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Linear Model of Prices, Spring 2001

Data

The following is an attempt to explain the prices of clamps, using data gathered at Brimfield, Spring 2001, on the Saturday. There are 25 clamps in the raw data. Please note the special nature of the sample, and do not over-generalize the findings.

Model

My basic assumption is that the price of a clamp can be estimated from the product of two functions.

Price = Size_Function * Condition_Function

I further assume that the Size_Function is linear, a constant plus so much per inch of jaw. An alternative is explored in the Fitted Model.

Also, I constrain the Condition_Function so that "good" is worth 1, and the function is monotonic, that is, better condition must not imply lower prices. The conditions are defined to be poor, fair, good, and very good. (This time, there were no clamps of very good condition.)

The Excel Solver function was set up, and run several times.

• minimize the absolute value of the average error, using all data.
• minimize the average squared error, using all data.
• minimize the average squared error, after eliminating the two worst outliers.

Findings, using all data to minimize absolute average error

The result was that the average of the errors was 1%, and the mean square error was 18.4%.

Size Function

The size-function is $14.78 for existence, plus $0.40 per inch.

The tabulated value for each size is

Condition Function

The condition function has constant value 1 for poor, fair, and good, that is, condition does not impact price.

Findings, using all data to minimize average squared errors

The result was that the mean square error was reduced 14.94%, while the average of the errors was 16% .

Size Function

The size-function is $21.28 for existence, plus $0.26 per inch.

The tabulated value for each size is

Condition Function

The condition function has constant value 0.77 for poor, and value 1 for fair, and good, that is, condition does impact price, but only at the low end.

Findings, removing two outliers, to minimize average squared errors

Two items have relative errors of +81% and -72%. These were removed. The result was that the mean square error was reduced to 8.8%, while the average of the errors was 7% .

Size Function

The size-function is $23.53 for existence, minus $0.02 per inch.
That is, size has a negligible impact on price. Needless to say, this result is unexpected! Basically, it says that, except for a couple items, dealers figure one clamp is much like another.

The tabulated value for each size is

Condition Function

The condition function has value 0.81 for poor, and for fair, 1.00 for good, that is, condition does impact price, somewhat.

Fit

Basically, the fit to any of the models is very poor. If you take the raw data, and make scatter plots, you will find that you get a blobby cloud, with very low correlations between the price and the factor of your choice. This is regardless of whether you choose jaw length, clamp condition, clamp rarity, or anything else. Any straight line through the centroid is almost as good a fit as any other.

Then if you take out one or two data points, the math will derive another line, but it is nowhere close to the previous line.

Under these circumstances, I've decided not to make up the usual table of actuals versus estimates.

last revised and validated

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