Efficient Tests of the Seasonal Unit Root Hypothesis*

In this paper we derive, under the assumption of Gaussian errors with known error covariance matrix, asymptotic local power bounds for seasonal unit root tests for both known and unknown deterministic scenarios and for an arbitrary seasonal aspect. We demonstrate that the optimal test of a unit root at a given spectral frequency behaves asymptotically independently of whether unit roots exist at other frequencies or not. We also develop modified versions of the optimal tests which attain the asymptotic Gaussian power bounds under much weaker conditions. We further propose near-efficient regression-based seasonal unit root tests using pseudo-GLS de-trending and show that these have limiting null distributions and asymptotic local power functions of a known form. Monte Carlo experiments indicate that the regression-based tests perform well in finite samples.Point optimal invariant (seasonal) unit root tests, asymptotic local power bounds, near seasonal integration

Homotopy Lie Superalgebra in Yang-Mills Theory

The Yang-Mills equations are formulated in the form of generalized Maurer-Cartan equations, such that the corresponding algebraic operations are shown to satisfy the defining relations of homotopy Lie superalgebra.Comment: LaTeX2e, 10 page

Simple, Robust and Powerful Tests of the Breaking Trend Hypothesis*

In this paper we develop a simple procedure which delivers tests for the presence of a broken trend in a univariate time series which do not require knowledge of the form of serial correlation in the data and are robust as to whether the shocks are generated by an I(0) or an I(1) process. Two trend break models are considered: the first holds the level fixed while allowing the trend to break, while the latter allows for a simultaneous break in level and trend. For the known break date case our proposed tests are formed as a weighted average of the optimal tests appropriate for I(0) and I(1) shocks. The weighted statistics are shown to have standard normal limiting null distributions and to attain the Gaussian asymptotic local power envelope, in each case regardless of whether the shocks are I(0) or I(1). In the unknown break date case we adopt the method of Andrews (1993) and take a weighted average of the statistics formed as the supremum over all possible break dates, subject to a trimming parameter, in both the I(0) and I(1) environments. Monte Carlo evidence suggests that our tests are in most cases more powerful, often substantially so, than the robust broken trend tests of Sayginsoy and Vogelsang (2004). An empirical application highlights the practical usefulness of our proposed tests.Broken trend, power envelope, unit root, stationarity tests

Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

A number of recently published papers have focused on the problem of testing for a unit root in the case where the driving shocks may be unconditionally heteroskedastic. These papers have, however, assumed that the lag length in the unit root test regression is a deterministic function of the sample size, rather than data-determined, the latter being standard empirical practice. In this paper we investigate the finite sample impact of unconditional heteroskedasticity on conventional data-dependent methods of lag selection in augmented Dickey-Fuller type unit root test regressions and propose new lag selection criteria which allow for the presence of heteroskedasticity in the shocks. We show that standard lag selection methods show a tendency to over-fit the lag order under heteroskedasticity, which results in significant power losses in the (wild bootstrap implementation of the) augmented Dickey-Fuller tests under the alternative. The new lag selection criteria we propose are shown to avoid this problem yet deliver unit roots with almost identical finite sample size and power properties as the corresponding tests based on conventional lag selection methods when the shocks are homoskedastic.Unit root test, Lag selection, Information criteria, Wild bootstrap, Nonstationary volatility

Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

A number of recently published papers have focused on the problem of testing for a unit root inthe case where the driving shocks may be unconditionally heteroskedastic. These papers have,however, assumed that the lag length in the unit root test regression is a deterministic functionof the sample size, rather than data-determined, the latter being standard empirical practice. Inthis paper we investigate the finite sample impact of unconditional heteroskedasticity onconventional data-dependent methods of lag selection in augmented Dickey-Fuller type unit roottest regressions and propose new lag selection criteria which allow for the presence ofheteroskedasticity in the shocks. We show that standard lag selection methods show a tendency toover-fit the lag order under heteroskedasticity, which results in significant power losses in the(wild bootstrap implementation of the) augmented Dickey-Fuller tests under the alternative. Thenew lag selection criteria we propose are shown to avoid this problem yet deliver unit root testswith almost identical finite sample size and power properties as the corresponding tests based onconventional lag selection methods when the shocks are homoskedastic.econometrics;

Testing for a change in persistence in the presence of non-stationary volatility

In this paper we consider tests for the null of (trend-) stationarity against the alternative of a change in persistence at some (known or unknown) point in the observed sample, either from I(0) to I(1) behaviour or vice versa, of, inter alia, (Kim, J., 2000. Detection of change in persistence of a linear time series. Journal of Econometrics 95, 97–116). We show that in circumstances where the innovation process displays non-stationary unconditional volatility of a very general form, which includes single and multiple volatility breaks as special cases, the ratio-based statistics used to test for persistence change do not have pivotal limiting null distributions. Numerical evidence suggests that this can cause severe over-sizing in the tests. In practice it may therefore be hard to discriminate between persistence change processes and processes with constant persistence but which display time-varying unconditional volatility. We solve the identified inference problem by proposing wild bootstrap-based implementations of the tests. Monte Carlo evidence suggests that the bootstrap tests perform well in finite samples. An empirical illustration using US price inflation data is provided

Bootstrap determination of the co-integration rank in VAR models

This paper discusses a consistent bootstrap implementation of the likelihood ratio [LR] co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying VAR model which obtain under the reduced rank null hypothesis. A full asymptotic theory is provided which shows that, unlike the bootstrap procedure in Swensen (2006) where a combination of unrestricted and restricted estimates from the VAR model is used, the resulting bootstrap data are I(1) and satisfy the null co-integration rank, regardless of the true rank. This ensures that the bootstrap LR test is asymptotically correctly sized and that the probability that the bootstrap sequential procedure selects a rank smaller than the true rank converges to zero. Monte Carlo evidence suggests that our bootstrap procedures work very well in practice

Bootstrap determination of the co-integration rank in VAR models

This paper discusses a consistent bootstrap implementation of the likelihood ratio [LR] co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying VAR model which obtain under the reduced rank null hypothesis. A full asymptotic theory is provided which shows that, unlike the bootstrap procedure in Swensen (2006) where a combination of unrestricted and restricted estimates from the VAR model is used, the resulting bootstrap data are I(1) and satisfy the null co-integration rank, regardless of the true rank. This ensures that the bootstrap LR test is asymptotically correctly sized and that the probability that the bootstrap sequential procedure selects a rank smaller than the true rank converges to zero. Monte Carlo evidence suggests that our bootstrap procedures work very well in practice

Wild bootstrap of the mean in the infinite variance case

It is well known that the standard i.i.d. bootstrap of the mean is inconsistent in a location model with infinite variance (alfa-stable) innovations. This occurs because the bootstrap distribution of a normalised sum of infinite variance random variables tends to a random distribution. Consistent bootstrap algorithms based on subsampling methods have been proposed but have the drawback that they deliver much wider confidence sets than those generated by the i.i.d. bootstrap owing to the fact that they eliminate the dependence of the bootstrap distribution on the sample extremes. In this paper we propose sufficient conditions that allow a simple modification of the bootstrap (Wu, 1986, Ann.Stat.) to be consistent (in a conditional sense) yet to also reproduce the narrower confidence sets of the i.i.d. bootstrap. Numerical results demonstrate that our proposed bootstrap method works very well in practice delivering coverage rates very close to the nominal level and significantly narrower confidence sets than other consistent methods

On causality, unitarity and perturbative expansions

We present a pedagogical case study how to combine micro-causality and unitarity based on a perturbative approach. The method we advocate constructs an analytic extrapolation of partial-wave scattering amplitudes that is constrained by the unitarity condition. Suitably constructed conformal mappings help to arrive at a systematic approximation of the scattering amplitude. The technique is illustrated at hand of a Yukawa interaction. The typical case of a superposition of strong short-range and weak long-range forces is investigated.Comment: 12 pages, 12 figure