re: st: Use of Fixed Effects with State and National Data

[Christopher Baum <kit.baum@bc.edu>]

<>
Sam wrote 

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). 



A time fixed effect in a panel is an application of the within estimator w.r.t. t rather than i. That is, including a set of time FE is equivalent
to subtracting the mean of each variable over units for each point in time. When you do this for state-specific variables, no problem, as
they do not all have the same value of a state-level variable such as the unemployment rate at a point in time. But when you apply
this transformation to a macro (national) variable, you get a vector of zeros, as the 3 month Tbill rate is the same for every state at each
point in time. So I am puzzled that you can estimate coefficients on the national variables, unless Stata is automatically dropping all
of the time fixed effects. But as you're getting exactly the same coefficients on the other variables and the same r^2, that's probably not
the case.

If you include time fixed effects, you cannot separately estimate the effects of any factor that is constant over units at each t. Justina's suggestion--
that you should include time FE--reflects the concern that unobserved heterogeneity at the macro level is liable to be a problem 
no matter how many macro variables you explicitly include. Safer to include time FE. And a time trend is just a set of time FE with (T-1) constraints
on the coefficients. To test whether year FE are appropriate, include all but one of them, leave out any other national variables, and do a joint test on
the (T-1) coefficients that are estimated.

Kit

Kit Baum   |   Boston College Economics & DIW Berlin   |   http://ideas.repec.org/e/pba1.html
                             An Introduction to Stata Programming  |   http://www.stata-press.com/books/isp.html
  An Introduction to Modern Econometrics Using Stata  |   http://www.stata-press.com/books/imeus.html


*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/statalist/faq
*   http://www.ats.ucla.edu/stat/stata/