Testing for seasonal unit roots by frequency domain regression

This paper develops univariate seasonal unit root tests based on spectral regression estimators. An advantage of the frequency domain approach is that it enables serial correlation to be treated non-parametrically. We demonstrate that our proposed statistics have pivotal limiting distributions under both the null and near seasonally integrated alternatives when we allow for weak dependence in the driving shocks. This is in contrast to the popular seasonal unit root tests of, among others, Hylleberg et al. (1990) which treat serial correlation parametrically via lag augmentation of the test regression. Moreover, our analysis allows for (possibly infinite order) moving average behaviour in the shocks, while extant large sample results pertaining to the Hylleberg et al. (1990) type tests are based on the assumption of a finite autoregression. The size and power properties of our proposed frequency domain regression-based tests are explored and compared for the case of quarterly data with those of the tests of Hylleberg et al. (1990) in simulation experiments.Seasonal unit root tests; moving average; frequency domain regression; spectral density estimator; Brownian motion

Watching the watchmen::a statistical analysis of mark consistency across taught modules

AbstractVerifying that taught modules are marked and taught to a common standard is important but doing so by comparing mean module marks is inadequate when students’ ability is not uniform across these modules. For example, a module taken by a group of students of above average ability may justifiably result in a high mean mark, without implying that inconsistent standards have been applied. We propose a modified version of the fixed effects regression that provides direct estimates of module mark biases while conditioning for student composition and requiring no additional, potentially confidential, information on students or staff. We describe how this modified fixed effects regression can be implemented on a set of student marks and how the results can be interpreted. Increases in student numbers and tuition fees have increased the preoccupation with, and monitoring of, marks. We show how one can generate statistics that are more informative of the biases in marking, while being explicit about their limitations

CYCLICAL TRENDS IN CONTINUOUS TIME MODELS

It is undoubtedly desirable that econometric models capture the dynamic behavior, like trends and cycles, observed in many economic processes. Building models with such capabilities has been an important objective in the continuous time econometrics literature, for instance, the cyclical growth models of Bergstrom (1966); the economy-wide macroeconometric models of, for example, Bergstrom and Wymer (1976); unobserved stochastic trends of Harvey and Stock (1988 and 1993) and Bergstrom (1997); and differential-difference equations of Chambers and McGarry (2002). This paper considers continuous time cyclical trends, which complement the trend-plus-cycle models in the unobserved components literature but could also be incorporated into Bergstrom type systems of differential equations, as were stochastic trends in Bergstrom (1997).

On the Asymptotic Properties of a Feasible Estimator of the Continuous Time Long Memory Parameter

This paper considers a fractional noise model in continuous time and examines the asymptotic properties of a feasible frequency domain maximum likelihood estimator of the long memory parameter. The feasible estimator is one that maximises an approximation to the likelihood function (the approximation arises from the fact that the spectral density function involves the finite truncatin of an infinite summation). It is of interest therefore to explore the conditions required of this approximation to ensure the consistency and asymptotic normality of this estimator. It is shown that the truncation parameter has to be a function of the sample size and that the optimal rate is different for stocks and flows and is a function of the long memory parameter itself. The results of a simulation exercise are provided to assess the small sample properties of the estimator.Continuous time models, long memory processes

Cyclical Trends in Continuous Time Models

It is undoubtedly desirable that econometric models capture the dynamic behaviour,like trends and cycles, observed in many economic processes. Building models with such capabilities has been an important objective in the continuous time econometrics literature, see for instance the cyclical growth models of Bergstrom (1966), the complete economy-wide macroeconometric models of, for example, Bergstrom and Wymer (1976), unobserved stochastic trends of Harvey and Stock (1988 and 1993) and Bergstrom (1997), and differential-difference equations of Chambers and McGarry (2002). This paper’s contribution is to examine cyclical trends formulated in continuous time, which complement the trend-plus-cycle models that are frequently used in the unobserved components literature.Cyclical Trends, continuous time models, stochastic differential equations, differential-difference equations

ESTIMATION OF DIFFERENTIAL-DIFFERENCE EQUATION SYSTEMS WITH UNKNOWN LAG PARAMETERS

This paper considers the estimation of the parameters of general systems of stochastic differential-difference equations in which the lag parameters themselves are treated as unknown and are not restricted to be integers and therefore form part of the parameter vector to be estimated. The asymptotic properties of an infeasible frequency domain maximum likelihood estimator are established in addition to those of a feasible version based on truncating an infinite series that arises in the computation of the spectral density function of the observed discrete time series. Precise conditions that the truncation parameter must satisfy for the asymptotic results to hold are provided. © 2006 Cambridge University Press

Testing for seasonal unit roots by frequency domain regression

This paper develops univariate seasonal unit root tests based on spectral regression estimators. An advantage of the frequency domain approach is that it enables serial correlation to be treated non-parametrically. We demonstrate that our proposed statistics have pivotal limiting distributions under both the null and near seasonally integrated alternatives when we allow for weak dependence in the driving shocks. This is in contrast to the popular seasonal unit root tests of, among others, Hylleberg et al. (1990) which treat serial correlation parametrically via lag augmentation of the test regression. Our analysis allows for (possibly infinite order) moving average behaviour in the shocks. The size and power properties of our proposed frequency domain regression-based tests are explored and compared for the case of quarterly data with those of the tests of Hylleberg et al. (1990) in simulation experiments