The Effects of Sampling Frequency on Detrending Methods for Unit Root Tests

This paper analyses the effects of sampling frequency on detrending methods based on an underlying continuous time representation of the process of interest. Such an approach has the advantage of allowing for the explicit - and different - treatment of the ways in which stock and flow variables are actually observed. Some general results are provided before the focus turns to three particular detrending methods that have found widespread use in the conduct of tests for a unit root, these being GLS detrending, OLS detrending, and first differencing, and which correspond to particular values of the generic detrending parameter. In addition, three different scenarios concerning sampling frequency and data span, in each of which the number of observations increases, are considered for each detrending method. The limit properties of the detrending coeffcient estimates, as well as an invariance principle for the detrended variable, are derived. An example of the application of the techniques to testing for a unit root, using GLS detrending on an intercept, is provided and the results of a simulation exercise to analyse the size and power properties of the test in the three different sampling scenarios are reported

The Calculation of Some Limiting Distributions Arising in Near-Integrated Models with GLS Detrending

Many unit root test statistics are nowadays constructed using detrended data, with the method of GLS detrending being popular in the setting of a near-integrated model. This paper determines the properties of some associated limiting distributions when the GLS detrending is based on a linear time trend. A fundamental result for the moment generating function of two key functionals of the relevant stochastic process is provided and used to compute probability density functions and cumulative distribution functions, as well as means and variances, of the limiting distributions of some statistics of interest. Some further applications, including a comparison of limiting power functions and the consideration of a more complicated statistic, are also provided

Jackknife Estimation of Stationary Autoregressive Models

This paper reports the results of an extensive investigation into the use of the jackknife as a method of estimation in stationary autoregressive models. In addition to providing some general theoretical results concerning jackknife methods it is shown that a method based on the use of non-overlapping sub-intervals is found to work particularly well and is capable of reducing bias and root mean squared error (RMSE) compared to ordinary least squares (OLS), subject to a suitable choice of the number of sub-samples, rules-of-thumb for which are provided. The jackknife estimators also outperform OLS when the distribution of the disturbances departs from normality and when it is subject to autoregressive conditional heteroskedasticity. Furthermore the jackknife estimators are much closer to being median-unbiased than their OLS counterparts

Frequency Domain Estimation of Cointegrating Vectors with Mixed Frequency and Mixed Sample Data

This paper proposes a model suitable for exploiting fully the information contained in mixed frequency and mixed sample data in the estimation of cointegrating vectors. The asymptotic properties of easy-to-compute spectral regression estimators of the cointegrating vectors are derived and these estimators are shown to belong to the class of optimal cointegration estimators. Furthermore, Wald statistics based on these estimators have asymptotic chi-square distributions which enable inferences to be made straightforwardly. Simulation experiments suggest that the finite sample performance of a spectral regression estimator in an augmented mixed frequency model is particularly encouraging as it is capable of dramatically reducing the root mean squared error obtained in an entirely low frequency model to the levels comparable to an infeasible high frequency model. The finite sample size and power properties of the Wald statistic are also found to be good. An empirical example, to stock price and dividend data, is provided to demonstrate the methods in practice

Testing for Unit Roots with Flow Data and Varying Sampling Frequency

This paper considers tests for a unit root in a flow variable when the span of data and/or the sampling frequency are allowed to vary. The limiting distributions of the statistics are obtained under both the null and alternative hypotheses, thereby enabling an analysis of the consistency properties of the tests to be conducted. Contrary to the situation with a stock variable, it is found that it is possible to consistently test for a unit root in a flow variable even when the span of the data is fixed, and, furthermore, that increasing the span of the data is not in itself sufficient for consistent testing. Some new simulation results are provided while the theoretical results obtained help to explain recent simulation findings by other authors involving unit root tests with flow variables

Cointegration and Sampling Frequency

This paper analyses the effects of sampling frequency on the properties of spectral regression estimators of cointegrating parameters. Large sample asymptotic properties are derived under three scenarios concerning the span of data and sampling frequency, each scenario depending on whether span or frequency (or both) tends to infinity. The limiting distributions are shown to be different in each case. Furthermore, the asymptotic efficiency of the estimators obtained with a fixed sampling frequency is compared with that obtained with a continuous record of data, and it is shown that the only inefficiencies arise with respect to stock variables. Some simulation results and an empirical illustration are also provided

Jackknife Bias Reduction in the Presence of a Unit Root

This paper analyses the properties of jackknife estimators of the first-order autoregressive coefficient when the time series of interest contains a unit root. It is shown that, when the sub-samples do not overlap, the sub-sample estimators have different limiting distributions from the full-sample estimator and, hence, the jackknife estimator in its usual form does not eliminate fully the first-order bias as intended. The joint moment generating function of the numerator and denominator of these limiting distributions is derived and used to calculate the expectations that determine the optimal jackknife weights. Two methods of avoiding this procedure are proposed and investigated, one based on inclusion of an intercept in the regressions, the other based on adjusting the observations in the sub-samples. Extensions to more general augmented Dickey-Fuller (ADF) regressions are also considered. In addition to the theoretical results extensive simulations reveal the impressive bias reductions that can be obtained with these computationally simple jackknife estimators and they also highlight the importance of correct lag-length selection in ADF regressions

Continuous Time Modelling Based on an Exact Discrete Time Representation

This chapter provides a survey of methods of continuous time modelling based on an exact discrete time representation. It begins by highlighting the techniques involved with the derivation of an exact discrete time representation of an underlying continuous time model,providing specificc details for a second-order linear system of stochastic differential equations. Issues of parameter identification, Granger causality, nonstationarity, and mixed frequency data are addressed, all being important considerations in applications in economics and other disciplines. Although the focus is on Gaussian estimation of the exact discrete time model, alternative time domain (state space) and frequency domain approaches are also discussed. Computational issues are explored and two new empirical applications are included along with a discussion of applications in the field of macroeconometric modelling