An Example:
Now we're ready to look at a very simple empirical example. I'm going to use the data for U.S. and European natural gas prices that I made available as a second example in my post, Testing for Granger Causality. I didn't go through the details of testing for Granger causality with that set of data, but I mentioned near the end of the post, and the EViews file (which included a "read_me" object with comments about the results) is there on the code page for this blog (dated 29 April, 2011).

If you look back at that earlier file, you'll find that I used the Toda-Yamamoto (1995) testing procedure to determine that there is Granger causality running from the U.S. series to the European series, but not vice versa.

A new EViews file that uses the same data for our ARDL modelling is available on the code page, under the date for the current post. The data for the two time-series we'll be using are also available on the data page for this blog. The data are monthly, from 1995(01) to 2011(03). In terms of the notation that was introduced earlier, we have (k + 1) = 2 variables, so k = 1 when it comes to the bounds testing.

Here's a plot of the data we'll be using (remember that you can enlarge most of these inserts by clicking on them):

To complete Step 1, we need to check that neither of our time-series are I(2). Applying the ADF test to the levels of EUR and US, the p-values are 0.53 and 0.10 respectively. Applying the test to the first-differences of the series, the p-values are both 0.00. (The lag-lengths for the ADF regressions were chosen using the Schwarz criterion, SC.) Clearly, neither series is I(2).

Applying the KPSS test we  reject the null of stationarity, even at the 1% significance level, for both EUR and US, but cannot reject the null of I(1) against I(2). The p-value of 10% for the ADF test of I(1) vs. I(0) for the EUR series may leave us wondering if that series is stationary, or not. You'll know that apparent "conflicts" between the outcomes of tests such as these are very common in practice.

This is a great illustration of how the ARDL / Bounds Testing methodology can help us. In order for standard cointegration testing (such as that of Engle and Granger, or Johansen) to make any sense, we must be really sure that all of the series are integrated of the same order. In this instance, you might not be feeling totally sure that this is the case.

Step 2 is straightforward. Given that the Granger causality testing associated with my earlier post suggested that there is causality from US to EUR (but not vice versa), ΔEUR is going to be the dependent variable in my unrestricted ECM:

          ΔEURt = β0 + Σ βiΔEURt-i + ΣγjΔUSt-j + θ0EURt-1 + θ1USt-1 + et   ;     (5)

That's Step 2 out of the way!

To implement the information criteria for selecting the lag-lengths in an time-efficient way, I "tricked" EViews into providing lots of them at once by doing the following. I estimated a 1-equation VAR model for ΔEURt and I supplied the intercept, EURt-1, USt-1, and a fixed number of lags of ΔUSt as exogenous regressors. For example, when the fixed number of lags on ΔUSt was zero, here's how I specified the VAR:






After estimating this model, I then chose VIEW, LAG STRUCTURE, LAG LENGTH CRITERIA:



I then repeated this by adding  ΔUSt-1 to the list of exogenous variables, and got the following results:




I proceeded in this manner with additional lags of ΔUSt in the "exogenous" list. I also considered cases such as:




which resulted in the following information criteria values:



Looking at the SC values in these three tables of results, we see that a maximum lag of 4 is suggested for ΔEURt. (The AIC values suggest that 8 lags of ΔEURt may be appropriate, but some experimentation with this was not fruitful.)

There is virtually no difference between the SC values for the case where the model includes just USt as a regressor (0.8714), and the case where just ΔUSt-1 is included (0.8718). To get some dynamics into the model, I'm going to go with the latter case.

With Step 3 completed, and with this lag specification in mind, let's now look at the estimated unrestricted ECM:




Step 4 involves checking that the errors of this model are serially independent. Selecting VIEW, RESIDUAL DIAGNOSTICS, SERIAL CORRELATION LM TEST, I get the following results:

m           LM          p-value

1           0.079        0.779
2           2.878        0.237
3           5.380        0.146
4         11.753        0.019

O.K., we have a problem with serial correlation! To deal with it, I experimented with one or two additional lags of the dependent variable as regressors, and ended up with the following specification for the unrestricted ECM:



The serial independence results now look much more satisfactory:

m           LM          p-value

1           0.013        0.911
2           3.337        0.189
3           5.183        0.159
4           7.989        0.092
5           8.473        0.132
6         11.023        0.088
7         12.270        0.092
8         12.334        0.137

Next, Step 5 involves checking the dynamic stability of this ARDL model. Here are the inverse roots of the associated characteristic equation:

All seems to be well - these roots are all inside the unit circle.

Before proceeding to the Bounds Testing, let's take a look at the "fit" of our unrestricted ECM. The "Actual / Fitted / Residuals" plot looks like this:


When we "unscramble" these results, and look at the fit of the model in terms of explaining the level of EUR itself, rather than ΔEUR, things look pretty good:

We're now ready for Step 6 - the Bounds Test itself. We want to test if the coefficients of both EUR(-1) and US(-1) are zero in our estimated model (repeated below):