Separating Technology Growth and Its Persistent State in a State-Space Model
Summary
The document asks why a technology process is written as an exponential trend multiplied by a stochastic component, with the log component following an autoregressive process. It questions whether the trend could instead be included directly in the state equation and whether taking logarithms makes the stochastic term disappear. The response identifies the paired equations as a state-space representation, a common way to express dynamic models in financial econometrics and time-series analysis.
The answer points toward ARCH and GARCH formulations, Kalman filtering, and references on state-space models as relevant background. It offers a direction for understanding the notation but does not directly resolve the algebraic concern or explain the roles of the observed process and latent state in detail. The discussion is therefore a brief conceptual pointer, not a full derivation or an assessment of the cited model's empirical use.
Key ideas
- The technology process separates an exponential deterministic trend from a stochastic log component.
- The stochastic component is specified as an autoregressive state.
- The response interprets the two equations as a state-space representation.
- State-space methods are connected to financial time-series models and Kalman filtering.
- The answer does not work through the questioner's proposed alternative specification.
Tags
Full text
# Asset pricing - Technology
# Asset pricing - Technology
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.
## Answer by ahair (score 3)
https://quant.stackexchange.com/a/19033
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:
http://uk.mathworks.com/help/ident/ug/what-are-state-space-models.html?refresh=true
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.