NONPARAMETRIC ECONOMETRICS OR THE
MEANING OF ‘LINEAR’
starting
with a simple example. In economic theory, one can find “relationships” between
aggregated variables. For instance, with Philips curve, we want to visualize the
relationship between inflation and unemployment. An economic theory might tell
you that if unemployment decreases, inflation increases. From that economic
theory, we might try to fit an econometric model, describe that “relationship”
But usually, econometric models are based on a linear relationship. This is not
a prior reason to have a linear relationship. Even if you look at data, you to
find something linear.
The
Cowles Commission, which initiated Econometrics (by founding the Econometric
Society, and the prestigious journal Econometrica) had the postulate that an econometric model should be based on an economic model. That’s how you get SEM
(Structural Equation Model) like Klein model, in the 1950’s,
This
is a standard (linear) econometric model, with parameters and, and some linear
relationship among them. Nonparametric econometric models started, if models
might be nonlinear.
Splines is a rectangular key fitting into grooves in the hub
and shaft of a wheel, especially one formed integrally with the shaft which
allows movement of the wheel on the shaft., but also local regression is a method for fitting a smooth curve between two
variables, or fitting a smooth surface between an outcome and up to four
predictor variables. Which is very natural when you think about it if the goal
is to get a good estimation you look in the neighborhood. We do not want
necessarily a good global model, but a model good in that neighbourhood.
With
nonparametric models, we start to have numerical problems, since problems are
not as simple as linear ones. So, the first goal is to get an efficient
algorithm to solve it. So here we start to have connexions with machine
learning. The main difference I see is simple. In econometrics, seen as a
mathematical statistics problem, we seek asymptotic results, nice probabilistic
interpretations. In econometrics, seen as a machine learning problem, we focus
more on the algorithm. We do not care about the output or the interpretability,
we want a good algorithm. See for instance gradient boosting techniques against
spline regression.
The
blue line is a simple (linear) spline regression model, and the red line is a
boosted spline algorithm (it is a stepwise procedure). The blue line is simple,
easy to understand. The red line, after 200 iterations is a sum of 200
functions.
Author - Poonuraj Nadar
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