Cluster-robust standard errors in panel data analysis

Question

In a simple panel data analysis with data on 64 firms over 8 years, I use cluster-robust standard errors (at the firm level) to evaluate significance of coefficients. I observe important differences between clustered and non-clustered standard errors.

1) Does these differences necessarily mean that there is indeed serial correlations at the firm level in my data?

2) How could I test for serial correlation at the firm level to evaluate whether clustered standard errors are needed?

3) Say we observe that there is no serial correlation at the firm level, what is the impact of using clustered standard errors instead of non-clustered standard errors?

Answer 1

If, for each firm, time periods are independent then cluster-robust standard errors and standard errors without clustering will estimate the same thing (the same population standard deviation), but will use different strategies to do so. The standard errors without clustering will use more information (they don't have to infer that auto-correlations are zero from the data) and are expected to be a slightly better estimator. For large panels this will not make much of a difference but $N = 64$ is fairly small.

1. No, as explained some differences are expected. There may be a statistical test to explore this more formally. In practice, if differences between the two standard errors are observed, then one just uses the robust version. It is applicable under a larger set of models (this is what "robust" means in this context), and the "statistical cost" of choosing it over the alternative is very small.
2. Testing this is hard for two reasons
   1. Your time series is very short. Too short.
   2. Tests for auto-correlation look for specific forms of auto-correlation. Even if you could rule one form of auto-correlation (say AR-1), other forms may still be present.
3. Not much, as explained above you are still estimating the correct standard deviation. You are doing so in a slightly less efficient way but that shouldn't matter much (it will not matter at all if your panel is very large).

If you are wondering whether you should cluster: you probably should.