Finite Sample Power Research Articles

A nonparametric measure of heteroskedasticity

We introduce a nonparametric measure to quantify the degree of heteroskedasticity at a fixed quantile of the conditional distribution of a random variable. Our measure of heteroskedasticity is based on nonparametric quantile regressions and is expressed in terms of unrestricted and restricted expectations of quantile loss functions. It can be consistently estimated by replacing the unknown expectations by their nonparametric estimates. We derive a Bahadur-type representation for the nonparametric estimator of the measure. We provide the asymptotic distribution of this estimator, which one can use to build tests for the statistical significance of the measure. Thereafter, we establish the validity of a fixed regressor bootstrap that one can use in finite-sample settings to perform tests. A Monte Carlo simulation study reveals that the bootstrap-based test has a good finite sample size and power for a variety of data generating processes and different sample sizes. Finally, two empirical applications are provided to illustrate the importance of the proposed measure.

Tail Granger causalities and where to find them: Extreme risk spillovers vs spurious linkages

Identifying risk spillovers in financial markets is of great importance for assessing systemic risk and portfolio management. Granger causality in tail (or in risk) tests whether past extreme events of a time series help predicting future extreme events of another time series. The topology and connectedness of networks built with Granger causality in tail can be used to measure systemic risk and to identify risk transmitters. Here we introduce a novel test of Granger causality in tail which adopts the likelihood ratio statistic and is based on the multivariate generalization of a discrete autoregressive process for binary time series describing the sequence of extreme events of the underlying price dynamics. The proposed test has very good size and power in finite samples, especially for large sample size, allows inferring the correct time scale at which the causal interaction takes place, and it is flexible enough for multivariate extension when more than two time series are considered in order to decrease false detections as spurious effect of neglected variables. An extensive simulation study shows the performances of the proposed method with a large variety of data generating processes and it introduces also the comparison with the test of Granger causality in tail by Hong et al. (2009). We report both advantages and drawbacks of the different approaches, pointing out some crucial aspects related to the false detections of Granger causality for tail events. An empirical application to high frequency data of a portfolio of US stocks highlights the merits of our novel approach.

A new test of asset return predictability with an unstable predictor

This study constructs predictive regressions in which the predictable variable exhibits a level shift at some unknown date. We establish novel procedures to test asset return predictability via empirical likelihood (EL) methods based on weighted score equations. Monte Carlo simulations confirm that the EL-based tests perform well in terms of size and power in finite samples.

Testing for observation-dependent regime switching in mixture autoregressive models

Testing for regime switching when the regime switching probabilities are specified either as constants (‘mixture models’) or are governed by a finite-state Markov chain (‘Markov switching models’) are long-standing problems that have also attracted recent interest. This paper considers testing for regime switching when the regime switching probabilities are time-varying and depend on observed data (‘observation-dependent regime switching’). Specifically, we consider the likelihood ratio test for observation-dependent regime switching in mixture autoregressive models. The testing problem is highly nonstandard, involving unidentified nuisance parameters under the null, parameters on the boundary, singular information matrices, and higher-order approximations of the log-likelihood. We derive the asymptotic null distribution of the likelihood ratio test statistic in a general mixture autoregressive setting using high-level conditions that allow for various forms of dependence of the regime switching probabilities on past observations, and we illustrate the theory using two particular mixture autoregressive models. The likelihood ratio test has a nonstandard asymptotic distribution that can easily be simulated, and Monte Carlo studies show the test to have good finite sample size and power properties.

Lack-of-fit of a parametric measurement error AR(1) model

This paper proposes an asymptotically distribution free test for fitting a parametric model to the autoregressive function in the AR(1) model in the presence of measurement error. The test is based on a martingale transform of a certain marked residual empirical process. A simulation study assessing the finite sample level and power performance of the proposed test is also included.

Testing for shifts in a time trend panel data model with serially correlated error component disturbances

This paper studies testing of shifts in a time trend panel data model with serially correlated error component disturbances, without any prior knowledge of whether the error term is stationary or nonstationary. This is done in case the shift is known as well as unknown. Following the time series literature, we propose a Wald type test statistic that uses a fixed effects feasible generalized least squares (FE-FGLS) estimator. The proposed test has a chi-square limiting distribution and is valid for both I(0) and I(1) errors. The finite sample size and power of this Wald test is investigated using Monte Carlo simulations

A modified Wilcoxon test for change points in long-range dependent time series

This paper considers testing for structural change in long-memory time series. We modify the Wilcoxon two-sample rank test by standardizing it with a kernel-based fixed bandwidth long-run variance estimator. The corresponding test statistic converges to a well-defined distribution under the null hypothesis. In a Monte Carlo simulation we confirm that the test provides good finite sample size and power results and compare it with an existing approach.

Testing distributional assumptions using a continuum of moments

We propose specification tests for parametric distributions that compare the potentially complex theoretical and empirical characteristic functions using the continuum of moment conditions analogue to an overidentifying restrictions test, which takes into account the correlation between influence functions for different argument values. We derive its asymptotic distribution for fixed regularization parameter and when this vanishes with the sample size. We show its consistency against any deviation from the null, study its local power and compare it with existing tests. An extensive Monte Carlo exercise confirms that our proposed tests display good power in finite samples against a variety of alternatives.

Testing equality of autocovariance operators for functional time series

We consider strictly stationary stochastic processes of Hilbert space‐valued random variables and focus on fully functional tests for the equality of the lag‐zero autocovariance operators of several independent functional time series. A moving block bootstrap (MBB)‐based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomessome common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can be potentially applied to a broad range of test statistics of the hypotheses of interest. As an example, we establish validity for approximating the distribution under the null of a test statistic based on the Hilbert–Schmidt distance of the corresponding sample lag‐zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the MBB procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested MBB‐based testing procedure is illustrated through simulations and an application to a real‐life dataset is discussed.

Point optimal testing with roots that are functionally local to unity

Limit theory for regressions involving local to unit roots (LURs) is now used extensively in time series econometric work, establishing power properties for unit root and cointegration tests, assisting the construction of uniform confidence intervals for autoregressive coefficients, and enabling the development of methods robust to departures from unit roots. The present paper shows how to generalize LUR asymptotics to cases where the localized departure from unity is a time varying function rather than a constant. Such a functional local unit root (FLUR) model has much greater generality and encompasses many cases of additional interest that appear in practical work, including structural break formulations that admit subperiods of unit root, local stationary and local explosive behavior within a given sample. Point optimal FLUR tests are constructed in the paper to accommodate such cases and demonstrate how the power envelope changes in situations of practical interest. Against FLUR alternatives, conventional constant point optimal tests can be asymptotically infinitely deficient in power, with poor finite sample power performance particularly when the departure from unity occurs early in the sample period. New analytic explanation for this phenomenon is provided. Simulation results are reported and some implications for empirical practice are examined.

Testing serial correlations in high-dimensional time series via extreme value theory

This paper proposes a simple test for detecting serial correlations in high-dimensional time series. The proposed test makes use of the robust properties of Spearman’s rank correlation and the theory of extreme values. Asymptotic properties of the test statistics are derived under some minor conditions as both the sample size and dimension go to infinity. The test is not sensitive to the underlying distribution of the time series so long as the data are continuously distributed. In particular, the existence of finite-order moments of the underlying distribution is not required, and asymptotic critical values of the test statistics are available in closed form. In finite samples, we correct biases of the sample autocorrelations and conduct simulations to study the performance of the proposed test statistics. Simulation results show that the proposed test statistics enjoy good properties of size and power in finite samples. We apply the proposed test to a 92-dimensional series of asset returns. Finally, a simple R code is available to obtain finite-sample critical values of the test statistics if needed.

Frequency domain estimation of cointegrating vectors with mixed frequency and mixed sample data

This paper proposes a simple method for exploiting the information contained in mixed frequency and mixed sample data in the estimation of cointegrating vectors. The asymptotic properties of easy-to-compute spectral regression estimators of the cointegrating vectors are derived and these estimators are shown to belong to the class of optimal cointegration estimators. Furthermore, Wald statistics based on these estimators have asymptotic chi-square distributions which enable inferences to be made straightforwardly. Simulation experiments suggest that the spectral regression estimators considered perform well in finite samples and are at least as good as time domain fully modified estimators. The finite sample size and power properties of the spectral regression-based Wald statistic are also found to be good.

Tail Granger Causalities and Where to Find Them: Extreme Risk Spillovers vs Spurious Linkages

Identifying risk spillovers in financial markets is of great importance for assessing systemic risk and portfolio management. Granger causality in tail (or in risk) tests whether past extreme events of a time series help predicting future extreme events of another time series. The topology and connectedness of networks built with Granger causality in tail can be used to measure systemic risk and to identify risk transmitters. Here we introduce a novel test of Granger causality in tail which adopts the likelihood ratio statistic and is based on the multivariate generalization of a discrete autoregressive process for binary time series describing the sequence of extreme events of the underlying price dynamics. The proposed test has very good size and power in finite samples, especially for large sample size, allows inferring the correct time scale at which the causal interaction takes place, and it is flexible enough for multivariate extension when more than two time series are considered in order to decrease false detections as spurious effect of neglected variables. An extensive simulation study shows the performances of the proposed method with a large variety of data generating processes and it introduces also the comparison with the test of Granger causality in tail by [Hong et al., 2009]. We report both advantages and drawbacks of the different approaches, pointing out some crucial aspects related to the false detections of Granger causality for tail events. An empirical application to high frequency data of a portfolio of US stocks highlights the merits of our novel approach.

A Nonparametric Measure of Heteroskedasticity

We introduce a nonparametric measure to quantify the degree of heteroskedasticity at a fixed quantile of the conditional distribution of a random variable. Our measure of heteroskedasticity is based on nonparametric quantile regressions and is expressed in terms of unrestricted and restricted expectations of quantile loss functions. It can be consistently estimated by replacing the unknown expectations by their nonparametric estimates. We derive a Bahadur-type representation for the nonparametric estimator of the measure. We provide the asymptotic distribution of this estimator, which one can use to build tests for the statistical significance of the measure. Thereafter, we establish the validity of a fixed regressor bootstrap that one can use in finite-sample settings to perform tests. A Monte Carlo simulation study reveals that the bootstrap-based test has a good finite sample size and power for a variety of data generating processes and different sample sizes. Finally, two empirical applications are provided to illustrate the importance of the proposed measure.

A Projection-Based Nonparametric Test of Conditional Quantile Independence

This paper proposes a nonparametric procedure for testing conditional quantile independence using projections. Relative to existing smoothed nonparametric tests, the resulting test statistic: (i) detects the high frequency local alternatives that converge to the null hypothesis in probability at faster rate and, (ii) yields improvements in the finite sample power when a large number of variables are included under the alternative. In addition, it allows the researcher to include qualitative information and, if desired, direct the test against specific subsets of alternatives without imposing any functional form on them. We use the weighted Nadaraya-Watson (WNW) estimator of the conditional quantile function avoiding the boundary problems in estimation and testing and prove weak uniform consistency (with rate) of the WNW estimator for absolutely regular processes. The procedure is applied to a study of risk spillovers among the banks. We show that the methodology generalizes some of the recently proposed measures of systemic risk and we use the quantile framework to assess the intensity of risk spillovers among individual financial institutions.

Comparing two nonparametric regression curves in the presence of long memory in covariates and errors

This paper discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives in the presence of long memory in the common covariate and errors. The proposed test is based on a marked empirical process of the differences between the response variables. We discuss asymptotic null distribution of this process and consistency of the test for a class of general alternatives. We also conduct a Monte Carlo simulation study to evaluate the finite sample level and power behavior of the test at some alternatives.

Asymmetric adjustment, non-linearity and housing price bubbles: New international evidence

In this paper, we detect the housing price bubbles of eighteen OECD countries under assumptions of asymmetric adjustment and non-linearity. To this end, we adopt the momentum threshold autoregressive (MTAR) and exponential smooth transition (ESTR) approaches in this study. In order to consider the possibility of asymmetry between regimes of positive and negative changes in the price-rent ratio, we modify the Kilic (2011) and Park and Shintani (2005, 2016) tests and provide their critical values, finite sample size and power via Monte Carlo simulations. Among the main results, first, it is found that the price-rent ratios of eighteen OECD nations are integrated processes of order one by using the traditional unit root tests. Second, the property of asymmetric adjustment is empirically justified for the housing markets of Belgium, Germany and the US by using the MTAR unit root test. Third, the results of the ESTR-type unit root tests point to the conclusion that the housing markets of thirteen countries are globally stationary ESTR processes, implying that the further the price-rent ratio deviates from its fundamental, the faster will be the speed of mean reversion. Finally, the results of the modified Kilic (2011) and Park and Shintani (2005, 2016) tests are in line with the ESTR unit root tests, indicating that the bubble hypothesis is not accepted.

Testing normality via a distributional fixed point property in the Stein characterization

We propose two families of tests for the classical goodness-of-fit problem to univariate normality. The new procedures are based on $L^2$-distances of the empirical zero-bias transformation to the normal distribution or the empirical distribution of the data, respectively. Weak convergence results are derived under the null hypothesis, under fixed alternatives as well as under contiguous alternatives. Empirical critical values are provided and a comparative finite-sample power study shows the competitiveness to classical procedures.

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.

Robust tests for deterministic seasonality and seasonal mean shifts

We develop tests for the presence of deterministic seasonal behaviour and seasonal mean shifts in a seasonally observed univariate time series. These tests are designed to be asymptotically robust to the order of integration of the series at both the zero and seasonal frequencies. Motivated by the approach of Hylleberg, Engle, Granger and Yoo [1990, Journal of Econometrics vol. 44, pp. 215-238], we base our approach on linear filters of the data which remove any potential unit roots at the frequencies not associated with the deterministic component(s) under test. Test statistics are constructed using the filtered data such that they have well defined limiting null distributions regardless of whether the data are either integrated or stationary at the frequency associated with the deterministic component(s) under test. In the same manner as Vogelsang [1998, Econometrica vol. 66, pp. 123-148], Bunzel and Vogelsang [2005, Journal of Business and Economic Statistics vol. 23, pp. 381-394] and Sayginsoy and Vogelsang [2011, Econometric Theory vol. 27, pp. 992-1025], we scale these statistics by a function of an auxiliary seasonal unit root statistic. This allows us to construct tests which are asymptotically robust to the order of integration of the data at both the zero and seasonal frequencies. Monte Carlo evidence suggests that our proposed tests have good finite sample size and power properties. An empirical application to U.K. GDP indicates the presence of seasonal mean shifts in the data.