David I. Harvey

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.

THE PREBISCH-SINGER HYPOTHESIS: FOUR CENTURIES OF EVIDENCE

We employ a unique data set and new time-series techniques to reexamine the existence of trends in relative primary commodity prices. The data set comprises 25 commodities and provides a new historical perspective, spanning the seventeenth to the twenty-first centuries. New tests for the trend function, robust to the order of integration of the series, are applied to the data. Results show that eleven price series present a significant and downward trend over all or some fraction of the sample period. In the very long run, a secular, deteriorating trend is a relevant phenomenon for a significant proportion of primary commodities.

Forecast combination and encompassing

Forecast combination is often found to improve forecast accuracy. This chapter considers different types of forecast combination and tests of forecast encompassing. The latter indicate when a combination is more accurate than an individual forecast ex post, in a range of circumstances: when the forecasts themselves are the objects of interest; when the forecasts are derived from models with unknown parameters; and when the forecast models are nested. We consider forecast encompassing tests which are framed in terms of the model’s estimated parameters and recognize that parameter estimation uncertainty affects forecast accuracy, as well as conditonal tests of encompassing. We also look at the conditions under which forecast encompassing can be established irrespective of the form of the loss function.

Tests for multiple forecast encompassing

In the evaluation of economic forecasts, it is frequently the case that comparisons are made between a number of competing predictors. A natural question to ask in such contexts is whether one forecast encompasses its competitors, in the sense that they contain no useful information not present in the superior forecast. We develop tests for this notion of multiple forecast encompassing which are robust to properties expected in the forecast errors, and apply the tests to forecasts of UK growth and inflation. Copyright

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.

TESTING FOR A UNIT ROOT IN THE PRESENCE OF A POSSIBLE BREAK IN TREND

In this paper we consider the issue of testing a time series for a unit root in the possible presence of a break in a linear deterministic trend at some unknown point in the series. We propose a break fraction estimator which, in the presence of a break in trend, is consistent for the true break fraction at rate Op(T^-1) when there is either a unit root or near-unit root in the stochastic component of the series. In contrast to other estimators available in the literature, when there is no break in trend, our proposed break fraction estimator converges to zero at rate Op(T^-1/2). Used in conjunction with a quasi difference (QD) detrended unit root test that incorporates a trend break regressor in the deterministic component, we show that these rates of convergence ensure that known break fraction null critical values are applicable asymptotically. Unlike available procedures in the literature this holds even if there is no break in trend (the true break fraction is zero), in which case the trend break regressor is dropped from the deterministic component and standard QD detrended unit root test critical values then apply. We also propose a second testing procedure which makes use of a formal pre-test for a trend break in the series, including a trend break regressor only where the pre-test rejects the null of no break. Both procedures ensure that the correctly sized (near-) efficient unit root test that allows (does not allow) for a break in trend is applied in the limit when a trend break does (does not) occur.

Testing for time series linearity

process, and is consistent against non-linearity of either form. Finite sample simulation evidence, together with empirical evidence from an application to US Dollar real exchange rates, suggests that our procedure should work well in practice. Copyright Royal Economic Society 2007

A Powerful Test for Linearity When the Order of Integration is Unknown

In this paper we propose a test of the null hypothesis of time series linearity against a nonlinear alternative, when uncertainty exists as to whether or not the series contains a unit root. We provide a test statistic that has the same limiting null critical values regardless of whether the series under consideration is generated from a linear I(0) or linear I(1) process, and is consistent against nonlinearity of either form, being asymptotically equivalent to the efficient test in each case. Finite sample simulations show that the new procedure has better size control and offers substantial power gains over the recently proposed robust linearity test of Harvey and Leybourne (2007).

Innovational Outlier Unit Root Tests with an Endogenously Determined Break in Level

We show that a standard unit root test that permits an endogenously determined break in level can generate spurious rejections in practically interesting sample sizes when a large break occurs under the null hypothesis. This problem, which occurs for breaks of the innovational outlier type, can be corrected through a simple modification of the test procedure. Copyright 2001 by Blackwell Publishing Ltd