Why my correlation is negative when the estimated coefficients are positive?

[Guest]

Dear Statalist Community

This should be my very last questions for statalist community. Thank you every experts who have helped me this far. I appreciate it a lot.

I am trying to explain the reason why the correlation between my dependent variable (government expenditure as a % of GDP) and independent variable (immigration) is negatively correlated, while the estimated coefficients are positive (in all cases when I include or exclude each control variables.)

1. I try to investigate by using

extremes  govex_gdp immi

and found no outliers. I wonder whether there are other ways in which I can find the reason for this?
2. I also experiment with another regression that use log on all of my variables. I would like to know whether I can interpret that in terms of one percentage change in standard deviation? eg. one standard deviation in independent variable contributes to ___% of one standard deviation increase increase in dependent variable.

. extremes  govex_gdp immi

+------------------------+
| obs: govex_~p immi |
|------------------------|
| 901. 10.20876 . |
| 586. 10.2687 . |
| 585. 10.3964 . |
| 902. 10.46668 . |
| 722. 10.53066 . |
+------------------------+

+-------------------------+
| 879. 27.4907 61872 |
| 871. 27.63227 32272 |
| 369. 27.68548 . |
| 182. 27.69099 28223 |
| 205. 27.93502 51800 |
+-------------------------+

regression without log

 xtreg  govex_gdp immi  depratio unem_perlab urbanpop_pertot pop femalepop_pertot i.year ,fe cluster (country)

Fixed-effects (within) regression Number of obs = 657
Group variable: country Number of groups = 33

R-sq: within = 0.2989 Obs per group: min = 2
between = 0.0395 avg = 19.9
overall = 0.0627 max = 29

F(32,32) = .
corr(u_i, Xb) = -0.7913 Prob > F = .

(Std. Err. adjusted for 33 clusters in country)
----------------------------------------------------------------------------------
| Robust
govex_gdp | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-----------------+----------------------------------------------------------------
immi | 1.14e-06 5.01e-07 2.28 0.029 1.24e-07 2.16e-06
depratio | .0329021 .0603795 0.54 0.590 -.0900868 .1558911
unem_perlab | .1200443 .0409341 2.93 0.006 .0366643 .2034244
urbanpop_pertot | .2526613 .0645373 3.91 0.000 .1212032 .3841194
pop | 7.90e-08 1.57e-07 0.50 0.618 -2.40e-07 3.98e-07
femalepop_pertot | -.9128621 1.047256 -0.87 0.390 -3.046053 1.220329
|
year |
1982 | -.0275117 .1441005 -0.19 0.850 -.3210348 .2660114
1983 | .0287356 .3412937 0.08 0.933 -.666457 .7239282
1984 | -.672402 .4074508 -1.65 0.109 -1.502352 .1575481
1989 | -1.164444 .5968829 -1.95 0.060 -2.380255 .0513666
1990 | -.7825131 .6641094 -1.18 0.247 -2.13526 .5702335
1991 | -.3335672 .7901333 -0.42 0.676 -1.943016 1.275882
1992 | -.4937047 .800642 -0.62 0.542 -2.124559 1.13715
1993 | -.1212492 .7635012 -0.16 0.875 -1.67645 1.433952
1994 | -.8561286 .7242583 -1.18 0.246 -2.331394 .6191373
1995 | -.5426904 .8170002 -0.66 0.511 -2.206865 1.121485
1996 | -.8923355 .8015554 -1.11 0.274 -2.52505 .7403794
1997 | -1.156174 .8138863 -1.42 0.165 -2.814007 .5016577
1998 | -1.249869 .8070755 -1.55 0.131 -2.893828 .3940899
1999 | -1.177472 .8301466 -1.42 0.166 -2.868425 .5134814
2000 | -1.617467 .8595532 -1.88 0.069 -3.36832 .1333854
2001 | -1.411849 .8676217 -1.63 0.113 -3.179137 .3554382
2002 | -1.143768 .8905438 -1.28 0.208 -2.957747 .6702102
2003 | -.8068445 .9449567 -0.85 0.400 -2.731658 1.117969
2004 | -1.323295 .8930714 -1.48 0.148 -3.142422 .4958319
2005 | -1.32398 .9220267 -1.44 0.161 -3.202087 .5541268
2006 | -1.543219 .9286263 -1.66 0.106 -3.434768 .3483313
2007 | -1.860672 .9506458 -1.96 0.059 -3.797074 .0757304
2008 | -1.130692 .9602998 -1.18 0.248 -3.086759 .8253748
2009 | .1310738 .9863769 0.13 0.895 -1.87811 2.140258
2010 | -.3822114 .9845546 -0.39 0.700 -2.387684 1.623261
2011 | -1.07942 1.007688 -1.07 0.292 -3.132013 .9731725
2012 | -1.391096 1.023083 -1.36 0.183 -3.475048 .6928561
2013 | -1.530977 1.035502 -1.48 0.149 -3.640225 .5782712
|
_cons | 44.80689 54.43654 0.82 0.417 -66.0767 155.6905
-----------------+----------------------------------------------------------------
sigma_u | 4.7367037
sigma_e | 1.38224
rho | .92152662 (fraction of variance due to u_i)
----------------------------------------------------------------------------------

regression with log



Thank you
Guest

[Guest]

Do anyone know by any chances?

[Clyde Schechter]

It is simply not true that all your coefficients are positive. Look at femalepop_pertot. That one's negative. Whether the overall correlation between fitted and observed values will be positive or negative in this situation depends on the distributions of the different predictors. It could come out either way and there is no simple way to predict which way it will be.

[Guest]

Dear Clyde

Truly sorry for the misunderstanding. I mean the coefficient of immi is always positive regardless of whether the regressions have control variables or not.
This contradicts the simple scatter plot of the correlation between govex_gdp and immi.

Thank you
Guest

[Clyde Schechter]

OK, that's a different question from what I had understood. Nonetheless, it is still your expectation that is off, and there is nothing wrong with the analyses, nor anything surprising in the results. It is, in fact, quite common that the relationship between variables X and Y will be different when examined in isolation (the correlation between X and Y) and when adjusted for covariates ("control variables"--I hate that term, because except in experiments, you don't actually "control" anything--you just adjust for their effects). Indeed, it is not uncommon for the relationship even to change sign, as it has here. This is a very general phenomenon known as Simpson's paradox (or also called the Yule-Simpson paradox.) Wikipedia has an excellent page on Simpson's paradox and I recommend you read it. The explanation there relates only to discrete variables, but exactly the same phenomenon applies with continuous variables. In the context of continuous variables it is sometimes referred to as Lord's paradox.

So do read that Wikipedia page and take it to heart.