From: John Conover <john@email.johncon.com>

Subject: Re: General Equilibrium Model

Date: 29 Dec 1998 21:58:53 GMT

Donald L. Libby writes: > jim blair wrote: > > > > G o e t z K l u g e wrote: > > > > > > Markets at equilibrium and dynamic changes within these > > > markets are no contradiction. > > > > > > Roughly speaking, a system is in equilibrium with regard to > > > a variable, as long as the average of this variable does not > > > change. > > > > > > > So what economic variables have not changed their long term averages? > > The DEFINITION of "long run" is that short-run variables are constant. > I kind of struggled through the numbers, using the DJIA data as an example. I used the daily closes for the last century as the data set. In very rough numbers, the statistics are that the day-to-day likelihood of an up movement is 51%, (and the chances of a down movement are 49%.) The volatility, (ie., the root mean square of the day-do-day marginal growth,) is about 1%. The average of the day-to-day marginal growth is about 0.02%. There are two issues. The first is the measurement of a long term average growth of 0.02% per day that is rattling around at 1%, rms, per day. If it is desired that the accuracy of the measurement is 1%, then standard statistical estimation gives a data set size of 12,100 trading days, or about 48 calendar years. The second issue concerns the chances that the data set interval measured was predominately up, or down, by serendipity of the choice of the interval. (The durations of the up and down fluctuations, assuming a random walk model, have a frequency distribution of erf (1 / sqrt (t)) which for t > 1 is about 1 / sqrt (t), ie., it has large swings, above or below the long term average, that can last for literally decades.) Again, if a 1% accuracy is chosen as an arbitrary requirement, the measurement interval is 13,890 trading days, or about 55 years. Taken together, for the 1% arbitrary accuracy requirement, the data set size jumps to 33,000 days, or about 130 years. What's the point? Many do not conceptualize what the data set size requirements are to determine the value of "long run" variables with reasonable accuracy in random walk models. Note that it not necessarily the data set size, per se, but data over a large enough interval that is the issue-the 1 / sqrt (t) frequency distribution function decays rapidly-at first-then has long tails that decay very slowly to small values. John -- John Conover, john@email.johncon.com, http://www.johncon.com/

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