Stationarity - Exogenous Process

Dear all,

I encountered some difficulties by incorporating a specific process and do not know whether I need to de-trend it or not, and if so, how to do it.

I have the following exogenous processes in logs:

da  = mu+x(-1)+zeta*w(-1) + exp(v(-1))*e_a; 
x   = rho_x*x(-1)+exp(v(-1))*e_x;
v   = rho_v*v(-1)  - sigma_v/sigma_x*e_x; 
dw  = mu+(rho_w-1)*(w(-1)-a(-1)-w_bar)+kappa*x(-1)+e_w; 

da = Productivity growth process (labor augmenting)
x = Long run risk component
v = Stochastic vola
dw= Oil supply growth process

The problem arises, I guess, since da and dw are conintegrated.
Due to ‘da’, consumption, output etc have a trend. I can de-trend them by normalizing with e.g. A_t-1, i.e.

C_head = C / A_t-1

Due to conintegration, I also need to de-trend the oil resource constraint:

where G is oil consumption and O is oil as a production factor.
If I do so, I have W_head.

The questions now are:

1)How do I de-trend dw in order to define W_head?

  • I probably ‘just’ need to find a way to recursively write W_head in terms of dw, right? But how?
  • Or do I need to de-trend the whole process somehow?
  1. I also need to specify a and w in log levels since they appear in the processes. How do I do that?
    Can I just write:
a(-1)  = a - da
w(-1) = w -dw

BUT: wouldn’t arise the problem that I still dont have the log levels of ‘a’ and ‘w’?

  1. Do I also need to think about a trend which arises separately from ‘dw’? Or is it taken care of if I de-trend with A_t-1 because of conintegration? :question:

Many many thanks in advance.

Problem is solved.

For anyone who is interested, one ‘simply’ needs to de-trend the dw-process in order to get the de-trended w (w = ln(W/A)):
The result is:

w   = w(-1)-da(-1)+mu-(1-rho_w)*(w(-1)-wbar)+kappa*x(-1)+e_w;