12.6 Summing Up

In your own research you might ask the following questions:

  • Are my observations independent? Do I have constant error variance?
    • If independent, you might be fine with a standard model
    • If worried about the error variance, you could be fine with implementing some type of robust standard errors. See Gary King’s video for guidance and section 8.5.1 of the course notes.
  • Alternatively, is there some grouping structure to my data?
    • What are the different levels?
  • Are observations repeated over time?
    • Large N, few T, generally considered panel data
    • Large T, fewer N, generally considered time series cross-sectional

Then, you might consider the following choices if you have a grouping structure:

  • Consider clustering standard errors by group in a standard model
  • Or, incorporating the grouping structure into a multilevel random effects model
    • Do you meet the random effects assumptions?
    • Do you need random effects? Perhaps use the ICC to help with this.
    • Should you add varying slopes?
  • Or, incorporating fixed effects for the groups and/or time
    • How much data do you have within groups?
    • Do you need to model “level-2” factors?

What we have not yet discussed are the following issues. See links for further study.

  • Using fixed effects models for causal inference, specifically
  • Incorporating dynamics into longitudinal data: \(Y_{it} = \alpha_i + \rho Y_{i t-1} + \beta x_{it} + \epsilon_{it}\)
    • Outcome of today is a function of the past outcome modified by new information. Consider including if you think past outcomes influence future outcomes.
    • \(\rho\) is an “autocorrelation” term such that \(|\rho| < 1\).
    • Problem: Unless \(\rho = 0\), correlation created between regressors and error term \(\rightarrow\) strict exogeneity violated \(\rightarrow\) Nickell Bias. In random effects models, \(y_{it-1}\) also correlated with any group-level effects \(\alpha_i\).
    • Video explanation.
    • Concern greatest in samples with small \(T\).
    • A lot of debates on the inclusion of an LDV and “dynamic” models in general. See here
      • Achen C. H. (2001) Why lagged dependent variables can suppress the explanatory power of other independent variables
      • Keele, L. and Kelly N. J. (2005) Dynamic models for dynamic theories: the ins and outs of lagged dependent variables
      • Arjun Wilkins. (2017). To Lag or Not to Lag?: Re-Evaluating the Use of Lagged Dependent Variables in Regression Analysis
      • Arellano-Bond and Anderson-Hsiao estimators