Modeling repeated measures of cross-sectional county level data I'm currently preparing an analysis project and am wondering people's thoughts on how to best model these data. I have 8 waves of data from a statewide random sample of 6, 8, 10, & 12 graders that repeat the same questions at each wave. I'd like to do a county-level analysis (39 counties) of changes in adolescent behavior following some state-level health policy changes. There are also county-level policies to consider so my plan is to model those as well. I'm still gathering data but there will be at least a few time-varying independent variables and the ultimate dependent variable is time-varying.
I've seen difference in difference a lot for these type of data and although I haven't used it before, I have enough resources around to help with that.  I also know some profs who are using Joinpoint models but I'm not familiar with that as of yet.  Finally, I've been looking at some of Paul Allison's work recently on SEM but I'm not sure if that's applicable to the data structure I have.  I plan to use Mplus but am also comfortable in R (if that matters to someone's answer).
I'm a phd student so I'm just learning some of these methods. Any thoughts would be much appreciated!
 A: When I was in doctoral coursework many years ago, my econometrics professor, a brilliant Russian lady with several degrees in mathematical economics, used to tell us, "First, start with OLS."  Of course, not OLS is not always appropriate, but her point was that you should always start simple and avoid making things complicated and put off causing misery for yourself unless you absolutely have to.
There are a number of ways you can model your outcome. Cameron and Trivedi's (2009) Microeconometrics using Stata is an excellent resource even for non-users of Stata, because they throw econometric theory in the explanations to their examples.  The simplest I could think of would be to pool all 8 waves of data and throw in the wave as dummy variables, excluding one wave (the reference category),e.g.:
$y_{it}=\beta_0+\beta_1Wave1_{it}+\beta_2Wave2_{it}+...+\beta_7Wave7_{it}+X\beta+\epsilon_{it}$
It doesn't matter which one is excluded as the reference (in my example, it was Wave0), since the joint significance of the wave variables will be the same regardless.  Be sure to use robust standard errors and cluster your standard errors around the county, so that your statistical package can make the appropriate adjustments to the standard error.  This method is suitable if you're interested in seeing how the outcome changes over waves.  As an added bonus, you can perform the Chow Test for structural changes over time for specific covariates.  For example, if you want to know whether boys and girls perform differently as a result of the policy change, you can interact the wave dummies with child's sex and test the significance of those dummies, which is possible since your data are at the individual student level:
$y_{it}=\beta_0+\beta_1Wave1_{it}+\beta_2Wave2_{it}+...+\beta_7Wave7_{it}+\beta_8Sex_{it}+\beta_9Wave1*Sex_{it}+\beta_{10}Wave2*Sex_{it}+...+\beta_{15}Wave7*Sex_{it}+X\beta+\epsilon_{it}$
Note, I made the suggestion to cluster the standard errors around the county and not the individual, because as I understand it, you have a panel of counties, but not students.  If you have a panel of students, then you need to cluster around the individual identifier, instead of the county identifier.
Another possibility is the difference-in-differences (DiD) estimator, which incidentally, is the weapon of choice of another former econometrics professor.  Traditionally, DiD is done using 2 waves of data, though I imagine it is easily adaptable to >2 waves.  You just need to make sure you use robust standard errors and cluster around the county as with above.  As the name suggests, DiD takes the difference of two differences:
$DiD = (Treat_{Policy=1}-Treat_{Policy=0})-(Control_{Policy=1}-Control_{Policy=0})$
$Policy=0$ and $Policy=1$ refer to the time point(s) where the policy has not been/has been implemented.  Estimating this using a regression model, you are fitting the following:
$y_{it}=\beta_0+\beta_1YesPolicy_{it}+\beta_2After_{it}+\beta_3YesPolicy*After_{it}+x\beta+\epsilon_{it}$
Where $YesPolicy$ is a binary variable indicating whether the individual in County X ever had the policy implemented, $After$ is a binary variable indicating whether the policy had already been implemented at Time N, and $YesPolicy*After$ is an interaction term of $YesPolicy$ and $After$ and is the impact indicator of interest.  If the $\beta_3$ is significant you can conclude that the impact of the policy is statistically significant.  As you can guess, if the person/county never got the policy, the impact indicator will remain 0, whereas if the person/county got the policy, the impact indicator will equal 1 in the wave(s) following implementation.  To illustrate, supposing the policy was rolled out after Wave 3, your data may look something like this:

Other common estimation strategies for panel data include fixed effects models (if you care only about differences within counties over time), random effects models (if you care about differences between counties over time), and random effects models.  I like the IDRE at UCLA help pages on panel data analysis and the Centre for Multilevel Modeling at University of Bristol's training videos for beginners' overviews.  Gertler et al. (2010) provides good basic overview of impact evaluation techniques.  Wooldridge (2009) and Wooldridge (2002) discuss these issues in greater detail.
References:


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*Cameron AC, Trivedi PK. Microeconometrics Using Stata. College Station, TX, USA: Stata Press; 2009.

*Gertler PJ, Martinez S, Premand P, Rawlings LB, Vermeersch CMJ. Impact Evaluation in Practice. The World Bank; 2010. doi:10.1596/978-0-8213-8541-8.

*Wooldridge JM. Introductory Econometrics: A Modern Approach. 4th ed. Mason, OH, USA: South-Western, Cengage Learning; 2009.

*Wooldridge JM. Econometric Analysis of Cross Section and Panel Data. Cambridge, MA, USA: The MIT Press; 2002. 
