How to evaluate time invariant independent variables in a panel data regression?

[Oscar Fernholm]

Hi!

My and my thesis partner have encountered some problem with our panel data regression and would really appreciate some help / guidance.

We are evaluating a property tax reform and have downloaded panel data for the entire population affected by the reform, before and after the reform.
Our problem is to find a proper regression model.

Our independent variables that we want to evaluate are income-level(discrete variable from 1-26), geographical region(discrete variable from 1-21), age-groups(discrete variable from 1-3) and gender(dummy variable).
For example, we want to evaluate whether the income-level had an significant impact on the size of the change tax payments (which is the dependent variable) due to the reform.
The problem is that our independent variables of interest all are time invariant. This makes the fixed effects model not applicable since there are no variance within the independent variables of interest. Neither the random effects model seems like a good way to go since our data not is a sample but the entire population of interest.

One solution might be to use interaction terms between the independent variables of interest and a dummy for the time (which is 0 for the time before the reform and 1 after the reform).

xtreg y t*i.x1 t*i.x2 t*i.x3 t*i.x4, fe

This works in Stata but it don't seem like a good model since it still is the fixed effect model, which isn't really applicable to time invariant variables.

If someone can help us out with this problem it would help us a lot.

Thanks!

[Phil Bromiley]

You'll increase your chances of a useful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex.
Your code is clearly not usable Stata code - you need # not *, and you probably need to include the main effects. Once you include the main effects, your problem with fixed effects returns. One obvious approach would be a random effects model. There are also xthtaylor and the Mundlak approaches. However, with all your iv's time invariant, I'm not sure they work for you.

When you say "discrete variables", are you planning to estimate them all as individual dummy variables? That gives you a lot of parameters - I hope you have a large data set. It is also sometimes hard to interpret when you have that many dummy variable parameter estimates.

The population vs sample issue is a bit tricky - I'm not sure it really creates a preference for fixed effects. Almost all our statistical techniques assume a sample from a population - folks routinely use sample-justified tools with population data. Others on the listserve may be better qualified to help you - a better posed question may induce them to step in.

[Oscar Fernholm]

Thanks for your reply Phil!
Yes, I realise that my first is really not so clear.

I'll try to give some more insights about my data here below.
It is a short panel data set (T=17, N=3 276).
A total of 55 692 observations.

Here is a small sample of the variables of interest, plus the time and id variable at the end (ar & id).
Sorry about all the decimals in i natural logarithm variable, which is our dependent variable. It is both specific for T and N.
"inkomstbasbelopp" is a control variable which is time specific (a measure of inflation).
The rest of the independent variables are time-invariant.

lnFSAmedelpersskatt inkomstgrupp lan alder kon inkomstbasbelopp ar id
 7.727474212646484 1 1 1 0 37300  1 1
 7.683006286621094 1 1 1 0 37700  2 1
 7.947241306304932 1 1 1 0 38800  3 1
 7.874024391174316 1 1 1 0 40900  4 1
 7.936743259429932 1 1 1 0 42300  5 1
 7.981675148010254 1 1 1 0 43300  6 1
 7.735753536224365 1 1 1 0 44500  7 1
 7.702230930328369 1 1 1 0 45900  8 1
 7.240899562835693 1 1 1 0 48000  9 1
 7.553473949432373 1 1 1 0 50900 10 1
 7.581099987030029 1 1 1 0 51100 11 1
 7.581099987030029 1 1 1 0 52100 12 1
 7.552112102508545 1 1 1 0 54600 13 1
 7.695213317871094 1 1 1 0 56600 14 1
 7.627931118011475 1 1 1 0 56900 15 1
 7.706263065338135 1 1 1 0 58100 16 1
 8.087035179138184 1 1 1 0 59300 17 1
 7.740164756774902 2 1 1 0 37300  1 2
 7.875339508056641 2 1 1 0 37700  2 2
 8.166536331176758 2 1 1 0 38800  3 2
 7.902007579803467 2 1 1 0 40900  4 2
 8.189689636230469 2 1 1 0 42300  5 2
 8.134201049804688 2 1 1 0 43300  6 2
 7.836624622344971 2 1 1 0 44500  7 2
  8.21708869934082 2 1 1 0 45900  8 2
 7.250245571136475 2 1 1 0 48000  9 2
7.2788190841674805 2 1 1 0 50900 10 2
 7.236259460449219 2 1 1 0 51100 11 2
 7.684284210205078 2 1 1 0 52100 12 2
 7.662777900695801 2 1 1 0 54600 13 2
7.7752556800842285 2 1 1 0 56600 14 2
 7.929406642913818 2 1 1 0 56900 15 2
 7.929406642913818 2 1 1 0 58100 16 2
                 . 2 1 1 0 59300 17 2

The research question is, as mentioned in #1, to evaluate whether lnFSAmedelpersskatt have changed different after the reform (starting in time period 9) depending on the time-invariant variables.

Our plan is to evaluate the time-invariant variable (especially "inkomstgrupp") not as a dummy for each level of income but rather the pattern of how lnFSAmedelpersskatt changes depending on the the level of income.

I insert stata output for a regression with the random effects model.

. xtreg lnFSAmedelpersskatt inkomstgrupp lan alder kon inkomstbasbelopp, re

Random-effects GLS regression                   Number of obs     =     36,160
Group variable: id                              Number of groups  =      2,241

R-sq:                                           Obs per group:
     within  = 0.0707                                         min =          1
     between = 0.5627                                         avg =       16.1
     overall = 0.4880                                         max =         17

                                                Wald chi2(5)      =    5482.23
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

----------------------------------------------------------------------------------
lnFSAmedelpers~t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
    inkomstgrupp |   .0313853   .0007834    40.06   0.000     .0298498    .0329209
             lan |  -.0296706   .0009735   -30.48   0.000    -.0315787   -.0277626
           alder |   .1864012   .0108257    17.22   0.000     .1651833    .2076192
             kon |  -.0366704   .0118168    -3.10   0.002     -.059831   -.0135099
inkomstbasbelopp |  -7.72e-06   1.51e-07   -51.02   0.000    -8.02e-06   -7.42e-06
           _cons |   8.157062   .0312267   261.22   0.000     8.095859    8.218265
-----------------+----------------------------------------------------------------
         sigma_u |  .27454049
         sigma_e |  .20709423
             rho |  .63734281   (fraction of variance due to u_i)
----------------------------------------------------------------------------------

However, according to our economic intuition we think that the unobserved component is correlated with our regressors, which makes the fixed effect more proper to use.

Would it be plausible to use a fixed effect model interacting the time-invariant variables with a time dummy variable which is 0 for the years before the reform and then 1 the years after the reform?

Again, if anyone has some suggestion of what model to use or some other help we are very thankful.

Thanks!