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6/04/2004
The Cobweb Theorem: A True Application and Precursor of "Reflexivity"

Generic soybean prices at year end:

                 1975 4.5 1976 7.0
                 1977 6.0 1978 6.8
                 1979 6.5 1980 7.8
                 1981 6.1 1982 5.8
                 1983 8.0 1984 5.8
                 1985 5.4 1986 5.0
                 1987 6.0 1988 8.2
                 1989 5.8 1990 5.6
                 1991 5.6 1992 5.8
                 1993 7.0 1994 5.5
                 1995 7.4 1996 6.9
                 1997 6.7 1998 5.4
                 1999 4.6 2000 5.0
                 2001 4.2 2002 5.7
                 2003 7.9
                 Apr month end 2004 10.3
                 May month end 2004  8.1

On a trip to Chicago this week, I visited Jim Lorie, who wrote his PhD thesis on the cobweb theorem in economics. The theorem memorialized the dynamic process whereby farmers increase their supply of a commodity in the year after an increase in price of the commodity, thus causing a price drop in the next year.

There were four occasions where soybean prices increased by more than $1 a bushel in one year, and the average decline in price the next year was some 25%. The power of substitution, the choice that suppliers have in deploying their resources and their recovery techniques is underlined and provides a caution for those that haven't a copy of a recent economics book close at hand, or who are not anticipating a similar reductomy in oil.

A Sun-Baked Spec adds:

When I went to Louisville this spring and spent time at Orbis Farm in Corydon, Indiana, I noted some very densely planted rows of something in all of the fields.  I thought it odd how close the plants were. When queried, my hosts kindly educated me about "Roundup beans". Apparently they now have a genetically modified bean that is immune to Roundup, therefore weedkiller can be liberally applied to the fields to suppress weeds with no harm to the crop, thereby increasing density of the bean crop by 10-15%. So I would add that once again technological innovation is also a factor.

Professor Pennington adds:

                 For the equation y(t)=a+by(t-1)...

                 If there is a solution y(t) that converges to a
                 constant y_c, then by definition y_c will satisfy the
                 equation y_c=a+bY_c. Therefore y_c=a/(1-b).
                 
                 With that definition for a quantity y_c, the equation
                 can be rewritten as:
                 
                 (y(t)-y_c)=b(y(t-1)-y_c)
                 
                 Now define a quantity Y(t)=y(t)-y_c, and Y obeys the
                 equation:
                 
                 Y(t)=bY(t-1).

                 Assuming b is positive.. if b>1, then Y will grow
                 exponentially. This is what happens when you're
                 short. If b<1 then Y will decay exponentially (and
                 y(t) will approach y_c).

                 If b is negative, we'll get the same kinds of things
                 EXCEPT that the sign of Y will alternate on each time
                 step. For -1<b<0, y will oscillate around y_c, but
                 with the oscillations dying away over time with decay
                 rate b.

6/8/2004
More on the Cobweb Theorem: A Fast Hypothesis Generator, from Victor Niederhoffer:

Many simplified models of the cobweb theorem where price in one year is related to the price in the previous through a linear supply and demand schedule end up with the path of price following a first order difference equation of form current price = constant + a multiple of last period's price or the normal y (t) = a + b x y ( t-1). The equilibrium values in such a system depend on how b compares to 1 and can take 3 qualitative forms: stable, explosive, and cyclical.  The path is usually visualized with a cob web diagram which is just a regular curve fitting augmented by a 45 degree line to move from one period to the next on the regular curve. All this covered nicely in such texts as the highly recommended "Mathematical techniques" By DW Jordan, 1995, Oxford.  The hypothesis generator is how can you tell from a series of prices whether future prices seems to be more aptly predicted by one of the 3 forms: stable, explosive, or cyclical. and what improvements could be made to such efforts.

Philip J. McDonnell amplifies:

Perhaps many missed this excellent discussion thread. Sensing that the topic was very deep, I quickly did some research and uncovered a very rich field of extensions to the Chair's basic idea.

The family of recurrence relations are sometimes called the cobweb because like a spider designing his cobweb they move from one fixed point to another. In this case the fixed points are on the recurrence relation itself and the 45 degree line mentioned by the Chair.

A little counting on the QQQs for the last 1299 days shows that the following is a reasonable fit for the linear difference equation:

y(t) = .098 + .9977 * y(t-1)

with Rsq = 99.8% (too good to be true because the data has not been detrended) The tstat for b is 524. The tstat for a is only .94 and thus is not significant.

Refitting with the differenced versions (where dy(t) is change for day t) gives:

dy(t) = 9e-5 -.0306 * dy(t-1)

with Rsq a very insignificant .09% and nothing else significant.

The specific forms of the above recurrence relations come from the assumption of linear supply and demand functions. Implicitly we assume that eating 10 steaks is 10 times as satisfying as eating 1 steak. If we allow a quadratic utility curve for supply and demand we quickly get into logistic functions, quadratic maps and perhaps higher order functions as candidates for our recurrence relation. As a student of the Random Walk one has to be intrigued by the historical fact that the great mathematician John von< Neumann first proposed using the logistic equation with coefficient 4 as a pseudo random number generator on computers in the 1940s.

The equation for the logistic function used is:

dy(t) = a * dy(t-1) * ( 1- dy(t-1) ) = a * dy(t-1) - a * dy(t-1)^2

Because each day depends on the next it seemed intuitively clear that an exponentially smoothed average would be the way to go. I chose a 5-day smoothed average for the above logistic function and a 10 day average for the logistic function squared.

The resulting equation is:

dy(t) = sqrt( a x^2 + bx + c )

where x represents the exponentially smoothed realization of the logistic equation at time t-1. The theoretical reason for the sqrt term comes from the solution to the assumed quadratic supply demand curves. For the above equation the stepwise regression results were:

dy(t)^2 = .8316 * x^2 + .000131 with Rsq of 10.9%, sig at the e-34 level
t-stat
a 12.59
c 2.09

Using the residuals from the first step the following fit was obtained:

res = -.0367 * x with Rsq of 4.28%, sig at the e-14  level

the tstat for the b was -7.62

Overall the above regression explains about 15.2% of the variance with a multiple correlation of 39%.

A Few Other Notes:

  1. Simple correlations with individual days were strongest at lag 2.
  2. When quadratic recurrence forms are lagged they are implicitly
  3. multiplied. Doing the algebra shows that they quickly blossom to polynomials of order 9 (and greater) within 2 or 3 days.
  4. The above theoretical result was confirmed empirically. The correlations between changes lagged 2 days averaged about .09 for all powers from x^2 through x^9. The significant level is about .06 for the 5% level.
  5. Qualitatively one can conclude that the dominance of the change^2 terms implies that the driving force behind the solution to the supply demand equation is change in perceived variance.

The above just barely scratches the surface. There is a lot more work to do in this very fruitful area.