Near-unit Root Research Articles

Improved tests for stock return predictability

– Predictive regression methods are widely used to examine the predictability of (excess) stock returns by lagged financial variables characterized by unknown degrees of persistence and endogeneity. We develop a new hybrid test for predictability in these circumstances based on simple regression t-statistics. Where the predictor is endogenous, the optimal, but infeasible, test for predictability is based on the t-statistic on the lagged predictor in the basic predictive regression augmented with the current period innovation driving the predictor. We propose a feasible version of this augmented test, designed for the case where the predictor is an endogenous near-unit root process, using a GLS-based estimate of the innovation used in the infeasible test regression. The limiting null distribution of this statistic depends on both the endogeneity correlation parameter and the local-to-unity parameter characterizing the predictor. A method for obtaining asymptotic critical values is discussed and response surfaces are provided. We compare the asymptotic power properties of the feasible augmented test with those of a (non augmented) t-test recently considered in Harvey et al. and show that the augmented test is more powerful in the strongly persistent predictor case. We then propose using a weighted combination of the augmented statistic and the t-statistic of Harvey et al., where the weights are obtained using the p-values from a unit root test on the predictor. We find this can further improve asymptotic power in cases where the predictor has persistence at or close to that of a unit root process. Our final hybrid testing procedure then embeds the weighted statistic within a switching-based procedure which makes use of a standard predictive regression t-test, compared with standard normal critical values, when there is evidence for the predictor being weakly persistent. Monte Carlo simulations suggest that overall our new hybrid test displays superior finite sample performance to comparable extant tests.

ESTIMATION AND INFERENCE WITH NEAR UNIT ROOTS

New methods are developed for identifying, estimating, and performing inference with nonstationary time series that have autoregressive roots near unity. The approach subsumes unit-root (UR), local unit-root (LUR), mildly integrated (MI), and mildly explosive (ME) specifications in the new model formulation. It is shown how a new parameterization involving a localizing rate sequence that characterizes departures from unity can be consistently estimated in all cases. Simple pivotal limit distributions that enable valid inference about the form and degree of nonstationarity apply for MI and ME specifications and new limit theory holds in UR and LUR cases. Normalizing and variance stabilizing properties of the new parameterization are explored. Simulations are reported that reveal some of the advantages of this alternative formulation of nonstationary time series. A housing market application of the methods is conducted that distinguishes the differing forms of house price behavior in Australian state capital cities over the past decade.

Simple tests for stock return predictability with good size and power properties

We develop easy-to-implement tests for return predictability which, relative to extant tests in the literature, display attractive finite sample size control and power across a wide range of persistence and endogeneity levels for the predictor. Our approach is based on the standard regression t-ratio and a variant where the predictor is quasi-GLS (rather than OLS) demeaned. In the strongly persistent near-unit root environment, the limiting null distributions of these statistics depend on the endogeneity and local-to-unity parameters characterising the predictor. Analysis of the asymptotic local power functions of feasible implementations of these two tests, based on asymptotically conservative critical values, motivates a switching procedure between the two, employing the quasi-GLS demeaned variant unless the magnitude of the estimated endogeneity correlation parameter is small. Additionally, if the data suggests the predictor is weakly persistent, our approach switches to the standard t-ratio test with reference to standard normal critical values.

Optimal Taxation with Endogenous Default under Incomplete Markets

How are the optimal tax and debt policies affected if the government has the option to default on its debt? We address this question from a normative perspective in an economy with noncontingent government debt, domestic default and labor taxes. On one hand, default prevents the government from incurring future tax distortions that would come along with the service of the debt. On the other hand, default risk gives rise to endogenous credit limits that hinder the government's ability to smooth taxes. We characterize the fiscal policy and show how the option to default alters the near-unit root component of taxes in the economy with risk-free borrowing. When we allow the government to default and calibrate the model to Spain, fiscal policies are more volatile, borrowing costs are higher, indebtness and welfare are both lower than in two alternatives economies, one with only risk-free debt available and the other with government's commitment to the default strategy.

Asymptotic theory for near integrated processes driven by tempered linear processes

In an early article on near-unit root autoregression, Ahtola and Tiao (1984) studied the behavior of the score function in a stationary first order autoregression driven by independent Gaussian innovations as the autoregressive coefficient approached unity from below. The present paper develops asymptotic theory for near-integrated random processes and associated regressions including the score function in more general settings where the errors are tempered linear processes. Tempered processes are stationary time series that have a semi-long memory property in the sense that the autocovariogram of the process resembles that of a long memory model for moderate lags but eventually diminishes exponentially fast according to the presence of a decay factor governed by a tempering parameter. When the tempering parameter is sample size dependent, the resulting class of processes admits a wide range of behavior that includes both long memory, semi-long memory, and short memory processes. The paper develops asymptotic theory for such processes and associated regression statistics thereby extending earlier findings that fall within certain subclasses of processes involving near-integrated time series. The limit results relate to tempered fractional processes that include tempered fractional Brownian motion and tempered fractional diffusions of the second kind. The theory is extended to provide the limiting distribution for autoregressions with such tempered near-integrated time series, thereby enabling analysis of the limit properties of statistics of particular interest in econometrics, such as unit root tests, under more general conditions than existing theory. Some extensions of the theory to the multivariate case are reported.

A Near Unit Root Test for High-Dimensional Nonstationary Time Series

This paper proposes a new near-unit root test for a class of high-dimensional nonstationary time series. A central limit theorem for the proposed test is proposed and then evaluated by an extensive simulation study.

Change‐Point Detection in the Conditional Correlation Structure of Multivariate Volatility Models

ABSTRACTWe propose semiparametric CUSUM tests to detect a change-point in the correlation structures of nonlinear multivariate models with dynamically evolving volatilities. The asymptotic distributions of the proposed statistics are derived under mild conditions. We discuss the applicability of our method to the most often used models, including constant conditional correlation (CCC), dynamic conditional correlation (DCC), BEKK, corrected DCC, and factor models. Our simulations show that, our tests have good size and power properties. Also, even though the near-unit root property distorts the size and power of tests, de-volatizing the data by means of appropriate multivariate volatility models can correct such distortions. We apply the semiparametric CUSUM tests in the attempt to date the occurrence of financial contagion from the US to emerging markets worldwide during the great recession. Supplementary materials for this article are available online.

Jackknife Bias Reduction in the Presence of a Near-Unit Root

This paper considers the specification and performance of jackknife estimators of the autoregressive coefficient in a model with a near-unit root. The limit distributions of sub-sample estimators that are used in the construction of the jackknife estimator are derived, and the joint moment generating function (MGF) of two components of these distributions is obtained and its properties explored. The MGF can be used to derive the weights for an optimal jackknife estimator that removes fully the first-order finite sample bias from the estimator. The resulting jackknife estimator is shown to perform well in finite samples and, with a suitable choice of the number of sub-samples, is shown to reduce the overall finite sample root mean squared error, as well as bias. However, the optimal jackknife weights rely on knowledge of the near-unit root parameter and a quantity that is related to the long-run variance of the disturbance process, which are typically unknown in practice, and so, this dependence is characterised fully and a discussion provided of the issues that arise in practice in the most general settings.

Change-Point Detection in the Covariance Structure of Multivariate Volatility Models

We propose semi-parametric CUSUM tests to detect a change point in the covariance structure of non-linear multivariate models with dynamically evolving volatilities and correlations. The asymptotic distributions of the proposed statistics are derived under mild conditions. We discuss the applicability of our method to the most often used models, including constant conditional correlation (CCC), dynamic conditional correlation (DCC), factor, asymmetric DCC and BEKK. Our simulations show that, even though the near-unit root property distorts the size and power of tests, de-volatizing the data by means of appropriate multivariate GARCH models can correct such distortions. We apply the semi-parametric CUSUM tests in the attempt to date the occurrence of financial contagion during the great recession.

Robust Inference for Near-Unit Root Processes with Time-Varying Error Variances

The autoregressive Cauchy estimator uses the sign of the first lag as instrumental variable (IV); under independent and identically distributed (i.i.d.) errors, the resulting IV t-type statistic is known to have a standard normal limiting distribution in the unit root case. With unconditional heteroskedasticity, the ordinary least squares (OLS) t statistic is affected in the unit root case; but the paper shows that, by using some nonlinear transformation behaving asymptotically like the sign as instrument, limiting normality of the IV t-type statistic is maintained when the series to be tested has no deterministic trends. Neither estimation of the so-called variance profile nor bootstrap procedures are required to this end. The Cauchy unit root test has power in the same 1/T neighborhoods as the usual unit root tests, also for a wide range of magnitudes for the initial value. It is furthermore shown to be competitive with other, bootstrap-based, robust tests. When the series exhibit a linear trend, however, the null distribution of the Cauchy test for a unit root becomes nonstandard, reminiscent of the Dickey-Fuller distribution. In this case, inference robust to nonstationary volatility is obtained via the wild bootstrap.

Maximum likelihood estimation for nearly non-stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near-unit root autoregressive process Yt = ?nYt-1 + ?t with a-stable noise {?t} is studied in this paper. Herein ?n = 1 - ?/n, ? = 0 is a constant, Y0 is a fixed random variable and et is an a-stable random variable with characteristic function f(t,?) for some parameter ?. It is shown that when 0 1 and E?1 = 0, the limit distribution of the MLE of ?n and ? are mixtures of a stable process and Gaussian processes. On the other hand, when a > 1 and E?1 ? 0, the limit distribution of the MLE of ?n and ? are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index a is less than 1.

European incomplete catching-up

We study in this paper pairwise convergence of aggregate output dynamics by applying a known exact, finite-sample, most powerful and invariant (MPI) test statistic in a new analytical, conditional inverted beta setup. Besides being useful for finding interval estimates of a persistence parameter when needed, it turns out that our IB-MPI test has a comparable power to discriminate near-unit roots. By using standard, time-series definitions of divergence, incomplete or complete catching-up, and long-run convergence, the empirical application of the test shows that pairwise differences in per capita income levels between European countries overwhelmingly follow a trend-stationary, incomplete catching-up process.

A note on unit root testing in the presence of level shifts

In this note we discuss the properties of Augmented-Dickey-Fuller [ADF] unit root tests for autoregressive processes with a unit or near-unit root in the presence of multiple level shifts of large size. Due to the presence of level shifts, the ADF tests experience severe power losses. We consider new modified ADF unit root tests which require no knowledge of either the location or the number of level shifts. The tests are based on a two-step procedure where possible level shifts are initially detected using the level shift indicator estimators suggested by Chen and Tiao (1990, Journal of business and Economics Statistics) and Chen and Liu (1993, Journal of the American Statistical Association), and later removed by a novel procedure which is denoted as “de-jumping”. Using a Monte Carlo experiment we show that the new tests, although partially oversized in samples of moderate size, have much higher power than standard ADF tests.

TESTING FOR UNIT ROOTS IN AUTOREGRESSIONS WITH MULTIPLE LEVEL SHIFTS

The asymptotic distributions of augmented Dickey–Fuller (ADF) unit root tests for autoregressive processes with a unit or near-unit root are discussed in the presence of multiple stochastic level shifts of large size occurring independently in time. The distributions depend on a Brownian motion and a Poisson-type jump process. Due to the latter, tests based on standard critical values experience power losses increasing rapidly with the number and the magnitude of the shifts. A new approach to unit root testing is suggested which requires no knowledge of either the location or the number of level shifts, and which dispenses with the assumption of independent shift occurrence. It is proposed to remove possible shifts from a time series by weighting its increments according to how likely it is, with respect to an ad hoc postulated distribution, a shift to have occurred in each period. If the number of level shifts is bounded in probability, the limiting distributions of the proposed test statistics coincide with those of ADF statistics under standard conditions. A Monte Carlo experiment shows that, despite their generality, the new tests perform well in finite samples.We are grateful to Pentti Saikkonen (the co-editor) and two referees for their helpful and constructive comments on earlier versions of this paper. We also thank Soren Johansen, Helmut Lutkepohl, Anders Rahbek, Robert Taylor, and seminar participants at the European University Institute, the University of Copenhagen, and the URCT Conference held in Faro, Portugal, September 29 to October 1, 2005, for useful comments. Both authors are grateful to the Danish Social Sciences Research Council, project 2114-04-001, for financial support.

A New Application of Exact Nonparametric Methods to Long-Horizon Predictability Tests

Empirical results from long-horizon regression tests have been influential in the finance literature. Yet, it has come to be understood that traditional long-horizon tests may be unreliable in finite samples when regressors are persistent and when the horizon is long relative to sample size. Recent research has provided valid alternative inference procedures in long-horizon regression in the case for which the regressor follows a near-unit root autoregressive process. However, in small samples, such processes may sometimes be difficult to distinguish with confidence from other persistent data generating processes, such as those displaying long-memory or structural breaks. In this paper, we demonstrate a simple means by which existing nonparametric sign and signed rank tests may be applied to provide exact inference in long-horizon predictive tests, without requiring any modeling assumptions on the regressor. Employing this robust approach, we find evidence of stock return predictability at moderate horizons using short-term interest rates, but little evidence of either short or long-run predictability using dividend-price ratios.

Testing forward rate unbiasedness allowing for persistent regressors

Abstract Standard spot return/forward premium regressions have long been known to provide a strong rejection of unbiasedness. However, due to the strong autocorrelation in the forward premium, which shows estimated autoregressive roots close and in some cases statistically indistinguishable from one, recent literature has cast doubt on the finite sample accuracy of these tests. In fact, finite sample size distortion has now come to be considered as one of several possible explanations behind the forward premium puzzle. In order to pursue this possibility further, we revisit the unbiasedness hypothesis using more appropriate inference procedures. In particular, rather than relying on standard stationarity-based asymptotics, we model the forward premium as a near-unit root process and then test unbiasedness using the bounds tests of Cavanagh et al. (1995) [Cavanagh, C.L., Elliott, G., Stock, J.H., 1995. Inference in models with nearly integrated regressors. Econometric Theory 11, 1131-1147.], which are explicitly designed to provide accurate size under near-unit root assumptions. To summarize our empirical findings, confidence intervals on the largest root confirm uncertainty regarding the stationarity/nonstationarity of the forward premium. However, estimates of the error correlation suggest only modest simultaneity bias. Consequently, we can still reject unbiasedness at the 5% level, even when using appropriately sized bounds tests. This evidence tends to suggest that the forward premium puzzle is more robust than previously imagined. It would be interesting in further work to explore to what extent such conclusions extend to alternative characterizations of the persistence in the forward premium, such as long-memory and structural break models.

Testing for sign and amplitude asymmetries using threshold autoregressions

This paper uses threshold autoregressions to characterize asymmetries in adjustment dynamics and develops likelihood ratio tests to detect them. A robust bootstrap technique is proposed to circumvent the problem that the asymptotic distribution of the test statistics is non-standard. Monte Carlo simulations show that the bootstrap tests are correctly sized and are robust to near-unit roots and non-Gaussian errors. Their power is reasonable, improves sharply with the time series length and remains satisfactory for smooth transition autoregressions. Our approach combined with nonparametric tests can discriminate between asymmetries in adjustment dynamics and in innovations. An application to four monthly US dollar spot returns provides evidence of amplitude asymmetry.

Forecasting autoregressive time series with bias-corrected parameter estimators

The parameter estimators of autoregressive (AR) models are biased in small samples, and these biases can adversely affect their forecast accuracy. The purpose of this paper is to evaluate the effect of bias-correction for AR parameter estimators on forecast accuracy. The bias-corrected parameter estimators considered include a bootstrap mean bias-corrected estimator similar to [ The Review of Economics and Statistics, 80 (1998) 218], the bootstrap approximately median bias-corrected estimator, the modified estimator of [ Journal of Business & Economic Statistics, 19(4) (2001) 482], and the approximately median-unbiased estimator of [ Journal of Business & Economic Statistics, 12 (1994) 187]. Monte Carlo simulations are conducted for AR models with linear time trend. It is found that all bias-corrected estimators can deliver a substantial gain of forecast accuracy for unit root or near-unit root AR models, especially when the sample size is small. Overall, the bootstrap mean bias-corrected estimator is found to provide more accurate forecasts than the other alternatives over a wider range of the parameter space.

Bootstrap prediction intervals for autoregressive models of unknown or infinite lag order

AbstractRecent studies on bootstrap prediction intervals for autoregressive (AR) model provide simulation findings when the lag order is known. In practical applications, however, the AR lag order is unknown or can even be infinite. This paper is concerned with prediction intervals for AR models of unknown or infinite lag order. Akaike's information criterion is used to estimate (approximate) the unknown (infinite) AR lag order. Small‐sample properties of bootstrap and asymptotic prediction intervals are compared under both normal and non‐normal innovations. Bootstrap prediction intervals are constructed based on the percentile and percentile‐t methods, using the standard bootstrap as well as the bootstrap‐after‐bootstrap. It is found that bootstrap‐after‐bootstrap prediction intervals show small‐sample properties substantially better than other alternatives, especially when the sample size is small and the model has a unit root or near‐unit root. Copyright © 2002 John Wiley & Sons, Ltd.

New macroeconomic modeling approaches: Hierarchical dynamics and mean-field approximation

This paper proposes hierarchical structures and mean-field approximation schemes as two potentially useful ways of modeling collective behavior in macroeconomics. Economics systems with hierarchically organized state spaces or agents can exhibit the sluggish response patterns typical of economic time series with unit or near-unit roots. We describe how such hierarchies may arise in several ways, and how dynamics can exhibit behavior with multiple time scales. Then, we describe the use of the mean-field approximation method to model a system composed of many (heterogeneous) agents subject to nonlocalized externalities, and give two brief examples.