Sign on coefficient changes between pooled OLS and FE regression?

[Danielle Watson]

Hi there,
I have panel data on 100+ countries between 1984-2011. I am running a very simple static model where log(GDPpc)= log(capital stockpc), human capital, Corruption variable.

When running a pooled OLS model I get the expected sign on my variable of interest (corruption). However, when running a model with fixed effects the coefficient switches signs but remains significant.

I'd be really grateful if anyone could tell me whether this is 'possible' or if it is more likely that there is something going on that I have not accounted for.

Thanks in advance!

[Alfonso Sánchez-Peñalver]

Well, let's see. The use of fixed effects is justified when unobserved effects are correlated with the explanatory variables. In this case pooled OLS would produce biased estimates of the coefficients, whereas fixed effects would produce consistent estimates of the coefficients. Therefore, it is possible that you get different signs on the coefficients between the two estimators, because biased estimates can have a different sign.

[Danielle Watson]

Thanks so much for your reply! I think my worry was that the sign on the coefficient in the FE regression not only changed, but is also 'wrong' in terms of what my theory would predict.

[Rich Goldstein]

here are a couple of citations that may help:

Kennedy, PE (2005), "Oh No! I got the wrong sign! What should I do?", _The Journal of Economic Education_, 36(1): 77-90

Schuit, E, et al. (2013), "Unexpected predictor-outcome associations in clinical prediction research: causes and solutions," _Canadian Medical Association Journal_, 185(10): E499-E505

[Clyde Schechter]

Consider the data underlying the graph above:

id y x
1 4 1
1 3 2
1 2 3
2 7 4
2 6 5
2 5 6
3 10 7
3 9 8
3 8 9
4 13 10
4 12 11
4 11 12

You can see that the within-id slope of y vs x is negative, and yet the overall association of y with x is positive.

And if you do OLS vs FE you see the signs change:

 . regress y x
        Source |       SS       df       MS              Number of obs =      12
-------------+------------------------------           F(  1,    10) =   37.34
       Model |   112.79021     1   112.79021           Prob > F      =  0.0001
    Residual |  30.2097902    10  3.02097902           R-squared     =  0.7887
-------------+------------------------------           Adj R-squared =  0.7676
       Total |         143    11          13           Root MSE      =  1.7381
  ------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   .8881119   .1453469     6.11   0.000     .5642588    1.211965
       _cons |   1.727273   1.069724     1.61   0.137    -.6562217    4.110767
------------------------------------------------------------------------------
  . xtset id
       panel variable:  id (balanced)
  . xtreg y x, fe
  Fixed-effects (within) regression               Number of obs      =        12
Group variable: id                              Number of groups   =         4
  R-sq:  within  = 1.0000                         Obs per group: min =         3
       between = 1.0000                                        avg =       3.0
       overall = 0.7887                                        max =         3
                                                  F(1,7)             =         .
corr(u_i, Xb)  = -0.9716                        Prob > F           =         .
  ------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |         -1          .        .       .            .           .
       _cons |         14          .        .       .            .           .
-------------+----------------------------------------------------------------
     sigma_u |  7.7459667
     sigma_e |          0
         rho |          1   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0:     F(3, 7) =        .                Prob > F =      .
  .

This is an example of a Simpson's paradox.

[Maggio Marco]

Sorry I posted this in the wrong place

[Matthew Shelley]

[ATTACH=CONFIG]n113831[/ATTACH]

Consider the data underlying the graph above:

id y x
1 4 1
1 3 2
1 2 3
2 7 4
2 6 5
2 5 6
3 10 7
3 9 8
3 8 9
4 13 10
4 12 11
4 11 12

You can see that the within-id slope of y vs x is negative, and yet the overall association of y with x is positive.

And if you do OLS vs FE you see the signs change:

 . regress y x
Source | SS df MS Number of obs = 12
-------------+------------------------------ F( 1, 10) = 37.34
Model | 112.79021 1 112.79021 Prob > F = 0.0001
Residual | 30.2097902 10 3.02097902 R-squared = 0.7887
-------------+------------------------------ Adj R-squared = 0.7676
Total | 143 11 13 Root MSE = 1.7381
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x | .8881119 .1453469 6.11 0.000 .5642588 1.211965
_cons | 1.727273 1.069724 1.61 0.137 -.6562217 4.110767
------------------------------------------------------------------------------
. xtset id
panel variable: id (balanced)
. xtreg y x, fe
Fixed-effects (within) regression Number of obs = 12
Group variable: id Number of groups = 4
R-sq: within = 1.0000 Obs per group: min = 3
between = 1.0000 avg = 3.0
overall = 0.7887 max = 3
F(1,7) = .
corr(u_i, Xb) = -0.9716 Prob > F = .
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x | -1 . . . . .
_cons | 14 . . . . .
-------------+----------------------------------------------------------------
sigma_u | 7.7459667
sigma_e | 0
rho | 1 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(3, 7) = . Prob > F = .
.

This is an example of a Simpson's paradox.

Hi Clyde, I'm experiencing this issue in my fixed effects and OLS regressions looking at the effects of trust on GDP per capita. I have a situation which you described where countries that have high GDP per capita typically have high levels of trust, but trust is decreasing across countries, whilst countries are still growing. Is there anyway to overcome this paradox?

[Clyde Schechter]

Is there anyway to overcome this paradox?

No, it cannot be overcome. In a situation where you have substantial differences between within-country and between-country effects, you need to either state your research goals in a way that makes it crystal clear which of these effects you are trying to estimate, or you need to estimate both effects, and present and discuss both of them separately.