he associated F-test is obtained as follows:

With the result:

The value of our F-statistic is 5.827, and we have (k + 1) = 2 variables (EUR and US) in our model. So, when we go to the Bounds Test tables of critical values, we have k = 1.

Table CI (iii) on p.300 of Pesaran et al. (2001) is the relevant table for us to use here. We haven't constrained the intercept of our model, and there is no linear trend term included in the ECM.  The lower and upper bounds for the F-test statistic at the 10%, 5%, and 1% significance levels are [4.04 , 4.78], [4.94 , 5.73], and [6.84 , 7.84] respectively.

As the value of our F-statistic exceeds the upper bound at the 5% significance level, we can conclude that there is evidence of a long-run relationship between the two time-series (at this level of significance or greater).

In addition, the t-statistic on EUR(-1) is -2.926. When we look at Table CII (iii) on p.303 of Pesaran et al. (2001), we find that the I(0) and I(1) bounds for the t-statistic at the 10%, 5%, and 1% significance levels are [-2.57 , -2.91], [-2.86 , -3.22], and [-3.43 , -3.82] respectively. At least at the 10% significance level, this result reinforces our conclusion that there is a long-run relationship between EUR and US.

So, here we are at Step 7 and Step 8.

Recalling our preferred unrestricted ECM:

we see that the long-run multiplier between US and EUR is -(0.047134 / (-0.030804)) = 1.53. In the long run, an increase of 1 unit in US will lead to an increase of 1.53 units in EUR.

If we estimate the levels model,

                   EURt = α0 + α1USt + vt       ,

by OLS, and construct the residuals series, {zt}, we can fit a regular (restricted) ECM:

Notice that the coefficient of the error-correction term, zt-1, is negative and very significant. This is what we'd expect if there is cointegration between EUR and US. The magnitude of this coefficient implies that nearly 3% of any disequilibrium between EUR and US is corrected within one period (one month).

This final ECM is dynamically stable:

As none of the roots lie on the X (real) axis, it's clear that we have three complex conjugate pairs of roots. Accordingly, the short-run dynamics associated with the model are quite complicated. This can be seen if we consider the impulse response function associated with a "shock" of one (sample) standard deviation:

Finally, the within-sample fit (in terms of the levels of EUR) is exceptionally good:

In fact, the simple correlations between EUR and the "fitted" EUR series from the unrestricted and regular ECM's are each 0.994, and the correlation between the two fitted series is 0.9999.

So, there we have it - bounds testing with an ARDL model.

[Note: For an important update of this post, relating to EViews 9, see my 2015 post, here



References

Pesaran, M. H. and Y. Shin, 1999. An autoregressive distributed lag modelling approach to cointegration analysis. Chapter 11 in S. Strom (ed.), Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium. Cambridge University Press, Cambridge. (Discussion Paper version.)


Pesaran, M. H., Shin, Y. and Smith, R. J., 2001. Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16, 289–326.

Pesaran, M. H. and R. P. Smith, 1998. Structural analysis of cointegrating VARs. Journal of Economic Surveys, 12, 471-505.

Toda, H. Y and T. Yamamoto (1995). Statistical inferences in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66, 225-250.

© 2013, David E. Giles