Unit Root Behavior Research Articles

Testing the unit root of social, environmental, and governance stock price indexes using a multivariate ARDL model

ABSTRACT The establishment of a green stock market (GSM) has significant implications for the advancement of sustainable development. However, it is susceptible to fluctuations caused by fluctuations in fossil fuel prices, economic policy and geopolitical uncertainties, and monetary policy. This study evaluates the persistence of shocks to five social, environmental, and governance (ESG) stock price indexes using the novel multivariate ARDL unit root test. The Federal Fund Effective Rate (FFER) and West Texas Intermediate Crude Oil Price (WTIOIL) were separately included as covariates in the ARDL model and tested for the null hypothesis of a unit root using a multivariate ARDL unit root test. The results of this study indicate that the ESG stock price indexes exhibit unit root behaviour, suggesting that policy shifts and shocks have a lasting impact on the dynamics of the green stock market and sustainable development.

Impact of European egg marking system in France: empirical time series analysis between 2017 and 2022

Voluntary labeling strategies, such as Organic label or Label Rouge, have long been considered as a potential solution to address environmental and social issues in the food sector. As a complement, the European political authorities developed a mandatory marking system for fresh eggs. This article questions the effectiveness of public intervention to support sustainable practices using analysis of the demand for fresh eggs in France. Unit root (augmented Dickey Fuller) tests, stationarity (KPSS) tests, and Seasonal Autoregressive Integrated Moving Average (SARIMA) models are used to investigate the unit root behavior of the prices and expenditure shares of fresh eggs in France between 2017 and 2022. We use the Almost Ideal Demand System model on scanner data to analyze the demand of six eggs’ categories, including mandatory egg codes and two labels (Organic and Label Rouge). The results suggest that a low price does not compensate for low sustainability involvement in eggs from caged farming, favoring free range eggs. Label Rouge shows market weaknesses, while the organic label shows promising results with both a voluntary and a specific mandatory mark. The lack of elasticity observed, except between cage and free range eggs, implies that consumers who choose high-priced products with voluntary labeling strategies are less inclined to switch to alternatives. The mandatory marking system brings more transparency than voluntary labeling initiative, in favor of sustainable products. The case of eggs is a relevant example of how market intervention can push sustainable consumption and production without forbidding products in the market.

Tempered functional time series

We propose a broad class of models for time series of curves (functions) that can be used to quantify near long‐range dependence or near unit root behavior. We establish fundamental properties of these models and rates of consistency for the sample mean function and the sample covariance operator. The latter plays a role analogous to sample cross‐covariances for multivariate time series, but is far more important in the functional setting because its eigenfunctions are used in principal component analysis, which is a major tool in functional data analysis. It is used for dimension reduction of feature extraction. We also establish a central limit theorem for functions following our model. Both the consistency rates and the normalizations in the Central Limit Theorem (CLT) are nonstandard. They reflect the local unit root behavior and the long memory structure at moderate lags.

Point optimal testing with roots that are functionally local to unity

Limit theory for regressions involving local to unit roots (LURs) is now used extensively in time series econometric work, establishing power properties for unit root and cointegration tests, assisting the construction of uniform confidence intervals for autoregressive coefficients, and enabling the development of methods robust to departures from unit roots. The present paper shows how to generalize LUR asymptotics to cases where the localized departure from unity is a time varying function rather than a constant. Such a functional local unit root (FLUR) model has much greater generality and encompasses many cases of additional interest that appear in practical work, including structural break formulations that admit subperiods of unit root, local stationary and local explosive behavior within a given sample. Point optimal FLUR tests are constructed in the paper to accommodate such cases and demonstrate how the power envelope changes in situations of practical interest. Against FLUR alternatives, conventional constant point optimal tests can be asymptotically infinitely deficient in power, with poor finite sample power performance particularly when the departure from unity occurs early in the sample period. New analytic explanation for this phenomenon is provided. Simulation results are reported and some implications for empirical practice are examined.

The Convergence of Sulphur Dioxide (SO2) Emissions Per Capita in China

As the third-largest SO2 emitter in the world, China is facing mounting domestic and external pressure to tackle the increasingly serious SO2 pollution. Figuring out the convergence and persistence of sulfur dioxide (SO2) emissions matters much for environmental policymakers in China. This study mainly utilizes the Fourier quantile unit root test to survey the convergence of the SO2 emissions per capita in 74 cities of China during the period of December 2014 to June 2019, by conducting five traditional unit root tests and a quantile root unit test as a comparative analysis. The empirical results indicate that the SO2 emissions per capita in 72 out of 74 cities in China are convergent in the sample period. The results also suggest that the unit root behavior of the SO2 emissions per capita in these cities is asymmetrically persistent at different quantiles. For the cities with the convergent SO2 emissions, the government should consider the asymmetric mean-reverting pattern of SO2 emissions when implementing environmental protection policies at different stages. For Hefei and Nanjing, the local governments need to enact stricter environmental protection policies to control the emission of sulfur dioxide.

(Structural) VAR Models with Ignored Changes in Mean and Volatility

We discuss estimation of so-called long vector autoregressions for multivariate series exhibiting possibly time-varying mean and (co)variances. In applied work, such changes often escape undetected, and we ask how standard tools (least squares estimation, point forecasts, and estimated impulse responses) are affected when ignoring the changes altogether. Keeping the order of the autoregression fixed is known to lead to asymptotic bias in autoregressive parameter estimators in the presence of ignored changes in the mean. Yet we show that allowing the complexity of the model to increase with the sample size leads to consistent estimators of the AR coefficient matrices individually. The fitted long VAR models appear to have unit root behavior, in spite of the absence of any stochastic trend in the model, and may even mimic cointegration; but, in spite of the structural change in the data generating process, out-of-sample forecasts based on long VARs are consistent. These findings hold under constant as well as under time-varying covariances. In what concerns estimated impulse responses, their sampling behavior depends primarily on whether the residual covariance matrix is employed for identification of the structural shocks or not. While MA coefficient matrices obtained by inversion of the fitted long VAR model are consistent for the true coefficients under mild additional restrictions even under time-varying error (co)variances, the residual covariance matrix estimator converges to an average covariance matrix, such that localized estimators may be more suitable for a precise identification. Monte Carlo simulations and empirical illustration support our theoretical findings. Empirical relevance of the theory is illustrated through two illustrations: (i) international dynamics of inflation and (ii) uncertainty and economics activity.

Sequential monitoring for changes from stationarity to mild non-stationarity

We develop and study sequential testing procedures á la Chu et al. (1996) for on-line detection of changes in a time series from stationarity to mild forms of non-stationarity. The proposed tests are based on sequential CUSUM and KPSS-type detector processes, and are shown to provide consistent detection under a wide range of change point models, including changes in the parameters of ARMA and GARCH series from values within the model’s stationarity parameter region to values close (converging) to the stationarity boundary. Local asymptotic results are established giving precise descriptions of the time to detection under several of these models, which show that such procedures are powerful to detect a wide range of non-stationary characteristics, including changes in mean, volatility, and unit root behaviour. The proposed methods are investigated by means of a simulation study and in applications to monitoring for changes in trend and unit root behaviour in macroeconomic production series, and to detect changes in volatility of the S&P-500 stock market index.

Sequential Monitoring for Speculative Bubbles in Financial Time Series

We develop and study sequential testing procedures a la Chu et al. (1996) for on-line detection of changes in a time series from stationarity to mild forms of non-stationarity. The proposed tests are based on sequential CUSUM and KPSS-type detector processes, and are shown to provide consistent detection under a wide range of change point models, including changes in the parameters of ARMA and GARCH series from values within the model's stationarity parameter region to values close (converging) to the stationarity boundary. Local asymptotic results are established giving precise descriptions of the time to detection under several of these models, which show that such procedures are powerful to detect a wide range of non-stationary characteristics, including changes in mean, volatility, and unit root behaviour. The proposed methods are investigated by means of a simulation study and in applications to monitoring for changes in trend and unit root behaviour in macroeconomic production series, and to detect changes in volatility of the S&P-500 stock market index.

Random coefficient continuous systems: Testing for extreme sample path behavior

This paper studies a continuous time dynamic system with a random persistence parameter. The exact discrete time representation is obtained and related to several discrete time random coefficient models currently in the literature. The model distinguishes various forms of unstable and explosive behavior according to specific regions of the parameter space that open up the potential for testing these forms of extreme behavior. A two-stage approach that employs realized volatility is proposed for the continuous system estimation, asymptotic theory is developed, and test statistics to identify the different forms of extreme sample path behavior are proposed. Simulations show that the proposed estimators work well in empirically realistic settings and that the tests have good size and power properties in discriminating characteristics in the data that differ from typical unit root behavior. The theory is extended to cover models where the random persistence parameter is endogenously determined. An empirical application based on daily real S&P 500 index data over 1928–2018 reveals strong evidence against parameter constancy over the whole sample period leading to a long duration of what the model characterizes as extreme behavior in real stock prices.

Point Optimal Testing with Roots That Are Functionally Local to Unity

Limit theory for regressions involving local to unit roots (LURs) is now used extensively in time series econometric work, establishing power properties for unit root and cointegration tests, assisting the construction of uniform confidence intervals for autoregressive coefficients, and enabling the development of methods robust to departures from unit roots. The present paper shows how to generalize LUR asymptotics to cases where the localized departure from unity is a time varying function rather than a constant. Such a functional local unit root (FLUR) model has much greater generality and encompasses many cases of additional interest, including structural break formulations that admit subperiods of unit root, local stationary and local explosive behavior within a given sample. Point optimal FLUR tests are constructed in the paper to accommodate such cases. It is shown that against FLUR alternatives, conventional constant point optimal tests can have extremely low power, particularly when the departure from unity occurs early in the sample period. Simulation results are reported and some implications for empirical practice are examined.

Random Coefficient Continuous Systems: Testing for Extreme Sample Path Behaviour

This paper studies a continuous time dynamic system with a random persistence parameter. The exact discrete time representation is obtained and related to several discrete time random coefficient models currently in the literature. The model distinguishes various forms of unstable and explosive behaviour according to specific regions of the parameter space that open up the potential for testing these forms of extreme behaviour. A two-stage approach that employs realized volatility is proposed for the continuous system estimation, the asymptotic theory is developed, and test statistics to identify the different forms of extreme sample path behaviour are proposed. Simulations show that the proposed estimators work well in empirically realistic settings and that the tests have good size and power properties in discriminating characteristics in the data that differ from typical unit root behaviour. The theory is extended to cover models where the random persistence parameter is endogenously determined. An empirical application based on daily real S&P500 index data over 1928-2018 reveals strong evidence against parameter constancy over the whole sample period leading to a long duration of what the model characterizes as extreme behaviour in real stock prices.

TIME-VARYING COEFFICIENT TAYLOR RULE AND CHINESE MONETARY POLICY: EVIDENCE FROM THE TIME-VARYING COINTEGRATION

(ProQuest: ... denotes formulae omitted.)1.INTRODUCTIONThe Chinese economy, known as the miracle, has maintained steady and fast growth for three decades, and attracts much attention from scholars and practitioners. Unlike Eastern European countries, China undergoing a gradual market-oriented reform in which its economic structure and regime are changing slowly and smoothly. For this reason, it is of great importance to study whether existing economic theory can be applied to China. In macroeconomics, the proposed by Taylor (1993), is an important monetary policy that specifies how the US Federal Reserve should adjust its targeted interest rate to the inflation and output gaps. Studying the validity of the Taylor rule for China is theoretically and practically important.There is a vast literature on the Taylor rule. For example, Taylor (1999) studied the validity of the Taylor rule using historical data covering 1897 to 1914 and 1955 to 1997, finding that the deviation of the actual interest rate from the targeted interest rate could help reflect the effectiveness of monetary policy. Kim and Nelson (2006) proposed a time-varying parameter forward-looking monetary policy rule and provided an efficient estimation using the Kalman filter. Yuksel, Metin-Ozcan, and Hatipoglu (2013) introduced an interest rate pass-through specification of the monetary transmission process in a general Taylor model. Xie and Luo (2002) were the first to check the validity of the Taylor rule for China, finding that it was a good measure for Chinese monetary policy and provided a reference for Chinese monetary policy implementation. However, Bian (2006) showed that the Taylor rule was unstable in China by using the GMM and cointegration test approach. Zheng and Liu (2010) developed a regime-switching Taylor rule with a time-varying inflation target, and showed that the Chinese monetary policy rule could be significantly divided into passive and active regimes.As Osterholm (2005) pointed out, however, previous studies ignored the nonstationarity of the variables in the Taylor rule model, despite the enormous body of literature. On one hand, the variables might not be stationary processes. Given the variable's unit root behavior, the cointegration relationships among them become essential for regression modeling. In the absence of cointegation, the model would be a spurious regression. On the other hand, the traditional fixed coefficient cointegration approach assumes a constant long-run equilibrium relationship among the variables, thus failing to capture the dynamic structure of the equilibrium relationship. China's gradual reform process is leading to a smooth structural change in the relevant economic variable and market, such as stock markets (Huang et al., 2000) and interest rate. For example, China is gradually carrying out interest rate liberalization. First, inter-bank lending and bond markets were deregulated in 1996; then, the ceiling on deposit rates and the floor on lending rates of commercial banks were relaxed; then, the Shanghai Interbank Offered Rate (Shibor) became China's benchmark rate in 2007; and then, the lending interest rate was completely decontrolled in 2013. Accordingly, this paper considers the smooth and gradual structural change occurring in China rather than abrupt structural breaks.This paper develops a time-varying coefficient Taylor rule to study China's monetary policy. Given the nonstationarity of the model's variables, the paper employs the time-varying cointegration approach of Park and Hahn (1999). To the best of our knowledge, this paper is the first to examine the Taylor rule through the time-varying cointegration approach. Kim and Nelson (2006) also considered the time-varying coefficient Taylor rule, but failed to consider the nonstationarity of the variables. By using several unit root tests, we found that the interest rate, the inflation gap and the output gap are indeed nonstationary. …

The joint dynamics of sovereign ratings and government bond yields

Abstract Can a negative shock to sovereign ratings invoke a vicious cycle of increasing government bond yields and further downgrades, ultimately pushing a country toward default? The narratives of public and political discussions, as well as of some widely cited papers, suggest this possibility. In this paper, we will investigate the possible existence of such a vicious cycle. We find no evidence of a bad long-run equilibrium and cannot confirm a feedback loop leading into default as a transitory state for all but the very worst ratings. We use a bivariate semiparametric dynamic panel model to reproduce the joint dynamics of sovereign ratings and government bond yields. The individual equations resemble Pesaran-type cointegration models, which allow for valid interference regardless of whether the employed variables display unit-root behavior. To incorporate most of the empirical features previously documented (separately) in the literature, we allow for different long-run relationships in both equations, nonlinearities in the level effects of ratings, and asymmetric effects in changes of ratings and yields. Our finding of a single good equilibrium implies the slow convergence of ratings and yields toward this equilibrium. However, the persistence of ratings is sufficiently high that a rating shock can have substantial costs if it occurs at a highly speculative rating or lower. Rating shocks that drive the rating below this threshold can increase the interest rate sharply, and for a long time. Yet, simulation studies based on our estimations show that it is highly improbable that rating agencies can be made responsible for the most dramatic spikes in interest rates.

Artifactual unit root behavior of Value at risk (VaR)

An effective model for time-varying quantiles of a time series is of considerable practical importance across various disciplines. In particular, in financial risk management, computation of Value-at-risk (VaR), one of the most popular risk measures, involves knowledge of quantiles of portfolio returns. This paper examines the random walk behavior of VaRs constructed under two most common approaches, viz. historical simulation and the parametric approach using GARCH models. We find that sequences of historical VaRs appear to follow a unit root model, which can be an artifact under some settings, whereas its counterpart constructed via the parametric approach does not follow a random walk model by default.

Cyclical non-stationarity in commodity prices

This paper applies the Hylleberg et al. (J Econom 44(1):215–238, 1990) parametric seasonal unit root test to test for cyclical non-stationarity in commodity prices. The testing procedure is simple and involves evaluating various linear restrictions on lagged price levels in an error correction model of prices, equivalent to the Augmented Dickey–Fuller test. Unit root behaviour at low frequencies implies cyclical non-stationarity. In our empirical application, we fail to reject unit roots at frequencies associated with 3- to 5-year-long price cycles for 7 of 16 major commodities. Our results suggest that longer cycles in many commodity prices are highly stochastic, and care should be taken when interpreting the regularity and out-of-sample predictability of such cycles using historical price movements.

Public debt sustainability in Serbia before and during the global financial crisis

We have analyzed the behaviour of primary fiscal balance and public debt in Serbia before and in the aftermath of the global financial crisis. The results of our analysis are: i) public debt to GDP ratioexhibits (near) unit root behaviour with an overall upward time trend; ii) the response of primary fiscal balance to public debt has been insufficient to mean revert the upward trend in government debt; iii) the efforts of the Serbian government to repay the debt principal after the fiscal rule breach have not been persistent, providing empirical support to the fiscal fatigue hypothesis; iv) the government budget constraint has deteriorated since the beginning of the global financial crisis; v) the response of primary fiscal balance to public debt from the onset of the global financial crisis has dropped more severely in comparison to other European economies.

The Joint Dynamics of Sovereign Ratings and Government Bond Yields

Can a negative shock to sovereign ratings invoke a vicious cycle of increasing government bond yields and further downgrades, ultimately pushing a country toward default? The narratives of public and political discussions, as well as of some widely cited papers, suggest this possibility. In this paper, we will investigate the possible existence of such a vicious cycle. We find no evidence of a bad long-run equilibrium and cannot confirm a negative feedback loop leading into default as a transitory state for all but the very worst ratings. We use a bivariate semiparametric dynamic panel model to reproduce the joint dynamics of sovereign ratings and government bond yields. The individual equations resemble Pesaran-type cointegration models, which allow for valid interference regardless of whether the employed variables display unit-root behavior. To incorporate most of the empirical features previously documented (separately) in the literature, we allow for different long-run relationships in both equations, nonlinearities in the level effects of ratings, and asymmetric effects in changes of ratings and yields. Our finding of a single good equilibrium implies the slow convergence of ratings and yields toward this equilibrium. However, the persistence of ratings is sufficiently high that a rating shock can have substantial costs if it occurs at a highly speculative rating or lower. Rating shocks that drive the rating below this threshold can increase the interest rate sharply, and for a long time. Yet, simulation studies based on our estimations show that it is highly improbable that rating agencies can be made responsible for the most dramatic spikes in interest rates.

(WHEN) DO LONG AUTOREGRESSIONS ACCOUNT FOR NEGLECTED CHANGES IN PARAMETERS?

To construct forecasts for time series exhibiting breaks, the paper examines long autoregressions, where the number of lags is growing with T, and possible breaks are simply ignored. The paper shows that the OLS estimators are still elementwise consistent for the true autoregressive coefficients when neglecting a break in mean, but the sum of the estimators converges to unity. Thanks to this unit-root like behavior of the fitted model, the resulting conditional forecasts are consistent for the true values. As long as the dynamic structure is invariant, the robustness property of the forecasts holds a) under data-dependent lag length selection, b) for a piecewise smoothly varying mean function, and c) under general autoregressive dynamics of possibly infinite order including stationary long memory. Under breaks in the dynamic structure, however, estimators are asymptotically biased, and the forecasts from long autoregressions are biased themselves even in the limit.

Persistence of Shocks in the US Natural Gas Market: A Sector-Level Analysis

This paper tests the integration property of US natural gas consumption at the sector level and finds a heterogenous behavior. Residential, commercial, and power consumption are stationary. However, even after allowing for endogenous structural breaks, tests fail to reject unit-root in the industrial consumption of natural gas. The results highlight the value of conducting sector-level analysis of unit-root behavior in energy consumption.

Structural Breaks, Long Memory, or Unit Roots in Stock Prices: Evidence from Emerging Markets

This paper investigates whether daily stock price indices from fourteen emerging markets are random walk (unit root) or mean reverting long memory processes. We use an efficient statistical framework that tests for random walks in the presence of multiple structural breaks at unknown dates. This approach allows us to investigate a broader range of persistence than that allowed by the I(0)/I(1) paradigm about the order of integration, which is usually implemented for testing the random walk hypothesis in stock market indices. Our approach extends Robinson’s (1994) efficient test of unit root against fractional integration to allow for multiple endogenously determined structural breaks. For almost all countries, we find support for the random walk hypothesis, with the exception of four stock markets, where weak evidence of mean reverting long memory exist. Structural breaks have impact on the unit root behavior only for Mexico; for all other 11 markets unit roots exist even when structural breaks are not taken into account. In order to check the robustness of our results, we use the two-step feasible exact local Whittle (FELW2ST) estimator of Shimotsu (2010), which allows for polynomial trends, non-normal distributions, and non-stationarity. The results from the semi-parametric FELW2ST approach shows that, except for Mexico, stock price indices of 13 emerging markets are not mean reverting.