This article introduces and analyzes a framework that accommodates general heterogeneity in regression modeling. It demonstrates that regression models with fixed or time-varying parameters can be estimated using the ordinary least squares (OLS) and time-varying OLS methods, respectively, across a broad class of regressors and noise processes not covered by existing theory. The proposed setting facilitates the development of asymptotic theory and the estimation of robust standard errors. The robust confidence interval estimators accommodate substantial heterogeneity in both regressors and noise. The resulting robust standard error estimates coincide with White's (1980, Econometrica 48, 817-838) heteroskedasticity-consistent estimator but are applicable to a broader range of conditions, including models with missing data. They are computationally simple and perform well in Monte Carlo simulations. Their robustness, generality, and ease of implementation make them highly suitable for empirical applications. Finally, the article provides a brief empirical illustration.

A number of economic, financial, and climatic time series exhibit persistent cycles which are characterized by time-dependence patterns and peaks in the spectrum. In this article, we introduce a class of semiparametric cyclical–memory processes which enable the modeling of random cyclical patterns in stationary and non-stationary time series. We develop a theoretical background and asymptotic estimation theory for the frequency of a cycle represented by the location of a peak in the spectrum. The estimation procedure is easy to implement and allows for the construction of narrow confidence intervals around the location point. Monte Carlo simulations confirm the good finite sample performance of our estimator. We illustrate our method with three empirical applications. We uncover (quasi-)periodic cycles in macroeconomic series, both nominal and real (U.S. nominal GDP and real industrial production), and CO 2 $\mathrm {CO}_{2}$ upper C upper O 2 concentration levels.

Time series models are often fitted to the data without preliminary checks for stability of the mean and variance, conditions that may not hold in much economic and financial data, particularly over long periods. Ignoring such shifts may result in fitting models with spurious dynamics that lead to unsupported and controversial conclusions about time dependence, causality, and the effects of unanticipated shocks. In spite of what may seem as obvious differences between a time series of independent variates with changing variance and a stationary conditionally heteroskedastic (GARCH) process, such processes may be hard to distinguish in applied work using basic time series diagnostic tools. We develop and study some practical and easily implemented statistical procedures to test the mean and variance stability of uncorrelated and serially dependent time series. Application of the new methods to analyze the volatility properties of stock market returns leads to some unexpected surprising findings concerning the advantages of modeling time varying changes in unconditional variance.

Hypothesis testing in models allowing for trending processes that are possibly nonstationary and non-Gaussian is considered. Using semiparametric estimators, joint hypothesis testing for these processes is developed, taking into account the one-sided nature of typical hypotheses on the persistence parameter in order to gain power. The results are applicable for a wide class of processes and are easy to implement. They are illustrated with an application to the dynamics of GDP.

Considerable evidence in past research shows size distortion in standard tests for zero autocorrelation or zero cross-correlation when time series are not independent identically distributed random variables, pointing to the need for more robust procedures. Recent tests for serial correlation and cross-correlation in Dalla, Giraitis, and Phillips (2022) provide a more robust approach, allowing for heteroskedasticity and dependence in uncorrelated data under restrictions that require a smooth, slowly-evolving deterministic heteroskedasticity process. The present work removes those restrictions and validates the robust testing methodology for a wider class of innovations and regression residuals allowing for heteroscedastic uncorrelated and non-stationary data settings. The updated analysis given here enables more extensive use of the methodology in practical applications. Monte Carlo experiments confirm excellent finite sample performance of the robust test procedures even for extremely complex white noise processes. The empirical examples show that use of robust testing methods can materially reduce spurious evidence of correlations found by standard testing procedures.

In many different fields realizations of stationary time series might be recorded at irregular points in time, resulting in observed unevenly spaced samples. These missing observations can happen for several reasons, depending on the mechanisms that record the data or external conditions that force the missing observations. In this article, we first focus on the question if we can estimate the mean of a stationary time series when data are not equally spaced. We show that any unevenly spaced sample can be used to estimate the mean of an underlying stationary linear time series. Specifically, we do not impose any restrictions on sampling structure and times, as long as they are independent of the underlying time series. We provide an expression for the sample mean estimator and we establish its asymptotic properties and the central limit theorem. Subsequently we studentize estimation which allows to build confidence intervals for the mean. Finite sample properties of the estimator for the mean are investigated in a Monte Carlo study which confirms good performance of such estimation procedure.

The use of a non-i.i.d. noise in parametric modeling of stationary time series can lead to unexpected distortions of the standard errors and confidence intervals in parameter estimation. We consider AR(1) and MA(1) models and motivate the need for correction of standard errors when these are generated by a non-i.i.d. noise. The impact of the noise on the standard errors and confidence intervals is illustrated with Monte Carlo simulations using various types of noise.

This paper suggests a multiplicative volatility model where volatility is decomposed into a stationary and a nonstationary persistent part. We provide a testing procedure to determine which type of volatility is prevalent in the data. The persistent part of volatility is associated with a nonstationary persistent process satisfying some smoothness and moment conditions. The stationary part is related to stationary conditional heteroskedasticity. We outline theory and conditions that allow the extraction of the persistent part from the data and enable standard conditional heteroskedasticity tests to detect stationary volatility after persistent volatility is taken into account. Monte Carlo results support the testing strategy in small samples. The empirical application of the theory supports the persistent volatility paradigm, suggesting that stationary conditional heteroskedasticity is considerably less pronounced than previously thought.

Time variation is a fundamental problem in statistical and econometric analysis of macroeconomic and financial data. Recently, there has been considerable focus on developing econometric modelling that enables stochastic structural change in model parameters and on model estimation by Bayesian or nonparametric kernel methods. In the context of the estimation of covariance matrices of large dimensional panels, such data requires taking into account time variation, possible dependence and heavy-tailed distributions. In this paper, we introduce a nonparametric version of regularization techniques for sparse large covariance matrices, developed by Bickel and Levina (2008) and others. We focus on the robustness of such a procedure to time variation, dependence and heavy-tailedness of distributions. The paper includes a set of results on Bernstein type inequalities for dependent unbounded variables which are expected to be applicable in econometric analysis beyond estimation of large covariance matrices. We discuss the utility of the robust thresholding method, comparing it with other estimators in simulations and an empirical application on the design of minimum variance portfolios.

The R package testcorr implements standard and robust procedures for testing the significance of the autocorrelation in univariate data and the cross-correlation in bivariate data. It also includes tests for the significance of pairwise Pearson correlation in multivariate data and the i.i.d. property for univariate data. The standard testing procedures on significance of correlation are used commonly by practitioners while their robust versions were developed in Dalla, Giraitis, and Phillips (2020), where the tests for i.i.d. property can be also found. This document briefly outlines the testing procedures and provides simple examples.

We develop non-parametric instrumental variable estimation and inferential theory for econometric models with possibly endogenous regressors whose coefficients can vary over time either deterministically or stochastically, and the time-varying and uniform versions of the standard Hausman exogeneity test. After deriving the asymptotic properties of the proposed procedures, we assess their finite sample performance by means of a set of Monte Carlo experiments, and illustrate their application by means of an empirical example on the Phillips curve.

The paper develops point estimation and asymptotic theory with respect to a semiparametric model for time series with moving mean and unconditional heteroscedasticity. These two features are modelled nonparametrically, whereas autocorrelations are described by a short memory stationary parametric time series model. We first study the usual least squares estimate of the coefficient of the first-order autoregressive model based on constant but unknown mean and variance. Allowing for both the latter to vary over time in a general way we establish its probability limit and a central limit theorem for a suitably normed and centred statistic, giving explicit bias and variance formulae. As expected mean variation is the main source of inconsistency and heteroscedasticity the main source of inefficiency, though we discuss circumstances in which the estimate is consistent for, and asymptotically normal about, the autoregressive coefficient, albeit inefficient. We then consider standard implicitly-defined Whittle estimates of a more general class of short memory parametric time series model, under otherwise more restrictive conditions. When the mean is correctly assumed to be constant, estimates that ignore the heteroscedasticity are again found to be asymptotically normal but inefficient. Allowing a slowly time-varying mean we resort to trimming out of low frequencies to achieve the same outcome. Returning to finite order autoregression, nonparametric estimates of the varying mean and variance are given asymptotic justification, and forecasting formulae developed. Finite sample properties are studied by a small Monte Carlo simulation, and an empirical example is also included.

Commonly used tests to assess evidence for the absence of autocorrelation in a univariate time series or serial cross-correlation between time series rely on procedures whose validity holds for i.i.d. data. When the series are not i.i.d., the size of correlogram and cumulative Ljung-Box tests can be significantly distorted. This paper adapts standard correlogram and portmanteau tests to accommodate hidden dependence and non-stationarities involving heteroskedasticity, thereby uncoupling these tests from limiting assumptions that reduce their applicability in empirical work. To enhance the Ljung-Box test for non-i.i.d. data a new cumulative test is introduced. Asymptotic size of these tests is unaffected by hidden dependence and heteroskedasticity in the series. Related extensions are provided for testing cross-correlation at various lags in bivariate time series. Tests for the i.i.d. property of a time series are also developed. An extensive Monte Carlo study confirms good performance in both size and power for the new tests. Applications to real data reveal that standard tests frequently produce spurious evidence of serial correlation.

Model selection and estimation are important topics in econometric analysis which can become considerably complicated in high dimensional settings, where the set of possible regressors can become larger than the set of available observations. For large scale problems the penalized regression methods (e.g. Lasso) have become the de facto benchmark that can effectively trade off parsimony and fit. In this paper we introduce a regularized estimation and model selection approach that is based on sparse large covariance matrix estimation, introduced by Bickel and Levina (2008) and extended by Dendramis, Giraitis, and Kapetanios (2018). We provide asymptotic and small sample results that indicate that our approach can be an important alternative to the penalized regression. Moreover, we also introduce a number of extensions that can improve the asymptotic and small sample performance of the proposed method. The usefulness of what we propose is illustrated via Monte Carlo exercises and an empirical application in macroeconomic forecasting.

This paper extends Whittle estimation to linear processes with a general stationary ergodic martingale difference noise. We show that such estimation is valid for standard parametric time series models with smooth bounded spectral densities, e.g., ARMA models. Furthermore, we clarify the impact of the hidden dependence in the noise on such estimation. We show that although the asymptotic normality of the Whittle estimates may still hold, the presence of dependence in the noise impacts the limit variance. Hence, the standard errors and confidence intervals valid under i.i.d. noise may not be applicable and thus require correction. The goal of this paper is to raise awareness to the impact of a non-i.i.d. noise in applied work.

Recently there has been considerable focus on methods that improve the estimation of the parameters which are essential for the mean-variance optimal portfolio frontier. As it is dictated by the financial and macroeconomic series it is essential for these estimators to account for time variation, possible dependence, and heavy tailed distributions. In this paper we introduce estimation methods which are able to account for these characteristics, as well as the large dimensionality feature of the available investment opportunities,which characterize the financial markets today. We apply our methods on both the minimum variance and the mean variance optimal portfolios and obtain a new set of portfolio strategies. A comparative empirical analysis on a rich dataset that comprises diverse risk characteristics, shows that the new strategies generally offer higher risk-adjusted returns and lower levels of risk, in a large out of sample forecasting exercise.

In this article, we introduce the general setting of a multivariate time series autoregressive model with stochastic time‐varying coefficients and time‐varying conditional variance of the error process. This allows modelling VAR dynamics for non‐stationary time series and estimation of time‐varying parameter processes by the well‐known rolling regression estimation techniques. We establish consistency, convergence rates, and asymptotic normality for kernel estimators of the paths of coefficient processes and provide pointwise valid standard errors. The method is applied to a popular seven‐variable dataset to analyse evidence of time variation in empirical objects of interest for the DSGE (dynamic stochastic general equilibrium) literature.

We establish the asymptotic normality of a quadratic form \(Q_n\) in martingale difference random variables \(\eta _t\) when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables \(\eta _t\), asymptotic normality holds under condition \(||A||_{sp}=o(||A||) \), where \(||A||_{sp}\) and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences \(\eta _t\) has been an important open problem. We provide such sufficient conditions in this paper.