Panel : trend mixed with one time dummy vs full set of time dummy

[Jerome Archambault]

Hi,

I'm currently estimating the following model y_it = x_it B +O_t+e_it

with T=7

The parameter of the time trend is of interest to me.

In my first estimation I decomposed the time effect into time dummies, every dummies is significant but 2008 and 2011 are much higher thus the dummy do not completely follow a trend.

In the second estimation I used a time trend, and the B are now far from my first estimation.

I know that there is in fact a similar shock during the years 2008 and 2011.

In my third estimation I used a time trend and one dummy for 2008 or 2011. Now my B are very close to the first one, and the dummy is significant and have the expected sign.

Is my third estimation alright? and could I use an Hausman test between the first and third regression to assess if the difference between B is significant? What should I do since I use clusters for SE and Hausman do not allow it?

Thanks!

Jerome

[Carlo Lazzaro]

Jerome:
as per FAQ, please post what you typed and what Stata gave you back. Thanks.

[Jerome Archambault]

. xtreg y x1 x2 i.years , fe cluster(indid)

Fixed-effects (within) regression Number of obs = 956445
Group variable: indid Number of groups = 136635

R-sq: within = 0.0215 Obs per group: min = 7
between = 0.0054 avg = 7.0
overall = 0.0020 max = 7

F(8,136634) = 2063.61
corr(u_i, Xb) = -0.3289 Prob > F = 0.0000

(Std. Err. adjusted for 136635 clusters in indid)
------------------------------------------------------------------------------
| Robust
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .0040815 .0000851 47.97 0.000 .0039147 .0042483
x2 | -.000029 5.74e-07 -50.47 0.000 -.0000301 -.0000278
|
years |
2008 | .109485 .0014934 73.31 0.000 .1065579 .112412
2009 | .0762434 .0015167 50.27 0.000 .0732708 .0792161
2010 | .0613979 .0015669 39.18 0.000 .0583268 .0644689
2011 | .2647616 .002558 103.50 0.000 .259748 .2697752
2012 | .1871743 .0026023 71.93 0.000 .1820739 .1922748
2013 | .2361567 .0027295 86.52 0.000 .2308069 .2415064
|
_cons | .1363798 .0031749 42.96 0.000 .1301569 .1426026
-------------+----------------------------------------------------------------
sigma_u | .2445819
sigma_e | .42651714
rho | .24746009 (fraction of variance due to u_i)
------------------------------------------------------------------------------

. xtreg y x1 x2 trend, fe cluster(indid)

Fixed-effects (within) regression Number of obs = 956445
Group variable: indid Number of groups = 136635

R-sq: within = 0.0087 Obs per group: min = 7
between = 0.0058 avg = 7.0
overall = 0.0035 max = 7

F(3,136634) = 1503.64
corr(u_i, Xb) = -0.0605 Prob > F = 0.0000

(Std. Err. adjusted for 136635 clusters in indid)
------------------------------------------------------------------------------
| Robust
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .0014556 .0000731 19.92 0.000 .0013123 .0015988
x2 | -8.91e-06 4.66e-07 -19.11 0.000 -9.82e-06 -8.00e-06
trend | .0233577 .0004121 56.68 0.000 .02255 .0241654
_cons | .175378 .0033544 52.28 0.000 .1688034 .1819525
-------------+----------------------------------------------------------------
sigma_u | .22624306
sigma_e | .42929863
rho | .21736546 (fraction of variance due to u_i)
------------------------------------------------------------------------------

. xtreg y x1 x2 trend et ey, fe cluster(indid)

Fixed-effects (within) regression Number of obs = 956445
Group variable: indid Number of groups = 136635

R-sq: within = 0.0192 Obs per group: min = 7
between = 0.0054 avg = 7.0
overall = 0.0026 max = 7

F(5,136634) = 2897.09
corr(u_i, Xb) = -0.2595 Prob > F = 0.0000

(Std. Err. adjusted for 136635 clusters in indid)
------------------------------------------------------------------------------
| Robust
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .003388 .0000794 42.68 0.000 .0032324 .0035435
x2 | -.0000237 5.25e-07 -45.02 0.000 -.0000247 -.0000226
trend | .0346596 .0004731 73.27 0.000 .0337324 .0355867
et | .1033682 .0011771 87.82 0.000 .1010612 .1056753
ey | .0159174 .0018851 8.44 0.000 .0122226 .0196122
_cons | .0967299 .0034172 28.31 0.000 .0900324 .1034275
-------------+----------------------------------------------------------------
sigma_u | .2380697
sigma_e | .42701454
rho | .23712457 (fraction of variance due to u_i)
------------------------------------------------------------------------------

. Actually i have two dummies, et (2008 - 2011) and ey (2012-2013).

Using model 2 alone would be good, but I don't think it is a good specification because the B change too much compare to 1. Now model 3 is much closer but still is probably significantly different. But I only care about the trend and having the beta much nearer to model 1 seems to me to be more robust... (also I don't really care about the specific coefficient of the trend between 2 and 3, so I am not pushing for 3)

Thanks

[Carlo Lazzaro]

Jerome:
some remarks about your model:
- all the Rs are very low; are you sure that -fe- is the way to go?
- models 1 and 2 are different because the predictors are different. Hence, I would base any comparison between them on the literature in your research field;
- the previous remark holds for your last statement, too: you need to provide your audience with some theoretical undepinning to justify which model is "more robust" (and, by the way, "more robust" to what?)

[Jerome Archambault]

Hi, thanks for the answer,

R-square are very low because of the kind of dependent variable which is mainly a binary variable with a lot of 0. Unfortunately I cannot use random effect in a nonlinear model because of the bias due to incidental parameter and there is correlation between the unobserved heterogeneity. Therefore the linear probability model is my best bet but r-square have little chance to be very indicative.

What I mean by being more robust is that using time fixed effect place no restriction on the relationship between the time effects and the x_it, this is why time fixed effects are usually preferred to trend or others. Thus if I used a trend and dummies consequent with the theory and I find similar B, it tells me that I'm probably modeling the time variable correctly, or at least the part that is correlated with the x_it no?

In any case I wonder if from a statistical point of view using a time trend and other time specific dummy variable is a blunder if we stay far from colinearty. It is probably a stupid question because I have looked a lot to find an answer with no success ...

Thanks a lot!

Jerome

[Carlo Lazzaro]

Jerome:
Thanks for providing more details.
I'm not very familiar with linear probability model; however, I fail to get whether you are intended to plug both trend and time dummies as predictors or else. Personally, I would chose one of those two alternatives.

[Jerome Archambault]

Well imagine there is a trend but a shock in two years. Using only the trend might be wrong because of the shock, and if you expect the shock to be correlated with the x_it indeed time fixed effect would seem better, but the relevant parameter to evaluate is the trend.
Earlier I made a mistake, just correcting it, i meant:

'Unfortunately I cannot use fixed effect in a nonlinear model because of the bias due to incidental parameter and there is correlation between the unobserved heterogeneity so random effect would be wrong. '

[Carlo Lazzaro]

Jerome:
thanks for providing (much) more details.
If the literature in your research field vouches that approach (i.e., trend + year dummies), well, follow that road.