Dummy variables to control for clustering I have a panel-data sample which is not too large (1,973 observations). The unit of analysis is x (credit cards), which is grouped by y (say, individuals owning different credit cards). I cannot used fixed effects because I have important dummy variables, and random effects is inadequate. Therefore, I am using OLS. To control clustering in y, I have introduced a dummy variable for each y. My question is whether this is fine, i.e., whether clustering is taken into consideration this way (instead of using straightforward y-clustered errors).
 A: So you are estimating an equation like
$$\text{use}_{jit} = \beta \text{rate}_{jit} + \text{quarter}_t \delta + \text{individual}_i\gamma + \epsilon_{ijt}$$
where $\text{use}_{jit}$ is the usage of credit card $j$ at time $t$ owned by individual $i$, and $\text{quarter}_t$ and $\text{individual}_i$ are sets of time and individual dummies. You are worried that credit cards owned by the same individual have common shocks/errors, which makes sense. Let me give you a quick refresher on clustered standard errors and then I will address your question regarding the benefits of having group dummies to account for clustering.
For the moment forget about the time component for simplicity. If you didn't include your individual dummies the error would be a composite of the individual error (the group error) and the card level error, $\epsilon_{ij} = \nu_{i} + \eta_{ij}$. With such an error structure your intraclass correlation is then $\rho = \frac{\sigma^2_\nu}{\sigma^2_\nu + \sigma^2_\eta}$. Comparing the variances of the usual standard error and the cluster-robust standard error (clustered at the individual level) you would get
$$\frac{Var^{clust}(\widehat{\beta})}{Var^{ols}(\widehat{\beta})} = 1 + (n-1)\rho $$
which would tell you by how much the usual standard errors overestimate precision by ignoring the within group correlation (see for instance slide 17 and 18 of these notes).
If you include individual dummies, i.e. individual fixed effects, this would address the problem. In that case you cannot include any time-invariant control variables at the individual level though. This only works if your control variables are at an aggregate level (here at the individual level). For this see page 8 bullet point 3 in these notes. However, the usage of credit cards of the same individual is probably very highly correlated. In this situation the individual dummies will take out too much information in which case it is better to not use those dummies but still the errors would need to be adjusted. That's bullet point four in the same notes. Correcting the errors is easily done by estimating the usual cluster-robust standard errors clustering at the individual level.
Another advantage of the clustered standard errors is the time component. Clustering at the individual level will not only allow for within individual correlation of the credit card usage, it will also account for heteroscedasticity between individuals and it will correct for serial correlation. Given that you have a time component serial correlation may be an issue. The error of an individual yesterday is likely to be correlated with the error of the same individual today - simply because she is the same person with the same habits, shocks, etc.
Long story short: your way works under certain assumptions. The more flexible standard error correction is to estimate cluster-robust standard errors.
