Stochastic Unit Root Process Research Articles

Testing the null of difference stationarity against the alternative of a stochastic unit root: A new test based on multivariate STUR

A multivariate stochastic unit root process is used to test a simple versus a stochastic unit root. The score statistic and its asymptotic distribution under the null are given, and the test is seen to have a very acceptable power function. Simulations are performed to assess the robustness of the procedure in two common circumstances: preselecting the autoregressive order for the series using Akaike’s Information Criterion, and allowing for an MA component. An empirical application to macroeconomic and financial series is given to illustrate the test and to compare it to the main alternatives in the literature.

Stochastic unit root processes: Maximum likelihood estimation, and new Lagrange multiplier and likelihood ratio tests

We show in this study that the maximum likelihood estimators of stochastic unit root (STUR) processes are consistent and asymptotically normally distributed. We also present two new tests for STUR. We first propose a Lagrange multiplier test and show that it has a standard χ2 distribution asymptotically. We also propose a likelihood ratio test and show that it has an asymptotic distribution of 50–50 mixture of χ2 and a point mass at 0. As an empirical example, we test the existence of STUR in the Canadian real exchange rate and explore the implication of STUR on the validity of purchasing power parity.

Nonlinear mean reversion in real exchange rates: threshold autoregressive models and stochastic unit root processes

Threshold-type nonlinear relations are pretty popular in modelling the deviations from purchasing power parity. This article shows that there is a close relation between the nonlinear band Threshold Autoregressive (TAR) models studied by Obstfeld and Taylor (1997) and Stochastic Unit Root (STUR) processes discussed by Granger and Swanson (1997) and McCabe and Tremayne (1995). Employing the real exchange rates from four major countries against the US dollar, this article finds little evidence for TAR but some support for STUR during the post-Bretton Woods sample periods.

Reassessment of the finance-growth nexus in the presence of structural breaks

This article investigates empirically the causality between financial development and economic growth for 11 sample countries in a structural break framework via minimum Lagrange multiplier unit-root test and Hsiao's causality test. We find most of the series can be more accurately characterized as a segmented trend stationary process around structural breaks as opposed to a stochastic unit-root process, which has important implications for policy-makers to formulate long-term development strategy and short-run stabilization policies as well as for academics to carry out cointegration analysis. Besides, we find different causality patterns in sample countries, which depart from those found in the existing literature, highlighting that it might be unreliable to apply only conventional unit-root test and inappropriate to take the first difference of the series to achieve stationarity, all of which might result in model misspecification. Our analysis also provides some evidence on the source of the mixed results on finance-growth nexus as they may differ due to the length of data span and the different econometric procedure. ‘One of the most important problems in the field of finance … is the effect that financial structure and development have on economic growth’. Goldsmith (1969, p. 390)

Nonlinear mean-reversion to purchasing power parity: exponential smooth transition autoregressive models and stochastic unit root processes

Nonlinear exponential smooth transition autoregressive (ESTAR) models are recently very popular in modelling the deviation from purchasing power parity. This article, shows that there is a close relation between the ESTAR models estimated in Taylor et al. (2001) and stochastic unit root (STUR) processes of Granger and Swanson (1997) and McCabe and Tremayne (1995). Also, for a post-Bretton Woods sample period, the real exchange rates from four major countries are tested if they are better described as I(1), ESTAR or STUR processes.

Asymptotic theory for explosive random coefficient autoregressive models and inconsistency of a unit root test against a stochastic unit root process

In this article, we develop the asymptotic theory of Hwang and Basawa (2005) for explosive random coefficient autoregressive (ERCA) models. Applying the theory, we prove that a locally best invariant (LBI) test in McCabe and Tremayne (1995), which is for the null of a unit root (UR) process against the alternative of a stochastic unit root (STUR) process, is inconsistent against a class of ERCA models. This class includes a class of STUR processes as special cases. We show, however, that the well-known Dickey–Fuller (DF) UR tests and a LBI test of Lee (1998) are consistent against a particular case of this class of ERCA models.

Testing for coefficient stability of AR(1) model when the null is an integrated or a stationary process

In this paper, we propose a new test for coefficient stability of an AR(1) model against the random coefficient autoregressive model of order 1 neither assuming a stationary nor a non-stationary process under the null hypothesis of a constant coefficient. The proposed test is obtained as a modification of the locally best invariant (LBI) test by Lee [(1998). Coefficient constancy test in a random coefficient autoregressive model. J. Statist. Plann. Inference 74, 93–101]. We examine finite sample properties of the proposed test by Monte Carlo experiments comparing with other existing tests, in particular, the LBI test by McCabe and Tremayne [(1995). Testing a time series for difference stationary. Ann. Statist. 23 (3), 1015–1028], which is for the null of a unit root process against the alternative of a stochastic unit root process.

Inconsistency of a Unit Root Test Against a Stochastic Unit Root Process

In this article, we develop the asymptotic theory of Hwang and Basawa (2005) for explosive random coefficient autoregressive (ERCA) models. Applying the theory, we prove that a locally best invariant (LBI) test in McCabe and Tremayne (1995), which is for the null of a unit root (UR) process against the alternative of a stochastic unit root (STUR) process, is inconsistent against a class of ERCA models. This class includes a class of STUR processes as special cases. We show, however, that the well-known Dickey-Fuller(DF) UR tests and a LBI test of Lee (1998) are consistent against a particular case of this class of ERCA models.

Testing for unit root bilinearity in the Brazilian stock market

In this paper a simple test for detecting bilinearity in a stochastic unit root process is used to test for the presence of nonlinear unit roots in Brazilian equity shares. The empirical evidence for a set of 53 individual stocks, after adjusting for GARCH effects, suggests that for more than 66%, the hypothesis of unit root bilinearity is accepted. Therefore, the dynamics of Brazilian share prices is in conformity with this type of nonlinearity. These nonlinearities in spot prices may emerge due to the sophistication of the derivatives market.

Unit roots and structural breakpoints in China’s macroeconomic and financial time series

This paper applies unit-root tests to 10 Chinese macroeconomic and financial time series that allow for the possibility of up to two endogenous structural breaks. We found that 6 of the series, i.e., GDP, GDP per capita, employment, bank credit, deposit liabilities and investment, can be more accurately characterized as a segmented trend stationarity process around one or two structural breakpoints as opposed to a stochastic unit root process. Our findings have important implications for policy-makers to formulate long-term growth strategy and short-run stabilization policies, as well as causality analysis among the series.

Testing for rational bubbles in banking indices

In this paper a simple test for detecting bilinearity in a stochastic unit root process is used to test for the presence of rational bubbles in banking equity indices. The empirical evidence for a set of 39 banking indices for different countries, after adjusting for GARCH effects, suggests that for more than 69% of these indices, the hypothesis of no unit root bilinearity is rejected. Therefore, the dynamics of banking asset prices in most countries are in conformity with rational bubbles and high speculation. This is true for both developed and emerging economies.

CORRELATION COEFFICIENTS, HETEROSKEDASTICITY AND CONTAGION OF FINANCIAL CRISES

A significant increase in the correlation coefficients of returns across countries during periods of high turbulence is regarded as evidence of the contagion of financial crises. However, heteroskedasticity is known to cause correlation coefficients to be biased upward. This note shows that correlation coefficients can be biased downward under heteroskedasticity when returns are following stochastic unit root processes. Further, returns are known to be nonlinear, so correlation coefficients might not be very useful in measuring relations between them. These results indicate that the time-series behavior of returns needs to be more thoroughly studied prior to measuring the contagion of financial crises with correlation coefficients.

Conditional testing for unit-root bilinearity in financial time series: some theoretical and empirical results

This paper introduces a t-ratio type test for detecting bilinearity in a stochastic unit root process. It appears that such a process is a realistic approximation for many economic and financial time series. It is shown that, under the null of no bilinearity, the test statistics are asymptotically normally distributed. Proofs of these asymptotic normality results require the Gihman and Skorohod theory for multivariate diffusion processes. Finite sample results describe speed of convergence, power of the tests and possible distortions to unit root testing which might appear due to the presence of bilinearity. It is concluded that the two-step testing procedure suggested here (the first step for the linear unit root and the second step for its bilinearity) is consistent in the sense that the size of the step one test is not affected by the possible detection of bilinearity at step two. The empirical part of the paper describes testing unit root bilinearity for 65 GARCH-adjusted stock market indices from mature and emerging markets. It is shown that for at least 70% of these markets, the hypothesis of no unit root bilinearity has to be rejected.

A Simple Model that Generates Stylized Facts of Returns

This note shows that a very simple model can generate returns that resemble most of the temporal and distributional behavior of long returns surprisingly well. The model is based on the stochastic unit root process introduced in Granger and Swanson (1997).

Can Tests for Stochastic Unit Roots Provide Useful Portmanteau Tests for Persistence?*

In this paper we investigate whether or not the recently developed class of tests of the unit root null against the alternative of a stochastic unit root forms a useful statistical tool in distinguishing between time series processes whose degree of persistence is no more than that of a unit root [I(1)] process and those which display a greater degree of persistence than I(1) series, the stochastic unit root process being an example of the latter. For a wide range of processes which have been put forward as serious competitors to the I(1) process, both of a greater and lesser degree of persistence, we find, via numerical simulation methods, that broadly speaking the stochastic unit root tests do indeed appear to provide an efficacious diagnostic tool in this regard.

Stochastic unit roots modelling of stock price indices

Recently developed methodology to allow the possibility of a stochastic unit root process as an alternative to a fixed parameter unit root model is applied to six national indices of stock market prices. Evidence supporting the stochastic unit root hypothesis is found. However, the implementation of this model generally leads to only very minuscule gains in the prediction of daily prices, except in the case of the Hang Seng index, where the predictive gain is somewhat larger.