[Sebastian Kripfganz]

If your variables are I(1) and you have more than one co-integrating relationship among them, the single-equation ARDL model would be misspecified as it can accommodate only one co-integrating relationship. In that case you would prefer to estimate a vector error-correction model (VECM).

If your variables are I(1) and you have exactly one co-integrating relationship, you can rewrite the ARDL model analytically in error-correction representation with first-differences of depvar on the left-hand side, the co-integrating relationship of the level variables as well as additional lags of first-differenced depvar and indepvars on the right-hand side. All those components are then I(0) which shows that you can safely estimate this ARDL model in levels.

If your variables are I(1) but you do not have any co-integrating relationship among them, estimation is still fine because there exist values for the population parameters such that the error term can be I(0) due to the inclusion of lags of the dependent variable (the sum of the coefficients for the lags of depvar would equal unity in the underlying data generating process such that the level term drops out in the error-correction representation of the model; similarly for indepvars that are I(1)). However, in this case it would be more efficient to estimate an ARDL model directly in first differences.

If all of your variables are I(0) then you obviously do not have any problem with the ARDL model.

The point that I want to make is the following: Testing for non-stationarity and co-integration of your variables is still useful as it guides you towards the optimal model choice (VECM, ARDL in levels, ARDL in first differences).

[danishussalam]

Thanks alot for answering the second question. I understand that now.

Are you saying that integrated order is not important for ARDL? which in other words imply that one need not to use augemented dickey fuller test to check for stationarity of variables?

[Sebastian Kripfganz]

1) I do not fully understand your first question. In essence, your variables do not have to be stationary. The ARDL model is appropriate whenever you have (at most) one co-integrating relationship among your variables.

2) The optimal lag length is usually decided on the basis of model selection criteria, like the Akaike oder Schwarz information criterion. You run the model for all possible lag combinations and eventually choose the model that delivers the smallest value of the respective criterion among all models. For example, with the Schwarz-Bayesian criterion (SBC), one independent variable, and a maximum lag length of 4:

local maxlag = 4
local sbcstar = .
local pstar = 0
local qstar = 0
local p = 1
while `p' <= `maxlag' {
                local q = 0
                while `q' <= `maxlag' {
                               reg L(0/`p').depvar L(0/`q').indepvar if time >= 1 + `maxlag'
                               estat ic
                               mat stats = r(S)
                               local sbc = el(stats, 1, 6)
                               if `sbc' < `sbcstar' {
                                               local sbcstar = `sbc'
                                               local pstar = `p'
                                               local qstar = `q'
                               }
                               local ++q
                }
                local ++p
}
reg L(0/`pstar').depvar L(0/`qstar').indepvar if time >= 1 + `maxlag'

Importantly, make sure to restrict the sample to be the same with all lag combinations such that the same number of observations is used in each case. Otherwise, the sample selection criteria would not be comparable.

[danishussalam]

Here's my question:

1) Does all lags of the variables in an ARDL model have to be stationary? The model I posted above has the variable pcwage (at level) and it first differences too. Is it okay if pcwage is non-stationary?

2) Is there a formal test to decide how much lags of each variable should be added in an ARDL model.

Thanks.

[danishussalam]

I was about to ask the same question. It would be very helpful if someone can post a step by step guide to ARDL

b/w ARDL in STATA is estimated by

regress infln pcwage L1.pcwage L2.pcwage L3.pcwage L1.infln L2.infln

The above model has p lags of the dependent variable, yt ,and q lags of the independent variable, xt The ARDL(2,3) model of inflation can be estimated using least squares.