Journal of Real Estate Finance and Economics, Vol. 17:1, 87±97 (1998)
# 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
A Primer on Piecewise Parabolic Multiple Regression
Analysis via Estimations of Chicago CBD Land Prices
PETER F. COLWELL
Department of Finance, University of Illinois, 1407 West Gregory Dr., 304 DKH, Urbana, IL 61801,
This article proposes a non-parametric method for estimating spatial price functions. Space is divided into
squares. The independent variables are barycentric coordinates that uniquely describe the location of observations
in space. The regression coef®cients are estimates of the height of the function directly over the vertices of the
square spatial units. Within each square the function has hyperbolic iso-price curves and parabolic sections. The
price function is continuous, but not differentiable, at the boundaries between contiguous squares. This method is
applied to the problem of describing the price per front foot of land in the Chicago CBD. A rather complex price
surface is revealed that would be dif®cult to estimate using other methodologies but was easily estimated by this
Key Words: nonparametric, piecewise, hedonics, Chicago, land, price
One of the most dif®cult problems in real estate hedonics is to model the structure of the
spatial price function.
Negative exponential functions are a frequently used option from
urban economics, and inverse functions seem sensible asymptotically but are ill behaved
as distance approaches zero. Very complicated, continuous, and sensible spatial price
functions can be estimated if the analyst has ex ante knowledge of value in¯uence centers
and lines (such as roads) and the spatial extent of the in¯uence. This is done by estimating
functions that interpolate between the value in¯uence centers and the edges of the impact
(O'Conner and Eichenbaum, 1988). High-degree polynomials have the potential to
represent spatial trend surfaces without ex ante knowledge of the surface but at the
expense of imparting artifacts, unwarranted bumps and dips, and being generally ill-
mannered at the edges.
A simple, non-parametric approach is neededÐone that ®ts any function with the
fewest possible restrictions. The purpose of this article is to describe a method for using a
single, standard OLS regression to estimate a continuous price function in space that can
approximate any shape. The cost of the method developed here is found in terms of
degrees of freedom. It achieves ¯exibility by requiring large numbers of observations.
It is becoming popular to show three-dimensional maps of price because many
computer graphics programs exist that facilitate this practice.
Certain assumptions are
common to all this software: the fundamental spatial unit appears to be a square, and