# December 2015: Regression between reputation rank and log reputation on Econ.SE

Just for kicks, I decided to run a regression on "log reputation" on "all time ranking of Econ.SE". I'm saying reputation affects ranking, and hopefully not the other way around, because I don't have any good instrumental variables I can really use (honestly though, there's probably some endogeneity problem here).

I just recorded the current top 400 all time users on this site (they all ended up being being bunched up towards the end, but all over 101 reputation) and then I played around with the observations.

. regress ranking reputation

Source |       SS       df       MS              Number of obs =     400
-------------+------------------------------           F(  1,   398) =   86.55
Model |  952649.784     1  952649.784           Prob > F      =  0.0000
Residual |  4380650.22   398  11006.6588           R-squared     =  0.1786
Total |     5333300   399  13366.6667           Root MSE      =  104.91

------------------------------------------------------------------------------
ranking |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
reputation |  -.0557962   .0059974    -9.30   0.000    -.0675868   -.0440056
_cons |   221.2032   5.698142    38.82   0.000      210.001    232.4054
------------------------------------------------------------------------------


Without the log transformation of reputation, the regression is pretty bad, so we try this:

. regress ranking ln_reputation

Source |       SS       df       MS              Number of obs =     400
-------------+------------------------------           F(  1,   398) =  798.27
Model |  3558904.05     1  3558904.05           Prob > F      =  0.0000
Residual |  1774395.95   398  4458.28128           R-squared     =  0.6673
Total |     5333300   399  13366.6667           Root MSE      =   66.77

------------------------------------------------------------------------------
ranking |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ln_reputat~n |   -117.024   4.141907   -28.25   0.000    -125.1667   -108.8812
_cons |   825.5056   22.37175    36.90   0.000     781.5241    869.4872
------------------------------------------------------------------------------


The $R^2 = 0.67$ here is much better. But when you take out the crazy clustered observations at the lower reputation end and the really volatile/outliers at the high end of reputation (so we regress only on observations 15 to 200, dummy variable did not work well here).

. regress ranking ln_reputation in 15/200

Source |       SS       df       MS              Number of obs =     186
-------------+------------------------------           F(  1,   184) = 1908.24
Model |  489064.928     1  489064.928           Prob > F      =  0.0000
Residual |  47157.5718   184  256.291151           R-squared     =  0.9121
Total |    536222.5   185      2898.5           Root MSE      =  16.009

------------------------------------------------------------------------------
ranking |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ln_reputat~n |  -80.80816    1.84986   -43.68   0.000    -84.45782    -77.1585
_cons |   566.7525   10.57853    53.58   0.000     545.8817    587.6233
------------------------------------------------------------------------------


The relationship between log reputation and all time ranking is very strongly linear. $R^2 = 0.91$ (Keeping in the top 15 users took away ~10% of explanatory power of the model, $R^2 \approx 0.8$)

What do you guys and gals conclude from this (if there are any interesting results)? I figure the obvious results are that

• there are diminishing returns to getting enough reputation to go up one rank
• intrinsic motivation probably explains more of the variance in the regression the higher up the ranks you go

One of the major things I suspect based on the data are that, on average the first mover on answers gets the most reputation for a given question (especially if it is a highly viewed one), because the public finds large diminishing returns on additional information. It's kind of as though the market for reputation is analogous to Cournot "competition" (heavy on the quotes).

A few other questions:

Is there a way to get data on how much time each user spends online on this site? I think a fun extension of this model would be to see if we could determine a sort of shadow price for reputation, where time is the price. Would we need other information to take on a task like that though (I'm a big newb at econometrics)?

How would you hypothesize/characterize this website's "market activity" for reputation?

(SN: I can give the dataset if you want, ahaha but who would want to peer review this garbageee)

• It would be a good idea to provide a link to the data. I have some ideas. – Alecos Papadopoulos Dec 20 '15 at 19:51
• I only give data if I can be credited as a co-author ^>^...just kidding. I copied the data onto a google spreadsheet: docs.google.com/spreadsheets/d/… – Kitsune Cavalry Dec 21 '15 at 5:22
• Wow, this totally changes the way I look at bounties. – user4207 Dec 29 '15 at 6:30