由于帖子长度限制，我分开两篇发。ARDL Models - Part I

I've been promising, for far too long, to provide a post on ARDL models and bounds testing. Well, I've finally got around to it!

"ARDL" stands for "Autoregressive-Distributed Lag". Regression models of this type have been in use for decades, but in more recent times they have been shown to provide a very valuable vehicle for testing for the presence of long-run relationships between economic time-series.

I'm going to break my discussion of ARDL models into two parts. Here, I'm going to describe, very briefly, what we mean by an ARDL model. This will then provide the background for a second post that will discuss and illustrate how such models can be used to test for cointegration, and estimate long-run and short-run dynamics, even when the variables in question may include a mixture of stationary and non-stationary time-series.

In its basic form, an ARDL regression model looks like this:

  yt = β0 + β1yt-1 + .......+ βkyt-p + α0xt + α1xt-1 + α2xt-2 + ......... + αqxt-q + εt

where εt is a random "disturbance" term.

The model is "autoregressive", in the sense that yt is "explained (in part) by lagged values of itself. It also has a "distributed lag" component, in the form of successive lags of the "x" explanatory variable. Sometimes, the current value of xt itself is excluded from the distributed lag part of the model's structure.

Let's describe the model above as being one that is ARDL(p,q), for obvious reasons.

Given the presence of lagged values of the dependent variable as regressors, OLS estimation of an ARDL model will yieldbiased coefficient estimates. If the disturbance term, εt, is autocorrelated, the OLS will also be an inconsistentestimator, and in this case Instrumental Variables estimation was generally used in applications of this model.

In the 1960's and 1970's we used distributed lag (DL(q), or ARDL(0,q)) models a lot. To avoid the adverse effects of the multicollinearity associated with including many lags of "x" as regressors, it was common to reduce the number of parameters by imposing restrictions on the pattern (or "distribution") of values that the α coefficients could take.

Perhaps the best known set of restrictions was that associated with the Koyck (1954) for the estimation of DL (∞) model. These restrictions imposed a polynomial rate of decay on the α coefficients. This enabled the model to be manipulated into a new one that was autoregressive, but with an error term that followed a moving average process. Today, we'd call this an ARMAX model. Again, Instrumental Variables estimation was often used to obtain consistent estimates of the model's parameters.

Frances and van Oest (2004) provide an interesting perspective of the Koyck model, and the associated "Koyck transformation", 50 years after its introduction into the literature.

Shirley Almon popularized another set of restrictions (Almon, 1965) for the coefficients in a DL(q) model. Her approach was based on Weierstrass's Approximation Theorem, which tells us that any continuous function can be approximated, arbitrarily closely, by a polynomial of some order. The only question is "what is the order", and this had to be chosen by the practitioner.

The Almon estimator could actually be re-written as a restricted least squares estimator. For example, see Schmidt and Waud (1973), and Giles (1975). Surprisingly, though, this isn't how this estimator was usually presented to students and practitioners.

Almon's approach allowed restrictions to be placed on the shape of the "decay path" of the gamma coefficients, as well as on the values and slopes of this decay path at the end-points, t=0 and t=q. Almon's estimator is still included in a number of econometrics packages, including EViews. A Bayesian analysis of the Almon estimator, with an application to New Zealand imports data, can be found in Giles (1977), and Shiller (1973) provides a Bayesian analysis of a different type of distributed lag model.

Dhrymes (1971) provides a thorough and very general discussion of DL models.

So, now we know what an ARDL model is, and where the term "Autoregressive-Distributed Lag" comes from. In the next post on this topic I'll discuss the modern application of such models in the context of non-stationary time-series data, with the emphasis on an illustrative application with real data.

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


References

Almon, S., 1965. The distributed lag between capital appropriations and net expenditures. Econometrica, 33, 178-196.

Dhrymes, P. J., 1971. Distributed Lags: Problems of Estimation and Formulation.Holden-Day, San Francisco.

Frances, P. H. & R. van Oest, 2004. On the econometrics of the Koyck model. Report 2004-07, Econometric Institute, Erasmus University, Rotterdam.

Giles, D. E. A., 1975. A polynomal approximation for distributed lags. New Zealand Statistician, 10, 22-26.

Giles, D. E. A., 1977. Current payments for New Zealand’s imports: A Bayesian analysis. Applied Economics, 9, 185-201.

Johnston, J., 1984. Econometric Methods, 3rd ed.. McGraw-Hill, New York.


Koyck, L. M., 1954. Distributed Lags and Investment Analysis. North-Holland, Amsterdam.

Schmidt, P. & R. N. Waud, 1973. The Almon lag technique and the monetary versus fiscal policy debate. Journal of the American Statistical Association, 68, 1-19.

Shiller, R. J., 1973. A distributed lag estimator derived from smoothness priors. Econometrica, 41, 775-788.