FE-estimation: R-squared lower with variables in first differences

[Louisa Krekel]

Hey everyone,

I am using the FE-estimtor, first with variables in log-levels and then with variables in log-first-differences. Using first differences, the R-sqaured is much lower.


IN LEVELS:


xtreg logC logP logY logU lim com $t, fe vce (cluster Country)

Fixed-effects (within) regression Number of obs = 586
Group variable: Country Number of groups = 28

R-sq: within = 0.6739 Obs per group: min = 16
between = 0.0585 avg = 20.9
overall = 0.1881 max = 23

F(27,27) = 337.45
corr(u_i, Xb) = 0.0408 Prob > F = 0.0000

(Std. Err. adjusted for 28 clusters in Country)

Robust
logC Coef. Std. Err. t P>t [95% Conf. Interval]

logP -.1530841 .0696668 -2.20 0.037 -.2960285 -.0101396
logY -.0613126 .170431 -0.36 0.722 -.4110082 .288383
logU -.0097524 .0394944 -0.25 0.807 -.0907883 .0712835
lim .0328359 .0352908 0.93 0.360 -.0395749 .1052467
com .0037624 .0574163 0.07 0.948 -.1140461 .1215708
year1991 -.0019222 .022572 -0.09 0.933 -.048236 .0443917
year1992 -.0421362 .0270943 -1.56 0.132 -.0977291 .0134567
year1993 -.1038218 .0422374 -2.46 0.021 -.1904858 -.0171578
year1994 -.0851309 .0362736 -2.35 0.027 -.1595582 -.0107036
year1995 -.0837885 .0439258 -1.91 0.067 -.1739169 .0063399
year1996 -.0994367 .0469642 -2.12 0.044 -.1957993 -.0030742
year1997 -.098517 .0495209 -1.99 0.057 -.2001256 .0030915
year1998 -.1148296 .0556212 -2.06 0.049 -.2289549 -.0007043
year1999 -.1095873 .057305 -1.91 0.066 -.2271675 .0079928
year2000 -.1149373 .0663373 -1.73 0.095 -.2510502 .0211756
year2001 -.1358396 .0697202 -1.95 0.062 -.2788936 .0072143
year2002 -.1241133 .0761523 -1.63 0.115 -.2803648 .0321383
year2003 -.1544152 .0768221 -2.01 0.055 -.3120411 .0032107
year2004 -.1766785 .0791 -2.23 0.034 -.3389783 -.0143786
year2005 -.2164842 .0785883 -2.75 0.010 -.377734 -.0552343
year2006 -.2378107 .0824245 -2.89 0.008 -.4069319 -.0686895
year2007 -.2700316 .0786583 -3.43 0.002 -.4314251 -.1086381
year2008 -.2959255 .0772521 -3.83 0.001 -.4544336 -.1374174
year2009 -.3276065 .0801136 -4.09 0.000 -.4919861 -.1632269
year2010 -.349545 .0857596 -4.08 0.000 -.5255091 -.1735809
year2011 -.372167 .0868558 -4.28 0.000 -.5503803 -.1939536
year2012 -.4626692 .0929163 -4.98 0.000 -.6533178 -.2720206
_cons 8.47939 1.719101 4.93 0.000 4.952086 12.00669

sigma_u .36944696
sigma_e .10837265
rho .92077056 (fraction of variance due to u_i)



IN FRIST DIFFERENCES


xtreg dlogC dlogP dlogY dlogU lim com $t, fe vce(cluster Country)
note: year1991 omitted because of collinearity

Fixed-effects (within) regression Number of obs = 558
Group variable: Country Number of groups = 28

R-sq: within = 0.0552 Obs per group: min = 15
between = 0.0256 avg = 19.9
overall = 0.0468 max = 22

F(26,27) = 25.33
corr(u_i, Xb) = -0.1310 Prob > F = 0.0000

(Std. Err. adjusted for 28 clusters in Country)

Robust
dlogC Coef. Std. Err. t P>t [95% Conf. Interval]

dlogP .0202744 .0317973 0.64 0.529 -.0449682 .0855169
dlogY .2614144 .2350752 1.11 0.276 -.2209201 .743749
dlogU -.0266392 .0244428 -1.09 0.285 -.0767917 .0235133
lim .0042371 .010939 0.39 0.702 -.0182079 .0266821
com .0124518 .0164479 0.76 0.456 -.0212965 .0462002
year1991 0 (omitted)
year1992 -.0285764 .0133649 -2.14 0.042 -.0559989 -.0011538
year1993 -.0454495 .0162131 -2.80 0.009 -.0787161 -.0121829
year1994 .0032717 .0196694 0.17 0.869 -.0370866 .04363
year1995 -.0245817 .0174325 -1.41 0.170 -.0603502 .0111868
year1996 -.0244237 .0167583 -1.46 0.157 -.0588089 .0099616
year1997 -.0198456 .0142809 -1.39 0.176 -.0491475 .0094564
year1998 -.0214831 .016989 -1.26 0.217 -.0563417 .0133755
year1999 -.0075324 .0125627 -0.60 0.554 -.0333089 .0182441
year2000 -.0272667 .0247038 -1.10 0.279 -.0779547 .0234212
year2001 -.0331424 .0113053 -2.93 0.007 -.0563391 -.0099458
year2002 -.0004829 .0162197 -0.03 0.976 -.033763 .0327972
year2003 -.0267522 .013323 -2.01 0.055 -.0540888 .0005844
year2004 -.0371501 .0199465 -1.86 0.073 -.0780769 .0037767
year2005 -.0466669 .0160688 -2.90 0.007 -.0796374 -.0136964
year2006 -.0361674 .0177728 -2.03 0.052 -.0726342 .0002993
year2007 -.0400949 .0171534 -2.34 0.027 -.0752908 -.004899
year2008 -.0367722 .0208067 -1.77 0.088 -.079464 .0059196
year2009 -.0294165 .0303251 -0.97 0.341 -.0916384 .0328055
year2010 -.0377984 .0201746 -1.87 0.072 -.0791933 .0035966
year2011 -.0476827 .0337909 -1.41 0.170 -.1170159 .0216506
year2012 -.0748234 .0530716 -1.41 0.170 -.1837172 .0340704
_cons -.0063753 .009883 -0.65 0.524 -.0266535 .013903

sigma_u .01506935
sigma_e .0825537
rho .03224637 (fraction of variance due to u_i)




How come that the R-squared is that much lower in the first-difference euqation comparing to the levels equation?


Thanks a lot!

Louisa

[Maarten Buis]

You can only compare R-squares when you leave the dependent/left-hand-side/explained/y-variable unchanged. The R-square is the proportion of the variance in y explained by your model. By changing y (differencing in your case) you also changed the total variance, i.e. the denominator in R-square.

[Louisa Krekel]

Thanks for the answer, Maarten!

In this context, what does the lower R-squared tell me and how can it be interpreted? That only a small porportion of the variance in y can be explained?

[Phil Bromiley]

You often get high r-squares because everything is going up. The change model drops such trends. There was a literature on government budgeting that looked like it explained .95 of the variance, but when you differences the variables, it wasn't explaining much of the variance at all. So, ignoring general trends in the data, you have a low (but not surprising) level of explanation.