Help interpreting my results

[Darcy Hill]

Hi,

I ran the following regression:


xtreg recycling loginc logpopden md11 md12 md13 md14 md15 md16 md17 md18 md19 md20 md21 md22 md23 md24 md25 md26 md27 md28 md29 md291 wasteavg dryavg quarter2 quarter3 quarter4 year2 year3 year4 year5, fe vce(cluster acode)

This gave me the results below. I am struggling to interpret them. 'recycling' is a percentage (0-100%) of waste recycled, and 'income' is gross annual disposable income. loginc is just the logarithm of 'income'.

Am I correct in thinking a 1% increase in yearly income will increase recycling rates by 6.53/100 units, which is 0.0653, and because 'recycling' is a percentage it is 0.0653%.

Fixed-effects (within) regression Number of obs = 5,862
Group variable: acode Number of groups = 311

R-sq: Obs per group:
within = 0.3639 min = 4
between = 0.0634 avg = 18.8
overall = 0.0941 max = 20

F(31,310) = 53.22
corr(u_i, Xb) = -0.6908 Prob > F = 0.0000

(Std. Err. adjusted for 311 clusters in acode)
------------------------------------------------------------------------------
| Robust
recycling | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
loginc | 6.532939 6.030714 1.08 0.280 -5.333371 18.39925
logpopden | -3.643093 2.239148 -1.63 0.105 -8.048943 .7627563
md11 | .3323071 .4808336 0.69 0.490 -.6138031 1.278417
md12 | -.6383587 .408626 -1.56 0.119 -1.44239 .1656725
md13 | -.9090128 .5404645 -1.68 0.094 -1.972456 .15443
md14 | -.1001455 .4725626 -0.21 0.832 -1.029981 .8296903
md15 | .5641031 .5839921 0.97 0.335 -.5849866 1.713193
md16 | .1721142 .7144448 0.24 0.810 -1.23366 1.577889
md17 | -.2672899 .3879316 -0.69 0.491 -1.030602 .4960221
md18 | -.0376456 .6234474 -0.06 0.952 -1.264369 1.189078
md19 | .0423547 .4558731 0.09 0.926 -.8546421 .9393515
md20 | -1.560784 .5148514 -3.03 0.003 -2.57383 -.5477388
md21 | -.8955305 .4952328 -1.81 0.072 -1.869973 .0789123
md22 | -1.264937 .5061731 -2.50 0.013 -2.260907 -.268968
md23 | .1594418 .5278072 0.30 0.763 -.8790958 1.197979
md24 | .3258512 .4136664 0.79 0.431 -.4880977 1.1398
md25 | .1241636 .9183427 0.14 0.893 -1.68281 1.931137
md26 | .5464737 .3863866 1.41 0.158 -.2137983 1.306746
md27 | 1.420912 .3756702 3.78 0.000 .6817256 2.160097
md28 | .381145 .2994553 1.27 0.204 -.208077 .970367
md29 | .0880614 .6923709 0.13 0.899 -1.274279 1.450402
md291 | .4493586 .3743921 1.20 0.231 -.2873125 1.18603
wasteavg | 1.43095 .5684451 2.52 0.012 .312451 2.549448
dryavg | -.6606502 .6478801 -1.02 0.309 -1.935449 .6141484
quarter2 | -4.473236 .121817 -36.72 0.000 -4.712929 -4.233544
quarter3 | -4.172665 .1183665 -35.25 0.000 -4.405568 -3.939761
quarter4 | -2.505768 .1010889 -24.79 0.000 -2.704675 -2.306861
year2 | .1246461 .2125739 0.59 0.558 -.2936241 .5429162
year3 | -.4222133 .3448504 -1.22 0.222 -1.100757 .2563301
year4 | -1.007958 .6575602 -1.53 0.126 -2.301804 .2858874
year5 | -1.431394 .692197 -2.07 0.039 -2.793393 -.0693957
_cons | -30.59105 60.24127 -0.51 0.612 -149.1246 87.94244
-------------+----------------------------------------------------------------
sigma_u | 5.7402679
sigma_e | 2.5635018
rho | .83372539 (fraction of variance due to u_i)
------------------------------------------------------------------------------

.
TIA!!

[Clyde Schechter]

Am I correct in thinking a 1% increase in yearly income will increase recycling rates by 6.53/100 units, which is 0.0653, and because 'recycling' is a percentage it is 0.0653%.

Almost yes to the first, and almost yes to the second.

Actually, the first is based on an approximation that is good for small values, and it is adequate, but not exact, here. The exact change in the value of the recycling variable associated with a 1% increase in income is 6.532939*log(1.01) which works out to 0.06500049--close to 0.0653, probably close enough for practical purposes.

Now, given that the recycling variable itself increases by 0.0653 and the units of the recycling variable are already percentages, this means that the difference is 0.0653 (or, really 0.06500049) percentage points, not percents.

[Darcy Hill]

Hi Clyde, I am still a little confused about percentage points vs percentage where what is being measured IS a percentage? So recycling isn't increasing by 0.065%, but the recycling rate does increase by 0.065 percentage points?

I understand that moving up from 40% to 44% is a 4 percentage point increase, but is a 10 percent increase in what is being measured.

[Clyde Schechter]

I understand that moving up from 40% to 44% is a 4 percentage point increase, but is a 10 percent increase in what is being measured.

That's precisely right. A percentage point increase is additive and a percent increase is multiplicative.

Same thing with regard to your recycling output. Linear regression models give additive effects in association with additive changes in the predictors. Now, when you have a log-transformed predictor, an additive change in that predictor corresponds to a multiplicative change in the untransformed predictor. So that completes the chain: a multiplicative change in income (a 1% increase = multiply by 1.01) corresponds to an additive change of log(1.01) in log income, and then log1.01 gets multiplied by the coefficient 6.532939 to calculate an additive effect on percent recycling. As already noted, log(1.01)*6.532939 = 0.06500049, so we add 0.06500049 to the original percent recycling: which, in words, means an increase of .06500049 percentage points.