User’s Guide : Advanced Single Equation Analysis : Autoregressive Distributed Lag (ARDL) Models : Background
Representation 1: Intertemporal Dynamics Regression
Representation 2: Post-Regression Derivation of Long-Run Dynamics
Representation 3: Conditional Error Correction Form and the Bounds Test
ARDL models are linear time series models in which both the dependent and independent variables are related not only contemporaneously, but across historical (lagged) values as well. In particular, if is the dependent variable and are explanatory variables, a general ARDL model is given by:
where are the usual innovations, is a constant term, and , and are respectively the coefficients associated with a linear trend, lags of , and lags of the regressors for . Alternatively, let denote the usual lag operator and define and as the lag polynomials:
Equation (27.1) above can then be rewritten as:
Following this general formulation, three alternative representations can be made. While all three can be used for parameter estimation, the first is typically used for intertemporal dynamic estimation, the second for post-estimation derivation of the long-run (equilibrium) relationship, while the third is a reduction of (27.1) to the conditional error correction (CEC) representation in the PSS(2001) bounds test. All three alternative representations make use of the Beveridge-Nelson decomposition.
Representation 1: Intertemporal Dynamics Regression
The typical starting point for most ARDL applications is the estimation of intertemporal dynamics. In this form, one is interested in estimating the relationship between on both its own lags as well as the contemporaneous and lagged values of the regressors . This in fact the basis of the ARDL model studied in PS(1998). In particular, we cast (27.1) into the following representation:
where we have used the Beveridge-Nelson result to decompose into . Since this equation does not solve for , it is typically interpreted as a regression for the intertemporal dynamics of the model.
Representation 2: Post-Regression Derivation of Long-Run Dynamics
The second representation is in essence an attempt to derive the long-run relationship between and the regressors. As such, the representation solves for in terms of .
where we have again used the Beveridge-Nelson decomposition to express . Furthermore, from (27.1) we are typically interested in the long-run (trend) parameters captured by and  , for . In fact, given the one-to-one correspondence between the parameter estimates obtained in (27.1) and (27.5), it is possible to derive estimates of the long-run parameters post-estimation. In particular, if denote the relevant subset of estimated coefficients from the regression model corresponding to the model in (27.4), in particular,
then, a post-regression estimate of the long-run parameters is derived as follows:
Representation 3: Conditional Error Correction Form and the Bounds Test
The final representation is arguably the most interesting and one that typically receives the most attention in applied work. The objective here is to test for cointegration by reducing a typical vector autoregression framework to its corresponding conditional error correction (CEC) form. This CEC model is in fact an ARDL model with a one-to-one correspondence with the model in (27.1).
In particular, Equation (27.1) may be re-written as:
where we have used the convention
relying once again on the Beveridge-Nelson decomposition.
Equation (27.7) above is the CEC form derived from the ARDL model in (27.1). Rewriting this equation as:
it is readily verified that the error correction term, typically denoted as , is also the cointegrating relationship when and are cointegrated. PSS(2001) demonstrate that (27.7) is in fact (abstracting from differing lag values) the CEC of the VAR(p) model:
where is the (k +1)-vector and and are respectively the (k +1)-vectors of intercept and trend coefficients, and is the (k + 1) square matrix lag polynomial.
Traditionally, the cointegration tests of Engle-Granger (1987), Phillips and Ouliaris (1990), Park (1990), or Johansen (1991; 1995), typically require all variables in the VAR to be I(1). This clearly requires a battery pre-testing for the presence of a unit root in each of the variables under consideration, and is subject to misclassification. In contrast, PSS(2001) propose a test for cointegration that is robust to whether variables of interest are I(0), I(1), or mutually cointegrated. In this regard, PSS(2001) discuss a bounds test for cointegration as a test on parameter significance in the cointegrating relationship of the CEC model (27.7). In other words, the test is a standard F. or Wald test for the following null and alternative hypotheses:
Once the test statistic is computed, it is compared to two asymptotic critical values corresponding to polar cases of all variables being purely I(0) or purely I(1). When the test statistic is below the lower critical value, one fails to reject the null and concludes that cointegration is not possible. In contrast, when the test statistic is above the upper critical value, one rejects the null and concludes that cointegration is indeed possible. In either of these two cases, knowledge of the cointegrating rank is not necessary.
Alternatively, should the test statistic fall between the lower and upper critical values, testing is inconclusive, and knowledge of the cointegrating rank is required to proceed further.
Here it is also important to highlight that PSS(2001) offer five alternative interpretations of the CEC model (27.7), distinguished by whether deterministic terms integrate into the error correction term. When deterministic terms contribute to the error correction term, they are implicitly projected onto the span of the cointegrating vector. This implies that and in (27.7) must be restricted. Below are summaries of the regression (REG) models, for each of the five interpretations along with the appropriate cointegrating relationship and the bounds test null-hypothesis .
Case 1:
Case 2:
Case 3:
Case 4:
Case 5: