Beong o So

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.

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.

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.

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.

Recursive mean adjustment and tests for nonstationarities

Abstract Recursive mean adjustment of Shin and So [Journal of Time Series Analysis 22 (2001) 595] and So and Shin (Statistics and Probability Letters 43 (1999) 65] is considered for inference on nonstationarities. The approach is shown to be versatile in that it can be applied to a wide class of tests for nonstationarities such as testing unit roots in nonlinear time series models, testing cointegrations, testing double unit roots, and testing seasonal unit roots. In all of the testing problems, recursive mean adjustment gives us tests with substantially higher powers than existing tests based on the ordinary mean adjustment.

New tests for unit roots in autoregressive processes with possibly infinite variance errors

For autoregressive processes with possibly infinite variance innovations, tests for unit roots are constructed. The limiting null distributions of the test statistics are standard normal both for finite variance innovations and for infinite variance innovations. The test statistics are the pivotal statistics of modified M-estimators in which the signs of regressors rather than the regressors themselves are used as instrumental variables in estimating unit roots. A Monte-Carlo experiment compares the proposed tests favorably with tests based on the OLSE and tests based on the M-estimators for several innovations.

A new instrumental variable estimation for diffusion processes

We consider the problem of parametric inference from continuous sample paths of the diffusion processes {x(t)} generated by the system of possibly nonstationary and/or nonlinear Ito stochastic differential equations. We propose a new instrumental variable estimator of the parameter whose pivotal statistic has a Gaussian distribution for all possible values of parameter. The new estimator enables us to construct exact level-α confidence intervals and tests for the parameter in the possibly non-stationary and/or nonlinear diffusion processes. Applications to several non-stationary and/or nonlinear diffusion processes are considered as examples.

A new robust sign test for cointegration

We propose new tests for cointegration based on signs of the residuals of the conventional t-test. Our tests have the limiting normal distribution under the null hypothesis and are robust to heavy tailed disturbances. A Monte-Carlo simulation shows the new tests have a stable size property and are locally more powerful than that of Engle and Granger (1987) for heavy tailed error distribution.