ARDL Modelling in EViews 9

My previous posts relating to ARDL models (here and here) have drawn a lot of hits. So, it's great to see that EViews 9 (now in Beta release - see the details here) incorporates an ARDL modelling option, together with the associated "bounds testing".

This is a great feature, and I just know that it's going to be a "winner" for EViews.

It certainly deserves a post, so here goes!

First, it's important to note that although there was previously an EViews "add-in" for ARDL models (see here and here), this was quite limited in its capabilities. What's now available is a full-blown ARDL estimation option, together with bounds testing and an analysis of the long-run relationship between the variables being modelled.

Here, I'll take you through another example of ARDL modelling - this one involves the relationship between the retail price of gasoline, and the price of crude oil. More specifically, the crude oil price is for Canadian Par at Edmonton; and the gasoline price is that for the Canadian city of Vancouver. Although crude oil prices are recorded daily, the gasoline prices are available only weekly. So, the price data that we'll use are weekly (end-of-week), for the 4 January 2000 to 16 July 2013, inclusive.

The oil prices are measured in Candian dollars per cubic meter. The gasoline prices are in Canadian cents per litre, and they exclude taxes. Here's a plot of the raw data:

The data are available on the data page for this blog. The EViews workfile is on the code page.

I'm going to work with the logarithms of the data: LOG_CRUDE and LOG_GAS. There's still a clear structural break in the data for both of these series. Specifically there's a structural break that occurs over the weeks ended 8 July 2008 to 30 December 2008 inclusive. I've constructed a dummy variable, BREAK, that takes the value one for these observations, and zero everywhere else.

The break doesn't occur at just a single point in time. Instead, there's a change in the level and trend of the data that evolves over several periods. We call this an "innovational outlier", and in testing the two time series for unit roots, I've taken this into account.

In a recent post I discussed the new "Breakpoint Unit Root Test" options that are available in EViews 9. They're perfectly suited for our current situation. Here's how I've implemented the appropriate test of a unit root in the case of the LOG_CRUDE series:

The result is:

We wouldn't reject the hypothesis of a unit root at the 5% significance level, and the result is marginal at the 10% level. The corresponding result for the LOG_GAS series is:

In this case we'd reject the null hypothesis of a unit root at the 5% significance level, but not at the 1% level. Overall, the results are somewhat inconclusive, and this is precisely the situation that ARDL modelling and bounds testing is designed for. Applying the unit root tests to the first-differences of each series leads to a very clear rejection of the hypothesis that the data are I(2), which is important for the legitimate application of the bounds test below.

Now, let's go ahead with the specification and estimation of a basic ARDL model that explains the retail price of gasoline in terms of past values of that price, as well as the current and past values of the price of crude oil. We can do this in the same way that we'd estimate any equation in EViews, but we select the "Estimation Method" to be "ARDL" (see below):

Notice that I've set the maximum number of lags for both the dependent variable and the principal regressor to be 8. This means that 72 different model specifications will be considered, allowing for the fact that the current value of LOG_CRUDE can be considered as a regressor. Also, notice that I've included the BREAK dummy variable, as well as an intercept and linear trend as (fixed) regressors. (That is, they won't be lagged.)

Using the OPTIONS tab, let's select the Schwarz criterion (SC) as the basis for determining the lag orders for the regressors:

The model which minimizes SC will be chosen. This results in a rather parsimonious model specification, as you can see:

I mentioned in an earlier post on Information Criteria that SC tends to select a simpler model specification than some other information criteria. So, instead of SC, I'm going to use Akaike's Information Criterion (AIC) for selecting the lag structure in the ARDL model. There's a risk of "over-fitting" the model, but I definitely don't want to under-fit it. Here's what we get:

It's important that the errors of this model are serially independent - if not, the parameter estimates won't be consistent (because of the lagged values of the dependent variable that appear as regressors in the model. To that end, we can use the VIEW tab to choose, RESIDUAL DIAGNOSTICS; CORRELOGRAM - Q-STATISTICS, and this gives us the following results:

The p-values are only approximate, but they strongly suggest that there is no evidence of autocorrelation in the model's residuals. This is good news!

Now, recall that, in total, 72 ARDL model specifications were considered. Although an ARDL(4,2) was finally selected, we can also see how well some other specifications performed in terms of minimizing AIC. Selecting the VIEW tab in the regression output, and then choosing MODEL SELECTION SUMMARY; CRITERIA GRAPH from the drop-down, we see the "Top Twenty" results:

(You can get the full summary of the AIC, SC, Hannan-Quinn, and adjusted R² statistics for all 72 model specifications if you select CRITERIA TABLE, rather than CRITERIA GRAPH.)

One of the main purposes of estimating an ARDL model is to use it as the basis for applying the "Bounds Test". This test is discussed in detail in one of my earlier posts. The null hypothesis is that there is no long-run relationship between the variables - in this case, LOG_CRUDE and LOG_GAS.

In the estimation results, if we select the VIEW tab, and then from the drop-down menu choose COEFFICIENT DIAGNOSTICS; BOUNDS TEST, this is what we'll get:

We see that the F-statistic for the Bounds Test is 32.38, and this clearly exceeds even the 1% critical value for the upper bound. Accordingly, we strongly reject the hypothesis of "No Long-Run Relationship".

The output at this point also shows the modified ARDL model that was used to obtain this result. The form that this model takes will be familiar if you've read my earlier post on bounds testing.

In the estimation results for our chosen ARDL model, if we select the VIEW tab, and then from the drop-down menu choose COEFFICIENT DIAGNOSTICS; COINTEGRATION AND LONG RUN FORM, this is what we'll see:

The error-correction coefficient is negative (-0.2028), as required, and is very significant. Importantly, the long-run coefficients from the cointegrating equation are reported, with their standard errors, t-statistics, and p-values:

So, what do we conclude from all of this? 

First, not surprisingly, there's a long-run equilibrium relationship between the price of crude oil, and the retail price of gasoline. 

Second, there is a relatively quick adjustment in the price of gasoline when the price of crude oil changes. (Recall that the data are observed  weekly.) 

Third, a 10% change in the price of crude oil will result in a long-run change of 7% in the price of retail gasoline. 

Whether or not these responses are symmetric with respect to price increases and price decreases is the subject of some on-going work of mine.

© 2015, David E. Giles