Help with interpreting a log-log fixed effect model

[Viktor Oeberg]

Hello!

Thank you to all who takes the time to answer the questions in this forum, you are doing a great job. This place is a fantastic source of information for newcomers like myself.

So, to my problem. I am currently working on a bachelor thesis and i have gotten my thoughts all tangled up.
I am looking at the effect the change in the marginal tax between 2 years have on individuals number och sick days. I believe the models are fine as they are although i might add or remove a couple of controls. I am trying to get the elasticities out of this with a log log model.

I do not truly understand the fundementals of this type of model.
I am supposed to measure the difference between year 1990 and 1991, yet i have a hard time understanding the output here.

For year 1990 the lnmrtax coefficient is reasonable considering that when marginal tax increases by 10% it is more expensive to work so the sick days increases with 1.5% at that case.
But what i don't understand is how to interpret the coefficient for year 1991. What is good to know here is that a huge tax reform took place in 1991 and that the general marginal tax rates decreased, especially for high income earners.

The FE model. Where inkap = capital income.

Code:
xtreg lnsickdays lnmrtax i.year high_age inkap, fe vce(robust)


Fixed-effects (within) regression Number of obs = 268,541
Group variable: bidnr Number of groups = 139,677

R-sq: Obs per group:
within = 0.0190 min = 1
between = 0.0201 avg = 1.9
overall = 0.0004 max = 2

F(4,139676) = 624.11
corr(u_i, Xb) = -0.0691 Prob > F = 0.0000

(Std. Err. adjusted for 139,677 clusters in bidnr)
------------------------------------------------------------------------------
| Robust
lnsickdays | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lnmrtax | .1533314 .0264629 5.79 0.000 .1014646 .2051983
|
year |
1991 | -.1905544 .0076254 -24.99 0.000 -.2055 -.1756087
|
high_age | .0424174 . 0252012 1.68 0.092 -.0069765 .0918113
inkap | -1.05e-08 1.50e-08 -0.70 0.483 -3.98e-08 1.88e-08
_cons | 1.218533 .1009946 12.07 0.000 1.020586 1.41648
-------------+----------------------------------------------------------------
sigma_u | 1.3147138
sigma_e | 1.1483978
rho | .56721627 (fraction of variance due to u_i)
------------------------------------------------------------------------------


So, to the real question here: Can i consider the coefficient for year1991 to be the value for lnmrtax in 1991 with the same controls? Or is it to be interpreted as lnmrtax + year1991?
As you probably see i struggle to comprehend the basics of this model for comparing the effect over 2 years.


I apologize in advance if this port does not fully meet the standars of Statalist. I have read the FAQ before posting and tried my best.
Any answer will be much appreciated and possibly save me from huge anxiety during the holidays.

I wish you all a great christmas and a happy new year.
Respecfully
Viktor Öberg

[Clyde Schechter]

I wonder whether this is really the model you want. You have not stated your statistical hypothesis in precise terms here, so it's hard to tell. Anyway, here's an explanation of what the coefficient of 1991.year means in this model--then you can decide if it's what you were looking for or something else.

In the year 1991, compared to a person in 1991 of the same age, inkap (whatever that is), and with the same marginal tax rate, the expected value of the log of the number of sick days taken is 0.19 lower (to 2 decimal places). In terms of the number of sick days itself, this corresponds to a reduction by a factor of about 82.7%, which can also be called a 17.3% reduction.

It is important to understand that in your model, the effect of tax rates on sick days is stipulated to be the same in 1990 and 1991. Your model simply contemplates that, for reasons having nothing to do with an individual's tax rates, the use of sick days changes in 1991. If your purpose is to study whether the effect of tax rates on sick days changes between 1990 and 1991, you need a different model that includes a year#tax rate interaction.

[Viktor Oeberg]

First of all thank you very much Clyde. This was exactly what i was looking for.

My tutor told me he wanted this model for this type of research. I definitely see your point regarding the model type and this is why i have had trouble figuring out how to use this to measure the "effect". It's not really the effect of the change in marginal tax that i am after here, it is actually just the change in sick days since that can very well be explained by the marginal tax change.

Considering how much that changed during this reform, this is just a small part of it all and a way to keep it very simple for myself in this project.

I merely want to be able to show the difference in sick days between theese two years (1990-91) and also show the difference (none at all really) between earlier years so i can argue that the reform implemented here is the reason behind the change. Most work in this area takes on this problem from the labor force hours way and i thought it would be interesting to do it from sick days instead.

But with this said, if i had not run into alot of trouble on the way hence experiencing a shortage of time here, i would definitely had specified the model in another way as you suggest.

Sidenote. Inkap stand for income from capital i.e investments in contrast to labor income in this data set.

[Viktor Oeberg]

I'm sorry, I see now that i misunderstood the point of your reply.

This means that my tutor actually is actually pointing me in the wrong direction when he says that the above model is what he wants me to use in this case.
How could i specify a model in STATA that shows the effect of the marginal tax change on number of sick days?

I'm getting more and more confused here.

[Clyde Schechter]

To do this in a convincing manner, you need to have a control group: some people who were not subject to the change in marginal tax but are otherwise similar to the people who were. Of course, tax rate changes tend to be applied across the entire population, so such a group may not actually exist. In that case, it will be very difficult to identify a causal effect from the data. It may be that you can use the same model, but focus on the coefficient of lnmartax. That would be the simplest approach. It is rather limited in persuasiveness, but it may be the best that can be done.

[Viktor Oeberg]

Hello again, and thanks for your reply.
I have tried to create a better fitted model for my purpose by trying to make use of your suggestions.
Would be an idea to create an interaction variable between i.year (representing 1990 as base and 1991) and ln marginal tax?

As to your previous suggestion i can't really form or understand how i can separate the value of marginal tax between the two years and only focus on that.
Here is what i tried to do now:

Code:

xtreg lnsickdays high_age inkap c.lnmrtax##i.year , fe vce(robust)

output:

Fixed-effects (within) regression               Number of obs     =    268,541
Group variable: bidnr                           Number of groups  =    139,677

R-sq:                                           Obs per group:
     within  = 0.0193                                         min =          1
     between = 0.0200                                         avg =        1.9
     overall = 0.0005                                         max =          2

                                                F(5,139676)       =     500.52
corr(u_i, Xb)  = -0.0649                        Prob > F          =     0.0000

                              (Std. Err. adjusted for 139,677 clusters in bidnr)
--------------------------------------------------------------------------------
               |               Robust
    lnsickdays |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
      high_age |   .0356315   .0252355     1.41   0.158    -.0138295    .0850926
         inkap |  -1.20e-08   1.50e-08    -0.80   0.424    -4.13e-08    1.74e-08
       lnmrtax |   .0709157   .0300719     2.36   0.018     .0119754    .1298559
               |
          year |
         1991  |   -.695305   .0851988    -8.16   0.000    -.8622929    -.528317
               |
year#c.lnmrtax |
         1991  |   .1361542   .0227663     5.98   0.000     .0915326    .1807758
               |
         _cons |   1.534099    .114955    13.35   0.000     1.308789    1.759408
---------------+----------------------------------------------------------------
       sigma_u |  1.3142964
       sigma_e |  1.1482552
           rho |  .56712132   (fraction of variance due to u_i)
--------------------------------------------------------------------------------

Is this interpretation correct?

lnmrtax: The % impact on lnsickdays that a % change in lnmrtax has for year 1990.

year1991: As you helped me with earlier, for reasons having nothing to do with a change in tax, the use of sickdays still goes down.

year#c.lnmrtax: the % effect on sickdays a % change in tax has for year 1991. So the effect in 1991 is 0.071 (to 3 decimal places) + .136 (to 3 decimal places).

What i really am after is how big of a change happend in the use of sickdays between year 1990 and 1991 as a cause of the large tax reduction.


I am forever grateful for any response or help i can get in this matter.

Viktor Öberg

[Clyde Schechter]

I don't think having an interaction between marginal tax and year helps.

As for the interpretation of the results you show, it is not what you have said.

The coefficient of lnmartax represents the marginal effect of lnmartax on lnsickdays (which is also known as the elasticity of sick days with respect to marginal tax) in the year 1990 only. In that year, if the marginal tax rate increase by 1% (i.e. if it goes from 20% to 1.01*20% = 20.2%--NB not 21%), then lnmartax increases by ln(1.01) = 0.00995. So the expected value of lnsickdays increase by 0.0709157*0.00995 = 0.0070563, so sickdays themselves increase by a factor of exp(0.0070563) = 1.0007, so there is a 0.07% increase in sick days.

By contrast, in 1991, the marginal effect of lnmartax on lnsickdays is 0.0709157 + 0.1361542 = 0.2070699. Repeating the same calculations as above, this means that a 1% increase in lnmartax is associated with a 0.21% increase in sick days in 1991.

The -.695305 coefficient for 1991.year itself reflects a downward shift in sick days between 1990 and 1991 among those for whom lnmartax = 0 (i.e. martax itself = 1).

Unfortunately, none of this answers your primary question about the causal effect of the shift in marginal tax rates, because you have no control group. Your results suggest that the relationship between marginal tax rates and sick days changed between 1990 and 1991. But that is a different question entirely. To answer your causal question your data must include people for whom marginal tax rates did not change between 1990 and 1991. Since tax rate changes are generally applied across entire jurisdictions and not selectively to groups of people within them, this probably means gathering data on other jurisdictions that were reasonably comparable to the one(s) from which your current data have been gathered.