Vlieghe pointed out that “the idea persists, because of commonly adopted – but misleading –
practices in solving macro models.” Modern macroeconomics is based on highly
non-linear models but in order to make them more tractable for solution
purposes, we use logarithmic transformations to linearise them. Vlieghe uses
the example of how the linear transformation of the standard method of
discounting future consumer utility results in “a tight relationship between the real interest rates and growth, and
nothing else ... it kills off, mechanically, anything that might have been
interesting about risk … In the linearised world, there is no risk-free real
interest rate.” For those interested in the detail, the relevant part of
Vlieghe’s speech is reproduced in a footnote[1].
As it happens, this is perhaps an overly-rigorous
theoretical interpretation of the relationship between growth and interest
rates. Admittedly, the fact that there is a strong correlation between (real)
short-term interest rates and (real) GDP growth does not necessarily imply a
causal relationship. But the work of the late-nineteenth century economist Knut
Wicksell postulated that there is a ‘natural’ interest rate which is determined
by the real disturbances affecting the economy. To the extent that these
disturbances are manifest in output growth and inflation, it is clear that
nominal GDP growth and interest rates ought to be closely related. It is not a
1:1 relationship, but over the long-run the rate of return on real assets ought
to be similar to that on financial assets in order to satisfy equilibrium
conditions. Not for nothing have many economists argued that there is a strong
case for using nominal GDP growth as an anchor for monetary policy.
That said, Vlieghe’s point on how linear approximations can
result in specious outcomes was well made. All students of econometrics spend a
lot of time learning about the properties of linear regression models which
partly explains why grubby practitioners like me are comfortable applying
linear transformations in order to easily apply linear estimation techniques.
Another reason for preferring linearity in econometrics is that non-linear
solutions can be indeterminate because we do not know whether they are the
universally right answer, or whether they apply only under certain conditions.
The reason for this is that many of the common solution techniques rely on grid
searches conducted over a range of values. In the jargon, we do not know
whether we have found local maxima only within the range in which the search is
conducted or whether the “true” answer lies outside it. Our models may thus be
biased – in other words, deliver the wrong answers under certain conditions –
and as a result many economists stick to what they know in the form of
linearity.
But this bias towards linearity, tempting though it is, can
be applied to situations in which it is not appropriate. The authors of a paper
in experimental psychology[2]
assessed the accuracy of long-term growth estimates by panels of “experts” and
laypeople. Whilst both groups tended to underestimate growth at rates above 1%,
the degree of underestimation was greater for “experts” because they ignored
exponential effects more often than the group of laypeople. Another paper by DeBock et al (2013) is interesting because it provides a literature review of the reliance on
linearity and reports the findings of an experiment amongst business economics
students who were confronted with correct and incorrect statements on
linearity in economic situations. The authors concluded that many of the students showed
over-reliance on linearity in their analysis.
An
interesting paper on nonlinearity by Doyne Farmer (here) looks at various aspects of nonlinearity and complexity in economics. He makes
the point that DSGE models are too highly stylised to say anything useful about
the behaviour of economies in the real world. Instead, economics might start to
take lessons from areas such as meteorology which builds very complex data-based
nonlinear models. This has been enabled by the significant increase in
computing power which allows simulation analysis to be conducted much more
cheaply and effectively than in the past.
For policymakers, the fear is that linear approximations in
a nonlinear world lead to distorted policy conclusions. One problem is that
economics tends to focus on equilibrium solutions. But as noted above, there may not be a single equilibrium. Indeed, the impact of the financial crash of 2008 was an object lesson in how nonlinear feedbacks can produce outcomes far beyond our expectations.
The mathematician Stan Ulam used to give lectures on nonlinear
mathematics and apologise that the title was a misnomer, for all interesting
maths involves nonlinearity. As Farmer put it, “just as almost all mathematics is nonlinear, almost all economic
phenomena are complex … A more tractable topic would be whether there are any problems
it does not illuminate, or should not illuminate.”
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