Interpreting time-varying coefficients in a fixed effects regression using SEM (as in xtdpml)

[Oscar Eichler]

I've estimated a fixed effects model within a structural equation modeling (SEM) framework, inspired by the work by Bollen & Brand; Williams, Allison & Moral-Beniro; and others.

One aspect of this approach is that it allows time-varying coefficients of independent variables, in contrast to a traditional fixed effects model using, for example xtreg, which would estimate coefficients spanning all waves, for example:

xtset id year
xtreg score treat year, fe


------------------------------------------------------------------------------
       score | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       treat |  -,0240636   ,0065614    -3,67   0,000    -,0369239   -,0112033
        year |  -,0982931   ,0023614   -41,62   0,000    -,1029215   -,0936646
       _cons |    ,152554   ,0016699    91,36   0,000      ,149281     ,155827
-------------+----------------------------------------------------------------

(...)

My question is how time-varying coefficients can be understood in comparison with the estimates produced in a traditional fixed effects regression as above?

First, when I compare time-varying and time-constant coefficients, the time-constant coefficient are roughly similar to, and sometimes a bit like averages of, the time-varying coefficients. This does not entirely explain the relationship between the coefficients, though (as illustrated below). Is there a more accurate way to interpret the difference between the time-varying and time-constant coefficients?

	T1	T2	Time-constant
var1	-0.050	-0.040	-0.024
var2	-0.052	0.065	0.002

Second, are there situations in which time-constant or time-varying coefficients should be more trusted?

Best regards,
Oscar

Two references regarding time-varying coefficients in SEM approaches are:

• The results discussed in Figure 4 in the article by Bollen & Brand (2010), A General Panel Model with Random and Fixed Effects: A Structural Equations Approach.
• The xfree and xfree2 options in the user-developed Stata command xtdpml by Williams, Allison & Moral-Benito.