Fixed Effects or Dynamic Panel Model for Sluggish Independent Variable?

[alex severson]

I am estimating the effect of state-level budget transparency as well as its interactions with media market penetration and class bias in voter turnout on social welfare expenditures (measured as social welfare expenditures/capita) in 48 U.S. states between the years 1978-2000. The panel is balanced. However, I am struggling to identify the appropriate model specification for the data.

My initial hunch was to use a dynamic panel model with state and year fixed effects, specifically using the Arellano-Bond estimator. At first, it seemed theoretically defensible to assume that history matters and that previous social welfare expenditures would predict current expenditures. However, when I estimated these models, inclusion of the autoregressive term "dominated" the model and suppressed the explanatory power of all exogenous variables in my regression (relative to pooled OLS, and static FE/RE models). I proceeded to read [Achen 2000] (https://polmeth.wustl.edu/files/polmeth/achen00.pdf), and am now convinced that a dynamic specification is not appropriate due to serial correlation/stationarity concerns.

My next intuition was to defer to a simple static fixed-effects model. However, when I do an "eye-test" of the amount of within-variation in my "pet variable" - budget transparency - it appears to be relatively time-invariant - it very slowly changes within states over the 22 year span. As one potentially significant limitation of fixed effects models is that one cannot assess the effect of variables that have little within-group variation, I am left wondering two things.

First, how much within-variation is sufficient to proceed with a fixed-effects model? I know xtreg automatically (and sensibly) drops completely time-invariant variables from the model, but are sluggish independent variables ok? Are there any acceptable thresholds - that is, is there a predetermined level of within-variation that is necessary to proceed with fixed-effects? How does one measure this?

Second, if the sluggishness of the transparency variable is, in fact, problematic, which alternative model specification is most appropriate to use given the nature of my data?

Thanks!

*Cross-posted at: https://stackoverflow.com/questions/...ndent-variable

[Carlo Lazzaro]

Alex:
as far as I know, there's no such a measure like a threshold for within-variation.
You may take a look at -hausman- outcome (if your regression model assumes default standard errors) or the user-written programme -xtoverid- one (for robustified/clustered standard errors) ( type -search xtoverid-).

[Sebastian Kripfganz]

I have glanced through the linked paper by Achen (2000) and I tend to disagree with him to a certain extent.

It is true that the introduction of a lagged dependent variable (LDV) can create a bias if there is remaining serial correlation in the error term after the inclusion of the LDV. This can be tested for example with the Arellano-Bond test. The LDV is often introduced in the first place to account for serial correlation present in the static model. Moreover, Achen (2000) claims that the bias from introducing the LDV becomes strongest when the autoregressive coefficient of both the exogenous regressor and the error term approach unity; in other words: they will become non-stationary. But in such a situation with a unit-root in the regressor and the error term, the static model will be subject to the familiar "spurious regression" problem. The conclusion would then be that in such models with a strong serial correlation, the static model can lead to severe bias. The observed statistical significance for the coefficient of the exogenous regressor might just be spurious and the corresponding insignificant coefficient in the dynamic model would then give a better account of the state of the world.

Notice also that the interpretation of the coefficients in static versus dynamic models is different, and the interpretation of the coefficients in static models depends on whether the true data-generating process is actually dynamic (indicated by the presence of serial correlation in the static model) or not. See for example Egger and Pfaffermayr (2004).

• Egger, P., and M. Pfaffermayr (2004). Estimating Long and Short Run Effects in Static Panel Models. Econometric Reviews 23 (3), 199-214.