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 (y_t) and weakly exogenous regressors (x_t) 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 y_t and x_t.

📐 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 y_t and x_t 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.