[Paper Review] How to estimate time-varying Vector Autoregressive Models? A comparison of two methods

Jonas M B Haslbeck, Laura F. Bringmann+1 more|arXiv (Cornell University)|Nov 14, 2017

Mental Health Research Topics31 references13 citations

TL;DR

This paper compares two methods for estimating time-varying Vector Autoregressive (TV-VAR) models: spline-based and kernel-smoothing approaches. It demonstrates through simulations and a practical tutorial that both methods outperform stationary VAR models in capturing non-stationary dynamics in individual-level time series data, with kernel smoothing showing slightly better performance in finite samples.

ABSTRACT

The ubiquity of mobile devices led to a surge in intensive longitudinal (or time series) data of individuals. This is an exciting development because personalized models both naturally tackle the issue of heterogeneities between people and increase the validity of models for applications. A popular model for time series is the Vector Autoregressive (VAR) model, in which each variable is modeled as a linear function of all variables at previous time points. A key assumption of this model is that the parameters of the true data generating model are constant (or stationary) across time. The most straightforward way to check for time-varying parameters is to fit a model that allows for time-varying parameters. In the present paper we compare two methods to estimate time-varying VAR models: the first method uses a spline-approach to allow for time-varying parameters, the second uses kernel-smoothing. We report the performance of both methods and their stationary counterparts in an extensive simulation study that reflects the situations typically encountered in practice. We compare the performance of stationary and time-varying models and discuss the theoretical characteristics of all methods in the light of the simulation results. In addition, we provide a step-by-step tutorial for both methods showing how to estimate a time-varying VAR model on an openly available individual time series dataset.

Motivation & Objective

• To address the limitation of stationary VAR models in capturing time-varying dynamics in individual-level time series data.
• To evaluate the performance of two time-varying VAR estimation methods—spline-based and kernel-smoothing—under realistic simulation conditions.
• To provide a practical, step-by-step tutorial for applying time-varying VAR models to real individual-level data using open-source tools.
• To compare the theoretical and empirical properties of time-varying and stationary VAR models in the context of intensive longitudinal data.

Proposed method

• The spline-approach models time-varying coefficients using B-splines, allowing smooth, piecewise polynomial estimation of VAR parameters over time.
• The kernel-smoothing method estimates time-varying coefficients by weighting past observations using a kernel function, with bandwidth selection critical for performance.
• Both methods are applied to individual-level time series data, with model fitting performed using maximum likelihood or restricted maximum likelihood estimation.
• The simulation study replicates common features of intensive longitudinal data, including small sample sizes, non-stationarity, and heteroscedasticity.
• A real-world dataset from an open-access repository is used to demonstrate the implementation of both methods in practice.
• Model performance is evaluated using criteria such as mean squared error of parameter estimates and prediction accuracy.

Experimental results

Research questions

• RQ1 How do spline-based and kernel-smoothing methods compare in estimating time-varying VAR models under realistic data conditions?
• RQ2 To what extent do time-varying VAR models improve upon stationary VAR models in capturing non-stationary dynamics in individual time series?
• RQ3 Which method—spline or kernel—yields more accurate and stable parameter estimates in finite samples?
• RQ4 How do the theoretical properties of each method align with their empirical performance in simulations?
• RQ5 What are the practical steps for implementing time-varying VAR models using real individual-level data?

Key findings

• The kernel-smoothing method generally outperformed the spline-based approach in terms of parameter estimation accuracy, especially in small to moderate sample sizes.
• Time-varying VAR models significantly reduced prediction error compared to stationary VAR models when the true data-generating process exhibited time-varying parameters.
• The spline-based method showed higher variability in parameter estimates under high noise and low sample sizes, though it provided smoother coefficient trajectories.
• Both time-varying methods detected non-stationarities missed by stationary VAR models, improving model fit and validity in heterogeneous individual data.
• The simulation results confirmed that ignoring time-varying dynamics leads to biased parameter estimates and poor predictive performance.
• The tutorial demonstrated that both methods are implementable using standard statistical software, with clear guidance on bandwidth and knot selection.