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
Yt is wrongly detected to be I(0).
This estimation is spurious in long run
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.
- 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.