In this article, we highlight three points. First, we counter Grant and Lebo's claim that the error correction model (ECM) cannot be applied to stationary data. We maintain that when data are properly stationary, the ECM is an entirely appropriate model. We clarify that for a model to be properly stationary, it must be balanced. Second, we contend that while fractional integration techniques can be useful, they also have important weaknesses, especially when applied to many time series typical in political science. We also highlight two related but often ignored complications in time series: low power and overfitting. We argue that the statistical tests used in time-series analyses have little power to detect differences in many of the sample sizes typical in political science. Moreover, given the small sample sizes, many analysts overfit their time-series models. Overfitting occurs when a statical model describes random error or noise instead of the underlying relationship. We argue that the results in the Grant and Lebo replications could easily be a function of overfitting.

Pesaran, Shin, and Smith (2001) (PSS) proposed a bounds procedure for testing for the existence of long run cointegrating relationships between a unit root dependent variable ($y_{t}$) and a set of weakly exogenous regressors $\boldsymbol{x}_{t}$ when the analyst does not know whether the independent variables are stationary, unit root, or mutually cointegrated processes. This procedure recognizes the analyst’s uncertainty over the nature of the regressors but not the dependent variable. When the analyst is uncertain whether $y_{t}$ is a stationary or unit root process, the test statistics proposed by PSS are uninformative for inference on the existence of a long run relationship (LRR) between $y_{t}$ and $\boldsymbol{x}_{t}$. We propose the long run multiplier (LRM) test statistic as a means of testing for LRRs without knowing whether the series are stationary or unit roots. Using stochastic simulations, we demonstrate the behavior of the test statistic given uncertainty about the univariate dynamics of both $y_{t}$ and $\boldsymbol{x}_{t}$, illustrate the bounds of the test statistic, and generate small sample and approximate asymptotic critical values for the upper and lower bounds for a range of sample sizes and model specifications. We demonstrate the utility of the bounds framework for testing for LRRs in models of public policy mood and presidential success.

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.

Why does the public care more about some terrorist attacks than others? In recent years, there has been a wave of terrorist attacks carried out by similar terrorist organizations, but these attacks have produced disparate public responses. Existing research shows that terrorist attacks are more traumatic for people who live near terrorist targets, but this research cannot explain differences in public attitudes about attacks occurring in other countries. We argue that threat perceptions are shaped by the physical and personal proximity of terrorist attacks. The identities of the victims are rarely known. People impute the characteristics of victims based on the country where the attack occurred. These perceived identities determine the empathy people feel toward victims and affect perceptions of terrorist threats. People feel a greater sense of vulnerability when attacks occur near their borders. We test these arguments using a series of online experiments. We find that the location of the attack and the race and nationality of the victims drive threat perceptions.

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