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I am using cross-sectional data with the following OLS model:

$$ Y_{(i,j)} = \beta_{(0)} + \beta X_{(i)} + \beta X_{(i,j)} + \beta fixed\; effects_{(j-1)} + \varepsilon_{i,j} $$

where $i$ stands for individuals and $j$ stands for groups.

In my application, I am trying to predict survey respondent's satisfaction with democracy. I have the hypothesis that satisfaction with democracy is a function of a number of individual level predictors like age, education, etc ($\beta X_{(i)}$) but also of country level characteristics, like the quality of democracy, economic inequality etc. ($\beta X_{(i,j)}$).

Initially I wanted to run hierarchical models. But unfortunately the survey was only conducted in 7 countries. Therefore I settled for country fixed effects, with standard errors clustered by country.

My question is: what happens with the upper level coefficients in my model? I thought they reflect a precise estimate and the country fixed effects pick up any remaining upper level variance, not explained by $\beta X_{(i,j)}$. However, I was not sure, and I would appreciate if someone can help interpret them.

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  • $\begingroup$ You haven't told us anything about the survey design. I suspect the survey is a complex a sample survey. If so, do you plan to incorporate the complex standard errors from the survey design into your final standard error calcuations? $\endgroup$
    – StatsStudent
    Commented Dec 14, 2020 at 14:07

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This is more of a general question; but how exactly does including dummies for countries in such a model control for country fixed effects? Since fixed effects rely on within-transformation it only works when you observe the same unit more than once. Sorry if that does not answer your question.

Another point is that clustering survey respondents by country is seriously misleading and wrong. Usually clustered errors only work when you have above 40 clusters. Effectively you are treating your data set as if you have 7 observations when you cluster by country.

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    $\begingroup$ sorry for the confusion with the fixed effects name. This is a bit different than the panel data application, I should have referred to country dummies but in political science the name "fixed effects " often flies around for this type of model. I understand the limits of clustered standard errors, but surely not clustering them and assuming that respondents are independent observations is equally wrong/worse? $\endgroup$
    – Erdne Htábrob
    Commented Jun 1, 2017 at 8:24
  • $\begingroup$ Imagine a regression model with 7 observations. That would not be convincing. It is kind of the same you do when clustering on 7 countries. The standard errors from this will not produce anything you can rely on. you could still cluster, but instead try cluster over regions within countries (if you have data on that). You assume there is no correlation between regions (which might be wrong), but I guess you will still capture some of the clustering in your data. $\endgroup$
    – Jonathan Marin
    Commented Jun 1, 2017 at 8:46

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