We propose a covariance stationarity test for an otherwise dependent and possibly globally non-stationary time series. We work in a generalized version of the new setting in Jin, Wang and Wang (J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 893-922), who exploit Walsh (Amer. J. Math. 45 (1923) 5-24) functions in order to compare sub-sample covariances with the full sample counterpart. They impose strict stationarity under the null, only consider linear processes under either hypothesis in order to achieve a parametric estimator for an inverted high dimensional asymptotic covariance matrix, and do not consider any other orthonormal basis. Conversely, we work with a general orthonormal basis under mild conditions that include Haar wavelet and Walsh functions, and we allow for linear or nonlinear processes with possibly non-iid innovations. This is important in macroeconomics and finance where nonlinear feedback and random volatility occur in many settings. We completely sidestep asymptotic covariance matrix estimation and inversion by bootstrapping a max-correlation difference statistic, where the maximum is taken over the correlation lag h and basis generated sub-sample counter k (the number of systematic samples). We achieve a higher feasible rate of increase for the maximum lag and counter HT and KT. Of particular note, our test is capable of detecting breaks in variance, and distant, or very mild, deviations from stationarity.

We derive so-called weak and strong \textit{max-laws of large numbers} for $% \max_{1\leq i\leq k_{n}}|1/n\sum_{t=1}^{n}x_{i,n,t}|$ for zero mean stochastic triangular arrays $\{x_{i,n,t}$ $:$ $1$ $\leq $ $t$ $\leq n\}_{n\geq 1}$, with dimension counter $i$ $=$ $1,...,k_{n}$ and dimension $% k_{n}$ $\rightarrow $ $\infty $. Rates of convergence are also analyzed based on feasible sequences $\{k_{n}\}$. We work in three dependence settings: independence, Dedecker and Prieur's (2004) $τ$-mixing and Wu's (2005) physical dependence. We initially ignore cross-coordinate $i$ dependence as a benchmark. We then work with martingale, nearly martingale, and mixing coordinates to deliver improved bounds on $k_{n}$. Finally, we use the results in three applications, each representing a key novelty: we ($i$) bound $k_{n}$\ for a max-correlation statistic for regression residuals under $α$-mixing or physical dependence; ($ii$) extend correlation screening, or marginal regressions, to physical dependent data with diverging dimension $k_{n}$ $\rightarrow $ $\infty $; and ($iii$) test a high dimensional parameter after partialling out a fixed dimensional nuisance parameter in a linear time series regression model under $τ$% -mixing.

We study the probability tail properties of Inverse Probability Weighting (IPW) estimators of the Average Treatment Effect (ATE) when there is limited overlap between the covariate distributions of the treatment and control groups. Under unconfoundedness of treatment assignment conditional on covariates, such limited overlap is manifested in the propensity score for certain units being very close (but not equal) to 0 or 1. This renders IPW estimators possibly heavy tailed, and with a slower than sqrt(n) rate of convergence. Trimming or truncation is ultimately based on the covariates, ignoring important information about the inverse probability weighted random variable Z that identifies ATE by E[Z]= ATE. We propose a tail-trimmed IPW estimator whose performance is robust to limited overlap. In terms of the propensity score, which is generally unknown, we plug-in its parametric estimator in the infeasible Z, and then negligibly trim the resulting feasible Z adaptively by its large values. Trimming leads to bias if Z has an asymmetric distribution and an infinite variance, hence we estimate and remove the bias using important improvements on existing theory and methods. Our estimator sidesteps dimensionality, bias and poor correspondence properties associated with trimming by the covariates or propensity score. Monte Carlo experiments demonstrate that trimming by the covariates or the propensity score requires the removal of a substantial portion of the sample to render a low bias and close to normal estimator, while our estimator has low bias and mean-squared error, and is close to normal, based on the removal of very few sample extremes.

We propose a test of many zero parameter restrictions in a high dimensional linear iid regression model with $k$ $>>$ $n$ regressors. The test statistic is formed by estimating key parameters one at a time based on many low dimension regression models with nuisance terms. The parsimoniously parametrized models identify whether the original parameter of interest is or is not zero. Estimating fixed low dimension sub-parameters ensures greater estimator accuracy, it does not require a sparsity assumption nor therefore a regularized estimator, it is computationally fast compared to, e.g., de-biased Lasso, and using only the largest in a sequence of weighted estimators reduces test statistic complexity and therefore estimation error. We provide a parametric wild bootstrap for p-value computation, and prove the test is consistent and has non-trivial $\sqrt{n/\{\ln (n)\mathcal{M}% _{n}\}}$-local-to-null power where $\mathcal{M}_{n}$ is the $l_{\infty }$ covariate fourth moment.

We present a new test when there is a nuisance parameter λ under the alternative hypothesis. The test exploits the p-value occupation time [PVOT], the measure of the subset of λ on which a p-value test based on a test statistic Tn(λ) rejects the null hypothesis. The PVOT has only been explored in Hill and Aguilar (2013) and Hill (2012) as a way to smooth over a trimming parameter for heavy tail robust test statistics. Our key contributions are: (i) we show that a weighted average local power of a test based on Tn(λ) is identically a weighted average mean PVOT, and the PVOT used for our test is therefore a point estimate of the weighted average probability of PV test rejection, under the null; (ii) an asymptotic critical value upper bound for our test is the significance level itself, making inference easy (as opposed to supremum and average test statistic transforms which typically require a bootstrap method for p-value computation); (iii) we only require Tn(λ) to have a known or bootstrappable limit distribution, hence we do not require √n-Gaussian asymptotics as is nearly always assumed, and we allow for some parameters to be weakly or non-identified; and (iv) a numerical experiment, in which local asymptotic power is computed for a test of omitted nonlinearity, reveals the asymptotic critical value is exactly the significance level, and the PVOT test is virtually equivalent to a test with the greatest weighted average power in the sense of Andrews and Ploberger (1994)We give examples of PVOT tests of omitted nonlinearity, GARCH effects and a one time structural break. A simulation study demonstrates the merits of PVOT test of omitted nonlinearity and GARCH effects, and demonstrates the asymptotic critical value is exactly the significance level.

We propose a covariance stationarity test for an otherwise dependent and possibly globally non-stationary time series. We work in a generalized version of the new setting in Jin, Wang and Wang (2015), who exploit Walsh (1923) functions in order to compare sub-sample covariances with the full sample counterpart. They impose strict stationarity under the null, only consider linear processes under either hypothesis in order to achieve a parametric estimator for an inverted high dimensional asymptotic covariance matrix, and do not consider any other orthonormal basis. Conversely, we work with a general orthonormal basis under mild conditions that include Haar wavelet and Walsh functions; and we allow for linear or nonlinear processes with possibly non-iid innovations. This is important in macroeconomics and finance where nonlinear feedback and random volatility occur in many settings. We completely sidestep asymptotic covariance matrix estimation and inversion by bootstrapping a max-correlation difference statistic, where the maximum is taken over the correlation lag $h$ and basis generated sub-sample counter $k$ (the number of systematic samples). We achieve a higher feasible rate of increase for the maximum lag and counter $\mathcal{H}_{T}$ and $\mathcal{K}_{T}$. Of particular note, our test is capable of detecting breaks in variance, and distant, or very mild, deviations from stationarity.

We present a new robust bootstrap method for a test when there is a nuisance parameter under the alternative, and some parameters are possibly weakly or non-identified. We focus on a Bierens (1990)-type conditional moment test of omitted nonlinearity for convenience, and because of difficulties that have been ignored to date. Existing methods include the supremum p-value which promotes a conservative test that is generally not consistent, and test statistic transforms like the supremum and average for which bootstrap methods are not valid under weak identification. We propose a new wild bootstrap method for p-value computation by targeting specific identification cases. We then combine bootstrapped p-values across polar identification cases to form an asymptotically valid p-value approximation that is robust to any identification case. The wild bootstrap does not require knowledge of the covariance structure of the bootstrapped processes, whereas Andrews and Cheng's (2012, 2013, 2014) simulation approach generally does. Our method allows for robust bootstrap critical value computation as well. Our bootstrap method (like conventional ones) does not lead to a consistent p-value approximation for test statistic functions like the supremum and average. We therefore smooth over the robust bootstrapped p-value as the basis for several tests which achieve the correct asymptotic level, and are consistent, for any degree of identification. They also achieve uniform size control. A simulation study reveals possibly large empirical size distortions in non-robust tests when weak or non-identification arises. One of our smoothed p-value tests, however, dominates all other tests by delivering accurate empirical size and comparatively high power.

This paper proposes a max-test for testing (possibly infinitely) many zero parameter restrictions in an extremum estimation framework. The test statistic is formed by estimating key parameters one at a time based on many empirical loss functions that map from a low dimension parameter space, and choosing the largest in absolute value from these individually estimated parameters. The parsimoniously parametrized loss identify whether the original parameter of interest is or is not zero. Estimating fixed low dimension sub-parameters ensures greater estimator accuracy, does not require a sparsity assumption, and using only the largest in a sequence of weighted estimators reduces test statistic complexity and therefore estimation error, ensuring sharper size and greater power in practice. Weights allow for standardization in order to control for estimator dispersion. In a nonlinear parametric regression framework we provide a parametric wild bootstrap for p-value computation without directly requiring the max-statistic's limit distribution. A simulation experiment shows the max-test dominates a conventional bootstrapped test.

This paper presents a bootstrapped p-value white noise test based on the maximum correlation, for a time series that may be weakly dependent under the null hypothesis. The time series may be prefiltered residuals. The test statistic is a normalized weighted maximum sample correlation, where the maximum lag increases at a rate slower than the sample size. We only require uncorrelatedness under the null hypothesis, along with a moment contraction dependence property that includes mixing and non-mixing sequences. We show Shao's (2011) dependent wild bootstrap is valid for a much larger class of processes than originally considered. It is also valid for residuals from a general class of parametric models as long as the bootstrap is applied to a first order expansion of the sample correlation. We prove the bootstrap validity without exploiting extreme value theory (standard in the literature) or recent Gaussian approximation theory. Finally, we extend Escanciano and Lobato's (2009) automatic maximum lag selection to our setting with an unbounded choice set, and find it works strikingly well in controlled experiments. Our proposed test achieves accurate size under various white noise null hypotheses and high power under various alternative hypotheses including distant serial dependence.

This paper proposes a new test for a large set of zero restrictions in regression models based on a seemingly overlooked, but simple, dimension reduction technique. The procedure involves multiple parsimonious regression models where key regressors are split across simple regressions. Each parsimonious regression model has one key regressor and other regressors not associated with the null hypothesis. The test is based on the maximum of the squared parameters of the key regressors. Parsimony ensures sharper estimates and therefore improves power in small sample. We present the general theory of our test and focus on mixed frequency Granger causality as a prominent application involving many zero restrictions.

Weak form efficiency of stock markets implies unpredictability of stock returns in a time series sense, and the latter is tested predominantly under a serial independence or martingale difference assumption. Since these properties rule out weak dependence that may exist in stock returns, it is of interest to test whether returns are white noise. We perform white noise tests assisted by Shao's (2011) blockwise wild bootstrap. We reveal that, in rolling windows, the block structure inscribes an artificial periodicity in bootstrapped confidence bands. We eliminate the periodicity by randomizing a block size. The white noise hypothesis is accepted for Chinese and Japanese markets, suggesting that those markets are weak form efficient. The white noise hypothesis is rejected for U.K. and U.S. markets during the Iraq War and the subprime mortgage crisis due to significantly negative autocorrelations, suggesting that those markets are inefficient in crisis periods.

We provide methods to robustly estimate the parameters of stationary ergodic short-memory time series models in the potential presence of additive low-frequency contamination. The types of contamination covered include level shifts (changes in mean) and monotone or smooth time trends, both of which have been shown to bias parameter estimates toward regions of persistence in a variety of contexts. The estimators presented here minimize trimmed frequency domain quasi-maximum likelihood (FDQML) objective functions without requiring specification of the low-frequency contaminating component. When proper sample size-dependent trimmings are used, the FDQML estimators are consistent and asymptotically normal, asymptotically eliminating the presence of any spurious persistence. These asymptotic results also hold in the absence of additive low-frequency contamination, enabling the practitioner to robustly estimate model parameters without prior knowledge of whether contamination is present. Popular time series models that fit into the framework of this article include autoregressive moving average (ARMA), stochastic volatility, generalized autoregressive conditional heteroscedasticity (GARCH), and autoregressive conditional heteroscedasticity (ARCH) models. We explore the finite sample properties of the trimmed FDQML estimators of the parameters of some of these models, providing practical guidance on trimming choice. Empirical estimation results suggest that a large portion of the apparent persistence in certain volatility time series may indeed be spurious. Supplementary materials for this article are available online.

Apart from the human suffering it causes, growth volatility is a major factor that retards growth. Recently several studies provided empirical evidence that democratic political institutions generate less volatile growth. In the economics literature fluctuations in major macroeconomic processes, like aggregate investment, are considered fundamental factors influencing growth volatility. However, the studies on democracy and growth volatility do not provide any link between democracy and investment volatility. Here, instead of democracy, we consider another institutional variable in explaining growth volatility: we focus on the specific channel that links individualistic societies and low growth volatility. In our theoretical model, it turns out that in a collectivistic society agents choose to invest together or choose not to invest together. In an individualistic society, on the other hand, there are also some parameter values at which the agents with more wealth choose to invest while the agents with less wealth do not find it worth investing. Hence, investment volatility and consequently growth volatility are lower in an individualistic society than in a collectivistic society. This is because in an individualistic society, agents are able to reap the entire benefits of their individual investments themselves and make their investment decisions regardless of the other agent’s investment decision. Whereas in a collectivistic society, individuals are not able to reap the entire benefits of their individual investments themselves and cannot make their investment decisions regardless of the other agent’s investment decision. We test the theoretical model’s prediction by constructing a two-equation system of investment and income growth volatility, allowing various measures of individualism to influence growth volatility both directly and indirectly. Using standard controls, we importantly control for the development of democratic institutions. Based on a battery of sensitivity tests, we find individualism significantly directly and indirectly influences growth volatility negatively. We also find that, unlike individualism, democracy’s influence on investment depends on the measure of democracy and econometric specification used. ______________________________________________________________________ * We would like to thank Geert Hofstede, Cem Karayalcin, Dani Rodrik and Mehmet Ali Ulubasoglu for their many helpful comments and suggestions at various stages of this project. ** Nejat Anbarci, Department of Economics, Florida International University, University Park, Miami, FL 33186, phone: 305.348.2735; fax: 305.348.1524; e-mail: anbarcin@fiu.edu (corresponding author). Jonathan Hill, Department of Economics, Florida International University, University Park, Miami, FL 33186, phone: 305.348.2682; fax: 305.348.1524; e-mail: hilljona@fiu.edu. Hasan Kirmanoglu, Department of Economics, Bilgi University, Inonu cad., No. 28, Sisli, Istanbul 80130, Turkey; phone: +90.212.311.6329; fax:+90.212.216.8478; e-mail: hkirman@bilgi.edu.tr.

present a new general asymptotic theory for the maximum of a random array {𝒳_n(i) : 1 ≤ i ≤ ℒ}_n≥ 1, where each 𝒳_n(i) is assumed to converge in probability as n → ∞. The array dimension ℒ is allowed to increase with the sample size n. Existing extreme value theory arguments focus on observed data 𝒳_n(i), and require a well defined limit law for max_1≤ i≤ℒ|𝒳_n(i)| by restricting dependence across i. The high dimensional central limit theory literature presumes approximability by a Gaussian law, and also restricts attention to observed data. do not require max_1≤ i≤ℒ_n|𝒳_n(i)| to have a well defined limit nor be approximable by a Gaussian random variable, and we do not make any assumptions about dependence across i. We apply the theory to filtered data when the variable of interest 𝒳_n(i,θ _0) is not observed, but its sample counterpart 𝒳_n(i,θ̂_n) is observed where θ̂_n estimates θ _0. The main results are illustrated by looking at unit root tests for a high dimensional random variable, and a residuals white noise test.

We develop Granger causality tests that apply directly to data sampled at different frequencies. We show that taking advantage of mixed frequency data allows us to better recover causal relationships when compared to the conventional common low frequency approach. We also show that the new causality tests have higher local asymptotic power as well as more power in finite samples compared to conventional tests. In an empirical application involving U.S. macroeconomic indicators, we show that the mixed frequency approach and the low frequency approach produce very different causal implications, with the former yielding more intuitively appealing result.

We construct a Generalized Empirical Likelihood estimator for a GARCH(1, 1) model with a possibly heavy tailed error. The estimator imbeds tail-trimmed estimating equations allowing for over-identifying conditions, asymptotic normality, efficiency and empirical likelihood based confidence regions for very heavy-tailed random volatility data. We show the implied probabilities from the tail-trimmed Continuously Updated Estimator elevate weight for usable large values, assign large but not maximum weight to extreme observations, and give the lowest weight to non-leverage points. We derive a higher order expansion for GEL with imbedded tail-trimming (GELITT), which reveals higher order bias and efficiency properties, available when the GARCH error has a finite second moment. Higher order asymptotics for GEL without tail-trimming requires the error to have moments of substantially higher order. We use first order asymptotics and higher order bias to justify the choice of the number of trimmed observations in any given sample. We also present robust versions of Generalized Empirical Likelihood Ratio, Wald, and Lagrange Multiplier tests, and an efficient and heavy tail robust moment estimator with an application to expected shortfall estimation. Finally, we present a broad simulation study for GEL and GELITT, and demonstrate profile weighted expected shortfall for the Russian Ruble–US Dollar exchange rate. We show that tail-trimmed CUE-GMM dominates other estimators in terms of bias, mse and approximate normality.

An empirical likelihood method was proposed in Hill and Peng (2014) to construct a unified interval estimation for the coefficient in an AR(1) model, regardless of whether the sequence was stationary or near integrated. The error term, however, was assumed independent, and this method fails when the errors are dependent. Testing for a unit root in an AR(1) model has been studied in the literature for dependent errors, but existing methods cannot be used to test for a near unit root. In this paper, assuming the errors are governed by an AR(p) process, we exploit the efficient empirical likelihood method to give a unified interval for the coefficient by taking the structure of errors into account. Furthermore, a jackknife empirical likelihood method is proposed to reduce the computation of the empirical likelihood method when the order in the AR errors is not small. A simulation study is conducted to examine the finite sample behavior of the proposed methods.

This paper presents a variety of tests of volatility spillover that are robust to heavy tails generated by large errors or GARCH-type feedback. The tests are couched in a general conditional heteroskedasticity framework with idiosyncratic shocks that are only required to have a finite variance if they are independent. We negligibly trim test equations, or components of the equations, and construct heavy tail robust score and portmanteau statistics. Trimming is either simple based on an indicator function, or smoothed. In particular, we develop the tail-trimmed sample correlation coefficient for robust inference, and prove that its Gaussian limit under the null hypothesis of no spillover has the same standardization irrespective of tail thickness. Further, if spillover occurs within a specified horizon, our test statistics obtain power of one asymptotically. We discuss the choice of trimming portion, including a smoothed p-value over a window of extreme observations. A Monte Carlo study shows our tests provide significant improvements over extant GARCH-based tests of spillover, and we apply the tests to financial returns data. Finally, based on ideas in Patton (2011) we construct a heavy tail robust forecast improvement statistic, which allows us to demonstrate that our spillover test can be used as a model specification pre-test to improve volatility forecasting.

We present a robust Generalized Empirical Likelihood estimator and confidence region for the parameters of an autoregression that may have a heavy tailed heteroscedastic error. The estimator exploits two transformations for heavy tail robustness: a redescending transformation of the error that robustifies against innovation outliers, and weighted least squares instruments that ensure robustness against heavy tailed regressors. Our estimator is consistent for the true parameter and asymptotically normally distributed irrespective of heavy tails.

This paper proposes a new mixed frequency Granger causality test that achieves high power even when we have a small sample size and a large ratio of sampling frequencies. We postulate multiple parsimonious regression models where each model regresses a low frequency variable onto only one individual lag or lead of a high frequency variable. We then formulate what we call a max test statistic by picking the largest squared estimator among all parsimonious regression models. We show via Monte Carlo simulations that the max test is more powerful than existing mixed frequency Granger causality tests in small sample. In empirical application, the max test yields a plausible result that weekly U.S. interest rate spread used to cause U.S. real growth until about the year 2000 but such causality has vanished more recently.