We explore how information from covariates can be incorporated into the CUSUM-based real-time monitoring procedure for explosive asset price bubbles developed in Homm and Breitung (2012, Journal of Financial Econometrics 10, 198-231). Where dynamic covariates are present in the data generating process (DGP), the false positive rate (FPR) of the basic CUSUM procedure, which is based on the assumption that prices follow a univariate DGP, under the null of no explosivity will not, in general, be properly controlled, even asymptotically. In contrast, accounting for these relevant covariates in the construction of the CUSUM statistics leads to a procedure whose FPR can be controlled using the same asymptotic crossing function as employed by Homm and Breitung (2012). Doing so is also shown to have the potential to significantly increase the chance of detecting an emerging bubble episode in finite samples. We additionally allow for time-varying volatility in the innovations driving the model through the use of a kernel-based variance estimator.

We propose a nonparametric portmanteau test for detecting changes in the unconditional mean of a univariate time series which may display either long or short memory. Our approach is designed to have power against, among other things, cases where the mean component of the series displays abrupt level shifts, deterministic trending behaviour, or is subject to some form of time-varying, continuous change. The test we propose is simple to compute, being based on ratios of periodogram ordinates, has a pivotal limiting null distribution of known form which reduces to the multiple of a x(2)(2) random variable in the case where the series is short memory, and has power against a wide class of time-varying mean models. A Monte Carlo simulation study into the finite sample behaviour of the test shows it to have both good size properties under the null for a range of long and short memory series and to exhibit good power against a variety of plausible time-varying mean alternatives. Because of its simplicity, we recommend our periodogram ratio test as a routine portmanteau test for whether the mean component of a time series can reasonably be treated as constant.

We develop new tests for predictability at a given quantile, based on the Lagrange Multiplier [LM] principle, in the context of quantile regression [QR] models which allow for persistent and endogenous predictors driven by heteroskedastic errors. Of the extant predictive QR tests in the literature, only the moving blocks bootstrap implementation, due to Fan and Lee (2019) , of the Wald-type test of Lee (2016) can allow for conditionally heteroskedastic errors in the context of a QR model with persistent predictors. In common with all other tests in the literature these tests cannot, however, allow for unconditionally heteroskedastic behaviour in the errors. The LM-based approach we adopt in this paper is obtained from a simple auxiliary linear test regression which facilitates inference based on established instrumental variable methods. We demonstrate that, as a result, the tests we develop, based on either conventional or heteroskedasticity-consistent standard errors in the auxiliary regression, are robust under the null hypothesis of no predictability to conditional heteroskedasticity and to unconditional heteroskedasticity in the errors driving the predictors, with no need for bootstrap implementation. We also propose tests for joint predictability across a set of multiple distinct quantiles. Simulation results for both conditionally and unconditionally heteroskedastic errors highlight the superior finite sample properties of our proposed LM tests over the tests of Lee (2016) and Fan and Lee (2019) and the recent variable addition tests of Cai et al. (2023). An empirical application to the equity premium for the S&P 500 highlights the practical usefulness of our proposed tests, uncovering significant evidence of predictability in the left and right tails of the returns distribution for a number of predictors containing information on market or firm risk.

We propose a new class of modified regression-based tests for detecting asset price bubbles designed to be robust to the presence of general forms of both conditional and unconditional heteroskedasticity in the price series. This modification, based on the approach developed in Beare (2018) in the context of conventional unit root testing, is achieved by purging the impact of unconditional heteroskedasticity from the data using a kernel estimate of volatility before the application of the bubble detection methods proposed in Phillips, Shi and Yu (2015) (PSY). The modified statistic is shown to achieve the same limiting null distribution as the corresponding (heteroskedasticity-uncorrected) statistic from PSY would obtain under homoskedasticity, such that the usual critical values provided in PSY may still be used. Versions of the test based on regressions including either no intercept or a (redundant) intercept are considered. Representations for asymptotic local power against a single bubble model are also derived. Monte Carlo simulation results highlight that neither one of these tests dominates the other across different bubble locations and magnitudes, and across different models of time-varying volatility. Accordingly, we also propose a test based on a union of rejections between the with- and without-intercept variants of the modified PSY test. The union procedure is shown to perform almost as well as the better of the constituent tests for a given DGP, and also performs very well compared to existing heteroskedasticity-robust tests across a large range of simulation DGPs.

We develop tests for predictability that are robust to both the magnitude of the initial condition and the degree of persistence of the predictor. While the popular Bonferroni Q test of Campbell and Yogo displays excellent power properties for strongly persistent predictors with an asymptotically negligible initial condition, it can suffer from severe size distortions and power losses when either the initial condition is asymptotically non-negligible or the predictor is weakly persistent. The Bonferroni t test of Elliott, and Stock, although displaying power well below that of the Bonferroni Q test for strongly persistent predictors with an asymptotically negligible initial condition, displays superior size control and power when the initial condition is asymptotically nonnegligible. In the case where the predictor is weakly persistent, a conventional regression t test comparing to standard normal quantiles is known to be asymptotically optimal under Gaussianity. Based on these properties, we propose two asymptotically size controlled hybrid tests that are functions of the Bonferroni Q, Bonferroni t, and conventional t tests. Our proposed hybrid tests exhibit very good power regardless of the magnitude of the initial condition or the persistence degree of the predictor. An empirical application to the data originally analyzed by Campbell and Yogo shows our new hybrid tests are much more likely to find evidence of predictability than the Bonferroni Q test when the initial condition of the predictor is estimated to be large in magnitude.

The Bonferroni Q$$ Q $$ test is widely used in empirical studies investigating predictability in asset returns by strongly persistent and endogenous predictors. Its formulation, however, only allows for a constant mean in the predictor, seemingly at odds with many of the predictors used in practice. We establish the asymptotic size and local power properties of the Q$$ Q $$ test, and the corresponding Bonferroni t$$ t $$-test, under a local-to-zero specification for a linear trend in the predictor, revealing that size and power depend on the magnitude of the trend for both. To rectify this, we develop with-trend variants of the operational Bonferroni Q$$ Q $$ and t$$ t $$ tests. However, where a trend is not present in the predictor, we show that these tests lose (both finite sample and asymptotic local) power relative to the extant constant-only versions of the tests. In practice, uncertainty will necessarily exist over whether a linear trend is genuinely present in the predictor or not. To deal with this, we also develop hybrid tests based on union-of-rejections and switching mechanisms to capitalise on the relative power advantages of the constant-only tests when a trend is absent (or very weak) and the with-trend tests otherwise. A further extension allows the use of a conventional t$$ t $$-test where the predictor appears to be weakly persistent. We show that, overall, our recommended hybrid test can offer excellent size and power properties regardless of whether or not a linear trend is present in the predictor, or the predictor's degrees of persistence and endogeneity. An empirical application illustrates the practical relevance of our new approach.JEL Classifications: C22, C12, G14

We are inviting submissions for a special issue of the Journal of Time Series Analysis on methods for detecting bubbles and crashes in economic and financial time series. This is a field of research that has grown rapidly in importance in recent times, and the aim of the special issue is to bring together new contributions in the areas of modelling, testing and dating of bubble-type time series processes. We welcome contributions that contain significant developments in theory, methodology or application. All submissions must contain original unpublished work that is not being considered for publication elsewhere. Any given person can only be an author on one paper submitted to the special issue (without explicit permission from the Editor). The deadline for submissions is 31 December 2023. However, papers can be submitted at any time and once they are received, they will enter the editorial system immediately. Submissions will be refereed according to standard procedures for the Journal of Time Series Analysis. Please see the Journal of Time Series Analysis Author Guidelines for how to submit: https://onlinelibrary.wiley.com/page/journal/14679892/homepage/forauthors.html During the submission process, you should choose the option “Special Issue Paper” and also the name of the intended issue when prompted: “Bubbles and Crashes”. For any information about JTSA, please contact the Editor-in-Chief, Robert Taylor, University of Essex, [email protected]

Standard methods, such as sequential procedures based on Johansen's (pseudo-)likelihood ratio (PLR) test, for determining the co-integration rank of a vector autoregressive (VAR) system of variables integrated of order one can be significantly affected, even asymptotically, by unconditional heteroskedasticity (non-stationary volatility) in the data. Known solutions to this problem include wild bootstrap implementations of the PLR test or the use of an information criterion, such as the BIC, to select the co-integration rank. Although asymptotically valid in the presence of heteroskedasticity, these methods can display very low finite sample power under some patterns of non-stationary volatility. In particular, they do not exploit potential efficiency gains that could be realized in the presence of non-stationary volatility by using adaptive inference methods. Under the assumption of a known autoregressive lag length, Boswijk and Zu develop adaptive PLR test based methods using a non-parametric estimate of the covariance matrix process. It is well-known, however, that selecting an incorrect lag length can significantly impact on the efficacy of both information criteria and bootstrap PLR tests to determine co-integration rank in finite samples. We show that adaptive information criteria-based approaches can be used to estimate the autoregressive lag order to use in connection with bootstrap adaptive PLR tests, or to jointly determine the co-integration rank and the VAR lag length and that in both cases they are weakly consistent for these parameters in the presence of non-stationary volatility provided standard conditions hold on the penalty term. Monte Carlo simulations are used to demonstrate the potential gains from using adaptive methods and an empirical application to the U.S. term structure is provided.

We propose new tests for long-horizon predictability based on IVX estimation of a transformed regression which explicitly accounts for the over-lapping nature of the dependent variable in the long-horizon regression arising from temporal aggregation. To improve efficiency, we moreover incorporate the residual augmentation approach recently used in the context of short-horizon predictability testing by Demetrescu and Rodrigues (2022). Our proposed tests improve on extant tests in the literature in a number of ways. First, they allow practitioners to remain ambivalent over the strength of the persistence of the predictors. Second, they are valid under much weaker conditions on the innovations than extant long-horizon predictability tests; in particular, we allow for general forms of conditional and unconditional heteroskedasticity in the innovations, neither of which are tied to a parametric model. Third, unlike the popular Bonferroni-based methods in the literature, our proposed tests can handle multiple predictors, and can be easily implemented as either one or two-sided hypotheses tests. Monte Carlo analysis suggests that our preferred tests offer improved finite sample properties compared to the leading tests in the literature. We report results from an empirical application investigating the use of real exchange rates for predicting nominal exchange rates and inflation.

We explore how information additional to a specific price series can be used to improve the power of popular univariate autoregressive-based methods for detecting and dating speculative price bubble episodes. Following Phillips et al. (2011, 2015) we base our approach on sequences of sub-sample regression-based augmented Dickey-Fuller [ADF] statistics. Our point of departure from these extant procedures is to allow for additional information in the testing and dating procedures. To do so we follow the approach of Hansen (1995) and augment the sub-sample ADF regressions with covariate regressors. The limiting null distributions of the resulting statistics depend on the long-run squared correlation between the covariates and the regression error. We show that this dependence can be accounted for by using a residual bootstrap re-sampling method. Simulation evidence shows that including relevant covariates can significantly improve the efficacy of both the resulting bubble detection tests and the associated date-stamping procedure, relative to using standard sub-sample ADF statistics. An empirical application of the proposed methodology to monthly S&P 500 data is considered, using a variety of candidate covariates. Using these covariates, the onset of the dotcom bubble and the bubble associated with Black Monday are both identified significantly earlier than when using standard methods.

Our dear friend and world-leading econometrician, Professor Michael John McAleer, passed away on 8 July 2021 after a long and graceful fight with cancer. Mike’s father was Irish and his mother Japanese. His formative years were spent in Japan, with fluency in Japanese and a lifelong affinity with Asian cultures. His grace and good humor during his battle with cancer is an example to all, and a true model of resilience and the power of a positive mental attitude. Mike continued to be a highly active researcher right up until his untimely death. Mike obtained his PhD from Queens University, Canada. Mike spent most of his professional career in Australia, including appointments at the Australian National University and the University of Western Australia. He also held distinguished positions at a number of higher educational institutions spanning several continents. Mike was a passionate debater and thought deeply and argued effectively on the pros and cons of various econometric methods and approaches. His research interests ranged widely in econometrics, financial econometrics, finance, energy economics, economics of patents, bibliometrics, tourism, and lastly COVID-19-related research. Mike was generous and easy to work with and was given to respect and kindness toward his collaborators and students. Michael McAleer is one of the most published econometricians in the world, in a record of scholarly collaboration that is unique in its breadth and width, involving many coauthors, especially younger scholars. In particular, Mike coauthored 415 publications, with 6786 citations, as indicated on Publons, and 1270 referenced pieces on Google Scholar, with 23,273 citations. He was ranked 62 on REPEC for work in economics over a recent 10 year period, 46 in econometrics globally on Google Scholar, and 8 in Financial Econometrics. Mike was also an outstanding Associate Editor of Econometric Reviews, with one of the longest years of service for the journal since the late 1980s. He was the Editor-in-Chief of six international journals and was a member of the editorial boards of forty international journals. Among others, Mike edited and coedited numerous special issues of the Journal of Econometrics, providing timely state of art collections of contributions to the latest topics, some under-covered were it not for his tireless efforts. He contributed to the launching of several journals and showed special sensitivity to the needs of younger scholars. Mike was also a superb host and a great friend, always generous and graceful. He is sorely missed by all those of us who were privileged to call him a friend. This special issue of Econometric Reviews is dedicated to Mike’s memory and honors his contributions as scholar, author, teacher, mentor, and editor. We now provide a short summary of each of the papers (in alphabetical order of the first author) that comprise this special issue, Vol. 42; 9-10. Each was anonymously reviewed in accordance with the usual standards required for publication in Econometric Reviews. In the first paper, Andreou, Anyfantaki, Maasoumi, and Sala demonstrate how extreme value theory can identify stock price crashes, featuring low-probability events that produce large, firmspecific negative outliers in the conditional distribution returns. This is an alternative to the imprecise

Predictive regression methods are widely used to examine the predictability of (excess) stock returns by lagged financial variables characterized by unknown degrees of persistence and endogeneity. We develop a new hybrid test for predictability in these circumstances based on simple regression t-statistics. Where the predictor is endogenous, the optimal, but infeasible, test for predictability is based on the t-statistic on the lagged predictor in the basic predictive regression augmented with the current period innovation driving the predictor. We propose a feasible version of this augmented test, designed for the case where the predictor is an endogenous near-unit root process, using a GLS-based estimate of the innovation used in the infeasible test regression. The limiting null distribution of this statistic depends on both the endogeneity correlation parameter and the local-to-unity parameter characterizing the predictor. A method for obtaining asymptotic critical values is discussed and response surfaces are provided. We compare the asymptotic power properties of the feasible augmented test with those of a (non augmented) t-test recently considered in Harvey et al. and show that the augmented test is more powerful in the strongly persistent predictor case. We then propose using a weighted combination of the augmented statistic and the t-statistic of Harvey et al., where the weights are obtained using the p-values from a unit root test on the predictor. We find this can further improve asymptotic power in cases where the predictor has persistence at or close to that of a unit root process. Our final hybrid testing procedure then embeds the weighted statistic within a switching-based procedure which makes use of a standard predictive regression t-test, compared with standard normal critical values, when there is evidence for the predictor being weakly persistent. Monte Carlo simulations suggest that overall our new hybrid test displays superior finite sample performance to comparable extant tests.

The contribution of this paper is threefold. First, we demonstrate that, provided either a suitable bootstrap implementation is employed or heteroskedasticity-consistent stan-dard errors are used, the IVX-based predictability tests of Kostakis et al. (2015) retain asymptotically valid inference under the null hypothesis under considerably weaker as-sumptions on the innovations than are required by Kostakis et al. (2015). Second, under the same assumptions, we develop asymptotically valid bootstrap implementations of the IVX tests. Monte Carlo simulations show that the bootstrap tests deliver considerably more accurate finite sample inference than the asymptotic implementations of the tests under certain problematic parameter constellations, most notably for one-sided testing, and where multiple predictors are included. Third, we show how sub-sample implementations of the IVX approach can be used to develop asymptotically valid one-sided and two-sided tests for the presence of temporary windows of predictability. (c) 2022 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We consider estimation and inference in fractionally integrated time series models driven by shocks which can display conditional and unconditional heteroskedasticity of unknown form. Although the standard conditional sum-of-squares (CSS) estimator remains consistent and asymptotically normal in such cases, unconditional heteroskedasticity inflates its variance matrix by a scalar quantity, \lambda > 1, thereby inducing a loss in efficiency relative to the unconditionally homoskedastic case, \lambda = 1. We propose an adaptive version of the CSS estimator, based on non-parametric kernel-based estimation of the unconditional volatility process. We show that adaptive estimation eliminates the factor \lambda from the variance matrix, thereby delivering the same asymptotic efficiency as that attained by the standard CSS estimator in the unconditionally homoskedastic case and, hence, asymptotic efficiency under Gaussianity. Importantly, the asymptotic analysis is based on a novel proof strategy, which does not require consistent estimation (in the sup norm) of the volatility process. Consequently, we are able to work under a weaker set of assumptions than those employed in the extant literature. The asymptotic variance matrices of both the standard and adaptive CSS estimators depend on any weak parametric autocorrelation present in the fractional model and any conditional heteroskedasticity in the shocks. Consequently, asymptotically pivotal inference can be achieved through the development of confidence regions or hypothesis tests using either heteroskedasticity-robust standard errors and/or a wild bootstrap. Monte Carlo simulations and empirical applications illustrate the practical usefulness of the methods proposed.

Lobato and Robinson developed semiparametric tests for the null hypothesis that a series is weakly autocorrelated, or I(0), about a constant level, against fractionally integrated alternatives. These tests have the advantage that the user is not required to specify a parametric model for any weak autocorrelation present in the series. We extend this approach in two distinct ways. First, we show that it can be generalized to allow for testing of the null hypothesis that a series is I(delta) for any delta lying in the usual stationary and invertible region of the parameter space. The second extension is the more substantive and addresses the well-known issue in the literature that long memory and level breaks can be mistaken for one another, with unmodeled level breaks rendering fractional integration tests highly unreliable. To deal with this inference problem, we extend the Lobato and Robinson approach to allow for the possibility of changes in level at unknown points in the series. We show that the resulting statistics have standard limiting null distributions, and that the tests based on these statistics attain the same asymptotic local power functions as infeasible tests based on the unobserved errors, and hence there is no loss in asymptotic local power from allowing for level breaks, even where none is present. We report results from a Monte Carlo study into the finite-sample behavior of our proposed tests, as well as several empirical examples.

We develop easy-to-implement tests for return predictability which, relative to extant tests in the literature, display attractive finite sample size control and power across a wide range of persistence and endogeneity levels for the predictor. Our approach is based on the standard regression t-ratio and a variant where the predictor is quasi-GLS (rather than OLS) demeaned. In the strongly persistent near-unit root environment, the limiting null distributions of these statistics depend on the endogeneity and local-to-unity parameters characterising the predictor. Analysis of the asymptotic local power functions of feasible implementations of these two tests, based on asymptotically conservative critical values, motivates a switching procedure between the two, employing the quasi-GLS demeaned variant unless the magnitude of the estimated endogeneity correlation parameter is small. Additionally, if the data suggests the predictor is weakly persistent, our approach switches to the standard t-ratio test with reference to standard normal critical values.

T + log T mγ → 0 as T →∞, for any γ > 0. For ELW the range of permissible values in the optimisation in (3) is such that d2 − d1 ≤ 9 2 , although note that this does not restrict the value of d itself. The full set of required conditions for consistency are given in Assumptions A1 to A4 of Robinson (1995) for the LW estimator and in Assumptions 1 to 5 of Phillips and Shimotsu (2005) for the ELW estimator. Assumption 2, however, requires the stronger concept of convergence in mean square. This would need to be verified for any given estimator. Given convergence in distribution, a sufficient condition for Assumption 2 to hold is that d̂2 is uniformly integrable.

We generalize the Homm and Breitung (2012) CUSUM-based procedure for the real-time detection of explosive autoregressive episodes in financial price data to allow for time-varying volatility. Such behavior can heavily inflate the false positive rate (FPR) of the CUSUM-based procedure to spuriously signal the presence of an explosive episode. Our modified procedure involves replacing the standard variance estimate in the CUSUM statistics with a nonparametric kernel-based spot variance estimate. We show that the sequence of modified CUSUM statistics has a joint limiting null distribution which is invariant to any time-varying volatility present in the innovations and that this delivers a real-time monitoring procedure whose theoretical FPR is controlled. Simulations show that the modification is effective in controlling the empirical FPR of the procedure, yet sacrifices only a small amount of power to detect explosive episodes, relative to the standard procedure, when the shocks are homoskedastic. An empirical illustration using Bitcoin price data is provided.

SummaryWe introduce a new joint test for the order of fractional integration of a multivariate fractionally integrated vector autoregressive (FIVAR) time series based on applying the Lagrange multiplier principle to a feasible generalised least squares estimate of the FIVAR model obtained under the null hypothesis. A key feature of the test we propose is that it is constructed using a heteroskedasticity‐robust estimate of the variance matrix. As a result, the test has a standardχ2limiting null distribution under considerably weaker conditions on the innovations than are permitted in the extant literature. Specifically, we allow the innovations driving the FIVAR model to follow a vector martingale difference sequence allowing for both serial and cross‐sectional dependence in the conditional second‐order moments. We also do not constrain the order of fractional integration of each element of the series to lie in a particular region, thereby allowing for both stationary and non‐stationary dynamics, nor do we assume any particular distribution for the innovations. A Monte Carlo study demonstrates that our proposed tests avoid the large oversizing problems seen with extant tests when conditional heteroskedasticity is present in the data. We report an empirical case study for a sample of major US stocks investigating the order of fractional integration in trading volume and different measures of volatility in returns, including realised variance. Our results suggest that both return volatility and trading volume are fractionally integrated, but with the former generally found to be more persistent (having a higher fractional exponent) than the latter, when more reliable proxies for volatility such as the range or realised variance are used.