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We propose new real-time monitoring procedures for the emergence of end-of-sample predictive regimes using sequential implementations of standard (heteroskedasticity-robust) regression t-statistics for predictability applied over relatively short time periods. The procedures we develop can also be used for detecting historical regimes of temporary predictability. Our proposed methods are robust to both the degree of persistence and endogeneity of the regressors in the predictive regression and to certain forms of heteroskedasticity in the shocks. We discuss how the monitoring procedures can be designed such that their false positive rate can be set by the practitioner at the start of the monitoring period using detection rules based on information obtained from the data in a training period. We use these new monitoring procedures to investigate the presence of regime changes in the predictability of the U.S. equity premium at the one-month horizon by traditional macroeconomic and financial variables, and by binary technical analysis indicators. Our results suggest that the one-month ahead equity premium has temporarily been predictable, displaying so-called 'pockets of predictability', and that these episodes of predictability could have been detected in real-time by practitioners using our proposed methodology.

We propose new methods for the real-time detection of explosive bubbles in financial time series. Most extant methods are constructed for a fixed sample of data and, as such, are only appropriate when applied as one-shot tests. Sequential application of these, declaring the presence of a bubble as soon as one of these statistics exceeds the one-shot critical value, would yield a detection procedure with an unknown false positive rate likely to be far in excess of the nominal level. Our approach sequentially applies the one-shot tests of Astill et al. (2017), comparing sub-sample statistics calculated in real time during the monitoring period with corresponding sub-sample statistics obtained from a prior training period. We propose two procedures: one based on comparing the real time monitoring period statistics with the maximum statistic over the training period, and another which compares the number of consecutive exceedances of a threshold value in the monitoring and training periods, the threshold value obtained from the training period. Both allow the practitioner to determine the false positive rate for any given monitoring horizon, or to ensure this rate does not exceed a specified level by setting a maximum monitoring horizon. Monte Carlo simulations suggest that the finite sample false positive rates lie close to their theoretical counterparts, even in the presence of time-varying volatility and serial correlation in the shocks. The procedures are shown to perform well in the presence of a bubble in the monitoring period, offering the possibility of rapid detection of an emerging bubble in a real time setting. An empirical application to monthly stock market index data is considered.

We develop a test for the presence of nonlinear deterministic components in a univariate time series, approximated using a Fourier series expansion, designed to be asymptotically robust to the order of integration of the process and to any weak dependence present. Our approach is motivated by the Wald-based testing procedure of Harvey, Leybourne and Xiao (2010) [Journal of Time Series Analysis, vol. 31, p.379-391], but uses a function of an auxiliary unit root statistic to select between the asymptotic I(0) and I(1) critical values, rather than modifying the Wald test statistic as in Harvey et al.. We show that our proposed test has uniformly greater local asymptotic power than the test of Harvey et al. when the shocks are I(1), identical local asymptotic power when the shocks are I(0), and also improved .nite sample properties. We also consider the issue of determining the number of Fourier frequencies used to specify any nonlinear deterministic components, evaluating the performance of algorithmic- and information criterion-based model selection procedures.

In a recent paper, Harvey et al. (2013) [HLT] propose a new unit root test that allows for the possibility of multiple breaks in trend. Their proposed test is based on the infimum of the sequence (across all candidate break points) of local GLS detrended augmented Dickey-Fuller-type statistics. HLT show that the power of their unit root test is robust to the magnitude of any trend breaks. In contrast, HLT show that the power of the only alternative available procedure of Carrion-i-Silvestre et al. (2009), which employs a pre-test-based approach, can be very low indeed (even zero) for the magnitudes of trend breaks typically observed in practice. Both HLT and Carrion-i-Silvestre et al. (2009) base their approaches on the assumption of homoskedastic shocks. In this paper we analyse the impact of non-stationary volatility (for example single and multiple abrupt variance breaks, smooth transition variance breaks, and trending variances) on the tests proposed in HLT. We show that the limiting null distribution of the HLT unit root test statistic is not pivotal under non- stationary volatility. A solution to the problem, which does not require the practitioner to specify a parametric model for volatility, is provided using the wild bootstrap and is shown to perform well in practice. A number of dfferent possible implementations of the bootstrap algorithm are discussed.

In this paper, we propose a test for a break in the level of a fractionally integrated process when the timing of the putative break is not known. This testing problem has received considerable attention in the literature in the case where the time series is weakly autocorrelated. Less attention has been given to the case where the underlying time series is allowed to be fractionally integrated. Here, valid testing can only be performed if the limiting null distribution of the level break test statistic is well defined for all values of the fractional integration exponent considered. However, conventional sup-Wald type tests diverge when the data are strongly autocorrelated. We show that a sup-Wald statistic, which is standardized using a non-parametric kernel-based long-run variance estimator, does possess a well-defined limit distribution, depending only on the fractional integration parameter, provided the recently developed fixed-b asymptotic framework is applied. We give the appropriate asymptotic critical values for this sup-Wald statistic and show that it has good finite sample size and power properties.

We consider unit root testing allowing for a break in trend when partial information is available regarding the location of the break date. This takes the form of knowledge of a relatively narrow window of data within which the break takes place, should it occur at all. For such circumstances, we suggest employing a union of rejections strategy, which combines a unit root test that allows for a trend break somewhere within the window with a unit root test that makes no allowance for a trend break. Asymptotic and finite sample evidence shows that our suggested strategy works well, provided that, when a break does occur, the partial information is correct. An empirical application to UK interest rate data containing the 1973 ‘oil shock’ is also considered.

We consider testing for the presence of nonlinearities in the deterministic component of a time series, approximating the potential nonlinear behaviour using a Fourier function expansion. In contrast to procedures that are currently available, we develop tests that are robust to the order of integration, in the sense that they are asymptotically correctly sized regardless of whether the stochastic component of the series is stationary or contains a unit root. The tests we propose take the form of Wald statistics based on cumulated series, together with a correction factor to line up the asymptotic critical values across the I(0) and I(1) environments. The local asymptotic power and finite sample properties of the tests are evaluated using various different correction factors. We envisage that the testing procedure we recommend should be very useful to applied researchers wishing to draw robust inference regarding the presence of nonlinear deterministic components in a series.

Using standardized cumulative sums of squared sub-sample residuals, we propose a new ratio-based test of the null hypothesis that a time series exhibits no change in its persistence structure [specifically that it displays constant I(1) behaviour] against the alternative of a change in persistence from trend stationarity to difference stationarity, or vice versa. Neither the direction nor location of any possible change under the alternative hypothesis need be assumed known. A key feature of our proposed test which distinguishes it from extant tests for persistence change [certain of which test the null hypothesis of constant I(0) behaviour while others, like our proposed test, test the null hypothesis of constant I(1) behaviour] is that it displays no tendency to spuriously over-reject when applied to series which, although not constant I(1) series, do not display a change in persistence [specifically are constant I(0) processes]. Moreover, where our ratio test correctly rejects the null of no persistence change, the tail in which the rejection occurs can also be used to identify the direction of change since, even in relatively small samples, the test almost never rejects in the right [left] tail when there is a change from I(0) to I(1) [I(1) to I(0)]. Again this useful property is not shared by existing tests. As a by-product of our analysis, we also propose breakpoint estimators which are consistent where the timing of the change in persistence is unknown.

We show that the minimal forward (reverse) recursive unit tests of Banerjee, Lumsdaine and Stock [Journal of Business and Economics Statistics (1992) Vol. 10, pp. 271–288] are consistent against the alternative of a change in persistence from I(0) to I(1) [I(1) to I(0)]. However, these statistics are also shown to diverge for series which are I(0) throughout. Consequently, a rejection by these tests does not necessarily imply a change in persistence. We propose a further test, based on the ratio of these statistics, which is consistent against changes either from I(0) to I(1), or vice versa, yet does not over‐reject against constant I(0) series. Consistent breakpoint estimators are proposed.

It is now well known that how the initial observation is generated can have a significant effect on the power of a unit-root test. In this article, we show that by taking a simple data-dependent weighted average of the initial condition-robust test of Elliott and Muller [Journal of Econometrics (2006), forthcoming] and the standard augmented Dickey–Fuller test, we are able to produce a new unit-root test that can improve power, both asymptotically and in finite samples, over a wide range of possibilities governing the generation of the initial observation.

Although the t-ratio variant of the Dickey–Fuller test is the most commonly applied unit-root test in practical applications, it has been known for some time that readily implementable, more powerful modifications are available. We explore the large-sample properties of five of these modified tests, and the small-sample properties of these five plus six hybrids. As a result of this study we recommend two particular test procedures.

We analyse the case where a unit-root test is based on a Dickey–Fuller regression the only deterministic term of which is a fixed intercept. Suppose, however, as could well be the case, that the actual data-generating process includes a broken linear trend. It is shown theoretically, and verified empirically, that under the I(1) null and I(0) alternative hypotheses the Dickey–Fuller test can display a wide range of different characteristics depending on the nature and location of the break.

Assume that a time series is generated by an autoregression which has atmost one unit root. A correctly specified model, including linear time trend, is estimated by ordinary least squares, but no allowance is made for any unit root in the generating process. We investigate the impact of estimation error on the mean-squared error of forecasts calculated from the fitted model.

In this paper, we consider tests for a break in the level of a series at an unknown point in time. It is often the case that uncertainty exists concerning the order of integration of the series; consequently, we focus on tests that are applicable when the order of integration is not known. The size and power of existing tests are analysed, and a modification to one of the established sets of tests is proposed which offers improved performance in certain circumstances.

Augmented Dickey–Fuller regressions on pooled (but not individual) real exchange rates for the post–1973 period consistently reject the unit root null, even after accounting for cross–sectional dependence. The inference that the series is stationary, however, is not necessarily correct, because these tests strongly over–reject the null in certain circumstances, particularly when the series has a stochastic unit root. We find that bilateral real exchange rates against the US dollar have a stochastic unit root. Out–of–sample prediction exercises for an autoregressive model confirm these findings.

In this article, we consider estimating the timing of a break in level and/or trend when the order of integration and autocorrelation properties of the data are unknown. For stationary innovations, break point estimation is commonly performed by minimizing the sum of squared residuals across all candidate break points, using a regression of the levels of the series on the assumed deterministic components. For unit root processes, the obvious modification is to use a first differenced version of the regression, while a further alternative in a stationary autoregressive setting is to consider a GLS-type quasi-differenced regression. Given uncertainty over which of these approaches to adopt in practice, we develop a hybrid break fraction estimator that selects from the levels-based estimator, the first-difference-based estimator, and a range of quasi-difference-based estimators, according to which achieves the global minimum sum of squared residuals. We establish the asymptotic properties of the estimators considered, and compare their performance in practically relevant sample sizes using simulation. We find that the new hybrid estimator has desirable asymptotic properties and performs very well in finite samples, providing a reliable approach to break date estimation without requiring decisions to be made regarding the autocorrelation properties of the data.

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.

In this paper, we concentrate on the case of an exogeneously chosen break date, but entertain the possibility that an incorrect choice is made. In fact, the Perron test statistics considered are invariant to any break in the generating process at the assumed break date. Our results therefore apply equally to the case of a generating process with two breaks, only one of which is specifically accounted for in the analysis. As in Leybourne et al. (1998), we find that a neglected relatively early break can lead to spurious rejections of the unit root null hypothesis. Moreover, for all but one of the tests analyzed, spurious rejections now also arise if a true break occurs relatively soon after the assumed break date.

In this paper we test whether GDP series in 12 European Union countries are integrated or are stationary around a deterministic component that may change gradually and smoothly between two regimes over time. We find that in two-thirds of cases there appears to be a role for modelling with deterministic functions that allow smooth transitions, in some cases standing alone, in others in conjunction with additional integrated regressors. These findings constitute a challenge to traditional approaches to modelling breaking-trend behaviour in GDP, which typically impose the condition that breaks, when present, must occur instantaneously.

It has been found useful, as an alternative to a difference-stationary generating model, to consider the possibility of stationarity around a function that permits a smooth transition from one linear trend to another. Previous research has concentrated on functions that are symmetric around the point at which 50% of the transition is completed. Here we permit the possibility of asymmetry in the transition, allowing for example slow entry to and rapid exit from this evolutionary phase. It is shown through simulation and an interesting real data set that this can lead to the detection of further departures from difference stationarity.

We consider tests for the equality of prediction mean squared errors and for forecast encompassing. It is shown that, if forecast errors exhibit ARCH, size distortions are induced in the usual tests. Adjusted test statistics are suggested to alleviate this problem. Copyright © 1999 John Wiley & Sons, Ltd.

In this paper we suggest a new and consistent procedure designed to test for the presence of unit roots in quarterly time series, be they at the zero or seasonal frequencies. For each frequency the proposed statistic is based on the correlogram of a transformation (which removes unit roots that may exist at frequencies other than that of interest) of the residuals from a regression of the observed series on relevant deterministic components. Each statistic is proportional to the maximum lag length for which the residual autocorrelations at all lower lag lengths are strictly positive. Critical values for each test statistic are calculated using Monte Carlo simulation assuming a seasonal random walk data-generation process (DGP). The robustness properties of each statistic to different integrated DGPs are examined along with their respective power characteristics. These simulations indicate that the procedure suggested in this paper provides a useful complement to the set of test statistics proposed by Hylleberg et al. (‘Seasonal integration and cointegration’, Journal of Econometrics, Vol. 44 (1990), pp. 215–238).

The unit root tests of Phillips and Perron (Testing for a unit root in time series regression. Biometrika 75 (1988), 335–46) are frequently employed in applied econometric research. The chief attraction of these tests is that they are non-parametric, so it is unnecessary to specify a model, even as an approximation to the underlying generating process. The continued popularity of these tests is perhaps surprising since, beginning with Schwert (Tests for unit roots: a Monte Carlo investigation. J. Bus. Econ. Stat. 7 (1989), 147–59), simulation evidence has frequently suggested severe size distortions in specific low persistence generating models. It has been assumed that this size distortion is due to the requirement to estimate short- and long-run variances in practical implementation of the tests, and various alternative estimators have been analysed. However, the analysis of this paper suggests that part of the cause of the size distortion is more fundamental. Even in the ‘idealistic case’ where actual values of the true variances are used, serious size distortions remain for two simple generating models in sample sizes common in economic applications. We explore in detail the sources of this phenomenon for the ARIMA(0, 1, 1) generating model.

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.

I. INTRODUCTION There is a large and growing literature on liberalization and its effects, stimulated initially by the very extensive trade reform programs promoted in developing countries over the last fifteen years or so and sustained by the attempts to understand what is happening in the eastern and central European economies. The latter do not feature in this paper, though the techniques demonstrated here may in time prove to be useful for gaining insight into the effects of regime changes underway. Liberalization, in the sense of trade reforms which reduce anti-export bias in one way or another while improving the information content of relative price changes, has been promoted in some eighty or so developing countries in the 1980s and 1990s. Some of these liberalizations are unilateral, most are policy conditioned under the aegis of World Bank Structural Adjustment Loans (SALs).(1) All have been undertaken on the assumption that liberalization will ultimately improve export and growth performance. The evidence on liberalization and growth, however, is at best somewhat mixed. On the one hand, one has the exhaustive study of Papageorgiou, Michaely and Choksi [1991] which claims a close and direct association. On the other hand, Greenaway and Sapsford [1994] find only a limited role for liberalization, and a range of studies of SALs associate adjustment with a deterioration in growth performance.(2) These studies generally rely either on crude data inspection, simple correlation analysis, or multivariate regression. This paper takes a new approach to the question. It starts from the presumption that any changes in economic performance following a liberalization may be more appropriately modeled as a steady transition rather than a discrete change. To model the change in this way, we make use of the recent work of Granger and Terasvirta [1994], who explore the properties of a variety of nonlinear specifications that facilitate modeling structural change as a smooth transition between states. We estimate and test the adequacy of a number of such specifications. Our analysis is in two stages. First we take a novel approach to the modeling of growth which allows us to model deterministic change without, as other analysts have done, imposing discrete changes. Not only does this help clarify the statistical properties of the time series, it also challenges the assumptions that underpin much growth modeling. Having identified the transitions in the growth series we are investigating, we then explore the coincidence of these transitions with well-documented episodes of liberalization. We do not formally test whether liberalization results in growth. Our results are, however, informative in two respects. Firstly, they place a question mark against the widely held presumption that liberalization is a panacea for growth. Secondly, they also point the way to the appropriate econometric modeling of these processes. The remainder of the paper is structured as follows. Section II briefly describes the context against which our analysis is set. Section III sets out the details of our methodology and the models to be tested. Section IV reports our results, contrasts them with those of previous work and evaluates the implications for policy. Section V concludes and identifies possible extensions. II. LIBERALIZATION, (EXPORTS) AND GROWTH: PREVIOUS EVIDENCE There are essentially two strands of the literature which provide relevant background: one which relates exports and growth; a second which relates liberalization to exports and growth. The exports and growth literature is an extensive one, which goes back some years.(3) It starts from the presumption that exports and growth are directly related, with causality running from changes in exports to changes in growth.(4) Analysts have deployed a variety of empirical methods, usually of growth accounting type models, and covering a range of countries and time periods. …

A residual-based test for cointegration is proposed where a parametric adjustment is made to account for the possible stationarity of the disturbance vector. Allowance is also made for the regressor variables to be cointegrated among themselves. The parametric adjustment turns out to be more robust and powerful than tests based on long-run variance estimators according to theoretical and simulation evidence.

This article suggests a simple test for the null hypothesis that a process contains a unit root against a (trend) stationary alternative. It is based on the maximum value of the standard Dickey-Fuller tests when applied to both the forward and reverse data realizations. Thus, it requires no special computation and can easily be calculated from most existing econometric software packages. The null distribution of the maximum test is tabulated using Monte Carlo simulation and it is demonstrated to have considerably more power to reject the false null of a unit root than the standard Dickey-Fuller test. The tests are also shown to share very similar robustness properties. When applied to a number of well known U.S. macroeconomics time series, the new test rejects the unit root hypothesis more frequently than does the usual Dickey-Fuller test. Copyright 1995 by Blackwell Publishing Ltd

A linear regression model with random walk coefficients is extended to allow for linear restrictions between the coefficients to be satisfied at each point in time. Estimation in this model is shown to be no more involved than estimation in the standard model. It is also demonstrated how, after a slight modification to the testing problem, classical test procedures may be applied to the problem of testing for such restrictions. The performance of the Lagrange Multiplier test for a variety of different restrictions is then investigated via simulation. An empirical application involving testing for homogeneity in a random walk coefficient version of the AIDS model is given.

This article 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 on the use of trimmed data, which are shown to help reduce this sensitivity.

The paper investigates the question of whether the trend rate of growth of GDP slowed down in Britain or France in the nineteenth century. We establish that real GDP belongs to the trend-stationary process class of time-series in each case rather than the difference-stationary process and using Kalman filter techniques, analyse trends in a model incorporating time-varying parameters without specifying breakpoints in advance. We find, contrary to the literature, that neither country experienced a climacteric. Our results suggest that economic historians have been prone to exaggerated views of variations in trend economic growth.