Robust methods for detecting multiple level breaks in autocorrelated time series

In this paper we propose tests for the null hypothesis that a time series process displays a constant level against the alternative that it displays (possibly) multiple changes in level. Our proposed tests are based on functions of appropriately standardized sequences of the differences between sub-sample mean estimates from the series under investigation. The tests we propose differ notably from extant tests for level breaks in the literature in that they are designed to be robust as to whether the process admits an autoregressive unit root (the data are I(1)) or stable autoregressive roots (the data are I(0)). We derive the asymptotic null distributions of our proposed tests, along with representations for their asymptotic local power functions against Pitman drift alternatives under both I(0) and I(1) environments. Associated estimators of the level break fractions are also discussed. We initially outline our procedure through the case of non-trending series, but our analysis is subsequently extended to allow for series which display an underlying linear trend, in addition to possible level breaks. Monte Carlo simulation results are presented which suggest that the proposed tests perform well in small samples, showing good size control under the null, regardless of the order of integration of the data, and displaying very decent power when level breaks occur.Level breaks; unit root; moving means; long run variance estimation; robust tests; breakpoint estimation

The impact of the initial condition on robust tests for a linear trend

This paper examines the behaviour of some recently proposed robust (to the order of integration of the data) tests for the presence of a deterministic linear trend in a univariate times series in situations where the magnitude of the initial condition of the series is non-negligible. We demonstrate that the asymptotic size and/or local power properties of these tests are extremely sensitive to the initial condition. Straightforward modifications to the trend tests are suggested, based around the use of trimmed data, which are demonstrated to greatly reduce this sensitivity.Trend tests; initial condition; asymptotic local power

Seasonal unit root tests and the role of initial conditions

In the context of regression-based (quarterly) seasonal unit root tests, we examine the impact of initial conditions (one for each quarter) of the process on test power. We investigate the behaviour of the OLS detrended HEGY seasonal unit root tests of Hylleberg et al. (1990) and the corresponding quasi-differenced (QD) detrended tests of Rodrigues and Taylor (2007), when the initial conditions are not asymptotically negligible. We show that the asymptotic local power of a test at a given frequency depends on the value of particular linear (frequency-specific) combinations of the initial conditions. Consistent with previous findings in the non-seasonal case (see, inter alia, Harvey et al., 2008, Elliott and Muller, 2006), the QD detrended test at a given spectral frequency dominates on power for relatively small values of this combination, while the OLS detrended test dominates for larger values. Since, in practice, the seasonal initial conditions are not observed, in order to maintain good power across both small and large initial conditions, we extend the idea of Harvey et al. (2008) to the seasonal case, forming tests based on a union of rejections decision rule; rejecting the unit root null at a given frequency (or group of frequencies) if either of the relevant QD and OLS detrended HEGY tests rejects. This procedure is shown to perform well in practice, simultaneously exploiting the superior power of the QD (OLS) detrended HEGY test for small (large) combinations of the initial conditions. Moreover, our procedure is particularly adept in the seasonal context since, by design, it exploits the power advantage of the QD (OLS) detrended HEGY tests at a particular frequency when the relevant initial condition is small (large) without imposing that same method of detrending on tests at other frequencies.HEGY seasonal unit root tests; initial conditions; asymptotic local power; union of rejections decision rule

Testing for unit roots and the impact of quadratic trends, with an application to relative primary commodity prices

In practice a degree of uncertainty will always exist concerning what specification to adopt for the deterministic trend function when running unit root tests. While most macroeconomic time series appear to display an underlying trend, it is often far from clear whether this component is best modelled as a simple linear trend (so that long-run growth rates are constant) or by a more complicated non-linear trend function which may, for instance, allow the deterministic trend component to evolve gradually over time. In this paper we consider the effects on unit root testing of allowing for a local quadratic trend, a simple yet very flexible example of the latter. Where a local quadratic trend is present but not modelled we show that the quasi-differenced detrended Dickey-Fuller-type test of Elliott et al. (1996) has both size and power which tend to zero asymptotically. An extension of the Elliott et al. (1996) approach to allow for a quadratic trend resolves this problem but is shown to result in large power losses relative to the standard detrended test when no quadratic trend is present. We consequently propose a simple and practical approach to dealing with this form of uncertainty based on a union of rejections-based decision rule whereby the unit root null is rejected whenever either of the detrended or quadratic detrended unit root tests rejects. A modification of this basic strategy is also suggested which further improves on the properties of the procedure. An application to relative primary commodity price data highlights the empirical relevance of the methods outlined in this paper. A by-product of our analysis is the development of a test for the presence of a quadratic trend which is robust to whether or not the data admit a unit root.Unit root test; trend uncertainty; quadratic trends; asymptotic power; union of rejections decision rule

Unit root testing under a local break in trend

It is well known that it is vital to account for trend breaks when testing for a unit root. In practice, uncertainty exists over whether or not a trend break is present and, if it is, where it is located. Harris et al. (2009) and Carrion-i-Silvestre et al. (2009) propose procedures which account for both of these forms of uncertainty. Each uses what amounts to a pre-test for a trend break, accounting for a trend break (the associated break fraction estimated from the data) in the unit root procedure only where the pre-test signals a break. Assuming the break magnitude is fixed (independent of sample size) these authors show that their methods achieve near asymptotically ecient unit root inference in both trend break and no trend break environments. These asymptotic results are, however, somewhat at odds with the finite sample simulations reported in both papers. These show the presence of pronounced "valleys" in the finite sample power functions (when mapped as functions of the break magnitude) of the tests such that power is initially high for very small breaks, then decreases as the break magnitude increases, before increasing again. Here we show that treating the break magnitude as local to zero (in a Pitman drift sense) allows the asymptotic analysis to very closely approximate this finite sample effect, thereby providing useful analytical insights into the observed phenomenon. In response to this problem we propose practical solutions, based either on the use of a with break unit root test but with adaptive critical values, or on a union of rejections principle taken across with break and without break unit root tests. The former is shown to eliminate power valleys but at the expense of power when no break is present, while the latter considerably mitigates the valleys while not losing all the power gains available when no break exists.Unit root test; local trend break; asymptotic local power; union of rejections; adaptive critical values

Testing for a unit root when uncertain about the trend [Revised to become 07/03 above]

In this paper we consider the issue of testing for a unit root when it is uncertain as to whether or not a linear deterministic trend is present in the data. The Dickey-Fuller-type tests of Elliott, Rothenberg and Stock (1996), based on (local) GLS detrended (demeaned) data, are near asymptotically efficient when a deterministic trend is (is not) present in the data generating process. We consider a variety of strategies which aim to select the demeaned variant when a trend is not present and the detrended variant otherwise. Asymptotic and finite sample evidence demonstrates that some sophisticated strategies which involve auxiliary methods of trend detection are generally outperformed by a simple decision rule of rejecting the unit root null whenever either the GLS demeaned or GLS detrended Dickey-Fuller-type tests reject. We show that this simple strategy is asymptotically identical to a sequential testing strategy proposed by Ayat and Burridge (2000). Moreover, our results make it clear that any other unit root testing strategy, however elaborate, can at best only offer a rather modest improvement over the simple one.Unit root test; trend uncertainty; initial condition; asymtotic power; union of rejections decision rule

Provincetown--auditorium for music and experimental drama

Thesis (M. Arch.)--Massachusetts Institute of Technology, Dept. of Architecture, 1955.Accompanying drawings held by MIT Museum.Includes bibliographical references (leaf 38 ).by Stephen Shiowitz and Fred M. Taylor.M.Arch

A simple, robust and powerful test of the trend hypothesis

In this paper we develop a simple test procedure for a linear trend which does not require knowledge of the form of serial correlation in the data, is robust to strong serial correlation, and has a standard normal limiting null distribution under either I(0) or I(1) shocks. In contrast to other available robust linear trend tests, our proposed test achieves the Gaussian asymptotic local power envelope in both the I(0) and I(1) cases. For near- I(1) errors our proposed procedure is conservative and a modification for this situation is suggested. An estimator of the trend parameter, together with an associated confidence interval, which is asymptotically efficient, again regardless of whether the shocks are I(0) or I(1), is also provided.Linear trend; strong serial correlation; asymptotic normality; power enveloope; unit root tests; stationarity tests

Testing for unit roots in the presence of uncertainty over both the trend and initial condition

We provide a joint treatment of two major problems that surround testing for a unit root in practice, namely uncertainty as to whether or not a linear deterministic trend is present in the data, and uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. In earlier work [Harvey, Leybourne and Taylor, 2008] we proposed methods to deal with trend uncertainty when the initial condition is assumed to be (asymptotically) negligible, together with methods to deal with uncertainty over the initial condition when the form of the trend function was taken as known. In each case we recommended a simple union of rejections-based decision rule. In the first case rejecting the unit root null whenever either of the quasi-differenced (QD) detrended or QD demeaned augmented Dickey-Fuller [ADF] unit root tests yields a rejection, and in the second case if either of the QD and OLS detrended/demeaned ADF tests rejects. Both approaches were shown to work well. In this paper we extend these procedures to allow for both trend and initial condition uncertainty, proposing a four-way union of rejections decision rule based on the QD and OLS demeaned and the QD and OLS detrended ADF tests. This is shown to work well but to lack power, relative to the best available test, in some scenarios. A modification of the basic union, based on auxiliary information including linear trend pre-test statistics, is proposed and shown to deliver significant improvements. A by-product of our analysis is that the power functions of the associated trend function pre-tests are shown to be heavily dependent on the initial condition.Unit root tests; trend uncertainty; initial condition uncertainty; asymptotic power; union of rejections decision rule; trend tests