We establish theoretical results about the low frequency contamination (i.e., long memory effects) induced by general nonstationarity for estimates such as the sample autocovariance and the periodogram, and deduce consequences for heteroskedasticity and autocorrelation robust (HAR) inference. We present explicit expressions for the asymptotic bias of these estimates. We show theoretically that nonparametric smoothing over time is robust to low frequency contamination. Nonstationarity can have consequences for both the size and power of HAR tests. Under the null hypothesis there are larger size distortions than when data are stationary. Under the alternative hypothesis, existing LRV estimators tend to be inflated and HAR tests can exhibit dramatic power losses. Our theory indicates that long bandwidths or fixed-b HAR tests suffer more from low frequency contamination relative to HAR tests based on HAC estimators, whereas recently introduced double kernel HAC estimators do not suffer from this problem. We present second-order Edgeworth expansions under nonstationarity about the distribution of HAC and DK-HAC estimators and about the corresponding t-test in the regression model. The results show that the distortions in the rejection rates can be induced by time variation in the second moments even when there is no break in the mean.

New limit theory is provided for a wide class of sample variance and covariance functionals involving both nonstationary and stationary time series. Sample functionals of this type commonly appear in regression applications and the asymptotics are particularly relevant to estimation and inference in nonlinear nonstationary regressions that involve unit root, local unit root, or fractional processes. The limit theory is unusually general in that it covers both parametric and nonparametric regressions. Self-normalized versions of these statistics are considered that are useful in inference. Numerical evidence reveals interesting strong bimodality in the finite sample distributions of conventional self-normalized statistics similar to the bimodality that can arise in t-ratio statistics based on heavy tailed data. Bimodal behavior in these statistics is due to the presence of long memory innovations and is shown to persist for very large sample sizes even though the limit theory is Gaussian when the long memory innovations are stationary. Bimodality is shown to occur even in the limit theory when the long memory innovations are nonstationary. To address these complications, new self-normalized versions of the test statistics are introduced that deliver improved approximations that can be used for inference.

The Least Trimmed Squares (LTS) regression estimator is known to be very robust to the presence of “outliers”. It is based on a clear and intuitive idea: in a sample of size n, it searches for the h-subsample of observations with the smallest sum of squared residuals. The remaining $n-h$ observations are declared “outliers”. Fast algorithms for its computation exist. Nevertheless, the existing asymptotic theory for LTS, based on the traditional $\epsilon $-contamination model, shows that the asymptotic behavior of both regression and scale estimators depend on nuisance parameters. Using a recently proposed new model, in which the LTS estimator is maximum likelihood, we show that the asymptotic behavior of both the LTS regression and scale estimators are free of nuisance parameters. Thus, with the new model as a benchmark, standard inference procedures apply while allowing a broad range of contamination.

Relatively, recent work by Jeganathan (2008, Cowles Foundation Discussion Paper 1649) and Wang (2014, Econometric Theory, 30(3), 509–535) on generalized martingale central limit theorems (MCLTs) implicitly introduces a new class of instrument arrays that yield (mixed) Gaussian limit theory irrespective of the persistence level in the data. Motivated by these developments, we propose a new semiparametric method for estimation and inference in nonlinear predictive regressions with persistent predictors. The proposed method that we term chronologically trimmed least squares (CTLS) is comparable to the IVX method of Phillips and Magdalinos (2009, Econometric inference in the vicinity of unity. Mimeo, Singapore Management University) and yields conventional inference in regressions where the nature and extent of persistence in the data are uncertain. In terms of model generality, our contribution to the existing literature is twofold. First, our covariate model space allows for both nearly integrated (NI) and fractional processes (stationary and nonstationary) as a special case, while the vast majority of articles in this area only consider NI arrays. Second, we allow for nonlinear regression functions. The CTLS estimator is obtained by applying certain chronological trimming to the OLS instruments using appropriate kernel functions of time trend variables. In particular, the instruments under consideration are a generalized (averaged) version of those widely used for time-varying parameter (TVP) models. For the purposes of our analysis, we develop a novel asymptotic theory for sample averages of various processes weighted by such kernel functionals which is of independent interest and highly relevant to the TVP literature. Leveraging our nonlinear framework, we also provide an investigation on the effects of misbalancing on the predictability hypothesis. A new methodology is proposed to mitigate misbalancing effects. These methods are used for exploring the predictability of SP500 returns.

In this paper, we consider estimating spot/instantaneous volatility matrices of high-frequency data collected for a large number of assets. We first combine classic nonparametric kernel-based smoothing with a generalized shrinkage technique in the matrix estimation for noise-free data under a uniform sparsity assumption, a natural extension of the approximate sparsity commonly used in the literature. The uniform consistency property is derived for the proposed spot volatility matrix estimator with convergence rates comparable to the optimal minimax one. For high-frequency data contaminated by microstructure noise, we introduce a localized pre-averaging estimation method that reduces the effective magnitude of the noise. We then use the estimation tool developed in the noise-free scenario and derive the uniform convergence rates for the developed spot volatility matrix estimator. We further combine kernel smoothing with the shrinkage technique to estimate the time-varying volatility matrix of the high-dimensional noise vector. In addition, we consider large spot volatility matrix estimation in time-varying factor models with observable risk factors and derive the uniform convergence property. We provide numerical studies including simulation and empirical application to examine the performance of the proposed estimation methods in finite samples.

This paper proposes a consistent nonparametric test with good sampling properties to detect instantaneous causality between vector autoregressive (VAR) variables with time-varying variances. The new test takes the form of the U-statistic, and has a limiting standard normal distribution under the null. We further show that the test is consistent against any fixed alternatives, and has nontrivial asymptotic power against a class of local alternatives with a rate slower than $T^{-1/2}$. We also propose a wild bootstrap procedure to better approximate the finite sample null distribution of the test statistic. Monte Carlo experiments are conducted to highlight the merits of the proposed test relative to other popular tests in finite samples. Finally, we apply the new test to investigate the instantaneous causality relationship between money supply and inflation rates in the USA.

Since the implementation of the Basel III Accord, expected shortfall (ES) has gained increasing attention from regulators as a complement to value-at-risk (VaR). The problem of elicitability for ES makes jointly modeling VaR and ES a popular method to study ES. In this article, we develop model averaging for joint VaR and ES regression models that selects the two weight vectors by minimizing a jackknife criterion. We show the large sample properties of the estimators under potential model misspecification with increasing dimension of parameters and the asymptotic optimality of the selected weights in the sense of minimizing the out-of-sample excess final prediction error. Simulation studies and three empirical analyses reveal good finite sample performance.

This paper considers linear rational expectations models in the frequency domain. The paper characterizes existence and uniqueness of solutions to particular as well as generic systems. The set of all solutions to a given system is shown to be a finite-dimensional affine space in the frequency domain. It is demonstrated that solutions can be discontinuous with respect to the parameters of the models in the context of nonuniqueness, invalidating mainstream frequentist and Bayesian methods. The ill-posedness of the problem motivates regularized solutions with theoretically guaranteed uniqueness, continuity, and even differentiability properties.

This study proposes a test for coefficient randomness in autoregressive models where the autoregressive coefficient is local to unity, which is empirically relevant given earlier work. Under this specification, we analyze the effect of the correlation between the random coefficient and disturbance on the properties of tests, a matter that remains largely unexplored in the literature. Our analysis reveals that tests proposed in earlier studies can have poor power when the correlation is moderate to large. The test proposed here is designed to have power functions robust to the correlation. A modified version of the test is suggested that can be applied when the disturbance is serially correlated and conditionally heteroskedastic. The test is shown to have better power properties than existing ones in large and finite samples.

We consider bootstrap inference in predictive (or Granger-causality) regressions when the parameter of interest may lie on the boundary of the parameter space, here defined by means of a smooth inequality constraint. For instance, this situation occurs when the definition of the parameter space allows for the cases of either no predictability or sign-restricted predictability. We show that in this context constrained estimation gives rise to bootstrap statistics whose limit distribution is, in general, random, and thus distinct from the limit null distribution of the original statistics of interest. This is due to both (i) the possible location of the true parameter vector on the boundary of the parameter space and (ii) the possible non-stationarity of the posited predicting (resp. Granger-causing) variable. We discuss a modification of the standard fixed-regressor wild bootstrap scheme where the bootstrap parameter space is shifted by a data-dependent function in order to eliminate the portion of limiting bootstrap randomness attributable to the boundary and prove validity of the associated bootstrap inference under non-stationarity of the predicting variable as the only remaining source of limiting bootstrap randomness. Our approach, which is initially presented in a simple location model, has bearing on inference in parameter-on-the-boundary situations beyond the predictive regression problem.

A new explicit solution representation is provided for ARMA recursions with drift and either deterministically or stochastically varying coefficients. It is expressed in terms of the determinants of banded Hessenberg matrices and, as such, is an explicit function of the coefficients. In addition to computational efficiency, the proposed solution provides a more explicit analysis of the fundamental properties of such processes, including their Wold–Cramér decomposition, their covariance structure, and their asymptotic stability and efficiency. Explicit formulae for optimal linear forecasts based either on finite or infinite sequences of past observations are provided. The practical significance of the theoretical results in this work is illustrated with an application to U.S. inflation data. The main finding is that inflation persistence increased after 1976, whereas from 1986 onward, the persistence declines and stabilizes to even lower levels than the pre-1976 period.