Explosive Asset Price Bubble Detection with Unknown Bubble Length and Initial Condition

Recent research has proposed a method of detecting explosive processes that is based on forward recursions of OLS, right‐tailed, Dickey–Fuller [DF] unit root tests. In this paper, an alternative approach using GLS DF test statistics is considered. We derive limiting distributions for both mean‐invariant and trend‐invariant versions of OLS and GLS‐based Phillips, Wu and Yu (2011, International Economic Review 52, 201–226) [PWY] test statistics under a temporary, locally explosive alternative. These limits are shown to be dependent on both the value of the initial condition and the start and end points of the temporary explosive regime. Local asymptotic power simulations show that a GLS version of the PWY statistic offers superior power when a large proportion of the data is explosive, but that the OLS approach is preferred for explosive periods of short duration as a proportion of the total sample. These power differences are magnified by the presence of an asymptotically non‐negligible initial condition. We propose a union of rejections procedure that capitalizes on the respective power advantages of both OLS and GLS‐based approaches. This procedure achieves power close to the effective envelope provided by the two individual PWY tests across all settings of the initial condition and length of the explosive period considered in this paper. These results are shown to be robust to the point in the sample at which the temporary explosive regime occurs. An application of the union procedure to NASDAQ prices confirms the empirical value of this testing strategy.     

Testing for unit roots in the possible presence of multiple trend breaks using minimum Dickey–Fuller statistics

Trend breaks appear to be prevalent in macroeconomic time series, and unit root tests therefore need to make allowance for these if they are to avoid the serious effects that unmodelled trend breaks have on power. Carrion-i-Silvestre et al. (2009) propose a pre-test-based approach which delivers near asymptotically efficient unit root inference both when breaks do not occur and where multiple breaks occur, provided the break magnitudes are fixed. Unfortunately, however, the fixed magnitude trend break asymptotic theory does not predict well the finite sample power functions of these tests, and power can be very low for the magnitudes of trend breaks typically observed in practice. In response to this problem we propose a unit root test that allows for multiple breaks in trend, obtained by taking the infimum of the sequence (across all candidate break points in a trimmed range) of local GLS detrended augmented Dickey–Fuller-type statistics. We show that this procedure has power that is robust to the magnitude of any trend breaks, thereby retaining good finite sample power in the presence of plausibly-sized breaks. We also demonstrate that, unlike the OLS detrended infimum tests of Zivot and Andrews (1992), these tests display no tendency to spuriously reject in the limit when fixed magnitude trend breaks occur under the unit root null.     

The Flexible Fourier Form and Local Generalised Least Squares De‐trended Unit Root Tests*

In two recent papers Enders and Lee (2009) and Becker, Enders and Lee (2006) provide Lagrange multiplier and ordinary least squares de‐trended unit root tests, and stationarity tests, respectively, which incorporate a Fourier approximation element in the deterministic component. Such an approach can prove useful in providing robustness against a variety of breaks in the deterministic trend function of unknown form and number. In this article, we generalize the unit root testing procedure based on local generalized least squares (GLS) de‐trending proposed by Elliott, Rothenberg and Stock (1996) to allow for a Fourier approximation to the unknown deterministic component in the same way. We show that the resulting unit root tests possess good finite sample size and power properties and the test statistics have stable non‐standard distributions, despite the curious result that their limiting null distributions exhibit asymptotic rank deficiency.     

Unit root tests in panel data: asymptotic and finite-sample properties

We consider pooling cross-section time series data for testing the unit root hypothesis. The degree of persistence in individual regression error, the intercept and trend coefficient are allowed to vary freely across individuals. As both the cross-section and time series dimensions of the panel grow large, the pooled t-statistic has a limiting normal distribution that depends on the regression specification but is free from nuisance parameters. Monte Carlo simulations indicate that the asymptotic results provide a good approximation to the test statistics in panels of moderate size, and that the power of the panel-based unit root test is dramatically higher, compared to performing a separate unit root test for each individual time series.     

On the Power of GLS-Type Unit Root Tests

Elliott, Rothenberg and Stock (1996), (ERS), present a 'GLS' variant of the Dickey-Fuller (DF) unit root test. Their statistic is approximately point-optimal invariant at a chosen local alternative, and usually displays better finite sample power than the DF test. Following the usual efficiency motive for GLS estimation, the higher finite sample power of the ERS test has often been attributed to the greater accuracy of the estimate of the series' non-stochastic component under stationary alternatives close to the null. This paper shows that the GLS estimates of the non-stochastic component are not, in general, more accurate. The power gain arises from the fact that the GLS statistic's null distribution has a greater positive shift relative to the DF test, than its distribution under relevant alternatives, and this persists even when the GLS estimates of the non stochastics have higher variance than the OLS estimates.     

Testing for a unit root in the volatility of asset returns

It is now well established that the volatility of asset returns is time varying and highly persistent. One leading model that is used to represent these features of the data is the stochastic volatility model. The researcher may test for non-stationarity of the volatility process by testing for a unit root in the log-squared time series. This strategy for inference has many advantages, but is not followed in practice because these unit root tests are known to have very poor size properties. In this paper I show that new tests that are robust to negative MA roots allow a reliable test for a unit root in the volatility process to be conducted. In applying these tests to exchange rate and stock returns, strong rejections of non-stationarity in volatility are obtained. Copyright © 1999 John Wiley & Sons, Ltd.     

同期相关面板数据退势单位根检验的小样本性质

本文基于SUR回归将时间序列的两种单位根检验（ADF—GLS检验）推广到面板数据，得到了同期相关面板数据退势单位根检验，称为SUR—ADF—GLS检验。通过蒙特卡洛试验研究发现，SUR—ADF—GLS检验具有良好的小样本性质。并且，SUR—ADF—GLS检验关于面板数据的同期相关性结构存在着较强的“依存性”。     

Panel Unit‐root Tests for Cross‐sectionally Correlated Panels: A Monte Carlo Comparison*

This paper deals with the finite‐sample performance of a set of unit‐root tests for cross‐correlated panels. Most of the available macroeconomic time series cover short time periods. The lack of information, in terms of time observations, implies that univariate tests are not powerful enough to reject the null of a unit‐root while panel tests, by exploiting the large number of cross‐sectional units, have been shown to be a promising way of increasing the power of unit‐root tests. We investigate the finite sample properties of recently proposed panel unit‐root tests for cross‐sectionally correlated panels. Specifically, the size and power of Choi's [Econometric Theory and Practice: Frontiers of Analysis and Applied Research: Essays in Honor of Peter C. B. Phillips, Cambridge University Press, Cambridge (2001)], Bai and Ng's [Econometrica (2004), Vol. 72, p. 1127], Moon and Perron's [Journal of Econometrics (2004), Vol. 122, p. 81], and Phillips and Sul's [Econometrics Journal (2003), Vol. 6, p. 217] tests are analysed by a Monte Carlo simulation study. In synthesis, Moon and Perron's tests show good size and power for different values of T and N, and model specifications. Focusing on Bai and Ng's procedure, the simulation study highlights that the pooled Dickey–Fuller generalized least squares test provides higher power than the pooled augmented Dickey–Fuller test for the analysis of non‐stationary properties of the idiosyncratic components. Choi's tests are strongly oversized when the common factor influences the cross‐sectional units heterogeneously.     

Testing for Predictability in panels with General Predictors

The difficulty of predicting returns has recently motivated researchers to start looking for tests that are either more powerful or robust to more features of the data. Unfortunately, the way that these tests work typically involves trading robustness for power or vice versa. The current paper takes this as its starting point to develop a new panel‐based approach to predictability that is both robust and powerful. Specifically, while the panel route to increased power is not new, the way in which the cross‐section variation is exploited also to achieve robustness with respect to the predictor is. The result is two new tests that enable asymptotically standard normal and chi‐squared inference across a wide range of empirically relevant scenarios in which the predictor may be stationary, moderately non‐stationary, nearly non‐stationary, or indeed unit root non‐stationary. The type of cross‐section dependence that can be permitted in the predictor is also very general, and can be weak or strong, although we do require that the cross‐section dependence in the regression errors is of the strong form. What is more, this generality comes at no cost in terms of complicated test construction. The new tests are therefore very user‐friendly. Copyright © 2016 John Wiley & Sons, Ltd.     

Cross‐sectional Dependency and Size Distortion in a Small‐sample Homogeneous Panel Data Unit Root Test*

In this paper, we investigate the effects of cross‐sectional disturbance correlation in a homogeneous panel data unit root test. As reported by other authors, the unit root test has incorrect size in the presence of cross‐sectional correlation. We suggest that a previously known estimator can be used to reduce the size distortions. We supply response surface estimates for critical values and study the size characteristics of the proposed test. We find that the suggested estimator performs well in small‐sample homogeneous panel data unit root tests. The reduction in size distortion comes at a small cost of lower power against a stationary alternative.     

A robust version of the KPSS test based on indicators

This paper proposes a test of the null hypothesis of stationarity that is robust to the presence of fat-tailed errors. The test statistic is a modified version of the so-called KPSS statistic. The modified statistic uses the “sign” of the data minus the sample median, whereas KPSS used deviations from means. This “indicator” KPSS statistic has the same limit distribution as the standard KPSS statistic under the null, without relying on assumptions about moments, but a different limit distribution under unit root alternatives. The indicator test has lower power than standard KPSS when tails are thin, but higher power when tails are fat.     

Trend‐stationary GNP: evidence from a new exact pointwise most powerful invariant unit root test

There has been a substantial debate whether GNP has a unit root. However, statistical tests have had little success in distinguishing between unit‐root and trend‐reverting specifications because of poor statistical properties. This paper develops a new exact small‐sample, pointwise most powerful unit root test that is invariant to the unknown mean and scale of the time series tested, that generates exact small‐sample critical values, powers and p‐values, that has power which approximates the maximum possible power, and that is highly robust to conditional heteroscedasticity. This test decisively rejects the unit root null hypothesis when applied to annual US real GNP and US real per capita GNP series. This paper also develops a modified version of the test to address whether a time series contains a permanent, unit root process in addition to a temporary, stationary process. It shows that if these GNP series contain a unit root process in addition to the stationary process, then it is most likely very small. Copyright © 2001 John Wiley & Sons, Ltd.     

Tests for Stationarity in Series with Endogenously Determined Structural Change*

We consider tests of the null hypothesis of stationarity against a unit root alternative, when the series is subject to structural change at an unknown point in time. Three extant tests are reviewed which allow for an endogenously determined instantaneous structural break, and a related fourth procedure is introduced. We further propose tests which permit the structural change to be gradual rather than instantaneous, allowing the null hypothesis to be stationarity about a smooth transition in linear trend. The size and power properties of the tests are investigated, and the tests are applied to four economic time series.     

Convergence of Prices and Rates of Inflation*

We consider how unit‐root and stationarity tests can be used to study the convergence of prices and rates of inflation. We show how the joint use of these tests in levels and first differences allows the researcher to distinguish between series that are converging and series that have already converged, and we set out a strategy to establish whether convergence occurs in relative prices or just in rates of inflation. Special attention is paid to the issue of whether a mean should be extracted in carrying out tests in first differences and whether there is an advantage to adopting a (Dickey–Fuller) unit‐root test based on deviations from the last observation. The asymptotic distribution of this last test statistic is given and Monte Carlo simulation experiments show that the test yields considerable power gains for highly persistent autoregressive processes with ‘relatively large’ initial conditions. The tests are applied to the monthly series of the consumer price index in the Italian regional capitals over the period 1970–2003.     

Evaluating GARCH models

In this paper, a unified framework for testing the adequacy of an estimated GARCH model is presented. Parametric Lagrange multiplier (LM) or LM type tests of no ARCH in standardized errors, linearity, and parameter constancy are proposed. The asymptotic null distributions of the tests are standard, which makes application easy. Versions of the tests that are robust against nonnormal errors are provided. The finite sample properties of the test statistics are investigated by simulation. The robust tests prove superior to the nonrobust ones when the errors are nonnormal. They also compare favourably in terms of power with misspecification tests previously proposed in the literature.     

A versatile and robust metric entropy test of time-reversibility,and other hypotheses

We examine the performance of a metric entropy statistic as a robust test for time-reversibility (TR), symmetry, and serial dependence. It also serves as a measure of goodness-of-fit. The statistic provides a consistent and unified basis in model search, and is a powerful diagnostic measure with surprising ability to pinpoint areas of model failure. We provide empirical evidence comparing the performance of the proposed procedure with some of the modern competitors in nonlinear time-series analysis, such as robust implementations of the BDS and characteristic function-based tests of TR, along with correlation-based competitors such as the Ljung–Box Q-statistic. Unlike our procedure, each of its competitors is motivated for a different, specific, context and hypothesis. Our evidence is based on Monte Carlo simulations along with an application to several stock indices for the US equity market.     

Unit Root Testing Using Covariates: Some Theory and Evidence

This paper analyzes the conditions under which power gains can be achieved using the Covariate Augmented Dickey-Fuller test (CADF) rather than the conventional Augmented Dickey-Fuller (ADF), and argues that this method has the advantage, relative to univariate unit root tests, of increasing power without suffering from the large size distortions affecting the latter. The inclusion of covariates affects unit root testing by: (a) reducing the standard error of the estimate of the autoregressive parameter without affecting the estimate itself, and/or (b) reducing both the standard error and the absolute value of the estimate itself. Conditions in terms of contemporaneous correlation and Granger causality are derived for case (a) or (b) to arise. As an illustration, it is shown that applying the more powerful CADF (rather than the ADF) test reverses the finding of a unit root for many US macroeconomic series.     

The impact of a Hausman pretest on the size of a hypothesis test: The panel data case

The size properties of a two-stage test in a panel data model are investigated where in the first stage a Hausman (1978) specification test is used as a pretest of the random effects specification and in the second stage, a simple hypothesis about a component of the parameter vector is tested, using a t$t$-statistic that is based on either the random effects or the fixed effects estimator depending on the outcome of the Hausman pretest. It is shown that the asymptotic size of the two-stage test equals 1 for empirically relevant specifications of the parameter space. The size distortion is caused mainly by the poor power properties of the pretest. Given these results, we recommend using a t$t$-statistic based on the fixed effects estimator instead of the two-stage procedure.     

Exactly distribution-free inference in instrumental variables regression with possibly weak instruments

This paper introduces a rank-based test for the instrumental variables regression model that dominates the Anderson–Rubin test in terms of finite sample size and asymptotic power in certain circumstances. The test has correct size for any distribution of the errors with weak or strong instruments. The test has noticeably higher power than the Anderson–Rubin test when the error distribution has thick tails and comparable power otherwise. Like the Anderson–Rubin test, the rank tests considered here perform best, relative to other available tests, in exactly identified models.     

A study of several new and existing tests for heteroscedasticity in the general linear model1

Several optimum non-parametric tests for heteroscedasticity are proposed and studied along with the tests introduced in the literature in terms of power and robustness properties. It is found that all tests are reasonably robust to the Ordinary Least Squares (OLS) residual estimates, number and character of the regressors. Only a few are robust to both the distributional and independence assumptions about the errors. The power of tests can be improved with the OLS residual estimates, the increased sample size and the variability of the regressors. It can be substantially reduced if the observations are not normally distributed, and may increase or decrease if the errors are dependent. Each test is optimum to detect a specific form of heteroscedasticity and a serious power loss may occur if the underlying heteroscedasticity assumption in the data generation deviates from it.