# Presenting the indexation of a cross-sectional model ($i = 1,…,N$)

Many papers in applied econometrics present the indexation of a cross sectional OLS model as $i = 1,...N$. For example:

$(1) Y_i = a + b X_i + e_i$

$E[e_i] = 0$

$i = 1,...,N$.

Would it be okay to present it as follows?

$(1) Y_i = a + b X_i + e_i$

$E[e_i] = 0$

$i \in \{1,...,N\}$.

Perhaps researchers do the former because the latter doesn't necessitate that $(1)$ is estimated $\forall i$ (only that $i$ is a member of the set $\{1,...,N\}$, and $\{i\}$ could be some proper subset of $\{1,...,N\}$).

Want to know best practice. Thanks.

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The first one is the usual. The second is superfluous. And mathematically they are identical. – mpiktas Aug 22 '12 at 5:56
Thanks. I'll do the first one. However, I wouldn't say they're identical. We could estimate $i$ over $\{1,2,3\}$ but it would still be mathematically correct to notate it $i \in \{1,2,3,4,...,N\}$ because $\{1,2,3\} \subset \{1,2,3,4,...,N\}$. – user13253 Aug 22 '12 at 6:12
Why specify more indices than what you are going to use? – Michael Chernick Aug 22 '12 at 10:19
If the relationship holds only for subset of $\{1,...,N\}$, then you need to indicate that. In this case you do not have the usual model. When you write $i=1,...,N$ you mean that relationship holds for each $i$. This is the same as writing $\forall i \in\{1,...,N\}$. Since $i=1,...,N$ looks nicer, it is used more frequently. – mpiktas Aug 22 '12 at 12:13
@mpiktas But say you have an index set going for 1 to N but only k<N indices are used in the model. Why not just reindex the subset to be 1,2..,k and define the set that way? – Michael Chernick Aug 22 '12 at 20:45