Nonlinear Stationarity Research Articles

High Persistence and Nonlinear Behavior in Financial Variables: A More Powerful Unit Root Testing in the ESTAR Framework

In this study, we consider the hybrid nonlinear features of the Exponential Smooth Transition Autoregressive-Fractional Fourier Function (ESTAR-FFF) form unit root test. As is well known, when developing a unit root test for the ESTAR model, linearization is performed by the Taylor approximation, and thereby the nuisance parameter problem is eliminated. Although this linearization process leads to a certain amount of information loss in the unit root testing equation, it also causes the resulting test to be more accessible and consistent. The method that we propose here contributes to the literature in three important ways. First, it reduces the information loss that arises due to the Taylor expansion. Second, the research to date has tended to misinterpret the Fourier function used with the Kapetanios, Shin and Snell (2003) (KSS) unit root test and considers it to capture multiple smooth transition structural breaks. The simulation studies that we carry out in this study clearly show that the Fourier function only restores the Taylor residuals of the ESTAR type function rather than accounting forthe smooth structural break. Third, the new nonlinear unit root test developed in this paper has very strong power in the highly persistent near unit root environment that the financial data exhibit. The application of the Kapetanios Shin Snell- Fractional Fourier (KSS-FF) test to ex-post real interest rates data of 11 OECD countries for country-specific sample periods shows that the new test catches nonlinear stationarity in many more countries than the KSS test itself.

The Validity of Purchasing Power Parity in BRICS Countries

It is observed that purchasing power parity (PPP) as one of the controversial and most interesting topics of international macroeconomics literature is tested by using different econometric methods for certain countries and/or country groups by many researchers. The validity of PPP is important in terms of being a common exchange rate used in international comparison. In this context, the validity of relative purchasing power parities in the BRICS countries (Brazil, Russia, India, China and South Africa) were analysed for the January 1993-March 2015 period. Non-linear stationarity analysis was used in the study. According to the findings of the analysis, it was concluded that all of the BRICS countries have a non-linear structure; the PPP approach was valid for Brazil and South Africa, but not valid for Russia, India and China in the relevant period.

London calling: Nonlinear mean reversion across national stock markets

This paper revisits empirical evidence of mean reversion of relative stock prices in international stock markets. We implement a strand of univariate and panel unit root tests for linear and nonlinear models of 18 national stock indices from 1969 to 2016. Our major findings are as follows. First, we find strong evidence of nonlinear mean reversion of the relative stock price with the UK index as the reference, calling attention to the stock index in the UK, but not with the US index. Our results imply an important role of the local common factor in the European stock markets. Second, panel tests yield no evidence of linear and nonlinear stationarity when the cross-section dependence is considered, which provides conflicting results from those of the univariate tests. Last, we show how to understand these results via dynamic factor analysis. When the stationary common factor dominates nonstationary idiosyncratic components in small samples, panel tests that filter out the stationary common factor may yield evidence against the stationarity null hypothesis in finite samples. We corroborate this conjecture via extensive Monte Carlo simulations.

Testing for a unit root against ESTAR stationarity

Abstract In this paper we examine the local power of unit root tests against globally stationary exponential smooth transition autoregressive [ESTAR] alternatives under two sources of uncertainty: the degree of nonlinearity in the ESTAR model, and the presence of a linear deterministic trend. First, we show that the KSS test (Kapetanios, G., Y. Shin, and A. Snell. 2003. “Testing for a Unit Root in the Nonlinear STAR Framework.” Journal of Econometrics 112: 359–379) for nonlinear stationarity has local asymptotic power gains over standard Dickey-Fuller [DF] tests for certain degrees of nonlinearity in the ESTAR model, but that for other degrees of nonlinearity, the linear DF test has superior power. Second, we derive limiting distributions of demeaned, and demeaned and detrended KSS and DF tests under a local ESTAR alternative when a local trend is present in the DGP. We show that the power of the demeaned tests outperforms that of the detrended tests when no trend is present in the DGP, but deteriorates as the magnitude of the trend increases. We propose a union of rejections testing procedure that combines all four individual tests and show that this captures most of the power available from the individual tests across different degrees of nonlinearity and trend magnitudes. We also show that incorporating a trend detection procedure into this union testing strategy can result in higher power when a large trend is present in the DGP.

Are fluctuations in oil consumption permanent or transitory? Evidence from linear and nonlinear unit root tests

This paper examines the integration properties of the total oil consumption in 57 countries for the period of 1965–2012. A combination of new and powerful linear and nonlinear stationarity tests are employed to achieve the objectives of the study. We find that the oil consumption series in 21 countries follow a nonlinearity path while those in the other countries are linear in nature. Evidence of the presence of a unit root is found for the total oil consumption series in 38 countries while the series is stationary in the remaining 19 countries. An important insight is that the blueprints that were designed to reduce oil consumption are likely to have a permanent effect in most of the countries.

Stochastic convergence in per capita CO2 emissions. An approach from nonlinear stationarity analysis

This paper studies stochastic convergence of per capita CO2 emissions in 28 OECD countries for the 1901–2009 period. The analysis is carried out at two aggregation levels: first for the whole set of countries (joint analysis) and then separately for developed and developing states (group analysis). A powerful time series methodology - adapted to a nonlinear framework that allows for quadratic trends with possibly smooth transition between regimes - is applied. This approach provides more robust conclusions in convergence path analysis, enabling (a) robust detection of the presence, and if so, the number of changes in the level and/or slope of the trend of the series; (b) inferences on stationarity of relative per capita CO2 emissions, conditionally on the presence of breaks and smooth transitions between regimes; and (c) estimation of change locations in the convergence paths. Finally, as stochastic convergence is attained when both stationarity around a trend and β-convergence simultaneously hold, the linear approach proposed by Tomljanovich and Vogelsang (2002) is extended in order to allow for more general, quadratic models. Overall, joint analysis finds some evidence of stochastic convergence in per capita CO2 emissions. Some dispersion in terms of β-convergence is detected by the group analysis, particularly among developed countries. This is in accordance with per capita GDP not being the sole determinant of convergence in emissions, with factors like search for more efficient technologies, fossil fuel substitution, innovation, and possibly industry outsourcing, also having a crucial role.

Estimation and empirical performance of non-scalar dynamic conditional correlation models

A method capable of estimating richly parametrized versions of the dynamic conditional correlation (DCC) model that go beyond the standard scalar case is presented. The algorithm is based on the maximization of a Gaussian quasi-likelihood using a Bregman-proximal trust-region method that handles the various non-linear stationarity and positivity constraints that arise in this context. The general matrix Hadamard DCC model with full rank, rank equal to two and, additionally, two different rank one matrix specifications are considered. In the last mentioned case, the elements of the vectors that determine the rank one parameter matrices are either arbitrary or parsimoniously defined using the Almon lag function. Actual stock returns data in dimensions up to thirty are used in order to carry out performance comparisons according to several in- and out-of-sample criteria. Empirical results show that the use of richly parametrized models adds value with respect to the conventional scalar case.

Are stock prices stationary? Some new evidence from a panel data approach

Purpose– This paper investigates whether mean reversion holds for a panel of 16 OECD stock price indices for the period 1970 to 2011.Design/methodology/approach– We employ seemingly unrelated regression (SUR)-based linear and non-linear unit root tests which are not only able to exploit the power of panel data analysis but also account for cross sectional dependencies as well as identify which panel members are stationary.Findings– In contrast to a literature that offers mixed findings on stationarity, it was found that most of our sample is characterized as mean- or trend-reverting with approximated half-lives in the region of three to five years.Originality/value– In contrast to other panel unit root tests of stock prices, the authors identify which individual panel members are stationary and non-stationary using a SURADF test. A further novelty of our approach is that we also develop a SUR-based panel KSS test that allows us to explore the possibility that stock prices exhibit non-linear stationarity.

SU-E-J-261: Statistical Analysis and Chaotic Dynamics of Respiratory Signal of Patients in BodyFix

Purpose: To quantify respiratory signal of patients in BodyFix undergoing 4DCT scan with and without immobilization cover. Methods: 20 pairs of respiratory tracks recorded with RPM system during 4DCT scan were analyzed. Descriptive statistic was applied to selected parameters of exhale-inhale decomposition. Standardized signals were used with the delay method to build orbits in embedded space. Nonlinear behavior was tested with surrogate data. Sample entropy SE, Lempel-Ziv complexity LZC and the largest Lyapunov exponents LLE were compared. Results: Statistical tests show difference between scans for inspiration time and its variability, which is bigger for scans without cover. The same is for variability of the end of exhalation and inhalation. Other parameters fail to show the difference. For both scans respiratory signals show determinism and nonlinear stationarity. Statistical test on surrogate data reveals their nonlinearity. LLEs show signals chaotic nature and its correlation with breathing period and its embedding delay time. SE, LZC and LLE measure respiratory signal complexity. Nonlinear characteristics do not differ between scans. Conclusion: Contrary to expectation cover applied to patients in BodyFix appears to have limited effect on signal parameters. Analysis based on trajectories of delay vectors shows respiratory system nonlinear character and its sensitive dependence on initial conditions. Reproducibility of respiratory signal can be evaluated with measures of signal complexity and its predictability window. Longer respiratory period is conducive for signal reproducibility as shown by these gauges. Statistical independence of the exhale and inhale times is also supported by the magnitude of LLE. The nonlinear parameters seem more appropriate to gauge respiratory signal complexity since its deterministic chaotic nature. It contrasts with measures based on harmonic analysis that are blind for nonlinear features. Dynamics of breathing, so crucial for 4D-based clinical technologies, can be better controlled if nonlinear-based methodology, which reflects respiration characteristic, is applied. Funding provided by Varian Medical Systems via Investigator Initiated Research Project

Estimation and Empirical Performance of Non-Scalar Dynamic Conditional Correlation Models

A method capable of estimating richly parametrized versions of the dynamic conditional correlation (DCC) model that go beyond the standard scalar case is presented. The algorithm is based on the maximization of a Gaussian quasi-likelihood using a Bregman-proximal trust-region method that handles the various non-linear stationarity and positivity constraints that arise in this context. The general matrix Hadamard DCC model with full rank, rank equal to two and, additionally, two different rank one matrix specifications are considered. In the last mentioned case, the elements of the vectors that determine the rank one parameter matrices are either arbitrary or parsimoniously defined using the Almon lag function. Actual stock returns data in dimensions up to thirty are used in order to carry out performance comparisons according to several in- and out-of-sample criteria. Empirical results show that the use of richly parametrized models adds value with respect to the conventional scalar case.

Nonlinear stationarity of real interest rates in the EMU countries

PurposeThis paper aims to examine two hypotheses that have not been well investigated in the existing literature. One hypothesis is that the real interest rates of industrial countries tend to be mean‐reverting during the current floating exchange rate period. Another hypothesis is that the real interest rates of the countries involved in forming the Euro area are more likely to behave as nonlinear stationary series than those of other industrial countries.Design/methodology/approachThe study applies the conventional linear unit root tests and recently developed nonlinear unit root tests, as well as the tests of specifying nonlinearity in time series, to the short‐term real interest rates of 16 industrial countries.FindingsThe results of the study provide support for both hypotheses.Practical implicationsThe results imply that, having adopted target‐zone type stabilization policies for years, the central banks of European Monetary Union (EMU) countries were likely to have exercised monetary policies in a nonlinear way, especially in the process of meeting the requirements of joining EMU.Originality/valueThe study provides stronger evidence than previous studies for the theory that real interest rates of industrial countries tend to have mean‐reverting behavior. The study suggests that more active monetary policies for inflation control in the floating exchange rate period may have enhanced mean reversion in real interest rates.

Nonlinearity and stationarity of inflation rates: evidence from the euro-zone countries

Earlier studies hardly reject the hypothesis of a unit root in inflation. Few studies have examined the possibility of nonlinearity in inflation and tested nonlinear stationarity of the inflation rates. This study thus intends to fill the gap. This study utilizes the tests for nonlinearity along with the unit root tests that allow for nonlinearity in the variables to examine the stationarity of inflation rates of 12 European countries that formed the Euro Zone (EZ) later in the sample period. The results suggest that the majority of these countries’ inflation rates can be characterized by mean reversion during the floating exchange rate period. Many of them appear to be nonlinear stationary. This finding is essential in conducting applied economic studies for these countries, when constructing models whose validity relies on whether or not inflation is stationary. The results of this study also imply that shocks to inflation have a transitory effect on inflation in the euro area. Therefore, it would be less costly in exercising the policies of disinflation for the monetary authorities of the euro area than for those of the countries with nonstationary inflation.

Testing for unemployment hysteresis in nonlinear heterogeneous panels: International evidence

Nonlinear behavior of unemployment is well documented in the literature, and thus linear unit root tests may not be appropriate in this case. This paper tests for hysteresis of unemployment for 29 OECD countries through the use of a new nonlinear panel unit root test developed by Ucar and Omay (2009). The test examines the joint null hypothesis of linearity and a unit root against the alternative hypothesis of nonlinear stationarity. Large power gains are achieved by both combining cross-sectional information with nonlinearities in the data. In addition, after the unit root null being rejected, we use a sequential panel selection method suggested by Chortareas and Kapetanios (2009) to classify the whole panel into a group of stationary countries and a group of non-stationary countries. The empirical findings show that the nonlinear panel test gives strong evidence in favor of the natural rate hypothesis of unemployment for 23 of 29 OECD countries, in contrast to those obtained by Chang's (2002) linear panel test that 17 countries display evidence of stationary unemployment.

Is PPP sensitive to time-varying trade weights in constructing real effective exchange rates?

While the use of real effective exchange rates in stationarity tests of purchasing power parity (PPP) avoids the problems created using bilateral rates, these rates are often constructed using trade shares that are fixed at a single base year. This method fails to take into account the fact that trade shares can change drastically in parts of the world that are undergoing dramatic transformations. In this study, we apply linear as well as nonlinear stationarity tests to 52 currencies’ real effective rates, which were constructed using time-varying weights. Incorporating a time trend, we are also able to assess whether breakdowns in PPP are due to productivity differentials. We find that while nonlinear tests provide more evidence of “productivity bias” than do linear tests, they do not provide much more evidence of PPP. A comparison to a previous study that used fixed-weight data shows that there is somewhat more evidence of productivity bias using the new dataset, especially in Eastern Europe and Asia. We can conclude that PPP and a key cause of its breakdown are somewhat sensitive to the use of time-varying weights in these stationarity tests.

Do Real Exchange Rates Follow a Nonlinear Mean Reverting Process in Developing Countries?

In an effort to fight relatively high inflation, many developing countries try to manage their nominal exchange rates through official intervention. In addition, developing countries tend to have high transportation costs, tariffs, and nontariff barriers. These factors are among the sources of generating nonlinearity in real exchange rates and hence some nonlinear adjustment toward purchasing power parity (PPP) in developing countries. In this paper, we employ monthly real effective exchange rate (REER) data of 88 developing countries and test the null of nonstationarity versus an alternative of linear stationarity by the means of a conventional unit root test and compare the results with those obtained from a new test in which the null is the same but the alternative hypothesis is nonlinear stationarity. The latter test supports the PPP theory in more developing countries compared with the former test, suggesting that nonlinear adjustment toward PPP in developing countries is an important phenomenon. Reported country characterizations indicate that reversion in REER occurs more often for high‐inflation countries and for countries with high flexibility in their exchange rates.

A revisit to the nonlinear stationarity of real exchange rates

This article applies the test by Kapetanios et al. (2003) to re-examine the nonlinear stationarity of real exchange rates in which we emphasize both the real bilateral exchange rates and the real effective exchange rates. Our empirical findings provide weak evidence on the nonlinear mean-reverting adjustment of real exchange rates in both fixed and floating regimes, which is in sharp contrast to that in the existing literature.

Nonlinear trend stationarity of real exchange rates: the case of the Mediterranean countries

The aim of this article is to provide additional evidence on the fulfilment of the Purchasing Power Parity (PPP) hypothesis in the so-called Mediterranean countries. In order to test for the empirical validity of such a hypothesis, we have applied two types of unit root tests. The first group is due to Bierens (1997), who generalised the alternative hypothesis to nonlinear trend stationarity and the second is Leybourne et al.'s (1998) approach that uses a nonlinear specification for the intercept and slope in order to detrend the series. The results suggest that the evidence in favour of the Purchasing Power Parity hypothesis increases when we allow for nonlinear alternatives.

Unit roots or nonlinear stationarity in Turkish real exchange rates

The objective of this paper is to test if Turkish real exchange rates have a linear unit root or are generated by an Exponential Smooth Transition Autoregressive Model for the post-1980 period. Using two real exchange rates, one with the USA and the other with Germany, strong evidence of nonlinear stationarity was found for the US CPI-based series but no such evidence for the DM CPI-based series. When compared with earlier results in a previous paper where the alternative of the linear unit root test was stationarity with multiple shifts in the deterministic terms, it was found that similar results were obtained for the US CPI-based series but not for the DM CPI-based one, possibly implying that the multiple shifts approach may be more appropriate for the Turkish series.

Nonlinear Trends and Co-trending in Canadian Money Demand

Several authors have suggested that, instead of being unit root processes, some macro variables may actually be stationary around nonlinear deterministic trends (Perron, 1989, 1990, Bierens, 1997). This paper investigates this for four variables in a standard money demand specification, using Canadian data. Evidence is first presented that the null of unit root with drift (constant, linear, or nonlinear) can be rejected in favor of nonlinear trend stationarity for the variables. Then, Bierens' (2000) nonlinear co-trending test finds two common nonlinear trends among the variables. The trends are consistent with a standard money demand relationship. All unit root and co-trending test conclusions are based on size and power results from Monte Carlo simulations as well as on asymptotic critical values. The paper concludes with a discussion of how the observed nonlinear trending and co-trending might arise in a theoretical model, and with implications for further empirical tests.

Defects in the Sine-Gordon model: Statics

The one-dimensional Sine-Gordon model where the field couples linearly to localized defects is studied. This model describes the phase fluctuations of one-dimensional charge density waves if amplitude fluctuations are neglected. For isolated defects the nonlinear stationarity conditions can be solved analytically. Given these stationary solutions the fluctuation spectrum consisting of a bound state and the scattering states and the relevant correlation functions can be computed exactly. The stationary states for a system of two defects are presented. The free energies as a function of the separation of the defects is computed for the absolutely stable states and for the local minima of the free energy function. This allows us to consider the interaction of the defects induced by the Sine-Gordon field. Finally we compute the order parameter correlation function for a random distribution of defects for small concentrationn. Devising a cumulant expansion the correlation function is found exactly in the first order inn. Our result contains the combined effect of the defects on the stationary states and on the phonon Green's function. It has applicability beyond the present context.