Beyond the Unit Root Question: Uncertainty and Inference

Abstract

A fundamental challenge facing applied time‐series analysts is how to draw inferences about long‐run relationships (LRR) when we are uncertain whether the data contain unit roots. Unit root tests are notoriously unreliable and often leave analysts uncertain, but popular extant methods hinge on correct classification. Webb, Linn, and Lebo (WLL; 2019) develop a framework for inference based on critical value bounds for hypothesis tests on the long‐run multiplier (LRM) that eschews unit root tests and incorporates the uncertainty inherent in identifying the dynamic properties of the data into inferences about LRRs. We show how the WLL bounds procedure can be applied to any fully specified regression model to solve this fundamental challenge, extend the results of WLL by presenting a general set of critical value bounds to be used in applied work, and demonstrate the empirical relevance of the LRM bounds procedure in two applications.

Do COVID-19 Policies Affect Mobility Behaviour? Evidence from 75 Canadian and American Cities

“…We also include dummy variables for days of the week-excluding Sunday as a reference category-to account for predictable day-based patterns in mobility. Webb et al (2020 ) provide two critical value bounds for the long-run multipliers. If the test statistic for an LRM is below the lower bound, we fail to reject the null hypothesis of no long-run relationship.…”

Deficit or Austerity Bias? The Changing Nature of Canadians’ Opinion of Fiscal Policies

“…Hence, we are confident that the effect of changes to net lending does not simply reflect the impact of the economy on the budgetary balance. However, in the appendix, we run a long run multiplier bounds test (Webb et al, 2020) that indicates that we have a short-term, immediate relationship between net lending and approval but no long-term relationship between the two.…”

The Unsustainable State: Greenhouse Gas Emissions, Inequality, and Human Well-Being in the United States, 1913 to 2017

“…To address these challenges and limitations, we use the long-run multiplier (LRM) bounds approach to time-series analysis recently developed by Webb, Linn, and Lebo (2019, 2020). Rather than relying on pretesting for unit roots that have low statistical power, the LRM bounds approach draws inferences from the LRM t statistic according to critical value bounds reported by Webb et al (2019, 2020). This approach avoids the problem of pretesting all together, while accounting for the uncertainty inherent in time-series analysis.…”

“…The second step is to calculate the LRM, or long-run effect, ( − x t − 1 y t − 1 ) , and its standard error, which we estimate using the delta method (the nlcom command in Stata). The third step is to compare the absolute value of the t statistic to the bounds provided by Webb et al (2020). If the t statistic is above the upper bound, we conclude that there is a long-run relationship.…”