A Bounds Approach to Inference Using the Long Run Multiplier

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.