The selling prices of a number of houses in a particular section
of the city overlooking the bay are given in the following table,
along with the size of the lot and its elevation:
Selling price (Pi) | Lot size (sq. ft.) (Li) | Elevation (feet)
(Ei)
$155,000 | $12,000 | 350
120,000 | 10,000 | 300
100,000 | 9,000 | 100
70,000 | 8,000 | 200
60,000 | 6,000 | 100
100,000 | 9,000 | 200
A real-estate agent wishes to construct a model to forecast the
selling prices of other houses in this section of
the city from their lot sizes and elevations. The agent feels
that a linear model of the form
P = b0 + b1L + b2E
would be reasonably accurate and easy to use. Here b1 and b2
would indicate how the price varies with lot size and elevation,
respectively, while b0 would reflect a base price for this section
of the city.
The agent would like to select the ''best'' linear model in some
sense, but he is unsure how to proceed. It he knew the three
parameters b0, b1 and b2, the six observations in the table would
each provide a forecast of the selling price as follows:
Pˆi = b0 + b1Li + b2Ei, where i = 1, 2, . . . , 6.
However, since b0, b1, and b2 cannot, in general, be chosen so
that the actual prices Pi are exactly equal to the forecast prices
Pˆi for all observations, the agent would like to minimize the
absolute value of the residuals Ri = Pi ? Pˆi. Formulate
mathematical programs to find the ''best'' values of b0, b1, and b2
by minimizing the following criterion:
a) maximum of absolute value of Pi ? Pˆi for i = 1 to 6
(Maximum absolute residual)