling African inflation rates: nonlinear deterministic terms and long-range dependence

This article estimates a fractional integration model with nonlinear deterministic trends for the inflation rates of five African countries. The results indicate that nonlinearities are present in the case of Angola and Lesotho, but not in the case of Botswana, Namibia and South Africa. Moreover, the degrees of differentiation are higher in the latter group of countries.

Modelling African inflation rates: nonlinear deterministic terms and long-range dependence

This article estimates a fractional integration model with nonlinear deterministic trends for the inflation rates of five African countries. The results indicate that nonlinearities are present in the case of Angola and Lesotho, but not in the case of Botswana, Namibia and South Africa. Moreover, the degrees of differentiation are higher in the latter group of countries.

Semiparametric Model Selection in Panel Data Models with Deterministic Trending and Cross-Sectional Dependence

In this paper, we consider a model selection issue in semiparametric panel data models with fixed effects. The modelling framework under investigation can accommodate both nonlinear deterministic trends and cross-sectional dependence. And we consider the so-called "large panels" where both the time series and cross sectional sizes are very large. A penalised profile least squares method with first-stage local linear smoothing is developed to select the significant covariates and estimate the regression coefficients simultaneously. The convergence rate and the oracle property of the resulting semiparametric estimator are established by the joint limit approach. The developed semiparametric model selection methodology is illustrated by two Monte-Carlo simulation studies, where we compare the performance in model selection and estimation of three penalties, i.e., the least absolute shrinkage and selection operator (LASSO), the smoothly clipped absolute deviation (SCAD), and the minimax concave penalty (MCP).

Consistent co‐trending rank selection when both stochastic and non‐linear deterministic trends are present

This paper proposes a model-free cotrending rank selection procedure based on the eigenstructure of a multivariate version of the von Neumann ratio, in the presence of both stochastic and nonlinear deterministic trends. Our selection criteria are easily implemented, and the consistency of the rank estimator is established under very general conditions. Simulation results suggest good …nite sample properties of the new rank selection criteria. The proposed method is then illustrated through an application of the Japanese money demand function allowing for the cotrending relationship among money, income and interest rates.

Simple unit root testing in generally trending data with an application to precious metal prices in Asia

This paper proposes a new unit root test that is general enough to accommodate a potentially non-linear deterministic trend function, making it one of the most general tests around. However, the main advantage lies with its simple implementation. In particular, the asymptotic critical values are shown to be “almost” independent of the deterministic trend function, and as a result the test can be implemented without the need for model-specific critical values. The new test is applied to a sample consisting of monthly prices of four precious metals for a number of Asian countries.

Heterogeneous technology and the technological catching-up hypothesis: Theory and assessment in the case of MENA countries

Long run convergence implies that the convergence hypothesis will be rejected if the income differential is not stationary. However, this definition is valid only if the catching-up process between the two countries is already over. If we take into account catching-up dynamics, then poorest countries should obtain a faster growth than developed countries. Thus, income gaps should integrate decreasing time trends. We formalise this hypothesis theoretically using a stochastic neoclassical growth model with heterogeneous technology. We then apply this model to the issue of per-capita GDP catching-up of eight MENA countries towards the level of income in Europe. We approximate the nonlinear deterministic trend by a linear function with breaks and apply panel unit root tests with breaks. The analysis reveals firstly that the periods of divergence outnumber the periods of convergence. Secondly, since the year 2000 all countries but Syria have been converging toward the European per-capita income level.