Stochastic Unit Root Research Articles

Stochastic local and moderate departures from a unit root and its application to unit root testing

Local‐to‐unity and moderate‐deviations specifications have been popular alternatives to unit root modeling. This article considers another kind of departures from a unit root, of the form , where is random and determines the distance from a unit root. We classify the stochastic departures into two types: local and moderate. This classification task is completed by investigating the asymptotic behavior of unit root tests that assume the stochastic unit root (STUR) processes as the alternative hypothesis. The stochastic local‐to‐unity model arises when ; in this case, the test statistics have limiting distributions different from those under the unit root null, and their asymptotic powers are greater than size. Moderate deviations emerge when , in which case the test statistics diverge. We also propose new tests for a unit root against an STUR, whose construction is based on the limit theory developed in this article. To evaluate the performance of these new tests, we derive the limiting Gaussian power envelope under the local alternative from an approximate model.

Robust inference with stochastic local unit root regressors in predictive regressions

This paper explores predictive regression models with stochastic unit root (STUR) components and robust inference procedures that encompass a wide class of persistent and time-varying stochastically nonstationary regressors. The paper extends the mechanism of endogenously generated instrumentation known as IVX, showing that these methods remain valid for short and long-horizon predictive regressions in which the predictors have STUR and local STUR (LSTUR) generating mechanisms. Both mean regression and quantile regression methods are considered. The asymptotic distributions of the IVX estimators are new and require some new methods in their derivation. The distributions are compared to previous results and, as in earlier work, lead to pivotal limit distributions for Wald testing procedures that remain robust for both single and multiple regressors with various degrees of persistence and stochastic and fixed local departures from unit roots. Numerical experiments corroborate the asymptotic theory, and IVX testing shows good power and size control. The IVX methods are illustrated in an empirical application to evaluate the predictive capability of economic fundamentals in forecasting S&P 500 excess returns.

Estimation and Inference for the Threshold Model with Hybrid Stochastic Local Unit Root Regressors

In this paper, we study the estimation and inference of the threshold model with hybrid local stochastic unit root regressors. Our main contribution is to propose an estimator that generalizes the threshold model with various forms of nonstationary regressors and to obtain its limiting distribution theory. In particular, our proposed model generalizes the threshold model with unit root, local-to-unity, and stochastic unit root regressors. We provide the estimation strategy for the least squares estimator and derive the asymptotic results for the proposed estimator. Depending on the diminishing rate of the threshold effect, we find that the limiting distribution of the threshold estimator takes different forms. Monte Carlo simulations are used to assess our proposed estimator’s finite sample performance, which is found to perform well.

A note on nonlinear least squares estimation of multivariate stochastic unit root models

A note on nonlinear least squares estimation of multivariate stochastic unit root models

Testing for random coefficient autoregressive and stochastic unit root models

Abstract The random coefficient autoregressive model has been utilized for modeling financial time series because it possesses features that are often observed in financial time series. When the mean of the random coefficient is one, it is called the stochastic unit root model. This paper proposes two Lagrange multiplier tests for the null hypotheses of random coefficient autoregressive and stochastic unit root models against a more general model. We apply our Lagrange multiplier tests to several stock index data, and find that the stochastic unit root model is rejected, whereas the random coefficient autoregressive model is not. This result indicates that it is important to check the validity of the stochastic unit root model prior to applying it to financial time series data, which may be better modeled by the random coefficient autoregressive model with the mean being not equal to one.

Testing for Random Coefficient Autoregressive and Stochastic Unit Root Models

The random coefficient autoregressive model has been utilized for modeling financial time series because it possesses features that are often observed in financial time series. When the mean of the random autoregressive coefficient is one, it is called the stochastic unit root model. This paper proposes two Lagrange multiplier tests for the null hypothesis of random coefficient autoregressive and stochastic unit root models against a more general model. We apply our Lagrange multiplier tests to several stock index data, and find that the stochastic unit root model is rejected, whereas the random coefficient autoregressive model is not. This result indicates that it is important to check the validity of the stochastic unit root model prior to applying it to financial time series data, which may be better modeled by the random coefficient autoregressive model with the mean being not equal to one.

Understanding temporal aggregation effects on kurtosis in financial indices

Indices of financial returns typically display sample kurtosis that declines towards the Gaussian value 3 as the sampling interval increases. This paper uses stochastic unit root (STUR) and continuous time analysis to explain the phenomenon. Limit theory for the sample kurtosis reveals that STUR specifications provide two sources of excess kurtosis, both of which decline with the sampling interval. Limiting kurtosis is shown to be random and is a functional of the limiting price process. Using a continuous time version of the model under no-drift, local drift, and drift inclusions, we suggest a new continuous time kurtosis measure for financial returns that assists in reconciling these models with the empirical kurtosis characteristics of returns. Simulations are reported and applications to several financial indices demonstrate the usefulness of this approach.

IV and GMM Estimation and Testing of Multivariate Stochastic Unit Root Models

Lieberman and Phillips (2016; Journal of Econometrics; LP) introduced a multivariate stochastic unit root (STUR) model, which allows for random, time varying local departures from a unit root (UR) model, where nonlinear least squares (NLLS) may be used for estimation and inference on the STUR coefficient. In a structural version of this model where the driver variables of the STUR coefficient are endogenous, the NLLS estimate of the STUR parameter is inconsistent, as are the corresponding estimates of the associated covariance parameters. This paper develops a nonlinear instrumental variable (NLIV) as well as GMM estimators of the STUR parameter which conveniently addresses endogeneity. We derive the asymptotic distributions of the NLIV and GMM estimators and establish consistency under similar orthogonality and relevance conditions to those used in the linear model. An overidentification test and its asymptotic distribution are also developed. The results enable inference about structural STUR models and a mechanism for testing the local STUR model against a simple UR null, which complements usual UR tests. Simulations reveal that the asymptotic distributions of the the NLIV and GMM estimators of the STUR parameter as well as the test for overidentifying restrictions perform well in small samples and that the distribution of the NLIV estimator is heavily leptokurtic with a limit theory which has Cauchy-like tails. Comparisons of STUR coefficient and a standard UR coefficient test show that the one-sided UR test performs poorly against the one-sided STUR coefficient test both as the sample size and departures from the null rise.

LSTUR regression theory and the instability of the sample correlation coefficient between financial return indices

Summary It is well known that the sample correlation coefficient between many financial return indices exhibits substantial variation on any reasonable sampling window. This stylised fact contradicts a unit root model for the underlying processes in levels, as the statistic converges in probability to a constant under this modeling scheme. In this paper, we establish asymptotic theory for regression in local stochastic unit root (LSTUR) variables. An empirical application reveals that the new theory explains very well the instability, in both sign and scale, of the sample correlation coefficient between gold, oil, and stock return price indices. In addition, we establish spurious regression theory for LSTUR variables, which generalises the results known hitherto, as well as a theory for balanced regression in this setting.

Asymptotic theory for a stochastic unit root model

Lieberman and Phillips (Journal of Time Series Analysis) proposed a stochastic unit root model in which the source of the variation of the autoregressive coefficient is driven by a stationary process. More recently, Lieberman and Phillips (Journal of Econometrics) generalized this model to the multivariate case and a hybrid case. Their studies revealed that these stochastic unit root models lead to a generalization of the Black-Scholes formula for derivative pricing. Inspired by their studies, in this paper, we propose a new stochastic unit root model, in which the source of the variation of the autoregressive coefficient is driven by a (nearly) non-stationary process. The asymptotic theory for this model is established. Our study reveals some new findings which are different from those established by Lieberman and Phillips. Results of Monte Carlo simulations are given to illustrate the finite-sample performance of estimators in the model. Moreover, a comparison between the stochastic unit root model proposed by Lieberman and Phillips and that proposed in this paper is conducted via an empirical study.

Asymptotic Theory for a Stochastic Unit Root Model with Intercept and Under Mis-Specification of Intercept

Lieberman and Phillips (J Time Ser Anal 35(6):592–623 2014, J Econ 196(1):99–110 2017) proposed a stochastic unit root model, where the source of the variation of the autoregressive coefficient is driven by a stationary process. In this paper, we study a new stochastic unit root model, in which the source of the variation of the autoregressive coefficient is driven by a (nearly) non-stationary process. The asymptotic theory for this model is established not only in the presence of an intercept but also under a mis-specification of intercept. Our study reveals some new findings which are different from those established in Lieberman and Phillips (J Time Ser Anal 35(6):592–623 2014, J Econ 196(1):99–110 2017). Our theoretical results demonstrate that the statistical properties of the estimators of the model parameters vary from case to case, and depend not only on whether the parameters are zero or not, and the validity of the model specification, but also on the degree of the persistence of the source of the variation of the autoregressive coefficient. The application to a hypothesis testing with null hypothesis of a deterministic unit root model is briefly discussed. Monte Carlo simulations are conducted to examine the finite-sample performance for the estimators of model parameters. Our theoretical findings are supported by the simulation results.

Hybrid stochastic local unit roots

Two approaches have dominated formulations designed to capture small departures from unit root autoregressions. The first involves deterministic departures that include local-to-unity (LUR) and mildly (or moderately) integrated (MI) specifications where departures shrink to zero as the sample size n→∞. The second approach allows for stochastic departures from unity, leading to stochastic unit root (STUR) specifications. This paper introduces a hybrid local stochastic unit root (LSTUR) specification that has both LUR and STUR components and allows for endogeneity in the time varying coefficient that introduces structural elements to the autoregression. This hybrid model generates trajectories that, upon normalization, have non-linear diffusion limit processes that link closely to models that have been studied in mathematical finance, particularly with respect to option pricing. It is shown that some LSTUR parameterizations have a mean and variance which are the same as a random walk process but with a kurtosis exceeding 3, a feature which is consistent with much financial data. We develop limit theory and asymptotic expansions for the process and document how inference in LUR and STUR autoregressions is affected asymptotically by ignoring one or the other component in the more general hybrid generating mechanism. In particular, we show how confidence belts constructed from the LUR model are affected by the presence of a STUR component in the generating mechanism. The import of these findings for empirical research is explored in an application to the spreads on US investment grade corporate debt.

Understanding Temporal Aggregation Effects on Kurtosis in Financial Indices

Indices of financial returns typically display sample kurtosis that declines towards the Gaussian value 3 as the sampling interval increases. This paper uses stochastic unit root (STUR) and continuous time analysis to explain the phenomenon. Limit theory for the sample kurtosis reveals that STUR specifications provide two sources of excess kurtosis, both of which decline with the sampling interval. Limiting kurtosis is shown to be random and is a functional of the limiting price process. Using a continuous time version of the model under no-drift, local drift, and drift inclusions, we suggest a new continuous time kurtosis measure for financial returns that assists in reconciling these models with the empirical kurtosis characteristics of returns. Simulations are reported and applications to several financial indices demonstrate the usefulness of this approach.

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.

IV AND GMM INFERENCE IN ENDOGENOUS STOCHASTIC UNIT ROOT MODELS

Lieberman and Phillips (2017; LP) introduced a multivariate stochastic unit root (STUR) model, which allows for random, time varying local departures from a unit root (UR) model, where nonlinear least squares (NLLS) may be used for estimation and inference on the STUR coefficient. In a structural version of this model where the driver variables of the STUR coefficient are endogenous, the NLLS estimate of the STUR parameter is inconsistent, as are the corresponding estimates of the associated covariance parameters. This paper develops a nonlinear instrumental variable (NLIV) as well as GMM estimators of the STUR parameter which conveniently addresses endogeneity. We derive the asymptotic distributions of the NLIV and GMM estimators and establish consistency under similar orthogonality and relevance conditions to those used in the linear model. An overidentification test and its asymptotic distribution are also developed. The results enable inference about structural STUR models and a mechanism for testing the local STUR model against a simple UR null, which complements usual UR tests. Simulations reveal that the asymptotic distributions of the NLIV and GMM estimators of the STUR parameter as well as the test for overidentifying restrictions perform well in small samples and that the distribution of the NLIV estimator is heavily leptokurtic with a limit theory which has Cauchy-like tails. Comparisons of STUR coefficient and standard UR coefficient tests show that the one-sided UR test performs poorly against the one-sided STUR coefficient test both as the sample size and departures from the null rise. The results are applied to study the relationships between stock returns and bond spread changes.

Hybrid Stochastic Local Unit Roots

Two approaches have dominated formulations designed to capture small departures from unit root autoregressions. The first involves deterministic departures that include local-to-unity (LUR) and mildly (or moderately) integrated (MI) specifications where departures shrink to zero as the sample size n→∞. The second approach allows for stochastic departures from unity, leading to stochastic unit root (STUR) specifications. This paper introduces a hybrid local stochastic unit root (LSTUR) specification that has both LUR and STUR components and allows for endogeneity in the time-varying coefficient that introduces structural elements to the autoregression. This hybrid model generates trajectories that, upon normalization, have non-linear diffusion limit processes that link closely to models that have been studied in mathematical finance, particularly with respect to option pricing. It is shown that some LSTUR parameterizations have a mean and variance which are the same as a random walk process but with a kurtosis exceeding 3, a feature which is consistent with much financial data. We develop limit theory and asymptotic expansions for the process and document how inference in LUR and STUR autoregressions is affected asymptotically by ignoring one or the other component in the more general hybrid generating mechanism. In particular, we show how confidence belts constructed from the LUR model are affected by the presence of a STUR component in the generating mechanism. The import of these findings for empirical research are explored in an application to the spreads on US investment grade corporate debt.

A multivariate stochastic unit root model with an application to derivative pricing

This paper extends recent findings of Lieberman and Phillips (2014) on stochastic unit root (STUR) models to a multivariate case including asymptotic theory for estimation of the model’s parameters. The extensions are useful for applications of STUR modeling and because they lead to a generalization of the Black–Scholes formula for derivative pricing. In place of the standard assumption that the price process follows a geometric Brownian motion, we derive a new form of the Black–Scholes equation that allows for a multivariate time varying coefficient element in the price equation. The corresponding formula for the value of a European-type call option is obtained and shown to extend the existing option price formula in a manner that embodies the effect of a stochastic departure from a unit root. An empirical application reveals that the new model substantially reduces the average percentage pricing error of the Black–Scholes and Heston’s (1993) stochastic volatility (with zero volatility risk premium) pricing schemes in most moneyness-maturity categories considered.

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.

Cointegration with structural changes between per capita product and income inequality in Mexico

The long-run relationship between per capita product and income inequality in Mexico is assessed for the period 1963–2010. The methodology consists of stochastic unit root techniques with structural changes. The integration and cointegration tests suggest that it is not possible to understand this link if the possibility of structural changes is not taken into account. Causality, running from per capita product to income inequality, and a negative and significant overall effect are estimated. However, after the regime shifts – mainly estimated to occur in the 1980s – the connection is no longer meaningful. This result seems to be linked to the slow growth of the Mexican economy after these structural breaks, affecting, in turn, the reductions in income inequality.

NORMING RATES AND LIMIT THEORY FOR SOME TIME‐VARYING COEFFICIENT AUTOREGRESSIONS

A time‐varying autoregression is considered with a similarity‐based coefficient and possible drift. It is shown that the random‐walk model has a natural interpretation as the leading term in a small‐sigma expansion of a similarity model with an exponential similarity function as its AR coefficient. Consistency of the quasi‐maximum likelihood estimator of the parameters in this model is established, the behaviours of the score and Hessian functions are analysed and test statistics are suggested. A complete list is provided of the normalization rates required for the consistency proof and for the score and Hessian function standardization. A large family of unit root models with stationary and explosive alternatives is characterized within the similarity class through the asymptotic negligibility of a certain quadratic form that appears in the score function. A variant of the stochastic unit root model within the class is studied, and a large‐sample limit theory provided, which leads to a new nonlinear diffusion process limit showing the form of the drift and conditional volatility induced by sustained stochastic departures from unity. The findings provide a composite case for time‐varying coefficient dynamic modelling. Some simulations and a brief empirical application to data on international Exchange Traded Funds are included. Copyright © 2014 Wiley Publishing Ltd