Range unit root tests

Since the seminal paper by Dickey and Fuller in 1979, unit-root tests have conditioned the standard approaches to analyse time series with strong serial dependence, the focus being placed in the detection of eventual unit roots in an autorregresive model fitted to the series. In this paper we propose a completely different method to test for the type of "long-wave" patterns observed not only in unit root time series but also in series following more complex data generating mechanism. To this end, our testing device analyses the trend exhibit by the data, without imposing any constraint on the generating mechanism. We call our device the Range Unit Root (RUR) Test since it is constructed from running ranges of the series. These statistics allow a more general characterization of a strong serial dependence in the mean behavior, thus endowing our test with a number of desirable properties. Among these properties are the invariance to nonlinear monotonic transformations of the series and the robustness to the presence of level shifts and additive outliers. In addition, the RUR test outperforms the power of standard unit root tests on near-unit-root stationary time series

Bootstrap Unit Root Tests

We consider the bootstrap unit root tests based on autoregressive integrated models, with or without deterministic time trends. A general methodology is developed to approximate asymptotic distributions for the models driven by integrated time series, and used to obtain asymptotic expansions for the Dickey-Fuller unit root tests. The second-order terms in their expansions are of stochastic orders Op($n^{-1/4}$) and Op($n^{-1/2}$), and involve functionals of Brownian motions and normal random variates. The asymptotic expansions for the bootstrap tests are also derived and compared with those of the Dickey-Fuller tests. We show in particular that the usual nonparametric bootstrap offers asymptotic refinements for the Dickey-Fuller tests, i.e., it corrects their second-order errors. More precisely, it is shown that the critical values obtained by the bootstrap resampling are correct up to the second-order terms, and the errors in rejection probabilities are of order o($n^{-1/2}$) if the tests are based upon the bootstrap critical values. Through simulation, we investigate how effective is the bootstrap correction in small samples.

Bootstrap Unit Root Tests

We consider the bootstrap unit root tests based on finite order autoregressive integrated models driven by iid innovations, with or without deterministic time trends. A general methodology is developed to approximate asymptotic distributions for the models driven by integrated time series, and used to obtain asymptotic expansions for the Dickey-Fuller unit root tests. The second-order terms in their expansions are of stochastic orders Op(n-1/4) and Op(n-1/2), and involve functionals of Brownian motions and normal random variates. The asymptotic expansions for the bootstrap tests are also derived and compared with those of the Dickey-Fuller tests. We show in particular that the bootstrap offers asymptotic refinements for the Dickey-Fuller tests, i.e., it corrects their second-order errors. More precisely, it is shown that the critical values obtained by the bootstrap resampling are correct up to the second-order terms, and the errors in rejection probabilities are of order o(n-1/2) if the tests are based upon the bootstrap critical values. Through simulations, we investigate how effective is the bootstrap correction in small samples.

Nonlinear Mean Reversion across National Stock Markets: Evidence from Emerging Asian Markets

This paper seeks empirical evidence of nonlinear mean-reversion in relative national stock price indices for Emerging Asian countries. It is well known that conventional linear unit root tests suffer from low power against the stationary nonlinear alternative. Implementing the nonlinear unit root tests proposed by Kapetanios, et al. (2003) and Cerrato, et al. (2009) for the relative stock prices of Emerging Asian markets, we find strong evidence of nonlinear mean reversion, whereas linear tests fail to reject the unit root null for most cases. We also report some evidence that stock markets in China and Taiwan are highly localized.Linear Unit Root Test; Nonlinear Unit Root Test; Nonlinear Panel Unit Root Test; International Relative Stock Prices

Regression-based seasonal unit root tests

The contribution of this paper is three-fold. Firstly, a characterisation theorem of the sub-hypotheses comprising the seasonal unit root hypothesis is presented which provides a precise formulation of the alternative hypotheses against which regression-based seasonal unit root tests test. Secondly, it proposes regressionbased tests for the seasonal unit root hypothesis which allow a general seasonal aspect for the data and are similar both exactly and asymptotically with respect to initial values and seasonal drift parameters. Thirdly, limiting distribution theory is given for these statistics where, in contrast to previous papers in the literature, in doing so it is not assumed that unit roots hold at all of the zero and seasonal frequencies. This is shown to alter the large sample null distribution theory for regression t-statistics for unit roots at the complex frequencies, but interestingly to not affect the limiting null distributions of the regression t-statistics for unit roots at the zero and Nyquist frequencies and regression Fstatistics for unit roots at the complex frequencies. Our results therefore have important implications for how tests of the seasonal unit root hypothesis should be conducted in practice. Associated simulation evidence on the size and power properties of the statistics presented in this paper is given which is consonant with the predictions from the large sample theory.Seasonal unit root tests; seasonal drifts; characterisation theorem

Unit Roots, Level Shifts and Trend Breaks in Per Capita Output: A Robust Evaluation

Determining whether per capita output can be characterized by a stochastic trend is complicated by the fact that infrequent breaks in trend can bias standard unit root tests towards non-rejection of the unit root hypothesis. The bulk of the existing literature has focused on the application of unit root tests allowing for structural breaks in the trend function under the trend stationary alternative but not under the unit root null. These tests, however, provide little information regarding the existence and number of trend breaks. Moreover, these tests su¤er from serious power and size distortions due to the asymmetric treatment of breaks under the null and alternative hypotheses. This paper estimates the number of breaks in trend employing procedures that are robust to the unit root/stationarity properties of the data. Our analysis of the per-capita GDP for OECD countries thereby permits a robust classi?cation of countries according to the ?growth shift?, ?level shift? and ?linear trend? hypotheses. In contrast to the extant literature, unit root tests conditional on the presence or absence of breaks do not provide evidence against the unit root hypothesis.growth shift, level shift, structural change, trend breaks, unit root

Nonlinear IV Panel Unit Root Tests

This paper presents the nonlinear IV methodology as an effective inferential basis for nonstationary panels. The nonlinear IV method resolves the inferential difficulties in testing for unit roots arising from the intrinsic heterogeneities and cross-dependencies of panel models. Individual units are allowed to be dependent through correlations among innovations, interrelatedness of short-run dynamics and/or cross-sectional cointegrations. If based on the instrumental variables that are nonlinear transformations of the lagged levels, the usual IV estimation of the augmented Dickey-Fuller type regressions yields asymptotically normal unit root tests for panels with general dependencies and heterogeneities. Moreover, the nonlinear IV estimation allows for the use of covariates to further increase power, and order statistics to test for more flexible forms of hypotheses, which are especially important in heterogeneous panels.

PPP May not Hold for Agricultural Commodities

We use the well known USDA dataset of real exchange rates to address the question of whether PPP holds for agricultural commodities. Both unit root tests and the recently proposed more powerful class of panel unit root tests, which take into account cross-section correlation across the units in the panel, are used. Properties of unit roots and panel tests are analyzed by Monte Carlo simulation. Summarizing, our results show that during the post-Bretton-Woods period of flexible exchange rates, PPP does not hold for agricultural commodities.Key words : Purchasing Power Parity, Agricultural Commodities, Monte Carlo, Unit Root tests, Panel unit root tests.

Bootstrapping Unit Root Tests with Covariates

We consider the bootstrap method for the covariates augmented Dickey-Fuller (CADF) unit root test suggested in Hansen (1995) which uses related variables to improve the power of univariate unit root tests. It is shown that there are substantial power gains from including correlated covariates. The limit distribution of the CADF test, however, depends on the nuisance parameter that represents the correlation between the equation error and the covariates. Hence, inference based directly on the CADF test is not possible. To provide a valid inferential basis for the CADF test, we propose to use the bootstrap procedure to obtain critical values, and establish the asymptotic validity of the bootstrap CADF test. Simulations show that the bootstrap CADF test significantly improves the finite sample size performances of the CADF test, especially when the covariates are highly correlated with the error. Indeed, the bootstrap CADF test offers drastic power gains over the conventional ADF test. We apply our testing procedures to the extended Nelson-Plosser data set for the post-1929 samples as well as postwar annual CPI-based real exchange rates for 14 OECD countries.

Unit Root Tests with Markov-Switching

Diba and Grossman (1988) and Hamilton and Whiteman (1985) recommended unit root tests for rational bubbles. They argued that if stock prices are not more explosive than dividends, then it can be concluded that rational bubbles are not present. Evans (1991) demonstrated that these tests will fail to detect the class of rational bubbles which collapse periodically. When such bubbles are present, stock prices will not appear to be more explosive than the dividends on the basis of these tests, even though the bubbles are substantial in magnitude and volatility. Hall et al. (1999) show that the power of unit root test can be improved substantially when the underlying process of the sample observations is allowed to follow a first-order Markov process. Our paper applies unit root tests to the property prices of Hong Kong and Seoul, allowing for the data generating process to follow a three states Markov chain. The null hypothesis of unit root is tested against the explosive bubble or stable alternative. Simulation studies are used to generate the critical values for the one-sided test. The time series used in the tests are the monthly price and rent indices of Seoul's housing (1986:1 to 2003:6) and Hong Kong's retail premise (1980:12 to 2003:1). The investigations show that only one state appears to be highly likely in both cases. The switching unit root tests failed to find explosive bubbles in the price series, which might be due to the fact that the power of test is weak in the presence of heteroscedasticityunit root, three states markov switching, explosive rational bubbles