Asymptotic Local Power Research Articles

Simple tests for stock return predictability with good size and power properties

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.

Adaptive Testing for Cointegration With Nonstationary Volatility

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.

An inverse norm weight spatial sign test for high-dimensional directional data

In this paper, we focus on the high-dimensional location testing problem of directional data under the assumption of rotationally symmetric distributions, where the data dimension is potentially much larger than the sample size. We study the family of directional weighted spatial sign tests for this testing problem and establish the asymptotic null distributions and local power properties of this family. In particular, we find that the test based on the inverse norm weight, named as the inverse norm weight spatial sign test, has the maximum asymptotic power in this family. As demonstrated by extensive numerical results, the inverse norm weight spatial sign test has advantages in empirical power compared with some other members in the family as well as some existing tests.

Robust Inference by Sub-sampling

This paper provides a simple technique of carrying out inference robust to serial correlation, heteroskedasticity and spatial correlation on the estimators which follow an asymptotic normal distribution. The idea is based on the fact that the estimates from a larger sample tend to have a smaller variance which can be expressed as a function of the variance of the estimator from smaller subsamples. The major advantage of the technique other than the ease of application and simplicity is its finite sample performance both in terms of the empirical null rejection probability as well as the power of the test. It does not restrict the data in terms of structure in any way and works pretty well for any kind of heteroskedasticity, autocorrelation and spatial correlation in a finite sample. Furthermore, unlike theoretical HAC robust techniques available in the existing literature, it does not require any kernel estimation and hence eliminates the discretion of the analyst to choose a specific kernel and bandwidth. The technique outperforms the Ibragimov and Muller (2010) approach in terms of null rejection probability as well as the local asymptotic power of the test.

An Inverse Norm Sign Test of Location Parameter for High-Dimensional Data

We consider the one sample location testing problem for high-dimensional data, where the data dimension is potentially much larger than the sample size. We devise a novel inverse norm sign test (INST) that is consistent and has much improved power than many existing popular tests. We further construct a general class of weighted spatial sign tests which includes these existing tests, and show that INST is the optimal member within this class, in that it is consistent and is uniformly more powerful than all other members. We establish the asymptotic null distribution and local power property of the class of tests rigorously. Extensive numerical experiments demonstrate the superiority of INST in terms of both efficiency and robustness.

Nonparametric Inference of Jump Autocorrelation

Understanding the jump dynamics of market prices is important for asset pricing and risk management. Despite their analytical tractability, parametric models may impose unrealistic restrictions on the temporal dependence structure of jumps. In this paper, we introduce a nonparametric inference procedure for the presence of jump autocorrelation in the DGP. Our toolkit includes (i) an omnibus test that jointly detect the autocorrelation of stationary jumps over all lags, and (ii) a jump autocorrelogram that enables visualization and pointwise inference of jump autocorrelation. We establish asymptotic normality and local power of our procedure for a rich set of local alternatives (e.g., self-exciting and/or self-inhibitory jumps). Under a unified framework that combines infill and long-span asymptotics, the joint test for jump autocorrelations is robust to the choice of sampling scheme and different degree of jump activity. Simulation study confirms its robustness property and reveals its competitive size and power performance relative to existing tests. In an empirical study on high-frequency stock returns, our procedure uncovers a wide array of jump autocorrelation profiles for different stocks in different time periods.

Tests For the Parameters of Chirp Signal Model

In this paper, the testing of the hypothesis problem on the unknown parameters involved in one-dimensional chirp signal model is explored. To be precise, we here theoretically investigate whether the vector of unknown parameters is the same as the vector with specified parameters. For that purpose, we propose four tests based on the least squares and the least absolute deviation estimators of the unknown parameters using $L_1$ and $L_2$ distances. It is shown that the proposed tests are consistent (i.e., the power of the tests tend to one as the sample size tends to infinity). In addition, the asymptotic local power of the tests using contiguous (local) alternatives is also thoroughly studied. An extensive simulation study shows the satisfactory performance of the new tests, and the usefulness of the proposed tests is exhibited on a few benchmark real datasets that are closely associated with various chirp signal models.

Heteroskedasticity‐Robust Unit Root Testing for Trending Panels

Time‐varying volatility and linear trends are common features of several macroeconomic time series. Recent articles have proposed panel unit root tests (PURTs) that are pivotal in the presence of volatility shifts, excluding linear trends, however. This article proposes a new PURT that works well for data that is both heteroskedastic and trending. Under the null hypothesis, the test statistic has a limiting Gaussian distribution. We derive the local asymptotic power to underpin the consistency of the test statistic. Simulation results reveal that the test performs well in small samples. As an empirical illustration, we examine the stationarity of energy use per capita in OECD economies. While the series are in general difference stationary, they could also be considered as trend stationary for specific time spans.

Testing for independence in arbitrary distributions

Statistics are proposed for testing the hypothesis that arbitrary random variables are mutually independent. The tests are consistent and well behaved for any marginal distributions; they can be used, for example, for contingency tables which are sparse or whose dimension depends on the sample size, as well as for mixed data. No regularity conditions, data jittering, or binning mechanisms are required. The statistics are rank-based functionals of Cramér–von Mises type whose asymptotic behaviour derives from the empirical multilinear copula process. Approximate |$p$|-values are computed using a wild bootstrap. The procedures are simple to implement and computationally efficient, and maintain their level well in moderate to large samples. Simulations suggest that the tests are robust with respect to the number of ties in the data, can easily detect a broad range of alternatives, and outperform existing procedures in many settings. Additional insight into their performance is provided through asymptotic local power calculations under contiguous alternatives. The procedures are illustrated on traumatic brain injury data.

Improved inference in generalized mean-reverting processes with multiple change-points

In this paper, we consider inference problem about the drift parameter vector in generalized mean reverting processes with multiple and unknown change-points. In particular, we study the case where the parameter may satisfy uncertain restriction. As compared to the results in literature, we generalize some findings in five ways. First, we consider the model which incorporates the uncertain prior knowledge. Second, we derive the unrestricted estimator (UE) and the restricted estimator (RE) and we study their asymptotic properties. Third, we derive a test for testing the hypothesized restriction and we derive its asymptotic local power. We also prove that the proposed test is consistent. Fourth, we construct a class of shrinkage type estimators (SEs) which encloses the UE, the RE and classical SEs. Fifth, we derive the relative risk dominance of the proposed estimators. More precisely, we prove that the SEs dominate the UE. Finally, we present some simulation results which corroborate the established theoretical findings.

Local Asymptotic Equivalence of the Bai and Ng (2004) and Moon and Perron (2004) Frameworks for Panel Unit Root Testing

This paper considers unit-root tests in large n and large T heterogeneous panels with cross-sectional dependence generated by unobserved factors. We reconsider the two prevalent approaches in the literature, that of Moon and Perron (2004) and the PANIC setup proposed in Bai and Ng (2004). While these have been considered as completely different setups, we show that, in case of Gaussian innovations, the frameworks are asymptotically equivalent in the sense that both experiments are locally asymptotically normal (LAN) with the same central sequence. Using Le Cam's theory of statistical experiments we determine the local asymptotic power envelope and derive an optimal test jointly in both setups. We show that the popular Moon and Perron (2004) and Bai and Ng (2010) tests only attain the power envelope in case there is no heterogeneity in the long-run variance of the idiosyncratic components. The new test is asymptotically uniformly most powerful irrespective of possible heterogeneity. Moreover, it turns out that for any test, satisfying a mild regularity condition, the size and local asymptotic power are the same under both data generating processes. Thus, applied researchers do not need to decide on one of the two frameworks to conduct unit root tests. Monte-Carlo simulations corroborate our asymptotic results and document significant gains in finite-sample power if the variances of the idiosyncratic shocks differ substantially among the cross sectional units.

Adaptive Testing for Cointegration with Nonstationary Volatility

This paper generalises Boswijk and Zu (2018)'s adaptive unit root test for time series with nonstationary volatility to a multivariate context. 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 by Cavaliere et al. (2010, 2014). 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. Two empirical examples illustrate the applicability of the tests.

Asymptotic Distribution of the Score Test for Detecting Marks in Hawkes Processes

The asymptotic distribution of the score test of the null hypothesis that marks do not impact the intensity of a Hawkes marked self-exciting point process is shown to be chi-squared. For local asymptotic power, the distribution against local alternatives is also established as non-central chi-squared. These asymptotic results are derived using existing asymptotic results for likelihood estimates of the unmarked Hawkes process model together with mild additional conditions on the moments and ergodicity of the marks process and an additional uniform boundedness assumption, shown to be true for the exponential decay Hawkes process.

TESTING THE ORDER OF FRACTIONAL INTEGRATION OF A TIME SERIES IN THE POSSIBLE PRESENCE OF A TREND BREAK AT AN UNKNOWN POINT

We develop a test, based on the Lagrange multiplier [LM] testing principle, for the value of the long memory parameter of a univariate time series that is composed of a fractionally integrated shock around a potentially broken deterministic trend. Our proposed test is constructed from data which are de-trended allowing for a trend break whose (unknown) location is estimated by a standard residual sum of squares estimator applied either to the levels or first differences of the data, depending on the value specified for the long memory parameter under the null hypothesis. We demonstrate that the resulting LM-type statistic has a standard limiting null chi-squared distribution with one degree of freedom, and attains the same asymptotic local power function as an infeasible LM test based on the true shocks. Our proposed test therefore attains the same asymptotic local optimality properties as an oracle LM test in both the trend break and no trend break environments. Moreover, this asymptotic local power function does not alter between the break and no break cases and so there is no loss in asymptotic local power from allowing for a trend break at an unknown point in the sample, even in the case where no break is present. We also report the results from a Monte Carlo study into the finite-sample behaviour of our proposed test.

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

AbstractRecent 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.

Subvector inference when the true parameter vector may be near or at the boundary

Extremum estimators are not asymptotically normally distributed when the estimator satisfies the restrictions on the parameter space – such as the non-negativity of a variance parameter – and the true parameter vector is near or at the boundary. This possible lack of asymptotic normality makes it difficult to construct tests for testing subvector hypotheses that control asymptotic size in a uniform sense and have good local asymptotic power irrespective of whether the true parameter vector is at, near, or far from the boundary. We propose a novel estimator that is asymptotically normally distributed even when the true parameter vector is near or at the boundary and the objective function is not defined outside the parameter space. The proposed estimator allows the implementation of a new test based on the Conditional Likelihood Ratio statistic that is easy-to-implement, controls asymptotic size, and has good local asymptotic power properties. Furthermore, we show that the test enjoys certain asymptotic optimality properties when the parameter of interest is scalar. In an application of the random coefficients logit model (Berry, Levinsohn and Pakes, 1995) to the European car market, we find that, for most parameters, the new test leads to tighter confidence intervals than the two-sided t-test commonly used in practice.

Ratio tests under limiting normality

ABSTRACTWe propose a class of ratio tests that is applicable whenever a cumulation (of transformed) data is asymptotically normal upon appropriate normalization. The Karhunen–Loève theorem is employed to compute weighted averages. The test statistics are ratios of quadratic forms of these averages and hence scale-invariant, also called self-normalizing: The scaling parameter cancels asymptotically. Limiting distributions are obtained. Critical values and asymptotic local power functions can be calculated by standard numerical means. The ratio tests are directed against local alternatives and turn out to be almost as powerful as optimal competitors, without being plagued by nuisance parameters at the same time. Also in finite samples they perform well relative to self-normalizing competitors.

Heteroskedasticity-Robust Unit Root Testing for Trending Panels

Time‐varying volatility and linear trends are common features of several macroeconomic time series. Recent articles have proposed panel unit root tests (PURTs) that are pivotal in the presence of volatility shifts, excluding linear trends, however. This article proposes a new PURT that works well for data that is both heteroskedastic and trending. Under the null hypothesis, the test statistic has a limiting Gaussian distribution. We derive the local asymptotic power to underpin the consistency of the test statistic. Simulation results reveal that the test performs well in small samples. As an empirical illustration, we examine the stationarity of energy use per capita in OECD economies. While the series are in general difference stationary, they could also be considered as trend stationary for specific time spans.

Testing for a unit root against ESTAR stationarity

Abstract In this paper we examine the local power of unit root tests against globally stationary exponential smooth transition autoregressive [ESTAR] alternatives under two sources of uncertainty: the degree of nonlinearity in the ESTAR model, and the presence of a linear deterministic trend. First, we show that the KSS test (Kapetanios, G., Y. Shin, and A. Snell. 2003. “Testing for a Unit Root in the Nonlinear STAR Framework.” Journal of Econometrics 112: 359–379) for nonlinear stationarity has local asymptotic power gains over standard Dickey-Fuller [DF] tests for certain degrees of nonlinearity in the ESTAR model, but that for other degrees of nonlinearity, the linear DF test has superior power. Second, we derive limiting distributions of demeaned, and demeaned and detrended KSS and DF tests under a local ESTAR alternative when a local trend is present in the DGP. We show that the power of the demeaned tests outperforms that of the detrended tests when no trend is present in the DGP, but deteriorates as the magnitude of the trend increases. We propose a union of rejections testing procedure that combines all four individual tests and show that this captures most of the power available from the individual tests across different degrees of nonlinearity and trend magnitudes. We also show that incorporating a trend detection procedure into this union testing strategy can result in higher power when a large trend is present in the DGP.

Lag truncation and the local asymptotic distribution of the ADF test for a unit root

The issue of lag selection in ADF unit root testing is important, even asymptotically, for if the number of lags is not allowed to increase at a certain rate the test might not be correctly sized. However, size control is not the only concern. Indeed, simulations have repeatedly shown how increasing lag lengths tend to be associated with reductions in power, thus adding to the well-known low power problem when the alternative is local to the unit root. But while the simulation evidence is plentiful, there is as of yet almost no asymptotic results that can be used to ascertain whether lag length has any effect on the local asymptotic power of the ADF test. The purpose of the present paper is to fill this gap in the literature.