I am working a bit on this paper, which is about Long-run risk through Consumption Smoothing.

In equation (8) and (9) the authors define the stochastic process for the technology as:

$$Z_t = \exp(\mu t + z_t)$$

$$z_t = \varphi z_{t-1}+\epsilon_t$$

My question is straightforward, why do they specify these two equations? Wouldn't it be exactly the same to specify: $z_t = \mu_t + \varphi z_{t-1}+\epsilon_t$

Also, with their specification, if you take logs of the first equation, the $z_t$ cancel out, right?

I am assuming that $z_t = \ln(Z_t)$ which I believe is correct.

  • 1
    $\begingroup$ For the user who voted it down, it would be helpful to have some constructive comment... $\endgroup$
    – phdstudent
    Jul 26, 2015 at 10:54
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    $\begingroup$ I guess the problem is that you didn't link or name the paper. It would certainly help! $\endgroup$
    – Bob Jansen
    Jul 26, 2015 at 16:38
  • $\begingroup$ Why do you assume $z_t = \ln(Z_t)$, it should be $z_t = \ln(Z_t) - \mu t$... and how does that help? $Z$ and $z$ are different here, but I think the notation is pretty bad. Basically this "is" a geometric brownian motion with autoregressive local volatility. $\endgroup$
    – SRKX
    Oct 26, 2015 at 7:14

1 Answer 1


Well this is not my area of expertise but I have come across this sort of work before in Time Series Analysis/ Financial Econometrics. I don't know how much detail you want but from my understanding the author has written the two equation in State Space Form. I believe it is fairly common to write ARCH and GARCH models in this fashion. There are a lot of papers covering the basics and motivation behind State Space Models. There is a decent introduction into them on:


Some other further reading into State Space models and their construction can be found in such books as "System Dynamics in Economic and Financial Models" by Heij, Schumacher and Hanzon (1997) in Chapter 9 and "Journal of Basic Engineering" by Kalman (1960). This overlaps with the theory of Kalman Filters.

As I said above this isn't an area I have much exposure too but if you maybe know anyone who has studied Financial Econometrics they might be able to help since that is where I have encountered these sort of models before.

If I have missed the point of the question I am sorry, but I thought some direction may be better than nothing. Tbh I would have left this as a comment but not enough reputation.


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