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We develop inference methods about long-run comovement of two time series. The parameters of interest are defined in terms of population second moments of low-frequency transformations (\"low-pass\" filtered versions) of the data. We numerically determine confidence sets that control coverage over a wide range of potential bivariate persistence patterns, which include arbitrary linear combinations of I(0), I(1), near unit roots, and fractionally integrated processes. In an application to U.S. economic data, we quantify the long-run covariability of a variety of series, such as those giving rise to balanced growth, nominal exchange rates and relative nominal prices, the unemployment rate and inflation, money growth and inflation, earnings and stock prices, etc.

This study investigates whether post-Soviet countries are caught in the middle-income trap, using the Robertson and Ye (2013) approach. A comprehensive set of unit root tests was employed, including traditional tests (ADF), nonlinear tests (KSS, Kruse, Sollis), and advanced Fourier-based tests (FKPSS, FF-ADF, FADF, FKSS, FKruse, FSollis) to analyze the data spanning from 1990 to 2023. The results revealed a significant heterogeneity in convergence patterns across the countries. It can be concluded that Moldova, Kyrgyzstan, Tajikistan, Armenia, Azerbaijan, Belarus, and Georgia exhibited stationarity in most tests, indicating that these countries are in the middle-income trap. In contrast, Kazakhstan, Uzbekistan, Turkmenistan, Russia, and Ukraine displayed non-stationary results, suggesting that they are not in the trap and are converging toward higher income levels. In addition, the radar chart, coefficient of variation, and three different Multi-Criteria Decision Analysis techniques (Equal Weight Score, Discrimination Weighted Score, and Entropy Weighted Score) were used for robustness check. The results of these tests appear to be consistent with the outcomes of the unit root tests.

New methods are developed for identifying, estimating, and performing inference with nonstationary time series that have autoregressive roots near unity. The approach subsumes unit-root (UR), local unit-root (LUR), mildly integrated (MI), and mildly explosive (ME) specifications in the new model formulation. It is shown how a new parameterization involving a localizing rate sequence that characterizes departures from unity can be consistently estimated in all cases. Simple pivotal limit distributions that enable valid inference about the form and degree of nonstationarity apply for MI and ME specifications and new limit theory holds in UR and LUR cases. Normalizing and variance stabilizing properties of the new parameterization are explored. Simulations are reported that reveal some of the advantages of this alternative formulation of nonstationary time series. A housing market application of the methods is conducted that distinguishes the differing forms of house price behavior in Australian state capital cities over the past decade.

Many time series exhibit \"long memory\": Their autocorrelation function decays slowly with lag. This behavior has traditionally been modeled via unit roots or fractional Brownian motion and explained via aggregation of heterogeneous processes, nonlinearity, learning dynamics, regime switching, or structural breaks. This paper identifies a different and complementary mechanism for long-memory generation by showing that it can naturally arise when a large number of simple linear homogeneous economic subsystems with short memory are interconnected to form a network such that the outputs of the subsystems are fed into the inputs of others. This networking picture yields a type of aggregation that is not merely additive, resulting in a collective behavior that is richer than that of individual subsystems. Interestingly, the long-memory behavior is found to be almost entirely determined by the geometry of the network, while being relatively insensitive to the specific behavior of individual agents.

We compare the effectiveness of Classical, Bayesian, and Machine Learning (ML) methods for predicting the presence of a unit root in univariate time-series models. Framing the issue as a classification problem, we demonstrate how ML may be used to uncover structural features of a macroeconomic time series with small data. We use a Monte Carlo approach to evaluate the predictions from these approaches and find that ML outperforms both the Classical and Bayesian tests using prediction accuracy, and appears to be the most flexible for classifying unit roots when class imbalance is present. In data, we find broad consensus among the approaches for predicted nonstationary series, with some disagreement for predicted stationary series.

Our study investigates the relationship between technology innovation and human development in technologically advanced countries using data from quarterly observations from the last decade of the twentieth century to the first two decades of the twenty-first century. This objective of this study is to implement Quantile-on-Quantile regression (QQ) technique that as formulated by Sim and Zhou (J Bank Finance 55:1–8, 2015) and the renowned Granger-causality in quantiles as proposed by Troster (Econom Rev 37(8):850–866, 2018) examine the basic relationship between the given quantiles of technology innovation and their effects on the quantiles of human development. Therefore, the outcomes of this study explain the overall interdependence of technology innovation and affect the overall human development index. It is enumerated that the empirical results indicate that a significant positive relationship exists between technology innovation and human development in all selected technologically advanced countries, predominantly in both low and high tails. Moreover, the outcomes of Granger causality quantiles indicate a bi-directional fundamental relationship between these two variables in the dataset of all countries. The outcomes of the observations are extended to the recent findings on these two variables’ nexus and imply a differential impact on the technologically advanced countries. This causality guides us to offer some specific policy recommendations to each group of countries.

This study examines whether OECD countries have converged over time in terms of export diversification. Focusing on the period 1995–2023, the study employs a new Fourier function-enhanced panel unit root test that takes into account gradual changes instead of sudden breaks. The findings show that more than half of OECD countries have not converged in terms of export diversification. Export diversification plays an important role in achieving economic growth and development as well as long-term sustainability goals. In this context, the study provides critical data for policymakers in support of sustainable development goals. The results point to the need to reassess the economic and environmental impacts of export diversification policies in OECD countries.

Pesaran, Shin, and Smith (2001) (PSS) proposed a bounds procedure for testing for the existence of long run cointegrating relationships between a unit root dependent variable ( $y_{t}$ ) and a set of weakly exogenous regressors $\\boldsymbol{x}_{t}$ when the analyst does not know whether the independent variables are stationary, unit root, or mutually cointegrated processes. This procedure recognizes the analyst’s uncertainty over the nature of the regressors but not the dependent variable. When the analyst is uncertain whether $y_{t}$ is a stationary or unit root process, the test statistics proposed by PSS are uninformative for inference on the existence of a long run relationship (LRR) between $y_{t}$ and $\\boldsymbol{x}_{t}$ . We propose the long run multiplier (LRM) test statistic as a means of testing for LRRs without knowing whether the series are stationary or unit roots. Using stochastic simulations, we demonstrate the behavior of the test statistic given uncertainty about the univariate dynamics of both $y_{t}$ and $\\boldsymbol{x}_{t}$ , illustrate the bounds of the test statistic, and generate small sample and approximate asymptotic critical values for the upper and lower bounds for a range of sample sizes and model specifications. We demonstrate the utility of the bounds framework for testing for LRRs in models of public policy mood and presidential success.

A generalization of the Exact Local Whittle estimator in Shimotsu and Phillips (2005, Annals of Statistics 33, 1890–1933) is proposed for jointly estimating all the memory parameters in general long memory time series that possibly display standard, seasonal, and/or other cyclical strong persistence. Consistency and asymptotic normality are proven for stationary, nonstationary, and noninvertible series, permitting straightforward standard inference of interesting hypotheses such as the existence of unit roots and equality of memory parameters at some or all seasonal frequencies, which can be used as a prior test for the application of seasonal differencing filters. The effects of unknown deterministic terms are also discussed. Finally, the finite sample performance is analyzed in an extensive Monte Carlo exercise and an application to an U.S. Industrial Production index.

Confidence intervals are commonly used to describe parameter uncertainty. In nonstandard problems, however, their frequentist coverage property does not guarantee that they do so in a reasonable fashion. For instance, confidence intervals may be empty or extremely short with positive probability, even if they are based on inverting powerful tests. We apply a betting framework and a notion of bet-proofness to formalize the \"reasonableness\" of confidence intervals as descriptions of parameter uncertainty, and use it for two purposes. First, we quantify the violations of bet-proofness for previously suggested confidence intervals in nonstandard problems. Second, we derive alternative confidence sets that are bet-proof by construction. We apply our framework to several nonstandard problems involving weak instruments, near unit roots, and moment inequalities. We find that previously suggested confidence intervals are not bet-proof, and numerically determine alternative bet-proof confidence sets.