Stochastic Unit Root Research Articles

Stochastic Unit Roots in the Capital Asset Pricing Model?

Evidence is provided in this article for the existence of a stochastic unit root (STUR) in a proxy for the US risk-free interest rate, in preference to a standard fixed unit root. The implications of the existence of the STUR, on estimating and testing the capital asset pricing model, are also examined through simulations. The effects of the STUR in the risk-free interest rate, on conducting unit root tests for excess market returns and estimating the betas of assets, are found to be qualitatively similar to those of the standard (fixed) unit root. Thus, this article confirms the conjecture of Markellos and Mills (2001, Applied Economics Letters, 8, pp. 499–502) on the risk-free interest rate following near-integrated processes, at least for a STUR.

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.

Practical drift conditions for subgeometric rates of convergence

We present a new drift condition which implies rates of convergence to the stationary distribution of the iterates of a ψ-irreducible aperiodic and positive recurrent transition kernel. This condition, extending a condition introduced by Jarner and Roberts [Ann. Appl. Probab. 12 (2002) 224–247] for polynomial convergence rates, turns out to be very convenient to prove subgeometric rates of convergence. Several applications are presented including nonlinear autoregressive models, stochastic unit root models and multidimensional random walk Hastings–Metropolis algorithms.

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.

On the existence of expected utility with CRRA under STUR

This note shows that in a standard framework for choice under uncertainty, expected utility fails to exist with constant relative risk aversion [CRRA], when the endowment follows a stochastic unit root [STUR] process.

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).

Real Exchange Rate Dynamics Under The Current Float: A Re–Examination

Augmented Dickey–Fuller regressions on pooled (but not individual) real exchange rates for the post–1973 period consistently reject the unit root null, even after accounting for cross–sectional dependence. The inference that the series is stationary, however, is not necessarily correct, because these tests strongly over–reject the null in certain circumstances, particularly when the series has a stochastic unit root. We find that bilateral real exchange rates against the US dollar have a stochastic unit root. Out–of–sample prediction exercises for an autoregressive model confirm these findings.

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.

Tests of the Seasonal Unit-Root Hypothesis Against Heteroscedastic Seasonal Integration

This article considers the problem of testing for a nonstochastic seasonal unit root in a seasonally observed time series process against the alternative of a randomized seasonal root with mean unity; that is, the process displays heteroscedastic seasonal integration. The alternative hypothesis allows for potentially frequently occurring changes of regime in the process under investigation, allowing for more volatile forms of seasonal nonstationarity. We discuss a family of models that allow for a potentially smooth transition between the explosive and stationary phases of the seasonal model. To test this hypothesis we consider extensions to existing approaches developed to test against nonseasonal stochastic unit roots. Asymptotic representations of the test statistics are derived. An empirical application to a variety of quarterly measures of U.K. consumer's expenditure is also considered.

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.

Mean Reversion of Real Exchange Rates in High‐Inflation Countries

We test for mean reversion in real exchange rates using data from five countries, four of which have experienced episodes of high inflation. Using monthly data for Argentina, Brazil, Chile, Colombia, and Israel, we find that a stochastic unit root model is typically appropriate (Brazil is the exception). Kalman filter estimates of the stochastic unit roots show sharp deviations from unity associated with high inflation episodes. This suggests that stochastic unit root models are a more appropriate way to model mean reversion in real exchange rates for high inflation countries than models with fixed rates of mean reversion.

Mean Reversion of Real Exchange Rates in High-Inflation Countries

We test for mean reversion in real exchange rates using data from five countries, four of which have experienced episodes of high inflation. Using monthly data for Argentina, Brazil, Chile, Colombia, and Israel, we find that a stochastic unit root model is typically appropriate (Brazil is the exception). Kalman filter estimates of the stochastic unit roots show sharp deviations from unity associated with high inflation episodes. This suggests that stochastic unit root models are a more appropriate way to model mean reversion in real exchange rates for high inflation countries than models with fixed rates of mean reversion.

An introduction to stochastic unit-root processes

A class of nonlinear processes which have a root that is not constant, but is stochastic, and varying around unity is introduced. The process can be stationary for some periods, and mildly explosive for others. Stochastic unit roots are seen to arise naturally in economic theory, as well as in everyday macroeconomic applications. It is shown that standard tests, such as the augmented Dickey-Fuller test, cannot easily distinguish between exact unit roots and stochastic unit roots. An alternative test which has difference stationarity as the null suggests that exact unit-root models are often rejected in favor of more general nonlinear stochastic unit-root (STUR) models. Estimation is discussed, and, a forecast comparison of linear random walk and AR(p) models, time-varying parameter models, and STUR models suggests that this new class of processes is potentially useful, particularly when the objective is multi-step ahead forecasting.