We construct a spot volatility estimator for high-frequency financial data which contain market microstructure noise. We prove consistency and derive the asymptotic distribution of the estimator. A data-driven method is proposed to select the scale parameter and the bandwidth parameter in the estimator. In Monte Carlo simulations, we compare the finite sample performance of our estimator with some existing estimators. Empirical examples are given to illustrate the potential applications of the estimator.JEL Classification: C58, C13, C14.

Summary Recent research has emphasized that permanent changes in the innovation variance (caused by structural shifts or an integrated volatility process) lead to size distortions in conventional unit root tests. It has been shown how these size distortions can be resolved using the wild bootstrap. In this paper, we first derive the asymptotic power envelope for the unit root testing problem when the non-stationary volatility process is known. Next, we show that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that non-parametric estimation of the volatility process leads to the same asymptotic power envelope. Implementation of the resulting test involves cross-validation and the wild bootstrap. A Monte Carlo experiment shows that the asymptotic results are reflected in finite sample properties, and an empirical analysis of real exchange rates illustrates the applicability of the proposed procedures.

This article develops a class of adaptive cointegration tests for multivariate time series with nonstationary volatility. Persistent changes in the innovation variance matrix of a vector autoregressive model lead to size distortions in conventional cointegration tests, which may be resolved using the wild bootstrap, as shown in recent work by Cavaliere, Rahbek, and Taylor. We show that it also leads to the possibility of constructing tests with higher power, by taking the time-varying volatilities and correlations into account in the formulation of the likelihood function and the resulting likelihood ratio test statistic. We find that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that nonparametric volatility matrix estimation does not lead to a loss of asymptotic local power relative to the case where the volatilities are observed. The asymptotic null distribution of the test is nonstandard and depends on the volatility process; we show that various bootstrap implementations may be used to conduct asymptotically valid inference. Monte Carlo simulations show that the resulting test has good size properties, and higher power than existing tests. Empirical analyses of the U.S. term structure of interest rates and purchasing power parity illustrate the applicability of the tests.

This article considers the problem of testing for an explosive bubble in financial data in the presence of time-varying volatility. We propose a sign-based variant of the Phillips, Shi, and Yu (2015, International Economic Review 56, 1043–1077) test. Unlike the original test, the sign-based test does not require bootstrap-type methods to control size in the presence of time-varying volatility. Under a locally explosive alternative, the sign-based test delivers higher power than the original test for many time-varying volatility and bubble specifications. However, since the original test can still outperform the sign-based one for some specifications, we also propose a union of rejections procedure that combines the original and sign-based tests, employing a wild bootstrap to control size. This is shown to capture most of the power available from the better performing of the two tests. We also show how a sign-based statistic can be used to date the bubble start and end points. An empirical illustration using Bitcoin price data is provided.

This article considers the problem of testing for an explosive bubble in financial data in the presence of time-varying volatility. We propose a weighted least squares-based variant of the Phillips et al.) test for explosive autoregressive behavior. We find that such an approach has appealing asymptotic power properties, with the potential to deliver substantially greater power than the established OLS-based approach for many volatility and bubble settings. Given that the OLS-based test can outperform the weighted least squares-based test for other volatility and bubble specifications, we also suggest a union of rejections procedure that succeeds in capturing the better power available from the two constituent tests for a given alternative. Our approach involves a nonparametric kernel-based volatility function estimator for computation of the weighted least squares-based statistic, together with the use of a wild bootstrap procedure applied jointly to both individual tests, delivering a powerful testing procedure that is asymptotically size-robust to a wide range of time-varying volatility specifications. KEYWORDS Explosive autoregression; rational bubble; right-tailed unit root testing; timevarying volatility; weighted least squares JEL CLASSIFICATION C12; C14; C22 IntroductionEmpirical identification of explosive behavior in financial asset price series is closely related to the study of rational bubbles, with a rational bubble deemed to have occurred if explosive characteristics are manifest in the time path of prices, but not for the dividends. Consequently, methods for testing for explosive time series behavior have been a focus of much recent research. In a now seminal paper, Phillips et al. (2011) [PWY] model potential bubble behavior using an explosive autoregressive specification, and suggest testing for such a property using the supremum of a sequence of forward recursive right-tailed Dickey-Fuller unit root tests. The test has been widely applied, and has also prompted the development of related test procedures, such as those of Homm and Breitung (2012) who consider a supremum of backward recursive Chow-type Dickey-Fuller statistics, and Phillips et al. (2015) who consider a double supremum of forward and backward recursive statistics.While these original papers assumed constant unconditional volatility in the underlying error process, in practice time-varying volatility is a well-known stylized fact observed in empirical financial data. Harvey et al. (2016) [HLST] demonstrate that the asymptotic null distribution of the PWY test depends on the nature of the volatility, and if the test is implemented using critical values derived under a homoskedastic error assumption, size is not controlled for nonstationary volatility patterns. This raises the possibility of misleading inference when using the standard PWY test in the presence of time-varying volatility, with the potential for spurious identification of a bubble. HLST propose a wild bootstrap implementation of the PWY test, which ensures