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 effecting 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 does not change it means this variable is not effected by any variable in long run, especially those 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.

**Illustration**:

Consider a model

Y_{t} = a + b_{1} X_{t} + b_{2} W_{t} + b_{3} Z_{t} + e_{t} — (1)

Let Y_{t} ~ I(0), X_{t} ~ I(1), Z_{t} ~ I(1), W_{t} ~ I(0) and e_{t} ~ I(0)

Here in long run normally b_{1} and b_{2} must be insignificant as variables having time variant mean or variance (X_{t} and Z_{t}) cannot effect a variable with constant mean and variance (Y_{t}). Here b_{3} can be significant as in long run I(0) variable can effect I(0). If b_{1} and b_{2} are significant will indicate following

Case 1:

Y_{t} is wrongly detected to be I(0).

Case 2:

This estimation is spurious in long run

Case 3:

Special case where X_{t} and Z_{t} are cointegrated with each other forming resultant R_{t} ~ I(0)

Y_{t} = a + c_{1} R_{t} + b_{3} W_{t} + e_{t} — (2)

**Issue**:

The issue with this special case is that, now X_{t} = f(Z_{t}) or Z_{t} = (X_{t}) indicating presence of long run multicollinearity. Causing b_{1} or b_{2} to be biased, only b_{3} 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).

Y_{t} = a + b_{1} ΔX_{t} + b_{2} ΔW_{t} + b_{3} Z_{t} + e_{t} — (3)

- Nested Model:

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

X_{t} = d_{0} + d_{1} Z_{t} + d_{2} f(Y|X) + v_{t} – (4)

Then the resultant v_{t} should be used as independent variable.

Y_{t} = a + d v_{t} + b_{3} Z_{t} + e’_{t} – (5)

Here the effect of X_{t} and Z_{t} can be checked by calculation.

Y_{t} = a + d [X_{t} – d_{0} – d_{1} Z_{t} + d_{3} f(x)] + b_{3} Z_{t} + 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 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 Y_{t} through X_{t} theoretically.

**Criticisms and Suggestions**:

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

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

Yes. VECM can provide more information in this situation though

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

do it seperately

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.

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.