Multivariate Time Series Analysis Research Articles

Dependency, reciprocity, and informal mentorship in predicting long-term research collaboration: A co-authorship matrix-based multivariate time series analysis

In this study, we examine the roles of dependency, reciprocity, and informal mentorship in the prediction of long-term research collaboration in five disciplines. We use co-authorship matrix-based multivariate time series features and interpretable machine learning to train long-term collaboration prediction models and interpret the feature importance of trained models. Overall, long-term research collaboration that is defined using various standards was rare across the examined disciplines, and the prediction results were moderate to good. We found dependency, reciprocity, and informal mentorship to have different roles in different disciplines. Among the three, informal mentorship was important in predicting long-term research collaboration in Agriculture, Geology, and Library and Information Science. Reciprocity, which measures the interdependence between two researchers was important to prediction in the fields of Agriculture and Geology. Finally, dependency was important in all the disciplines with varying degrees of importance.

Improved climate time series forecasts by machine learning and statistical models coupled with signature method: A case study with El Niño

The different phases of ENSO (El Niño Southern Oscillation) directly influence the occurrence of natural disasters and global warming. To limit the socio-economic impact, it is essential to develop simple and fast numerical models that can predict these different cycles. Here, we aimed to improve the predictive performance and extracting relevant information from climatic events by applying the signature method to time series models. After transforming the data using this signature method, we performed a comparative analysis of the statistical and machine learning models. In addition, we used PDP (Partial Dependence Plot) and SPRC (Standard Partial Regression Coefficient) to better understand the interactions between different climate indices.Our results showed that the best predictive performance was obtained when we use the signature method with the LSTM (R2 = 0.74) and Lasso (R2 = 0.79) models. Two complementary methods were used to highlight the influence of the following climate indices on ENSO cycle changes: NINO3, NINO3.4 and NPI (North Pacific Index). These methodologies also enabled us to determine the switchover thresholds, and order temporal variations. With this first application of the signature method to this type of time series, we obtained accurate forecasts on a 6-month scale with reduced computation time. This suggests that our methodology can be applied to many other fields of research that use multivariate time-series analyses.

Multivariate Analysis of the Main Operational Variables Involved in Steel Producing on BOF Using Time Series Tools

There is significant interest in accurately modeling the operational variables of the steel-making process in LD converters. Despite this, the task is challenging due to the complex interactions between process variables, which are not entirely comprehended. Often, decisions in the industry are grounded in experience. This study aims to introduce a robust model that can effectively guide engineers and technicians by forecasting the future behavior of steelmaking variables in the BOF furnace. We employed multivariate time series analysis to reach this goal, utilizing tools like Vector Autoregression models, ElasticNet, K-Nearest-Neighbors, Multiple Linear Regression, and Long Short-Term Memory Neural networks. These models were tested on data from three distinct steel production campaigns. A successful model was identified, predicting 35 out of the 42 chosen variables, demonstrating the potential to correlate a majority of the selected parameters.

Identifying quasi-periodic variability using multivariate empirical mode decomposition: a case of the tropical Pacific

Abstract. A variety of statistical tools have been used in climate science to gain a better understanding of the climate system's variability on various temporal and spatial scales. However, these tools are mostly linear, stationary, or both. In this study, we use a recently developed nonlinear and nonstationary multivariate time series analysis tool – multivariate empirical mode decomposition (MEMD). MEMD is a powerful tool for objectively identifying (intrinsic) timescales of variability within a given spatio-temporal system without any timescale pre-selection. Additionally, a red noise significance test is developed to robustly extract quasi-periodic modes of variability. We apply these tools to reanalysis and observational data of the tropical Pacific. This reveals a quasi-periodic variability in the tropical Pacific on timescales ∼ 1.5–4.5 years, which is consistent with El Niño–Southern Oscillation (ENSO) – one of the most prominent quasi-periodic modes of variability in the Earth’s climate system. The approach successfully confirms the well-known out-of-phase relationship of the tropical Pacific mean thermocline depth with sea surface temperature in the eastern tropical Pacific (recharge–discharge process). Furthermore, we find a co-variability between zonal wind stress in the western tropical Pacific and the tropical Pacific mean thermocline depth, which only occurs on the quasi-periodic timescale. MEMD coupled with a red noise test can therefore successfully extract (nonstationary) quasi-periodic variability from the spatio-temporal data and could be used in the future for identifying potential (new) relationships between different variables in the climate system.

A Multivariate Time Series Analysis of Electrical Load Forecasting Based on a Hybrid Feature Selection Approach and Explainable Deep Learning

In the smart grid paradigm, precise electrical load forecasting (ELF) offers significant advantages for enhancing grid reliability and informing energy planning decisions. Specifically, mid-term ELF is a key priority for power system planning and operation. Although statistical methods were primarily used because ELF is a time series problem, deep learning (DL)-based forecasting approaches are more commonly employed and successful in achieving precise predictions. However, these DL-based techniques, known as black box models, lack interpretability. When interpreting the DL model, employing explainable artificial intelligence (XAI) yields significant advantages by extracting meaningful information from the DL model outputs and the causal relationships among various factors. On the contrary, precise load forecasting necessitates employing feature engineering to identify pertinent input features and determine optimal time lags. This research study strives to accomplish a mid-term forecast of ELF study load utilizing aggregated electrical load consumption data, while considering the aforementioned critical aspects. A hybrid framework for feature selection and extraction is proposed for electric load forecasting. Technical term abbreviations are explained upon first use. The feature selection phase employs a combination of filter, Pearson correlation (PC), embedded random forest regressor (RFR) and decision tree regressor (DTR) methods to determine the correlation and significance of each feature. In the feature extraction phase, we utilized a wrapper-based technique called recursive feature elimination cross-validation (RFECV) to eliminate redundant features. Multi-step-ahead time series forecasting is conducted utilizing three distinct long-short term memory (LSTM) models: basic LSTM, bi-directional LSTM (Bi-LSTM) and attention-based LSTM models to accurately predict electrical load consumption thirty days in advance. Through numerous studies, a reduction in forecasting errors of nearly 50% has been attained. Additionally, the local interpretable model-agnostic explanations (LIME) methodology, which is an explainable artificial intelligence (XAI) technique, is utilized for explaining the mid-term ELF model. As far as the authors are aware, XAI has not yet been implemented in mid-term aggregated energy forecasting studies utilizing the ELF method. Quantitative and detailed evaluations have been conducted, with the experimental results indicating that this comprehensive approach is entirely successful in forecasting multivariate mid-term loads.

2879. Utilization of vector autoregressive models for multivariable time-series analysis to evaluate the difference between days of therapy and days of antibiotic spectrum coverage in an inpatient antimicrobial stewardship program

Abstract Background Days of therapy (DOT), commonly used to estimate antimicrobial consumption, has some limitations. Days of antibiotic spectrum coverage (DASC), a novel metric, overcomes these limitations. This study examined the difference between these two metrics of inpatient intravenous antimicrobial consumption in assessing antimicrobial stewardship efficacy and antimicrobial resistance by using the vector autoregressive (VAR) model with time-series analysis, which has been used in macroeconomics. Methods Differences between DOT and DASC were investigated at a tertiary care center in Japan over eight years using VAR models with three variables in the following order: 1) the monthly proportion of prospective audit and feedback (PAF) acceptance as an index of antimicrobial stewardship efficacy; 2) monthly DOT and DASC adjusted by 1,000 days present as indices of antimicrobial consumption; and 3) the monthly incidence of five, drug-resistant organisms as an index of antimicrobial resistance (Clostridioides difficile infections (CDI), extended-spectrum β-lactamase (ESBL)-producing Enterobacterales, methicillin-resistant Staphylococcus aureus (MRSA), drug-resistant Pseudomonas aeruginosa, and drug-resistant Enterobacterales). Results The Granger-causality test, which evaluates whether incorporating lagged variables can help to predict other variables, found that PAF acceptance contributed to DOT and DASC; these in turn contributed to the incidence of drug-resistant P. aeruginosa. Notably, only DASC helped to predict the incidence of drug-resistant Enterobacterales. Another VAR analysis demonstrated that a high proportion of PAF acceptances was accompanied by decreased DASC in a given month while increased DASC was accompanied by an increased incidence of drug-resistant Enterobacterales, unlike with DOT. Conclusion VAR models with variables in the order of PAF acceptance, antimicrobial consumption, and antimicrobial resistance suggested that DASC, including antimicrobial spectrum information, may more accurately reflect the impact of PAF on antimicrobial consumption and be superior to DOT for predicting the incidence of drug-resistant Enterobacterales. Disclosures Hitoshi Honda, MD, Moderna: Honoraria Manabu Akazawa, MPH, PhD, Astellas Pharma: Advisor/Consultant|GSK: Advisor/Consultant|Jansen: Advisor/Consultant|Shionogi: Advisor/Consultant|Takeda: Advisor/Consultant

Predicting scoring trends in basketball games based on multivariate time series analysis

Abstract In this paper, we first study the evaluation framework of Granger causality between time series. Then, the multivariate time series prediction model is constructed based on transfer entropy and graph attention network. The multivariate time series prediction model architecture has causal branches with transfer entropy, and the prediction module is executed using graph attention network. Finally, the accuracy of basketball game score prediction is analyzed, as well as comparing the accuracy of the multivariate time series prediction model with other models. The number of offensive rounds in basketball games can be predicted with a high accuracy of 67.94% using the multivariate time series prediction model. For the prediction of the scoring trend of player rotation, the ABHJI combination rotating on the court is the most efficient in scoring, and the average value of scoring is as high as 85.463. The average accuracy of the TSP reaches 72.2%, and the model has high feasibility and is suitable for scoring prediction in basketball games.

MultiBEATS: Blocks of eigenvalues algorithm for multivariate time series dimensionality reduction

Multivariate Time Series are sequences of observations taken from multiple sources. The proliferation of environments in which data is collected by means of sensors and the adoption of data-based services reveals the importance of their efficient analysis. In smart environments, the analysis of the raw Multivariate Time Series is cumbersome. The algorithms are heavy, slow, and fail to extract all the knowledge available. In that sense, representation techniques reduce the data points but maintain the inner patterns so that the information present in data persists. There exist several approaches to represent Univariate Time Series, but for the multivariate case they are scarce. We have adapted the existent univariate representation algorithm BEATS, which is based on Discrete Cosine transformation and the extraction of eigenvalues to multivariate scenarios and we have called it MultiBEATS. Our method allows flexibility with regard to data reduction and does not require labelled data. To investigate the efficacy of MultiBEATS, we selected 8 open multivariate time series datasets and compared the running time and accuracy of their classification using raw data and MultiBEATS transformed data. We have also compared our results with the representation algorithm Seq2VAR, based on autoencoders, for having a baseline. The MultiBEATS transformation reduces the execution time of classification algorithms, has improved the results in accuracy in 2 of the datasets and matches a third one. With regards to the baseline, MultiBEATS is faster than Seq2VAR and more accurate in 7 out of the 8 datasets. In conclusion, MultiBEATS is a promising approach for efficiently reducing multivariate time series data, therefore helping decision-making in dynamic smart environments.

Emotional states predict cellular immune system activity under conditions of life as it is lived: A multivariate time-series analysis approach.

The relationship between emotional states and immune system activity is characterized by bidirectional influences; however, limited information is available regarding the temporal dynamics of these effects. The goal of this investigation was to examine how these psychoimmunological interdependencies unfold over time under conditions of "life as it is lived". For this purpose, three healthy women collected their entire urine over a period of approximately two months at 12-h intervals (8 am-8 pm, 8 pm-8 am), resulting in a total of 112 to 126 consecutive measurements per subject. In addition, among other regular psychological assessments, the subjects completed the EWL-60-S, an emotional state questionnaire, each morning and evening. To assess the extent of T-helper type 1 immune activation, the neopterin per creatinine concentration was measured in the urine samples using high-pressure liquid chromatography. The dynamic relationships between the time series of the six emotional states (performance-related activity, general inactivity, extraversion/introversion, general feeling of comfort, emotional irritation, anxiety/depressiveness) and urinary neopterin levels were estimated in vector-autoregressive models and evaluated using Granger-causality tests, impulse-response functions and forecast error variance decompositions. The findings showed that emotional states explained up to 20% of the variance of urinary neopterin per creatinine levels, whereby most of the effects occurred within a period of approximately three days. Across all subjects, increases in anxiety/depressiveness and extraversion led to increases in neopterin levels, while a general feeling of comfort led to decreases in neopterin. These results emphasize the importance of the interdependencies between emotional states and immune system activity and showcase the potential that intensive longitudinal study designs offer for psychoneuroimmunology.

A new approach for time domain analysis of multivariate and functional time series

We apply the classical finite Fourier transform to construct an embedded functional process to a given multivariate time series model. The basic properties of the embedded functional process are presented. It appears that the embedded functional process is quite useful for the model building and prediction. We also provide a new method for approximating an infinite functional process with a finite functional process. The methodology is different from the classical one. Indeed, the active underlying frequencies are taken into consideration rather than the significant eigenvectors. The performance of our method is illustrated through simulations. Interestingly, in doing prediction, it appears that for our proposal the computation time is much shorter compared to existing ones.

Kernel-based joint independence tests for multivariate stationary and non-stationary time series.

Multivariate time-series data that capture the temporal evolution of interconnected systems are ubiquitous in diverse areas. Understanding the complex relationships and potential dependencies among co-observed variables is crucial for the accurate statistical modelling and analysis of such systems. Here, we introduce kernel-based statistical tests of joint independence in multivariate time series by extending the d-variable Hilbert-Schmidt independence criterion to encompass both stationary and non-stationary processes, thus allowing broader real-world applications. By leveraging resampling techniques tailored for both single- and multiple-realization time series, we show how the method robustly uncovers significant higher-order dependencies in synthetic examples, including frequency mixing data and logic gates, as well as real-world climate, neuroscience and socio-economic data. Our method adds to the mathematical toolbox for the analysis of multivariate time series and can aid in uncovering high-order interactions in data.

Does the "Obesity Paradox" Have an Expiration Date? A Retrospective Cohort Study.

(1) Background: The "obesity paradox" refers to a protective effect of higher body mass index (BMI) on mortality in acute infectious disease patients. However, the long-term impact of this paradox remains uncertain. (2) Methods: A retrospective study of patients diagnosed with community-acquired acute infectious diseases at Shamir Medical Center, Israel (2010-2020) was conducted. Patients were grouped by BMI: underweight, normal weight, overweight, and obesity classes I-III. Short- and long-term mortality rates were compared across these groups. (3) Results: Of the 25,226 patients, diverse demographics and comorbidities were observed across BMI categories. Short-term (90-day) and long-term (one-year) mortality rates were notably higher in underweight and normal-weight groups compared to others. Specifically, 90-day mortality was 22% and 13.2% for underweight and normal weight respectively, versus 7-9% for others (p < 0.001). Multivariate time series analysis revealed underweight individuals had a significantly higher 5-year mortality risk (HR 1.41 (95% CI 1.27-1.58, p < 0.001)), while overweight and obese categories had a reduced risk (overweight-HR 0.76 (95% CI 0.72-0.80, p < 0.001), obesity class I-HR 0.71 (95% CI 0.66-0.76, p < 0.001), obesity class II-HR 0.77 (95% CI 0.70-0.85, p < 0.001), and obesity class III-HR 0.79 (95% CI 0.67-0.92, p = 0.003)). (4) Conclusions: In this comprehensive study, obesity was independently associated with decreased short- and long-term mortality. These unexpected results prompt further exploration of this counterintuitive phenomenon.

Dynamic relationship among immediate release fentanyl use and cancer incidence: A multivariate time-series analysis using vector autoregressive models

Background A substantial increase in the incidence of immediate release fentanyl (IRF) use was reported in Spain from 2012 to 2017. Purpose This study aimed to investigate the relationship dynamically with cancer incidence in order to provide empirical evidence of inappropriate use of IRF with respect to the pathology. Research design A vector autoregresive (VAR) model was constructed using data from a nationwide electronic healthcare record database in primary care in Spain (BIFAP) according to the following step procedure: (1) split data into training data for modelling and test for validation (2) assessing for time series stationarity; (3) selecting lag-length; (4) building the VAR model; (5) assessing residual autocorrelation; (6) checking stability of the VAR system; (7) evaluating Granger causality; (8) impulse response analysis and forecast error variance decomposition (9) prediction performance with validation data. Results The analysis showed a strong and linear correlation between IRF and cancer (Pearson correlation coefficient: 0.594 (95% CI: 0.420–0.726). Two VAR models, VAR (2) and VAR (11) were selected and compared. All tests performed for both models satisfied assumptions for stability, predictability and accuracy. Granger causality revealed cancer incidence is a good predictor for IRF use. VAR (2) seemed to be slightly more accurate, according to the RMSE of the test data. Conclusions This study demonstrates that using a robust and structured VAR modelling approach, is able to estimate dynamics associations, involving IRF use and cancer incidence.

Generalized limited penetrable vector visibility graph

The paper introduced a new way for converting the multivariate time series to complex network combining the vector visibility graph and limited penetrable theory, and the concept is discussed on the generalized limited penetrable range, termed as generalized limited penetrable vector visibility graph. The proposed method is verified by the Lorenz system. We use the four kinds of network structure characteristic to evaluate the noise resistance of the method quantitatively, and provide the strategy with guidance for choosing the optimal limited penetrable range. The results show that the limited penetrable vector visibility graph inherits the noise resistance of limited penetrable visibility graph in the multivariate time series analysis. The generalized limited penetrable range is discussed to clarify the effect on the network structure.

STTRE: A Spatio-Temporal Transformer with Relative Embeddings for multivariate time series forecasting

The prevalence of multivariate time series data across several disciplines fosters a demand and, subsequently, significant growth in the research and advancement of multivariate time series analysis. Drawing inspiration from a popular natural language processing model, the Transformer, we propose the Spatio-Temporal Transformer with Relative Embeddings (STTRE) to address multivariate time series forecasting. This work primarily focuses on developing a Transformer-based framework that can fully exploit the spatio-temporal nature of a multivariate time series by incorporating several of the Transformer’s key components, but with augmentations that allow them to excel in multivariate time series forecasting. Current Transformer-based models for multivariate time series often neglect the data’s spatial component(s) and utilize absolute position embeddings as their only means to detect the data’s temporal component(s), which we show is flawed for time series applications. The lack of emphasis on fully exploiting the spatio-temporality of the data can incur subpar results in terms of accuracy. We redesign relative position representations, which we rename to relative embeddings, to unveil a new method for detecting latent spatial, temporal, and spatio-temporal dependencies more effectively than previous Transformer-based models. We couple these relative embeddings with a restructuring of the Transformer’s primary sequence learning mechanism, multi-head attention, in a way that allows for full utilization of relative embeddings, thus achieving up to a 24% improvement in accuracy over other state-of-the-art multivariate time series models on a comprehensive selection of publicly available multivariate time series forecasting datasets.

Analyzing and forecasting service demands using human mobility data: A two-stage predictive framework with decomposition and multivariate analysis

Accurate service demand forecasts at critical facilities are fundamental for efficiently managing resources and promptly providing essential services to people and community. However, it has received little attention in the literature, mainly due to the unavailability of granular data and the lack of sophisticated forecasting methods. To address this gap, we provide a new perspective on sensing service demands at critical facilities leveraging fine-grained human mobility data, and propose a novel data-driven framework to forecast mobility patterns at the neighborhood level. Specifically, we develop a two-stage forecasting scheme to manage large-scale and complex human movement information. The first stage is to decompose the large-scale mobility data into spatial and temporal patterns, whereas the second stage is to model complex temporal dynamics using multivariate time series analysis. The proposed framework is implemented using real human mobility data obtained from mobile phone users. The results show that our model demonstrates the best predictive performance for varying forecast horizons, when compared to multiple benchmark methods including traditionally-used statistical and deep learning models. We also performed model robustness checks, showing that the proposed model is robust in making short-term and long-term forecasts. The proposed predictive framework could help businesses and local governments accurately forecast service demands for critical facilities for better allocating their resources.

Opening the Black Box of Family-Based Treatments: An Artificial Intelligence Framework to Examine Therapeutic Alliance and Therapist Empathy.

The evidence-based treatment (EBT) movement has primarily focused on core intervention content or treatment fidelity and has largely ignored practitioner skills to manage interpersonal process issues that emerge during treatment, especially with difficult-to-treat adolescents (delinquent, substance-using, medical non-adherence) and those of color. A chief complaint of "real world" practitioners about manualized treatments is the lack of correspondence between following a manual and managing microsocial interpersonal processes (e.g. negative affect) that arise in treating "real world clients." Although family-based EBTs share core similarities (e.g. focus on family interactions, emphasis on practitioner engagement, family involvement), most of these treatments do not have an evidence base regarding common implementation and treatment process problems that practitioners experience in delivering particular models, especially in mid-treatment when demands on families to change their behavior is greatest in treatment - a lack that characterizes the field as a whole. Failure to effectively address common interpersonal processes with difficult-to-treat families likely undermines treatment fidelity and sustained use of EBTs, treatment outcome, and contributes to treatment dropout and treatment nonadherence. Recent advancements in wearables, sensing technologies, multivariate time-series analyses, and machine learning allow scientists to make significant advancements in the study of psychotherapy processes by looking "under the skin" of the provider-client interpersonal interactions that define therapeutic alliance, empathy, and empathic accuracy, along with the predictive validity of these therapy processes (therapeutic alliance, therapist empathy) to treatment outcome. Moreover, assessment of these processes can be extended to develop procedures for training providers to manage difficult interpersonal processes while maintaining a physiological profile that is consistent with astute skills in psychotherapeutic processes. This paper argues for opening the "black box" of therapy to advance the science of evidence-based psychotherapy by examining the clinical interior of evidence-based treatments to develop the next generation of audit- and feedback- (i.e., systemic review of professional performance) supervision systems.

Multiscale adaptive multifractal cross-correlation analysis of multivariate time series

In the real world, most time series generated from complex systems are nonlinear. To effectively study their fractal properties, in this work, we first generalize the adaptive fractal analysis (AFA) to the adaptive multifractal cross-correlation analysis (AMFCCA), which can be used to study the multifractal cross-correlation between two time series. Considering the complexity of time series from complex systems, we extend AMFCCA to the case of multivariate time series, namely multivariate adaptive multifractal cross-correlation analysis (MV-AMFCCA). In order to detect multiscale multifractality, we propose multiscale multivariate adaptive multifractal cross-correlation analysis (MMV-AMFCCA). By the numerical simulation on synthetic multivariate processes, our method shows the theoretical validity. Furthermore, we applied these methods to the multifractal analysis of three pollutants in urban and suburban areas in Beijing. After removing the seasonal trend, we find that the urban and suburban systems both have multifractality, especially the multifractality of the urban system are more evident than the suburban systems, and the degree of multifractality in spring and winter stronger than that in summer and autumn. Therefore, our methods are suitable for systems with multiple outputs and provide more comprehensive methods for characterizing multivariate autocorrelation and cross-correlation.

Electrophysiological markers for anticipatory processing of nocebo-augmented pain.

Nocebo effects on pain are widely thought to be driven by negative expectations. This suggests that anticipatory processing, or some other form of top-down cognitive activity prior to the experience of pain, takes place to form sensory-augmenting expectations. However, little is known about the neural markers of anticipatory processing for nocebo effects. In this event-related potential study on healthy participants (n = 42), we tested whether anticipatory processing for classically conditioned nocebo-augmented pain differed from pain without nocebo augmentation using stimulus preceding negativity (SPN), and Granger Causality (GC). SPN is a slow-wave ERP component thought to measure top-down processing, and GC is a multivariate time series analysis used to measure functional connectivity between brain regions. Fear of pain was assessed with the Fear of Pain Questionnaire-III and tested for correlation with SPN and GC metrics. We found evidence that both anticipatory processing measured with SPN and functional connectivity from frontal to temporoparietal brain regions measured with GC were increased for nocebo pain stimuli relative to control pain stimuli. Other GC node pairs did not yield significant effects, and a lag in the timing of nocebo pain stimuli limited interpretation of the results. No correlations with trait fear of pain measured after the conditioning procedure were detected, indicating that while differences in neural activity could be detected between the anticipation of nocebo and control pain trials, they likely were not related to fear. These results highlight the role that top-down processes play in augmenting sensory perception based on negative expectations before sensation occurs.

(Structural) VAR models with ignored changes in mean and volatility

The paper discusses how standard forecasting tools in multivariate time series analysis are affected when ignoring possible changes in the mean and the (co)variance. We study the estimation, forecasts, and estimated impulse responses of so-called long vector autoregressions, for which the complexity of the model increases with the sample size. We prove that, in spite of structural change in the data generating process, coefficient estimates and out-of-sample forecasts based on such long vector autoregressions are consistent. The sampling behaviour of estimated impulse responses depends primarily on the residual covariance matrix, which converges to an “average” covariance matrix in the case of varying (co)variances. Localised estimators (also obtained by means of a suitable long vector autoregression) may be more suitable in this case. Monte Carlo simulations support our theoretical findings. The empirical relevance of the theory is illustrated in two applications: (i) the international dynamics of inflation, and (ii) uncertainty and economic activity.