. . . somewhere in phase space.
I have often used the the index of home prices in the United States (the Case Shiller index) as an example of a complex system showing multistability. The data are updated monthly here.
The multistable nature of such systems is demonstrated in reconstructed phase space portraits, which can be generated by two principal methods--the time-delay approach and the time-derivative approach.
For nearly 40 years, the system remained confined to the area of the yellow circle. In fact, house prices were confined to this small area for much longer than that--for the longer term chart I've presented previously shows that this yellow area has been occupied for a total of about 70 years.
It looks like there was some kind of redefinition or recalculation of the time series on the Shiller website sometime in the last year or so, as the current time series available on the site (from which the above chart is drawn) differs from the previous time series available (from which my older graphs are drawn). For instance, in my older figure, the house price appears to have been above 100 over the last 60 years--this does not appear to be the case in the graphs I have produced in previous years.
The overall story has not changed--after 50 years of relative stability (the bubble of 1989-90 looks benign in the above figure), the system broke out of its area of Lyapunov stability, and has been meandering through phase space ever since. Two years ago, it seemed to be on a trajectory to return to the yellow area of stability. In the last two years the system trajectory veered away from that target, and is now headed . . . where?
Normally we expect it to migrate to an area which has previously acted as an area of stability--but both such areas are at much lower prices than is currently the case.
It is possible for the system to carve out a new area of stability. For reasons of geometry, stable areas must be located on the y = x line. Since we are close to that line now, it implies that perhaps Yellen can engineer a soft landing for housing at close to the current price range. Unfortunately, the future level of the trajectory in state space is partially determined by the past trajectory--and in 2013, the housing index was in the low 130s (on the horizontal axis). In two years, therefore, the trajectory will dip to the same level on the vertical axis. If house prices remain where they are now, the trajectory will be far enough from the y = x line to be unstable, and a further decline in house prices would be indicated.
If house prices rise again, we will find ourselves in a bubble destined for collapse, just as we were in about 2004. If the yellin' wants to bring housing to stability, the thing to do would be to engineer a slight decline in house prices over the next two years. Unfortunately, I don't think she wants to do that.
I have often used the the index of home prices in the United States (the Case Shiller index) as an example of a complex system showing multistability. The data are updated monthly here.
The multistable nature of such systems is demonstrated in reconstructed phase space portraits, which can be generated by two principal methods--the time-delay approach and the time-derivative approach.
For nearly 40 years, the system remained confined to the area of the yellow circle. In fact, house prices were confined to this small area for much longer than that--for the longer term chart I've presented previously shows that this yellow area has been occupied for a total of about 70 years.
It looks like there was some kind of redefinition or recalculation of the time series on the Shiller website sometime in the last year or so, as the current time series available on the site (from which the above chart is drawn) differs from the previous time series available (from which my older graphs are drawn). For instance, in my older figure, the house price appears to have been above 100 over the last 60 years--this does not appear to be the case in the graphs I have produced in previous years.
The overall story has not changed--after 50 years of relative stability (the bubble of 1989-90 looks benign in the above figure), the system broke out of its area of Lyapunov stability, and has been meandering through phase space ever since. Two years ago, it seemed to be on a trajectory to return to the yellow area of stability. In the last two years the system trajectory veered away from that target, and is now headed . . . where?
Normally we expect it to migrate to an area which has previously acted as an area of stability--but both such areas are at much lower prices than is currently the case.
It is possible for the system to carve out a new area of stability. For reasons of geometry, stable areas must be located on the y = x line. Since we are close to that line now, it implies that perhaps Yellen can engineer a soft landing for housing at close to the current price range. Unfortunately, the future level of the trajectory in state space is partially determined by the past trajectory--and in 2013, the housing index was in the low 130s (on the horizontal axis). In two years, therefore, the trajectory will dip to the same level on the vertical axis. If house prices remain where they are now, the trajectory will be far enough from the y = x line to be unstable, and a further decline in house prices would be indicated.
If house prices rise again, we will find ourselves in a bubble destined for collapse, just as we were in about 2004. If the yellin' wants to bring housing to stability, the thing to do would be to engineer a slight decline in house prices over the next two years. Unfortunately, I don't think she wants to do that.
Please update!
ReplyDeleteThank you.
I don't update this very frequently because it evolves so slowly that you need 6-12 months to see anything interesting. But I will update this at some point.
Delete