st: Use of Fixed Effects with State and National Data

[Samuel Finkelstein <s.finkelstein73@gmail.com>]

Hi everyone,

I am in the process of working with a data set that includes both
yearly state-level independent variables as well as yearly
national-level independent variables, while the dependent variable is
a state-specific variable.  I have received mixed feedback regarding
whether or not the use of yearly fixed effects is appropriate.  The
model I have been estimating is:

State-specific demand i,t  = state-specific control variables i,t +
national-level control variable t + e

where the national-level control variable is the rate on various
T-bills and bonds, ranging from 3 months to 20 years, or inflation.
Each model I estimate includes 5 state-level variables and only one
national-level (interest rate or inflation) variable.  The state-level
and national-level variables all vary by time, but the state-level
variables also vary by state.

In each model, I have included state-specific fixed effects.  However,
I am trying to determine whether or not the inclusion of a time
(yearly) fixed effect is also appropriate.  When I estimate my models
using yearly fixed-effects, the coefficients on the state-level
variables do not change across the different models and the r-squared
does not change across models.  When I estimate my models when
excluding yearly fixed effects, the r-squared varies across the models
and the coefficients on the state-level variables also vary across the
different models.  I am inclined to go with that particular set of
models (excluding the yearly fixed effects), but I have been warned by
some that by excluding yearly fixed effects, the national-level
interest rate variable is picking up variation that should be picked
up by time fixed effects.  In other words, the national-level
variables are replacing the excluded year fixed effects and I cannot
then make any valuable interpretation of the national-level variable
(i.e., how do interest rates relate to demand).  Below are my results
(all non-interest variables vary by year and state, interest variables
only vary by year and not by state since they are national variables):

3 month interest rate with year and state FE
                         coeff.         std. error
3monthinterest|   .0937955   .0299048
unemploy |   .0003255   .0233223
income |  -3.74e-06   .0000128
age |   .0292728   .0448224
newbuy |   .4022719   .3591874
homes|   .0254584   .0107563
r2 = 0.6689

20 year interest rate with year and state FE
                        coeff.        std. error
20yrinterest|   .1819766   .0580196
unemploy |   .0003255   .0233223
income |  -3.74e-06   .0000128
age |   .0292728   .0448224
newbuy |   .4022719   .3591874
homes |   .0254584   .0107563
r2 = 0.6689

Below are the results when including state fixed effects without year
fixed effects:

3 month interest rate with state FE only
                              coeff.      std. error
3monthinterest |    .141143    .015493
unemploy |   .0073605   .0220101
income |  -.0000461   8.91e-06
age |   .0342493   .0515669
newbuy |   1.801027   .4193282
homes|   .0244424   .0118203
r2 = 0.5306

20 year interest rate with state FE only
                        coeff.         std. error
20yrinterest|   .2815332   .0498411
unemploy |  -.0621116   .0218478
income|  -.0000226    .000013
age |   .0785964   .0519008
newbuy|   .9568831   .3819117
homes|   .0156787   .0123501
r2 = 0.4906

For what it is worth, I have also re-estimated the models excluding
the year fixed effects and including additional national-level
variables (such as annualized market return) along with the interest
rate variable and the result on the interest rate variable is similar
to the version without the annualized market return variable, while
the annualized market return variable is insignificant.

So, my question is: Is it appropriate or inappropriate to include year
fixed effects given that I am including a variable that varies by year
but does not vary by state within a given year (i.e., the interest
rate variable)?

If you have any thoughts regarding this issue I would greatly
appreciate it, as I have received mixed feedback and I haven't found a
study that explicitly deals with this particular issue.

Any assistance would be greatly appreciated.

Respectfully,

Samuel Finkelstein
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