ARDL - Trend Specification

[mickeykozzi]

Hello,

I am running my ARDL and ECM models and I am having some confusion surrounding Trend Specification and which option to choose (pictured).
Looking at guidelines from Eviews and other authors, option 2 and 3 seem to be the most popular.
I have read what each option means, however its difficult to understand why some choose option 2 when their data has a trend.

I have noticed that if you choose option 2, the ARDL model and the long run model (with coefficients) show a C (constant) term, however the ECM does not show a C (constant). With option 3, the ARDL and ECM model show the C (constant), however the long-run model shows no C (constant). Why is this the case?

With these confusions, which option should an author choose and why?

The data I am using is macro data: investment, interest rates, government spending, net exports etc, annual series, 36 years 1980 - 2015.

Thank you

picture 1.jpg (151.12 KiB) Viewed 22732 times

[mickeykozzi]

I worked out the answer.
Message me if you want to know, cheers

[igor]

See Johansen cointegration procedure, 5 cases of deterministic terms specification. Here is pretty the same interpretation.

[mickeykozzi]

I dont understand. However I found the solution.

[EnjoFaes]

I really don’t like eviews for this. Literally all papers report a constant in both the long run as in the ecm. Are we really forced to combine both case 2 restricted constant to get a constant in the long run and case 3 unrestricted constant to get a constant in the ecm?

[EViews Mirza]

In the last 7 days, you seem to have been involved in more than 6 different EViews Forum threads related to ARDL questions. All are related or revolving around the same general idea. Most of the questions you've been asking have been answered by myself and/or Igor, or, you have been guided to the original papers and/or our ARDL blog series to seek the answers yourself.

Furthermore, the majority of your questions sound like homework exercises or are meant to delve into some discussion of what methodology is better, when to use a particular procedure, what can we conclude from this, and so on.

While there is certainly room in this forum for questions of your nature, be aware that you're not typically going to receive answers to such questions because the answers you seek are typically gained by reading many papers, gaining years of experience, and are just frankly next to impossible to answer most of the time. You will rarely receive an answer on this forum that says that one methodology is better than another, etc... because each has its own merits and each is used under the circumstances and assumptions that motivated their construction in the first place.

Finally, to get back to your last comment in this thread:

I really don’t like eviews for this. Literally all papers report a constant in both the long run as in the ecm. Are we really forced to combine both case 2 restricted constant to get a constant in the long run and case 3 unrestricted constant to get a constant in the ecm?

In another post you also wrote this:

Given that the long run relationship is not equal to the eviews levels equation this must be incorrect. However in Eviews there is no possibility to choose a case where AND a constant in the levels (long run) equation is given AND a constant in the EC-model, while most papers like for example the good author Narayan (2005), best in the field, does display both..

I'm sorry to be this blunt, but I have no idea what you're talking about! You keep mentioning some papers? You keep mentioning other resources that do something EViews doesn't. And that may very well be the case, but when it comes to the original ARDL paper by Pesaran, Shin, and Smith (2001), there are only 5 cases and not ONE of them involve a constant both inside and outside the error correction term.

pss.png (83.51 KiB) Viewed 22605 times

You also seem to mention Narayan a lot. Well, either we're reading different papers or our level of comprehension is very different, but Narayan (2005) does NOT have a model where the constant is both inside and outside the error correction term. Here's Narayan's (2005) Appendix with finite sample critical values for the ARDL bounds test:

narayan.png (839 KiB) Viewed 22605 times

I guess what I'm trying to say is that you may not like what EViews does, but what EViews does is correct, and entirely in accordance with the important branches of literature.

[EnjoFaes]

In the last 7 days, you seem to have been involved in more than 6 different EViews Forum threads related to ARDL questions. All are related or revolving around the same general idea. Most of the questions you've been asking have been answered by myself and/or Igor, or, you have been guided to the original papers and/or our ARDL blog series to seek the answers yourself.

Furthermore, the majority of your questions sound like homework exercises or are meant to delve into some discussion of what methodology is better, when to use a particular procedure, what can we conclude from this, and so on.

While there is certainly room in this forum for questions of your nature, be aware that you're not typically going to receive answers to such questions because the answers you seek are typically gained by reading many papers, gaining years of experience, and are just frankly next to impossible to answer most of the time. You will rarely receive an answer on this forum that says that one methodology is better than another, etc... because each has its own merits and each is used under the circumstances and assumptions that motivated their construction in the first place.

Finally, to get back to your last comment in this thread:

I really don’t like eviews for this. Literally all papers report a constant in both the long run as in the ecm. Are we really forced to combine both case 2 restricted constant to get a constant in the long run and case 3 unrestricted constant to get a constant in the ecm?

In another post you also wrote this:

Given that the long run relationship is not equal to the eviews levels equation this must be incorrect. However in Eviews there is no possibility to choose a case where AND a constant in the levels (long run) equation is given AND a constant in the EC-model, while most papers like for example the good author Narayan (2005), best in the field, does display both..

I'm sorry to be this blunt, but I have no idea what you're talking about! You keep mentioning some papers? You keep mentioning other resources that do something EViews doesn't. And that may very well be the case, but when it comes to the original ARDL paper by Pesaran, Shin, and Smith (2001), there are only 5 cases and not ONE of them involve a constant both inside and outside the error correction term.

pss.png

You also seem to mention Narayan a lot. Well, either we're reading different papers or our level of comprehension is very different, but Narayan (2005) does NOT have a model where the constant is both inside and outside the error correction term. Here's Narayan's (2005) Appendix with finite sample critical values for the ARDL bounds test:

narayan.png

I guess what I'm trying to say is that you may not like what EViews does, but what EViews does is correct, and entirely in accordance with the important branches of literature.

Hi Mirza,

Thank you for your response. I am just a student, trying to do some research.. I must admit, time series analysis: never done before so multivariate time series is even a step further. So yes, our knowledge is not on the same level concerning the modelling, mathematics, implications. To refer to the homework remark, I totally understand it may feel like this, it is just that I am a bit a perfectionist for my research. And okay a master dissertation is something serious. But if I don't understand, I want to understand. Co-integration is not a thing that most students do as a dissertation and I am on my one, because it is quite advanced econometrics.

What I was trying to say is the following, in attached picture: Narayan (2004) used the following tables. And while Michael also does similar research, we have based our finding (among many bad papers, of course) that the best to use paper (along to his supervisor with 20 years of econometric experience) that "Fiji's Tourism Demand: The ARDL Approach to Cointegration" was the best paper for ARDL to use.

When looking at the original PSS paper of 2001, I did not get wiser of the cases they propose and why. Igor gave a good explanation, however it is still not so clear for me. Every case is different.

Case 2 and case 3 gave exactly the same implication for the significance level of variables on short term (small difference, significance level stays the same) and long term (exactly the same) SO if I use 2 or 3 it does not matter a thing for my results. Most papers don't even discuss the constants.. so yeah, help me understand it better. It cannot be I am the only scholar that feels this way.

[igor]

Hi EnjoFaes,

I think it's a time to end this discussion. Mirza answered you in great details and very politely. Perhaps, my comments were harsher, but I live in a free country, where there is still no political correctness. P.S. It's a joke

Now seriously. I would not like to comment on specific papers on the forum. We are all not perfect. But I have already wrote that a huge amout of the papers on this subject is total trash. In particular, please take note that this approach requires the right-hand variables should be at least weakly exogenous (all these issues you can read in PSS (2001) and Eviews blog). Who cares about this, and so forth?

Welcome to economic science, dear colleague!

[mickeykozzi]

Hi,

The comments to this post are very disappointing, as EnjoFaes and I have a strong point.
As shown by a very strong author, Narayan (2004), there is a reported Constant (C) for the long-run model and the ECM.
In fact, this is common in the literature.

An here lies the issue and hence the original question. We have read Pesaran, Shin, and Smith (2001) and understand the 5 cases, but confusion lies when so many authors report Constant (C) for the long-run and ECM. We wanted to ensure that if we follow Narayan (2004), we were able to include the (C) by using option 2 and 3 in the ARDL Eviews process.

We are young researchers and we are thinking outside the box to find help and answers for our studies.
I am more than happy to follow the rules and regulations of the forum and I am not here to cause problems.
But it can be very frustrating when the original topic is sidelined by other things and people like Igor are trolling young researchers online.

I know that when I see questions from my peers online, I offer help, not put them down and create barriers and confusion.

Something to think about.

[igor]

Dear mickeykozzi,

I don't trolling anyone. I just want to force you to think yourself and read the original papers, and not to blindly follow derivative papers.
You've get all answers here on ARDL Bounds Test. All other issues, including Constant issue you mention again and again are outside the prcedure implemented in Eviews. So you can ask your superviser, I think he/she must help you.

[mickeykozzi]

Thanks Igor

[EViews Mirza]

I think Igor is right. It's time to end this discussion. Before we do however, there is something to be said about peddling "strong" authors and "common in the literature" practices.

The reason @EnjoFaes and @mickeykozzi are confused is because they are reading articles without having the proper econometric foundations to read between the lines when authors make assumptions in their papers without explaining to their readers as to why they are doing so. As I will make clear below, Narayan (2004) is a prime example of this injustice and one I hope our future researchers will take heed of.

Let's begin with a VAR(p) model with a linear trend component:

(1) Y(t) = mu(t) + X(t)

where:

(1a) X(t) = [X1(t), ..., Xk(t)] ~ VAR(p)
(1b) mu(t) = mu0 + mu1*t

After a bunch of algebra, (1) can be written in its VEC form as follows:

(2) D(Y(t)) - mu1 = PI*(Y(t-1) - mu0 - mu1*(t-1)) + GAMMA(1)*(D(Y(t-1) - mu1) + ... + GAMMA(p-1)*(D(Y(t-p+1) - mu1)

Or simplified further still as:

(2*) D(Y(t)) = a0 + a1*t + PI*Y(t-1) + GAMMA(1)*D(Y(t-1) + ... + GAMMA(p-1)*D(Y(t-p+1)

where:

(r1) a0 = (I - GAMMA(1) - ... - GAMMA(p-1) + PI)*mu1 - PI*mu0
(r2) a1 = -PI*mu1

Following Pesaran, Shin, and Smith (2001), henceforth PSS(2001), the VEC form in (2*) can now be transformed into its conditional ECM form as follows:

(3) D(Y(t)) = b0 + b1*t + PI(1)(X1(t-1) + (PI(2)/PI(1))*X2(t-1) + ... + (PI(k)/PI(1))*Xk(t-1)) + PSI(1)D(X(t-1) + ... + PSI(p-1)*D(X(t-p+1)) + PHI(2)D(X2(t)) + ... + PHI(k)*D(Xk(t))

where b0 and b1 are some transformation of a0 and a1 in (2*)

This is in fact the starting point for analyzing the 5 cases presented in PSS(2001), although the same 5 cases can be observed in the work of Johansen. It's important to point out that what follows has also been thoroughly outlined in our blog series on ARDL models.

We now have the following assumptions:

Case 1: (No constant and no trend) => Assumption is a0=0 and a1=0
Case 2: (Restricted constant and no trend) => Assumption is a0=PI*mu0 and a1=0
Case 3: (Unrestricted constant and no trend) = > Assumption is a0\=0 and a1=0
Case 4: (Unrestricted constant and restricted trend) => Assumption is a0\=0 and a1=-PI*mu1
Case 5: (Unrestricted constant and trend) => Assumption is a0\=0 and a1\=0

Notice that whenever we have "restricted" deterministics, what's implied is that those deterministic terms are put inside the cointegrating equation.

Let's next turn to Narayan (2004), Tables 2 and 3 to be specific. Evidently, there is a constant inside and outside the cointegrating equation in the ECM. So what's going on?

Consider equation (2) above and notice that the only way we can actually have a constant both inside and outside the cointegrating equation in the ECM is if the determinstic dynamics mu(t) in the initial VAR model contain both a constant and a linear trend. In other words, mu(t) = mu0 + mu1*t. Hence, after differencing, mu1 is actually the term that contributes to the constant outside the cointegrating equation, and mu0 is the term that contributes to the constant inside the cointegrating equation.

Is this mentioned anywhere in Narayan (2004)? No, it's not!

Next, observe that equation (2) can in fact be written to look very similar to Case 4, as follows:

(4) D(Y(t)) = (I - GAMMA(1) - ... - GAMMA(p-1) + PI)*mu1 - PI*mu0 + PI*(Y(t-1) - mu1*t) + GAMMA(1)*D(Y(t-1) + ... + GAMMA(p-1)*D(Y(t-p+1)

or

(4) D(Y(t)) = a0 + PI*(Y(t-1) - mu1*t) + GAMMA(1)*D(Y(t-1) + ... + GAMMA(p-1)*D(Y(t-p+1)

This is in fact Case 4 with the added restriction that a0 must satisfy restriction (r1) above. However, from the perspective of estimation, we can in fact estimate Case 4, obtain estimates of a0, PI, mu1, and GAMMA(k) for k = 1 to p-1 -- say hat(a0), hat(PI), hat(mu1), and hat(GAMMA(k)) respectively -- and then manually impose (r1) and back out the part of hat(a0) which is the constant outside and inside, namely:

const outside EC term: (I - hat(GAMMA(1)) - ... - hat(GAMMA(p-1)) + hat(PI))*hat(mu1)
const inside EC term: hat(a0) - ((I - hat(GAMMA(1)) - ... - hat(GAMMA(p-1)) + hat(PI))*hat(mu1))

Well that takes care of the constant term inside and outside, but we still have the trend term mu1*t inside the cointegrating equation which is not present in Narayan (2004). This is where the real confusion begins.

Narayan (2004), like many in the industry, either out of ignorance or arrogance, are following "common in the literature" practices without justifying their choices. Furthermore, adding to this dangerous mix laziness or carelessness, they often fail to offer readers the literature necessary to understand the choices and assumptions they make. Hence, there is virtually no effort on Narayan's (2004) part to actually motivate and explain his modelling choices. The reader is therefore left scratching his/her head:

"if the only way of deriving an ECM with both a restricted and unrestricted constant is to start with a VAR with linear trend deterministics, where did mu1*t go?"

The answer lies in an important discussion buried in Lutkepohl's (2006) "New Introduction to Multiple Time Series Analysis". In particular, on page 258 of this work, Lutkepohl states:

"It is also possible, that the trend slope parameter mu1 is orthogonal to the cointegration matrix so that PI*µ1 = 0... and the trend term disappears from the cointegration relations. This situation can also occur if mu1\=0 and the variables actually have linear trends in their means... It represents a situation where a linear trend appears in the variables but not in the cointegration relations."

Mystery solved. What Narayan (2004) is doing is in fact starting with a VAR with linear trend deterministics, and then manually imposing two additional restrictions:

1) PI*mu1 = 0
2) Restriction (r1) above.

In practice, this is simpler still. One simply can estimate the following model:

(5) D(Y(t)) = b0 + PI(1)(X1(t-1) + (PI(2)/PI(1))*X2(t-1) + ... + (PI(k)/PI(1))*Xk(t-1)) + PSI(1)D(X(t-1) + ... + PSI(p-1)*D(X(t-p+1)) + PHI(2)D(X2(t)) + ... + PHI(k)*D(Xk(t))

and impose restriction (r1) above to isolate the constant on the inside and outside of the EC term.

As mentioned earlier, no discussion of this ever takes places in the Narayan (2004) paper, nor in so many others like it.

As a final remark, observe that if the starting VAR equation has a linear trend deterministic component, one can never entirely restrict the constant to be on the inside of the cointegrating equation because in this case, integrating the VEC model back to its VAR counterpart will render a constant term at most, and never a trend term.

I think this now answers all your questions and hopefully we can put this to bed. I sincerely hope that our budding econometric researchers have gained more appreciation for the breadth of the science they love so much and will henceforth meticulously study the subject matter from proven and reliable sources.