# Stationarity - Exogenous Process

**URL:** <https://forum.dynare.org/t/stationarity-exogenous-process/4532>\
**Category:** Dynare help (legacy posts)\
**Created:** [4 March 2015 09:57 UTC](https://forum.dynare.org/t/stationarity-exogenous-process/4532 "2015-03-04T09:57:44Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![salkiewi](https://forum.dynare.org/letter_avatar_proxy/v4/letter/s/8797f3/32.png) [@salkiewi](https://forum.dynare.org/u/salkiewi)\
**Post date:** [4 March 2015 09:57 UTC](https://forum.dynare.org/t/stationarity-exogenous-process/4532/1 "2015-03-04T09:57:44Z")

</div>

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:

```auto
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.

```auto
C_head = C / A_t-1
```

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

```auto

```

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:

```auto
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? ❓

Many many thanks in advance.

---

<div class="post-metadata">

**Author:** ![salkiewi](https://forum.dynare.org/letter_avatar_proxy/v4/letter/s/8797f3/32.png) [@salkiewi](https://forum.dynare.org/u/salkiewi)\
**Post date:** [6 March 2015 10:00 UTC](https://forum.dynare.org/t/stationarity-exogenous-process/4532/2 "2015-03-06T10:00:11Z")

</div>

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:

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