How To Test Long-Run Relationships When Stationarity Is Unknown

🔎 Background and Problem

Pesaran, Shin, and Smith (2001) (PSS) introduced a bounds procedure to test for long-run cointegrating relationships between a unit-root dependent variable (yt) and weakly exogenous regressors (xt) when the stationarity properties of the regressors are uncertain. That procedure allows uncertainty over the regressors but assumes the dependent variable is a unit-root process. If the dependent variable might instead be stationary, the PSS test statistics become uninformative for deciding whether a long-run relationship (LRR) exists between yt and xt.

📐 New Test: Long-Run Multiplier (LRM)

A long-run multiplier (LRM) test statistic is proposed to detect LRRs without prior knowledge of whether the series are stationary or unit-root processes. The LRM approach is designed to operate under uncertainty about the univariate dynamics of both the dependent variable and the regressors.

🧪 Simulation Design and What Was Computed

• Stochastic simulations examine the behavior of the LRM test when the univariate dynamics of yt and xt are uncertain (stationary, unit-root, or mutually cointegrated).
• The simulations illustrate the bounds that the LRM statistic can take under alternative data-generating processes.
• Small-sample and approximate asymptotic critical values were generated for both the upper and lower bounds across a range of sample sizes and model specifications.

🔍 Key Findings

• The LRM statistic yields informative tests for long-run relationships even when it is unknown whether y_t is stationary or contains a unit root—situations in which PSS statistics fail to inform inference.
• A bounds framework for the LRM provides upper and lower decision thresholds that accommodate uncertainty about the series’ order of integration.
• Tabulated critical values (small-sample and approximate asymptotic) are provided for various sample sizes and model setups, improving applied inference.
• The bounds framework is demonstrated in empirical applications to models of public policy mood and presidential success, showing practical utility for political time-series analysis.

💡 Why It Matters

The LRM-based bounds approach enables reliable inference on long-run relationships without needing to know the stationarity of series in advance. This advances applied time-series methods in political science, offering a practical solution for researchers studying long-run links in public opinion and executive politics where unit-root properties are often ambiguous.