This paper proposes a very tractable approach to modeling changes in regime. The parameters of an autoregression are viewed as the outcome of a discrete-state Markov process. For example, the mean growth rate of a nonstationary series may be subject to occasional, discrete shifts. The econometrician is presumed not to observe these shifts directly, but instead must draw probabilistic inference about whether and when they may have occurred based on the observed behavior of the series. The paper presents an algorithm for drawing such probabilistic inference in the form of a nonlinear iterative filter

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.

This paper considers tests for parameter instability and structural change with unknown change point. The results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures. The asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null. The tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant. The tests are found to perform quite well in a Monte Carlo experiment reported elsewhere. Copyright 1993 by The Econometric Society.

/pdf/tests-for-parameter-instability-and-structural-change-with-26amgrdoht.pdf

Tests for Parameter Instability and Structural Change with Unknown Change Point.

The asymptotic power envelope is derived for point-optimal tests of a unit root in the autoregressive representation of a Gaussian time series under various trend specifications. We propose a family of tests whose asymptotic power functions are tangent to the power envelope at one point and are never far below the envelope. When the series has no deterministic component, some previously proposed tests are shown to be asymptotically equivalent to members of this family. When the series has an unknown mean or linear trend, commonly used tests are found to be dominated by members of the family of point-optimal invariant tests. We propose a modified version of the Dickey-Fuller t test which has substantially improved power when an unknown mean or trend is present. A Monte Carlo experiment indicates that the modified test works well in small samples.

/pdf/efficient-tests-for-an-autoregressive-unit-root-15aexwgsnj.pdf

Efficient Tests for an Autoregressive Unit Root

This book is downloadable from http://www.irisa.fr/sisthem/kniga/. Many monitoring problems can be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. The main goal of this book is to describe a unified framework for the design and the performance analysis of the algorithms for solving these change detection problems. Also the book contains the key mathematical background necessary for this purpose. Finally links with the analytical redundancy approach to fault detection in linear systems are established. We call abrupt change any change in the parameters of the system that occurs either instantaneously or at least very fast with respect to the sampling period of the measurements. Abrupt changes by no means refer to changes with large magnitude; on the contrary, in most applications the main problem is to detect small changes. Moreover, in some applications, the early warning of small - and not necessarily fast - changes is of crucial interest in order to avoid the economic or even catastrophic consequences that can result from an accumulation of such small changes. For example, small faults arising in the sensors of a navigation system can result, through the underlying integration, in serious errors in the estimated position of the plane. Another example is the early warning of small deviations from the normal operating conditions of an industrial process. The early detection of slight changes in the state of the process allows to plan in a more adequate manner the periods during which the process should be inspected and possibly repaired, and thus to reduce the exploitation costs.

Detection of abrupt changes: theory and application

The scan statistic is commonly used to test if a one dimensional point process is purely random, or if any clusters can be detected. Here it is simultaneously extended in three directions:(i) a spatial scan statistic for the detection of clusters in a multi-dimensional point process is proposed, (ii) the area of the scanning window is allowed to vary, and (iii) the baseline process may be any inhomogeneous Poisson process or Bernoulli process with intensity pro-portional to some known function. The main interest is in detecting clusters not explained by the baseline process. These methods are illustrated on an epidemiological data set, but there are other potential areas of application as well.

A spatial scan statistic

SUMMARY We wish to test a simple hypothesis against a family of alternatives indexed by a one-dimensional parameter, 0. We use a test derived from the corresponding family of test statistics appropriate for the case when 0 is given. Davies (1977) introduced this problem when these test statistics had normal distributions. The present paper considers the case when their distribution is chi-squared. The results are applied to the detection of a discrete frequency component of unknown frequency in a time series. In addition quick methods for finding approximate significance probabilities are given for both the normal and chi-squared cases and applied to the two-phase regression problem in the normal case.

Hypothesis testing when a nuisance parameter is present only under the alternative

SUMMARY We consider the power of tests for distinguishing between fractional Gaussian noise and white noise of a first-order autoregressive process. Our tests are based on the beta-optimal principle (Davies, 1969), local optimality and the rescaled range test.

https://www.jstor.org/stable/pdfplus/2336024.pdf

Tests for Hurst effect

Hypothesis Testing when a Nuisance Parameter is Present Only Under the Alternatives

The results of Davies (1977, 1987) are extended to a linear model situation with unknown residual variance.

Hypothesis testing when a nuisance parameter is present only under the alternative: Linear model case

SUMMARY A method is described for finding a bound on the error when a version of the usual characteristic function inversion formula is evaluated by numerical integration. The method is applied to the calculation of the distribution function of a quadratic form in normal random variables.

Robert B. Davies

Papers

Hypothesis testing when a nuisance parameter is present only under the alternative

Tests for Hurst effect

Hypothesis Testing when a Nuisance Parameter is Present Only Under the Alternatives

Hypothesis testing when a nuisance parameter is present only under the alternative: Linear model case

Numerical inversion of a characteristic function