Finite Sample Power Research Articles

Bonferroni‐Type Tests for Return Predictability With Possibly Trending Predictors

ABSTRACTThe Bonferroni 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 test, and the corresponding Bonferroni ‐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 and 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 ‐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

ASYMPTOTICALLY UNIFORMLY MOST POWERFUL TESTS FOR UNIT ROOTS IN GAUSSIAN PANELS WITH CROSS-SECTIONAL DEPENDENCE GENERATED BY COMMON FACTORS

This paper considers testing for unit roots in Gaussian panels with cross-sectional dependence generated by common factors. Within our setup, we can analyze restricted versions of the two prevalent approaches in the literature, that of Moon and Perron (2004, Journal of Econometrics 122, 81–126), who specify a factor model for the innovations, and the PANIC setup proposed in Bai and Ng (2004, Econometrica 72, 1127–1177), who test common factors and idiosyncratic deviations separately for unit roots. We show that both frameworks lead to locally asymptotically normal experiments with the same central sequence and Fisher information. Using Le Cam’s theory of statistical experiments, we obtain the local asymptotic power envelope for unit-root tests. We show that the popular Moon and Perron (2004, Journal of Econometrics 122, 81–126) and Bai and Ng (2010, Econometric Theory 26, 1088–1114) tests only attain the power envelope in case there is no heterogeneity in the long-run variance of the idiosyncratic components. We develop a new test which is asymptotically uniformly most powerful irrespective of possible heterogeneity in the long-run variance of the idiosyncratic components. 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.

Testing Homoskedasticity in Spatial Panel Data Models

A robust test statistic for testing homoskedasticity in spatial panel data models that have entity and time fixed effects is introduced in a quasi maximum likelihood estimation setting. A size-correction approach is introduced to ensure that the score functions have an asymptotic distribution centered around zero in the local presence of certain nuisance parameters. The outer-product-of-martingale-difference (OPMD) method is used to formulate an estimator for the asymptotic variance of the score functions. The OPMD estimator and the adjusted score functions are used to formulate a computationally simple robust test statistic. The suggested test statistic does not require knowing the presence of spatial dependence in the outcome variable and/or the disturbance terms. The asymptotic distribution of the test statistic is established under the null and local alternative hypotheses. Through Monte Carlo simulations, the finite sample size and power properties of the proposed test statistic are investigated. Finally, two empirical applications are provided to illustrate the practical use of the proposed test statistic.

Highly irregular serial correlation tests

Tests are developed for neglected serial correlation when the information matrix is repeatedly singular under the null hypothesis. Specifically, consideration is given to white noise against a multiplicative seasonal Ar model, and a local-level model against a nesting Ucarima one. The proposed tests, which involve higher-order derivatives, are asymptotically equivalent to the likelihood ratio test but only require estimation under the null. It is shown that the tests effectively check that certain autocorrelations of the observations are zero, so their asymptotic distribution is standard. Monte Carlo exercises examine finite sample size and power properties, with comparisons made to alternative approaches.

Adaptive information-based methods for determining the co-integration rank in heteroskedastic VAR models

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.

A New Test on Asset Return Predictability with Structural Breaks

This article considers predictive regressions in which a structural break is allowed on an unknown date. We establish novel testing procedures for asset return predictability using empirical likelihood (EL) methods based on weighted score equations. The theoretical results are useful in practice because our unified framework does not require distinguishing whether the predictor variables are stationary or non-stationary. Monte Carlo simulation studies show that the EL-based tests perform well in terms of size and power in finite samples. Finally, as an empirical analysis, we test the predictability of the monthly S&P 500 value-weighted log excess return using various predictor variables.

Logistic or not Logistic?

We propose a new class of goodness‐of‐fit tests for the logistic distribution based on a characterization related to the density approach in the context of Stein's method. This characterization‐based test is a first of its kind for the logistic distribution. The asymptotic null distribution of the test statistic is derived and it is shown that the test is consistent against fixed alternatives. The finite sample power performance of the newly proposed class of tests is compared to various existing tests by means of a Monte Carlo study. It is found that this new class of tests are especially powerful when the alternative distributions are heavy tailed, like Student's t and Cauchy, or for skew alternatives such as the log‐normal, gamma and chi‐square distributions.

Asymptotic Behavior of Temporal Aggregation in Mixed‐Frequency Datasets

AbstractHere, we present an unexplored issue regarding temporal aggregation. When a model contains frequency‐dependent coefficients, such as a distinct long‐ and short‐term coefficient, temporal aggregation leads to inconsistent least squares estimates. Because the sub‐sampled variable's spectrum is equal to its folded original spectrum, the low‐frequency variable may exhibit a mixture of distinct linear relations for a given frequency. We propose a new method to disentangle the frequencies superposition based on band spectrum regression, thus avoiding the inconsistency problem. As a result, we can test for the presence of frequency‐dependent coefficients. We use stationary and non‐stationary linear semi‐parametric models to demonstrate our findings. Our Monte Carlo simulations show good finite sample size and power properties. Finally, our empirical study rejects the presence of a single coefficient for all frequencies between quarterly GDP and monthly US indicators.

Testing Linear Operator Constraints in Functional Response Regression with Incomplete Response Functions.

Hypothesis testing procedures are developed to assess linear operator constraints in function-on-scalar regression when incomplete functional responses are observed. The approach enables statistical inferences about the shape and other aspects of the functional regression coefficients within a unified framework encompassing three incomplete sampling scenarios: (i) partially observed response functions as curve segments over random sub-intervals of the domain; (ii) discretely observed functional responses with additive measurement errors; and (iii) the composition of former two scenarios, where partially observed response segments are observed discretely with measurement error. The latter scenario has been little explored to date, although such structured data is increasingly common in applications. For statistical inference, deviations from the constraint space are measured via integrated -distance between the model estimates from the constrained and unconstrained model spaces. Large sample properties of the proposed test procedure are established, including the consistency, asymptotic distribution and local power of the test statistic. Finite sample power and level of the proposed test are investigated in a simulation study covering a variety of scenarios. The proposed methodologies are illustrated by applications to U.S. obesity prevalence data, analyzing the functional shape of its trends over time, and motion analysis in a study of automotive ergonomics.

A new omnibus test of fit based on a characterization of the uniform distribution

ABSTRACT In this paper, we revisit the classical goodness-of-fit problems for univariate distributions; we propose a new testing procedure based on a characterization of the uniform distribution. Asymptotic theory for the simple hypothesis case is provided in a Hilbert-Space setting, including the asymptotic null distribution as well as values for the first four cumulants of this distribution, which are used to fit a Pearson system of distributions as an approximation to the limit distribution. Numerical results indicate that the null distribution of the test converges quickly to its asymptotic distribution, making the critical values obtained using the Pearson system particularly useful. Consistency of the test is shown against any fixed alternative distribution and we derive the limiting behaviour under fixed alternatives with an application to power approximation. We demonstrate the applicability of the newly proposed test when testing composite hypotheses. A Monte Carlo power study compares the finite sample power performance of the newly proposed test to existing omnibus tests in both the simple and composite hypothesis settings. This power study includes results related to testing for the uniform, normal, Pareto and gamma distributions. The empirical results obtained indicate that the test is competitive. An application of the newly proposed test in financial modelling is also included.

Tests of serial dependence for multivariate time series with arbitrary distributions

In this paper, one studies tests of serial independence using a fixed number p of consecutive observations from a stationary time series, first in the univariate case, and then in the multivariate case, where even high-dimensional vectors can be used. The common distribution function is not assumed to be continuous, and the test statistics are based on the multilinear copula process. A bootstrap procedure based on multipliers is also proposed and shown to be valid. Tests based on Spearman’s rho and Kendall’s tau statistics are also considered, extending the results known for the case of continuous distributions. Contiguity results are obtained for some specific models and sufficient conditions for consistency of test statistics are stated, as well as a data-driven procedure to select p. Also, numerical experiments are performed to assess the finite sample level and power of the proposed tests. A case study using a time series of Arctic sea ice extent images is used to illustrate the usefulness of the proposed methodologies. The R package MixedIndTests (Nasri et al., 2022) includes all the methodologies proposed in this article.

Theory of evolutionary spectra for heteroskedasticity and autocorrelation robust inference in possibly misspecified and nonstationary models

The literature on heteroskedasticity and autocorrelation robust (HAR) inference is extensive but its usefulness relies on stationarity of the relevant process, say Vt, usually a function of the data and estimated model residuals. Yet, a large body of work shows widespread evidence of various forms of nonstationarity in the latter. Also, many testing problems are such that Vt is stationary under the null hypothesis but nonstationary under the alternative. In either case, the consequences are possible size distortions and, especially, a reduction in power which can be substantial (e.g., non-monotonic power), since all such estimates are based on weighted sums of the sample autocovariances of Vt, which are inflated. We propose HAR inference methods valid under a broad class of nonstationary processes, labeled Segmented Local Stationary, which possess a spectrum that varies both over frequencies and time. It is allowed to change either slowly and continuously and/or abruptly at some time points, thereby encompassing most nonstationary models used in applied work. We introduce a double kernel estimator (DK-HAC) that applies a smoothing over both lagged autocovariances and time. The optimal kernels and bandwidth sequences are derived under a mean-squared error criterion. The data-dependent bandwidths rely on the “plug-in” approach using approximating parametric models having time-varying parameters estimated with standard methods applied to local data. Our method yields tests with good size and power in finite-samples. In particular, the power gains are achieved without notable size distortions.

Goodness-of-fit tests for a logistic regression model with missing covariates

Model checking for logistic regression with covariates missing at random is considered. Based on the ideas of Copas (1989) and Osius and Rojek (1992) and studies of Homser et al. (1997), proposed are the two-type goodness-of-fit tests, Pearson chi-squared and unweighted residual sum-of-squares tests, in which their test statistics are centralized by subtracting their estimated mean to be mean-zero-form test statistics via the inverse probability weighting (IPW) and nonparametric multiple imputation (MI) methods to solve the missing value problem. The asymptotic properties of these test statistics are established under the null hypothesis and some regularity conditions. The test statistics conducted by using the IPW and MI estimators are asymptotically equivalent. Proposed are the IPW method and two bootstrap re-sampling approaches for estimation of the variances of the proposed test statistics to solve the issue of underestimating their variances by the MI method of Rubin (1987). Simulation studies are carried out to assess the finite-sample power performances of these proposed tests. Two real data examples are used to illustrate the applicability of the proposed tests.

ON MULTIPLE STRUCTURAL BREAKS IN DISTRIBUTION: AN EMPIRICAL CHARACTERISTIC FUNCTION APPROACH

We estimate and test for multiple structural breaks in distribution via an empirical characteristic function approach. By minimizing the sum of squared generalized residuals, we can consistently estimate the break fractions. We propose a sup-F type test for structural breaks in distribution as well as an information criterion and a sequential testing procedure to determine the number of breaks. We further construct a class of derivative tests to gauge possible sources of structural breaks, which is asymptotically more powerful than the smoothed nonparametric tests for structural breaks. Simulation studies show that our method performs well in determining the appropriate number of breaks and estimating the unknown breaks. Furthermore, the proposed tests have reasonable size and excellent power in finite samples. In an application to exchange rate returns, our tests are able to detect structural breaks in distribution and locate the break dates. Our tests also indicate that the documented breaks appear to occur in variance and higher-order moments, but not so often in mean.

Testing for Co‐explosive Behaviour in Financial Time Series

AbstractThis article proposes a test to determine if two price series that each contain an explosive autoregressive regime consistent with the presence of a bubble are related in the sense that a linear combination of them is integrated of order zero. We refer to such a phenomenon as ‘co‐explosive behaviour’, and propose a test based on a stationarity testing framework. The test allows the explosive episode in one series to lead (or lag) that in the other by a number of time periods. We establish the asymptotic properties of the test statistic and propose a wild bootstrap procedure for obtaining critical values that are robust to heteroskedasticity. Simulations show that the proposed test has good finite sample size and power performance. An empirical application to detect whether co‐explosive behaviour exists among a set of precious and non‐ferrous metals is presented.

On robust testing for trend

This paper provides a simple approach for robust testing for the trend function in the time series under uncertainty over the order of integration of the error term. The proposed approach relies on the asymptotic normality of the trend coefficient estimator and utilizes t-statistic approach of Ibragimov and Müller (2010) based on splitting the sample. The Monte-Carlo results demonstrate that the approach has the correct finite sample size and favorable finite sample power properties for all data generating processes considered. The proposed approach is robust to very general assumptions on the error term including various forms of non-stationary volatility and heavy tails

Testing for individual and time effects in unbalanced panel data models with time-invariant regressors

<abstract><p>In this paper, we use a moment-based method to test the existence of the individual and time effects in unbalanced panel data models with time-invariant regressors. Based on the difference of two variance estimators of idiosyncratic errors, three test statistics are proposed. The test statistics for individual (time) effect is robust when the time (individual) effect exists, and is robust for the correlation between explanatory variables and individual or time effect. Additionally, they do not require prior distributional assumptions on the error term. The asymptotic properties of estimators and the test statistics are given in this paper. The Monte Carlo simulations show that the test statistics have good power in finite samples at various situations and a real example is studied for illustration.</p></abstract>

Testing for a Moderately Explosive Process with Structural Change in Drift*

AbstractThis paper studies large sample properties of a moderately explosive autoregression with a structural change in the unobservable drift term, and develops asymptotic tests for the null of moderate explosiveness under different dependence structures. When the innovation sequence is independently and identically distributed (i.i.d.), we show that the t statistic is asymptotically standard normal. When the innovations are weakly dependent in the form of homoskedasticity or conditional heteroskedasticity, we invoke the fixed‐smoothing asymptotics to construct the heteroskedasticity and autocorrelation robust standard error, under which the t statistic follows Student’s t distribution in large samples. Monte Carlo simulations show that our tests have small size distortion and high power in finite samples. As we impose no restrictions on the occurrence time and magnitude of the drift, our proposed asymptotic tests enjoy strong robustness and applicability.

A family of nonparametric unit root tests for processes driven by infinite variance innovations

Abstract This paper presents extensions to the family of nonparametric fractional variance ratio (FVR) unit root tests of Nielsen (2009. “A Powerful Test of the Autoregressive Unit Root Hypothesis Based on a Tuning Parameter Free Statistic.” Econometric Theory 25: 1515–44) under heavy tailed (infinite variance) innovations. In this regard, we first develop the asymptotic theory for these FVR tests under this setup. We show that the limiting distributions of the tests are free of serial correlation nuisance parameters, but depend on the tail index of the infinite variance process. Then, we compare the finite sample size and power performance of our FVR unit root tests with the well-known parametric ADF test under the impact of the heavy tailed shocks. Simulations demonstrate that under heavy tailed innovations, the nonparametric FVR tests have desirable size and power properties.

Testing Homoskedasticity in Cross-sectional Spatial Autoregressive Models

In this paper, we suggest an outer-product-of-gradient (OPG) variant of Lagrange multiplier (LM) test statistic for testing homoskedasticity in cross-sectional spatial autoregressive models. We use the OPG method to estimate the asymptotic variance of the score functions in a quasi-maximum likelihood (QML) setting. We use the OPG variance estimate to develop a robust test statistic in the local presence of spatial parameters. Under some general assumptions, we establish the asymptotic distribution of our test statistic under the null and local alternative hypotheses. In a Monte Carlo simulation, we investigate the finite sample size and power properties of our test. Our simulation results show that our tests work well in finite samples.