[Note: For an important update of this post, relating to EViews 9, see my 2015 post, here.]
Well, I finally got it done! Some of these posts take more time to prepare than you might think.
The first part of this discussion was covered in a (sort of!) recent post, in which I gave a brief description of Autoregressive Distributed Lag (ARDL) models, together with some historical perspective. Now it's time for us to get down to business and see how these models have come to play a very important role recently in the modelling of non-stationary time-series data.
In particular, we'll see how they're used to implement the so-called "Bounds Tests", to see if long-run relationships are present when we have a group of time-series, some of which may be stationary, while others are not. A detailed worked example, using EViews, is included.
First, recall that the basic form of an ARDL regression model is:
where εt is a random "disturbance" term, which we'll assume is "well-behaved" in the usual sense. In particular, it will be serially independent.
We're going to modify this model somewhat for our purposes here. Specifically, we'll work with a mixture of differences and levels of the data. The reasons for this will become apparent as we go along.
Let's suppose that we have a set of time-series variables, and we want to model the relationship between them, taking into account any unit roots and/or cointegration associated with the data. First, note that there are three straightforward situations that we're going to put to one side, because they can be dealt with in standard ways:
We know that all of the series are I(0), and hence stationary. In this case, we can simply model the data in their levels, using OLS estimation, for example.
We know that all of the series are integrated of the same order (e.g., I(1)), but they are not cointegrated. In this case, we can just (appropriately) difference each series, and estimate a standard regression model using OLS.
We know that all of the series are integrated of the same order, and they are cointegrated. In this case, we can estimate two types of models: (i) An OLS regression model using the levels of the data. This will provide the long-run equilibrating relationship between the variables. (ii) An error-correction model (ECM), estimated by OLS. This model will represent the short-run dynamics of the relationship between the variables.
Now, let's return to the more complicated situation mentioned above. Some of the variables in question may bestationary, some may be I(1) or even fractionally integrated, and there is also the possibility of cointegration among some of the I(1) variables. In other words, things just aren't as "clear cut" as in the three situations noted above.
What do we do in such cases if we want to model the data appropriately and extract both long-run and short-run relationships? This is where the ARDL model enters the picture.
The ARDL / Bounds Testing methodology of Pesaran and Shin (1999) and Pesaran et al. (2001) has a number of features that many researchers feel give it some advantages over conventional cointegration testing. For instance:
It can be used with a mixture of I(0) and I(1) data.
It involves just a single-equation set-up, making it simple to implement and interpret.
Different variables can be assigned different lag-lengths as they enter the model.
We need a road map to help us. Here are the basic steps that we're going to follow (with details to be added below):
Make sure than none of the variables are I(2), as such data will invalidate the methodology.
Formulate an "unrestricted" error-correction model (ECM). This will be a particular type of ARDL model.
Determine the appropriate lag structure for the model in step 2.
Make sure that the errors of this model are serially independent.
Make sure that the model is "dynamically stable".
Perform a "Bounds Test" to see if there is evidence of a long-run relationship between the variables.
If the outcome at step 6 is positive, estimate a long-run "levels model", as well as a separate "restricted" ECM.
Use the results of the models estimated in step 7 to measure short-run dynamic effects, and the long-run equilibrating relationship between the variables.
We can see from the form of the generic ARDL model given in equation (1) above, that such models are characterised by having lags of the dependent variable, as well as lags (and perhaps the current value) of other variables, as the regressors. Let's suppose that there are three variables that we're interested in modelling: a dependent variable, y, and two other explanatory variables, x1 and x2. More generally, there will be (k + 1) variables - a dependent variable, and k other variables.
Before we start, let's recall what a conventional ECM for cointegrated data looks like. It would be of the form:
Notice that this is almost like a traditional ECM. The difference is that we've now replaced the error-correction term, zt-1 with the terms yt-1, x1t-1, and x2t-1. From (3), we can see that the lagged residuals series would be zt-1 = (yt-1 - a0 - a1x1t-1 - a2x2t-1), where the a's are the OLS estimates of the α's. So, what we're doing in equation (4) is including the same lagged levels as we do in a regular ECM, but we're not restricting their coefficients.
This is why we might call equation (4) an "unrestricted ECM", or an "unconstrained ECM". Pesaran et al. (2001) call this a "conditional ECM".
Step 3:
The ranges of summation in the various terms in (4) are from 1 to p, 0 to q1, and 0 to q2 respectively.We need to select the appropriate values for the maximum lags, p, q1, and q2. Also, note that the "zero lags" on Δx1 and Δx2 may not necessarily be needed. Usually, these maximum lags are determined by using one or more of the "information criteria" - AIC, SC (BIC), HQ, etc. These criteria are based on a high log-likelihood value, with a "penalty" for including more lags to achieve this. The form of the penalty varies from one criterion to another. Each criterion starts with -2log(L), and then penalizes, so the smaller the value of an information criterion the better the result.
I generally use the Schwarz (Bayes) criterion (SC), as it's a consistent model-selector. Some care has to be taken not to "over-select" the maximum lags, and I usually also pay some attention to the (apparent) significance of the coefficients in the model. Step 4:
A key assumption in the ARDL / Bounds Testing methodology of Pesaran et al. (2001) is that the errors of equation (4) must be serially independent. As those authors note (p.308), this requirement may also be influential in our final choice of the maximum lags for the variables in the model.
Once an apparently suitable version of (4) has been estimated, we should use the LM test to test the null hypothesis that the errors are serially independent, against the alternative hypothesis that the errors are (either) AR(m) or MA(m), for m = 1, 2, 3,...., etc.
Step 5:
We have a model with an autoregressive structure, so we have to be sure that the model is "dynamically stable". For full details of what this means, see my recent post, When is an Autoregressive Model Dynamically Stable? What we need to do is to check that all of the inverse roots of the characteristic equation associated with our model lie strictly inside the unit circle. That recent post of mine showed how to trick EViews into giving us the information we want in order to check that this condition is satisfied. I won't repeat that here. Step 6:
Now we're ready to perform the "Bounds Testing"!
All that we're going to do is preform an "F-test" of the hypothesis, H0: θ0 = θ1 = θ2 = 0 ; against the alternative that H0 is not true. Simple enough - but why are we doing this?
As in conventional cointegration testing, we're testing for the absence of a long-run equilibrium relationship between the variables. This absence coincides with zero coefficients for yt-1, x1t-1 and x2t-1 in equation (4). A rejection of H0 implies that we have a long-run relationship.
There is a practical difficulty that has to be addressed when we conduct the F-test. The distribution of the test statistic is totally non-standard (and also depends on a "nuisance parameter", the cointegrating rank of the system) even in the asymptotic case where we have an infinitely large sample size. (This is somewhat akin to the situation with the Wald test when we test for Granger non-causality in the presence of non-stationary data. In that case, the problem is resolved by using the Toda-Yamamoto (1995) procedure, to ensure that the Wald test statistic is asymptotically chi-square, as discussed here.)
Exact critical values for the F-test aren't available for an arbitrary mix of I(0) and I(1) variables. However, Pesaran et al. (2001) supply bounds on the critical values for the asymptotic distribution of the F-statistic. For various situations (e.g., different numbers of variables, (k + 1)), they give lower and upper bounds on the critical values. In each case, the lower bound is based on the assumption that all of the variables are I(0), and the upper bound is based on the assumption that all of the variables are I(1). In fact, the truth may be somewhere in between these two polar extremes.
If the computed F-statistic falls below the lower bound we would conclude that the variables are I(0), so no cointegration is possible, by definition. If the F-statistic exceeds the upper bound, we conclude that we have cointegration. Finally, if the F-statistic falls between the bounds, the test is inconclusive.
Does this remind you of the old Durbin-Watson test for serial independence? It should!
As a cross-check, we should also perform a "Bounds t-test" of H0 : θ0 = 0, against H1 : θ0 < 0. If the t-statistic for yt-1 in equation (4) is greater than the "I(1) bound" tabulated by Pesaran et al. (2001; pp.303-304), this would support the conclusion that there is a long-run relationship between the variables. If the t-statistic is less than the "I(0) bound", we'd conclude that the data are all stationary.
Step 7:
Assuming that the bounds test leads to the conclusion of cointegration, we can meaningfully estimate the long-run equilibrium relationship between the variables:
where zt-1 = (yt-1 -a0 - a1x1t-1 - a2x2t-1), and the a's are the OLS estimates of the α's in (5).
Step 8:
We can "extract" long-run effects from the unrestricted ECM. Looking back at equation (4), and noting that at a long-run equilibrium, Δyt = 0, Δx1t = Δx2t = 0, we see that the long-run coefficients for x1 and x2 are -(θ1/ θ0) and -(θ2/ θ0) respectively.
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