What is the T-Test in ARDL ECM used for?

[Alina Klein]

Hi,

I am new to using STATA. For my Bachelors thesis I am analyzing the effect of income, own price and price in another destination on the demand for tourism. In my literature, only the F-statistic of the bounds test is used to determine wether there is cointegration. So i am wondering what the t-statistic is used for and if I have to use it.

[Sebastian Kripfganz]

The bounds test t-statistic tests whether the speed-of-adjustment coefficient equals zero (null hypothesis) or less than zero (alternative hypothesis). We cannot use the conventional critical values and p-values directly reported in the regression output due to the nonstationarity of the data under the null hypothesis.

For the existence of a long-run relationship, it is a necessary requirement that this speed-of-adjustment coefficient is less than zero. The F-statistic jointly tests for insignificance of this speed-of-adjustment coefficient and the coefficients of the regressors in the assumed long-run relationship. Thus, a rejection of the F-statistic could mean that the latter coefficients are nonzero, while the speed-of-adjustment coefficient could still be zero. To rule out this special case, we also need to look at the t-statistic.

More information:

• Kripfganz, S. and D. C. Schneider (2018). ardl: Estimating autoregressive distributed lag and equilibrium correction models. Proceedings of the 2018 London Stata Conference.
• ARDL: updated Stata command for the estimation of autoregressive distributed lag and error correction models

[John Costopoulos]

In Kripfganz and Schneider (2018), both the long-run and the short-run coefficients of the ARDL model are estimated in a single step though the unrestricted EC model. In this (single-step) estimation approach, the speed of convergence is given by the long-run coefficient of the first lag of the dependent variable. As explicitly stated by the authors in page 18, there is statistical evidence for the existence of a long-run relationship only if all the following conditions are met:

1. A successful bounds F-test has been conducted.
2. A successful bounds t-test has been conducted.
3. At least one of the long-run coefficients of the independent variables is statistically significantly different from zero.

In the ARDL empirical literature there are many authors who instead prefer to first perform a bounds F-test and then follow a two-step estimation approach by first estimating a long-run levels model in order to extract the long-run coefficients, and then a restricted ECM in order to extract both the short-run coefficients and the coefficient of the first lag of the error correction term, i.e. ECT(-1). In this two-step estimation approach, would it be correct to say that there is statistical evidence for the existence of a long-run relationship only if all the following conditions are met:

1. A successful bounds F-test has been conducted.
2. The coefficient of ECT(-1) is statistically significant and negative.
3. At least one of the long-run coefficients of the independent variables is statistically significantly different from zero.

Do we also need to perform a bounds t-test or it is rather redundant in the case of the above two-step estimation approach?

[Sebastian Kripfganz]

The two-step approach you are describing is essentially the Engle-Granger approach implemented in Mark Schaffer's Stata command egranger; available from SSC. The coefficient of the error correction term still has a nonstandard distribution under this two-step procedure. The egranger command uses the appropriate critical values. Note that the Engle-Granger approach assumes that your variables are I(1). This is as if you only consider the upper bound from the bounds test. See also slides 5 and 6 of our 2016 Stata Conference Presentation.

[John Costopoulos]

Dear Sebastian,

Thank you very much for your prompt reply. I readily agree with you that the two-step approach I described in my previous post (#3) is essentially the Engle-Granger approach. I must confess that whenever I came across articles that incorrectly used this approach in the ARDL framework, I was not surprised because I was sure that I had seen an application of a two-step approach in PSS (2001, pp. 306-314).

This time, I read the above example much more carefully. I noticed that Equation (31) on page 313 reports the estimates of a “levels relationship”, while the last term in this equation is an “equilibrium correction term”. As I realized this time, I hope correctly, the coefficients displayed in Equation (31) had not been derived from the estimation of a "levels equation" (as in the first step of the Engle-Granger approach) but they actually represented the estimated normalized long-run coefficients of the ARDL model described in Pesaran and Sin (1999). As PSS (2001, p. 313, footnote 29) point out, the ARDL approach advanced in Pesaran and Shin(1999) is applicable irrespective of whether the regressors are purely I(0), purely I(1) or mutually cointegrated.
In the second step, PSS (2001, p. 313) use the equilibrium correction term, calculated by Equation (31), in a conditional (unrestricted) EC model to compute both the coefficient of the equilibrium correction term and the short-run coefficients.
Based on the above, it they are absolutely correct, I would like to ask you if in the above two-step approach we need to perform a bounds t-test, or it is actually redundant.

[Sebastian Kripfganz]

Yes, equation (31) in PSS is not estimated as such. Instead, the conditional EC model in equation (30) - or the equivalent ARDL representation - is estimated, and all coefficients (long-run, short-run, equilibrium adjustment) are inferred from that single estimation step. This is exactly what the ardl command does in Stata. The estimated equilbrium correction term is not actually used as a regressor in a second step, even though Table III in PSS might give that impression.

That means, the ARDL approach pursued by PSS is not a two-step procedure. The bounds test is still useful. If there is no evidence for a long-run relationship, then you could simplify the model and run a regression in first differences only, i.e. as in Table III but excluding the error correction term. Unless the bounds F-test lets you already conclude that there is no evidence of a long-run relationship, you also need to check the t-test. (Note that the coefficient of the error correction term has a non-standard distribution, which is why we need to use the critical values from the bounds test.) Only a rejection of both the F- and the t-test can be seen as evidence for the existence of a long-run relationship. For further discussion, our recent OBES paper might also be of interest:
Kripfganz, S., and D.C. Schneider (2020). Response surface regressions for critical value bounds and approximate p-values in equilibrium correction models. Oxford Bulletin of Economics and Statistics 82 (6), 1456-1481.

[John Costopoulos]

Dear Sebastian,

I am grateful for your prompt. Indeed, Table III in PSS gave me the impression that they followed a two-step approach. I have also read many times both your 2018 presentation and the 2020 paper. Each time, your enlightening answers on Statalist help me to see them in a new light. I apologize for any inconvenience caused to you, but I would appreciate it very much if you could give two clarifications:

1. If PSS had followed a two-step approach, as I initially thought, should they have to perform a bounds t-test or a statistically significant and negative ECT(-1) coefficient would make the bounds t-test redundant?

2. In the example provided on page 14 of Kripfganz and Schneider (2018), is the coefficient of the variable L.ln_consump directly produced by the regression or it is the result of a transformation? I ask this because I consider the coefficient of the variable L.ln_consump as being the same with the –α mentioned in Slide 12, where α is (as I see it) the coefficient of the error correction part of the EC models in Slide 12. If it is so, could we report the label “ECT(-1)” instead of “ln_consump.L1” in the ADJ section?

[Sebastian Kripfganz]

1. You cannot infer the statistical significance of the ECT(-1) coefficient from the standard regression output because the conventional critical values are not valid for this coefficient. The bounds test t-statistic is the same test statistic as the one you would use for that significance test. The difference is just the critical values.

2. Yes, this is correct. This is the coefficient of the ECT(-1) term.

[John Costopoulos]

Dear Sebastian,

Thank you again. If I understood well, the bounds t-test is mandatory in the two-step approach in order to test for the statistical significance of the ECT(-1) coefficient.

[John Costopoulos]

Dear Sebastian,
I would like to retract my previous post (#9). As I thought of it again, based on your answer in post #4, the statistical significance of the ECT(-1) coefficient can be checked using the MacKinnon (1990, 2010) critical values, as in egranger. This means that the bounds t-test is redundant in a two-step approach.

[Sebastian Kripfganz]

the statistical significance of the ECT(-1) coefficient can be checked using the MacKinnon (1990, 2010) critical values, as in egranger. This means that the bounds t-test is redundant in a two-step approach.

This is correct if the model is indeed estimated in two stages (i.e. if the Engle-Granger procedure is applied) and all of the exogenous variables are I(1).

If the model is estimated in one stage (i.e. the PSS approach) and it is known that all of the variables are I(1), then the correct critical values are those for the upper bound from the bounds test.

[John Costopoulos]

Dear Sebastian,
I apologize for making you spend so much of your time to help me with your enlightening answers. However, I cannot still understand if there is a two-step approach which can reliably be used in the usual case where the ARDL model includes both I(1) and I(0) independent variables. In this case, it would inappropriate to use an “Engle-Granger style” approach. Α suitable approach might be the one I have encountered in a good-quality article, which includes the following steps:

1. Estimate the unrestricted ECM to extract the long-run coefficients.
2. Normalize the extracted long-run coefficients and use them to compute ECT(-1).
3. Create a restricted ECM using the ECT(-1) term computed in step (2) above.
4. Estimate the restricted ECM to get the short-run coefficients and the ECT(-1) coefficient.

Based on the above, I have three questions:

1. Is the about approach correct?
2. If yes, are the conventional critical values valid for the ECT(-1) coefficient?
3. Is the bounds t-test compulsory in this case? Or useful only? Or actually redundant?

[Sebastian Kripfganz]

I have seen people doing this, but I never understood the reason for creating the ECT(-1) term from an unrestricted EC model and including it in a second-step restricted EC regression. I do not think that the conventional critical values will be correct for the ECT(-1) coefficient. I am not even sure whether any of the critical values for either the Engle-Granger two-step approach or the PSS bounds test would be appropriate here.

[John Costopoulos]

Sear Sebastian,

Here are some possible answers (in italics) to the question about the use of the term ECT(-1) term:

a. I have seen the equilibrium correction term (ut) in Table III in PSS(2001).
How many people have your very deep knowledge as to know that PSS actually followed a single-step approach?

b. I have seen it in the Professor Giles’ influential article “ARDL Models - Part II - Bounds Tests” (https://davegiles.blogspot.com/2013/...nds-tests.html).
Personally speaking, this is the place where I came across the ARDL bounds test for the first time.

c. I have seen it in some previous articles.
If someone studied carefully some of these “previous articles” he would possibly see: (1) a lengthy description of the bounds test approach, (2) no information about the steps followed in order to estimate the model, (3) a table with the long-run coefficients, and (4) a table with the short-run coefficients and not, so surprisingly, the coefficient of the ECT( -1) term.

d. I have not seen anywhere that it is forbidden.
This is possibly the strongest argument!

After all, I am really grateful for your patience and your great help in my journey to the ARDL world. Your students are very lucky to have you as a teacher!

[Sebastian Kripfganz]

b. In Dave Giles' blog post, he estimates the EC model in a single step and applies the bounds test (steps 2 to 6). He subsequently applies the Engle-Granger two-step procedure (step 7). Assuming that all variables are found to be I(1) (step 1), the MacKinnon critical values can be applied. I still do not see why we would want to do this step 7. As Dave Giles mentiones himself, long-run effects are readily available from the single-step EC model (step 8). Yet, many people might indeed follow his suggested procedure.

d. As long as you use the correct critical values, the procedure in Dave Giles' blog is certainly not forbidden.