The following flow-chart summarizes the steps:
Lastly, if cointegration does exist, one can estimate and conduct inference on the speed of convergence to equilibrium. Thus, in estimating the CECM (ARDL), one can simultaneously test for cointegration and estimate the equilibrating relationship. The advantage of the procedure is that it uses the CECM (ARDL) as a platform. It is here where the Bounds test comes into the limelight: it is a way of statistically detecting the presence of cointegration. What we expound on here, is that this estimate may not always be defined or sensible, and even if it is, it may be degenerate that is, seemingly stable in the short-run, but dissipates in the long-run.
However, recall from Part 1 of this series that one of the major advantages of the ARDL model is due to its ability to estimate the long-run or cointegrating relationship. In this regard, we will demonstrate that the ARDL model is in fact a special case of the CECM. In other words, we hope to derive a conditional ECM (CECM), which formalizes an ECM model for some variable conditional on all the others, but at the same time, isolates the cointegrating relationship among them. How does one variable in the VAR behave conditional on a all the others, which are themselves endogenously determined, and is their any cointegrating relationship among them? However, applications in Economics typically ask: Nevertheless, like the VAR, the VECM models simultaneous interactions among several endogenous variables. In this regard, the vector error correction model (VECM), which is a reparameterization of the VAR to isolate the equilibrating relationships, if they exist, is of central importance. As such, it lends itself to the analysis of simultaneous interactions between variables - namely, their short-run dynamics, but more importantly, their long-run (equilibrating) or cointegrating behaviour. Recall that a VAR is a natural extension of the univariate autoregressive model to multivariate series, and is often interpreted as an autoregressive system-of-equations regression model with multiple endogenous variables. In Part 1 of this series, we mentioned that the ARDL framework is a one-to-one reparameterization of the conditional error correction model (ECM) representation of the underlying vector auto-regression (VAR). While the ARDL approach to cointegration is typically considered synonymous with the Pesaran, Shin, and Smith (2001) Bounds test for cointegration, in this post we emphasize that correct inference is in fact rooted in cointegration theory.
Whilst the discussion is by its nature quite technical, it is important that practitioners of the Bounds test have a grasp of the background behind its inferences. In this post we outline the correct theoretical underpinning of the inference behind the Bounds test for cointegration in an ARDL model. For Part 1, please go here, and for Part 3, please visit here.
This is the second part of our AutoRegressive Distributed Lag (ARDL) post.