Dong Wan Shin

recursive Mean Adjustment for Unit Root Tests

For unit root tests, we propose a new mean adjustment scheme, called recursive mean adjustment. For adjusting the mean of an observation at a time t, instead of the global sample mean, we use the recursive sample mean which is the sample mean of the observations up to the time t. The approach is simple and can be applied to a wide class of unit root tests. The recursive mean adjustment gives us tests with substantially higher powers compared with the tests based on the ordinary mean adjustment.

Recursive mean adjustment in time-series inferences

When time-series data are positively autocorrelated, mean adjustment using the overall sample mean causes biases for sample autocorrelations and parameter estimates, which decreases the coverage probabilities of confidence intervals. A new method for mean adjustment is proposed, in which a datum at a time is adjusted for the mean through the partial sample mean, the average of data up to the time point. The method is simple and reduces the biases of the parameter estimators and the sample autocorrelations when data are positively autocorrelated. The empirical coverage probabilities of the confidence intervals of the autoregressive coefficient become quite close to the nominal level.

Unit root tests for time series with outliers

Effects of additive and innovational outliers on unit root tests in ARIMA(p, 1, q) models are investigated. The limiting distribution of the ordinary least-squares estimator of the unit root parameter in the AR(1) model is affected by additive outliers but is unaffected by innovational outliers. To test for a unit root in ARIMA(p, 1, q) models in the presence of outliers, a very simple, easy-to-compute procedure is given that detects additive outliers and adjusts the observations accordingly. The detection method performed well in our numerical experiment. Our unit root tests based on the adjusted data are shown to have very good empirical sizes and powers in AR(1), AR(2) and ARMA(1, 1) models.

CAUCHY ESTIMATORS FOR AUTOREGRESSIVE PROCESSES WITH APPLICATIONS TO UNIT ROOT TESTS AND CONFIDENCE INTERVALS

For autoregressive processes, we propose new estimators whose pivotal statistics have the standard normal limiting distribution for all ranges of the autoregressive parameters. The proposed estimators are approximately median unbiased. For seasonal time series, the new estimators give us unit root tests that have limiting normal distribution regardless of period of the seasonality. Using the estimators, confidence intervals of the autoregressive parameters are constructed. A Monte-Carlo simulation for first-order autoregressions shows that the proposed tests for unit roots are locally more powerful than the tests based on the ordinary least squares estimators. It also shows that the proposed confidence intervals have shorter average lengths than those of Andrews (1993, Econometrica 61, 139–165) based on the ordinary least squares estimators when the autoregressive coefficient is close to one.

Tests for Asymmetry in Possibly Nonstationary Time Series Data

Tests for asymmetric adjustment in possibly nonstationary, nearly nonstationary, or stationary time series data are developed. The asymmetry is modeled by the momentum threshold autoregressive model of Enders and Granger and an extension of it. The tests are t-type tests and Wald tests based on instrumental-variable estimators and are asymptotically normal or chi-squared regardless of stationarity/nonstationarity of data-generating processes. This is in contrast to the fact that the ttests and the Wald tests based on the ordinary least squares estimator (OLSE) are asymptotically normal and chisquared, respectively, only under stationarity and are thus statistically invalid under nonstationarity. A Monte Carlo simulation shows that the proposed tests have stable sizes. Powers of the proposed tests against stationary alternatives are comparable to those of the OLSE-based tests. The Monte Carlo study also shows that the new estimators are less biased than the OLSE when data-generating processes are random walks. The proposed tests are applied to a monthly U.K. interest-rate dataset to find evidences for asymmetry in directions of adjustments as well as in amounts of adjustments.

An invariant sign test for random walks based on recursive median adjustment

Abstract We propose a new invariant sign test for random walks against general stationary processes and develop a theory for the test. In addition to the exact binomial null distribution of the test, we establish various important properties of the test: the consistency against a wide class of possibly nonlinear stationary autoregressive conditionally heteroscedastic processes and/or heavy-tailed errors; a local asymptotic power advantage over the classical Dickey–Fuller test; and invariance to monotone data transformations, to conditional heteroscedasticity and to heavy-tailed errors. Using the sign test, we also investigate various interrelated issues such as M-estimator, exact confidence interval, sign test for serial correlation, robust inference for a cointegration model, and discuss possible extensions to models with autocorrelated errors. Monte-Carlo experiments verify that the sign test has not only very stable sizes but also locally better powers than the parametric Dickey–Fuller test and the nonparametric tests of Granger and Hallman (1991. Journal of Time Series Analysis 12, 207–224) and Burridge and Guerre (1996. Econometric Theory 12, 705–719) for heteroscedastic and/or heavy tailed errors.

UNIT ROOT TESTS BASED ON ADAPTIVE MAXIMUM LIKELIHOOD ESTIMATION

Adaptive maximum likelihood estimators of unit roots in autoregressive processes with possibly non-Gaussian innovations are considered. Unit root tests based on the adaptive estimators are constructed. Limiting distributions of the test statistics are derived, which are linear combinations of two functionals of Brownian motions. A Monte Carlo simulation reveals that the proposed tests have improved powers over the classical Dickey–Fuller tests when the distribution of the innovation is not close to normal. We also compare the proposed tests with those of Lucas (1995, Econometric Theory 11, 331–346) based on M-estimators.

Unit Root Tests Based on Unconditional Maximum Likelihood Estimation for the Autoregressive Moving Average

Unconditional maximum likelihood estimation is considered for an autoregressive moving average that may contain an autoregressive unit root. The limiting distribution of the normalized maximum likelihood estimator of the unit root is shown to be the same as that of the estimator for the first-order autoregressive process. A likelihood ratio test based on unconditional maximum likelihood estimation is proposed. In a Monte Carlo study for the autoregressive moving-average model of order (1, 1), the new test is shown to have better size and power than those of several other tests.

Gaussian tests for seasonal unit roots based on Cauchy estimation and recursive mean adjustments

Abstract We propose tests for seasonal unit roots whose limiting null distributions are always standard normal regardless of the period of seasonality and types of mean adjustments. The seasonal models of Dickey, Hasza and Fuller (1984. Journal of American Statistical Association 79, 355–367) (DHF) and Hylleberg, Engle, Granger and Yoo (1990. Journal of Econometrics 44, 215–238) (HEGY) are considered. For estimating parameters related to the seasonal unit roots, regressor signs are used as instrumental variables while recursive sample means are used for adjusting the seasonal means. In addition to normality of the limiting null distributions, in seasonal mean models, the recursive mean adjustment provides the new tests with locally higher powers than those of the existing tests of DHF and HEGY based on the ordinary least-squares estimators. If data have a strong linear time trend, the recursive mean adjustment is a source of both power gains of some tests for local alternatives and power losses of all tests for other alternatives. Limiting normality allow evaluation of p-values and testing joint significance of subsets of seasonal unit roots.

An algorithm for generating correlated random variables in a class of infinitely divisible distributions

A simple algorithm is proposed for generating a set of nonnegatively correlated variables having a specified correlation structure in a certain class of infinitely divisible distributions. These distributions are characterized by a property that they are closed under summation. The correlated variables are expressed as sums of other independent random variables. The algorithm produces simple explicit expressions for typical correlation structures such as AR(1) correlation, banded correlation and symmetric correlation.