Stephen J. Leybourne

Testing the equality of prediction mean squared errors

Abstract Given two sources of forecasts of the same quantity, it is possible to compare prediction records. In particular, it can be useful to test the hypothesis of equal accuracy in forecast performance. We analyse the behaviour of two possible tests, and of modifications of these tests designed to circumvent shortcomings in the original formulations. As a result of this analysis, a recommendation for one particular testing approach is made for practical applications.

Tests for Forecast Encompassing

We consider the situation in which two forecasts of the same variable are available. The possibility exists of forming a combined forecast as a weighted average of the individual ones and estimating the weights that should be optimally attached to each forecast. If the entire weight should optimally be associated with one forecast, that forecast is said to encompass the other. A natural test for forecast encompassing is based on least squares regression. We find, however, that the null distribution of this test statistic is not robust to nonnormality in the forecast errors. We discuss several alternative tests that are robust.

Unit roots and smooth transitions

It is common practice in time series econometrics to test the null hypothesis that the generating function is integrated—i.e. that a series is stationary only after differencing—against the alternative of stationarity about either a fixed mean or a linear trend. However, there has been considerable recent interest in the possibility of stationarity around a linear trend with an abrupt break. Here we broaden this class of alternatives to allow for a smooth transition from one trend function to another. Dickey–Fuller type tests against this alternative are developed, and their properties are explored.

A Consistent Test for a Unit Root

This article investigates several U.S. macroeconomic time series for the presence of a unit root using a newly developed test. This test has stationarity as its null hypothesis, and the alternative is a unit-root process. The test is shown to be consistent, and its asymptotic null distribution is determined. Our findings contrast sharply with those obtained via the standard unit-root tests.

Spurious rejections by Dickey-Fuller tests in the presence of a break under the null

Abstract It is well known that if a series is generated by a process that is stationary around a broken trend, conventional Dickey–Fuller tests can have very low power. In this paper, the converse phenomenon is studied and illustrated. Suppose that the true generating process is integrated of order one, but with a break. Then it is shown that, if the break occurs early in the series, routine application of standard Dickey–Fuller tests can lead to a very serious problem of spurious rejection of the unit root null hypothesis.

Spurious rejections by cointegration tests induced by structural breaks

The effects on three cointegration tests are examined when the series analysed are independent integrated processes, each with a structural break. Although there are differences in detail among the tests, the results indicate in all cases that, when structural breaks are neglected in the analysis, spurious rejections, indicating the presence of cointegration, can occur.

Can Economic Time Series Be Differenced to Stationarity

This article considers a class of nonstationary varying-coefficient autoregressive models that allow stochastic variability in the autoregressive root. It is argued that such models provide a better description of the behavior of macroeconomic variables than fixed-unit-root autoregressive models because they allow more general forms of nonstationarity. We construct a test of the null hypothesis of a fixed unit root against the alternative of a randomized root with unit mean and derive its asymptotic distribution. The test is applied to several U.S. macroeconomic series generally considered to contain fixed unit roots. We find that for about half of the series the fixed-unit-root null is rejected.

Unit root tests with a break in innovation variance

Abstract It is shown that an abrupt change in the innovation variance of an integrated process can generate spurious rejections of the unit root null hypothesis in routine applications of Dickey–Fuller tests. We develop and investigate modified test statistics, based on unit root tests of Perron for a time series with a changing level, or changing intercept and slope, which are applicable when there is a change in innovation variance of an unknown magnitude at an unknown location.

Unit root testing in practice: dealing with uncertainty over the trend and initial condition

In this paper we focus on two major issues that surround testing for a unit root in practice, namely, (i) uncertainty as to whether or not a linear deterministic trend is present in the data and (ii) uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. In each case simple testing procedures are proposed with the aim of maintaining good power properties across such uncertainties. For the first issue, if the initial condition is negligible, quasi-differenced (QD) detrended (demeaned) Dickey–Fuller-type unit root tests are near asymptotically efficient when a deterministic trend is (is not) present in the data generating process. Consequently, we compare a variety of strategies that aim to select the detrended variant when a trend is present, and the demeaned variant otherwise. Based on asymptotic and finite-sample evidence, we recommend a simple union of rejections-based decision rule whereby the unit root null hypothesis is rejected whenever either of the detrended or demeaned unit root tests yields a rejection. Our results show that this approach generally outperforms more sophisticated strategies based on auxiliary methods of trend detection. For the second issue, we again recommend a union of rejections decision rule, rejecting the unit root null if either of the QD or ordinary least squares (OLS) detrended/demeaned Dickey–Fuller-type tests rejects. This procedure is also shown to perform well in practice, simultaneously exploiting the superior power of the QD (OLS) detrended/demeaned test for small (large) initial conditions.

Simple, Robust, And Powerful Tests Of The Breaking Trend Hypothesis

In this paper we develop a simple procedure that delivers tests for the presence of a broken trend in a univariate time series that do not require knowledge of the form of serial correlation in the data and are robust as to whether the shocks are generated by an I (0) or an I (1) process. Two trend break models are considered: the first holds the level fixed while allowing the trend to break, while the latter allows for a simultaneous break in level and trend. For the known break date case, our proposed tests are formed as a weighted average of the optimal tests appropriate for I (0) and I (1) shocks. The weighted statistics are shown to have standard normal limiting null distributions and to attain the Gaussian asymptotic local power envelope, in each case regardless of whether the shocks are I (0) or I (1). In the unknown break date case, we adopt the method of Andrews (1993) and take a weighted average of the statistics formed as the supremum over all possible break dates, subject to a trimming parameter, in both the I (0) and I (1) environments. Monte Carlo evidence suggests that our tests are in most cases more powerful, often substantially so, than the robust broken trend tests of Sayginsoy and Vogelsang (2004). An empirical application highlights the practical usefulness of our proposed tests.