A Bounds Approach to Inference Using the Long Run Multiplier

Abstract

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.

Beyond the Unit Root Question: Uncertainty and Inference

“…As Webb, Linn, and Lebo () demonstrate, this ambiguity also affects the interpretation of alternative tests for LRRs, including the widely cited tests proposed by Pesaran, Shin, and Smith (). The PSS testing procedure applies critical value bounds to the conventional t ‐test on , as well as an F‐test for cointegration, to accommodate pretest uncertainty in X .…”

“…Webb, Linn, and Lebo () offer a solution to the problem that builds on the logic of the PSS approach. Specifically, they establish critical value bounds to reflect pretest uncertainty in tests for LRRs based on the long‐run multiplier.…”

“…Webb, Linn, and Lebo () propose using the LRM to test for the existence of LRRs. The logic can be easily understood by considering the LRM in the simple bivariate GECM presented in Equation , where .…”

“…De Boef and Keele () recommend that practitioners use the LRM to draw inferences about LRRs but do not consider how the dynamic properties of the data affect the appropriate critical values for hypothesis tests. However, as Webb, Linn, and Lebo () demonstrate, the distribution of the LRM t ‐statistic, and thus the critical values for the hypothesis test on the LRM, depends on the univariate characteristics of the variables in the model, as well as the deterministic features of the DGP for y , the number of independent variables, and the sample size. The authors use this information to establish critical value bounds for the LRM t ‐test that reflect pretesting uncertainty.…”

“…Philips () recommends adopting the cointegration testing strategy developed by Pesaran, Shin, and Smith (PSS; ) that allows the analyst to be agnostic about the properties of their regressors. While this procedure represents an important innovation, the approach developed by PSS requires that the analyst know the univariate properties of the regressand and breaks down if y is not a unit root (Webb, Linn, and Lebo ).…”