Bootstrapping realized multivariate volatility measures

We propose a bootstrap method for statistics that are a function of multivariate high frequency returns such as realized regression, covariance and correlation coefficients. We show that the finite sample performance of the bootstrap is superior to the existing first-order asymptotic theory. Nevertheless, and contrary to the existing results in the bootstrap literature for regression models subject to error heteroskedasticity, the Edgeworth expansion for the pairs bootstrap that we develop here shows that this method is not second-order accurate. We argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility.     

Median unbiased forecasts for highly persistent autoregressive processes

This paper considers the construction of median unbiased forecasts for near-integrated autoregressive processes. It derives the appropriately scaled limiting distribution of the deviation of the forecast from the true conditional mean. The dependence of the limiting distribution on nuisance parameters precludes the use of the standard asymptotic and bootstrap methods for bias correction. We propose a bootstrap method that generates samples backward in time and approximates the median function of the predictive distribution on a grid of values for the nuisance parameter. The method can be easily adapted to approximate any quantile of the conditional predictive distribution.     

Asymptotic and bootstrap tests for linearity in a TAR-GARCH(1,1) model with a unit root

This paper derives the limiting distribution of the Lagrange Multiplier (LM) test for threshold nonlinearity in a TAR model with GARCH errors when one of the regimes contains a unit root. It is shown that the asymptotic distribution is nonstandard and depends on nuisance parameters that capture the degree of conditional heteroskedasticity and non-Gaussian nature of the process. We propose a bootstrap procedure for approximating the exact finite-sample distribution of the test for linearity and establish its asymptotic validity.     

A nonparametric test for equality of distributions with mixed categorical and continuous data

In this paper we consider the problem of testing for equality of two density or two conditional density functions defined over mixed discrete and continuous variables. We smooth both the discrete and continuous variables, with the smoothing parameters chosen via least-squares cross-validation. The test statistics are shown to have (asymptotic) normal null distributions. However, we advocate the use of bootstrap methods in order to better approximate their null distribution in finite-sample settings and we provide asymptotic validity of the proposed bootstrap method. Simulations show that the proposed tests have better power than both conventional frequency-based tests and smoothing tests based on ad hoc smoothing parameter selection, while a demonstrative empirical application to the joint distribution of earnings and educational attainment underscores the utility of the proposed approach in mixed data settings.     

Testing for Linearity

The problem of testing for linearity and the number of regimes in the context of self‐exciting threshold autoregressive (SETAR) models is reviewed. We describe least‐squares methods of estimation and inference. The primary complication is that the testing problem is non‐standard, due to the presence of parameters which are only defined under the alternative, so the asymptotic distribution of the test statistics is non‐standard. Simulation methods to calculate asymptotic and bootstrap distributions are presented. As the sampling distributions are quite sensitive to conditional heteroskedasticity in the error, careful modeling of the conditional variance is necessary for accurate inference on the conditional mean. We illustrate these methods with two applications — annual sunspot means and monthly U.S. industrial production. We find that annual sunspots and monthly industrial production are SETAR(2) processes.     

Testing conditional independence via empirical likelihood

We construct two classes of smoothed empirical likelihood ratio tests for the conditional independence hypothesis by writing the null hypothesis as an infinite collection of conditional moment restrictions indexed by a nuisance parameter. One class is based on the CDF; another is based on smoother functions. We show that the test statistics are asymptotically normal under the null hypothesis and a sequence of Pitman local alternatives. We also show that the tests possess an asymptotic optimality property in terms of average power. Simulations suggest that the tests are well behaved in finite samples. Applications to some economic and financial time series indicate that our tests reveal some interesting nonlinear causal relations which the traditional linear Granger causality test fails to detect.     

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.     

Co‐integration Rank Determination in Partial Systems Using Information Criteria

We investigate the asymptotic and finite sample properties of the most widely used information criteria for co‐integration rank determination in ‘partial’ systems, i.e. in co‐integrated vector autoregressive (VAR) models where a sub‐set of variables of interest is modelled conditional on another sub‐set of variables. The asymptotic properties of the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Hannan–Quinn information criterion (HQC) are established, and consistency of BIC and HQC is proved. Notably, the consistency of BIC and HQC is robust to violations of weak exogeneity of the conditioning variables with respect to the co‐integration parameters. More precisely, BIC and HQC recover the true co‐integration rank from the partial system analysis also when the conditional model does not convey all information about the co‐integration parameters. This result opens up interesting possibilities for practitioners who can now determine the co‐integration rank in partial systems without being concerned about the weak exogeneity of the conditioning variables. A Monte Carlo experiment based on a large dimensional data generating process shows that BIC and HQC applied in partial systems perform reasonably well in small samples and comparatively better than ‘traditional’ methods for co‐integration rank determination. We further show the usefulness of our approach and the benefits of the conditional system analysis in two empirical illustrations, both based on the estimation of VAR systems on US quarterly data. Overall, our analysis shows the gains of combining information criteria with partial system analysis.     

Bootstrap and Asymptotic Tests of Long-run Relationships in Cointegrated Systems

Hypothesis testing on cointegrating vectors based on the asymptotic distributions of the test statistics are known to suffer from severe small sample size distortion. In this paper an alternative bootstrap procedure is proposed and evaluated through a Monte Carlo experiment, finding that the Type I errors are close to the nominal signficance levels but power might be not entirely adequate. It is then shown that a combined test based on the outcomes of both the asymptotic and the bootstrap tests will have both correct size and low Type II error, therefore improving the currently available procedures.     

External bootstrap tests for parameter stability

This article considers tests for parameter stability over time in general econometric models, possibly nonlinear-in-variables. Existing test statistics are commonly not asymptotically pivotal under nonstandard conditions. In such cases, the external bootstrap tests proposed in this paper are appealing from a practical viewpoint. We propose to use bootstrap versions of the asymptotic critical values based on a first-order asymptotic expansion of the test statistics under the null hypothesis, which consists of a linear transformation of the unobserved “innovations” partial sum process. The nature of these transformations under nonstandard conditions is discussed for the main testing principles. Also, we investigate the small sample performance of the proposed bootstrap tests by means of a small Monte Carlo experiment.     

Testing cointegration relationship in a semiparametric varying coefficient model

In this paper, we develop two cointegration tests for two varying coefficient cointegration regression models, respectively. Our test statistics are residual based. We derive the asymptotic distributions of test statistics under the null hypothesis of cointegration and show that they are consistent against the alternative hypotheses. We also propose a wild bootstrap procedure companioned with the continuous moving block bootstrap method proposed in  Paparoditis and Politis (2001) and  Phillips (2010) to rectify severe distortions found in simulations when the sample size is small. We apply the proposed test statistic to examine the purchasing power parity (PPP) hypothesis between the US and Canada. In contrast to the existing results from linear cointegration tests, our varying coefficient cointegration test does not reject that PPP holds between the US and Canada.     

Testing the parametric form of the volatility in continuous time diffusion models—a stochastic process approach

We present new tests for the form of the volatility function which are based on stochastic processes of the integrated volatility. We prove weak convergence of these processes to centered processes whose conditional distributions are Gaussian. In the case of testing for a constant volatility the limiting process are standard Brownian bridges. As a consequence an asymptotic distribution free test and bootstrap tests (for testing of a general parametric form) can easily be implemented. It is demonstrated that the new tests are more than the currently available procedures. The new approach is also demonstrated by means of a simulation study.     

System‐Equation ADL Test for Threshold Cointegration with an Application to the Term Structure of Interest Rates

This paper proposes a new system‐equation test for threshold cointegration based on a threshold vector autoregressive distributed lag (ADL) model. The new test can be applied when the cointegrating vector is unknown and when weak exogeneity fails. The asymptotic null distribution of the new test is derived, critical values are tabulated and finite‐sample properties are examined. In particular, the new test is shown to have good size, so the bootstrap is not required. The new test is illustrated using the long‐term and short‐term interest rates. We show that the system‐equation model can shed light on both asymmetric adjustment speeds and asymmetric adjustment roles. The latter is unavailable in the single‐equation testing strategy.     

Two-stage stationary bootstrapping for bivariate average realized volatility matrix under market microstructure noise and asynchronicity

Under the two important modern financial market features of noise and non-synchronicity for multiple assets, for consistent estimators of the integrated covariations, we adopt the two-time scale average realized volatility matrix (ARVM) which is a matrix extension of the two-time scale realized volatilities of Zhang et al. (2005). An asymptotic normal theory is provided for the two-time scale ARVM and resulting realized covariations. The asymptotic normality is not directly applicable in practice to construct statistical methods owning to nuisance parameters. To bypass the nuisance parameter problem, two-stage stationary bootstrapping is proposed. We establish consistencies of the bootstrap distributions, and construct confidence intervals and hypothesis tests for the integrated covariance, regression coefficient and correlation coefficient. The validity of the stationary bootstrap for the high frequency heterogeneous returns is proved by showing that there exist parameters of the stationary bootstrap blocks so that the bootstrap consistencies hold. The proposed bootstrap methods extend the i.i.d. bootstrapping methods for realized covariations by Dovonon et al. (2013), that are confined to synchronous noise-free sampling. For high frequency noisy asynchronous samples, a Monte-Carlo experiment shows better finite sample performances of the proposed stationary bootstrap methods based on the two-time scale ARVM estimator than the wild blocks of blocks bootstrap methods of Hounyo (2017), based on pre-averaged truncated estimator.     

A unified approach to validating univariate and multivariate conditional distribution models in time series

Modeling conditional distributions in time series has attracted increasing attention in economics and finance. We develop a new class of generalized Cramer–von Mises (GCM) specification tests for time series conditional distribution models using a novel approach, which embeds the empirical distribution function in a spectral framework. Our tests check a large number of lags and are therefore expected to be powerful against neglected dynamics at higher order lags, which is particularly useful for non-Markovian processes. Despite using a large number of lags, our tests do not suffer much from loss of a large number of degrees of freedom, because our approach naturally downweights higher order lags, which is consistent with the stylized fact that economic or financial markets are more affected by recent past events than by remote past events. Unlike the existing methods in the literature, the proposed GCM tests cover both univariate and multivariate conditional distribution models in a unified framework. They exploit the information in the joint conditional distribution of underlying economic processes. Moreover, a class of easy-to-interpret diagnostic procedures are supplemented to gauge possible sources of model misspecifications. Distinct from conventional CM and Kolmogorov–Smirnov (KS) tests, which are also based on the empirical distribution function, our GCM test statistics follow a convenient asymptotic $N(0,1)$ distribution and enjoy the appealing “nuisance parameter free” property that parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistics. Simulation studies show that the tests provide reliable inference for sample sizes often encountered in economics and finance.     

Testing the compounding structure of the CP-INARCH model

A statistical test to distinguish between a Poisson INARCH model and a Compound Poisson INARCH model is proposed, based on the form of the probability generating function of the compounding distribution of the conditional law of the model. For first-order autoregression, the normality of the test statistics’ asymptotic distribution is established, either in the case where the model parameters are specified, or when such parameters are consistently estimated. As the test statistics’ law involves the moments of inverse conditional means of the Compound Poisson INARCH process, the analysis of their existence and calculation is performed by two approaches. For higher-order autoregressions, we use a bootstrap implementation of the test. A simulation study illustrating the finite-sample performance of this test methodology in what concerns its size and power concludes the paper.     

Goodness-of-fit tests based on series estimators in nonparametric instrumental regression

This paper proposes several tests of restricted specification in nonparametric instrumental regression. Based on series estimators, test statistics are established that allow for tests of the general model against a parametric or nonparametric specification as well as a test of exogeneity of the vector of regressors. The tests’ asymptotic distributions under correct specification are derived and their consistency against any alternative model is shown. Under a sequence of local alternative hypotheses, the asymptotic distributions of the tests are derived. Moreover, uniform consistency is established over a class of alternatives whose distance to the null hypothesis shrinks appropriately as the sample size increases. A Monte Carlo study examines finite sample performance of the test statistics.     

Exogeneity in structural equation models

The practical relevance of several concepts of exogeneity of treatments for the estimation of causal parameters based on observational data are discussed. We show that the traditional concepts, such as strong ignorability and weak and super-exogeneity, are too restrictive if interest lies in average effects (i.e. not on distributional effects of the treatment). We suggest a new definition of exogeneity, KL-exogeneity. It does not rely on distributional assumptions and is not based on counterfactual random variables. As a consequence it can be empirically tested using a proposed test that is simple to implement and is distribution-free.     

GEL statistics under weak identification

The central concern of this paper is the provision in a time series moment condition framework of practical recommendations of confidence regions for parameters whose coverage probabilities are robust to the strength or weakness of identification. To this end we develop Pearson-type test statistics based on GEL implied probabilities formed from general kernel smoothed versions of the moment indicators. We also modify the statistics suggested in Guggenberger and Smith (2008) for a general kernel smoothing function. Importantly for our conclusions, we provide GEL time series counterparts to GMM and GEL conditional likelihood ratio statistics given in Kleibergen (2005) and Smith (2007). Our analysis not only demonstrates that these statistics are asymptotically (conditionally) pivotal under both classical asymptotic theory and weak instrument asymptotics of Stock and Wright (2000) but also provides asymptotic power results in the weakly identified time series context. Consequently, the empirical null rejection probabilities of the associated tests and, thereby, the coverage probabilities of the corresponding confidence regions, should not be affected greatly by the strength or otherwise of identification. A comprehensive Monte Carlo study indicates that a number of the tests proposed here represent very competitive choices in comparison with those suggested elsewhere in the literature.