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.

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.

Testing for a Unit Root in a Near-Integrated Model with Skip-Sampled Data

This article examines tests for a unit root in skip‐sampled data. A generalization of the usual discrete time framework that allows for a continuous time detrending procedure prior to estimation of the resulting discrete time dynamic model that embodies exactly the restrictions imposed by the process of temporal aggregation is proposed. A simulation study reveals that taking these restrictions into account can yield improved size and power properties compared to a statistic based on a model that ignores the temporal aggregation, and an empirical illustration of the methods using monthly producer price data for the UK and the USA is provided. Further avenues for investigation in future work are also highlighted

Jackknife bias reduction in the presence of a near-unit root

This paper considers the specification and performance of jackknife estimators of the autoregressive coefficient in a model with a near-unit root. The limit distributions of sub-sample estimators that are used in the construction of the jackknife estimator are derived and the joint moment generating function (MGF) of two components of these distributions is obtained and its properties are explored. The MGF can be used to derive the weights for an optimal jackknife estimator that removes fully the first-order finite sample bias from the estimator. The resulting jackknife estimator is shown to perform well in finite samples and, with a suitable choice of the number of sub-samples, is shown to reduce the overall finite sample root mean squared error as well as bias. However, the optimal jackknife weights rely on knowledge of the near-unit root parameter, which is typically unknown in practice, and so an alternative, feasible, jackknife estimator is proposed which achieves the intended bias reduction but does not rely on knowledge of this parameter. This feasible jackknife estimator is also capable of substantial bias and root mean squared error reductions in finite samples across a range of values of the near-unit root parameter and across different sample sizes

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

Time-Varying Parameters in Continuous and Discrete Time

We consider models for both deterministic one-time and continuous stochastic parameter change in a continuous time autoregressive model around a deterministic trend function. For the latter we focus on the case where the autoregressive parameter itself follows a first-order autoregression. Exact discrete time analogue models are detailed in each case and compared to corresponding parameter change models adopted in the discrete time literature. The relationships between the parameters in the continuous time models and their discrete time analogues are also explored. For the one- time parameter change model the discrete time models used in the literature can be justified by the corresponding continuous time model, with a only a minor modification needed for the (most likely) case where the changepoint does not coincide with one of the discrete time observation points. For the stochastic parameter change model considered we show that the resulting discrete time model is characterised by an autoregressive parameter the logarithm of which follows an ARMA(1,1) process. We discuss how this relates to models which have been proposed in the discrete time stochastic unit root literature. The implications of our results for a number of extant discrete time models and testing procedures are discussed

Deterministic Parameter Change Models in Continuous and Discrete Time

We consider a model of deterministic one-time parameter change in a continuous time autoregressive model around a deterministic trend function. The exact discrete time analogue model is detailed and compared to corresponding parameter change models adopted in the discrete time literature. The relationships between the parameters in the continuous time model and the discrete time analogue model are also explored. Our results show that the discrete time models used in the literature can be justified by the corresponding continuous time model, with a only a minor modification needed for the (most likely) case where the changepoint does not coincide with one of the discrete time observation points. The implications of our results for a number of extant discrete time models and testing procedures are discussed

The estimation of continuous time models with mixed frequency data

This paper derives exact representations for discrete time mixed frequency data generated by an underlying multivariate continuous time model. Allowance is made for different combinations of stock and flow variables as well as deterministic trends, and the variables themselves may be stationary or nonstationary (and possibly cointegrated). The resulting discrete time representations allow for the information contained in high frequency data to be utilised alongside the low frequency data in the estimation of the parameters of the continuous time model. Monte Carlo simulations explore the finite sample performance of the maximum likelihood estimator of the continuous time system parameters based on mixed frequency data, and a comparison with extant methods of using data only at the lowest frequency is provided. An empirical application demonstrates the methods developed in the paper and it concludes with a discussion of further ways in which the present analysis can be extended and refined

The Exact Discretisation of CARMA Models with Applications in Finance

The problem of estimating a continuous time model using discretely observed data is common in empirical finance. This paper uses recently developed methods of deriving the exact discrete representation for a continuous time ARMA (autoregressive moving average) system of order p, q to consider three popular models in finance. Our results for two benchmark term structure models show that higher order ARMA processes provide a significantly better fit than standard Ornstein-Uhlenbeck processes. We then explore present value models linking stock prices and dividends in the presence of cointegration. Our methods enable us to take account of the fact that the two variables are observed in fundamentally different ways by explicitly modelling the data as mixed stock-flow type, which we then compare with the (more common, but incorrect) treatment of dividends as a stock variable

Methodologies in Development Studies:An Overview

The interdisciplinary nature of the field of Development Studies (DS) makes it hard to point towards a ‘signature’ methodology. Different development challenges bring different ideas about what the problem is (ontology) and how researchers can know about it (epistemology), as well as different research methods. The rationale for choosing a method can be ideological or pragmatic. In the field of DS, this often entails knowing what methods research commissioners see as credible and what types of evidence they find persuasive. The weight placed on the data generated by certain methods and the lack of critical attention to how it was actually produced shows the importance of a focus on methodology. In looking at, or for, the defining methodologies of DS, this chapter focuses on methodology in a relatively narrow sense: what types of sample and what combinations of methods are typically used by researchers within DS to construct credible arguments around questions of policy or practice. It describes which methodologies constitute the bulk of DS research through analysis of projects and outputs. Finally, it asks what people who generate and use DS research could do to increase its rigour and relevance (Gujit and Roche, 2014; see also Oswald, Leach and Gaventa, this volume) and how the political economy of development research funding might militate against this