Issues with ARDL Cointegrating bounds model with I(0) variable as dependent

Recently I had discussion that in ARDL cointegrating bounds model, dependent variable must be I(1). This model is illustrated previously in Microfit, Eviews and STATA. Following is the reasons for it.

I(1) variable has time variant mean or variance it means this variable is or has been effect by other variables. These affecting variables can be exogenous variables or the policy intervention variables.

But when the variable is I(0) it means that time invariant mean or variance. If mean and variance do not change it means this variable is not affected by any variable in long run, especially that variable whose mean and variance change (i.e. I(1) variables). Yes it can be effected in short run only which is actually deviations around the constant mean.

So in ARDL cointegrating bounds model I(0) variable can only be dependent only for the case where all independent variables are cointegrated with each other without the dependent variable forming resultant I(0) variable. But still, it does not solve the problem rather it indicates towards a bigger issue.


Consider a model

Yt = a + b1 Xt + b2 Wt + b3 Zt + et  — (1)

Let Yt ~ I(0), Xt ~ I(1), Zt ~ I(1), Wt ~ I(0) and et ~ I(0)

Here in long run normally b1 and b2 must be insignificant as variables having time variant mean or variance (Xt and Zt) cannot affect a variable with constant mean and variance (Yt). Here b3 can be significant as in long run I(0) variable can affect I(0). If b1 and b2 are significant will indicate following

Case 1:

Yt is wrongly detected to be I(0).

Case 2:

This estimation is spurious in long run

Case 3:

Special case where Xt and Zt are cointegrated with each other forming resultant Rt ~ I(0)

Yt = a + c1 Rt + b3 Wt + et  — (2)


The issue with this special case is that now Xt = f(Zt) or Zt = (Xt) indicating the presence of long run multicollinearity.  Causing b1 or b2 to be biased, only b3 will be unbiased.

Possible solution:

  • Short Run Model:

In this case, all I(1) variables should be converted to first difference so that they become I(0).

Yt = a + b1 ΔXt + b2 ΔWt + b3 Zt + et  — (3)

  • Nested Model:

In such case, the endogenous variable out of all I(1) variables must be estimated and tested for cointegration (i.e. vt ~ I(0)). Here f(Y|X) shows any other variables which are effecting Yt but their effect is passing through Xt.

Xt = d0 + d1 Zt + d2 f(Y|X) + vt – (4)

Then the resultant vt should be used as independent variable.

Yt = a + d vt + b3 Zt + e’t – (5)

Here the effect of Xt and Zt can be checked by calculation.

Yt = a + d [Xt – d0 – d1 Zt + d3 f(x)] + b3 Zt + e’t – (6)

Above is the long run model where the dependent is I(0) and all independent are directly or indirectly I(0). So this model is modified form of ordinary least square model which can incorporate any number of I(1) variables. Note that once the cointegration is confirmed in equation 4, any of the I(1) variables can be made dependent in equation 5, the coefficient will adjust to the specification.

The advantage of this nested model is that if there is any I(0) variable which is causing conceptual multicollinearity, that I(0) variable can be moved from equation 1 as an exogenous variable to be used as nested variables in equation 4.

Only need to confirm that the effect of the nested variables (independent variables in equation 4) are effecting Yt through Xt theoretically.

Criticisms and Suggestions:

Above arguments are my observations and are fully open for criticisms and suggestions.


9 thoughts on “Issues with ARDL Cointegrating bounds model with I(0) variable as dependent

  1. Iriogbe Pamela says:

    Hello Noman, please can i proceed with ARDL if all my variables are integrated of the same order i.e I(1)?

  2. Ubangida Shuaibu says:

    Hello Dr. Norman. please can I can I use the ECM part my ARDL model to test for Toda and yamamota Granger non causality?

  3. Qamar says:

    I have 6 variables which are integrated of order I(0) and I(1). The order of integration of dependent variables is I(o).
    Now what is the suitable method for cointegration and causality.

    1. Noman Arshed says:

      Usually, the dependent variable whose literature is available is not I(0), so try using more than one type of unit root test to confirm I(0). If it is confirmed try only estimating the model with I(0) independent variable using OLS.

  4. youssef says:

    thank you so much, Dr.Noman Arshed
    some tests of unit root test confirm the dependent variable I(0) some of them confirm the dependent variable I(I) ….so it is possible to this variable as a dependent variable.

    your response is fully appreciated…

  5. Abdul Rehman says:

    Hello Guys, i need your previous views about ARDL long run results of ECM are given below
    CointEq(-1) coffecient= -1.049220 Standarad dev.= 0.145911, t-stat= -7.190806 and probability0.0000 is value of cofficient is correct and model is stable? thanks in advance

    1. Noman Arshed says:

      if all of your variables in log form then it is correct, while stability is tested using CUSUM or CUSUM sq graph. If atleast one of your variable is not in log form then this value is over correcting.

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