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:

y

"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:

y

_{t}= β_{0}+ β_{1}y_{t-1}+ .......+ β_{k}y_{t-p}+ α_{0}x_{t}+ α_{1}x_{t-1}+ α_{2}x_{t-2}+ ......... + α_{q}x_{t-q}+ ε_{t}
where ε

The model is "autoregressive", in the sense that y

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 yield

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

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

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

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.

_{t}is a random "disturbance" term.The model is "autoregressive", in the sense that y

_{t}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 x_{t}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 yield

*biased*coefficient estimates. If the disturbance term, ε_{t}, is autocorrelated, the OLS will also be an*inconsistent*estimator, 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.

**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.

**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.

© 2013, David E. Giles

If the disturbance term is not autocorrelated, why would OLS produce biased estimates?

ReplyDeleteEconometrics 101 - e.g., Y(t) = a + bY(t-1) + e(t), where e(t) is i.i.d. and homoskedastic. Clearly, E[e(t).Y(t)] is not zero; and so E[e(t).Y(t+s)] is not zero, for all non-negative s, and for all t. Hence OLS is biased in finite samples, but the bias vanishes asymptotically (as long as the errors are indeed serially independent). In the case of no drift (a = 0), the bias of the OLS estimator of 'b' is -sb/n, to O(n^-2), where 'n' is the sample size. I believe this was first established by J. S. White, in "Biometrika", 1961.

DeleteLooking forward to the second part of this and modern uses of ARDL models. I was under the impression that they were relatively old-school models that were put into the dustbin once ARIMA and ARIMAX models became easy to fit.

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

ReplyDeleteSorry to ask the obvious, but why aren't p and q the same? What would be an example of using a different number of lags for the y term and the x term.

And, any model that includes y (even where p=1) And an x explanatory variable, but also any number of other explanatory variables (z, w, etc.) also with any number of lags would be considered an ARDL?

And finally, the term autoregressive seems descriptive, but the term distributed lag to describe the other regressors which may only have a lag of 0 (?) isn't intuitively descriptive to me? Not central question to be sure, but it would be helpful to understand the term.

Thanks!

Dan

Dan - it's the same as in a VAR model - we don't necessarily want the same lag length on all variables. We need to think about the economics of the problem too.

DeleteYes, we can have additional explanatory variables with their own maximum lag lengths.

Usually, we'd have several lags of the x (and other explanatory) variables. IN this case the "shape" of the distribution of the weights (coefficients) as we go back through the lags is of interest.

Is it possible to study the log-run relationship between three macro-economic variables using ARDL model? Should Engle-Granger Johansen co-integration technique be preferred over ARDL model when we are dealing with three variables?

ReplyDeleteYes, you can certainly use the ARDL methodology with three or more varaibles, possibly integrated of different orders. The example that's coming in Part II of this post will do just that.

DeleteSir

ReplyDeleteEagerly waiting for Part II

Abhijit

Hi, so in essence, this method can be used instead of a VECM approach, where the variables show cointegration, but aren't all I(1)?

ReplyDeleteLooking forward to part 2!

Yes. Part 2 really is coming!

DeleteDear Sir,

ReplyDeletewhat is the minimum required observations for ARDL estimation?

There's no simple answer - it depends on the frequency of your data (monthly, quarterly, annual) and on the number of lags you need to properly specify the model.

DeleteDG

Thank you. This has given me a better understanding to ARDL but is there a main difference between this and ARIMA .

DeleteCan't wait for the Part 2

ARIMA is for a single time-series. For Part II, see http://davegiles.blogspot.ca/2013/06/ardl-models-part-ii-bounds-tests.html

DeleteSir,

DeleteFor ARDL if the frequency of data is monthly what would be the minimum number of observation required?

I'd be wanting at least 10 years of data - n>120. See below - the tests are only asymptotically valid.

DeleteCan we apply ardl approach when number of observations are 10?

ReplyDeleteThe tests have only asymptotic (large n) relevance, so "no". More generally, an time-series models using only n=10 is unlikely to be of much use.

Deletesir u are very affable and full of knowledge.i have read many of ur posts. sir plz tell me that whether we have to find out Durdin h stat in ARDL to cheak out serial Auto correlation or not?if not then DW-stat value detect serial auto correlation correctly and LM test?plz comment sir.

ReplyDeleteThe DW test is inappropriate, given the lagged dependent variables. The LM test can be used to test for various orders of AR() and MA() processes in the errors. Keep in mind that this is only an asymptotically valid test, as is the h-test for the AR(1) case.

DeleteCan Y(t) be I(0) ? or do have to run a unit root to be sure that Y(t) is not I(0)for ARDL?

ReplyDeleteThanks

Marc

See an earlier query - it will usually be I(1), but it need not be.

DeleteHi Sir, thanks for your prompt reply.

DeleteI did notice though that some authors are quoting that the dependent variable has to be I(1), (Trade Liberalisation, Financial Development and Economic Growth: Evidence from Pakistan (1980-2009)

Rao Muhammad Atif , Abida Jadoon, Khalid Zaman , Aisha Ismail, Rabia Seemab,

Journal of International Academic Research (2010) Vol.10, No.2.).

I went back and read the referenced Pesaran et al. (2001), article but could not really find that the dependent variable has to be I(1).

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

I am a meteorologist, so I might have missed something in the Pesaran article, but it looks like I have to concur with you too that Yt can be I(0) also.

Regards

Mark

This comment has been removed by the author.

DeleteI can see nothing in the Pesaran paper on the bounds testing that required the Y variable to be I(1). However, if you proceeded with a Y variable that appeared to be I(0) and then the bounds test gave a clear outcome of cointegration, then you have a conflict. You can't have cointegration unless the variables are non-stationary to begin with.

DeleteSo, if you are really, really confident in your unit root testing, and you feel that Y is I(0), in that case it would make little sense to do the bounds testing in the first place!

can i use ARDL model for 21 observations ?

ReplyDeleteThat's up to you, but in my view that's a very short sample, especially if you are testing for cointegration along the way.

DeleteDr. Giles,

ReplyDeleteHave you encountered any sort of ARDL of the form?:

y(t) = a + b*y(t-1) + c(t)*x(t) + e(t)

with c(t) = k + p*c(t-1) + i(t) (c is an unobservable random coefficient)

x(t) is observable co-variate. The correlation between x(t) and i(t) may be non-zero, but no correlation between e(t) and either i(t) or x(t)

thanks!

Nope - can't say that I have. Sorry!

DeleteOnce we decided the lag length of distributed lag model (lag length of explanatory variable), and if the coefficients of a lagged explanatory variable are changing signs, some are positive and some negative. If we add them up, can we say that it is the "net effect" on that explanatory variable.

ReplyDeleteThanks

No, we can't because they're measured at different points in time. That's the point of the dynamics.

DeleteWould it be possible to post on Nonlinear ARDLs. There is a lot of two-step Engle-Granger stuff on asymmetric adjustment in ECMs but I don't see much on asymmetric adjustment in ARDLS or one-step error correction models.

ReplyDeleteGood suggestion - I'll see what I can do. Just need more hours in the day..... :-)

DeleteIn that spirit...this blog is truly great and many of these intricate posts must take a pretty long time to create. Thanks for sharing all your knowledge, I truly appreciate it.

DeleteTom, thanks fr the comment and the sentiment. They do take time, but it's fun, and it's nice to "give back".

Deletethank u dr, i have a question.

Deletehow can we solve the low degree of freedom risk in ARDL for short-time series?

You can't - except by getting more data!

DeleteThank You Sir,,, Can I use ARDL Bounds Test even when my data fractionally integrated such as I(1) AND I(2) or more???

ReplyDeleteRegardsss

I've already made it clear in the post that you can't use it if any of the data are I(2). (BTW, I(1) and I(2) are different from "fractionally integrated".)

DeleteDear Sir, I want to investigate this relation y = f(x,d) where y and x two series , y~ I(2) and x~ I(1) by ADF test, d = dummy veraible, from Jan 2003 to June 2011 =0 , and from July 2011 to Dec 2013 = 1. My observations are 132.

ReplyDeleteI want to examine the long equilibrium relationship between y and x, and the impact of the dummy veriable on y.

I got no cointegration between y and x, using cointegration method an error correction model.

Please how can i analyz this data using dynamic models such as ARDL "remember one series is ~I (2)" or any other model.

Thank You

Ahmed - you could second-difference the Y variable, first-difference the X variable, and use OLS.

DeleteThank You Dr. Giles...

DeleteDear Sir,

ReplyDeleteI am currently conducting a forecast on the demand on petroleum( and its allied products, petrol and diesel) in India. Can this model be used. Eagerly awaiting your inputs.

regards

Aditya

Yes, as long as none of your series are I(2).

DeleteThank you sir.

DeleteDear Sir

ReplyDeleteam working on the determinants of electricity consumption, can i use ardl methodology and how?

See here: http://davegiles.blogspot.ca/2013/06/ardl-models-part-ii-bounds-tests.html

DeleteMake sure that none of your variables are I(2).

Dear sir,

ReplyDeleteam working on income inequality, can i use ARDL as i have only 27 annual observations. Also does ARDL itself takes care of problem of endogeinity. and what about, if there is multicollinearity among explanatory variables, can we still use ARDL. is any eviews code available to run ARDL.

thanku

See: http://davegiles.blogspot.ca/2014/06/some-questions-about-ardl-models.html

ReplyDeleteDear Dr.Dave,

ReplyDeleteI am working on time series data and I found one of my variables though it is I(1) in intercept and trend and intercept, it is I(2) in None. Is it possible to run ARDL?

Many thanks,

You can't use the ARDL bounds if any of the series are I(2).

DeleteThank you so much.

DeleteDear sir, i want to investigate the causal relationship bt/w two time series apart from my independent and dependent variables i have to use some control variables also but have no idea how to use them in granger causality test is there any other method through which i also include my control variables.

ReplyDeleteYou just enter them as additional "exogenous variables" and they won't be included in the Granger causality test.

DeleteSir, this is incredible, what you are doing for the world. Thanks again.

ReplyDeleteJust to find out if you have or whether you can run for us and example with the Nalove Distribution lag model.

I am Ateh Thomson Pepeah, from Cameroon-Africa.

https://www.facebook.com/ateh.pepeah

Thanks!

You've made it clear that any of the series shouldn't be I(2) in order to run ARDL.

ReplyDeleteBut, what'd be your suggestion if I need to test for cointegration and my Y~I(1) and X~I(2)?

Thanks in advance, I really appreciate what you've done with this blog!

In this case they can't be cointegrated. If you have 2 or more I(2) series, and one or more I(1) series, you can have cointegration. In this case it's possible for the I(2) to cointegrate to an I(1) series, and the latter can then cointegrate with the other I(1) data. See http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2145744

DeleteThanks for your quick response.

DeleteActually, the issue is more simple: my variables have different integration order:

GDP per capita: I(1); Y

Population: I(2); X

Period: 1960-2012

Would Haldrup (1998) still be valid?

For the sake of specification, my I(1) variable is GDP per capita and my I(2) variable is Population for Mexico from 1960 to 2012.

DeleteLogical sense dictates both variables could be cointegrated. I read that "hypothesis testing of the I(2) model" and/or running DOLS could be a solution for cointegration an I(1) variable with an I(2) variable.

Is this correct? Thanks in advance, sir.

No. If you have 2 variables, one I(1) and the other I(2), they can't be cointegrated, BY DEFINITION. Also, ARDL bounds testing can't be used with I(2) data, but that's a different point.

DeleteYou need at least 2 I(2) variables for Haldrup's results.

DeleteHi,

ReplyDeleteI would like to study the impact the advertising of a product on its sales (weekly data for 5 years).

The final aim is to forecast what would be the impact on sales of a change in the advertising spending. I was considering including also other variables (such as competitors prices, macroeconomics variables,…)

As I know that advertising has a long lasting effect and that, for example, at time t I would be only able to know prices at t-1 I need to include lags.

Would ARIMAX model be appropriate? ARDL?

Thank you.